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THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


SCIENTIFIC    PAPEES 


Eonbon:   C.  J.    CLAY   AND  SONS, 
CAMBRIDGE    UNIVERSITY   PRESS   WAREHOUSE, 

AVE    MARIA   LANE. 
©laaaoto:    30,  WELLINGTON  STREET. 


F.  A.  BROCKHAUS. 
lorfe:    THE  MACMILLAN  COMPANY. 
anto  Calcutta:    MACMILLAN  AND  CO.,  LTD. 


[All  Rights  reserved.] 


SCIENTIFIC   PAPEBS 


BY 


JOHN  WILLIAM   STEUTT, 

BARON    RAYLEIGH, 

O.M.,  D.Sc.,  F.R.S., 

HONORARY    FELLOW    OF    TRINITY    COLLEGE,    CAMBRIDGE, 
PROFESSOR   OF    NATURAL    PHILOSOPHY   IN   THE   ROYAL   INSTITUTION. 


VOL.    IV. 
1892-1901. 


CAMBRIDGE: 

AT  THE   UNIVERSITY   PRESS. 
1903 


dambrtogc: 

PRINTED    BY   J.    AND    C.    F.    CLAY, 
AT    THE    UNIVERSITY   PRESS. 


Engineering 
Library 

(ZO 


PBEFACE. 


the  present  volume  the  Collection  of  Papers  is  brought  down  to 
the  end  of  1901.  The  diversity  of  subjects — many  of  them,  it  is  to 
be  feared,  treated  in  a  rather  fragmentary  manner — is  as  apparent  as  ever, 
and  is  perhaps  intensified  by  the  occurrence  of  papers  recording  experi- 
mental work  on  gases.  The  memoir  on  Argon  (Art.  214)  by  Sir  W.  Ramsay 
and  myself  is  included  by  special  permission  of  my  colleague. 

A  Classified  Table  of  Contents  and  an  Index  of  Names  are  appended. 
The  large  number  of  references  to  the  works  of  Sir  George  Stokes, 
Lord  Kelvin  and  Maxwell,  as  well  as  of  Helmholtz  and  some  other 
investigators  abroad,  will  shew  to  whom  I  have  been  most  indebted  for 
inspiration. 

I  desire  also  to  record  my  obligations  to  the  Syndics  and  Staff  of  the 
University  Press  for  the  efficient  and  ever  courteous  manner  in  which  they 
have  carried  out  my  wishes  in  the  republication  of  this  long  series  of 


TERLING  PLACE,  WITHAM, 
December  1902. 


782201 


The  works  of  the  Lord  are  great, 

Sought  out  of  all  them  that  have  pleasure  therein. 


CONTENTS. 

ART.  PAGE 

197.  Density  of  Nitrogen 1 

[Nature,  XLVI.  pp.  512,  513,  1892.] 

198.  On  the  Intensity  of  Light  reflected  from  Water  and  Mercury 

at  nearly  Perpendicular  Incidence          .....  3 

Appendix     ..........         13 

[Philosophical  Magazine,  xxxiv.  pp.  309 — 320,  1892.] 

199.  On   the  Interference  Bands   of  Approximately    Homogeneous 

Light;   in  a  Letter  to  Prof.  A.  Michelson     .         .         .         .         15 
[Philosophical  Magazine,  xxxiv.  pp.  407 — 411,  1892.] 

200.  On  the  Influence  of  Obstacles  arranged  in  Rectangular  Order 

upon  the  Properties  of  a  Medium 19 

[Philosophical  Magazine,  xxxiv.  pp.  481—502,  1892.] 

201.  On  the  Densities  of  the  Principal  Gases .....         39 

The  Manometer 40 

Connexions  with  Pump  and  Manometer    ....  43 

The  Weights       .         .      ...         .         .         .         .         .         .  44 

The  Water  Contents  of  the  Globe     .         .         .         .         .  45 

Air      .        .        ..       .        ....        .        .        .  46 

Oxygen        .         .         .         ....         .         .         .  47 

Nitrogen     ...        .        ...         .         .         .  48 

Reduction  to  Standard  Pressure          .....  50 

Note  A.    On  the  Establishment  of  Equilibrium  of  Pressure 

in  Two  Vessels  connected  by  a  Constricted  Channel     .  53 
[Proceedings  of  the  Royal  Society,  LIII.  pp.  134—149,  1893.] 

202.  Interference  Bands  and  their  Applications        ....         54 
[Proc.  Roy.  Inst.  xiv.  pp.  72—78,  1893 ;  Nature,  XLVIII.  pp.  212—214,  1893.] 

203.  On  the  Theory  of  Stellar  Scintillation     .         ...         .         60 

[Philosophical  Magazine,  xx xvi,  pp.  129—142,  1893.] 

204.  Astronomical  Photography          .         ...  ,.     «n      *         ..}?..         73 

[Nature,  XLVIII.  p.  391,  1893.) 


Vlll  CONTENTS. 

ART.  PAGE 

205.  Grinding  and  Polishing  of  Glass  Surfaces         ....         74 
[British  Association,  Sept.  14,  1893,  from  a  report  in  Nature,  XLVIII.  p.  526,  1893.] 

206.  On    the    Reflection    of   Sound    or    Light    from    a    Corrugated 

Surface 75 

[British  Association  Report,  pp.  690,  691,  1893.] 

207.  On  a  Simple  Interference  Arrangement 76 

[British  Association  Report,  pp.  703,  704,  1893.] 

208.  On  the  Flow  of  Viscous  Liquids,  especially  in  Two  Dimensions         78 

[Philosophical  Magazine,  xxxvi.  pp.  354—372,  1893.] 

209.  The  Scientific  Work  of  Tyndall 94 

[Proceedings  of  the  Royal  Institution,  xiv.  pp.  216—224,  1894.] 

210.  On  an  Anomaly  encountered  in  Determinations  of  the  Density 

of  Nitrogen  Gas '.""'.         .104 

[Proceedings  of  the  Royal  Society,  LV.  pp.  340—344,  April,  1894.] 

211.  On  the  Minimum  Current  audible  in  the  Telephone       .         .       109 

[Philosophical  Magazine,  xxxvm.  pp.  285 — 295,  1894.] 

212.  An  Attempt  at  a  Quantitative  Theory  of  the  Telephone         .       119 

[Philosophical  Magazine,  xxxvm.  pp.  295 — 301,  1894.] 

213.  On  the  Amplitude  of  Aerial  Waves  which  are  but  just  Audible       125 

[Philosophical  Magazine,  xxxvm.  pp.  365 — 370,  1894.] 

214.  Argon,   a   New    Constituent   of  the   Atmosphere.      By   LOKJ> 

RAYLEIGH,  Sec.  R.S.,  and  PROFESSOR  WILLIAM  RAMSAY,  F.R.S.       130 

1.  Density  of  Nitrogen  from  Various  Sources  .         .         .130 

2.  Reasons  for  Suspecting  a  hitherto  Undiscovered  Con- 

stituent in  Air          .         .         .         .         .         .         .135 

3.  Methods  of  Causing  Free  Nitrogen  to  Combine .         .       138 

4.  Early  Experiments  on  sparking  Nitrogen  with  Oxygen 

in  presence  of  Alkali         .         .         .         .         .         .141 

5.  Early  Experiments  on  Withdrawal  of  Nitrogen   from 

Air  by  means  of  Red-hot  Magnesium      .         .         .144 

6.  Proof  of    the    Presence   of    Argon   in   Air,   by   means 

of  Atmolysis      .         .         .         .         .         .         .         .       150 

7.  Negative    Experiments    to   prove    that   Argon   is   not 

derived  from  Nitrogen  or  from  Chemical  Sources   .       153 

8.  Separation  of  Argon  on  a  Large  Scale         .         .         .155 

9.  Density  of  Argon  prepared  by  means  of  Oxygen         .       105 

10.  Density  of  Argon  prepared  by  means  of  Magnesium  .       167 

11.  Spectrum  of  Argon        .         .  .         .         .         .       168 

12.  Solubility  of  Argon  in  Water 170 


CONTENTS.  ix 

ART.  PAGE 

13.  Behaviour  at  Low  Temperatures.         .         .  .  „     .       173 

14.  The  ratio  of  the  Specific  Heats  of  Argon   .  .  .       174 

15.  Attempts  to  induce  Chemical  Combination  .  .       176 

16.  General  Conclusions      .         .         .         .         .  .  .180 

Addendum,  March  20  (by  PROFESSOR  W.  RAMSAY)  .  .       184 

Addendum,  April  9       .         .         .         .         .         .  .  .187 

[Phil.  Trans.  186  A,  pp.  187—241,  1895.] 

215.  Argon 188 

[Royal  Institution  Proceedings,  xiv.  pp.  524 — 538,  April  1895.] 

216.  On  the  Stability  or  Instability  of  Certain  Fluid  Motions.    III.       203 

Addendum,  January  1896 .209 

[Proceedings  of  the  London  Mathematical  Society,  xxvu.  pp.  5—12,  1895.] 

217.  On  the  Propagation  of  Waves  upon  the  Plane  Surface  separ- 

ating Two  Portions  of  Fluid  of  Different  Vorticities    .         .       210 
[Proceedings  of  the  London  Mathematical  Society,  xxvu.  pp.  13 — 18,  1895.] 

218.  On  some  Physical  Properties  of  Argon  and  Helium        .         .       215 

Density  of  Argon        .         .         .         .         .         .  .215 

The  Refractivity  of  Argon  and  Helium     ....  218 

Viscosity  of  Argon  and  Helium          .....  222 

Gas  from  the  Bath  Springs        ......  223 

Buxton  Gas 223 

Is  Helium  contained  in  the  Atmosphere  ?          .         .         .  224 
[Proceedings  of  the  Royal  Society,  LIX.  pp.  198—208,  Jan.  1896.] 

219.  On  the  Amount  of  Argon  and  Helium  contained  in  the  Gas 

from  the  Bath  Springs  .         .         .         .         ...         .       225 

[Proceedings  of  -the  Royal  Society,  LX.  pp.  56,  57,  1896.] 

220.  The  Reproduction  of  Diffraction  Gratings         ....       226 

[Nature,  mv.  pp.  332,  333,  1896.] 

221.  The  Electrical  Resistance  of  Alloys 232 

[Nature,  LIV.  pp.  154,  155,  1896.] 

222.  On   the   Theory   of  Optical   Images,   with    Special    Reference 

to  the  Microscope  .........       235 

[Philosophical  Magazine,  XLII.  pp.  167 — 195,  1896.] 

223.  Theoretical  Considerations  respecting  the.  Separation  of  Gases 

by  Diffusion  and  Similar  Processes    -    .  •    •'.'        .         .         .       261 
[Philosophical  Magazine,  XLII.  pp.  493—498,  1896.] 

224.  The  Theory  of  Solutions  .        ..--.•      ;  .  ••" :  •."•:••  .    '     .         .       267 

[Nature,  T,V.  pp.  253,  254,  1897.] 


x  CONTENTS. 

AET.  PAGE 

225.  Observations  on  the  Oxidation  of  Nitrogen  Gas      .     :    .  •       .  270 

[Chemical  Society's  Journal,  71,  pp.  181 — 186,  1897.] 

226.  On    the    Passage    of  Electric    Waves   through   Tubes,  or  the 

Vibrations  of  Dielectric  Cylinders  .         .         .                  .         .  276 
General  Analytical  Investigation         .         .         .         .         .276 

Rectangular  Section 279 

Circular  Section.         ........  280 

[Philosophical  Magazine,  XLIII.  pp.  125—132,  1897.] 

227.  On  the  Passage  of  Waves  through  Apertures  in  Plane  Screens, 

and  Allied  Problems 283 

Perforated  Screen. — Boundary  Condition  d<f>/dn  =  0  .         .  284 

Boundary  Condition  <£  =  0          .         .         .         .         ;         .  286 

Reflecting  Plate.— d<j>/dn  =  0 288 

Reflecting  Plate.— 0  =  0 289 

Two-dimensional  Vibrations         ......  290 

Narrow  Slit. — Boundary  Condition  d<j>/dn  =  0  .         .         .  291 
Narrow  Slit.— Boundary  Condition  0=0  .         .         .         .293 

Reflecting  Blade.— Boundary  Condition  d<j>/dn  =  0     .         .  294 

Reflecting  Blade. — Boundary  Condition  <f>  =  0   .         .         .  295 

Various  Applications  ........  295 

[Philosophical  Magazine,  XLIII.  pp.  259 — 272,  1897.] 

228.  The  Limits  of  Audition 297 

[Royal  Institution  Proceedings,  xv.  pp.  417—418,  1897.] 

229.  On    the    Measurement    of    Alternate   Currents   by   means   of 

an  obliquely  situated  Galvanometer  Needle,  with  a  Method 

of  Determining  the  Angle  of  Lag          .         .         .         .         .  299 
[Philosophical  Magazine,  XLIII.  pp.  343—349,  1897.] 

230.  On  the  Incidence  of  Aerial  and  Electric  Waves  upon   Small 

Obstacles   in   the   Form   of  Ellipsoids  or  Elliptic  Cylinders, 
and  on  the   Passage   of  Electric  Waves  through  a  Circular 

Aperture  in  a  Conducting  Screen          ,         .                  .         .  305 

Obstacle  in  a  Uniform  Field      ...         .         .         .         .  306 

In  Two  Dimensions    .         .         .         ....         .         .  309 

Aerial  Waves      .         .         ...         .         .         .         .         .  310 

Waves  in  Two  Dimensions         .        .         .         .         .         .314 

Electrical  Applications        .         .         .         .         .^        .         .317 

Electric  Waves  in  Three  Dimensions          .         .         .         .318 

Obstacle  in  the  Form  of  an  Ellipsoid        ....  323 

Circular  Aperture  in  Conducting  Screen  ....  324 

[Philosophical  Magazine,  XLIV.  pp.  28—52,  1897.] 


CONTENTS.  xi 

ART.  PAGE 

231.  On    the    Propagation    of    Electric   Waves    along    Cylindrical 

Conductors  of  any  Section     .         .         ...         .         .       327 

[Philosophical  Magazine,  XLIV.  pp.  199 — 204,  1897.] 

232.  The  Electro-Chemical  Equivalent  of  Silver    •„,       .         .         .       332 

[Nature,  LVI.  p.  292,  1897.] 

233.  On  an  Optical  Device  for  the  Intensification  of  Photographic 

Pictures 333 

[Philosophical  Magazine,  XLIV.  pp.  282—285,  1897.] 

234.  On  the  Viscosity  of  Hydrogen  as  affected  by  Moisture  .         .       336 

[Proceedings  of  the  Royal  Society,  LXII.  pp.  112 — 116,  1897.] 

235.  On   the    Propagation    of  Waves   along  Connected  Systems  of 

Similar  Bodies         .         .         .         .  .        '.         .         .       340 

[Philosophical  Magazine,  XLIV.  pp.  356—362,  1897.] 

236.  On   the   Densities   of    Carbonic   Oxide,   Carbonic    Anhydride, 

and  Nitrous  Oxide          .        .        ......        .         .  347 

Carbonic  Oxide  .         .        .        .        .        .        ...        .-  347 

Carbonic  Anhydride 349 

Nitrous  Oxide     .........  350 

[Proceedings  of  the  Royal  Society,  LXII.  pp.  204—209,  1897.] 

237.  Rontgen  Rays  and  Ordinary  Light 353 

[Nature,  LVII.  p.  607,  1898.] 

238.  Note    on    the    Pressure    of  Radiation,  showing   an  Apparent 

Failure  of  the  Usual  Electromagnetic  Equations  .         .         .       354 
[Philosophical  Magazine,  XLV.  pp.  522—525,  1898.] 

239.  Some  Experiments  with  the  Telephone    .         .         .         .         .357 
[Roy.  Inst.  Proc.  xv.  pp.  786—789,  1898  ;  Nature,  LVIII.  pp.  429—430,  1898.] 

240.  Liquid  Air  at  one  Operation 360 

[Nature,  LVIII.  p.  199,  1898.] 

241.  On  the  Character   of  the   Impurity   found   in   Nitrogen   Gas 

Derived  from  Urea   [with   an   Appendix   containing   details 

of  Refractometer] 361 

Details  of  Refractometer 364 

[Proceedings  of  the  Royal  Society,  LXIV.  pp.  95 — 100,  -1898.] 

242.  On  Iso-periodic  Systems  .         .        .        .        .        .        .     "  .      367 

[Philosophical  Magazine,  XLVI.  pp.  567—569,  1898.] 

243.  On  James  Bernoulli's  Theorem  in  Probabilities       .        •.         .       370 

Magazine,  XLVII.  pp.  246—251,  1899.] 


Xll  CONTENTS. 

ART.  PAGE 

244.     On   the   Cooling  of  Air   by  Radiation   and   Conduction,   and 

on  the  Propagation  of  Sound         .         .         .         .         .         .376 

[Philosophical  Magazine,  XLVII.  pp.  308—314,  1899.] 

'245.     On    the    Conduction   of  Heat    in   a   Spherical    Mass    of    Air 

confined  by  Walls  at  a  Constant  Temperature      .         .         .       382 
[Philosophical  Magazine,  XLVII.  pp.  314—325,  1899.] 

246.  Transparency  and  Opacity -•..-...       392 

[Proc.  Roy.  Inst.  xvi.  pp.  116—119,  1899;    Nature,  LX.  pp.  64,  65,  1899.] 

247.  On  the  Transmission  of  Light  through   an    Atmosphere    con- 

taining  Small    Particles   in    Suspension,  and  on  the  Origin 
of  the  Blue  of  the  Sky .         .         .         .     -    .         .         .         .       397 
[Philosophical  Magazine,  XLVII.  pp.  375—384,  1899.] 

248.  The  Interferometer 406 

[Nature,  LIX.  p.  533,  1899.] 

249.  On  the  Calculation  of  the  Frequency  of  Vibration  of  a  System 

in  its  Gravest  Mode,  with  an  Example  from  Hydrodynamics       407 
[Philosophical  Magazine,  XLVII.  pp.  566 — 572,  1899.] 

250.  The  Theory  of  Anomalous  Dispersion       .....       413 

[Philosophical  Magazine,  XLVIII.  pp.  151,  152,  1899.] 

251.  Investigations  in  Capillarity      .         .         .         .         .         .         .415 

The  Size  of  Drops 415 

The  Liberation  of  Gas  from  Supersaturated  Solutions       .       420 

Colliding  Jets 421 

The  Tension  of  Contaminated  Water-Surfaces  .         .         .425 

A  Curious  Observation 430 

[Philosophical  Magazine,  XLVIII.  pp.  321—337,  1899.] 

252.  The  Mutual  Induction  of  Coaxial  Helices        .         .         .         .431 

[British  Association  Report,  pp.  241,  242,  1899.] 

253.  The  Law  of  Partition  of  Kinetic  Energy         -.        w         .         .       433 

[Philosophical  Magazine,  XLIX.  pp.  98—118,  1900.] 

254.  On  the  Ariscosity  of  Argon  as  affected  by  Temperature  .         .       452 

[Proceedings  of  the  Royal  Society,  LXVI.  pp.  68—74,  1900.] 

255.  On  the  Passage  of  Argon  through  Thin  Films  of  Indiarubber       459 

[Philosophical  Magazine,  XLIX.  pp.  220,  221,  1900.] 

256.  On  the  Weight  of  Hydrogen  desiccated  by  Liquid  Air  .         .       461 

[Proceedings  of  the  Royal  Society,  LXVI.  p.  344,  1900.] 


CONTENTS.  xiii 

ART.  PAGE 

257.  The  Mechanical  Principles  of  Flight         .         .        ^       '.      •  :.       462 

[Manchester  Memoirs,  XLIV.  pp.  1 — 26,  1900.] 

258.  On  the  Law  of  Reciprocity  in  Diffuse  Reflexion     .         .         .       480 

[Philosophical  Magazine,  XLIX.  pp.  324,  325,  1900.] 

259.  On  the    Viscosity  of  Gases  as  Affected  by  Temperature          .       481 

[Proceedings  of  the  Royal  Society,  LXVII.  pp.  137—139,  1900.] 

260.  Remarks  upon  the  Law  of  Complete  Radiation       .         .         .       483 

[Philosophical  Magazine,  XLIX.  pp.  539,  540,  1900.] 

261.  On  Approximately  Simple  Waves    ......       486 

[Philosophical  Magazine,  L.  pp.   135 — 139,  1900.] 

262.  On  a  Theorem  analogous  to  the  Virial  Theorem     .         .         .       491 

[Philosophical  Magazine,  L.   pp.  210—213,   1900.] 

263.  On    Balfour    Stewart's    Theory    of    the    Connexion    between 

Radiation   and   Absorption     .......       494 

[Philosophical  Magazine,  i.  pp.   98—100,  1901.] 

264.  Spectroscopic  Notes  concerning  the  Gases  of  the  Atmosphere       496 

On  the  Visibility  of  Hydrogen  in  Air      ....       496 

Demonstration   at   Atmospheric   Pressure   of  Argon   from 

very  small  quantities  of  Air.         .....       499 

Concentration  of  Helium  from  the  Atmosphere         .         .       500 
[Philosophical  Magazine,  I.  pp.  100—105,  1901.] 

265.  On  the   Stresses   in   Solid   Bodies   due   to   Unequal  Heating, 

and  on  the  Double  Refraction  resulting  therefrom       .         .       502 
[Philosophical  Magazine,   I.  pp.  169—178,  1901.] 

266.  On  a  New  Manometer,  and  on  the  Law  of  the  Pressure  of 

Gases  between  1'5  and  O'Ol  Millimetres  of  Mercury    .         .       511 
Introduction         .         .         .         .         .         .         .         .         .511 

Improved  Apparatus  for  Measuring  very  small  Pressures.       514 
Experiments  to  determine  the  Relation  of  Pressure  and 

Volume  at  given  Temperature 519 

[Philosophical  Transactions,  cxcvi  A.  pp.  205 — 223,  1901.] 

267.  On  a  Problem  relating  to  the  Propagation  of  Sound  between 

Parallel  Walls 532 

[Philosophical  Magazine,  I.  pp.  301 — 311,  1901.] 

268.  Polish .       542 

[Proceedings  of  the  Royal  Institution,  xvi.  pp.  563 — 570,  1901  ; 
Nature,  LXIV.  pp.  385—388,   1901.] 


XIV  CONTENTS. 

ART.  PAGE 

269.  Does  Chemical  Transformation  influence  Weight  t .         .         .       549 

[Nature,  LXIV.  p.   181,  June,   1901.] 

270.  Acoustical  Notes.  VI.        .         .         ...         .         .         .       550 

Forced  Vibrations       . 550 

Vibrations  of  Strings  .         .         .         .         .         .         .551 

Beats  of  Sounds  led  to  the  Two  Ears  separately    .         .       553 
Loudness  of  Double  Sounds       .         .         .       ..         .         .       554 

[Philosophical  Magazine,  u.  pp.  280—285,  1901.] 

271.  On   the    Magnetic   Rotation   of  Light   and   the    Second   Law 

of  Thermodynamics         .         ...         .         .         .         .       555 

[Nature,  LXIV.  pp.  577,  578,   1901.] 

272.  On  the  Induction-Coil      .         .         .         .         ...         .         .       557 

[Philosophical  Magazine,  u.  pp.  581—594,  1901.] 

CLASSIFIED  TABLE  OF  CONTENTS 569 

INDEX  OF  NAMES     ....  599 


ILLUSTRATIONS. 

Portrait  of  LORD  RAYLEIGH         .        .  .  .  Frontispiece 

Plate  I  (Figs.  1  and  2)        .        .        .  .  .  To  face  p.  545 

Plate  II  (Figs.  3  and  4)      ...  .  .  548 


197. 

DENSITY   OF   NITROGEN. 
[Nature,  XLVI.  pp.  512,  513,  1892.] 

I  AM  much  puzzled  by  some  recent  results  as  to  the  density  of  nitrogen, 
and  shall  be  obliged  if  any  of  your  chemical  readers  can  offer  suggestions  as 
to  the  cause.  According  to  two  methods  of  preparation  I  obtain  quite  distinct 
values.  The  relative  difference,  amounting  to  about  1/1000  part,  is  small  in 
itself;  but  it  lies  entirely  outside  the  errors  of  experiment,  and  can  only  be 
attributed  to  a  variation  in  the  character  of  the  gas. 

In  the  first  method  the  oxygen  of  atmospheric  air  is  removed  in  the 
ordinary  way  by  metallic  copper,  itself  reduced  by  hydrogen  from  the  oxide. 
The  air,  freed  from  C02  by  potash,  gives  up  its  oxygen  to  copper  heated  in 
hard  glass  over  a  large  Bunsen,  and  then  passes  over  about  a  foot  of  red-hot 
copper  in  a  furnace.  This  tube  was  used  merely  as  an  indicator,  and  the 
copper  in  it  remained  bright  throughout.  The  gas  then  passed  through  a 
wash-bottle  containing  sulphuric  acid,  thence  again  through  the  furnace 
over  copper  oxide,  and  finally  over  sulphuric  acid,  potash  and  phosphoric 
anhydride. 

In  the  second  method  of  preparation,  suggested  to  me  by  Prof.  Ramsay, 
everything  remained  unchanged,  except  that  the  first  tube  of  hot  copper  was 
replaced  by  a  wash-bottle  containing  liquid  ammonia,  through  which  air  was 
allowed  to  bubble.  The  ammonia  method  is  very  convenient,  but  the  nitrogen 
obtained  by  means  of  it  was  1/1000  part  lighter  than  the  nitrogen  of  the  first 
method.  The  question  is,  to  what  is  the  discrepancy  due  ? 

The  first  nitrogen  would  be  too  heavy,  if  it  contained  residual  oxygen. 
But  on  this  hypothesis,  something  like  1  per  cent,  would  be  required.  I 
could  detect  none  whatever  by  means  of  alkaline  pyrogallate.  It  may  be 
remarked  that  the  density  of  the  nitrogen  agrees  closely  with  that  recently 
obtained  by  Leduc  using  the  same  method  of  preparation. 
6  7  R.  iv.  1 


2  DENSITY   OF  NITROGEN.  [197 

On  the  other  hand,  can  the  ammonia-made  nitrogen  be  too  light  from  the 
presence  of  impurity  ?  There  are  not  many  gases  lighter  than  nitrogen,  and 
the  absence  of  hydrogen,  ammonia,  and  water  seems  to  be  fully  secured.  On 
the  whole  it  seemed  the  more  probable  supposition  that  the  impurity  was 
hydrogen,  which  in  this  degree  of  dilution  escaped  the  action  of  the  copper 
oxide.  But  a  special  experiment  seems  to  exclude  this  explanation. 

Into  nitrogen  prepared  by  the  first  method,  but  before  its  passage  into 
the  furnace  tubes,  one  or  two  thousandths  by  volumes  of  hydrogen  were 
introduced.  To  effect  this  in  a  uniform  manner  the  gas  was  made  to  bubble 
through  a  small  hydrogen  generator,  which  would  be  set  in  action  under  its 
own  electro-motive  force  by  closing  an  external  contact.  The  rate  of  hydrogen 
production  was  determined  by  a  suitable  galvanometer  enclosed  in  the 
circuit.  But  the  introduction  of  hydrogen  had  not  the  smallest  effect  upon 
the  density,  showing  that  the  copper  oxide  was  capable  of  performing  the 
part  desired  of  it. 

Is  it  possible  that  the  difference  is  independent  of  impurity,  the  nitrogen 
itself  being  to  some  extent  in  a  different  (dissociated)  state  ? 

I  ought  to  have  mentioned  that  during  the  fillings  of  the  globe,  the  rate 
of  passage  of  gas  was  very  uniform,  and  about  2/3  litre  per  hour. 


198. 


ON    THE    INTENSITY    OF    LIGHT    REFLECTED    FROM    WATER 
AND  MERCURY  AT  NEARLY   PERPENDICULAR  INCIDENCE. 

[Philosophical  Magazine,  xxxiv.  pp.  309 — 320,  1892.] 

IN  a  former  paper*  I  gave  an  account  of  some  experiments  upon  the 
reflexion  from  glass  surfaces  tending  to  show  that  "recently  polished  glass 
surfaces  have  a  reflecting-power  differing  not  more  than  1  or  2  per  cent,  from 
that  given  by  Fresnel's  formula;  but  that  after  some  months  or  years  the 
reflexion  may  fall  off  from  10  to  30  per  cent.,  and  that  without  any  apparent 
tarnish."  Results  in  the  main  confirmatory  have  been  published  by  Sir  John 
Conroyf. 

The  accurate  comparison  of  Fresnel's  formula  with  observation  is  a  matter 
of  great  interest  from  the  point  of  view  of  optical  theory,  but  it  seems  scarcely 
possible  to  advance  the  matter  much  further  in  the  case  of  solids.  Apart 
from  contamination  with  foreign  bodies  of  a  greasy  nature,  and  disintegration 
under  atmospheric  influences,  we  can  never  be  sure  that  the  results  are 
unaffected  by  the  polishing-powder  which  it  is  necessary  to  employ.  For 
these  reasons  I  have  long  thought  it  desirable  to  institute  experiments 
with  liquids,  of  which  the  surfaces  are  easily  renewed ;  and  the  more  since 
I  succeeded  in  proving  that  (in  the  case  of  water  at  any  rate)  the  deviation 
from  Fresnel's  formula  found  by  Jamin  in  the  neighbourhood  of  the  polarizing 
angle  is  due  to  greasy  contamination.  The  very  close  verification  of  the 
theoretical  formula  in  this  critical  case  seemed  to  render  its  applicability  to 
perpendicular  incidence  in  a  high  degree  probable.  I  was  thus  induced  to 
attack  the  somewhat  troublesome  problem  of  designing  a  photometric  method 
capable  of  dealing  with  the  reflexion  from  a  horizontal  surface.  The  details 
of  the  apparatus  and  of  the  measures  will  be  given  presently;  but  in  the 
meantime  it  may  be  well  to  consider  rather  closely  what  is  to  be  expected 
upon  the  supposition  that  Fresnel's  formulas  are  really  applicable.  Fresnel's 
formulas  are  spoken  of,  because  although  at  strictly  perpendicular  incidence 
we  should  have  to  do  only  with  Young's  expression  (JJL  —  1)2/G*  -f  I)2,  in 

*  Proc.  Roy.  Soc.  November,  1886.     [Vol.  n.  p.  522.] 
t  Phil.  Trans,  1889  A,  p.  245. 

1—2 


4  ON    THE    INTENSITY    OF    LIGHT   REFLECTED   FROM   WATER  [198 

practice  we  are  forced  to  work  at  finite  angles  of  incidence.  It  is  thus 
important  to  examine  the  march  of  Fresnel's  expressions,  when  the  angle  of 
incidence  (0)  is  small. 

Writing 

sin  (0-  ft)  tan  (0-  ft) 

sin  (0  +  ft)  '  tan  (0  +  ft)  ' 

where 

sin  ft  =  sin  0//i, 
we  find 


Thus  S2  and  T2  differ   from   the  value   appropriate  to   0  =  0  in   opposite 
directions  and  by  quantities  of  the  order  0*.     But  on  addition  we  get 


differing  from  the  value  appropriate  to  0  =  0  by  a  quantity  of  the  fourth  order 
only  in  0.  When  therefore  the  circumstances  are  such  that  it  is  unnecessary 
to  distinguish  the  two  polarized  components,  the  intensity  of  reflexion  at 
small  incidences  is  in  a  high  degree  independent  of  the  precise  angle.  If  fi  is 
nearly  equal  to  unity,  we  have 


simply.     Again,  if  p,  =  |, 

(5) 


A  few  calculations  from  the  original  expressions  will  serve  to  indicate  the 
field  of  these  approximations. 

^  =  f,  0  =  10°,  ft  =  7°  29', 

S»--jLx  1-0467,  r*  =  ^x 

S*  +  T2  =  2  x  ~  x  1-0004. 
4y 

From  (5)  we  get  as  the  last  factor  1-00050. 

/*  =  f,  0  =  20°,  ft  =  1 

S2  =  -!gX  1-2021,  T2  = 

8*  +  T2  =  2  x  ^  x  1-0090. 
By  (5)  the  last  factor  is  T0080. 


1892]  AND    MERCURY   AT   NEARLY   PERPENDICULAR   INCIDENCE. 

Again, 


e  =  30°, 


=  JLX  1-5189, 


0!  =  22°  l'-4, 
=4  x -5866, 
'  =  2x^x  1-0527. 
According  to  (5)  the  last  factor  is  here  1*0405. 

Fig.  l. 


It  appears  that  in  the  case  of  water  the  aggregate  reflexion  scarcely 
begins  to  vary  sensibly  from  its  value  for  0  =  0  until  6  =  20°,  a  property  of 
some  importance  for  our  present  purpose,  as  it  absolves  us  from  the  necessity 
of  striving  after  very  small  angles  of  incidence. 

I  will  now  describe  the  actual  arrangement  adopted  for  the  experi- 
ments. The  source  of  light  at  A  (Fig.  1)  is  a  small  incandescent  lamp,  the 


6  ON   THE    INTENSITY   OF    LIGHT   REFLECTED    FROM   WATER  [198 

current  through  which  is  controlled  with  the  aid  of  a  galvanometer.  It  is 
so  mounted  that  its  equatorial  plane  coincides  with  the  (vertical)  plane  of 
the  diagram.  Underneath,  upon  the  floor,  is  placed  the  liquid  (B)  whose 
reflecting  power  is  to  be  examined.  At  C,  just  under  the  roof,  the  direct 
ray  AC  and  the  reflected  ray  BC  are  turned  into  the  same  horizontal 
direction  by  two  mirrors  silvered  in  front  and  meeting  one  another  at  C 
under  a  small  angle.  The  eye  situated  opposite  to  the  edge  C  and  looking 
into  the  double  mirror  thus  sees  the  direct  and  reflected  images  superposed, 
so  far  as  the  different  apparent  magnitudes  allow.  D  represents  a  diaphragm 
and  E  a  photographic  portrait-lens  of  about  3  inches  aperture  which  forms 
an  image  of  A  and  A'  on  or  near  the  plane  F.  At  F  is  placed  a  screen 
perforated  with  a  hole  sufficiently  large  to  make  sure  of  including  all  the 
rays  from  A,  A'  which  pass  D.  To  determine  this  point  an  eye-piece  is 
focused  upon  F,  so  that  the  images  of  A,  A'  are  seen  nearly  in  focus.  Some 
margin  is  necessary  because  the  images  of  A,  A'  cannot  (both)  be  accurately 
in  focus  at  F. 

These  adjustments  being  made,  an  eye  placed  behind  F  and  focused 
upon  C  sees  the  upper  mirror  illuminated  by  the  direct  light  (from  A),  and 
the  lower  illuminated  by  the  reflected  light  (from  A).  And  if  the  aperture 
at  F  is  less  than  that  of  the  pupil  of  the  eye,  the  apparent  brightnesses 
of  the  two  parts  of  the  field  are  in  the  same  proportion  as  would  be  the 
illuminations  on  a  diffusing  screen  at  C  due  to  the  two  sources.  The 
advantage  of  the  present  arrangement,  as  compared  for  example  with  the 
double-shadow  method,  lies  in  the  immense  saving  of  light.  In  the  case 
of  water  there  is  a  great  disproportion  (of  about  50  to  1)  in  the  illuminations 
as  seen  from  F.  In  order  to  reduce  the  direct  light  to  at  least  approximate 
equality  with  the  reflected,  Talbot's  device*  of  a  revolving  disk  was  employed. 
This  is  shown  in  section  at  7,  and  in  plan  at  /'.  The  angular  opening  may  be 
chosen  so  as  to  allow  for  the  loss  in  reflexion,  and  for  the  further  disadvantage 
under  which  the  reflected  light  acts  in  respect  of  distance.  The  disk  finally 
employed  was  of  zinc,  stiffened  with  wood,  and  covered  on  both  faces  with 
black  velvet. 

It  was  at  first  proposed  to  work  as  above  described  by  eye  estimations ; 
but  the  necessity  for  a  ready  adjustment  capable  of  introducing  small  relative 
changes  of  brightness  leads  to  further  complications.  Moreover,  the  large 
disk  which  it  is  advisable  to  use  for  the  sake  of  accurate  measurement  of  the 
angular  opening,  cannot  well  be  rotated  at  the  necessary  speed  of  20  or  25 
revolutions  per  second.  For  this  reason,  and  also  for  the  sake  of  obtaining 
a  record  capable  of  being  examined  at  leisure,  it  was  decided  to  work  by 
photography.  This  involves  no  change  of  principle.  The  photographic 
plate  H  simply  takes  the  place  of  the  retina  of  the  eye.  But  now  the 

*  Phil.  Mag.  Vol.  v.  p.  327  (1834). 


1892]  AND   MERCURY   AT  NEARLY   PERPENDICULAR   INCIDENCE.  7 

integration  of  the  effect  over  a  somewhat  prolonged  exposure  (of  several 
minutes)  dispenses  with  the  necessity  for  a  rapid  rotation  of  the  Talbot  disk, 
and  allows  us  to  obtain  at  will  a  fine  adjustment  by  screening  one  or  the 
other  light  from  the  plate  for  a  measured  interval  of  time.  In  practice  the 
direct  light  was  thus  partially  cut  off,  a  mechanically  held  screen  being 
advanced  a  little  above  the  plane  of  the  revolving  disk.  The  reader  will  not 
fail  to  observe  that  the  incomplete  coincidence  of  the  times  of  exposure  has 
the  disadvantage  of  rendering  the  calculation  dependent  upon  the  assumption 
that  the  light  is  uniform  over  the  duration  of  an  experiment.  Error  that 
might  otherwise  enter  is,  however,  in  great  degree  obviated  by  the  precaution 
of  choosing  the  middle  of  the  total  period  of  exposure  as  the  time  for 
screening. 

The  above  is  a  sufficient  explanation  of  the  general  scheme,  but  there 
are  many  points  of  importance  still  to  be  described.  With  respect  to  the 
source  of  light,  it  was  at  first  supposed  that  even  if  the  radiation  upwards 
and  downwards  could  not  be  assumed  to  be  equal,  at  any  rate  a  reversal  by 
rotation  of  the  lamp  through  180°  in  the  plane  of  the  diagram  would  suffice 
to  eliminate  error.  On  examination,  however,  it  appeared  that  owing  to 
veins  in  the  glass  bulb  the  radiation  in  various  directions  was  very  irregular, 
so  much  so  that  it  was  feared  that  mere  reversal  might  prove  an  insufficient 
precaution.  The  difficulty  thus  arising  was  met  by  covering  the  bulb,  or  at 
least  an  equatorial  belt  of  sufficient  width,  with  thin  tissue-paper,  by  which 
anything  like  sudden  variations  of  radiation  with  direction  would  be  prevented, 
and  by  causing  the  lamp  to  revolve  slowly  about  its  axis  during  the  whole 
time  of  exposure.  The  diameter  of  the  bulb  was  about  1J  inch,  and  the 
illuminating-power  rather  less  than  that  of  one  candle. 

Another  point  of  great  importance  is  to  secure  that  the  light  regularly 
reflected  from  the  upper  surface  of  the  liquid,  which  we  wish  to  measure, 
shall  be  free  from  admixture.  It  must  be  remembered  that  by  far  the  greater 
part  of  the  light  incident  upon  the  liquid  penetrates  into  the  interior,  and 
must  be  annulled  or  at  any  rate  diverted  into  a  harmless  direction.  To  this 
end  it  is  necessary  that  the  liquid  be  free  from  turbidity  and  that  proper 
provision  be  made  for  the  disposal  of  the  light  after  its  passage.  It  is  not 
sufficient  merely  to  blacken  the  bottom  of  the  dish  in  which  the  water  is 
contained.  But  the  desired  object  is  attained  by  the  insertion  into  the  water 
of  a  piece  of  opaque  glass,  held  at  such  a  slight  inclination  to  the  horizon 
that  the  light  from  the  lamp  regularly  reflected  at  its  upper  surface  is  thrown 
to  one  side.  As  additional  precautions  the  disk  and  its  mountings  were 
blackened,  as  were  also  the  walls  and  ceiling  of  the  room  in  which  the 
experiments  were  made. 

The  surface  of  water  must  be  large  enough  to  avoid  curvature  due  to 
capillarity.  Shortly  before  an  experiment  it  is  cleansed  with  the  aid  of 


8 


ON   THE    INTENSITY   OF   LIGHT   REFLECTED    FROM    WATER 


[198 


a  hoop  of  thin  sheet-brass  about  2  inches  wide.  The  hoop  is  deposited  upon 
the  water  so  doubled  up  that  it  includes  but  an  insensible  area,  and  is  then 
opened  out  into  a  circle.  In  this  way  not  only  is  the  greasy  film  usually 
present  upon  the  surface  greatly  attenuated,  but  also  dust  is  swept  away. 
The  avoidance  of  dust,  especially  of  a  fibrous  character,  is  important.  Other- 
wise the  resulting  deformation  of  the  surface  causes  the  field  of  the  reflected 
light  to  become  patchy  and  irregular. 

We  come  now  to  the  silvered  glass  reflectors,  which  are  assumed  to  reflect 
the  direct  and  reflected  lights  equally  well.  It  seems  safe  to  suppose  that  no 
appreciable  error  can  enter  depending  upon  the  slightly  differing  angles  at 
which  the  reflexion  takes  place  in  the  two  cases.  But  the  mirrors  are  liable 
to  tarnish,  and,  indeed,  in  the  earlier  experiments  soon  showed  signs  of  being 
affected.  The  influence  of  this  tarnish  would  be  much  greater  in  photographs 
done  upon  ordinary  plates,  sensitive  principally  to  blue  light,  than  in  the 
estimation  of  the  eye ;  and  it  was  thought  desirable  to  eliminate  once  for  all 
any  question  of  the  effect  of  differential  tarnishing  by  interchanging  the 
mirrors  in  the  middle  of  each  exposure.  For  this  purpose  a  somewhat 
elaborate  mounting  had  to  be  contrived.  It  was  executed  by  Mr  Gordon 
and  answered  its  purpose  extremely  well. 

The  mirrors  are  carried  by  a  brass  tube  B  (Fig.  2),  which  revolves  in  an 

Fig.  2. 


external  tube  A  A  rigidly  attached  to  the  stand  of  the  apparatus.  A  lateral 
arm  G,  some  inches  in  length,  projects  from  B,  and  near  its  extremity  bears 
against  one  or  other  of  two  screw-stops  D.  The  lower  end  of  B  carries 


1892] 


AND   MERCURY    AT   NEARLY   PERPENDICULAR   INCIDENCE. 


perpendicular  to  itself  a  brass  plate  EE  (Fig.  3).     The  mirrors  GG  are  of 
plate-glass  and  are  fixed  by  cement  to  two  brass  plates  FF.     The  latter 


plates  are  attached  by  friction  only  to  EE,  being  on  the  one  hand  pushed 
away  by  adjusting-screws  HH,  and  on  the  other  held  up  by  four  steel 
springs  /.  The  edges  of  the  reflecting  surfaces  meet  accurately  in  a  line 
passing  through  the  axis  of  rotation,  and  the  stops  D  are  so  adjusted  that 
the  transition  from  the  one  bearing  to  the  other  corresponds  to  a  rotation 
through  precisely  180°,  so  that  on  reversal  the  common  edge  of  the  reflectors 
recovers  its  position.  The  two  mirrors  were  originally  silvered  in  one  piece, 
and  the  common  edge  corresponds  to  the  division  made  by  a  diamond-cut  at 
the  back.  These  arrangements  were  so  successful  that  in  spite  of  the  reversal 
between  the  two  parts  of  the  exposure  the  division-line  appears  sharp  in  the 
photographs  and  exhibits  no  appearance  of  duplicity. 

When  not  in  use  the  reflecting-surfaces  are  protected  by  a  sort  of  cap  of 
tin-plate,  which  fits  loosely  over  them.  The  improvement  thus  obtained  was 
very  remarkable,  the  mirrors  not  suffering  so  much  in  a  month  as  they 
formerly  did  in  a  day  before  the  protection  was  provided. 

The  following  are  the  measures  of  distances  required  for  the  calculation. 
From  the  division-line  C  to  the  axis  of  rotation  of  the  lamp  A  (Fig.  1), 

AC  =82-21  inches; 
45=11-28,  50=93-15, 


so  that 


45  +  50=104-43. 


The  factor  expressing   the  ratio  of  the   squares  of  the  distances   is   thus 
1-6137. 

The  angle  of  incidence  is  best  obtained  from  a  measurement  of  the 
horizontal  distance  between  G  and  A.  This  proved  to  be  11£  inches;  so 
that 

Sin*  =  ^3  =  'n'     and     e  =  W' 
This  applies  to  all  the  experiments  referred  to  in  the  present  paper. 


10  ON   THE   INTENSITY   OF   LIGHT   REFLECTED   FROM   WATER  [198 

The  estimation  of  the  angular  opening  in  the  disk  used  for  the  water 
experiments  depended  upon  measurements  of  corresponding  chord  and 
diameter.  The  chord,  measured  by  means  of  the  screw  of  a  travelling- 
microscope,  was  '7574  inch.  The  radius,  expressed  in  terms  of  the  same 
unit,  was  found  to  be  7'79.  Hence,  if  a  be  the  angular  opening, 

•7574 


or  £or  =  2°  47'  =167'. 

The  ratio  in  which  the  direct  light  is  reduced  is  thus 
167  167 


180  x  60     10800 


=  -01546. 


It  will  now  be  necessary  to  give  some  details  with  respect  to  the  actual 
matches  as  determined  photographically.  At  first  the  intention  was  to 
employ  ordinary  plates  (Ilford),  which  worked  very  satisfactorily.  But  when 
the  attempt  was  made  to  compare  the  result  with  theory,  the  comparison 
was  found  to  be  embarrassed  by  uncertainty  as  to  the  effective  wave-length 
of  the  light  in  operation.  Moreover,  as  these  plates  are  scarcely  sensitive 
to  yellow  and  green  light,  the  effective  wave-length  is  liable  to  considerable 
variation  with  the  current  used  to  ignite  the  lamp.  Photographs  were  in- 
deed taken  of  the  spectrum  of  the  lamp  as  actually  employed,  but  the 
unsymmetrical  character  of  the  falling  off  at  the  two  ends  made  it  difficult 
to  fix  upon  the  centre  of  activity.  Recourse  was  then  had  to  Edwards' 
"  isochromatic "  plates.  The  spectrum  of  the  lamp,  as  photographed  upon 
these  plates  after  passing  through  a  pale  yellow  glass,  was  very  well  defined, 
lying  with  almost  perfect  symmetry  between  the  sodium  and  the  thallium 
lines.  It  was,  therefore,  determined  to  use  these  plates  and  the  same  yellow 
glass  in  the  actual  experiments,  so  that 

X  =  |  (5892  +  5349)  =  5620 
could  be  taken  as  the  representative  wave-length. 

The  only  disadvantage  arising  from  this  change  was  in  the  necessary 
prolongation  of  the  exposure,  which  became  somewhat  tedious.  Although 
no  dense  image  is  required  or  indeed  desirable,  the  exposure  should  be  such 
that  the  development  does  not  need  to  be  forced.  Two  photographs,  with 
different  times  of  screening,  were  usually  taken  upon  the  same  plate,  the 
object  being  to  obtain  a  reversal  of  relative  intensity,  so  that  in  one  image 
the  semicircle  representing  the  direct  light  should  be  more  intense  and  in 
the  other  image  the  semicircle  representing  the  reflected  light.  The  best 
way  of  examining  the  pictures  depended  somewhat  upon  circumstances. 
When  the  exposure  and  development  had  been  suitable,  the  most  effective 
view  for  the  detection  of  a  feeble  difference  was  obtained  by  placing  the  dry 
picture,  film  downwards,  upon  a  piece  of  opal  glass.  The  light  returned  to 


1892]  AND    MERCURY    AT   NEARLY    PERPENDICULAR   INCIDENCE.  11 

the  eye  had  then  for  the  most  part  traversed  the  film  twice,  with  the  effect 
of  doubling  any  feeble  difference  which  would  occur  on  simple  transmission. 
Under  favourable  circumstances  it  was  possible  to  detect  a  reversal  between 
the  two  images  when  the  difference  amounted  to  3|  per  cent.  A  few  such 
experiments  might  therefore  be  expected  to  give  the  required  result  accurate 
to  less  than  one  per  cent. 

With  the  Edwards'  plates  an  exposure  of  12  minutes  was  found  to  be 
necessary.  This  was  divided  into  two  parts  of  6  minutes  each,  with  an 
interval  of  one  minute  during  which  the  mirrors  were  reversed.  About  the 
middle  of  each  period  of  6  minutes  the  direct  light  was  screened  off  for  a  time 
which  varied  from  picture  to  picture.  For  example,  on  June  6,  the  time  of 
screening  for  one  picture  was  71  seconds,  and  for  the  second  picture  48  seconds. 
This  means  that  while  in  both  pictures  the  exposure  for  the  reflected  light 
was  12  minutes  or  720  seconds,  the  exposures  for  the  direct  light  were 
respectively  720  -  2  x  71  =  578  seconds,  and  720  -  2  x  48  =  624  seconds.  The 
water  was  distilled,  and  its  temperature  was  17°'7  C.  The  examination  of 
the  finished  pictures  showed  that  the  contrast  was  reversed,  so  that  the 
total  exposure  (to  the  direct  light)  required  for  a  balance  was  intermediate 
between  578  and  624,  and,  further,  that  the  first  mentioned  was  the  nearer 
to  the  mark. 

The  general  conclusion  derived  from  a  large  number  of  photographs  was 
that  the  balance  corresponded  to  a  total  screening  of  121  seconds,  viz.,  to  an 
exposure  of  720  -  121  =  599  seconds.  This  is  for  the  direct  light,  the  exposure 
to  the  reflected  light  being  always  720  seconds.  The  ratio  of  exposures 
required  for  a  balance  is  thus 

599  :  720; 

and  this  may  be  considered  to  correspond  to  a  temperature  of  18°  C. 

We  can  now  calculate  the  observed  reflexion  for  6|°  incidence,  reckoned  as 
a  fraction  of  the  incident  light.  We  have 

599     167     /104-4ay 
720'  10800  \82-2l)  ~ 

The  above  relates  to  the  impression  upon  Edwards'  plates  after  the  light 
had  been  transmitted  through  a  yellow  glass.  When  Ilford  plates  were 
substituted  and  the  yellow  glass  omitted,  the  reflexion  appeared  decidedly 
more  powerful,  and  the  ratio  of  exposures  necessary  for  a  balance  was  about 
425  :  480,  or  637  :  720.  It  appears,  therefore,  that  the  reflexion  of  the  light 
operative  in  this  case  is  some  6  per  cent,  more  than  before,  or  about  '0220  of 
the  incident  light.  As  to  a  large  increase  of  reflexion  there  was  no  doubt ; 
but,  owing  perhaps  to  variations  in  the  quality  of  the  light,  the  agreement 
between  individual  results  was  not  so  good  as  before. 


12  ON   THE   INTENSITY   OF   LIGHT  REFLECTED   FROM   WATER  [198 

It  now  remains  to  calculate  the  reflexion  as  given  by  Fresnel's  formulae  ; 
and  it  appears  from  the  discussion  at  the  commencement  of  this  paper  that 
we  may  ignore  the  small  angle  of  incidence  (6^°)  and  take  the  formula  in  the 
simple  form  given  by  Young,  viz.  :  — 


As  to  the  value  of  /A  for  water,  Wiillner*  gives 

^  =  1-326067  -  -000099  1  +  '30531  \~2, 

t  denoting  the  temperature  in  Centigrade  degrees.     Applied  to  18°  and  to 
\  =  5620,  this  gives 

p  =  1-333951, 
whence 


The  reflexion  actually  found  is  accordingly  about  1|-  per  cent,  greater  than 
that  given  by  Fresnel's  formulae. 

In  order  to  estimate  the  effect,  according  to  the  formula,  of  a  change  in 
index,  we  may  use 

SR_   4fy, 
R  ~"j#^I' 
or,  in  the  case  of  water, 

8R  /R  =  5Sfj,  nearly. 

To  cause  a  variation  of  1^  per  cent,  in  the  reflexion,  £//,  would  have  to  be 
•003,  and  to  cause  6  per  cent.  S/z  would  have  to  be  "012.  The  latter  exceeds 
the  variation  of  /j,  in  passing  between  the  lines  D  and  H. 

The  agreement  with  Fresnel's  formula  is  thus  pretty  good,  but  the 
question  arises  whether  it  ought  not  to  be  better.  Apart  from  a  priori 
ideas  as  to  the  result  to  be  expected,  I  should  have  estimated  the  errors 
of  experiment  as  not  likely  to  exceed  one-half  per  cent.,  and  certainly  no 
straining  of  judgment  in  respect  of  the  photometric  pictures  would  bring 
about  agreement.  On  the  other  hand,  it  must  be  remembered  that  one  per 
cent,  is  not  a  large  error  in  photometry,  and  that  in  the  present  case  a 
one  per  cent,  error  in  the  reflexion  is  but  one  in  5000  reckoned  as  a  fraction 
of  the  incident  light.  While,  therefore,  the  disagreement  may  be  real,  it  is 
too  small  a  foundation  upon  which  to  build  with  any  confidence. 

It  only  remains  to  record  the  results  of  some  observations  upon  the 
reflexion  from  mercury.  In  these  experiments  the  revolving  disk  was  dis- 
pensed with,  and  the  photographs  were  taken  upon  Edwards'  plates  through 
yellow  glass.  The  angle  of  incidence  and  all  the  other  arrangements  remained 
as  before.  In  order  to  obtain  a  balance  it  appeared  that  the  direct  light 

*  Fogg.  Ann.  Bd.  cxxxiu. 


1892]  AND   MERCURY   AT   NEARLY   PERPENDICULAR   INCIDENCE.  13 

required  to  be  screened  for  64  seconds  out  of  120  seconds.     The  reflexion  is 
accordingly 

56  /104-43y 


The  mercury  was  of  good  quality,  and  was  filtered  into  a  glass  vessel  just 
before  use.  The  level  was  adjusted  to  be  the  same  as  that  adopted  for  the 
observations  upon  water.  A  surface  thus  obtained  would  not  be  free  from 
a  greasy  layer,  but  it  is  not  probable  that  this  would  sensibly  influence  the 
reflexion. 


APPENDIX. 

The  calculation  of  the  reflexion  depends  upon  the  assumption  that  the 
reflecting  surface  is  plane  ;  and  a  very  moderate  concavity  would  suffice  to 
explain  the  small  excess  in  the  observed  number  for  water  over  that  calculated 
from  Fresnel's  formula?.  It  is  thus  of  importance  to  assure  ourselves  that 
the  concavity  due  to  capillarity  is  really  small  enough  to  be  neglected.  For 
this  purpose  an  estimate  founded  upon  the  capillary  surface  applicable  in  two 
dimensions  will  suffice. 

If  6  be  the  inclination  to  the  horizon  at  any  point,  x  the  horizontal  and  y 
the  vertical  coordinate,  the  equations  to  the  surface  are  :  — 


where 

a* 

At  a  great  distance  from  the  edge, 

0  =  0,  y  =  0,  x  =  oo  . 

At  the  vertical  edge  of  a  wetted  vessel,     0  =  ^tr. 
The  origin  of  x  corresponds  to 

0  =  TT,  y=2a. 

In  the  case  of  water  T  =  74,    p  =  l,    and    g  =  981  C.G.s.  ;  so  that 
a  =  '274  centim. 

In  the  experiments  upon  reflexion  the  part  of  the  surface  in  action  was 
about  11  centim.  away  from  the  boundary,  so  that  x\  a  =  40,  and  6  is  very 
small. 


14  INTENSITY  OF   LIGHT  REFLECTED   FROM  WATER    AND   MERCURY.      [198 

For  the  curvature 

or  for  our  present  purpose 

To  find  0  we  have  approximately, 

cot  £  0  =  e 3a,     or     0  =  4e-38. 
Accordingly 


p     '274  xe38' 

This  may  be  multiplied  by  4  to  represent  the  increase  of  effect  in  the  actual 
circumstances  as  compared  with  what  is  supposed  in  the  two-dimensional 
problem ;  but  it  remains  absolutely  insensible  in  comparison  with  the  other 
curvatures  involved. 


199. 

ON  THE  INTERFERENCE   BANDS  OF  APPROXIMATELY  HOMO- 
GENEOUS  LIGHT;  IN   A   LETTER  TO   PROF.   A.   MICHELSON. 

[Philosophical  Magazine,  xxxiv.  pp.  407—411,  1892.] 

WHEN  we  were  discussing  together  the  results  of  your  interesting  work 
upon  high  interference,  you  asked  my  opinion  upon  one  or  two  questions 
connected  therewith.  I  have  delayed  answering  until  I  had  the  opportunity 
of  seeing  your  paper  in  print  (Phil.  Mag.  Sept.  1892),  but  now  I  may  as  well 
send  you  what  I  have  to  say. 

First,  as  to  the  definiteness  with  which  the  character  of  the  spectral  line 
(f>(x)  can  be  deduced  from  the  "  visibility-curve."  By  Fourier's  theorem, 

i  r°°     f         f+o°  r+o0  .  ) 

<£(#)=  —  I    dulcosuxl      cosuv<f>(v)dv  +  sm  ux  I      sin  uv<b(o)dv\ ; 

TTJo  (  J-oo  J  -oo  J 

or  in  your  notation,  if  we  identify  u  with  27T.D, 


=  -  I    du  I G  cos  ux  +  S  sin  ux 

Hence,  if  C  and  S  are  both  given  as  functions  of  u,  <f>(x)  is  absolutely,  and 
uniquely,  determined.  However,  the  visibility-curve  by  itself  gives,  not  both 
C  and  S,  but  only  C*  +  S*;  so  that  we  must  conclude  that  in  general  an 
indefinite  variety  of  structures  is  consistent  with  a  visibility-curve  given  in 
all  its  parts. 

But  if  we  may  assume  that  the  structure  is  symmetrical,  S  =  0  ;  and  <f>  is 
then  determined  by  means  of  (7.  And,  since  F2=  (72/P2,  the  visibility-curve 
determines  C,  or  at  least  C2.  In  practice,  considerations  of  continuity  would 
always  fix  the  choice  of  the  square  root.  Thus,  in  the  case  of  a  spectral  band 
of  uniform  brightness,  where 

we  are  to  take 
and  not 


16  ON  THE   INTERFERENCE   BANDS  [199 

In  order  to  determine  both  C  and  S,  observations  would  have  to  be  made 
not  only  upon  the  visibility,  but  also  upon  the  situation  of  the  bands.  You 
remark  that  "  it  is  theoretically  possible  by  this  means  to  determine,  in  case 
of  an  unequal  double,  or  a  line  unsymmetrically  broadened,  whether  the 
brighter  side  is  towards  the  blue  or  the  red  end  of  the  spectrum."  But  I 
suppose  that  a  complete  determination  of  both  C  and  S,  though  theoretically 
possible,  would  be  an  extremely  difficult  task. 

If  the  spectral  line  has  a  given  total  width,  the  "  visibility  "  begins  to  fall 
away  from  the  maximum  (unity)  most  rapidly  when  the  brightness  of  the  line 
is  all  concentrated  at  the  edges,  so  as  to  constitute  a  double  line. 

It  is  interesting  to  note  that  in  several  simple  cases  the  bands  seen  with 
ever  increasing  retardation  represent  the  character  of  the  luminous  vibration 
itself.  In  the  case  of  a  mathematical  spectral  line,  the  waves  are  regular  to 
infinity,  and  the  bands  are  formed  without  limit  and  with  maximum  visibility 
throughout.  Again,  in  the  case  of  a  double  line  (with  equal  components)  the 
waves  divide  themselves  into  groups  with  intermediate  evanescences,  and 
this  is  also  the  character  of  the  interference  bands.  Thirdly,  if  the  spectral 
line  be  a  band  of  uniform  brightness,  and  if  the  waves  at  the  origin  be 
supposed  to  be  all  in  one  phase,  the  actual  compound  vibration  will  be 
accurately  represented  by  the  corresponding  interference  bands.  But  this 
law  is  not  general  for  the  reason  that  in  one  case  we  have  to  deal  with 
amplitudes  and  in  the  other  with  intensities.  The  accuracy  of  correspondence 
thus  requires  that  the  finite  amplitudes  involved  be  all  of  one  magnitude.  A 
partial  exception  to  this  statement  occurs  in  the  case  of  a  spectral  line  in 
which  the  distribution  of  brightness  is  exponential. 

Another  question  related  to  the  effect  of  the  gradual  loss  of  energy,  from 
communication  to  the  ether,  upon  the  homogeneity  of  the  light  radiated  from 
freely  vibrating  molecules.  In  illustration  of  this  we  may  consider  the 
analysis  by  Fourier's  theorem  of  a  vibration  in  which  the  amplitude  follows 
the  exponential  law,  rising  from  zero  to  a  maximum,  and  afterwards  falling 
again  to  zero.  It  is  easily  proved  that 

du  COS  UX  ( 


in  which  the  second  member  expresses  an  aggregate  of  trains  of  waves,  each 
individual  train  being  absolutely  homogeneous.  If  a  be  small  in  comparison 
with  r,  as  will  happen  when  the  amplitude  on  the  left  varies  but  slowly, 
e-<M+r)'/4a«  may  ke  neglected,  and  e~(u~r^^  is  sensible  only  when  u  is  very 
nearly  equal  to  r. 

As  an  example  in  which  the  departure  from  regularity  consists  only  in  an 
abrupt  change  of  phase,  let  us  suppose  that 


1892]  OF   APPROXIMATELY   HOMOGENEOUS   LIGHT.  17 

the  sign  being  reversed  at  every  interval  of  ml,  so  that  the  positive  sign 
applies  from  0  to  ml,  2  ml  to  3  ml,  4  ml  to  5  ml,  &c.,  and  the  negative  sign 
from  ml  to  2  ml,  3  ml  to  4  ml,  &c.  As  the  analysis  into  simple  waves  we  find 

, 
^' 

the  summation  extending  to  odd  values  1,  3,  5,  ...  of  n.  The  fundamental 
component  cos(27nc/2raZ)  and  every  odd  harmonic  occur,  but  not  to  the  same 
extent.  When  n  is  nearly  equal  to  2m,  the  terms  rise  to  great  relative 
magnitude.  The  most  important  are  thus 

27raj/1         1  \  27nc/n         2  \ 

cos— r  (l  +  a—    ,       cos  -j-  1 1  -I-  ^-    ,   &c.; 
I    \    ~  2m/  I     \    ~  2mJ 

and  it  is  especially  to  be  remarked  that  what  might  at  first  sight  be  regarded 
as  the  principal,  if  not  the  solitary,  wave-length,  viz.  I,  does  not  occur  at  all. 

Besides  communication  of  energy  to  the  ether,  and  disturbance  during 
encounters  with  neighbours,  the  motion  of  the  molecule  itself  has  to  be  con- 
sidered as  hostile  to  homogeneity  of  radiation.  The  effect,  according  to 
Doppler's  principle,  of  motion  in  the  line  of  sight  was  calculated  by  me  on  a 
former  occasion  and  is  fully  regarded  in  your  paper.  But  there  is  another, 
and  perhaps  more  important,  consequence  of  molecular  motion,  which  does 
not  appear  to  have  been  remarked.  Besides  the  motion  of  translation  there 
is  the  motion  of  rotation  to  be  reckoned  with.  The  effect  of  the  latter  will 
depend  upon  the  law  of  radiation  in  various  directions  from  a  stationary 
molecule.  As  to  this  we  do  not  know  much,  but  enough  to  exclude  the  case 
of  radiation  alike  in  all  directions,  as  from  an  ideal  source  of  sound.  Such  a 
symmetry  is  indeed  inconsistent  with  the  law  of  transverse  vibrations.  The 
simplest  supposition  is  that  the  radiation  is  like  that  generated  in  an  elastic 
solid,  at  one  point  of  which  there  acts  a  periodic  force  in  a  given  direction. 
In  this  case  the  amplitude  in  any  direction  varies  as  the  sine  of  the  angle 
between  the  ray  and  the  force,  and  the  direction  of  (transverse)  vibration  lies 
in  the  plane  containing  these  two  lines.  A  complete  investigation  of  the 
radiation  from  such  molecules  vibrating  and  rotating  about  all  possible  axes 
would  be  rather  complicated,  but  from  one  or  two  particular  cases  it  is  easy 
to  recognize  the  general  character  of  the  effect  produced.  Suppose,  for 
example,  that  the  axis  of  rotation  is  perpendicular  to  the  axis  of  vibration, 
and  consider  the  radiation  in  a  direction  perpendicular  to  the  former  axis. 
If  o>  be  the  angular  velocity,  the  amplitude  varies  as  costot,  and  the  vibration 
may  be  represented  by 

2  cos  cat .  cos  nt  =  cos  (n  +  o>)  t  +  cos  (n  —  &>)  t. 

The  spectrum  would  thus  show  a  double  line,  whose  components  are  separated 
by  a  distance  proportional  to  o>. 

R.    iv.  2 


18      INTERFERENCE   BANDS   OF   APPROXIMATELY   HOMOGENEOUS   LIGHT.      [199 

Again,  if  the  ray  be  parallel  to  the  axis  of  rotation,  the  amplitude  is 
indeed  constant  in  magnitude,  but  its  direction  rotates.  The  plane-polarized 
rays  into  which  the  vibration  may  be  resolved  are  represented  as  before  by 
cos  ait .  cos  nt.  There  is  of  course  one  case  in  which  these  complications  fail  to 
occur,  i.e.  when  the  axis  of  rotation  coincides  with  the  axis  of  vibration ; 
but  with  axes  distributed  at  random  we  must  expect  vibrations  (n  ±  <u)  to  be 
almost  as  important  as  the  vibration  n.  The  law  of  distribution  of  brightness 
in  the  spectral  line  would  probably  be  exponential,  as  when  the  widening  is 
due  to  motion  of  molecules  as  wholes  in  the  line  of  sight. 

It  will  be  of  interest  to  compare  the  magnitudes  of  the  two  effects.  If  v 
be  the  linear  velocity  of  a  molecule  and  V  that  of  light,  the  comparison  is 
between  a>  and  nv/  V,  or  between  «o  and  v/\.  If  r  be  the  radius  of  a  molecule, 
the  circumferential  velocity  of  rotation  is  o>r,  and  we  may  compare  o>r  with 
vr/\.  Now,  according  to  Boltzmann's  theorem,  ra>  would  be  of  the  same  order 
of  magnitude  as  v,  so  that  the  importance  of  the  rotatory  and  linear  effects 
would  be  somewhat  as  X :  r.  There  is  every  reason  to  suppose  that  X  is  much 
greater  than  r,  and  thus  (if  Boltzmann's  relation  held  good)  to  expect  that 
the  disturbance  of  homogeneity  due  to  rotation  would  largely  outweigh  that 
due  to  translation. 

Your  results  seem  already  to  interpose  serious  obstacles  in  the  way  of 
accepting  such  a  conclusion;  and  the  fact  that  light  may  thus  be  thrown 
upon  a  much  controverted  question  in  molecular  physics  is  only  another  proof 
of  the  importance  of  the  research  upon  which  you  are  engaged. 


200. 


ON  THE  INFLUENCE  OF  OBSTACLES  ARRANGED  IN 
RECTANGULAR  ORDER  UPON  THE  PROPERTIES  OF 
A  MEDIUM. 

[Philosophical  Magazine,  xxxiv.  pp.  431 — 502,  1892.] 

THE  remarkable  formula,  arrived  at  almost  simultaneously  by  L.  Lorenz* 
and  H.  A.  Lorentzf,  and  expressing  the  relation  between  refractive  index 
and  density,  is  well  known ;  but  the  demonstrations  are  rather  difficult  to 
follow,  and  the  limits  of  application  are  far  from  obvious.  Indeed,  in  some 
discussions  the  necessity  for  any  limitation  at  all  is  ignored.  I  have  thought 
that  it  might  be  worth  while  to  consider  the  problem  in  the  more  definite 
form  which  it  assumes  when  the  obstacles  are  supposed  to  be  arranged  in 
rectangular  or  square  order,  and  to  show  how  the  approximation  may  be 
pursued  when  the  dimensions  of  the  obstacles  are  no  longer  very  small  in 
comparison  with  the  distances  between  them. 

Taking,  first,  the  case  of  two  dimensions,  let  us  investigate  the  con- 
ductivity for  heat,  or  electricity,  of  an  otherwise  uniform  medium  interrupted 
by  cylindrical  obstacles  which  are  arranged  in  rectangular  order.  The  sides 
of  the  rectangle  will  be  denoted  by  a,  /3,  and  the  radius  of  the  cylinders  by  a. 
The  simplest  cases  would  be  obtained  by  supposing  the  material  composing 
the  cylinders  to  be  either  non-conducting  or  perfectly  conducting;  but  it 
will  be  sufficient  to  suppose  that  it  has  a  definite  conductivity  different  from 
that  of  the  remainder  of  the  medium. 

By  the  principle  of  superposition  the  conductivity  of  the  interrupted 
medium  for  a  current  in  any  direction  can  be  deduced  from  its  conductivities 
in  the  three  principal  directions.  Since  conduction  parallel  to  the  axes  of 
the  cylinders  presents  nothing  special  for  our  consideration,  we  may  limit 

*  Wied.  Ann.  xi.  p.  70  (1880). 
t  Wied.  Ann.  ix.  p.  641  (1880). 

2—2 


20 


ON   THE   INFLUENCE   OF   OBSTACLES 


[200 


our  attention  to  conduction  parallel  to  one  of  the  sides  (a)  of  the  rectangular 
structure.     In  this  case  lines  parallel  to  a,  symmetrically  situated  between 

Fig-  l. 


o. 

B 

0 

a 

o 

0 

o 

A 

O 

o 

o 

0 

the  cylinders,  such  as  AD,  BC,  are  lines  of  flow,  and  the  perpendicular  lines 
AB,  CD  are  equipotential. 

If  we  take  the  centre  of  one  of  the  cylinders  P  as  origin  of  polar  co- 
ordinates, the  potential  external  to  the  cylinder  may  be  expanded  in  the 
series 

V=  A0  +  (^V  +  ^r-1)  cos  6  +  (A3r>  +  B3r~3) cos  30  +  ... ,    (1) 

and  at  points  within  the  cylinder  in  the  series 

F '  =  <70  +  <7j  r  cos  0  +  <73 r3  cos  3 0  +  . . . ,  ( 2 ) 

0  being  measured  from  the  direction  of  a.  The  sines  of  0  and  its  multiples 
are  excluded  by  the  symmetry  with  respect  to  0=0,  and  the  cosines  of  the 
even  multiples  by  the  symmetry  with  respect  to  0  =  |TT.  At  the  bounding 
surface,  where  r  =  a,  we  have  the  conditions 

F=F',         vdV'/dr  =  dV/dr, 

v  denoting  the  conductivity  of  the  material  composing  the  cylinders  in  terms 
of  that  of  the  remainder  reckoned  as  unity.  The  application  of  these  con- 
ditions to  the  term  in  cosn0  gives 

7?    —  ^    2w  A  (^\ 

In  the  case  where  the  cylinders  are  perfectly  conducting,  v  =  x  .  If  they 
are  non-conducting,  v  =  0. 

The  values  of  the  coefficients  .4 1}  .B^  A3,B3...  are  necessarily  the  same 
for  all  the  cylinders,  and  each  may  be  regarded  as  a  similar  multiple  source 
of  potential.  The  first  term  A0,  however,  varies  from  cylinder  to  cylinder,  as 
we  pass  up  or  down  the  stream. 

Let  us  now  apply  Green's  theorem, 


an 


an 


.(4) 


1892]  IN   RECTANGULAR   ORDER   UPON    A   MEDIUM.  21 

to  the  contour  of  the  region  between  the  rectangle  A  BCD  and  the  cylinder  P. 
Within  this  region  V  satisfies  Laplace's  equation,  as  also  will  U,  if  we 
assume 

U  =  x  =  r  cos  6  ..................................  (5) 

Over  the  sides  BC,  AD,  dU/dn,  dV/dn  both  vanish.  On  CD,  $dV/dn.ds 
represents  the  total  current  across  the  rectangle,  which  we  may  denote  by  C. 
The  value  of  this  part  of  the  integral  over  CD,  AB  is  thus  aC.  The  value 
of  the  remainder  of  the  integral  over  the  same  lines  is  —  F,$,  where  V^ 
is  the  fall  in  potential  corresponding  to  one  rectangle,  as  between  CD 
and  AB. 

On  the  circular  part  of  the  contour, 


and  thus  the  only  terms  in  (1)  which  will  contribute  to  the  result  are  those 
in  cos  0.     Thus  we  may  write 


dV/dn  =  -  (A1  - 

so  that  this  part  of  the  integral  is  2-rrB^  The  final  result  from  the  application 
of  (4)  is  thus 

«C-/9F1  +  2irB1  =  0  ............................  (6) 

If  #1  =  0,  we  fall  back  upon  the  uninterrupted  medium  of  which  the  con- 
ductivity is  unity.  For  the  case  of  the  actual  medium  we  require  a  further 
relation  between  Bl  and  Vl. 

The  potential  V  at  any  point  may  be  regarded  as  due  to  external  sources 
at  infinity  (by  which  the  flow  is  caused)  and  to  multiple  sources  situated 
on  the  axes  of  the  cylinders.     The  first  part  may  be  denoted  by  Hx.     In 
considering  the  second  it  will  conduce  to  clearness  if  we  imagine  the  (infinite) 
region  occupied  by  the  cylinders  to  have  a  rectangular  boundary  parallel  to 
a  and  /3.     Even  then  the  manner  in  which  the  infinite  system  of  sources 
is  to  be  taken  into  account  will  depend  upon  the  shape  of  the  rectangle. 
The  simplest  case,  which  suffices  for  our  purpose,  is  when  we  suppose  the 
rectangular  boundary  to  be  infinitely  more  extended  parallel  to  a  than  parallel 
to  /3.     It  is  then  evident  that  the  periodic  difference  Vl  may  be  reckoned 
as  due  entirely  to  Hx,  and  equated  to  Ha.     For  the  difference  due  to  the 
sources  upon  the  axes  will  be  equivalent  to  the  addition  of  one  extra  column 
at  +  QO  ,  and  the  removal  of  one  at   —  oo  ,  and  in   the  case  supposed  such 
a  transference  is  immaterial*.     Thus 

V,  =  Ha  ....................................  (7) 

simply,  and  it  remains  to  connect  H  with  BI. 

*  It  would  be  otherwise  if  the  infinite  rectangle  were  supposed  to  be  of  another  shape,  e.g.  to 
be  square. 


22  j.     ON   THE   INFLUENCE   OF   OBSTACLES  [200 

This  we  may  do  by  equating  two  forms  of  the  expression  for  the  potential 
at  a  point  x,  y  near  P.  The  part  of  the  potential  due  to  Hx  and  to  the 
multiple  sources  Q  (P  not  included)  is 


or,  if  we  subtract  Hx,  we  may  say  that  the  potential  at  x,  y  due  to  the 
multiple  sources  at  Q  is  the  real  part  of 

A0  +  (A1-H)(x  +  iy)  +  A3(x  +  iy)*  +  As(x  +  iy)*  +  ..........  (8) 

But  if  x',  y'  are  the  coordinates  of  the  same  point  when  referred  to  the  centre 
of  one  of  the  Q's,  the  same  potential  may  be  expressed  by 

2{B1(x'  +  iy')->  +  B3(x'  +  iy')-*+...},    ..................  (9) 

the  summation  being  extended  over  all  the  Q's.     If  £,  17  be  the  coordinates 
of  a  Q  referred  to  P, 

x'  =  x-£,     y'  =  y-r); 
so  that 

Bn(x'  +  iy')~n  =  Bn(x  +  iy-£  -117)-*. 

Since  (8)  is  the  expansion  of  (9)  in  rising  powers  of  (x  +  iy),  we  obtain, 
equating  term  to  term, 


-1.2.3^3=1.2.35^4  +  3.4.55326  +  ...  ...(10) 

-  1  .  2  .  3  .  4  .  5  4  6  =  1  .  2  .  3  .  4  .  5  #!  26  +  3  .  4  .  5  .  6  .  7  55  28  +  .  .  .  J 
and  so  on,  where 

2,n  =  2(f  +  t17)-»,    ..............................  (11) 

the  summation  extending  over  all  the  Q's. 

By  (3)  each  B  can  be  expressed  in  terms  of  the  corresponding  A.     For 
brevity,  we  will  write 

An  =  v'a-mBn,     ....  ..........................  (12) 

where 

*/=(!  +  *)/(!-*)  .............................  (13) 

We  are  now  prepared  to  find  the  approximate  value  of  the  conductivity. 
From  (6)  the  conductivity  of  the  rectangle  is 


so  that  the  specific  conductivity  of  the  actual  medium  for  currents  parallel 
to  a  is 


and  the  ratio  of  H  to  Bl  is  given  approximately  by  (10)  and  (12). 

In  the  first  approximation  we  neglect  24,  26  ...,  so  that  AS,AS...  B3,  Bs  ... 
vanish.     In  this  case 

(15) 


1892]  IN   RECTANGULAR  ORDER  UPON   A    MEDIUM.  23 

and  the  conductivity  is 

2-Tra2 
"££(,/  +  a%)  ............................  (16) 

The  second  approximation  gives 

^W  +  a^  —  a'S,',    ........................  (17) 

and  the  series  may  be  continued  as  far  as  desired. 

The  problem  is  thus  reduced  to  the  evaluation  of  the  quantities  2a,  24,.... 
We  will  consider  first  the  important  particular  case  which  arises  when  the 
cylinders  are  in  square  order,  that  is  when  f3  =  a.  £  and  77  in  (11)  are  then 
both  multiples  of  a,  and  we  may  write 

2n=a-nSn,  .................................  (18) 

where 

Sn=2(m'  +  im)-n;  ...........................  (19) 

the  summation  being  extended  to  all  integral  values  of  m,  m',  positive  or 
negative,  except  the  pair  m  =  0,  m'  =  0.  The  quantities  S  are  thus  purely 
numerical,  and  real. 

The  next  thing  to  be  remarked  is  that,  since  m,  m'  are  as  much  positive 
as  negative,  Sn  vanishes  for  every  odd  value  of  n.  This  holds  even  when 
a  and  ft  are  unequal. 

Again, 

Sm  =  2  (TO'  +  tw)-*1  =  i-™  2  (-  im' 


-  m 
Whenever  n  is  odd,  S^  =  —  S^,  or  $m  vanishes.     Thus  for  square  order, 

S.  =  8U  =  -SU=  ......  =  0  .........................  (20) 

This  argument  does  not,  without  reservation,  apply  to  8,.  In  that  case 
the  sum  is  not  convergent  ;  and  the  symmetry  between  m  and  m',  essential 
to  the  proof  of  evanescence,  only  holds  under  the  restriction  that  the  infinite 
region  over  which  the  summation  takes  place  is  symmetrical  with  respect 
to  the  two  directions  a  and  ft  —  is,  for  example,  square  or  circular.  On  the 
contrary,  we  have  supposed,  and  must  of  course  continue  to  suppose,  that  the 
region  in  question  is  infinitely  elongated  in  the  direction  of  a. 

The  question  of  convergency  may  be  tested  by  replacing  the  parts  of 
the  sum  relating  to  a  great  distance  by  the  corresponding  integral.  This  is 

[f    dxdy     _  [[cos2n0rdrd0  . 

])(x  +  iyr~M  »*» 

and  herein 

fr-^+1dr  =  r-^+2l(-  2w  +  2)  ; 

so  that  if  7i  >  1  there  is  convergency,  but  if  n  —  1  the  integral  contains  an 
infinite  logarithm. 


24  ON   THE   INFLUENCE   OF   OBSTACLES  [200 

We  have  now  to  investigate  the  value  of  S2  appropriate  to  our  purpose ; 
that  is,  when  the  summation  extends  over  the  region  bounded  by  x  —  ±  u, 
y  =  ±v,  where  u  and  v  are  both  infinite,  but  so  that  v/u  =  Q.  If  we  suppose 
that  the  region  of  summation  is  that  bounded  by  &•  =  +  v,  y—±v,  the  sum 
vanishes  by  symmetry.  We  may  therefore  regard  the  summation  as  ex- 
tending over  the  region  bounded  externally  by  x  =  +  00  ,  y  =  ±v,  and  internally 

Fig.  2. 


by  ac  =  ±  v  (Fig.  2).    When  v  is  very  great,  the  sum  may  be  replaced  by  the 
corresponding  integral.     Hence 


the  limits  for  y  being  ±  v,  and  those  for  x  being  v  and  oo  .     Ultimately  v  is  to 
be  made  infinite. 

We  have 

dy  j  i  2V 


/*+* 
J  _„  (a; 


=  = 

+  iyf     x  +  iv     x  —  iv     x*  +  vz  ' 
and 


Accordingly 

S.-T (22) 

In  the  case  of  square  order,  equations  (10),  (12)  give 
Ha?  3    _  7     . 


=  ,,'  +  ^._±!l^4»_l/  _£82-  ...;  (23) 

and  by  (14) 

Conductivity  =1--^  .  ^ (24) 

If  p  denote  the  proportional  space  occupied  by  the  cylinders, 

P  —  TTO^IO?;    (25) 

and 

Conductivity  =  1 ^-^ - (26) 


1892]  IN   RECTANGULAR   ORDER   UPON    A   MEDIUM.  25 

Of  the    double   summation   indicated  in  (19)  one  part  can  be  effected 
without  difficulty.     Consider  the  roots  of 

sin  (f  —  irmr)  =  0. 
They  are  all  included  in  the  form 

'        innr, 


where  m    is  any  integer,  positive,  negative,  or  zero.      Hence  we  see  that 
sin  (f  —  irmr)  may  be  written  in  the  form 


-^  .  ... 

irmr/  \        irmr  +  TrJ\        irmr  —  irj  \        irmr  +  2-rrJ 

in  which 

A  =  —  sin  irmr. 
Thus 

log  fcos  £  —  cot  ivmr  sin  £]  =  log  (  1  —  ^—  ]  +  log  (  1  —  -:  —  -  —  )  +  .  . 
s\        irmrj         ^  \        irmr  +  Tr/ 

If  we  change  the  sign  of  m,  and  add  the  two  equations,  we  get 


whence,  on  expansion  of  the  logarithms, 
sm2  sin4£ 

I  __  2  __  I  __  » 

' 


_  __   __  __  __ 

sin2im7r      2snrH'w7r     3  sin6  i  rmr 


'          *  """        ' 


(irmr)2     (irmr  4-  7r)2      (irmr  —  7r)2 
+  W  \ ,_.__.    x,  +  (t-m7r  +  ^4  +  ^-m ^ _  ^4  +  ••• 


+  ^a{J_  1  1  ]    , 

^?    ((irmr)6     (irmr  +  TT)"     (imw  -  vr)6  j 

By  expanding  the  sines  on  the  left  and  equating  the  corresponding  powers 
of  |,  we  find 

1  1  1  1  7T2 


(tm)2     (im  +  I)2     (im-1)2     (im  +  2)2      ......  ~  son"  tin  w 


l      I  ^^  * 

(im)4     (im  +  1)4  3sin2im7T     si"'-' 


__  __        __ 

8        '  6  15sin2tm7r     s4  8' 


26 


ON   THE   INFLUENCE   OF   OBSTACLES 


[200 


In  the  summation  with  respect  to  m,  required  in  (19),  we  are  to  take 
all  positive  and  negative  integral  values.  But  in  the  case  of  m  =  0  we  are 
to  leave  out  the  first  term,  corresponding  to  m'  =  0.  When  m  =  0, 


sm2ira7r     (im)2      3  ' 
which,  as  is  well  known,  is  the  value  of 
1111 


Hence 


and  in  like  manner 


»l  =  oo 
2    2    SUrtWTT +  J7T2; 


.(30) 


?4  =  ^  +  27r4   2  {- 1  snrtW  -f  sin-Hm-Tr},    (31) 


27r8      4.9    e"!00 

7i 7^~~  ~T  ATT      £* 


(32) 


We  have  seen  already  that  86  =  0,  and  that  S2  =  TT.     The  comparison  of  the 
latter  with  (30)  gives 

"-"  '  -"'  1-J.  ...(33) 


We    will   now    apply  (31)    to    the    numerical    calculation  of  S4.       We 


find: 


m 

—  sin~2  irmr 

sin"4  imx 

I 

•00749767 

•0000562150 

2 

v       1395 

2 

3 

3 

Sum 

•00751165 

•0000562152 

so  that 

S4=7r4x.  03235020 (34) 

In  the  same  way  we  may  verify  (33),  and  that  (32)  =  0. 

If  we  introduce  this  value  into  (26),  taking  for  example  the  case  where 
the  cylinders  are  non-conductive  (y'=  1),  we  get 


1- 


.(35) 


From  the  above  example  it  appears  that  in  the  summation  with  respect 
to  m  there  is  a  high  degree  of  convergency.  The  reason  for  this  will  appear 
more  clearly  if  we  consider  the  nature  of  the  first  summation  (with  respect 


1892]  IN   RECTANGULAR   ORDER   UPON   A    MEDIUM.  27 

to  m).  In  (19)  we  have  to  deal  with  the  sum  of  (x  +  iy)~n,  where  y  is  for 
the  moment  regarded  as  constant,  while  x  receives  the  values  x  =  m.  If 
instead  of  being  concentrated  at  equidistant  points,  the  values  of  x  were 
uniformly  distributed,  the  sum  would  become 

dx 


Now,  n  being  greater  than  1,  the  value  of  this  integral  is  zero.  We  see, 
then,  that  the  finite  value  of  the  sum  depends  entirely  upon  the  discontinuity 
of  its  formation,  and  thus  a  high  degree  of  convergency  when  y  increases  may 
be  expected. 

The  same  mode  of  calculation  may  be  applied  without  difficulty  to  any 
particular  case  of  a  rectangular  arrangement.     For  example,  in  (11) 

22  =  2  (ma.  +  tra/3)-2  =  cr22(w'  +im/3/a)-'. 
If  m  be  given,  the  summation  with  respect  to  mf  leads,  as  before,  to 


l  , 
a.) 

and  thus 


(36) 


The  numerical  calculation  would  now  proceed  as  before,  and  the  final 
approximate  result  for  the  conductivity  is  given  by  (16).  Since  (36)  is  not 
symmetrical  with  respect  to  a  and  ft,  the  conductivity  of  the  medium  is 
different  in  the  two  principal  directions. 

When  /3  =  a,  we  know  that  a~222  =  TT.  And  since  this  does  not  differ 
much  from  |7r2,  it  follows  that  the  series  on  the  right  of  (36)  contributes 
but  little  to  the  total.  The  same  will  be  true,  even  though  ft  be  not  equal 
to  a,  provided  the  ratio  of  the  two  quantities  be  moderate.  We  may  then 
identify  a~2S2  with  TT,  or  with  ^7r2,  if  we  are  content  with  a  very  rough 
approximation. 

The  question  of  the  values  of  the  sums  denoted  by  JLm  is  intimately 
connected  with  the  theory  of  the  0-  functions  *,  inasmuch  as  the  roots  of  H(u), 
or  O^TTu/ZK),  are  of  the  form 

2m  K  +  2m'iK'. 

The  analytical  question  is  accordingly  that  of  the  expansion  of  log  #j(#) 
in  ascending  powers  of  x.  Now,  Jacobif  has  himself  investigated  the  ex- 
pansion in  powers  of  x  of 

#!(#)  =  2  fa174  sm#-g9/4  sin  Stf  +  g26''4  sin  5#-  ...},  ............  (37) 


*  Cayley's  Elliptic  Functions,  p.  300.     The  notation  is  that  of  Jacobi. 
t  Crelle,  Bd.  LIV.  p.  82. 


28  ON  THE   INFLUENCE   OF   OBSTACLES  [200 

where  q  =  e-«K'tK.  .................................  (38) 

So  far  as  the  cube  of  x  the  result  is 


D  being  a  constant  which  it  is  not  necessary  further  to  specify.  K  and  E 
are  the  elliptic  functions  of  k  usually  so  denoted.  By  what  has  been  stated 
above  the  roots  of  Ox  are  of  the  form 


(40) 
so  that 

Z-k*)K'},     ............  (41) 


the  summation  on  the  left  being  extended  to  all  integral  values  of  m  and  m, 
except  m  =  0,  m'  =  0. 

This  is  the  general  solution  for  22.     If  K'  =  K,     k?  =  %,     and 

2  {m  +  im'}-*  =  2  {2KE  -  K*}  =  TT, 
since  in  general*, 

EK'  +  E'K-KK'  =  £TT. 

In  proceeding   further   it   is  convenient  to  use  the  form  in  which  an 
exponential  factor  is  removed  from  the  series.     This  is 

..  .,...  (42) 


-3 
5!  7! 

in  which 

(43) 


7T  7T  7T 


the  law  of  formation  of  s  being 

sm+l  =  2m  (2m  +  1)  ps^  +  aj3dsm/d/3  -  8^dsm/da,  ......  (44) 


while 

a  =  #»-*»,       f3  =  ^(kk')  ............................  (45) 

I  have  thought  it  worth  while  to  quote  these  expressions,  as  they  do 
not  seem  to  be  easily  accessible  ;  but  I  propose  to  apply  them  only  to  the 
case  of  square  order,  K'  =  K,  k'2  =  k?=%.  Thus 


(46) 

and 


*  Cayley's  Elliptic  Functions,  p.  49. 


1892]  IN   RECTANGULAR   ORDER   UPON    A   MEDIUM. 

Hence 

,      e^x)          a?      AW          Aao? 
l0^  5^  =-2^-275-!- i^35X!- 

If  +XD  ±X.j,  ...  are  the  roots  of  0l(as)/x  =  Q,  we  have 
- 


*•* /v        —  f\       >         •*•'  'v       —   K  i   ?        •** /v       —  v,  —  HT      (r        i       K   i  • 

2-7T  5!  7.5.4.5! 

Now  by  (40)  the  roots  in  question  are  TT  (m,  +  ira'),  and  thus 

™         7T4      .  rt  7TSA 


in  which 


Leaving  the  two-dimensional  problem,  I  will  now  pass  on  to  the  case 
of  a  medium  interrupted  by  spherical  obstacles  arranged  in  rectangular  order. 
As  before,  we  may  suppose  that  the  side  of  the  rectangle  in  the  direction 
of  flow  is  a,  the  two  others  being  /3  and  7.  The  radius  of  the  sphere  is  a. 

The  course  of  the  investigation  runs  so  nearly  parallel  to  that  already 
given,  that  it  will  suffice  to  indicate  some  of  the  steps  with  brevity.  In  place 
of  (1)  and  (2)  we  have  the  expansions 


*)Yn+...,     ............  (50) 

V'=C0+C1Y1r+...+CnYnrn+...,    ............  (51) 

Yn  denoting  the  spherical  surface  harmonic  of  order  n.     And  from  the  surface 
conditions 

V=V, 
we  find 


We    must   now  consider   the    limitations  to  be  imposed  upon  Yn.     In 
general, 

Yn  =  ^  ®n  <*>  (Hs  cos  s<f>  +  Ks  sin  s<f>),  ..................  (53) 

*  =  0 

where 

©„<">  =  sin*0  (cosw-*0  -  ^-sKrc-*-1)  cos«-«-20  +  ......  \       .  .  .(54) 

0  being  supposed  to  be  measured  from  the  axis  of  a;  (parallel  to  a),  and  <f> 
from  the  plane  of  xz.  In  the  present  application  symmetry  requires  that 
s  should  be  even,  and  that  Yn  (except  when  n  =  0)  should  be  reversed  when 


30  ON   THE   INFLUENCE   OF   OBSTACLES  [200 

(tr—d)  is  written  for  0.    Hence  even  values  of  n  are  to  be  excluded  altogether. 
Further,  no  sines  of  s<f>  are  admissible.     Thus  we  may  take 


(55) 
F3  =  cos30-f  cos  0+H2sm*0  cos  0  cos  2<f>,  .....................  (56) 

F5  =  cos8  6  -  J£  cos30  +  ^  cos  (9 

sin20  (cos30  -  £  cos  0)  cos  20 

40cos0cos4<£  ...............................  (57) 

In  the  case  where  ft  =  y  symmetry  further  requires  that 

#2  =  0,     Z2=0  ...............................  (58) 

In  applying  Green's  theorem  (4)  the  only  difference  is  that  we  must  now 
understand  by  s  the  area  of  the  surface  bounding  the  region  of  integration. 
If  C  denote  the  total  current  flowing  across  the  faces  /3<y,  V1  the  periodic 
difference  of  potential,  the  analogue  of  (6)  is 

aC-ftjV,  +  4-^  =  0  ............................  (59) 

We  suppose,  as  before,  that  the  system  of  obstacles,  extended  without 
limit  in  every  direction,  is  yet  infinitely  more  extended  in  the  direction 
of  a  than  in  the  directions  of  ft  and  7.  Then,  if  Hx  be  the  potential  due  to 
the  sources  at  infinity  other  than  the  spheres,  Vl  =  Ha,  and 


so  that  the  specific  conductivity  of  the  compound  medium  parallel  to  a  is 


We  will  now  show  how  the  ratio  B1/H  is  to  be  calculated  approximately, 
limiting  ourselves,  however,  for  the  sake  of  simplicity  to  the  case  of  cubic 
order,  where  a  =  j3  =  y.  The  potential  round  P,  viz. 


may  be  regarded  as  due  to  Hx  and  to  the  other  spheres  Q  acting  as  sources 
of  potential.  Thus,  if  we  revert  to  rectangular  coordinates  and  denote  the 
coordinates  of  a  point  relatively  to  P  by  a,  y,  z,  and  relatively  to  one  of  the 
Q's  by  a/,  y",  z',  we  have 


in  which 

a'  =  x  -  £,     y'  =  y-i1,     z'  =  z-%, 

^  %y  *?>  £  be  the  coordinates  of  Q  referred  to  P.     The  left  side  of  (61)  is  thus 
the  expansion  of  the  right  in  ascending  powers  of  x,  y,  z.      Accordingly, 


1892]  IN   RECTANGULAR   ORDER   UPON    A   MEDIUM.  31 

Al—H  is  found  by  taking  dfdx  of  the  right-hand  member  and  then  making 
x,  y,  z  vanish.  In  like  manner  6  A3  will  be  found  from  the  third  differential 
coefficient.  Now,  at  the  origin, 


_  =  ___=  = 

dx  r'3  ~     d%  r'3         d%  p*  p* 

in  which 

p*  =  ?  +  i)*  +  ?. 

It  will  be  observed  that  we  start  with  a  harmonic  of  order  1  and  that 
the  differentiation  raises  the  order  to  2.  The  law  that  each  differentiation 
raises  the  order  by  unity  is  general  ;  and,  so  far  as  we  shall  proceed,  the 
harmonics  are  all  zonal,  and  may  be  expressed  in  the  usual  way  as  functions 
Pn(fji)  of  p  where  /*  =  £/?.  Thus 


In  like  manner, 
and 


The  comparison  of  terms  in  (61)  thus  gives 

^3=  -45J  So-5  P\  +  5  I <62) 

...      =      j 

In  each  of  the  quantities,  such  as  2p~3P2>  the  summation  is  to  be  ex- 
tended to  all  the  points  whose  coordinates  are  of  the  form  la,  mx,  not,  where 
I,  m,  n  are  any  set  of  integers,  positive  or  negative,  except  0,  0,  0.  If  we 
take  a  =  1,  and  denote  the  corresponding  sums  by  $2,  S4,  ... ,  these  quantities 
will  be  purely  numerical,  and 

V"-1^  =  «-*-'£„ (63) 

From  (52),  (62)  we  now  obtain 


which  with  (60)  gives  the  desired  result  for  the  conductivity  of  the  medium. 
We  now  proceed  to  the  calculation  of  $2.     We  have 


By  the  symmetry  of  a  cubical  arrangement,  it  follows  that 


32  ON   THE   INFLUENCE   OF   OBSTACLES  [200 

so  that  if  $  were  calculated  for  a  space  bounded  by  a  cube,  it  would 
necessarily  vanish.  But  for  our  purpose  $2  is  to  be  calculated  over  the  space 
bounded  by  f  =  ±  oc  ,  i)  =  ±v,  £=±v,  where  v  is  finally  to  be  made  infinite  ; 
and,  as  we  have  just  seen,  we  may  exclude  the  space  bounded  by 


so  that  £$2  will  be  obtained  from  the  space  bounded  by 
^  —  v,     f=oo,     ?)=  +  v,     £=  ±v. 

Now  when  p  is  sufficiently  great,  the  summation  may  be  replaced  by  an 
integration;  thus 


In  this, 

(" 

J, 


and  finally 

1  -v  (*>2  +  r9(20*  +  f1)*  =  Jo  V(2  +  tan20)  =  Jo      >/(2  -  52)  =  3  ' 

Thus 

c      ^7r  /«K\ 

^2  =  -g- V65) 

If  we  neglect  a10/a10,  and  write  p  for  the  ratio  of  volumes,  viz. 


we  have  by  (60)  for  the  conductivity 

-- 

' 


or  in  the  particular  case  of  non-conducting  obstacles  (v  =  0) 


In  order  to  carry  on  the  approximation  we  must  calculate  S4  &c.  Not 
seeing  any  general  analytical  method,  such  as  was  available  in  the  former 
problem,  I  have  calculated  an  approximate  value  of  $4  by  direct  summation 
from  the  formula 


Compare  Maxwell's  Electricity,  §  314. 


1892] 


IN    RECTANGULAR   ORDER   UPON    A    MEDIUM. 


33 


We  may  limit  ourselves  to  the  consideration  of  positive  and  zero  values  of 
f  ,  i],  £  Every  term  for  which  (-,  17,  £  are  finite  is  repeated  in  each  octant, 
that  is  8  times.  If  one  of  the  three  coordinates  vanish,  the  repetition  is 
fourfold,  and  if  two  vanish,  twofold. 

The  following  table  contains  the  result  for  all  points  which  lie  within 
p*  =  18.  This  repetition  in  the  case,  for  example,  of  p2  =  9  represents  two 
kinds  of  composition.  In  the  first 


and  in  the  second 


=  32  +  O2  +  O2  =  9. 


The  success  of  the  approximation  is  favoured  by  the  fact  that  P  vanishes 
when  integrated  over  the  complete  sphere,  so  that  the  sum  required  is  only 
a  kind  of  residue  depending  upon  the  discontinuity  of  the  summation. 


The  result  is 


.(69) 


P2 

P2 

0,  0,  1 

1 

+  3-5000 

0,  0,  3 

9 

+  -0144 

0,  1,  1 

2 

-  -3094 

0,  1,  3 

10 

+  -0243 

1,  1,  1 

3 

-  -1996 

1,  1,  3 

11 

+  -0075 

0,  0,  2 

4 

+  -1094 

2,  2,  2 

12 

-  -0062 

0,  1,  2 

5 

+  -0501 

0,  2,  3 

13 

-  -0015 

1,  1,  2 

6 

-  -0397 

1,  2,  3 

14 

-  -0095 

0,  2,  2 

8 

-  -0097 

0,  0,  4 

16 

+  -0034 

1,  2,  2 

9 

-  -0277 

2,  2,  3 

17 

-  -0061 

0,  1,  4 

17 

+  '0085 

1 

The  results  of  our  investigation  have  been  expressed  for  the  sake  of 
simplicity  in  electrical  language  as  the  conductivity  of  a  compound  medium, 
but  they  may  now  be  applied  to  certain  problems  of  vibration.  The  simplest 
of  these  is  the  problem  of  wave-motion  in  a  gaseous  medium  obstructed  by 
rigid  and  fixed  cylinders  or  spheres.  It  is  assumed  that  the  wave-length 
is  very  great  in  comparison  with  the  period  (a,  ft,  7)  of  the  structure.  Under 
these  circumstances  the  flow  of  gas  round  the  obstacles  follows  the  same 
law  as  that  of  electricity,  and  the  kinetic  energy  of  the  motion  is  at  once 
given  by  the  expressions  already  obtained.  In  fact  the  kinetic  energy 
corresponding  to  a  given  total  flow  is  increased  by  the  obstacles  in  the  same 
proportion  as  the  electrical  resistances  of  the  original  problem,  so  that  the 
influence  of  the  obstacles  is  taken  into  account  if  we  suppose  that  the 

B.     IV.  3 


34  ON   THE   INFLUENCE   OF   OBSTACLES  [200 

density  of  the  gas  is  increased  in  the  above  ratio  of  resistances.      In  the 
case  of  cylinders  in  square  order  (35),  the  ratio  is  approximately 


and  in  the  case  of  spheres  in  cubic  order  by  (68)  it  is  approximately 
(1 


But  this  is  not  the  only  effect  of  the  obstacles  which  we  must  take 
into  account  in  considering  the  velocity  of  propagation.  The  potential 
energy  also  undergoes  a  change.  The  space  available  for  compression 
or  rarefaction  is  now  (1  -  p)  only  instead  of  1;  and  in  this  proportion 
is  increased  the  potential  energy  corresponding  to  a  given  accumulation  of 
gas*.  For  cylindrical  obstruction  the  square  of  the  velocity  of  propagation  is 
thus  altered  in  the  ratio 


so  that  if  fj,  denote  the  refractive  index,  referred  to  that  of  the  unobstructed 
medium  as  unity,  we  find 


(p?-l)jp  =  constant,     ........................  (70) 

which  shows  that  a  medium  thus  constituted  would  follow  Newton's  law 
as  to  the  relation  between  refraction  and  density  of  obstructing  matter.  The 
same  law  (70)  obtains  also  in  the  case  of  spherical  obstacles  ;  but  reckoned 
absolutely  the  effect  of  spheres  is  only  that  of  cylinders  of  halved  density. 
It  must  be  remembered,  however,  that  while  the  velocity  in  the  last  case 
is  the  same  in  all  directions,  in  the  case  of  cylinders  it  is  otherwise.  For 
waves  propagated  parallel  to  the  cylinders  the  velocity  is  uninfluenced  by 
their  presence.  The  medium  containing  the  cylinders  has  therefore  some 
of  the  properties  which  we  are  accustomed  to  associate  with  double  refraction, 
although  here  the  refraction  is  necessarily  single.  To  this  point  we  shall 
presently  return,  but  in  the  meantime  it  may  be  well  to  apply  the  formulae 
to  the  more  general  case  where  the  obstacles  have  the  properties  of  fluid, 
with  finite  density  and  compressibility. 

To  deduce  the  formula  for  the  kinetic  energy  we  have  only  to  bear  in 
mind  that  density  corresponds  to  electrical  resistance.  Hence,  by  (26),  if 
a  denote  the  density  of  the  cylindrical  obstacle,  that  of  the  remainder  of 
the  medium  being  unity,  the  kinetic  energy  is  altered  by  the  obstacles  in  the 
approximate  ratio 

(<r  +  l)/(<r-l)+ff 

(<r  +  l)/(cr-l)-p- 

*  Theory  of  Sound,  §  303. 


1892]  IN   RECTANGULAR   ORDER   UPON   A    MEDIUM.  35 

The  effect  of  this  is  the  same  as  if  the  density  of  the  whole  medium  were 
increased  in  the  like  ratio. 

The  change  in  the  potential  energy  depends  upon  the  "  compressibility  " 
of  the  obstacles.  If  the  material  composing  them  resists  compression  m  times 
as  much  as  the  remainder  of  the  medium,  the  volume^?  counts  only  as p/m, 
and  the  whole  space  available  may  be  reckoned  asl—  p  +p/m  instead  of  1. 
In  this  proportion  is  the  potential  energy  of  a  given  accumulation  reduced. 
Accordingly,  if  p  be  the  refractive  index  as  altered  by  the  obstacles, 

^  =  (n)x(l-p+plm) (72) 

The  compressibilities  of  all  actual  gases  are  nearly  the  same,  so  that  if  we 
suppose  ourselves  to  be  thus  limited,  we  may  set  m  =  l,  and 


(  } 


or,  as  it  may  also  be  written, 

At2-  1  1 

^-y  -  =  constant  ............................  (74) 

In  the  case  of  spherical  obstacles  of  density  a-  we  obtain  in  like  manner 

-- 
or 

^Ji  =  constant  ............................  (76) 

In  the  general  case,  where  m  is  arbitrary,  the  equation  expressing  p  in 
terms  of  p?  is  a  quadratic,  and  there  are  no  simple  formulae  analogous  to 
(74)  and  (76). 

It  must  not  be  forgotten  that  the  application  of  these  formulae  is  limited 
to  moderately  small  values  of  p.  If  it  be  desired  to  push  the  application 
as  far  as  possible,  we  must  employ  closer  approximations  to  (26),  &c.  It 
may  be  remarked  that  however  far  we  may  go  in  this  direction,  the  final 
formula  will  always  give  p?  explicitly  as  a  function  of  p.  For  example,  in  the 
case  of  rigid  cylindrical  obstacles,  we  have  from  (35) 


It  will  be  evident  that  results  such  as  these  afford  no  foundation  for 
a  theory  by  which  the  refractive  properties  of  a  mixture  are  to  be  deduced 
by  addition  from  the  corresponding  properties  of  the  components.  Such 
theories  require  formulae  in  which  p  occurs  in  the  first  power  only,  as 
in  (76). 

3—2 


36  ON   THE   INFLUENCE   OF   OBSTACLES  [200 

If  the  obstacles  are  themselves  elongated,  or  even,  though  their  form 
be  spherical,  if  they  are  disposed  in  a  rectangular  order  which  is  not  cubic, 
the  velocity  of  wave-propagation  becomes  a  function  of  the  direction  of  the 
wave-normal.  As  in  Optics,  we  may  regard  the  character  of  the  refraction  as 
determined  by  the  form  of  the  wave-surface. 

The  seolotropy  of  the  structure  will  not  introduce  any  corresponding 
property  into  the  potential  energy,  which  depends  only  upon  the  volumes 
and  compressibilities  concerned.  The  present  question,  therefore,  reduces 
itself  to  the  consideration  of  the  kinetic  energy  as  influenced  by  the  direction 
of  wave-propagation.  And  this,  as  we  have  seen,  is  a  matter  of  the  electrical 
resistance  of  certain  compound  conductors,  on  the  supposition,  which  we 
continue  to  make,  that  the  wave-length  is  very  large  in  comparison  with  the 
periods  of  the  structure.  The  theory  of  electrical  conduction  in  general 
has  been  treated  by  Maxwell  (Electricity,  §  297).  A  parallel  treatment  of 
the  present  question  shows  that  in  all  cases  it  is  possible  to  assign  a  system 
of  principal  axes,  having  the  property  that  if  the  wave-normal  coincide  with 
any  one  of  them  the  direction  of  flow  will  also  lie  in  the  same  direction, 
whereas  in  general  there  would  be  a  divergence.  To  each  principal  axis 
corresponds  an  efficient  "  density,"  and  the  equations  of  motion,  applicable  to 
the  medium  in  the  gross,  take  the  form 

d°-%          dB  d*-n          dS  d*£          dS 

<rx  -r-  =  ml  j-  ,         a-v  jTr-  =  Wa  -3-  ,        <rz  -3-2.  =  m 
dtz          dx  y  dt*  dy  dt2          dz 

where  £,  17,  £  are  the  displacements  parallel  to  the  axes,  ml  is  the  compressi- 
bility, and 

d£     dq     d£ 

o  —  -j  —  (-  -j—  -|-  -j—  . 
dx     dy     dz 

If  X,  fi,  v  are  the  direction-cosines  of  the  displacement,  I,  m,  n  of  the 
wave-normal,  we  may  take 

£=X0,       7?  =  /i0,       £=v0, 
where 

0  _  gi(lx+my+nz  -  Vt) 

Thus 

d8/da;  =  -W(l\  +  mfji  +  nv),  &c. 
and  the  equations  become 

<rx\Vz  =  mj(l\  +  mfi  +  nv), 

a-yfiV2  =  TT^m^X  +  w/i  +  nv), 

crzvV*  =  mjn  (l\  +  m/j.  +  nv), 
from  which,  on  elimination  of  X  :  /*  :  v,  we  get 

V*  =mi(l-  +  m\  ^]  =  an'~+  Km?  +  c%2,  .  .  .(78) 

- 


if  a,  b,  c  denote  the  velocities  in  the  principal  directions  x,  y,  z. 


1892]  IN   RECTANGULAR  ORDER  UPON   A   MEDIUM  37 

The  wave-surface  after  unit  time  is  accordingly  the  ellipsoid  whose  axes 
are  a,  b,  c. 

As  an  example,  if  the  medium,  otherwise  uniform,  be  obstructed  by  rigid 
cylinders  occupying  a  moderate  fraction  (p)  of  the  whole  space,  the  velocity 
in  the  direction  z,  parallel  to  the  cylinders,  is  unaltered  ;  so  that 


In  the  application  of  our  results  to  the  electric  theory  of  light  we  con- 
template a  medium  interrupted  by  spherical,  or  cylindrical,  obstacles,  whose 
inductive  capacity  is  different  from  that  of  the  undisturbed  medium.  On 
the  other  hand,  the  magnetic  constant  is  supposed  to  retain  its  value  un- 
broken. This  being  so,  the  kinetic  energy  of  the  electric  currents  for  the 
same  total  flux  is  the  same  as  if  there  were  no  obstacles,  at  least  if  we  regard 
the  wave-length  as  infinitely  great*.  And  the  potential  energy  of  electric 
displacement  is  subject  to  the  same  mathematical  laws  as  the  resistance  of 
our  compound  electrical  conductor,  specific  inductive  capacity  in  the  one 
question  corresponding  to  electrical  conductivity  in  the  other. 

Accordingly,  if  v  denote  the  inductive  capacity  of  the  material  composing 
the  spherical  obstacles,  that  of  the  undisturbed  medium  being  unity,  then 
the  approximate  value  of  p?  is  given  at  once  by  (67).  The  equation  may 
also  be  written  in  the  form  given  by  Lorentz, 


(79) 


and,  indeed,  it  appears  to  have  been  by  the  above  argument  that  (79)  was 
originally  discovered. 

The  above  formula  applies  in  strictness  only  when  the  spheres  are 
arranged  in  cubic  order  f,  and,  further,  when  p  is  moderate.  The  next 
approximation  is 


(80) 


If  the  obstacles  be  cylindrical,  and  arranged  in  square  order,  the  compound 
medium  is  doubly  refracting,  as  in  the  usual  electric  theory  of  light,  in  which 
the  medium  is  supposed  to  have  an  inductive  capacity  variable  with  the 
direction  of  displacement,  independently  of  any  discontinuity  in  its  structure. 
The  double  refraction  is  of  course  of  the  uniaxal  kind,  and  the  wave-surface  is 
the  sphere  and  ellipsoid  of  Huygens. 

*  See  Prof.  Willard  Gibbs's  "  Comparison  of  the  Elastic  and  Electric  Theories  of  Light," 
Am.  Journ.  Sci.  xxxv.  (1888). 

t  An  irregular  isotropic  arrangement  would,  doubtless,  give  the  same  result. 


38     INFLUENCE  OF  OBSTACLES  IN  RECTANGULAR  ORDER  UPON  A  MEDIUM.    [200 

For  displacements  parallel  to  the  cylinders  the  resultant  inductive  capacity 
(analogous  to  conductivity  in  the  conduction  problem)  is  clearly  1  —p  +  vp: 
so  that  the  value  of  p?  for  the  principal  extraordinary  index  is 

l*=I  +  (»-l)p,    - (81) 

giving  Newton's  law  for  the  relation  between  index  and  density. 

For  the  ordinary  index  we  have 

^=(26), 

in  which  v  =  (1  +  z>)/(l  —  v),  while  $4,  S8 ...  have  the  values  given  by  (49). 
If  we  omit  p*,  &c.  we  get 

(82) 


v+p' 
-11  1        v-l 


/A2  +  1  P  V          V+l' 


.(83) 


The  general  conclusion  as  regards  the  optical  application  is  that,  even 
if  we  may  neglect  dispersion,  we  must  not  expect  such  formulae  as  (79) 
to  be  more  than  approximately  correct  in  the  case  of  dense  fluid  and  solid 
bodies. 


201. 

ON  THE  DENSITIES   OF  THE  PRINCIPAL  GASES. 

[Proceedings  of  the  Royal  Society,  LIII.  pp.  134 — 149,  1893.] 

IN  former  communications  *  I  have  described  the  arrangements  by  which 
I  determined  the  ratio  of  densities  of  oxygen  and  hydrogen  (1 5*882).  For 
the  purpose  of  that  work  it  was  not  necessary  to  know  with  precision  the 
actual  volume  of  gas  weighed,  nor  even  the  pressure  at  which  the  containing 
vessel  was  filled.  But  I  was  desirous,  before  leaving  the  subject,  of  ascertain- 
ing not  merely  the  relative,  but  also  the  absolute,  densities  of  the  more 
important  gases,  that  is,  of  comparing  their  weights  with  that  of  an  equal 
volume  of  water.  To  effect  this  it  was  necessary  to  weigh  the  globe,  used  to 
contain  the  gases,  when  charged  with  water,  an  operation  not  quite  so  simple 
as  at  first  sight  it  appears.  And,  further,  in  the  corresponding  work  upon  the 
gases,  a  precise  absolute  specification  is  required  of  the  temperature  and 
pressure  at  which  a  filling  takes  place.  To  render  the  former  weighings 
available  for  this  purpose,  it  would  be  necessary  to  determine  the  errors  of 
the  barometers  then  employed.  There  would,  perhaps,  be  no  great  difficulty 
in  doing  this ;  but  I  was  of  opinion  that  it  would  be  an  improvement  to  use  a 
manometer  in  direct  connexion  with  the  globe,  without  the  intervention  of 
the  atmosphere.  In  the  latter  manner  of  working,  there  is  a  doubt  as  to 
the  time  required  for  full  establishment  of  equilibrium  of  pressure,  especially 
when  the  passages  through  the  taps  are  partially  obstructed  by  grease. 
When  the  directly  connected  manometer  is  employed,  there  is  no  temptation 
to  hurry  from  fear  of  the  entrance  of  air  by  diffusion,  and,  moreover  (Note  A), 
the  time  actually  required  for  the  establishment  of  equilibrium  is  greatly 
diminished.  With  respect  to  temperature,  also,  it  was  thought  better  to 
avoid  all  further  questions  by  surrounding  the  globe  with  ice,  as  in  Regnault's 
original  determinations.  It  is  true  that  this  procedure  involves  a  subsequent 
cleaning  and  wiping  of  the  globe,  by  which  the  errors  of  weighing  are  con- 
siderably augmented ;  but,  as  it  was  not  proposed  to  experiment  further  with 
hydrogen,  the  objection  was  of  less  force.  In  the^case  of  the  heavier  gases, 
unsystematic  errors  of  weighing  are  less  to  be  feared  than  doubts  as  to  the 
actual  temperature. 

*  Roy.  Soc.  Proc.  February,  1888  [Vol.  in.  p.  37] ;  February,  1892  [Vol.  in.  p.  524]. 


40  ON   THE   DENSITIES   OF   THE   PRINCIPAL   GASES.  [201 

In  order  to  secure  the  unsystematic  character  of  these  errors,  it  is 
necessary  to  wash  and  wipe  the  working  globe  after  an  exhaustion  in  the 
same  manner  as  after  a  filling.  The  dummy  globe  (of  equal  external  volume, 
as  required  in  Regnault's  method  of  weighing  gases)  need  not  be  wiped 
merely  to  secure  symmetry,  but  it  was  thought  desirable  to  do  so  before  each 
weighing.  In  this  way  there  would  be  no  tendency  to  a  progressive  change. 
In  wiping  the  globes  the  utmost  care  is  required  to  avoid  removing  any 
loosely  attached  grease  in  the  neighbourhood  of  the  tap.  The  results  to  be 
given  later  will  show  that,  whether  the  working  globe  be  full  or  empty,  the 
relative  weights  of  the  two  globes  can  usually  be  recovered  to  an  accuracy  of 
about  0'3  milligramme.  As  in  the  former  papers,  the  results  were  usually 
calculated  by  comparison  of  each  "  full "  weight  with  the  mean  of  the 
immediately  preceding  and  following  empty  weights.  The  balance  and  the 
arrangements  for  weighing  remained  as  already  described. 

The  Manometer. 

The  arrangements  adopted  for  the  measurement  of  pressure  must  be 
described  in  some  detail,  as  they  offer  several  points  of  novelty.  The  apparatus 
actually  used  would,  indeed,  be  more  accurately  spoken  of  as  a  manometric 
gauge,  but  it  would  be  easy  so  to  modify  it  as  to  fit  it  for  measurements 
extending  over  a  small  range. 

The  object  in  view  was  to  avoid  certain  defects  to  which  ordinary 
barometers  are  liable,  when  applied  to  absolute  measurements.  Of  these 
three  especially  may  be  formulated  : — 

a.     It  is  difficult  to  be  sure  that  the  vacuum  at  the  top  of  the  mercury  is 

suitable  for  the  purpose. 

6.     No  measurements  of  a  length  can  be  regarded  as  satisfactory  in  which 
different  methods  of  reading  are  used  for  the  two  extremities. 

c.  There  is  necessarily  some  uncertainty  due  to  irregular  refraction  by 

the  walls  of  the  tube.     The  apparent   level   of  the  mercury  may 
deviate  from  the  real  position. 

d.  To  the   above   may  be  added  that  the  accurate  observation  of  the 
barometer,  as  used  by  Regnault  and  most  of  his  successors,  requires 
the  use  of  a  cathetometer,  an  expensive  and  not  always  satisfactory 
instrument. 

The  guiding  idea  of  the  present  apparatus  is  the  actual  application  of  a 
measuring  rod  to  the  upper  and  lower  mercury  surfaces,  arranged  so  as  to  be 
vertically  superposed.  The  rod  A  A,  Fig.  1,  is  of  iron  (7  mm.  in  diameter), 
pointed  below,  B.  At  the  upper  end,  C,  it  divides  at  the  level  of  the  mercury 
into  a  sort  of  fork,  and  terminates  in  a  point  similar  to  that  at  B,  and,  like  it, 
directed  downwards.  The  coincidence  of  these  points  with  their  images 


1893]  ON   THE    DENSITIES   OF   THE   PRINCIPAL  GASES.  41 

reflected  in  the  mercury  surfaces,  is  observed  with  the  aid  of  lenses  of  about 
30  mm.  focus,  held  in  position  upon  the  wooden  framework  of  the  apparatus. 
It  is,  of  course,  independent  of  any  irregular  refraction  which  the  tube  may 
exercise.  The  vertically  of  the  line  joining  the  points  is  tested  without 
difficulty  by  a  plumb-line. 

Fig.  l. 


The  upper  and  lower  chambers  C,  B  are  formed  from  tubing  of  the  same 
diameter  (about  21  mm.  internal).  The  upper  communicates  through  a  tap, 
D,  with  the  Toppler,  by  means  of  which  a  suitable  vacuum  can  at  any  time 
be  established  and  tested.  In  ordinary  use,  D  stands  permanently  open,  but 
its  introduction  was  found  useful  in  the  preliminary  arrangements  and  in 
testing  for  leaks.  The  connexion  between  the  lower  chamber  B  and  the 
vessel  in  which  the  pressure  is  to  be  verified  takes  place  through  a  side 
tube,  E. 

The  greater  part  of  the  column  of  mercury  to  which  the  pressure  is  due  is 
contained  in  the  connecting  tube  FF,  of  about  3  mm.  internal  diameter.  The 
temperature  is  taken  by  a  thermometer  whose  bulb  is  situated  near  the 


42  ON   THE    DENSITIES   OF   THE    PRINCIPAL   GASES.  [201 

middle  of  FF.  Towards  the  close  of  operations  the  more  sensitive  parts  are 
protected  by  a  packing  of  tow  or  cotton-wool,  held  in  position  between  two 
wooden  boards.  The  anterior  board  is  provided  with  a  suitable  glass  window, 
through  which  the  thermometer  may  be  read. 

It  is  an  essential  requirement  of  a  manometer  on  the  present  plan  that 
the  measuring  rod  pass  air-tight  from  the  upper  and  lower  chambers  into  the 
atmosphere.  To  effect  this  the  glass  tubing  is  drawn  out  until  its  internal 
diameter  is  not  much  greater  than  that  of  the  rod.  The  joints  are  then  made 
by  short  lengths  of  thick-walled  india-rubber  H,  G,  wired  on  and  drowned 
externally  in  mercury.  The  vessels  for  holding  the  mercury  are  shown  at  /, 
K.  There  is  usually  no  difficulty  at  all  in  making  perfectly  tight  joints 
between  glass  tubes  in  this  manner ;  but  in  the  present  case  some  trouble 
was  experienced  in  consequence  apparently  of  imperfect  approximation  be- 
tween the  iron  and  the  mercury.  At  one  time  it  was  found  necessary  to 
supplement  the  mercury  with  vaseline.  When  tightness  is  once  obtained, 
there  seems  to  be  no  tendency  to  deterioration,  and  the  condition  of  things  is 
under  constant  observation  by  means  of  the  Toppler. 

The  distance  between  the  points  of  the  rod  is  determined  under 
microscopes  by  comparison  with  a  standard  scale,  before  the  apparatus  is  put 
together.  As  the  rod  is  held  only  by  the  rubber  connexions,  there  is  no  fear 
of  its  length  being  altered  by  stress. 

The  adjustment  of  the  mercury  (distilled  in  a  vacuum)  to  the  right  level 
is  effected  by  means  of  the  tube  of  black  rubber  LM,  terminating  in  the 
reservoir  N.  When  the  supply  of  mercury  to  the  manometer  is  a  little  short 
of  what  is  needed,  the  connexion  with  the  reservoir  is  cut  off  by  a  pinch -cock 
at  0,  and  the  fine  adjustment  is  continued  by  squeezing  the  tube  at  P 
between  a  pair  of  hinged  boards,  gradually  approximated  by  a  screw.  This 
plan,  though  apparently  rough,  worked  perfectly,  leaving  nothing  to  be 
desired. 

It  remains  to  explain  the  object  of  the  vessel  shown  at  Q.  In  the  early 
trials,  when  the  rubber  tube  was  connected  directly  to  R,  the  gradual  fouling 
of  the  mercury  surface,  which  it  seems  impossible  to  avoid,  threatened  to 
interfere  with  the  setting  at  B.  By  means  of  Q,  the  mercury  can  be  discharged 
from  the  measuring  chambers,  and  a  fresh  surface  constituted  at  B  as  well 
as  at  (7. 

The  manometer  above  described  was  constructed  by  my  assistant, 
Mr  Gordon,  at  a  nominal  cost  for  materials ;  and  it  is  thought  that  the  same 
principle  may  be  applied  with  advantage  in  other  investigations.  In  cases 
where  a  certain  latitude  in  respect  of  pressure  is  necessary,  the  measuring  rod 
might  be  constructed  in  two  portions,  sliding  upon  one  another.  Probably  a 
range  of  a  few  millimetres  could  be  obtained  without  interfering  with  the 
india-rubber  connexions. 


1893] 


ON   THE    DENSITIES   OF   THE   PRINCIPAL   GASES. 


43 


The  length  of  the  iron  rod  was  obtained  by  comparison  under  microscopes 
with  a  standard  bar  R  divided  into  millimetres.  In  terms  of  R  the  length 
at  15°  C.  is  762*248  mm.  It  remains  to  reduce  to  standard  millimetres. 
Mr  Chaney  has  been  good  enough  to  make  a  comparison  between  R  and  the 
iridio-platmum  standard  metre,  1890,  of  the  Board  of  Trade.  From  this  it 
appears  that  the  metre  bar  R  is  at  15°  C.  0'3454  mm.  too  long;  so  that  the 
true  distance  between  the  measuring  points  of  the  iron  rod  is  at  15°  C. 

762-248  x  1-0003454  =  762-511  mm. 


Connexions  with  Pump  and  Manometer. 

Some  of  the  details  of  the  process  of  filling  the  globe  with  gas  under 
standard  conditions  will  be  best  described  later  under  the  head  of  the 
particular  gas;  but  the  general  arrangement  and  the  connexions  with  the 
pump  and  the  manometer  are  common  to  all.  They  are  sketched  in  Fig.  2,  in 

Fig.  2. 


which  S  represents  the  globe,  T  the  inverted  bell-glass  employed  to  contain 
the  enveloping  ice.  The  connexion  with  the  rest  of  the  apparatus  is  by  a 
short  tube  U  of  thick  rubber,  carefully  wired  on.  The  tightness  of  these 
joints  was  always  tested  with  the  aid  of  the  Toppler  X,  the  tap  V  leading  to 
the  gas-generating  apparatus  being  closed.  The  side  tube  at  D  leads  to  the 
vacuum  chamber  of  the  manometer,  while  that  at  E  leads  to  the  pressure 
chamber  B.  The  wash-out  of  the  tubes,  and  in  some  cases  of  the  generator, 
was  aided  by  the  Toppler.  When  this  operation  was  judged  to  be  complete, 
V  was  again  closed,  and  a  good  vacuum  made  in  the  parts  still  connected  to 
the  pump.  W  would  then  be  closed,  and  the  actual  filling  commenced  by 
opening  V,  and  finally  the  tap  of  the  globe.  The  lower  chamber  of  the 
manometer  was  now  in  connexion  with  the  globe,  and  through  a  regulating 


44  ON   THE    DENSITIES   OF   THE   PRINCIPAL   GASES.  [201 

tap  (not  shown)  with  the  gas-generating  apparatus.  By  means  of  the  Toppler 
the  vacuum  in  the  manometer  could  be  carried  to  any  desired  point.  But 
with  respect  to  this  a  remark  must  be  made.  It  is  a  feature  of  the  method 
employed*  that  the  exhaustions  of  the  globe  are  carried  to  such  a  point  that 
the  weight  of  the  residual  gas  may  be  neglected,  thus  eliminating  errors  due 
to  a  second  manometer  reading.  There  is  no  difficulty  in  attaining  this  result, 
but  the  delicacy  of  the  Toppler  employed  as  a  gauge  is  so  great  that  the 
residual  gas  still  admits  of  tolerably  accurate  measurement.  Now  in  exhaust- 
ing the  head  of  the  manometer  it  would  be  easy  to  carry  the  process  to  a 
point  much  in  excess  of  what  is  necessary  in  the  case  of  the  globe,  but  there 
is  evidently  no  advantage  in  so  doing.  The  best  results  will  be  obtained  by 
carrying  both  exhaustions  to  the  same  degree  of  perfection. 

At  the  close  of  the  filling  the  pressure  has  to  be  adjusted  to  an  exact 
value,  and  it  might  appear  that  the  double  adjustment  required  (of  pressure 
and  of  mercury)  would  be  troublesome.  Such  was  not  found  to  be  the  case. 
After  a  little  practice  the  manometer  could  be  set  satisfactorily  without  too 
great  a  delay.  When  the  pressure  was  nearly  sufficient,  the  regulating  tap 
was  closed,  and  equilibrium  allowed  to  establish  itself.  If  more  gas  was  then 
required,  the  tap  could  be  opened  momentarily.  The  later  adjustments  were 
effected  by  the  application  of  heat  or  cold  to  parts  of  the  connecting  tubes. 
At  the  close,  advantage  was  taken  of  the  gradual  rise  in  the  temperature 
which  was  usually  met  with.  The  pressure  being  just  short  of  what  was 
required,  and  V  being  closed,  it  was  only  necessary  to  wait  until  the  point 
was  reached.  In  no  case  was  a  reading  considered  satisfactory  when  the 
pressure  was  changing  at  other  than  a  very  slow  rate.  It  is  believed  that  the 
comparison  between  the  state  of  things  at  the  top  and  at  the  bottom  of 
the  manometer  could  be  effected  with  very  great  accuracy,  and  this  is  all  that 
the  method  requires.  At  the  moment  when  the  pressure  was  judged  to  be 
right,  the  tap  of  the  globe  was  turned,  and  the  temperature  of  the  manometer 
was  read.  The  vacuum  was  then  verified  by  the  Toppler. 

The  Weights. 

The  object  of  the  investigation  being  to  ascertain  the  ratio  of  densities  of 
water  and  of  certain  gases  under  given  conditions,  the  absolute  values  of 
the  weights  employed  is  evidently  a  matter  of  indifference.  This  is  a  point 
which  I  think  it  desirable  to  emphasise,  because  v.  Jolly,  in  his,  in  many 
respects,  excellent  work  upon  this  subject -f-,  attributes  a  discrepancy  between 
his  final  result  for  oxygen  and  that  of  Regnault  to  a  possible  variation  in  the 
standard  of  weight.  On  the  same  ground  we  may  omit  to  allow  for  the 
buoyancy  of  the  weights  as  used  in  air,  since  only  the  variations  of  buoyancy, 

*  Due  to  von  Jolly. 

t  Munich  Acad.  Trans.  Vol.  xm.  Part  n.  p.  49,  1880. 


1893]  ON   THE   DENSITIES   OF   THE   PRINCIPAL   GASES.  45 

due,  for  example,  to  changing  barometer,  could  enter;  and  these  affect  the 
result  so  little  that  they  may  safely  be  neglected*. 

But,  while  the  absolute  values  of  the  weights  are  of  no  consequence,  their 
relative  values  must  be  known  with  great  precision.  The  investigation  of 
these  over  the  large  range  required  (from  a  kilogramme  to  a  centigramme)  is 
a  laborious  matter,  but  it  presents  nothing  special  for  remark.  The  weights 
quoted  in  this  paper  are,  in  all  cases,  corrected,  so  as  to  give  the  results  as 
they  would  have  been  obtained  from  a  perfectly  adjusted  system. 

The  Water  Contents  of  the  Globe. 

The  globe,  packed  in  finely-divided  ice,  was  filled  with  boiled  distilled 
water  up  to  the  level  of  the  top  of  the  channel  through  the  plug  of 
the  tap,  that  is,  being  itself  at  0°,  was  filled  with  water  also  at  0°.  Thus 
charged  the  globe  had  now  to  be  weighed ;  but  this  was  a  matter  of  some 
difficulty,  owing  to  the  very  small  capacity  available  above  the  tap.  At  about 
9°  there  would  be  a  risk  of  overflow.  Of  course  the  water  could  be  retained 
by  the  addition  of  extra  tubing,  but  this  was  a  complication  that  it  was 
desired  to  avoid.  In  February,  1892,  during  a  frost,  an  opportunity  was 
found  to  effect  the  weighing  in  a  cold  cellar  at  a  temperature  ranging  from  4° 
to  7°.  The  weights  required  (on  the  same  side  of  the  balance  as  the  globe 
and  its  supports)  amounted  to  O1822  gram.  On  the  other  side  were  other 
weights  whose  values  did  not  require  to  be  known  so  long  as  they  remained 
unmoved  during  the  whole  series  of  operations.  Barometer  (corrected) 
758-9  mm.;  temperature  6'3°. 

A  few  days  later  the  globe  was  discharged,  dried,  and  replaced  in  the 
balance  with  tap  open.  1834'!  701  grams  had  now  to  be  associated  with  it  in 
order  to  obtain  equilibrium.  The  difference, 

1834170  -  0182  =  1833-988, 

represents  the  weight  of  the  water  less  that  of  the  air  displaced  by  it.  The 
difference  of  atmospheric  conditions  was  sufficiently  small  to  allow  the  neglect 
of  the  variation  in  the  buoyancy  of  the  glass  globe  and  of  the  brass  counter- 
poises. 

It  remains  to  estimate  the  actual  weight  of  the  air  displaced  by  the  water 
under  the  above  mentioned  atmospheric  conditions.  It  appears  that,  on  this 
account,  we  are  to  add  2'314,  thus  obtaining 

1836-30 

as  the  weight  of  the  water  at  0°  which  fills  the  globe  at  0°. 

*  In  v.  Jolly's  calculations  the  buoyancy  of  the  weights  seems  to  be  allowed  for  in  dealing 
with  the  water,  and  neglected  in  dealing  with  the  gases.  If  this  be  so,  the  result  would  be  affected 
with  a  slight  error,  which,  however,  far  exceeds  any  that  could  arise  from  neglecting  buoyancy 
altogether. 


46 


ON   THE    DENSITIES   OF   THE   PRINCIPAL   GASES. 


[201 


A  further  small  correction  is  required  to  take  account  of  the  fact  that  the 
usual  standard  density  is  that  of  water  at  4°  and  not  at  0°.  According  to 
Broch  (Everett's  C.  G.  S.  System  of  Units),  the  factor  required  is  0-99988,  so 
that  we  have 


as  the  weight  of  water  at  4°  which  would  fill  the  globe  at  0°. 

Air. 

Air  drawn  from  outside  (in  the  country)  was  passed  through  a  solution  of 
potash.  On  leaving  the  regulating  tap  it  traversed  tubes  filled  with  frag- 
ments of  potash,  and  a  long  length  of  phosphoric  anhydride,  followed  by  a 
filter  of  glass  wool.  The  arrangements  beyond  the  regulating  tap  were  the 
same  for  all  the  gases  experimented  upon.  At  the  close  of  the  filling  it  was 
necessary  to  use  a  condensing  syringe  in  order  to  force  the  pressure  up  to  the 
required  point,  but  the  air  thus  introduced  would  not  reach  the  globe.  It 
may  be  well  to  give  the  results  for  air  in  some  detail,  so  as  to  enable  the 
reader  to  form  a  judgment  as  to  the  degree  of  accuracy  attained  in  the 
manipulations. 


Date 

Globe 
empty 

Globe 
full 

Temp,  of 
manometer 

Correction 
to  15° 

Corrected 
to  15° 

1892 
September  24  

2-90941 

27  
28  
29 

2-90867 

0-53327 
0-53271 

17-8 
15-7 

-0-00112 
-0-00028 

0-53219 
0-53243 

October     1 

2-90923 

»           3 

0-53151 

12-7 

+  0-00093 

0-53244 

„           4.  .      .  . 

2-90872 

„           7  
8 

2-91036 

Tap  regi 
0-53296 

•eased 
12'4 

+  0-00105 

0-53401 

10 

2-91056 

11 

0-53251 

11-8 

+  0-00129 

0-53380 

»         12  
„         13  

2-91039 

0-53201 

11-0 

+  0-00161 

0-53362 

„         14  

2-91043 

15 

0-53219 

10*6 

+  0-00177 

0-53396 

The  column  headed  "globe  empty"  gives  the  (corrected)  weights,  on  the 
side  of  the  working  globe,  required  for  balance.  The  third  column  gives  the 
corresponding  weights  when  the  globe  was  full  of  air,  having  been  charged  at 
0°  and  up  to  the  pressure  required  to  bring  the  mercury  in  the  manometer 
into  contact  with  the  two  points  of  the  measuring  rod. 


1893]  ON   THE    DENSITIES    OF    THE   PRINCIPAL   GASES.  47 

This  pressure  was  not  quite  the  same  on  different  occasions,  being  subject 
to  a  temperature  correction  for  the  density  of  mercury  and  for  the  expansion 
of  the  iron  rod.  The  correction  is  given  in  the  fifth  column,  and  the  weights 
that  would  have  been  required,  had  the  temperature  been  15°,  in  the  sixth. 
The  numbers  in  the  second  and  sixth  columns  should  agree,  but  they  are 
liable  to  a  discontinuity  when  the  tap  is  regreased. 

In  deducing  the  weight  of  the  gas  we  compare  each  weighing  "  full "  with 
the  mean  of  the  preceding  and  following  weights  "  empty,"  except  in  the  case 
of  October  15,  when  there  was  no  subsequent  weighing  empty.  The  results 
are 

September  27   2'37686 

29   2-37651 

October         3   2'37653 

8  2-37646 

11   2-37668 

13  2-37679 

15  .,  .    2-37647 


Mean   2-37661 

There  is  here  no  evidence  of  the  variation  in  the  density  of  air  suspected 
by  Regnault  and  v.  Jolly.  Even  if  we  include  the  result  for  September  27th, 
obviously  affected  by  irregularity  in  the  weights  of  the  globe  empty,  the 
extreme  difference  is  only  0'4  milligram,  or  about  l/6000th  part. 

To  allow  for  the  contraction  of  the  globe  (No.  14)  when  weighed  empty, 
discussed  in  my  former  papers,  we  are  to  add  0-00056  to  the  apparent  weight, 
so  that  the  result  for  air  becomes 

2-37717. 

This  is  the  weight  of  the  contents  at  0°  and  under  the  pressure  defined  by 
the  manometer  gauge  at  15°  of  the  thermometer.  The  reduction  to  standard 
conditions  is,  for  the  present,  postponed. 

Oxygen. 

This  gas  has  been  prepared  by  three  distinct  methods:  (a)  from  chlorates, 
(6)  from  permanganate  of  potash,  (c)  by  electrolysis. 

In  the  first  method  mixed  chlorates  of  potash  and  soda  were  employed, 
as  recommended  by  Shenstone,  the  advantage  lying  in  the  readier  fusibility. 
The  fused  mass  was  contained  in  a  Florence  flask,  afid  during  the  wash-out 
was  allowed  slowly  to  liberate  gas  into  a  vacuum.  After  all  air  had  been 
expelled,  the  regulating  tap  was  closed,  and  the  pressure  allowed  gradually 
to  rise  to  that  of  the  atmosphere.  The  temperature  could  then  be  pushed 
without  fear  of  distorting  the  glass,  and  the  gas  was  drawn  off  through  the 


48  ON  THE   DENSITIES   OF  THE   PRINCIPAL  GASES.  [201 

regulating  tap.  A  very  close  watch  over  the  temperature  was  necessary  to 
prevent  the  evolution  of  gas  from  becoming  too  rapid.  In  case  of  excess,  the 
superfluous  gas  was  caused  to  blow  off  into  the  atmosphere,  rather  than  risk 
imperfect  action  of  the  potash  and  phosphoric  anhydride.  Two  sets  of  five 
fillings  were  effected  with  this  oxygen.  In  the  first  set  (May,  1892)  the 
highest  result  was  2'6272,  and  the  lowest  2'6266,  mean  2'62691.  In  the 
second  set  (June,  July,  1892)  the  highest  result  was  2*6273  and  the  lowest 
2-6267,  mean  2'62693. 

The  second  method  (6)  proved  very  convenient,  the  evolution  of  gas  being 
under  much  better  control  than  in  the  case  of  chlorates.  The  recrystallised 
salt  was  heated  in  a  Florence  flask,  the  wash-out,  in  this  case  also,  being 
facilitated  by  a  vacuum.  Three  fillings  gave  satisfactory  results,  the  highest 
being  2'6273,  the  lowest  2'6270,  and  the  mean  2*62714.  The  gas  was  quite 
free  from  smell. 

By  the  third  method  I  have  not  as  many  results  as  I  could  have  wished, 
operations  having  been  interrupted  by  the  breakage  of  the  electrolytic 
generator.  This  was,  however,  of  less  importance,  as  I  had  evidence  from 
former  work  that  there  is  no  material  difference  between  the  oxygen  from 
chlorates  and  that  obtained  by  electrolysis.  The  gas  was  passed  over  hot 
copper  [oxide],  as  detailed  in  previous  papers.  The  result  of  one  filling, 
with  the  apparatus  as  here  described,  was  2'6271.  To  this  may  be  added  the 
result  of  two  fillings  obtained  at  an  earlier  stage  of  the  work,  when  the  head 
of  the  manometer  was  exhausted  by  an  independent  Sprengel  pump,  instead 
of  by  the  Toppler.  The  value  then  obtained  was  2'6272.  The  results  stand 
thus  :— 

Electrolysis  (2),  May,  1892   2'6272 

(1)          „  2-6271 

Chlorates  (5),  May,  1892  2'6269 

(5),  June,  1892 2'6269 

Permanganate  (3),  January,  1893     ...  2'6271 

Mean 2-62704 

Correction  for  contraction  . . .     0*00056 


2-62760 

It  will  be  seen  that  the  agreement  between  the  different  methods  is  very 
good,  the  differences,  such  as  they  are,  having  all  the  appearance  of  being 
accidental.  Oxygen  prepared  by  electrolysis  is  perhaps  most  in  danger  of 
being  light  (from  contamination  with  hydrogen),  and  that  from  chlorates  of 
being  abnormally  heavy. 

Nitrogen. 

This  gas  was  prepared,  in  the  usual  manner,  from  air  by  removal  of  oxygen 
with  heated  copper.  Precautions  are  required,  in  the  first  place,  to  secure  a 


1893]  ON   THE    DENSITIES    OF   THE    PRINCIPAL   GASES.  49 

sufficient  action  of  the  reduced  copper,  and,  secondly,  as  was  shown  by  v.  Jolly, 
and  later  by  Leduc,  to  avoid  contamination  with  hydrogen  which  may  be 
liberated  from  the  copper.  I  have  followed  the  plan,  recommended  by  v.  Jolly, 
of  causing  the  gas  to  pass  finally  over  a  length  of  unreduced  copper.  The 
arrangements  were  as  follows : — 

Air  drawn  through  solution  of  potash  was  deprived  of  its  oxygen  by 
reduced  copper,  contained  in  a  tube  of  hard  glass  heated  by  a  large  flame.  It 
then  traversed  a  U-tube,  in  which  was  deposited  most  of  the  water  of  combus- 
tion. The  gas,  practically  free,  as  the  event  proved,  from  oxygen,  was  passed, 
as  a  further  precaution,  over  a  length  of  copper  heated  in  a  combustion 
furnace,  then  through  strong  sulphuric  acid*,  and  afterwards  back  through 
the  furnace  over  a  length  of  oxide  of  copper.  It  then  passed  on  to  the  regu- 
lating tap,  and  thence  through  the  remainder  of  the  apparatus,  as  already 
described.  In  no  case  did  the  copper  in  the  furnace,  even  at  the  end  where 
the  gas  entered,  show  any  sign  of  losing  its  metallic  appearance. 

Three  results,  obtained  in  August,  1892,  were — 

August    8    2-31035 

10    2-31026 

15    2-31024 

Mean 2'31028 

To  these  may  be  added  the  results  of  two  special  experiments  made  to 
test  the  removal  of  hydrogen  by  the  copper  oxide.  For  this  purpose  a  small 
hydrogen  generator,  which  could  be  set  in  action  by  closing  an  external 
contact,  was  included  between  the  two  tubes  of  reduced  copper,  the  gas 
being  caused  to  bubble  through  the  electrolytic  liquid.  The  quantity  of 
hydrogen  liberated  was  calculated  from  the  deflection  of  a  galvanometer 
included  in  the  circuit,  and  was  sufficient,  if  retained,  to  alter  the  density 
very  materially.  Care  was  taken  that  the  small  stream  of  hydrogen  should 
be  uniform  during  the  whole  time  (about  2|  hours)  occupied  by  the  filling, 
but,  as  will  be  seen,  the  impurity  was  effectually  removed  by  the  copper 
oxide  f .  Two  experiments  gave — 

September  17  2-31012 

20  2-31027 

Mean 2-31020 

We  may  take  as  the  number  for  nitrogen — 

2-31026 
Correction  for  contraction...  56 


2-31082 

*  There  was  no  need  for  this,  but  the  acid  was  in  position  for  another  purpose, 
t  Much  larger  quantities  of  hydrogen,  sufficient  to  reduce  the  oxide  over  several  centimetres, 
have  been  introduced  without  appreciably  altering  the  weight  of  the  gas. 


50  ON   THE    DENSITIES   OF   THE   PRINCIPAL   GASES.  [201 

Although  the  subject  is  not  yet  ripe  for  discussion,  I  cannot  omit  to 
notice  here  that  nitrogen  prepared  from  ammonia,  and  expected  to  be  pure, 
turned  out  to  be  decidedly  lighter  than  the  above.  When  the  oxygen  of  air 
is  burned  by  excess  of  ammonia,  the  deficiency  is  about  I/ 1000th  part*. 
When  oxygen  is  substituted  for  air,  so  that  all  (instead  of  about  one-seventh 
part)  of  the  nitrogen  is  derived  from  ammonia,  the  deficiency  of  weight  may 
amount  to  ^  per  cent.  It  seems  certain  that  the  abnormal  lightness  cannot 
be  explained  by  contamination  with  hydrogen,  or  with  ammonia,  or  with 
water,  and  everything  suggests  that  the  explanation  is  to  be  sought  in  a 
dissociated  state  of  the  nitrogen  itself.  Until  the  questions  arising  out  of 
these  observations  are  thoroughly  cleared  up,  the  above  number  for  nitrogen 
must  be  received  with  a  certain  reserve.  But  it  has  not  been  thought 
necessary,  on  this  account,  to  delay  the  presentation  of  the  present  paper, 
more  especially  as  the  method  employed  in  preparing  the  nitrogen  for  which 
the  results  are  recorded  is  that  used  by  previous  experimenters. 

Reduction  to  Standard  Pressure. 

The  pressure  to  which  the  numbers  so  far  given  relate  is  that  due  to 
762-511  mm.  of  mercury  at  a  temperature  of  14-85°f,  and  under  the  gravity 
operative  in  my  laboratory  in  latitude  51°  47'.     In  order  to  compare  the 
results  with  those  of  other  experimenters,  it  will  be  convenient  to  reduce 
them  not  only  to  760  mm.  of  mercury  pressure  at  0°,  but  also  to  the  value  of 
gravity  at  Paris.     The  corrective  factor  for  length  is  760/762'511.     In  order 
to  correct  for  temperature,  we  will  employ  the  formula  J 
1  +  0-0001818 1+  0-00000000017 1* 
for  the  volume  of  mercury  at  t°.     The  factor  of  correction  for  temperature  is 

thus  1-002700.     For  gravity  we  may  employ  the  formula 

g  =  980-6056  -  2-5028  cos  2X, 
\  being  the  latitude.     Thus,  for  my  laboratory— 

#  =  981193, 
and  for  Paris — 

g  =  980-939, 

the  difference  of  elevation  being  negligible.     The  factor  of  correction  is  thus 
0-99974. 

The  product  of  the  three  factors,  corrective  for  length,  for  temperature, 
and  for  gravity,  is  accordingly  0'99914.  Thus  multiplied,  the  numbers  are  as 
follows : — 

Air  Oxygen  Nitrogen 

2-37512      2-62534      2-30883 

*  Nature,  Vol.  XLVI.  p.  512.     [Vol.  iv.  p.  1.] 

t  The  thermometer  employed  with  the  manometer  read  0-15°  too  high. 

t  Everett,  p.  142. 


1893] 


ON   THE   DENSITIES   OF   THE   PRINCIPAL   GASES. 


51 


and   these  may   now   be   compared  with   the  water  contents  of  the  globe, 
viz.,  1836-52. 

The  densities  of  the  various  gases  under  standard  conditions,  referred  to 
that  of  distilled  water  at  4°,  are  thus  : — 


Air 

0-00129327 


Oxygen 

0-00142952 


Nitrogen 

0-00125718 


With  regard  to  hydrogen,  we  may  calculate  its  density  by  means  of  the 
ratio  of  densities  of  oxygen  and  hydrogen  formerly  given  by  me,  viz.,  15'882. 
Hence 

Hydrogen 

0-000090009 

The  following  table  shows  the  results  arrived  at  by  various  experimenters. 
Von  Jolly  did  not  examine  hydrogen.  The  numbers  are  multiplied  by  1000 
so  as  to  exhibit  the  weights  in  grams  per  litre : — 


Air 

Oxygen 

Nitrogen 

Hydrogen 

Re°iiault    1847 

1-29319 

1  '42980 

1-25617 

0'08958 

Corrected  by  Crafts.    .    .    . 

1-29349 

1-43011 

1-25647 

0-08988 

Von  Jolly,  1880    

1-29351 

1-42939 

1-25787 

Ditto  corrected 

1-29383 

1-42971 

1-25819 

Leduc,  1891*     .    . 

1-29330 

1-42910 

1-25709 

0-08985 

Eayleigh,  1893  

1-29327 

1-42952 

1-25718 

0-09001 

The  correction  of  Regnault  by  Crafts  f  represents  allowance  for  the  con- 
traction of  Regnault's  globe  when  exhausted,  but  the  data  were  not  obtained 
from  the  identical  globe  used  by  Regnault.  In  the  fourth  row  I  have 
introduced  a  similar  correction  to  the  results  of  von  Jolly.  This  is  merely  an 
estimate  founded  upon  the  probability  that  the  proportional  contraction 
would  be  about  the  same  as  in  my  own  case  and  in  that  of  M.  Leduc. 

In  taking  a  mean  we  may  omit  the  uncorrected  numbers,  and  also  that 
obtained  by  Regnault  for  nitrogen,  as  there  is  reason  to  suppose  that  his  gas 
was  contaminated  with  hydrogen.  Thus 

Mean  Numbers. 


Air 
1-29347 


Oxygen 

1-42961 


Nitrogen 

1-25749 


Hydrogen 

0-08991 


The  evaluation  of  the  densities  as  compared  with  water  is  exposed  to 
many  sources  of  error  which  do  not  affect  the  comparison  of  one  gas  with 


*  Bulletin  des  Seances  de  la  Societe  de  Physique. 
t  Comptes  Rendug,  Vol.  cvi.  p.  1664. 


4—2 


52 


ON   THE   DENSITIES   OF   THE   PRINCIPAL  GASES. 


[201 


another.     It  may  therefore  be  instructive  to  exhibit  the  results  of  various 
workers  referred  to  air  as  unity*. 


Oxygen 

Nitrogen 

Hydrogen 

Regnault  (corrected)    

1-10562 

0-97138 

G'06949 

1-10502 

0-97245 

Leduc         

1-1050 

0-9720 

0-06947 

Eayleigh 

1-10535 

0-97209 

0-06960 

1-10525 

0-97218 

0*06952 

As  usually  happens  in  such  cases,  the  concordance  of  the  numbers 
obtained  by  various  experimenters  is  not  so  good  as  might  be  expected  from 
the  work  of  each  taken  separately.  The  most  serious  discrepancy  is  in  the 
difficult  case  of  hydrogen.  M.  Leduc  suggests  f  that  my  number  is  too  high 
on  account  of  penetration  of  air  through  the  blow-off  tube  (used  to  establish 
equilibrium  of  pressure  with  the  atmosphere),  which  he  reckons  at  1  m.  long 
and  1  cm.  in  diameter.  In  reality  the  length  was  about  double,  and  the 
diameter  one-half  of  these  estimates ;  and  the  explanation  is  difficult  to 
maintain,  in  view  of  the  fact,  recorded  in  my  paper,  that  a  prolongation  of 
the  time  of  contact  from  4m  to  30m  had  no  appreciable  ill  effect.  It  must  be 
admitted,  however,  that  there  is  a  certain  presumption  in  favour  of  a  lower 
number,  unless  it  can  be  explained  as  due  to  an  insufficient  estimate  of  the 
correction  for  contraction.  On  account  of  the  doubt  as  to  the  appropriate 
value  of  this  correction,  no  great  weight  can  be  assigned  to  Regnault's 
number  for  hydrogen.  If  the  atomic  weight  of  oxygen  be  indeed  15'88,  and 
the  ratio  of  densities  of  oxygen  and  hydrogen  be  15'90,  as  M.  Leduc  makes 
them,  we  should  have  to  accept  a  much  higher  number  for  the  ratio  of 
volumes  than  that  (2'0002)  resulting  from  the  very  elaborate  measurements 
of  Morley.  But  while  I  write  the  information  reaches  me  that  Mr  A.  Scott's 
recent  work  upon  the  volume  ratio  leads  him  to  just  such  a  higher  ratio, 
viz.,  2-00245,  a  number  a  priori  more  probable  than  2'0002.  Under  the 
circumstances  both  the  volume  ratio  and  the  density  of  hydrogen  must  be 
regarded  as  still  uncertain  to  the  I/ 1000th  part. 


*  [1902.     Cooke's  value  for  hydrogen,  viz.  -06958,  of  date  1889,  should  have  been  included  in 
the  above.] 

t  Comptes  Rendw,  July,  1892. 


1893]  ON   THE   DENSITIES   OF   THE   PRINCIPAL  GASES.  53 


NOTE  A. 

On  the  Establishment  of  Equilibrium  of  Pressure  in  Two  Vessels  connected  by 
a  Constricted  Channel. 

It  may  be  worth  while  to  give  explicitly  the  theory  of  this  process,  sup- 
posing that  the  difference  of  pressures  is  small  throughout,  and  that  the 
capacity  of  the  channel  may  be  neglected.  If  vlt  p^  denote  the  volume  and 
pressure  of  the  gas  in  the  first  vessel  at  time  t\  v2,  p2  the  corresponding 
quantities  for  the  second  vessel,  we  have 

vldp1/dt  +  c(pl  —  p2)  =  0, 


where  c  is  a  constant  which  we  may  regard  as  the  conductivity  of  the  channel. 
In  these  equations  inertia  is  neglected,  only  resistances  of  a  viscous  nature 
being  regarded,  as  amply  suffices  for  the  practical  problem.  From  the  above 
we  may  at  once  deduce 


showing  that  (PI  —  p2}  varies  as  e~qt,  where 

c      c      I 
q=~+-  =-  , 

V1        V2        T 

if  T  be  the  time  in  which  the  difference  of  pressures  is  reduced  in  the  ratio 
of  e  :  1. 

Let  us  now  apply  this  result  (a)  to  the  case  where  the  globe  of  volume 
v^  communicates  with  the  atmosphere,  (6)  to  the  case  where  the  globe  is  con- 
nected with  a  manometer  of  relatively  small  volume  vz.  For  (a)  we  have 

I/T  =  CK 

and  for  (6)  l/r  =  c/v2; 

so  that  r/r'=vl/v2. 

For  such  a  manometer  as  is  described  in  the  text,  the  ratio  v1/v2  is  at  least 
as  high  as  30  ;  and  in  this  proportion  is  diminished  the  time  required  for  the 
establishment  of  equilibrium  up  to  any  standard  of  perfection  that  may  be 
fixed  upon. 

[1902.  The  question  of  the  weight  of  nitrogen  is  further  treated  in 
Arts.  210,  214.  It  will  be  understood  that  the  results  given  in  the  present 
paper  relate  to  the  atmospheric  mixture  of  nitrogen  and  argon.] 


202. 


Fig.  1. 


INTERFERENCE  BANDS   AND  THEIR  APPLICATIONS. 

[Proceedings  of  the  Royal  Institution,  xiv.  pp.  72—78,  1893 ; 
Nature,  XLVIII.  pp.  212—214,  1893.] 

THE  formation  of  the  interference  bands,  known  as  Newton's  Rings, 
when  two  slightly  curved  glass  plates  are  pressed  into  contact,  was  illustrated 
by  an  acoustical  analogue.  A  high-pressure  flame  B  (Fig.  1)  is  sensitive  to 
sounds  which  reach  it  in  the  direction  EB,  but  is  insensitive  to  similar 
sounds  which  reach  it  in  the  nearly  perpendicular  direction  AB.  A  is 
a  "  bird-call,"  giving  a  pure  sound  (inaudible)  of  wave- 
length (A.)  equal  to  about  1  cm. ;  C  and  D  are  reflectors 
of  perforated  zinc.  If  C  acts  alone,  the  flame  is  visibly 
excited  by  the  waves  reflected  from  it,  though  by  far  the 
greater  part  of  the  energy  is  transmitted.  If  D,  held 
parallel  to  G,  be  then  brought  into  action,  the  result 
depends  upon  the  interval  between  the  two  partial  re- 
flectors. The  reflected  sounds  may  co-operate,  in  which 
case  the  flame  flares  vigorously;  or  they  may  interfere,  so 
that  the  flame  recovers,  and  behaves  as  if  no  sound  at  all 
were  falling  upon  it.  The  first  effect  occurs  when  the 
reflectors  are  close  together,  or  are  separated  by  any 
multiple  of  ^  V2 • ^  j  the  second  when  the  interval  is 
midway  between  those  of  the  above-mentioned  series,  that 
is,  when  it  coincides  with  an  odd  multiple  of  i\/2.X. 
depends  upon  the  obliquity  of  the  reflection. 

The  coloured  rings,  as  usually  formed  between  glass  plates,  lose  a  good 
deal  of  their  richness  by  contamination  with  white  light  reflected  from  the 
exterior  surfaces.  The  reflection  from  the  hindermost  surface  is  easily  got 
rid  of  by  employing  an  opaque  glass,  but  the  reflection  from  the  first  surface 
is  less  easy  to  deal  with.  One  plan,  used  in  the  lecture,  depends  upon  the 
use  of  slightly  wedge-shaped  glasses  (2°)  so  combined  that  the  exterior 
surfaces  are  parallel  to  one  another,  but  inclined  to  the  interior  operative 
surfaces.  In  this  arrangement  the  false  light  is  thrown  somewhat  to  one 


The   factor   V2 


1893]  INTERFERENCE   BANDS   AND   THEIR    APPLICATIONS.  55 

side,  and  can  be  stopped  by  a  screen  suitably  held  at  the  place  where  the 
image  of  the  electric  arc  is  formed. 

The  formation  of  colour  and  the  ultimate  disappearance  of  the  bands 
as  the  interval  between  the  surfaces  increases,  depends  upon  the  mixed 
character  of  white  light.  For  each  colour  the  bands  are  upon  a  scale 
proportional  to  the  wave-length  for  that  colour.  If  we  wish  to  observe 
the  bands  when  the  interval  is  considerable — bands  of  high  interference 
as  they  are  called — the  most  natural  course  is  to  employ  approximately 
homogeneous  light,  such  as  that  afforded  by  a  soda  flame.  Unfortunately, 
this  light  is  hardly  bright  enough  for  projection  upon  a  large  scale. 

A  partial  escape  from  this  difficulty  is  afforded  by  Newton's  observations 
as  to  what  occurs  when  a  ring  system  is  regarded  through  a  prism.  In  this 
case  the  bands  upon  one  side  may  become  approximately  achromatic,  and  are 
thus  visible  to  a  tolerably  high  order,  in  spite  of  the  whiteness  of  the  light. 
Under  these  circumstances  there  is,  of  course,  no  difficulty  in  obtaining 
sufficient  illumination;  and  bands  formed  in  this  way  were  projected  upon 
the  screen*. 

The  bands  seen  when  light  from  a  soda  flame  falls  upon  nearly  parallel 
surfaces  have  often  been  employed  as  a  test  of  flatness.  Two  flat  surfaces 
can  be  made  to  fit,  and  then  the  bands  are  few  and  broad,  if  not  entirely 
absent;  and,  however  the  surfaces  may  be  presented  to  one  another,  the 
bands  should  be  straight,  parallel,  and  equidistant.  If  this  condition  be 
violated,  one  or  other  of  the  surfaces  deviates  from  flatness.  In  Fig.  2, 
A  and  B  represent  the  glasses  to  be  tested,  and  C  is  a  lens  of  2  or  3  feet 
focal  length.  Rays  diverging  from  a  soda  flame  at  E  are  rendered  parallel  by 
the  lens,  and  after  reflection  from  the  surfaces  are  recombined  by  the  lens  at 
E.  To  make  an  observation,  the  coincidence  of  the  radiant  point  and  its 
image  must  be  somewhat  disturbed,  the  one  being  displaced  to  a  position 
a  little  beyond,  and  the  other  to  a  position  a  little  in  front  of,  the  diagram. 

The  eye,  protected  from  the  flame  by  a  suitable  screen,  is  placed  at  the 
image,  and  being  focused  upon  AB,  sees  the  field  traversed  by  bands.  The 
reflector  D  is  introduced  as  a  matter  of  convenience  to  make  the  line  of 
vision  horizontal. 

These  bands  may  be  photographed.  The  lens  of  the  camera  takes  the 
place  of  the  eye,  and  should  be  as  close  to  the  flame  as  possible.  With 
suitable  plates,  sensitised  by  cyanin,  the  exposure  required  may  vary  from 
ten  minutes  to  an  hour.  To  get  the  best  results,  the  hinder  surface  of  A 
should  be  blackened,  and  the  front  surface  of  B  should  be  thrown  out  of 
action  by  the  superposition  of  a  wedge-shaped  plate  of  glass,  the  intervening 
space  being  filled  with  oil  of  turpentine  or  other  fluid  having  nearly  the  same 

*  The  theory  is  given  in  a  paper  upon  "  Achromatic  Interference  Bands,"  Phil.  Mag.  Aug. 
1889.  [Vol.  in.  p.  288.] 


56 


INTERFERENCE   BANDS   AND   THEIR   APPLICATIONS. 


[202 


Fig.  2. 


refraction  as  glass.  Moreover,  the  light  should  be  purified  from  blue  rays  by 
a  trough  containing  solution  of  bichromate  of  potash.  With  these  pre- 
cautions the  dark  parts  of  the  bands  are  very  black,  and  the  exposure  may 
be  prolonged  much  beyond  what  would  otherwise  be  admissible. 

The  lantern  slides  exhibited  showed  the  elliptical  rings  indicative  of 
a  curvature  of  the  same  sign  in 
both  directions,  the  hyperbolic 
bands  corresponding  to  a  saddle- 
shaped  surface,  and  the  approxi- 
mately parallel  system  due  to  the 
juxtaposition  of  two  telescopic 
"flats,"  kindly  lent  by  Mr  Common. 
On  other  plates  were  seen  grooves 
due  to  rubbing  with  rouge  along 
a  defined  track,  and  depressions, 
some  of  considerable  regularity, 
obtained  by  the  action  of  diluted 
hydrofluoric  acid,  which  was  al- 
lowed to  stand  for  some  minutes  as  a  drop  upon  the  surface  of  the  glass. 

By  this  method  it  is  easy  to  compare  one  flat  with  another,  and  thus,  if 
the  first  be  known  to  be  free  from  error,  to  determine  the  errors  of  the 
second.  But  how  are  we  to  obtain  and  verify  a  standard  ?  The  plan 
usually  followed  is  to  bring  three  surfaces  into  comparison.  The  fact  that 
two  surfaces  can  be  made  to  fit  another  in  all  azimuths  proves  that  they  are 
spherical  and  of  equal  curvatures,  but  one  convex  and  the  other  concave,  the 
case  of  perfect  flatness  not  being  excluded.  If  A  and  B  fit  one  another,  and 
also  A  and  C,  it  follows  that  B  and  C  must  be  similar.  Hence,  if  B  and  G 
also  fit  one  another,  all  three  surfaces  must  be  flat.  By  an  extension  of  this 
process  the  errors  of  three  surfaces  which  are  not  flat  can  be  found  from 
a  consideration  of  the  interference  bands  which  they  present  when  combined 
in  three  pairs. 

But  although  the  method  just  referred  to  is  theoretically  complete,  its 
application  in  practice  is  extremely  tedious,  especially  when  the  surfaces  are 
not  of  revolution.  A  very  simple  solution  of  the  difficulty  has  been  found  in 
the  use  of  a  free  surface  of  water,  which,  when  protected  from  tremors  and 
motes,  is  as  flat  as  can  be  desired  *.  In  order  to  avoid  all  trace  of  capillary 
curvature  it  is  desirable  to  allow  a  margin  of  about  1£  inch.  The  surface  to 
be  tested  is  supported  horizontally  at  a  short  distance  (^  or  ^  inch)  below 
that  of  the  water,  and  the  whole  is  carried  upon  a  large  and  massive  levelling 
stand.  By  the  aid  of  screws  the  glass  surface  is  brought  into  approximate 

*  The  diameter  would  need  to  be  4  feet  in  order  that  the  depression  at  the  circumference, 
due  to  the  general  curvature  of  the  earth,  should  amount  to  ^  X. 


1893] 


INTERFEKENCE   BANDS    AND   THEIR   APPLICATIONS. 


57 


parallelism  with  the  water.  In  practice  the  principal  trouble  is  in  the 
avoidance  of  tremors  and  motes.  When  the  apparatus  is  set  up  on  the 
floor  of  a  cellar  in  the  country,  the  tremors  are  sufficiently  excluded,  but 
care  must  be  taken  to  protect  the  surface  from  the  slightest  draught.  To 
this  end  the  space  over  the  water  must  be  enclosed  almost  air-tight.  In 
towns,  during  the  hours  of  traffic,  it  would  probably  require  great  precaution 
to  avoid  the  disturbing  effects  of  tremors.  In  this  respect  it  is  advantageous 
to  diminish  the  thickness  of  the  layer  of  water;  but  if  the  thinning  be 
carried  too  far,  the  subsidence  of  the  water  surface  to  equilibrium  becomes 
surprisingly  slow,  and  a  doubt  may  be  felt  whether  after  all  there  may  not 
remain  some  deviation  from  flatness  due  to  irregularities  of  temperature. 

Fig.  3. 


With  the  aid  of  the  levelling  screws  the  bands  may  be  made  as  broad  as 
the  nature  of  the  surface  admits;  but  it  is  usually  better  so  to  adjust  the 
level  that  the  field  is  traversed  by  five  or  six  approximately  parallel  bands. 
Fig.  3  represents  bands  actually  observed  from  the  face  of  a  prism.  That 
these  are  not  straight,  parallel,  and  equidistant  is  a  proof  that  the  surface 
deviates  from  flatness.  The  question  next  arising  is  to  determine  the 
direction  of  the  deviation.  This  may  be  effected  by  observing  the  dis- 
placement of  the  bands  due  to  a  known  motion  of  the  levelling  screws ; 
but  a  simpler  process  is  open  to  us.  It  is  evident  that  if  the  surface  under 
test  were  to  be  moved  downwards  parallel  to  itself,  so  as  to  increase  the 
thickness  of  the  layer  of  water,  every  band  would  move  in  a  certain  direction, 
viz.  towards  the  side  where  the  layer  is  thinnest.  What  amounts  to  the 
same,  the  retardation  may  be  increased,  without  touching  the  apparatus,  by 
so  moving  the  eye  as  to  diminish  the  obliquity  of  the  reflection.  Suppose, 
for  example,  in  Fig.  3,  that  the  movement  in  question  causes  the  bands  to 
travel  downwards,  as  indicated  by  the  arrow.  The  inference  is  that  the 
surface  is  concave.  More  glass  must  be  removed  at  the  ends  of  the  bands 
than  in  the  middle  in  order  to  straighten  them.  If  the  object  be  to 
correct  the  errors  by  local  polishing  operations  upon  the  surface,  the  rule 
is  that  the  bands,  or  any  parts  of  them,  may  be  rubbed  in  the  direction  of 
the  arrow. 

A  good  many  surfaces  have  thus  been  operated  upon ;  and  although  a  fair 
amount  of  success  has  been  attained,  further  experiment  is  required  in  order 
to  determine  the  best  procedure.  There  is  a  tendency  to  leave  the  marginal 


58  INTERFERENCE   BANDS   AND  THEIR  APPLICATIONS.  [202 

parts  behind;  so  that  the  bands,  though  straight  over  the  greater  part  of 
their  length,  remain  curved  at  their  extremities.  In  some  cases  hydro- 
fluoric acid  has  been  resorted  to,  but  it  appears  to  be  rather  difficult  to 
control. 

The  delicacy  of  the  test  is  sufficient  for  every  optical  purpose. 
A  deviation  from  straightness  amounting  to  T'^  of  a  band  interval  could 
hardly  escape  the  eye,  even  on  simple  inspection.  This  corresponds  to 
a  departure  from  flatness  of  ^  of  a  wave-length  in  water,  or  about  3^  of 
the  wave-length  in  air.  Probably  a  deviation  of  -^X  could  be  made 
apparent. 

For  practical  purposes  a  layer  of  moderate  thickness,  adjusted  so  that 
the  two  systems  of  bands  corresponding  to  the  duplicity  of  the  soda  line  do 
not  interfere,  is  the  most  suitable.  But  if  we  wish  to  observe  bands  of  high 
interference,  not  only  must  the  thickness  be  increased,  but  certain  pre- 
cautions become  necessary.  For  instance,  the  influence  of  obliquity  must ' 
be  considered.  If  this  element  were  absolutely  constant,  it  would  entail  no 
ill  effect.  But  in  consequence  of  the  finite  diameter  of  the  pupil  of  the  eye, 
various  obliquities  are  mixed  up  together,  even  if  attention  be  confined  to 
one  part  of  the  field.  When  the  thickness  of  the  layer  is  increased,  it 
becomes  necessary  to  reduce  the  obliquity  to  a  minimum,  and  further  to 
diminish  the  aperture  of  the  eye  by  the  interposition  of  a  suitable  slit.  The 
effect  of  obliquity  is  shown  by  the  formula 

[2^t  cos  &  =  n\]. 

The  necessary  parallelism  of  the  operative  surfaces  may  be  obtained,  as  in 
the  above  described  apparatus,  by  the  aid  of  levelling.  But  a  much  simpler 
device  may  be  employed,  by  which  the  experimental  difficulties  are  greatly 
reduced.  If  we  superpose  a  layer  of  water  upon  a  surface  of  mercury,  the 
flatness  and  parallelism  of  the  surfaces  take  care  of  themselves.  The 
objection  that  the  two  surfaces  would  reflect  very  unequally  may  be  obviated 
by  the  addition  of  so  much  dissolved  colouring  matter,  e.g.  soluble  aniline 
blue,  to  the  water  as  shall  equalise  the  intensities  of  the  two  reflected  lights. 
If  the  adjustments  are  properly  made,  the  whole  field,  with  the  exception  of 
a  margin  near  the  sides  of  the  containing  vessel,  may  be  brought  to  one 
degree  of  brightness,  being  in  fact  all  included  within  a  fraction  of  a  band. 
The  width  of  the  margin,  within  which  rings  appear,  is  about  one  inch,  in 
agreement  with  calculation  founded  upon  the  known  values  of  the  capillary 
constants.  During  the  establishment  of  equilibrium  after  a  disturbance, 
bands  are  seen  due  to  variable  thickness,  and  when  the  layer  is  thin,  they 
persist  for  a  considerable  time. 

When  the  thickness  of  the  layer  is  increased  beyond  a  certain  point,  the 
difficulty  above  discussed,  depending  upon  obliquity,  becomes  excessive,  and 
it  is  advisable  to  change  the  manner  of  observation  to  that  adopted  by 


1893]  INTERFERENCE   BANDS   AND   THEIR   APPLICATIONS.  59 

Michelson*.  In  this  case  the  eye  is  focused,  not,  as  before,  upon  the 
operative  surfaces,  but  upon  the  flame,  or  rather  upon  its  image  at  E 
(Fig.  2).  For  this  purpose  it  is  only  necessary  to  introduce  an  eye-piece 
of  low  power,  which  with  the  lens  C  (in  its  second  operation)  may  be 
regarded  as  a  telescope.  The  bands  now  seen  depend  entirely  upon  obliquity 
according  to  the  formula  above  written,  and  therefore  take  the  form  of 
circular  arcs.  Since  the  thickness  of  the  layer  is  absolutely  constant,  there 
is  nothing  to  interfere  with  the  perfection  of  the  bands  except  want  of 
homogeneity  in  the  light. 

But,  as  Fizeau  found  many  years  ago,  the  latter  difficulty  soon  becomes 
serious.  At  a  very  moderate  thickness  it  becomes  necessary  to  reduce  the 
supply  of  soda,  and  even  with  a  very  feeble  flame  a  limit  is  soon  reached. 
When  the  thickness  was  pushed  as  far  as  possible,  the  retardation,  calculated 
from  the  volume  of  liquid  and  the  diameter  of  the  vessel,  was  found  to  be 
50,000  wave-lengths,  almost  exactly  the  limit  fixed  by  Fizeau. 

To  carry  the  experiment  further  requires  still  more  homogeneous  sources 
of  light.  It  is  well  known  that  Michelson  has  recently  observed  interference 
with  retardations  previously  unheard  of,  and  with  the  aid  of  an  instrument  of 
ingenious  construction  has  obtained  most  interesting  information  with  respect 
to  the  structure  of  various  spectral  lines. 

A  curious  observation  respecting  the  action  of  hydrofluoric  acid  upon 
polished  glass  surfaces  was  mentioned  in  conclusion.  After  the  operation  of 
the  acid  the  surfaces  appear  to  be  covered  with  fine  scratches,  in  a  manner 
which  at  first  suggested  the  idea  that  the  glass  had  been  left  in  a  specially 
tender  condition,  and  had  become  scratched  during  the  subsequent  wiping. 
But  it  soon  appeared  that  the  effect  was  a  development  of  scratches  previously 
existent  in  a  latent  state.  Thus  parallel  lines  ruled  with  a  knife-edge,  at  first 
invisible  even  in  a  favourable  light,  became  conspicuous  after  treatment  with 
acid.  Perhaps  the  simplest  way  of  regarding  the  matter  is  to  consider  the 
case  of  a  furrow  with  perpendicular  sides  and  a  flat  bottom.  If  the  acid  may 
be  supposed  to  eat  in  equally  in  all  directions,  the  effect  will  be  to  broaden 
the  furrow,  while  the  depth  remains  unaltered.  It  is  possible  that  this 
method  might  be  employed  with  advantage  to  intensify  (if  a  photographic 
term  may  be  permitted)  gratings  ruled  upon  glass  for  the  formation  of 
spectra. 

*  [1902.  The  influence  of  the  diameter  of  the  pupil  of  the  eye  in  lessening  the  visibility  of 
fringes  dependent  primarily  upon  variable  thickness,  seems  to  have  been  first  pointed  out  by 
Lummer  (Wied.  Ann.  xxm.  p.  4'J,  1884),  who  also  emphasised  the  advantages  attending  the  use  of 
a  plate  of  uniform  thickness  and  of  rings  dependent  solely  upon  obliquity,  whether  the  object 
be  the  investigation  of  high  interference  itself,  or  the  examination  for  uniformity  of  plates 
intended  to  be  plane-parallel. 

The  circular  ring  system  dependent  upon  obliquity  was  first  observed  by  Haidinger  (Pogg. 
Ann.  LXXVII.  p.  219,  1849)  and  explained  by  Mascart  (Ann.  de  Chim.  xxm.  p.  116,  1871).] 


203. 

ON  THE  THEORY   OF  STELLAR  SCINTILLATION. 

[Philosophical  Magazine,  xxxvi.  pp.  129—142,  1893.] 

ARAGO'S  theory  of  this  phenomenon  is  still  perhaps  the  most  familiar, 
although  I  believe  it  may  be  regarded  as  abandoned  by  the  best  authorities. 
According  to  it  the  momentary  disappearance  of  the  light  of  the  star  is  due 
to  accidental  interference  between  the  rays  which  pass  the  two  halves  of  the 
pupil  of  the  eye  or  the  object-glass  of  the  telescope.  When  the  relative 
retardation  amounts  to  an  odd  multiple  of  the  half  wave-length  of  any  kind 
of  light,  such  light,  it  is  argued,  vanishes  from  the  spectrum  of  the  star. 
,  But  this  theory  is  based  upon  a  complete  misconception.  "  It  is  as  far  as 
possible  from  being  true  that  a  body  emitting  homogeneous  light  would 
disappear  on  merely  covering  half  the  aperture  of  vision  with  a  half  wave 
plate.  Such  a  conclusion  would  be  in  the  face  of  the  principle  of  energy, 
which  teaches  plainly  that  the  retardation  in  question  would  leave  the 
aggregate  brightness  unaltered*."  It  follows  indeed  from  the  principle  of 
interference  that  there  will  be  darkness  at  the  precise  point  which  before  the 
introduction  of  the  half  wave  plate  formed  the  centre  of  the  image,  but  the 
light  missing  there  is  to  be  found  in  a  slightly  displaced  position  f. 

*  Enc.  Brit.,  "  Wave  Theory,"  p.  441.     [Vol.  ra.  p.  123.] 

t  Since  the  remarks  in  the  text  were  written  I  have  read  the  version  of  Arago's  theory  given 
by  Mascart  (Traite  d'Optique,  t.  in.  p.  348).  From  this  some  of  the  most  objectionable  features 
have  been  eliminated.  But  there  can  be  no  doubt  as  to  Arago's  meaning.  "  Supposons  que  les 
rayons  qui  tombent  a  gauche  du  centre  de  1'ohjectif  aient  rencontre,  depuis  les  limites  superieures 
de  I'atmosphere,  des  couches  qui,  a  cause  de  leur  densite,  de  leur  temperature,  ou  de  leur  etat 
hygrometrique,  etaient  douees  d'une  refringence  differente  de  celle  que  possedaient  les  conches 
traversees  par  les  rayons  de  droite ;  il  pourra  arriver,  qu'a  raison  de  cette  difference  de  refringence, 
les  rayons  rouges  de  droite  detrnisent  en  totalite  les  rayons  rouges  de  gauche,  et  que  le  foyer 

passe  du  blanc,  son  etat  normal,  an  vert Voila  done  le  resnltat  theorique  par fai lenient 

d'accord  avec  les  observations ;  voila  le  phenomena  de  la  scintillation  dans  nne  lunette  rat  t  ache 
d'une  maniere  intime  a  la  doctrine  des  interferences"  (I'Annuaire  du  Bureau  des  Longitude*  pour 
1852,  pp.  423,  425). 

That  the  difference  between  Arago's  theory  and  that  followed  in  the  present  paper  is  funda- 
mental will  be  recognized  when  it  is  noticed  that,  according  to  the  former,  the  colour  effects  of 
scintillation  would  be  nearly  independent  of  atmospheric  dupersion.  Arago  gives  an  interesting 
summary  of  the  views  held  by  early  writers. 


1893]  ON   THE   THEORY   OF  STELLAR  SCINTILLATION.  61 

The  older  view  that  scintillation  is  due  to  the  actual  diversion  of  light 
from  the  aperture  of  vision  by  atmospheric  irregularities  was  powerfully 
supported  by  Montigny*,  to  whom  we  owe  also  a  leading  feature  of  the  true 
theory,  that  is,  the  explanation  of  the  chromatic  effects  by  reference  to  the 
different  paths  pursued  by  rays  of  different  colours  in  virtue  of  regular 
atmospheric  dispersion.  The  path  of  the  violet  ray  lies  higher  than  that  of 
the  red  ray  which  reaches  the  eye  of  the  observer  from  the  same  star,  and  the 
separation  may  be  sufficient  to  allow  the  one  to  escape  the  influence  of  an 
atmospheric  irregularity  which  operates  upon  the  other.  In  Montigny 's  view 
the  diversion  of  the  light  is  caused  by  total  reflexion  at  strata  of  varying 
density. 

But  the  most  important  work  upon  this  subject  is  undoubtedly  that  of 
Respighi-f-,  who,  following  in  the  steps  of  Montigny  and  Wolf,  applied  the 
spectroscope  to  the  investigation  of  stellar  scintillation.  The  results  of  these 
observations  are  summed  up  under  thirteen  heads,  which  it  will  be  convenient 
to  give  almost  at  full  length. 

(I.)  In  spectra  of  stars  near  the  horizon  we  may  observe  dark  or  bright 
bands,  transversal  or  perpendicular  to  the  length  of  the  spectrum,  which  more 
or  less  quickly  travel  from  the  red  to  the  violet  or  from  the  violet  to  the  red, 
or  oscillate  from  one  to  the  other  colour ;  and  this  however  the  spectrum  may 
be  directed  from  the  horizontal  to  the  vertical. 

(II.)  In  normal  atmospheric  conditions  the  motion  of  the  bands  proceeds 
regularly  from  red  to  violet  for  stars  in  the  west,  and  from  violet  to  red  for 
stars  in  the  east ;  while  in  the  neighbourhood  of  the  meridian  the  movement 
is  usually  oscillatory,  or  even  limited  to  one  part  of  the  spectrum. 

(III.)  In  observing  the  horizontal  spectra  of  stars  more  and  more  elevated 
above  the  horizon,  the  bands  are  seen  sensibly  parallel  to  one  another,  but 
more  or  less  inclined  to  the  axis  of  the  spectrum,  passing  from  red  to  violet  or 
reversely  according  as  the  star  is  in  the  west  or  the  east. 

(IV.)  The  inclination  of  the  bands,  or  the  angle  formed  by  them  with 
the  axis  (?  transversal)  of  the  spectrum,  depends  upon  the  height  of  the  star ; 
it  reduces  to  0°  at  the  horizon  and  increases  rapidly  with  the  altitude  so  as  to 
reach  90°  at  an  elevation  of  30°  or  40°,  so  that  at  this  elevation  the  bands 
become  longitudinal. 

(V.)  The  inclination  of  the  bands,  reckoned  downwards,  is  towards  the 
more  refrangible  end  of  the  spectrum. 

(VI.)  The  bands  are  most  marked  and  distinct  when  the  altitude  of  the 
star  is  least.  At  an  altitude  of  more  than  40°  the  longitudinal  bands  are 
reduced  to  mere  shaded  streaks,  and  often  can  only  be  observed  upon  the 
spectrum  as  slight  general  variations  of  brightness. 

*  Mem.  de  VAcad.  d.  Bruxelles,  i.  xxvin.  (1856). 

t  Roma,  Atti  Nuovi  Lincei,  xxi.  (1868) ;  Assoc.  Fran$aise,  Compt.  Rend.  i.  (1872),  p.  169. 


62  ON   THE   THEORY   OF   STELLAK   SCINTILLATION.  [203 

(VII.)  As  the  altitude  increases,  the  movement  of  the  bands  becomes 
quicker  and  less  regular. 

(VIII.)  As  the  prism  is  turned  so  as  to  bring  the  spectrum  from  the 
horizontal  to  the  vertical  position,  the  inclination  of  the  bands  to  the 
transversal  of  the  spectrum  continually  diminishes  until  it  becomes  zero  when 
the  spectrum  is  nearly  vertical;  but  the  bands  then  become  less  marked, 
retaining,  however,  the  movement  in  the  direction  indicated  above  (III.)- 

(IX.)  Luminous  bands  are  less  frequent  and  less  regular  than  dark 
bands,  and  occur  well  marked  only  in  the  spectra  of  stars  near  the  horizon. 

(X.)  In  the  midst  of  this  general  and  violent  movement  of  bright  and 
dark  masses  in  the  spectra  of  stars,  the  black  spectral  lines  proper  to  the 
light  of  each  star  remain  sensibly  quiescent  or  undergo  very  slight  oscil- 
lations. 

(XI.)  Under  abnormal  atmospheric  conditions  the  bands  are  fainter  and 
less  regular  in  shape  and  movement. 

(XII.)  When  strong  winds  prevail  the  bands  are  usually  rather  faint 
and  ill  defined,  and  then  the  spectrum  exhibits  mere  changes  of  brightness, 
even  in  the  case  of  stars  near  the  horizon. 

(XIII.)  Good  definition  and  regular  movement  of  the  bands  seems  to  be 
a  sign  of  the  probable  continuance  of  fine  weather,  and,  on  the  other  hand, 
irregularity  in  these  phenomena  indicates  probable  change. 

These  results  show  plainly  that  the  changes  of  intensity  and  colour  in  the 
images  of  stars  are  produced  by  a  momentary  real  diversion  of  the  luminous 
rays  from  the  object-glass  of  the  telescope ;  that  in  the  neighbourhood  of  the 
horizon  rays  of  different  colours  are  affected  separately  and  successively,  and 
that  all  the  rays  of  a  given  colour  are  momentarily  withdrawn  from  the  whole 
of  the  object-glass. 

Most  of  his  conclusions  from  observation  were  readily  explained  by 
Respighi  as  due  to  irregular  refractions,  not  necessarily  or  usually  amounting 
(as  Montigny  supposed)  to  total  reflexions,  taking  place  at  a  sufficient  distance 
from  the  observer.  The  progress  of  the  bands  in  one  direction  along  the 
spectrum  (II.)  is  attributed  to  the  diurnal  motion.  In  the  case  of  a  setting 
star,  for  instance,  the  blue  rays  by  which  it  is  seen,  pursuing  a  higher  course 
through  the  atmosphere,  encounter  an  obstacle  somewhat  later  than  do  the 
red  rays.  Hence  the  band  travels  towards  the  violet  end  of  the  spectrum. 
In  the  neighbourhood  of  the  meridian  this  cause  of  a  progressive  movement 
ceases  to  operate. 

The  observations  recorded  in  (III.)  are  of  special  interest  as  establishing  a 
connexion  between  the  rates  with  which  various  parts  of  the  object-glass  and 
of  the  spectrum  are  affected.  Since  the  spectrum  is  horizontal,  various  parts 


1893]  ON   THE   THEORY   OF   STELLAR   SCINTILLATION.  63 

of  its  width  correspond  to  various  horizontal  sections  of  the  objective,  and 
the  existence  of  bands  at  a  definite  inclination  shows  that  at  the  moment 
when  the  shadow  of  the  obstacle  thrown  by  blue  rays  reaches  the  bottom 
of  the  glass  the  shadow  at  the  top  is  that  thrown  by  green,  yellow,  or  red 
rays  of  less  refrangibility.  When  the  altitude  of  the  star  reaches  30°  or 
40°,  the  difference  of  path  due  to  atmospheric  dispersion  is  insufficient  to 
differentiate  the  various  parts  of  the  spectrum.  The  bands  then  appear 
longitudinal. 

The  definite  obliquity  of  the  bands  at  moderate  altitudes,  reported  by 
Respighi,  leads  to  a  conclusion  of  some  interest,  which  does  not  appear  to 
have  been  noticed.  In  the  case  of  a  given  star,  observed  at  a  given  altitude, 
the  linear  separation  at  the  telescope  of  the  shadows  of  the  same  obstacle 
thrown  by  rays  of  various  colours  will  of  necessity  depend  upon  the  distance 
of  the  obstacle.  But  the  definiteness  of  the  obliquity  of  the  bands  requires 
that  this  separation  shall  not  vary,  and  therefore  that  the  obstacles  to  which 
the  effects  are  due  are  sensibly  at  one  distance  only.  It  would  seem  to  follow 
from  this  that,  under  "  normal  atmospheric  conditions,"  scintillation  depends 
upon  irregularities  limited  to  a  comparatively  narrow  horizontal  stratum 
situated  overhead.  A  further  consequence  will  be  that  the  distance  of  the 
obstacles  increases  as  the  altitude  of  the  star  diminishes,  and  this  according 
to  a  definite  law. 

The  principal  object  of  the  present  communication  is  to  exhibit  some 
of  the  consequences  of  the  theory  of  scintillation  in  a  definite  mathematical 
form.  The  investigation  may  be  conducted  by  simple  methods,  if,  as  suffices 
for  most  purposes,  we  regard  the  whole  refraction  as  small,  and  neglect  the 
influence  of  the  earth's  curvature.  When  the  object  is  to  calculate  with 
accuracy  the  refraction  itself,  further  approximations  are  necessary,  but  even 
in  this  case  the  required  result  can  be  obtained  with  more  ease  than  is 
generally  supposed. 

The  foundation  upon  which  it  is  most  convenient  to  build  is  the  idea  of 
James  Thomson*,  which  establishes  instantaneously  the  connexion  between 
the  curvature  of  a  ray  travelling  in  a  medium  of  varying  optical  constitution 
and  the  rate  at  which  the  index  changes  at  the  point  in  question.  The 
following  is  from  Everett's  memoir: — 

"  Draw  normal  planes  to  a  ray  at  two  consecutive  points  of  its  path. 
Then  the  distance  of  their  intersection  from  either  point  will  be  p,  the  radius 
of  curvature.  But  these  normal  planes  are  tangential  to  the  wave-front  in 
its  two  consecutive  positions.  Hence  it  is  easily  shown  by  similar  triangles 
that  a  very  short  line  dN  drawn  from  either  of  the  points  towards  the  centre 
of  curvature  is  to  the  whole  length  p,  of  which  it  forms  part,  as  do  the 

*  Brit.  Assoc.  Eep.  1872.     Everett,  Phil.  Mag.  March  1873. 


64  ON   THE   THEORY   OF   STELLAR   SCINTILLATION.  [203 

difference  of  the  velocities  of  light  at  its  two  ends  is  to  v  the  velocity  at 

either  end.     That  is 

dN/p  =  -  dv/v, 

the  negative  sign  being  used  because  the  velocity  diminishes  in  approaching 
the  centre  of  curvature.     But,  since  v  varies  inversely  as  //,,  we  have 

—  dv/v  =  dpi  p. 
Hence  the  curvature  1  /p  is  given  by  any  of  the  four  following  expressions  :  — 

1         1  dv  _     d  log  v  _  1  dfj,  _d  log  //, 
~P^~vdN  dN~=~jj,dN  =  ~dN~'    ' 

"  The  curvatures  of  different  rays  at  the  same  point  are  directly  as  the 
rates  of  increase  of  JJL  in  travelling  along  their  respective  normals."  If  6 
denote  the  angle  which  the  ray  makes  with  the  direction  of  most  rapid 
increase  of  index,  the  curvatures  will  be  directly  as  the  values  of  sin  6.  In 
fact,  if  dfi/dr  denote  the  rate  at  which  //,  increases  in  a  direction  normal  to 
the  surfaces  of  equal  index,  we  have 

da      dfj,    .     a 

-j^r  =  -r-  sin  6, 
dN     dr 

and  therefore 

1      1  da   .    n     d  log  a  .    f.  ,_, 

-  =  -  --sm#=        8"sm0  ......................  (2) 


, 

~  p     p.    r  dr 

Everett  shows  how  the  well-known  equation 

ftp  =  const  .....................................  (3) 

can  be  deduced  from  (2),  p  being  the  perpendicular  upon  the  ray  from  the 
centre  of  spherical  surfaces  of  equal  index.     In  general, 

I     I  dp  a     p 

-  =  -/-,        sin  6  =  *-  , 
p     r  dr  r 

and  thus 

1  dp  _  p  d  log  fj, 

r  dr      r     dr 
giving  (3)  on  integration. 

At  a  first  application  of  (2)  we  may  find  by  means  of  it  a  first  ap- 
proximation to  the  law  of  atmospheric  refraction,  on  the  supposition  that 
the  whole  refraction  is  small  and  that  the  curvature  of  the  earth  may  be 
neglected.  Under  these  limitations  B  in  (2)  may  be  treated  as  constant 
along  the  whole  path  of  the  ray  ;  and  if  dty  be  the  angle  through  which 
the  ray  turns  in  describing  the  element  of  arc  ds,  we  have 

d-*fr  =      °M  sin  6  ds  =  tan  6  .  d  log  p. 


1893]  ON   THE   THEORY   OF   STELLAR  SCINTILLATION.  65 

If  we   integrate   this   along   the   whole  course  of  the  ray  through  the 
atmosphere,  that  is  from  p,  =  1  to  /A  =  /i0,  we  get,  as  the  whole  refraction, 

^  =  log  /*„  tan  0  =  0*0-1)  tan  0,     .....................  (4) 

for  to  the  order  of  approximation  in  question  log  /*„  may  be  identified  with 
0"o-l). 

If  Si/r  denote  the  chromatic  variation  of  ^  corresponding  to  8jj,0,  we  have 
from  (4) 

-l)  ............................  (5) 


According  to  Mascart*  the  value  of  the  right-hand  member  of  (5)  in 
the  case  of  air  and  of  the  lines  B  and  H  is 

^0/0*0-1)  =  -024  ...............................  (6) 

We  will  now  take  a  step  further  and  calculate  the  linear  deviation  of 
a  ray  from  a  straight  course,  still  upon  the  supposition  that  the  whole 
refraction  is  small.  If  rj  denote  the  linear  deviation  (reckoned  perpen- 
dicularly) at  any  point  defined  by  the  length  s  measured  along  the  ray  0, 
we  have 


, 

ds 

so  that 

~  =  ltan0c?log/,t  =  tan0(yLt—  1)  +  a, 

a  being  a  constant  of  integration.     A  second  integration  now  gives 

17  =  tan  0j(fi-l)ds  +  as  +  {3,  .....................  (7) 

which  determines  the  path  of  the  ray.     If  y  be  the  height  of  any  point 
above  the  surface  of  the  earth,  ds  =  dy  sec  0;  so  that  (7)  may  also  be  written 


The  origin  of  s  is  arbitrary,  but  we  may  conveniently  take  it  at  the  point 
(A  )  where  the  ray  strikes  the  earth's  surface. 

We  will  now  consider  also  a  second  ray,  of  another  colour,  deviating 
from  the  line  0  by  the  distance  77  +  877,  and  corresponding  to  a  change  of  /* 
to  /i  +  8fji.  The  distance  between  the  two  rays  at  any  point  y  is 


(9) 


In  this  equation  &@  denotes  the  separation  of  the  rays  at  A  ,  where  y  =  0, 
*  Everett's  C,  G.  S.  System  of  Units. 


QQ  ON   THE   THEORY   OF   STELLAR   SCINTILLATION.  [203 

s  =  0.     And  8a  denotes  the  angle  between  the  rays  when  outside  the  atmo- 
sphere. 

Equation  (9)  may  be  applied  at  once  to  Montigny's  problem,  that  is  to 
determine  the  separation  of  two  rays  of  different  colours,  both  coming  from 
the  same  star,  and  both  arriving  at  the  same  point  A.  The  first  condition 
gives  Sot.  =  0,  and  the  second  gives  S/3  =  0.  Accordingly, 


is  the  solution  of  the  question. 

The  integral  in  (10)  may  be  otherwise  expressed  by  means  of  the  principle 
that  (/z  —  1)  and  Sp  are  proportional  to  the  density.  Thus,  if  I  denote  the 
"  height  of  the  homogeneous  atmosphere,"  and  h  the  elevation  in  such  an 
atmosphere  determined  by  the  condition  that  there  shall  be  as  much  air 
below  it  as  actually  exists  below  y, 


(11) 


S/AO  being  the  value  of  S/A  at  the  surface  of  the  earth.     Equation  (10)  thus 
becomes 


At  the  limits  of  the  atmosphere  and  beyond,  h  =  I,  and  the  separation  there  is 


cos'0     ............................... 

These  results  are  applicable  to  all  altitudes  higher  than  about  10°. 

The  formulae  given  by  Montigny  (loc.  cit.)  are  quite  different  from  the 
above.  That  corresponding  to  (13)  is 

By  =  S/i0asin  0,  ..............................  (14) 

a  being  the  radius  of  the  earth  !  The  substitution  of  a  for  I  increases  the 
calculated  result  some  800  times.  But  this  is  in  a  large  measure  compen- 
sated by  the  factor  sec2  6  in  (13),  for  at  low  altitudes  sec  0  is  large.  According 
to  Montigny  the  separation  at  moderately  low  altitudes  would  be  nearly  in- 
dependent of  the  altitude,  a  conclusion  entirely  wide  of  the  truth. 

The  value  of  (/*»-!)  for  air  at  0°  and  760  millim.  at  Paris  is  "0002927, 
so  that  S/*0  (for  the  lines  E  and  H)  is  "000007025.  The  height  of  the 
homogeneous  atmosphere  is  7'990  x  105  centim.,  and  thus  &?  reckoned  in 
centim.  is 


1893]  ON   THE   THEORY   OF   STELLAR   SCINTILLATION. 

The  following  are  a  few  corresponding  values  of  6  and  sin  0/cos20 


67 


e 

sin  0/cos2  6 

e 

sin  0/cos2  0 

0 

0 

o-ooo 

60° 

3-46 

20 

0-387 

70 

8-03 

40 

1-095 

80 

32-66 

Thus  at  the  limit  of  the  atmosphere  the  separation  of  rays  which  reach 
the  observer  at  an  apparent  altitude  of  10°  is  185  centim.  Nearer  the 
horizon  the  separation  would  be  still  greater,  but  its  value  cannot  well  be 
found  from  (15).  Although  these  estimates  are  considerably  less  than  those 
of  Montigny,  the  separation  near  the  horizon  seems  to  be  sufficient  to 
explain  the  vertical  position  of  the  bands  in  the  spectrum,  recorded  by 
Respighi  (I.).  The  fact  that  the  margin  is  not  very  great  suggests  that  the 
obstacles  to  which  scintillation  is  due  may  often  be  situated  at  a  considerable 
elevation. 

We  have  now  to  consider  the  effect  of  an  obstacle  situated  at  a  given 
point  B  at  level  y  on  the  course  of  the  ray.  And  the  first  desideratum  will 
be  the  estimation  of  the  separation  at  A,  the  object-glass  of  the  telescope, 
of  rays  of  various  colours  corning  from  the  same  star,  which  all  pass  through 
the  given  point  B.  It  will  appear  at  once  that  no  fresh  question  is  raised. 
For,  since  the  rays  come  from  the  same  star  at  the  same  time,  8a  —  0,  and 
thus  by  (9)  8-rjA  =  S/3.  The  value  of  £/3  is  given  at  once  by  the  condition 
that  &i)B  =  0.  Thus 


as  before.     The  discussion,  already  given  of  (15),  is  thus  immediately  ap- 
plicable. 

Equation  (16)  solves  the  problem  of  determining  the  inclination  of  the 
bands  seen  in  the  spectra  of  stars  not  very  low  (III.).  It  is  only  necessary 
to  equate  —  8rjA  to  the  aperture  of  the  telescope.  fy*0  then  gives  the  range 
of  refrangibility  covered  by  the  bands  as  inclined.  In  practice  h  would  not 
be  known  beforehand;  but  from  the  observed  inclination  of  the  bands  it 
would  be  possible  to  determine  it. 

In  a  given  state  of  the  atmosphere  h,  so  far  as  it  is  definite,  must  be 
constant  and  then  B/JLO  must  be  proportional  to  cos2  6  1  sin  6.  This  gives  the 
relation  between  the  altitude  of  the  star  and  the  inclination  of  the  bands. 

When  6  is  small,  fyi0  is  large  ;  that  is,  the  bands  become  longitudinal. 

5—2 


68  ON   THE   THEORY   OF   STELLAR   SCINTILLATION.  [203 

As  a  numerical  example,  let  us  suppose  that  the  aperture  of  the  telescope 
is  10  centim.,  and  that  at  an  altitude  of  10°  the  obliquity  of  the  bands  is 
such  that  the  vertical  diameter  of  the  object-glass  corresponds  to  the  entire 
range  from  B  to  H.  In  this  case  (15)  gives 

,      10  1 


indicating  that  the  obstacles  to  which  the  bands  are  due  are  situated  at  such 
a  level  that  about  -£$  of  the  whole  mass  of  the  atmosphere  is  below  them. 

The  next  question  to  which  (9)  may  be  applied  is  to  find  the  angle  Set 
outside  the  atmosphere  between  two  rays  of  different  colours  which  pass 
through  the  two  points  A  and  B.  Here  &r]A  =  0,  and  thus  8/3  =  0.  And 
further,  since  BrjB  =  0,  we  get 

sin  0    [u  ~    j       &/j,0h  tan  6 


If  the  height  of  the  obstacle  above  the  ground  be  so  small  that  the 
density  of  the  air  below  it  is  sensibly  uniform,  then  h  =  y,  and 

-  Sa  =  S/-iotan0  ...............................  (18) 

In  this  case  the  angle  is  the  same  as  that  of  the  spectrum  of  the  star 
observed  at  A,  as  appears  from  (4)  and  (5).  In  general,  y  is  greater  than  h, 
so  that  So.  is  somewhat  less  than  the  value  given  by  (18). 

The  interest  of  (18)  lies  in  the  application  of  it  to  find  the  time  occupied 
by  a  band  in  traversing  the  spectrum  in  virtue  of  the  diurnal  motion,  ac- 
cording to  Respighi's  observation  (II.).  The  time  required  is  that  necessary 
for  the  star  to  rise  or  fall  through  the  angle  of  its  dispersion-spectrum  at 
the  altitude  in  question.  At  an  altitude  of  10°,  this  angle  will  be  8",  being 
always  about  ^  of  the  whole  refraction.  The  rate  at  which  a  star  rises  or 
falls  depends  of  course  upon  the  declination  of  the  star  and  upon  the  latitude 
of  the  observer,  and  may  vary  from  zero  to  15°  per  hour.  At  the  latter 
maximum  rate  the  star  would  describe  8"  in  about  one  half  of  a  second, 
which  would  therefore  be  the  time  occupied  by  a  band  in  crossing  the 
spectrum  under  the  circumstances  supposed.  In  the  case  of  a  star  quite 
close  to  the  horizon,  the  progress  of  the  band  would  be  a  good  deal  slower. 

The  fact  that  the  larger  planets  scintillate  but  little,  even  under  favour- 
able conditions,  is  readily  explained  by  their  sensible  apparent  magnitude. 
The  separation  of  rays  of  one  colour  thus  arising  during  their  passage  through 
the  atmosphere  is  usually  far  greater  than  the  already  calculated  separation, 
due  to  chromatic  dispersion  ;  so  that  if  a  fixed  star  of  no  apparent  magnitude 
scintillates  in  colours,  the  different  parts  of  the  area  of  a  planet  must  a 
fortiori  scintillate  independently.  But  under  these  circumstances  the  eye 
perceives  only  an  average  effect,  and  there  is  no  scintillation  visible. 


1893]  ON   THE   THEORY    OF   STELLAR   SCINTILLATION.  69 

The  non-scintillation  of  small  stars  situated  near  the  horizon  may  be 
referred  to  the  failure  of  the  eye  to  appreciate  colour  when  the  light  is  faint. 

In  the  case  of  stars  higher  up,  the  whole  spectrum  is  affected  simul- 
taneously. A  momentary  accession  of  illumination,  due  to  the  passage  of 
an  atmospheric  irregularity,  may  thus  render  visible  a  star  which  on  account 
of  its  faintness  could  not  be  steadily  seen  through  an  undisturbed  atmo- 
sphere *. 

In  the  preceding  discussion  the  refracting  obstacles  have  for  the  sake  of 
brevity  been  spoken  of  as  throwing  sharp  shadows.  This  of  course  cannot 
happen,  if  only  in  consequence  of  diffraction ;  and  it  is  of  some  interest  to 
inquire  into  the  magnitude  of  the  necessary  diffusion.  The  theory  of 
diffraction  shows  that  even  in  the  case  of  an  opaque  screen  with  a  definite 
straight  boundary,  the  transition  of  illumination  at  the  edge  of  the  shadow 
occupies  a  space  such  as  ^/(b\),  where  X  is  the  wave-length  of  the  light,  and 
b  is  the  distance  across  which  the  shadow  is  thrown.  We  may  take  X  at 
6  x  10~5  centim.,  and  if  6  be  reckoned  in  kilometres,  we  have  as  the  space 
of  transition,  \'(6b).  Thus  if  b  were  4  kilometres,  the  space  of  transition 
would  amount  to  about  5  centim.  The  inference  is  that  the  various  parts  of 
the  aperture  of  a  small  telescope  cannot  be  very  differently  affected  unless 
the  obstacles  to  which  the  scintillation  is  due  are  at  a  less  distance  than 
4  kilometres. 

One  of  the  principal  outstanding  difficulties  in  the  theory  of  scintillation 
is  to  see  how  the  transition  from  one  index  to  another  in  an  atmospheric 
irregularity  can  be  sufficiently  sudden.  The  fact  that  the  various  parts  of  a 
not  too  small  object-glass  are  diversely  affected  seems  to  prove  that  the 
transitions  in  question  do  not  occupy  many  centimetres.  Now,  whether  the 
irregularity  be  due  to  temperature  or  to  moisture,  we  should  expect  that  a 
transition,  however  abrupt  at  first,  would  after  a  few  minutes  or  hours  be 
eased  off  to  a  greater  degree  than  would  accord  with  the  above  estimate. 
Perhaps  the  abruptness  of  transition  is,  as  it  were,  continually  renewed  by 
the  coming  into  contact  of  fresh  portions  of  light  and  dense  air  as  the 
ascending  and  descending  streams  proceed  in  their  courses.  The  speculations 
and  experiments  of  Jevons  on  the  Cirrus  form  of  Cloud  f  may  find  some 
application  here.  A  preliminary  question  requiring  attention  is  as  to  the 
origin  of  the  irregularities  which  cause  scintillation.  Is  it  always  at  the 
ground,  and  mainly  under  the  influence  of  sunshine  ?  Or  may  irregular 
absorption  of  solar  heat  in  the  atmosphere,  due  to  varying  proportions  of 
moisture,  give  rise  to  transitions  of  the  necessary  abruptness  ?  Again,  we 
may  ask  how  many  obstacles  are  to  be  supposed  operative  upon  the  same 

*  The  theory  of  Arago  leads  him  to  a  directly  opposite  conclusion  (loc.  cit.  p.  381). 
t  Phil.  Mag.  xiv.  p.  22,  1857.     For  a  mathematical  investigation,  by  the  author,  see  Math. 
Soc.  Proc.  xiv.  April  1883.     [Vol.  n.  p.  200.] 


70 


ON   THE  THEORY   OF   STELLAR  SCINTILLATION. 


[203 


ray  ?  Is  the  ultimate  effect  only  a  small  residue  from  many  causes  in  the 
main  neutralizing  one  another?  It  does  not  appear  that  in  the  present 
state  of  meteorological  science  satisfactory  answers  can  be  given  to  these 
questions. 

A  complete  investigation  of  atmospheric  refraction  can  only  be  made 
upon  the  basis  of  some  hypothesis  as  to  the  distribution  of  temperature ;  but, 
as  has  already  been  hinted,  a  second  approximation  to  the  value  of  the 
refraction  can  be  obtained  independently  of  such  knowledge  and  without 
difficulty.  In  Laplace's  elaborate  investigation  it  is  very  insufficiently  recog- 
nized, if  indeed  it  be  recognized  at  all,  that  the  whole  difficulty  of  the 
problem  depends  upon  the  curvature  of  the  earth.  If  this  be  neglected, 
that  is  if  the  strata  are  supposed  to  be  plane,  the  desired  result  follows  at 
once  from  the  law  of  refraction,  without  the  necessity  of  knowing  anything 
more  than  the  condition  of  affairs  at  the  surface.  For  in  virtue  of  the  law 
of  refraction, 

fju  sin  0  =  constant ; 

so  that  if  6  be  the  apparent  zenith  distance  of  a  star  seen  at  the  earth's 
surface,  and  80  the  refraction,  we  have  at  once 

/*o  sin  6  =  sin  (6  +  86), (19) 

from  which  the  refraction  can  be  rigorously  calculated.  If  an  expansion  be 
desired, 

B6  =  sin  Be  =  tan  0  (>0  -  cos  80) 

=  (/*„- 1)  tan  0{l+£O0-l)tan20} (20) 

is  the  second  approximation. 


When  the  curvature  of  the  earth  is  retained,  so  that  the  atmospheric 
strata  are  supposed  to  be  spheres  described  round  0  the  centre  of  the  earth, 
the  appropriate  form  of  the  law  of  refraction  is 

ftp  =  constant. 
Thus,  if  A   be   the  point  of  observation  at  the  earth's  surface  where  the 


1893]  ON   THE   THEORY   OF   STELLAR  SCINTILLATION.  71 

apparent  zenith  distance  is  0,  and  if  the  original  direction  of  the  ray  outside 
the  atmosphere  meet  the  vertical  OA  at  the  point  Q, 

1^.0  A.  sin  6  =  OQ  .  sin  (6  +  BO)  ; 
or  if  OJ.  =  a,  AQ  =  c, 

fjt^a  sin  6  =  (a  +  c)  sin  (0  +  86)  ......................  (21) 

If  c  be  neglected  altogether,  we  fall  back  upon  the  former  equations  (19), 
(20).     For  the  purposes  of  a  second  approximation  c,  though  it  cannot  be 
neglected,  may  be  calculated  as  if  the  refraction  were  small,  and  the  curvature 
of  the  strata  negligible.     If  77  be  the  whole  linear  deviation  of  the  ray  due 
to  the  refraction, 

c  =  f)fsm0,    .................................  (22) 

and,  as  in  (16), 

(23) 


so  that  c  =     °  ~  "      .  ....(24) 

cos2  6 

By  equations  (21),  (24)  the  value  of  80  may  be  calculated  from  the  trigono- 
metrical tables  without  further  approximation. 

To  obtain  an  expansion,  we  have 


+cja 


O/o  -  1)  tan  0   l  -  -  +  i  G-O  -  1)  tan' 

tatf0 (25) 


To  this  order  of  approximation  the  refraction  can  be  expressed  in  terms  of 
the  condition  of  things  at  the  earth's  surface,  and  (25)  is  equivalent  to  an 
expression  deduced  at  great  length  by  Laplace. 

From  the  value  of  I  already  quoted,  and  a  =  6'3709  x  108  centim.,  we  get 

I/a  =  -0012541 (26) 

If  further  we  take  as  the  value  under  standard  conditions  for  the  line  D 

^0-1  =-0002927,     ~ (27) 

we  find  as  the  refraction  expressed  in  seconds  of  arc 

86  =  60"-29  tan  0  -  0"'06688  tan80 (28) 

In  (28)  6  is  the  apparent  zenith  distance,  and  it  should  be  understood 
that  the  application  of  the  formula  must  not  be  pushed  too  close   to   the 


72 


ON   THE  THEORY   OF   STELLAR   SCINTILLATION. 


[203 

horizon.  If  the  density  of  the  air  at  the  surface  of  the  earth  differ  from 
the  standard  density  (0°  and  760  millim.)  the  numbers  in  (28)  must  be 
altered  proportionally.  It  will  be  observed  that  the  result  has  been  deduced 
entirely  a  priori  on  the  basis  of  data  obtained  in  laboratory  experiments. 

It  may  be  convenient  for  reference  to  give  a  few  values  calculated  from 
(28)  of  the  refraction,  and  of  the  dispersion,  reckoned  at  ^  of  the  refraction. 


Apparent  zenith 
distance 

Refraction 

Dispersion 
(B  to  H) 

6 

o-o 

o'-b 

20 

21-9 

•5 

40 

50-5 

1-3 

45 

i    0-2 

1-5 

60 

1  40-1 

2-5 

70 

2  44-2 

4-1 

75 

3  41  5 

5-5 

80 

5  29-7 

8-2 

85 

9  49-2 

14-7 

The  results  of  the  formula  (28)  agree  with  the  best  tables  up  to  a  zenith 
distance  of  75°,  at  which  point  the  value  of  the  second  term  is  3"*5.  For 
85°  the  number  usually  given  is  about  10'  16",  and  for  90°  about  36';  but 
at  these  low  altitudes  the  refraction  is  necessarily  uncertain  on  account  of 
irregularities  such  as  those  concerned  in  the  production  of  mirage. 

[1902.  When  the  above  memoir  was  written  I  was  unacquainted  with 
the  excellent  work  of  K.  Exner  upon  this  subject  (Wien.  Akad.  Sitzber.  84, 
p.  1038,  1881;  Uber  die  Scintillation.  Eine  Monographic.  Wien,  1891). 
Exner's  general  conclusions  are  very  similar  to  those  at  which  I  arrived 
independently,  and  his  work  includes  also  an  interesting  historical  discussion.] 


204. 

ASTRONOMICAL  PHOTOGRAPHY. 

[Nature,  XLVIII.  p.  391,  1893.] 

THE  announcement  (Nature,  August  10)  that  it  is  in  contemplation  to 
raise  a  sum  exceeding  £2000  for  the  establishment  of  a  special  photographic 
telescope  at  the  Cambridge  Observatory,  leads  me  to  ask  whether  astronomers 
have  duly  considered  the  facilities  afforded  by  modern  photography.  At  the 
time  of  my  early  experience  of  the  art,  thirty-five  years  ago,  it  would  have 
been  thought  a  great  feat  to  photograph  the  Fraunhofer  lines  in  the  yellow 
or  red  regions  of  the  spectrum,  although  even  then  the  statement  so 
commonly  made  that  chemical  activity  was  limited  to  the  blue  and  ultra- 
blue  rays  was  quite  unwarranted.  With  the  earlier  photographic  processes 
the  distinction  was  necessary  between  telescopes  to  be  used  with  the  eye  or 
for  photography.  In  the  former  case  the  focal  length  had  to  be  a  minimum 
for  the  yellow  rays,  in  the  latter  for  the  blue  rays  of  the  spectrum. 

But  the  situation  is  entirely  changed.  There  is  now  no  difficulty  in 
preparing  plates  sensitive  to  all  parts  of  the  spectrum,  witness  the  beautiful 
photographs  of  Rowland  and  Higgs.  I  have  myself  used  "  Orthochromatic  " 
plates  in  experiments  where  it  was  desirable  to  work  with  the  same  rays  as 
most  influence  the  eye.  The  interference  bands  of  sodium  light  may  be 
photographed  with  the  utmost  facility  on  plates  sensitised  in  a  bath  con- 
taining cyanin. 

The  question  I  wish  to  ask  is  whether  the  time  has  not  come  to 
accommodate  the  photographic  plates  to  the  telescopes  rather  than  the 
telescopes  to  the  plates.  It  is  possible  that  plates  already  in  the  market 
may  not  exactly  meet  the  requirements  of  the  case,  but  I  feel  sure  that 
a  tithe  of  the  sums  lavished  upon  instruments  would  put  us  in  possession  of 
plates  suitable  for  object-glasses  that  have  been  designed  for  visual  purposes. 
There  would  be  no  difficulty  even  in  studying  the  requirements  of  a  particular 
instrument,  over  or  under  corrected  as  the  case  might  be. 

A  doubt  may  arise  whether  plates  so  adjusted  would  be  as  sensitive  as 
those  now  in  use.  Probably  Captain  Abney,  or  some  other  authority,  could 
give  the  required  information.  For  some  astronomical  purposes  a  moderate 
loss  of  sensitiveness  could  hardly  be  of  much  consequence;  for  others 
doubtless  it  would  be  a  serious  matter. 


205. 


GRINDING  AND   POLISHING   OF   GLASS   SURFACES. 

[British  Association,  Sept.  14,  1893,  from  a  report  in 
Nature,  XLVIII.  p.  526,  1893.] 

LORD  RAYLEIGH  stated  that  he  had  been  investigating  the  nature  of  these 
processes,  and  gave  a  most  interesting  description  of  the  results.  He  first 
pointed  out  that  the  process  of  grinding  with  emery  is  not,  as  is  commonly 
supposed,  a  scratching  process.  The  normal  effect  is  the  production  of  isolated 
detached  pits — not  scratches.  The  glass  gives  way  under  the  emery ;  at  the 
same  time  the  emery  gives  way  under  the  glass  and  suffers  abrasion.  An 
image  seen  through  glass  which  has  been  finely  ground  (but  not  yet  polished) 
has  perfect  definition.  And  so  when  the  sun  is  viewed  through  a  cloud  the 
image  is  sharp  as  long  as  there  is  an  image ;  even  when  the  cloud  thickens, 
the  edge  appears  to  be  sharp  until  we  lose  the  image  altogether.  A  glass 
lens  finely  ground  gives  very  good  definition,  but  there  is  great  loss  of  light 
by  irregular  reflection.  To  obviate  this,  the  lens  is  polished,  and  examination 
under  the  microscope  shows  that  in  the  process  of  polishing  with  pitch  and 
rouge  the  polishing  goes  on  entirely  on  the  surface  or  plateau,  the  bottom 
of  each  pit  being  left  untouched  until  the  adjoining  surface  is  entirely 
worked  down  to  it.  It  appeared  interesting  to  investigate  the  amount  of 
glass  removed  during  the  process  of  polishing.  This  was  done  both  by 
weighing  and  interference  methods,  and  the  amount  removed  was  found  to 
be  surprisingly  small.  A  sufficiently  good  polish  was  obtained  when  a 
thickness  corresponding  to  2^-  wave-lengths  of  sodium  light  was  removed, 
and  the  polishing  was  complete  when  a  thickness  corresponding  to  4  wave- 
lengths was  removed.  Lord  Rayleigh  is  of  opinion  that  the  process  of 
polishing  is  not  continuous  with  that  of  grinding,  but  that  it  consists  of  a 
removal  of  molecular  layers  of  the  surface  of  the  glass.  Grinding  is  easy 
and  rapid,  whereas  polishing  is  tedious  and  difficult.  The  action  of  hydro- 
fluoric acid  in  dissolving  glass  was  also  investigated  and  was  found  to  be 
much  more  regular  than  it  has  generally  been  assumed  to  be  by  chemists.  It 
was  found  to  be  easy  to  remove  a  layer  corresponding  in  thickness  to  half  a 
wave-length  of  sodium  light ;  and  with  due  precautions  as  little  as  one-tenth 
of  a  wave-length.  [1902.  For  a  further  discussion  of  this  subject  see  Nature, 
LXIV.  p.  385,  1901.] 


206. 

ON  THE  REFLECTION   OF  SOUND   OR  LIGHT   FROM   A 
CORRUGATED  SURFACE. 

[British  Association  Report,  pp.  690,  691,  1893.] 

THE  angle  of  incidence  is  supposed  to  be  zero,  and  the  amplitude  of  the 
incident  wave  to  be  unity.     If  then 

£=ccosjtw?    .................................  (1) 

be  the  equation  of  the  surface,  the  problem  of  reflection  is  readily  solved 
so  long  as  p  in  (1)  is  small  relatively  to  k  or  2?r/X;  that  is  so  long  as  the 
wave-length  of  the  corrugation  is  large  in  comparison  with  that  of  the 
vibrations.  The  solution  assumes  a  specially  simple  form  when  the  second 
medium  is  impenetrable,  so  that  the  whole  energy  is  thrown  back  either  in 
the  perpendicularly  reflected  wave  or  in  the  lateral  spectra.  Of  this  two 
cases  are  notable  (a)  when  —  in  the  application  to  sound  —  the  second  medium 
is  gaseous  and  devoid  of  inertia,  as  in  the  theory  of  the  'open  ends'  of 
organ  pipes.  The  amplitude  A0  of  the  perpendicularly  reflected  wave,  so 
far  as  the  fourth  power  of  p/k  inclusive,  is  then  given  by 


-  A0  =    o  .         . 

in  which  there  is  no  limitation  upon  the  value  of  kc,  so  that  the  corrugation 
may  be  as  deep  as  we  please  in  relation  to  X.  If  p  be  very  small,  the  result, 
viz.  —  Jr0(2/cc),  is  the  same  as  would  be  obtained  by  the  methods  usual  in 
Optics;  and  it  appears  that  these  methods  cease  to  be  available  when  p 
cannot  be  neglected. 

The  second  case  (/3)  arises  when  sound  is  reflected  from  a  rigid  and  fixed 
wall.     We  find,  as  far  as  p*/k?, 


If  p,  instead  of  being  relatively  small,  exceeds  k  in  magnitude,  there  are  no 
lateral  spectra  in  the  reflected  vibrations;  and  if  the  second  medium  is 
impenetrable,  the  regular  reflection  is  necessarily  total.  It  thus  appears 
that  an  extremely  rough  wall  reflects  sounds  of  medium  pitch  as  well  as 
if  it  were  mathematically  smooth. 

The  question  arises  whether,  when  the  second  medium  is  not  impenetrable, 
the  regular  reflection  from  a  rough  wall  (p  >  k)  is  the  same  as  if  c  =  0. 
Reasons  are  given  for  concluding  that  the  answer  should  be  in  the  negative. 


207. 

ON  A  SIMPLE  INTERFERENCE  ARRANGEMENT. 

[British  Association  Report,  pp.  703,  704,  1893.] 

IF  a  point,  or  line,  of  light  be  regarded  through  a  telescope,  the  aperture 
of  which  is  limited  to  two  narrow  parallel  slits,  interference  bands  are  seen, 
of  which  the  theory  is  given  in  treatises  on  Optics.  The  width  of  the  bands 
is  inversely  proportional  to  the  distance  between  the  centres  of  the  slits, 
and  the  width  of  the  field,  upon  which  the  bands  are  seen,  is  inversely 
proportional  to  the  width  of  the  individual  slits.  If  the  latter  element  be 
given,  it  will  usually  be  advantageous  to  approximate  the  slits  until  only  a 
small  number  of  bands  are  included.  In  this  way  not  only  are  the  bands 
rendered  larger,  but  illumination  may  be  gained  by  the  then  admissible 
widening  of  the  original  source. 

Supposing,  then,  the  proportions  of  the  double  slit  to  be  given,  we  may 
inquire  as  to  the  effect  of  an  alteration  in  scale.  A  diminution  in  ratio  m 
will  have  the  effect  of  magnifying  m  times  the  field  and  the  bands  (fixed  in 
number)  visible  upon  it.  Since  the  total  aperture  is  diminished  m  times,  it 
might  appear  that  the  illumination  would  be  diminished  ?n,2  times,  but  the 
admissible  widening  of  the  original  source  m  times  reduces  the  loss,  so  that 
it  stands  at  m  times,  instead  of  m2  times. 

It  remains,  and  this  is  more  particularly  the  object  of  the  present  note, 
to  point  out  the  effect  of  the  telescope  upon  the  angular  magnitude  and 
illumination  of  the  bands.  If  the  magnifying  power  of  the  telescope  exceed 
the  ratio  of  aperture  of  object-glass  and  pupil,  its  introduction  is  prejudicial. 
And  even  if  the  above  limit  be  not  exceeded,  the  use  of  the  telescope  is 
without  advantage.  The  relation  between  the  greatest  brightness  and  the 
apparent  magnitude  of  the  bands  is  the  same  whether  a  telescope  be  used 
or  not,  the  loss  by  reflections  and  absorptions  being  neglected.  The  function 
of  the  telescope  is  merely  to  magnify  the  linear  dimensions  of  the  slit  system. 


1893]  ON   A   SIMPLE   INTERFERENCE   ARRANGEMENT.  77 

This  magnification  is  sometimes  important,  especially  when  it  is  desirable 
to  operate  separately  upon  the  interfering  pencils.  But  when  the  object  is 
merely  to  see  the  bands,  the  telescope  may  be  abolished  without  loss.  The 
only  difficulty  is  to  construct  the  very  diminutive  slit  system  then  required. 
In  the  arrangement  now  exhibited  the  slits  are  very  fine  lines  formed  by 
ruling  with  a  knife  upon  a  silver  film  supported  upon  glass.  This  double 
slit  is  mounted  at  one  end  of  a  tube  and  at  the  other  is  placed  a  parallel 
slit.  It  then  suffices  to  look  through  the  tube  at  a  candle  or  gas  flame  in 
order  to  see  interference  bands  in  a  high  degree  of  perfection. 

It  is  suggested  that  this  simple  apparatus  could  be  turned  out  very 
cheaply,  and  that  its  introduction  into  the  market  would  tend  to  popularise 
acquaintance  with  interference  phenomena. 


208. 


ON  THE   FLOW  OF  VISCOUS   LIQUIDS,  ESPECIALLY  IN  TWO 
DIMENSIONS. 

[Philosophical  Magazine,  xxxvi.  pp.  354—372,  1893.] 

THE  problems  in  fluid  motion  of  which  solutions  have  hitherto  been  given 
relate  for  the  most  part  to  two  extreme  conditions.  In  the  first  class  the 
viscosity  is  supposed  to  be  sensible,  but  the  motion  is  assumed  to  be  so  slow 
that  the  terms  involving  the  squares  of  the  velocities  may  be  omitted ;  in 
the  second  class  the  motion  is  not  limited,  but  viscosity  is  supposed  to  be 
absent  or  negligible. 

Special  problems  of  the  first  class  have  been  solved  by  Stokes  and  other 
mathematicians ;  and  general  theorems  of  importance  have  been  established 
by  v.  Helmholtz*  and  by  Kortewegf,  relating  to  the  laws  of  steady  motion. 
Thus  in  the  steady  motion  (M0)  of  an  incompressible  fluid  moving  with  velo- 
cities given  at  the  boundary,  less  energy  is  dissipated  than  in  the  case  of 
any  other  motion  (M)  consistent  with  the  same  conditions.  And  if  the 
motion  M  be  in  progress,  the  rate  of  dissipation  will  constantly  decrease  until 
it  reaches  the  minimum  corresponding  to  M0.  It  follows  that  the  motion  M0 
is  always  stable. 

It  is  not  necessary  for  our  purpose  to  repeat  the  investigation  of 
Korteweg;  but  it  may  be  well  to  call  attention  to  the  fact  that  problems 
in  viscous  motion  in  which  the  squares  of  the  velocities  are  neglected,  fall 
under  the  general  method  of  Lagrange,  at  least  when  this  is  extended  by 
the  introduction  of  a  dissipation  function  J.  In  the  present  application  there 
is  no  potential  energy  to  be  considered,  and  everything  depends  upon  the 
expressions  for  the  kinetic  energy  T  and  the  dissipation  function  F.  The 
conditions  to  be  satisfied  may  be  expressed  by  ascribing  given  constant 

*  Collected  Works,  i.  p.  223. 

t  Phil.  Mag.  xvi.  p.  112,  1883. 

t  Theory  of  Sound,  §  81.    [See  Vol.  I.  of  the  present  collection,  p.  176.] 


1893]  ON   THE   FLOW   OF   VISCOUS   LIQUIDS.  79 

values  to  some  of  the  generalized  velocities;  but  it  is  unnecessary  to  in- 
troduce more  than  one  into  the  argument,  inasmuch  as  any  others  may  be 
eliminated  beforehand  by  means  of  the  given  relations.  Suppose,  then,  that 
fa  is  given.  The  other  coordinates  fa,  fa,  ...  may  be  so  chosen  that  no 
product  of  their  velocities  enters  into  the  expressions  for  T7  and  F,  although 
products  with  fa,  such  as  fa  fa,  will  enter.  These  coordinates  are,  in  fact, 
the  normal  coordinates  of  the  system  when  fa  is  constrained  to  vanish. 
Thus  simplified  F  becomes 

F=lb1fa*+...+$bsfa+...+brsfafa  + (1) 

and  a  similar  expression  applies  to  T  with  a  written  for  b.  Lagrange's 
equation  is  now 

asfa  +  argfa  +  bsfa  +  brgfa  =  0, 

fa  being  any  one  of  the  coordinates  fa,  fa,  ....  In  this  equation  fa  =  Q, 
and  fa  has  a  prescribed  value;  so  that 

agfa+bsfa  =  -brsfa  (2) 

is  the  equation  giving  fa.  The  solution  of  (2)  is  well  known,  and  it  appears 
that  fa  settles  gradually  down  to  the  value  given  by 

bsfa  =  -brsfa,     (3) 

since  as,  bg  are  intrinsically  positive.     Further, 

^=  2  {bsfafa  +  brs(fafa  +  fa  fa)}, 

in  which  the  summation  extends  to  all  values  of  s  other  than  r.  In  this 
fa  =  ®,  so  that 

^  =  2fa{bsfa  +  brsfa}=-?.asfa*,  (4) 

by  (2).  The  last  expression  is  intrinsically  negative,  proving  that  until  the 
steady  motion  is  reached  F  continually  decreases.  Korteweg's  theorem  is 
thus  shown  to  be  of  general  application  to  systems  devoid  of  potential 
energy  for  which  T  and  F  can  be  expressed  as  quadratic  functions  of  the 
velocities  with  constant  coefficients. 

It  may  be  mentioned  in  passing  that  a  similar  theorem  holds  for  systems 
devoid  of  kinetic  energy,  for  which,  however,  F  and  V  (the  potential  energy) 
are  sensible,  and  may  be  proved  in  the  same  way.  If  such  a  system  be 
subjected  to  given  displacements,  it  settles  down  into  the  configuration  of 
minimum  V;  and  during  the  progress  of  the  motion  V  continually  decreases. 

The  theorem  of  Korteweg  places  in  a  clear  light  the  general  question  of 
the  slow  motion  of  a  viscous  liquid  under  given  boundary  conditions,  and  the 
only  remaining  difficulty  lies  in  finding  the  analytical  expressions  suitable 
for  special  problems.  It  is  proposed  to  consider  a  few  simple  cases  relating 
to  motion  in  two  dimensions. 


80  ON   THE   FLOW   OF   VISCOUS   LIQUIDS,  [208 

Under  the  above  restriction,  as  is  well  known,  the  motion  may  be  ex- 
pressed by  means  of  Earnshaw's  current  function  (•$•),  which  satisfies 

V4>/r  =  0,   (5) 

the  same  equation  as  governs  the  transverse  displacement  of  an  elastic  plate, 
when  in  equilibrium*.  Of  this  analogy  we  shall  avail  ourselves  in  the  sequel. 
At  a  fixed  wall  ^  retains  a  constant  value,  and,  further,  in  consequence  of 
the  friction  dty/dn,  representing  the  tangential  velocity,  is  evanescent.  The 
boundary  conditions  for  a  fixed  wall  in  the  fluid  problem  are  therefore  analo- 
gous to  those  of  a  clamped  edge  in  the  statical  problem. 

The  motion  within  a  simply  connected  area  is  determined  by  (5)  and  by 
the  values  of  the  component  velocities  over  the  boundary.  If  we  suppose 
that  two  such  motions  are  possible,  their  difference  constitutes  a  motion  also 
satisfying  (5),  and  making  i/r  and  d^rjdn  zero  over  the  boundary.  Consider- 
ations respecting  energy  in  this  or  in  the  analogous  problem  of  the  elastic 
plate  are  then  sufficient  to  show  that  i/r  must  vanish  throughout ;  and  an 
analytical  proof  may  readily  be  given  by  means  of  Green's  theorem.  For  if 
t/r  and  %  are  any  two  functions  of  x  and  y, 


the  integrations  being  taken  round  and  over  the  area  in  question.  If  we 
suppose  that  i/r  and  d\Jr/dn  are  zero  over  the  boundary,  the  left-hand  member 
vanishes.  If,  further,  %  =  V2-^,  we  have 

f 


of  which  the  right-hand  member  vanishes  by  (5).  Hence  V2-\Jr  vanishes  all 
over  the  area,  and  by  a  known  theorem,  as  ^  vanishes  on  the  contour,  this 
requires  that  i/r  vanish  throughout. 


We  will  now  investigate  in  detail  the  slow  motion  of  viscous  fluid  within 
a  circular  boundary.  In  virtue  of  (5)  V2-^,  which  represents  the  vorticity, 
satisfies  Laplace's  equation,  and  may  therefore  be  expanded  in  positive  and 
negative  integral  powers  of  r,  each  term  such  as  rn,  or  r~n,  being  accom- 
panied by  the  factor  cos  (nd  +  a).  But  if,  as  we  shall  suppose,  the  vorticity 
be  finite  at  the  centre  of  the  circle,  where  r  =  0,  the  negative  powers  are 
excluded,  and  we  have  to  consider  only  such  terms  as 


[1902.     If  w  be  the  displacement,  parallel  to  z,  at  any  point  of  a  plane  elastic  plate  in  the 
plane  of  xy,  the  differential  equation  of  equilibrium  is  V%  =  0,  impressed  forces  being  absent.] 


1893]  ESPECIALLY   IN   TWO   DIMENSIONS.  81 

The  solution  of  this  is  readily  obtained.     If  we  assume 

i/r  =  rm  cos  (nO  +  a),     ...........................  (9) 

we  find  m  =  n  +  <2.  To  this  may  be  added,  as  satisfying  V2i|r  =  0,  a  term 
corresponding  to  m  =  n;  so  that  the  type  of  solution  for  nQ  is 

^  =  Anrn+z  cos  (n0  +  a)  +  Bnrn  cos  (n0  +  (3)  .............  (10) 

By  differentiation, 

d-b 

-^  =  (n  +  2)  Anrn+>  cos  (n0  +  a)  +  nBnrn~l  cos  (nO  +  0)  .......  (11) 

The  first  problem  to  which  we  will  apply  these  equations  is  that  of  motion 
within  the  circle  r  =  1  under  the  condition  that  the  tangential  motion 
vanishes  at  every  part  of  the  circumference.  By  (11)  /8  =  a,  and 

(n  +  2)An  +  nBn  =  0  .........................  (12) 

The  normal  velocity  at  the  boundary  is  represented  by  d-^/dO,  and  we  might 
be  tempted,  in  our  search  after  simplicity,  to  suppose  that  this  is  sensible  in 
the  neighbourhood  of  one  point  only,  for  example  9  =  0.  But  in  that  case 
the  condition  of  incompressibility  would  require  that  the  total  flow  of  fluid 
at  the  place  in  question  should  be  zero.  If  the  total  quantity  of  fluid 
entering  the  enclosure  at  6  =  0  is  to  be  finite,  provision  must  be  made  for 
its  escape  elsewhere.  This  might  take  the  form  of  a  sink  at  the  centre  of 
the  circle  ;  but  it  will  come  to  much  the  same  thing,  and  be  more  in  harmony 
with  our  equations,  as  already  laid  down,  to  suppose  that  the  escape  takes 
place  uniformly  over  the  entire  circumference.  This  state  of  things  will  be 
represented  analytically  by  ascribing  to  ty  a  sudden  change  of  value  from 
—  1  to  +1  at  0  =  0,  with  a  gradual  passage  from  the  one  value  to  the  other 
as  6  increases  from  0  to  2?r,  or,  as  it  may  be  more  conveniently  expressed 
for  our  present  purpose,  ty  is  to  be  regarded  as  an  odd  function  of  6  such 
that  from  6  =  0  to  6  =  ir  its  value  is 

0  =  1-01-*  ..................................  (13) 

The  symmetry  with  respect  to  0  —  0  shows  that  we  are  concerned  in  (10) 
only  with  the  sines  of  multiples  of  0,  so  that  having  regard  to  (12)  we  may 
take  as  the  form  of  -\Jr  applicable  in  the  present  problem, 

..................  (14) 


in  which  n  is  any  integer  and   Cn  an  arbitrary  constant.     It  remains  to 
determine  the  coefficients  G  in  accordance  with  (13).    -When  r=l, 


and  this  must  hold  good  for  all  values  of  0  from  0  to  TT.     Multiplying  by 
sin  m0  and  integrating  as  usual,  we  find 


~;  .................................  (15) 


iv. 


82 


ON   THE   FLOW   OF   VISCOUS   LIQUIDS, 


that 


is  the  value  of  -$r  ex 


:2sinn0{(l-2/tt)rn-rn+2} 
in  series. 


[208 
.(16) 


These  series  may  be  summed.     In  the  first  place,  2r"sinn#  is  the  real 
partof-t2(re»)n,  or  of 


Thus 

Again,  Sri"1 
so  that 


r  sin  8 


1  -  2r  cos  6  +  r3 


..(17) 


is  the  real  part  of  -  iSn-1  (rew)n,  or  of  t  log  (1  -  r 

r  sm  ^ 


1  —  r  cos  a 
Thus,  as  the  expression  for  i/r  in  finite  terms,  we  have 

r  sin  6 


(18) 


.(19) 


In  (19)  the  separate  parts  admit  of  simple  geometrical  interpretation. 
The  second  represents  simply  twice  the  angle  PAG, 
Fig.  1,  which  is  known  to  constitute  a  solution  of 
V2i/r  =  0.     In  the  first  term, 

rsintf  PM      sinPAO 


Fig.  1. 


AP* 


AP 


which  is  also  obviously  a  solution  of  V2\|r  =  0.  The 
remaining  part  of  (19)  is  not  a  solution  of  V2\/r  =  0 ; 
but  it  satisfies  V4\Jr  =  0,  as  being  derived  from  a 
solution  of  V2i/r  =  0  by  multiplication  with  ?*2. 

On  the  foundation  of  (19)  we  may  build  up  by  simple  integration  the 
general  expression  for  i^,  subject  to  the  conditions  that  d-^r/dr  vanishes 
over  the  whole  circumference,  and  that  d^jdO  has  any  prescribed  values 
consistent  with  the  recurrence  of  -Jr. 


Fig.  2. 


A  simple  example  is  afforded  by  the  case  of  a  source  at  A  and  an  equal 
sink  at  B,  where  6  =  TT  (Fig.  2).     The  fluid  enters  and 
leaves  the  enclosure  by  two  perforations  situated  at 
opposite  ends  of  a  diameter,  the   walls   being   else- 
where  impenetrable.      The   solution   may   be   found 
independently,  or   from   (19),  by  changing   the  sign     , 
of  cos  6,  and  adding  the  equations  together.     Thus 


1893]  ESPECIALLY   IN   TWO   DIMENSIONS.  83 

In  this  case  the  walls  of  the  enclosure  are  of  necessity  stream-lines,  the 
value  of  -^  being  +  1  from  0  to  TT,  and  —  1  from  0  to  —  TT. 

When  6  =  %Tr,  that  is  along  OD  (Fig.  2), 

,  ........................  (21) 


From  (21)  we  obtain  by  interpolation  the  following  corresponding  values:  — 


•00         -25  -50  -75  1-00 

•00         -1330         -2800         -4698         I'OOOO 


In  the  neighbourhood  of  A  or  B,  Fig.  1,  (20)  assumes  a  special  form.   Thus 
in  the  former  case, 

1  -  2r2  cos  26  +  r4  -  (1  -  r2)2  +  4r2  sin2  (9  =  4  {^1M2  +  PM  "}, 


,  2r  sin  0  .     „  .  ,k 

tan-1  —  -  —  =  angle  PA  0. 

Thus  if  PAO  be  denoted  by  <f>,  the  value  of  -»Jr  in  the  neighbourhood  of 
A  is  given  by 

TT  .  ^  =  sin  2<£  +  2<£  ............................  (22') 


That  the  functions  of  <j>  which  occur  in  (22')   satisfy  the   fundamental 
equation  may  be  readily  seen. 

By  calculation  from  (22')  we  get  the  following  values  for  <f>  expressed  as 
fractions  of  degrees:  — 


0  -25  -50  -75  1-00 

0          11°'40          23°'83        39°-40          90C'00 


This  example  is  of  interest,  from  its  bearing  upon  the  laws  of  flow  at  a 
place  where  a  channel  is  enlarged.  In  actual  fluids  there  would  be  a  ten- 
dency to  shoot  directly  across  from  A  to  B,  the  region  about  C  being  occupied 
by  an  eddy,  or  backwater,  such  that  the  motion  of  the  fluid  near  the  wall  is 
reversed.  Nothing  of  the  kind  is  indicated  by  the  present  solution.  In 
(22)  d-^rfdr  represents  the  velocity  across  the  line  0  =  £?r,  and  we  see  that 
there  is  no  change  of  sign.  In  fact  the  velocity  decreases,  as  r  increases, 
all  the  way  from  r  =  0  to  r  =  l.  The  formation  of  a  backwater  may  thus 
be  connected  with  the  terms  involving  the  squares  of  the  velocities,  which 
are  neglected  in  the  present  solution.  And  we  may  infer  that  if  the  motion 
were  slow  enough,  or  if  the  fluid  were  viscous  enough,  the  backwater,  usually 
observed  in  practice,  would  disappear. 

6—2 


84  ON   THE   FLOW   OF   VISCOUS   LIQUIDS,  [208 

Another  particular  case  of  some  interest,  included  in  the  general  solution 
already  indicated,  would  be  obtained  by  supposing  similar  sources  to  be 
situated  at  0  =  0,  0  =  7T,  and  equal  sinks  at  #  =  ^TT,  O  =  ^TT. 

We  will  now  suppose  that  it  is  the  radial  velocity  which  vanishes  at 
every  point  of  the  circumference  r  =  l,  and  that  the  tangential  velocity  also 
vanishes  except  in  the  neighbourhood  of  0  =  0.  In  this  case,  by  the  sym- 
metry, \lr  in  (10)  reduces  to  a  series  of  cosines.  And 

-  d^/dO  =  2ra  sin  nB  (Anrn+2  +  Bnrn), 
which  is  to  vanish  when  r  =  1  for  all  values  of  6.     Hence 

An  +  Bn  =  0;    ..............................  (23) 

so  that 

^•=(1  -r*)2Bnrncosn6,  ........................  (24) 

d^r/dr  =  ^Bn  cos  n6  [nr^1  -  (n  +  2)  rn+l]  .............  (25) 

When  r  =  1, 

d^/dr  =  -2^Bncosn0,    .......................  (26) 

and  is  to  be  made  to  vanish  for  all  values  of  0  except  in  the  neighbourhood 
of  0  =  0.     If  we  suppose  that  the  integral  of  dty/dr  with  respect  to  d  over 
the  whole  region  where  dty/dr  is  sensible,  is  2,  we  find 

5.  =  -l/2w,        Bn  =  -l/7r,  .....................  (27) 

the  second  equation  applying  to  all  values  of  n  other  than  0.     Hence, 

-7r.>/r  =  -£(l-r2)  +  (l-r2)2"rwcosw0,  ...............  (28) 

or  in  finite  terms, 

-^  =  -l(l-0  +  (l-.%^co^  ..........  (29) 

The  equation  may  also  be  written 

-2-*=i 

In  (29), 

1  -  r  cos  6          AM      cosPAO 


~  AP2~      AP      ' 

which  is  a  solution  of  V2i/r  =  0.     When  multiplied  by  r2,  or  by  (1  -  r2),  it 
remains  a  solution  of  V4-^-  =  0. 

In  (30)  we  may  write  x  for  r  cos  0,  and  if  the  point  under  consideration 
lie  upon  the  axis,  a?  =  r2.     Hence  on  the  axis, 


-27T.A/T  =  (!+#)',  ...........................  (31) 

-•n-ctyr/cfo  =  (!+#),  ...........................  (32) 

equations  which  may  be  applied  at  all  points  except  near  #=  1.  It  appears 
from  (32)  that  the  velocity  transverse  to  the  axis  increases  continuously  from 
x  =  —  1  to  the  neighbourhood  of  a?  =  +  1. 


1893] 


ESPECIALLY   IN   TWO   DIMENSIONS. 


85 


The  lines  of  flow  are  readily  constructed  from  (30),  which  we  may  write 
in  the  form 


(33) 


showing  how  P  may  be  determined  by  the  intersection  of  circles  struck 
from  0  and  A.  A  few  of  the  lines  of  flow  are  shown  in  Fig.  3.  The  external 
circle  AB  corresponds  to  -^-  =  0;  AC,  AO,  AD  correspond  respectively  to 
-27TT/r  =  £,  1,  2.  As  appears  from  (31),  the  highest  value  of  -  27r>/r  is  4, 
and  gives  a  curve  at  A  of  infinitely  small  area. 

Fig.  3. 


In  the  neighbourhood  of  A  (Fig.  1),  (30)  reduces  to  a  simpler  form.    Thus 

(33') 

where  <}>  =  PAO.  The  second  term  here  satisfies  the  fundamental  equation 
as  being  derived  by  multiplication  with  AP2  from  a  solution,  AP~2cos2<j>, 
of  V2  =  0. 


86  ON  THE   FLOW   OF   VISCOUS   LIQUIDS,  [208 

Equations  (19),  (30)  give  the  means  of  expressing  the  stream-function 
subject  to  the  conditions  that  both  •$•  and  d-^r/dr  shall  have  values  arbitrarily 
given  at  all  points  of  the  circumference  of  the  circle.  It  is  not  necessary 
actually  to  write  down  the  formulae  :  but  it  may  be  well  to  notice  that  the 
same  solution  applies  to  the  question  of  determining  the  transverse  displace- 
ment w  of  a  thin  circular  plate  when  iv  and  dw/dr  have  arbitrarily  prescribed 
values  on  the  boundary. 

As  a  preliminary  to  further  questions,  it  will  be  desirable  to  consider  for 
a  moment  the  form  of  the  general  equations  of  viscous  motion.  In  the  usual 
notation, 

du        du        du         du      v     1  dp       _0 

-y-  +u-r+v-r+w-r  =  X--  -f-  +  vV2u,   ............  (34) 

dt        dx        dy         dz  p  dec 

with  two  similar  equations.     Further,  if  q2  denote  the  resultant  velocity,  and 
£,  i),  £  be  the  component  rotations, 


.....  (35) 

dy        dz      *  dx 

In  steady  motion  du/dt  =  0;  and  if  the  terms  of  the  second  order  in  velocity 
(35)  be  omitted  and  there  be  no  impressed  forces  except  such  as  have  a 
potential,  the  equations  reduce  to  the  form  already  considered.  A  solution 
thus  obtained  for  small  velocities  will  fail  to  satisfy  the  conditions  when  the 
velocities  are  increased;  but  the  equations  lead  readily  to  an  instructive 
expression  for  the  forces  X,  Y,  Z,  which  must  be  introduced  in  order  that 
the  solution  applicable  without  impressed  forces  to  small  velocities  may  still 
continue  to  hold  good.  From  (35)  we  see  that  the  necessary  forces  are 


(36) 


with  two  similar  equations.  In  this  the  term  \d(f\dx  need  not  be  regarded, 
as  it  tells  only  upon  the  pressure  and  does  not  influence  the  motion.  We 
may  therefore  write 


=<2v%-2ur]  .......  (37) 

These  equations  show  that 

uX  +  vY+wZ=0,         £X  +  -nY+ZZ  =  0,  ...............  (38) 

signifying  that  the  force  whose  components  are  X,  Y,  Z,  acts  at  every  point 
in  a  direction  perpendicular  both  to  the  velocity  and  to  the  axis  of  rotation. 
As  regards  its  magnitude, 

i(JT2  +  Y*  +  Z2)  =  (u2  +  v2  +  O(£2  +  7?2  +  £2)  -  (t*£  -vi)-  w%)\  .  .  .(39) 
If  the  motion  take  place  in  two  dimensions,  w  =  0,  f  =  77  =  0,  and 

(40) 


1893]  ESPECIALLY   IN   TWO   DIMENSIONS.  87 

In  the  case  of  symmetry  round  an  axis, 


=  0, 
and  (39)  reduces  to 

%(X*  +  Y*  +  Z*)  =  (u*  +  v*  +  w-)(Z*  +  rj*  +  f2)  ............  (41) 

These  expressions  for  the  forces  necessary  to  the  maintenance  of  a  motion 
similar  to  the  infinitely  small  motion  give  us  in  simple  cases  an  idea  of  the 
direction  in  which  the  law  is  first  departed  from  as  the  motion  increases. 

There  are  very  few  cases  in  which  the  problem  of  the  rapid  motion  of  a 
viscous  fluid  has  been  dealt  with.  When  the  motion  is  in  one  dimension, 
the  troublesome  terms  do  not  present  themselves,  and  the  same  solution 
holds  good  mathematically  for  the  steady  motion  at  all  velocities.  When 
the  motion  is  so  small  that  the  laws  appropriate  to  infinitely  small  motion 
hold  good  as  a  first  approximation,  a  correction  may  be  calculated.  This  has 
been  effected  by  Whitehead*,  and  in  an  unpublished  paper  by  Rowland,  for 
the  problem,  first  investigated  by  Stokes,  of  a  sphere  moving  with  velocity  V 
through  viscous  liquid.  For  infinitely  small  motion  the  velocity  of  the 
fluid  in  the  neighbourhood  of  the  sphere  is  of  order  V.  It  follows  from 
the  solution  referred  to,  or  may  be  proved  independently  by  considerations 
of  dimensions,  that  in  the  second  approximation  involving  F2,  the  terms 
are  of  the  order  V*a/v,  a  being  the  radius  of  the  sphere,  and  v,  equal  to 
//,/  p,  the  kinematic  coefficient  of  viscosity.  This  method  of  approximation 
is  thus  only  legitimate  when  Va/v  is  small,  a  condition  of  a  very  restricting 
character.  In  the  case  of  water  j/  =  '01  c.G.s.,  and  if  Fa/i>  =  'l,  it  is  required 
that  Fa  ='001. 

Thus  even  if  a  were  as  small  as  one  millimetre  (-1),  F  should  not  exceed 
'01  centimetre  per  second.  With  such  diameters  and  velocities  as  often 
occur  in  practice,  Va,\v  would  be  a  large,  instead  of  a  small,  quantity;  and 
a  solution  founded  upon  the  type  of  infinitely  slow  motion  is  wholly  in- 
applicable. 

We  will  now  recur  to  the  suppositions  that  the  motion  is  steady,  is  in 
two  dimensions,  and  that  its  square  may  be  neglected.  Thus,  writing  as  usual 

u  =  d^jdy,      v  =  -  d^r  /  dx, 
we  get  from  (34) 


Forces  derivable  from  a  potential  do  not  disturb  the  equation  V4-^  =  0. 
In  the  analogy  with  a  thin  elastic  plate,  already  referred  to,  a  place  where 
dY/dx-dX/dy  assumes  a  finite  value  in  the  fluid  problem  corresponds  to 
a  place  where  transverse  force  acts  upon  the  plate. 

*  Quart.  Journ.  of  Math.  Vol.  xxm.  p.  153  (1889). 


88  ON   THE   FLOW   OF    VISCOUS    LIQUIDS,  [208 

The  simplest  example  of  the  finiteness  of  the  second  member  of  (42) 
occurs  when  it  is  sensible  at  one  point  only.  This  is  the  case  of  forces 
derivable  from  a  potential  9,  where  6  denotes  the  angle  measured  round 
the  point  in  question.  It  is  to  be  observed  that  in  the  fluid  problem  the 
forces  themselves  are  not  limited  to  the  one  point,  but  they  have  no  "cir- 
culation" except  round  that  point.  In  the  elastic  problem,  on  the  other 
hand,  the  transverse  force  is  limited  to  the  one  point. 

The  circumstance  last  mentioned  renders  the  elastic  problem  the  easier  of 
the  two  to  deal  with  in  thought  and  expression,  and  we  will  accordingly 
avail  ourselves  of  the  analogy  in  the  investigation  which  follows.  It  is  pro- 
posed to  examine  the  infinitely  slow  motion  of  fluid  within  an  enclosure, 
which  is  maintained  by  forces  having  circulation  at  one  point  only,  with  the 
view  of  determining  whether  a  contrary  flow,  or  backwater,  is  possible.  In  the 
analogous  elastic  problem  we  have  to  consider  a  plate,  subject  at  the  boundary 
to  the  conditions  that  w  (the  transverse  displacement)  and  dw/dn  shall  every- 
where vanish,  and  disturbed  from  its  original  plane  condition  by  a  force 
acting  transversely  at  a  single  point  P.  For  distinctness  we  may  suppose 
that  the  plane  is  horizontal  and  that  the  force  at  P  acts  downwards,  in 
which  direction  the  displacements  are  reckoned  positive.  At  the  point  P 
itself  the  principle  of  energy  shows  that  the  displacement  is  positive,  and 
it  might  appear  probable  that  the  displacement  would  be  also  positive  at 
all  other  points  of  the  plate.  A  similar  conclusion  is  readily  proved  to  be 
true  in  the  case  of  a  stretched  membrane  of  any  shape  subjected  to  trans- 
verse force  at  any  point,  and  also  in  one  dimension  for  a  bar  resisting  flexure 
by  its  stiffness.  But  a  consideration  of  particular  cases  suffices  to  show  that 
the  theorem  cannot  be  generally  true  in  the  present  case. 

For  suppose  that  the  plate  (Fig.  4)  is  almost  divided  into  two  independent 
parts  by  a  straight  partition  CD  extending  across,  but  perforated        ^ 
by  an  aperture  AB;   and  that  the  force  is  applied  at  a  distance 
from  CD  on  the  left.     If  the  partition  were  complete,  w  and  dw/dn 
would  be  zero  over  the  whole,  and  the  displacement  in  the  neigh- 
bourhood on  the  left  would  be  simple  one-dimensional  bending,  with 
w  positive  throughout.     On  the  right  w  would  vanish  throughout. 
In  order  to  maintain  this  condition  of  things  a  certain  couple  acts 
upon  the  plate  in  virtue  of  the  supposed  constraints  along  CD. 

Along  the  perforated  portion  AB  the  couple  required  to  produce 
the  one-dimensional  bending  fails.  The  actual  deformation  accord- 
ingly differs  from  the  one-dimensional  bending  by  the  deformation 
that  would  be  produced  by  a  couple  over  AB  acting  upon  the  plate 
as  clamped  along  CA,  BD,  but  otherwise  free  from  force.  This 
deformation  is  evidently  symmetrical  with  change  of  sign  upon  the 
two  sides  of  CD,  w  being  positive  on  the  left,  negative  on  the  right,  and 


1893] 


ESPECIALLY   IN   TWO    DIMENSIONS. 


89 


Fig.  5. 


vanishing  on  AB  itself.  Thus  upon  the  whole  a  downward  force  acting  on 
the  left  gives  rise  to  an  upward  motion  on  the  right,  in  opposition  to  the 
general  law  proposed  for  examination. 

In  the  application  to  the  hydrodynamical  problem  we  see  that  the  fluid 
moving  on  the  left  from  D  to  B  passes  on  in  a 
straight  course  to  A,  and  thence  along  AC,  and 
that  on  the  right  an  eddy,  or  backwater,  is  formed. 
At  distances  from  the  aperture  large  in  com- 
parison with  AB  the  supplementary  motion  is  of 
the  character  expressed  in  (33'). 

A  similar  argument  may  be  applied  to  the  . 
case  (Fig.  5)  where  fluid  moves  along  a  wall  DC 
into  which  a  channel  AF  opens,  and  it  leads  to 
the  conclusion  that  the  fluid  on  arrival  at  B  will 
refuse  to  follow  the  wall  BF,  but  will  rather  shoot 
across  towards  A. 

These  examples  are  of  some  interest  as  estab- 
lishing that  the  formation  of  eddies  observed  in 

practice  is  not  wholly  due  to  the  influence  of  the  terms  involving  the  squares 
of  the  velocities,  but  would  persist  in  certain  cases  even  though  the  motion 
were  made  infinitely  slow. 


We  will  now  investigate  the  motion  in  two  dimensions  of  a   viscous 

Fig.  6. 


incompressible  fluid  past  a  corrugated  wall  AB  (Fig.  6),  whose  equation  may 
be  taken  to  be 

y  =  @coskx  ...............................  (43) 

In  this  kft  will  be  supposed  to  be  a  small  quantity;  in  other  words,  the 
depth  of  the  corrugations  small  in  comparison  with  their  wave-length 
(ITT  Ik).  Further  we  shall  suppose,  in  the  first  instance,  that  the  motion 
is  slow  enough  to  allow  the  terms  involving  squares  of  the  velocities  to 
be  neglected;  in  which  case  the  equation  for  the  stream-function  may  be 
written 

0  ..................................  (44) 


At  a  distance  from  the  wall  we  suppose  the  motion  to  take  place  in  plane 
strata,  as  defined  by 

+  =  Lf  ..................................  (45) 


90  ON  THE   FLOW   OF   VISCOUS   LIQUIDS,  [208 

In  the  absence  of  corrugations  this  value  of  \|r  might  hold  good  throughout, 
up  to  the  wall  at  y  =  0.  The  effect  of  the  corrugations  will  be  to  introduce 
terms  periodic  with  respect  to  x  ;  but  the  influence  of  these  will  be  confined 
to  the  neighbourhood  of  the  wall.  For  any  term  in  ty,  proportional  to 
cos  mar,  (44)  gives 


or  ^  =  A  e~my  +  By  e-™*  +  Cemy  +  Dyemv  ; 

but  the  condition  last  named  requires  that  of  the  four  arbitrary  constants  C 
and  D  vanish.  Also  for  our  present  purpose  m  is  limited  to  be  a  multiple 
of  k. 

The  form  of  i/r  applicable  to  our  present  purpose  is  accordingly 
^  =  A0  +  B0y  +  Lf  +  cos  kx  (A.e^  +  Biye~ky) 

...,  ......  (47) 


in  which  the  constants  A0,  B0,  Alt  ...  are  to  be  determined  from  the  con- 
ditions that  i|r  and  dty/dy  vanish  when  y  =  ftcosko;.  It  may  be  observed 
that  the  problem  is  mathematically  identical  with  that  of  an  elastic  plate 
clamped  at  a  sinuous  edge,  and  deformed  in  such  a  manner  that  if  there 
were  no  sinuosity  the  bending  would  be  one-dimensional. 

The  boundary  conditions  are 


cos  kx)  e~k^  °°»  kx 
+  cos  2kx  (A2  +  B2ft  cos  kx)  e-8tf  «»te 

+  ......  =0    ......................................................  (48) 

and 

B0  +  2Lj3  cos  kx 

+  cos  kx  (Bi  -kAl-Blkft  cos  kx)  e~k^  cos  ** 

+  cos  2kx(Ba  -2kA2-  2B2k0  cos  kx)  g-u*™** 

+  ......  =  0;  ......................................................  (49) 

or,  with  use  of  (48), 

kA0  +  B0  +  (BJcft  +  2L/3)  cos  kx  +  Lk/32  cos"kx 


+  (B.2  -  kAt  -  B2k/3  cos  kx)  e~^  °°skx 

+  ......  =0  .......................................................  (50) 

The  exponentials  in  (48),  (50)  could  be  expanded  in  Fourier's  series  by 
means  of  Bessel's  functions  of  an  imaginary  argument,  and  the  complete 


1893]  ESPECIALLY   IN   TWO   DIMENSIONS.  91 

equations  formed  which  express  the  evanescence  of  the  various  Fourier  terms. 
But  the  results  are  too  complicated  to  be  useful  in  the  general  case  ;  and,  if  we 
regard  kfi  as  small,  it  is  hardly  worth  while  to  introduce  the  Bessel's  functions 
at  all.  The  first  approximation,  in  which  /32  is  neglected  in  (48),  (50),  gives 


and  the  second  approximation,  in  which  ft2  is  retained,  gives 

' 


the  coefficients  with  higher  suffixes  than  2  vanishing  to  this  order  of  ap- 
proximation.    Thus 

TJr/L  =  /32  (I  -  2%)  +  7/2  -  2/3?/e-^  cos  kas 

+  /3*($-ky)  e~*y  cos  2kx,  .........  (53) 

i  ^  =  -  2  kft*  +  2y  -  2(3  (1  -  ky)  er*  cos  kx 

-2k/3-2(l-ky)e-aJcycos2kx,  ........................  (54) 

solutions  applicable  also  to  the  problem  of  the  elastic  plate,  if  ty  be  under- 
stood to  mean  the  transverse  displacement. 

In  the  above  investigation,  so  far  as  it  applies  to  the  hydrodynamical 
question,  L?  has  been  supposed  to  be  negligible.  We  will  now  retain  the 
square  of  L,  but  simplify  the  problem  in  another  direction  by  neglecting  the 
square  of  /3,  so  that  the  first  approximation  is 

(55) 


The  exact  equation  (derivable  from  (34))  for  the  motion  of  a  viscous  fluid 
in  two  dimensions  is 


v     dx        v    dy 
From  (55), 


2L  +  4>Lkfie-kv  cos  kx, 

V2-\//- 

(57) 


Using  this  in  (56)  we  have 

«k2/Q  T.2 

i (58) 


v 
The  solution  of 


<59> 


92  ON  THE   FLOW  OF   VISCOUS   LIQUIDS,  [208 

so  that  the  required  solution  of  (58),  correct  as  far  as  the  term  involving 
Z2,  is 

ty  =  Lf-  2L@ye-k*  cos  kx  -  —^  (y2  +  ±ky3)  e~^  sin  kx.    .  .  .(61) 

It  may  be  well  to  repeat  that,  though  Z2  is  retained.  /32  is  neglected  in 
(61);  that  is,  the  depth  of  the  corrugations  is  supposed  to  be  infinitely 
small. 

The  part  of  the  motion  proportional  to  Z2  is,  of  course,  independent  of 
the  direction  of  the  principal  motion  of  the  fluid,  and  is  thus  in  a  manner 
applicable  even  when  the  principal  motion  is  alternating.  With  regard  to 
the  relative  importance  of  the  third  and  second  terms  in  (61),  we  have  to 
consider  the  value  of 


and  the  conclusion  will  depend  upon  the  value  of  y.  If  we  suppose  that 
ky  =  \,  the  ratio  is  2L  :  3k'2  v,  or,  if  we  denote  by  V  the  undisturbed  velocity 
of  the  fluid  when  ky  =  l,  V/Skv,  or  VX/Girv,  X  being  the  wave-length  of 
the  corrugation.  With  ordinary  liquids  and  moderate  values  of  X,  V  would 
have  to  be  very  small  in  order  to  permit  the  success  of  the  method  of 
approximation. 

The  character  of  the  motion  proportional  to  L*  is  easily  seen  from  the 
value  of  v.     We  have 


(62) 


indicating  a  motion  directed  outwards  from  the  wall  over  the  places  where 
the  sinuosities  encroach  upon  the  fluid,  and  an  inward  motion  where  the 
sinuosities  recede. 

The  application  of  the  results  towards  the  explanation  of  such  phenomena 
as  ripple-mark  and  wave-formation  requires  a  calculation  of  the  forces  operative 
upon  the  boundary.  We  will  confine  ourselves  to  the  first  term  in  /3  and  L, 
in  (55). 


The  normal  stress,  parallel  to  y,  is  given  by 

q~-*+*%~j*-%8i'' (63) 

and  the  tangential  stress,  parallel  to  x,  is 
dv 

(64) 


1893]  ESPECIALLY   IN   TWO   DIMENSIONS.  93 

From  (34),  (55)  we  find 

p  =  —  4/iA^e-*^  sin  kx, 

or  when  y  =  0, 

£>  =  —  4<k/3  sin  <r,   simply. 

Also,  when  y  =  0, 

2/A  -?-•£•  =  -  4fc/3  sin  kx  ; 
r  dxdy 

so  that  Q  =  0  .....................................  (65) 

In  like  manner,  when  y  =  0, 

sAw}  .........................  (66) 


So  far  as  the  first  power  of  /3  the  action  upon  the  boundary  is  thus  purely 
tangential,  and  of  magnitude  given  by  (66).  The  periodic  part  has  the  same 
sign  as  the  constant  part  at  the  places  where  the  boundary  encroaches  upon 
the  fluid. 

This  result  finds  immediate  application  to  the  question  of  wave-formation 
under  the  action  of  wind,  especially  if  we  suppose  that  the  waves  move  very 
slowly,  as  they  would  do  if  gravity  (and  cohesion)  were  small.  The  main- 
tenance or  augmentation  of  the  waves  requires  that  the  forces  operative  at 
the  surface  be  of  suitable  phase.  Thus  pressures  acting  upon  the  retreating 
shoulders  are  favourable,  as  are  also  tangential  forces  acting  forwards  at  the 
crests  of  the  waves,  where  the  internal  motion  is  itself  in  the  forward  direction. 
Equation  (65)  shows  that  the  pressures  produce  no  effect,  and  that  we  have 
only  to  consider  the  action  of  the  tangential  stress.  We  see  from  (66)  that 
when  the  waves  move  in  the  same  direction  as  the  wind,  the  effect  of  the 
latter  is  to  favour  the  development  of  the  former.  Whether  the  waves  will 
actually  increase  depends  upon  whether  the  supply  of  energy,  proportional 
to  yS2,  is  greater  or  less  than  the  loss  from  internal  dissipation,  itself  propor- 
tional to  the  same  quantity.  If  the  waves  are  moving  against  the  wind,  the 
tendency  is  to  a  more  rapid  subsidence  than  would  occur  in  a  calm. 


209. 

THE  SCIENTIFIC   WORK  OF  TYNDALL. 
[Proceedings  of  the  Royal  Institution,  xiv.  pp.  216—224,  1894.] 

IT  is  fitting  that  the  present  season  should  not  pass  without  a  reference 
on  these  evenings  to  the  work  of  him  whose  tragic  death  a  few  months  since 
was  felt  as  a  personal  grief  and  loss  by  every  member  of  the  Royal  Institution. 
With  much  diffidence  I  have  undertaken  the  task  to-night,  wishing  that  it 
had  fallen  to  one  better  qualified  by  long  and  intimate  acquaintance  to  do 
justice  to  the  theme.  For  Tyndall  was  a  personality  of  exceeding  interest. 
He  exercised  an  often  magical  charm  upon  those  with  whom  he  was  closely 
associated,  but  when  his  opposition  was  aroused  he  showed  himself  a  keen 
controversialist.  My  subject  of  to-night  is  but  half  the  story. 

Even  the  strictest  devotion  of  the  time  at  my  disposal  to  a  survey  of  the 
scientific  work  of  Tyndall  will  not  allow  of  more  than  a  very  imperfect  and 
fragmentary  treatment.  During  his  thirty  years  of  labour  within  these 
walls  he  ranged  over  a  vast  field,  and  accumulated  results  of  a  very  varied 
character,  important  not  only  to  the  cultivators  of  the  physical  sciences,  but 
also  to  the  biologist.  All  that  I  can  hope  to  do  is  to  bring  back  to  your 
recollection  the  more  salient  points  of  his  work,  and  to  illustrate  them  where 
possible  by  experiments  of  his  own  devising. 

In  looking  through  the  catalogue  of  scientific  papers  issued  by  the  Royal 
Society,  one  of  the  first  entries  under  the  name  of  Tyndall  relates  to  a  matter 
comparatively  simple,  but  still  of  some  interest.  It  has  been  noticed  that 
when  a  jet  of  liquid  is  allowed  to  play  into  a  receiving  vessel,  a  good  deal 
of  air  is  sometimes  carried  down  with  it,  while  at  other  times  this  does  not 
happen.  The  matter  was  examined  experimentally  by  Tyndall,  and  he  found 
that  it  was  closely  connected  with  the  peculiar  transformation  undergone  by 
a  jet  of  liquid  which  had  been  previously  investigated  by  Savart.  A  jet  as 
it  issues  from  the  nozzle  is  at  first  cylindrical,  but  after  a  time  it  becomes 
what  the  physiologists  call  varicose;  it  swells  in  some  places  and  contracts 
in  others.  This  effect  becomes  more  exaggerated  as  the  jet  descends,  until 


1894]  THE   SCIENTIFIC   WORK   OF   TYNDALL.  95 

the  swellings  separate  into  distinct  drops,  which  follow  one  another  in  single 
file.  Savart  showed  that  under  the  influence  of  vibration  the  resolution  into 
drops  takes  place  more  rapidly,  so  that  the  place  of  resolution  travels  up 
closer  to  the  nozzle. 

Tyndall's  observation  was  that  the  carrying  down  of  air  required  a  jet 
already  resolved  into  drops  when  it  strikes  the  liquid.  I  hope  to  be  able  to 
show  you  the  experiment  by  projection  upon  the  screen.  At  the  present 
moment  the  jet  is  striking  the  water  in  the  tank  previously  to  resolution 
into  drops,  and  is  therefore  carrying  down  no  air.  If  I  operate  on  the  nozzle 
with  a  vibrating  tuning-fork,  the  resolution  occurs  earlier,  and  the  drops  now 
carry  down  with  them  a  considerable  quantity  of  air. 

Among  the  earlier  of  Tyndall's  papers  are  some  relating  to  ice,  a  subject 
which  attracted  him  much,  probably  from  his  mountaineering  experiences. 
About  the  time  of  which  I  am  speaking  Faraday  made  interesting  observa- 
tions upon  a  peculiar  behaviour  of  ice,  afterwards  called  by  the  name  of 
regelation.  He  found  that  if  two  pieces  of  ice  were  brought  into  contact 
they  stuck  or  froze  together.  The  pressure  required  to  produce  this  effect 
need  not  be  more  than  exceedingly  small.  Tyndall  found  that  if  fragments 
of  ice  are  squeezed  they  pack  themselves  into  a  continuous  mass.  We  have 
here  some  small  ice  in  a  mould,  where  it  can  be  subjected  to  a  powerful 
squeeze.  The  ice  under  this  operation  will  be  regelated,  and  a  mass  obtained 
which  may  appear  almost  transparent,  and  as  if  it  had  never  been  fractured 
at  all.  The  flow  of  glaciers  has  been  attributed  to  this  action,  the  fractures 
which  the  stresses  produce  being  mended  again  by  regelation.  I  should  say, 
perhaps,  that  the  question  of  glacier  motion  presents  difficulties  not  yet 
wholly  explained.  There  can  be  no  doubt,  however,  that  regelation  plays  an 
important  part. 

Another  question  treated  by  Tyndall  is  the  manner  in  which  ice  first 
begins  to  melt  under  the  action  of  a  beam  of  light  passing  into  it  from  an 
electric  lamp.  Ice  usually  melts  by  conducted  heat,  which  reaches  first  the 
outside  layers.  But  if  we  employ  a  beam  from  an  electric  lamp,  the  heat 
will  reach  the  ice  not  only  outside  but  internally,  and  the  melting  will  begin 
at  certain  points  in  the  interior.  Here  we  have  a  slab  of  ice  which  we 
project  upon  the  screen.  We  see  that  the  melting  begins  at  certain  points, 
which  develop  a  crystallised  appearance  resembling  flowers.  They  are  points 
in  the  interior  of  the  ice,  not  upon  the  surface.  Tyndall  found  that  when 
the  ice  gives  way  at  these  internal  points  there  is  a  formation  of  apparently 
empty  space.  He  carefully  melted  under  water  such  a  piece  of  ice,  and 
found  that  when  the  cavity  was  melted  out  there  was  no  escape  of  air, 
proving  that  the  cavity  was  really  vacuous. 

Various  speculations  have  been  made  as  to  the  cause  of  this  internal 
melting  at  definite  points,  but  here  again  I  am  not  sure  if  the  difficulty  has 


96  THE   SCIENTIFIC   WORK   OF   TYNDALL.  [209 

been  altogether  removed.  One  point  of  importance  brought  out  by  Tyndall 
relates  to  the  plane  of  the  flowers.  It  is  parallel  to  the  direction  in  which 
the  ice  originally  froze,  that  is,  parallel  to  the  original  surface  of  the  water 
from  which  it  was  formed. 

I  must  not  dwell  further  upon  isolated  questions,  however  interesting; 
but  will  pass  on  at  once  to  our  main  subject,  which  may  be  divided  into 
three  distinct  parts,  relating  namely  to  heat,  especially  dark  radiation,  sound, 
and  the  behaviour  of  small  particles,  such  as  compose  dust,  whether  of  living 
or  dead  matter. 

The  earlier  publications  of  Tyndall  on  the  subject  of  heat  are  for  the 
most  part  embodied  in  his  work  entitled  Heat  as  a  Mode  of  Motion.  This 
book  has  fascinated  many  readers.  I  could  name  more  than  one  now  distin- 
guished physicist  who  drew  his  first  scientific  nutriment  from  it.  At  the 
time  of  its  appearance  the  law  of  the  equivalence  of  heat  and  work  was 
quite  recently  established  by  the  labours  of  Mayer  and  Joule,  and  had  taken 
firm  hold  of  the  minds  of  scientific  men ;  and  a  great  part  of  Tyndall's  book 
may  be  considered  to  be  inspired  by  and  founded  upon  this  first  law  of 
thermodynamics.  At  the  time  of  publication  of  Joule's  labours,  however, 
there  seems  to  have  been  a  considerable  body  of  hostile  opinion,  favourable 
to  the  now  obsolete  notion  that  heat  is  a  distinct  entity  called  caloric. 
Looking  back,  it  is  a  little  difficult  to  find  out  who  were  responsible  for 
this  reception  of  the  theory  of  caloric.  Perhaps  it  was  rather  the  popular 
writers  of  the  time  than  the  first  scientific  authorities.  A  scientific  worker, 
especially  if  he  devotes  himself  to  original  work,  has  not  time  to  examine 
for  himself  all  questions,  even  those  relating  to  his  own  department,  but 
must  take  something  on  trust  from  others  whom  he  regards  as  authorities. 
One  might  say  that  a  knowledge  of  science,  like  a  knowledge  of  law,  consists 
in  knowing  where  to  look  for  it.  But  even  this  kind  of  knowledge  is  not 
always  easy  to  obtain.  It  is  only  by  experience  that  one  can  find  out  who 
are  most  entitled  to  confidence.  It  is  difficult  now  to  understand  the  hesita- 
tion that  was  shown  in  fully  accepting,  the  doctrine  that  heat  is  a  mode  of 
motion,  for  all  the  great  authorities,  especially  in  England,  seem  to  have 
favoured  it.  Not  to  mention  Newton  and  Cavendish,  we  have  Rumford 
making  almost  conclusive  experiments  in  its  support,  Davy  accepting  it, 
and  Young,  who  was  hardly  ever  wrong,  speaking  of  the  antagonistic  theory 
almost  with  contempt.  On  the  Continent  perhaps,  and  especially  among 
the  French  school  of  chemists  and  physicists,  caloric  had  more  influential 
support. 

As  has  been  said,  a  great  part,  though  not  the  whole  of  Tyndall's  work 
was  devoted  to  the  new  doctrine.  Much  relates  to  other  matters,  such  as 
radiant  heat.  Objection  has  been  taken  to  this  phrase,  not  altogether 
without  reason;  for  it  may  be  said  that  when  heat  it  is  not  radiant,  and 


1894]  THE   SCIENTIFIC   WORK   OF   TYNDALL.  97 

while  radiant  it  is  not  heat.  The  term  dark  radiation,  or  dark  radiance  as 
Newcomb  calls  it,  is  preferable,  and  was  often  used  by  Tyndall.  If  we 
analyse,  as  Newton  did,  the  components  of  light,  we  find  that  only  certain 
parts  are  visible.  The  invisible  parts  produce,  however,  as  great,  or  greater, 
effects  in  other  ways  than  do  the  visible  parts.  The  heating  effect,  for 
example,  is  vastly  greater  in  the  invisible  region  than  in  the  visible.  One 
of  the  experiments  that  Tyndall  devised  in  order  to  illustrate  this  fact  I 
hope  now  to  repeat.  He  found  that  it  was  possible  by  means  of  a  solution 
of  iodine  in  bisulphide  of  carbon  to  isolate  the  invisible  rays.  This  solution 
is  opaque  to  light ;  even  the  sun  could  not  be  seen  through  it ;  but  it  is  very 
fairly  transparent  to  the  invisible  ultra-red  radiation.  By  means  of  a  concave 
reflector  I  concentrate  the  rays  from  an  arc  lamp.  In  their  path  is  inserted 
the  opaque  solution,  but  in  the  focus  of  invisible  radiation  the  heat  developed 
is  sufficient  to  cause  the  inflammation  of  a  piece  of  gun-cotton. 

Tyndall  varied  this  beautiful  experiment  in  many  ways.  By  raising  to 
incandescence  a  piece  of  platinum  foil,  he  illustrated  the  transformation  of 
invisible  into  visible  radiation. 

The  most  important  work,  however,  that  we  owe  to  Tyndall  in  connexion 
with  heat  is  the  investigation  of  the  absorption  of  invisible  radiation  by 
gaseous  bodies.  Melloni  had  examined  the  behaviour  of  solid  and  liquid 
bodies,  but  not  of  gases.  He  found  that  transparent  bodies  like  glass  might 
be  very  opaque  to  invisible  radiation.  Thus,  as  we  all  know,  a  glass  screen 
will  keep  off  the  heat  of  a  fire,  while  if  we  wish  to  protect  ourselves  from 
the  sun,  the  glass  screen  would  be  useless.  On  the  other  hand  rock-salt 
freely  transmitted  invisible  radiation.  But  nothing  had  been  done  on  the 
subject  of  gaseous  absorption,  when  Tyndall  attacked  this  very  difficult 
problem.  Some  of  his  results  are  shown  in  the  accompanying  table.  The 
absorption  of  the  ordinary  non-condensable,  or  rather,  not  easily  condensable 
gases — for  we  must  not  talk  of  non-condensable  gases  now,  least  of  all  in 
this  place — the  absorption  of  these  gases  is  very  small ;  but  when  we  pass 
to  the  more  compound  gases,  such  as  nitric  oxide,  we  find  the  absorption 
much  greater — and  in  the  case  of  olefiant  gas  we  see  that  the  absorbing 
power  is  as  much  as  6000  times  that  of  the  ordinary  gases. 

Relative  Absorption  at 
1  inch  Pressure 

Air 1 

Oxygen 1 

Nitrogen 1 

Hydrogen       .' 1 

Carbonic  acid 972 

Nitric  oxide 1590 

Ammonia       5460 

Olefiant  gas    G030 


98  THE  SCIENTIFIC  WORK   OF  TYNDALL.  [209 

There  is  one  substance  as  to  which  there  has  been  a  great  diversity  of 
opinion — aqueous  vapour.  Tyndall  found  that  aqueous  vapour  exercises  a 
strong  power  of  absorption — strong  relatively  to  that  of  the  air  in  which  it 
is  contained.  This  is  of  course  a  question  of  great  importance,  especially 
in  relation  to  meteorology.  Tyndall's  conclusions  were  vehemently  contested 
by  many  of  the  authorities  of  the  time,  among  whom  was  Magnus,  the 
celebrated  physicist  of  Berlin.  With  a  view  to  this  lecture  I  have  gone 
somewhat  carefully  into  this  question,  and  I  have  been  greatly  impressed 
by  the  care  and  skill  showed  by  Tyndall,  even  in  his  earlier  experiments 
upon  this  subject.  He  was  at  once  sanguine  and  sceptical — a  combination 
necessary  for  success  in  any  branch  of  science.  The  experimentalist  who  is 
not  sceptical  will  be  led  away  on  a  false  tack  and  accept  conclusions  which 
he  would  find  it  necessary  to  reject  were  he  to  pursue  the  matter  further;  if 
not  sanguine,  he  will  be  discouraged  altogether  by  the  difficulties  encountered 
in  his  earlier  efforts,  and  so  arrive  at  no  conclusion  at  all.  One  criticism, 
however,  may  be  made.  Tyndall  did  not  at  first  describe  with  sufficient 
detail  the  method  and  the  precautions  which  he  used.  There  was  a  want  of 
that  precise  information  necessary  to  allow  another  to  follow  in  his  steps. 
Perhaps  this  may  have  been  due  to  his  literary  instinct,  which  made  him 
averse  from  overloading  his  pages  with  technical  experimental  details. 

The  controversy  above  referred  to  I  think  we  may  now  consider  to  be 
closed.  Nobody  now  doubts  the  absorbing  power  of  aqueous  vapour.  In- 
deed the  question  seems  to  have  entered  upon  a  new  phase ;  for  in  a  recent 
number  of  Wiedemann's  Annalen,  Paschen  investigates  the  precise  position 
in  the  spectrum  of  the  rays  which  are  absorbed  by  aqueous  vapour. 

I  cannot  attempt  to  show  you  here  any  of  the  early  experiments  on  the 
absorption  of  vapours.  But  some  years  later  Tyndall  contrived  an  experi- 
ment, which  will  allow  of  reproduction.  It  is  founded  on  some  observations 
of  Graham  Bell,  who  discovered  that  various  bodies  become  sonorous  when 
exposed  to  intermittent  radiation. 

The  radiation  is  supplied  from  incandescent  lime,  and  is  focused  by  a 
concave  reflector.  In  the  path  of  the  rays  is  a  revolving  wheel  provided 
with  projecting  teeth.  When  a  tooth  intervenes,  the  radiation  is  stopped; 
but  in  the  interval  between  the  teeth  the  radiation  passes  through,  and 
falls  upon  any  object  held  at  the  focus.  The  object  in  this  case  is  a  small 
glass  bulb  containing  a  few  drops  of  ether,  and  communicating  with  the  ear 
by  a  rubber  tube.  Under  the  operation  of  the  intermittent  radiation,  the 
ether  vapour  expands  and  contracts  ;  in  other  words  a  vibration  is  established, 
and  a  sound  is  heard  by  the  observer.  But  if  the  vapour  were  absolutely 
diathermanous,  no  sound  would  be  heard. 

I  have  repeated  the  experiment  of  Tyndall  which  allowed  him  to  distin- 
guish between  the  behaviour  of  ordinary  air  and  dry  air.  If,  dispensing  with 


1894]  THE   SCIENTIFIC    WORK   OF   TYNDALL.  99 

ether,  we  fill  the  bulb  with  air  in  the  ordinary  moist  state,  a  sound  is  heard 
with  perfect  distinctness,  but  if  we  drop  in  a  little  sulphuric  acid,  so  as  to 
dry  the  air,  the  sound  disappears. 

According  to  the  law  of  exchanges,  absorption  is  connected  with  radiation ; 
so  that  while  hydrogen  or  oxygen  do  not  radiate,  from  ammonia  we  might 
expect  to  get  considerable  radiation.  In  the  following  experiment  I  aim  at 
showing  that  the  radiation  of  hot  coal  gas  exceeds  the  radiation  of  equally 
hot  air. 

The  face  of  the  thermopile,  protected  by  screens  from  the  ball  itself,  is 
exposed  to  the  radiation  from  the  heated  air  which  rises  from  a  hot  copper 
ball.  The  effect  is  manifested  by  the  [spot  of]  light  reflected  from  a  galva- 
nometer mirror.  When  we  replace  the  air  by  a  stream  of  coal  gas,  the 
galvanometer  indicates  an  augmentation  of  heat,  so  that  we  have  before  us  a 
demonstration  that  coal  gas  when  heated  does  radiate  more  than  equally  hot 
air,  from  which  we  conclude  that  it  would  exercise  more  absorption  than  air. 

I  come  now  to  the  second  division  of  my  subject,  that  relating  to  Sound. 
Tyndall,  as  you  know,  wrote  a  book  on  Sound,  founded  on  lectures  delivered 
in  this  place.  Many  interesting  and  original  discoveries  are  there  embodied. 
One  that  I  have  been  especially  interested  in  myself,  is  on  the  subject  of 
sensitive  flames.  Professor  Leconte  in  America  made  the  first  observations 
at  an  amateur  concert,  but  it  was  Tyndall  who  introduced  the  remarkable 
high-pressure  flame  now  before  you.  It  issues  from  a  pin-hole  burner,  and 
the  sensitiveness  is  entirely  a  question  of  the  pressure  at  which  the  gas  is 
supplied.  Tyndall  describes  the  phenomenon  by  saying  that  the  flame 
under  the  influence  of  a  high  pressure  is  like  something  on  the  edge  of  a 
precipice.  If  left  alone,  it  will  maintain  itself ;  but  under  the  slightest  touch 
it  will  be  pushed  over.  The  gas  at  high  pressure  will,  if  undisturbed,  burn 
steadily  and  erect,  but  if  a  hiss  is  made  in  its  neighbourhood  it  becomes  at 
once  unsteady,  and  ducks  down.  A  very  high  sound  is  necessary.  Even  a 
whistle,  as  you  see,  does  not  act.  Smooth  pure  sounds  are  practically  without 
effect  unless  of  very  high  pitch. 

I  will  illustrate  the  importance  of  the  flame  as  a  means  of  investigation 
by  an  experiment  in  the  diffraction  of  sound.  I  have  here  a  source  of  sound, 
but  of  pitch  so  high  as  to  be  inaudible.  The  waves  impinge  perpendicularly 
upon  a  circular  disc  of  plate  glass.  Behind  the  disc  there  is  a  sound  shadow, 
and  you  might  expect  that  the  shadow  would  be  most  complete  at  the  centre. 
But  it  is  not  so.  When  the  burner  occupies  this  [central]  position  the  flame 
flares ;  but  when  by  a  slight  motion  of  the  disc  the  position  of  the  flame  is 
made  eccentric,  the  existence  of  the  shadow  is  manifested  by  the  recovery 
of  the  flame.  At  the  centre  the  intensity  of  sound  is  the  same  as  if  no 
obstacle  were  interposed. 

7—2 


100  THE   SCIENTIFIC   WORK   OF   TYNDALL.  [209 

The  optical  analogue  of  the  above  experiment  was  made  at  the  suggestion 
of  Poisson,  who  had  deduced  the  result  theoretically,  but  considered  it  so 
unlikely  that  he  regarded  it  as  an  objection  to  the  undulatory  theory  of 
light.  Now,  I  need  hardly  say,  it  is  regarded  as  a  beautiful  confirmation. 

It  is  of  importance  to  prove  that  the  flame  is  not  of  the  essence  of  the 
matter,  that  there  is  no  need  to  have  a  flame,  or  to  ignite  it  at  the  burner. 
Thus,  it  is  quite  possible  to  have  a  jet  of  gas  so  arranged  that  ignition  does 
not  occur  until  the  jet  has  lost  its  sensitiveness.  The  sensitive  part  is  that 
quite  close  to  the  nozzle,  and  the  flame  is  only  an  indicator.  But  it  is  not 
necessary  to  have  any  kind  of  flame  at  all.  Tyndall  made  observations  on 
smoke-jets,  showing  that  a  jet  of  air  can  be  made  sensitive  to  sound.  The 
difficulty  is  to  see  it,  and  to  operate  successfully  upon  it ;  because,  as  Tyndall 
soon  found,  a  smoke-jet  is  much  more  difficult  to  deal  with  than  flames,  and 
is  sensitive  to  much  graver  sounds.  I  doubt  whether  I  am  wise  in  trying  to 
exhibit  smoke-jets  to  an  audience,  but  I  have  a  special  means  of  projection 
by  which  I  ought  at  least  to  succeed  in  making  them  visible.  It  consists  in 
a  device  by  which  the  main  part  of  the  light  from  the  lamp  is  stopped  at 
the  image  of  the  arc,  so  that  the  only  light  which  can  reach  the  screen  is 
light  which  by  diffusion  has  been  diverted  out  of  its  course.  Thus  we  shall 
get  an  exhibition  of  a  jet  of  smoke  upon  the  screen,  showing  bright  on  a 
dark  ground.  The  jet  issues  near  the  mouth  of  a  resonator  of  pitch  256. 
When  undisturbed  it  pursues  a  straight  course,  and  remains  cylindrical.  But 
if  a  fork  of  suitable  pitch  be  sounded  in  the  neighbourhood,  the  jet  spreads 
out  into  a  sort  of  fan,  or  even  bifurcates,  as  you  see  upon  the  screen.  The 
real  motion  of  the  jet  cannot  of  course  be  ascertained  by  mere  inspection. 
It  consists  in  a  continuously  increasing  sinuosity,  leading  after  a  while  to 
complete  disruption.  If  two  forks  slightly  out  of  unison  are  sounded  to- 
gether, the  jet  expands  and  re-collects  itself,  synchronously  with  the  audible 
beats.  I  should  say  that  my  jet  is  a  very  coarse  imitation  of  Tyndall's. 
The  nozzle  that  I  am  using  is  much  too  large.  With  a  proper  nozzle,  and 
in  a  perfectly  undisturbed  atmosphere — undisturbed  not  only  by  sounds,  but 
free  from  all  draughts — the  sensitiveness  is  wonderful.  The  slightest  noise 
is  seen  to  act  instantly  and  to  bring  the  jet  down  to  a  fraction  of  its  former 
height. 

Another  important  part  of  Tyndall's  work  on  Sound  was  carried  out  as 
adviser  of  the  Trinity  House.  When  in  thick  weather  the  ordinary  lights 
fail,  an  attempt  is  made  to  replace  them  with  sound  signals.  These  are 
found  to  vary  much  in  their  action,  sometimes  being  heard  to  a  very  great 
distance,  and  at  other  times  failing  to  make  themselves  audible  even  at  a 
moderate  distance.  Two  explanations  have  been  suggested,- depending  upon 
acoustic  refraction  and  acoustic  reflection. 

Under  the  influence  of  variations  of  temperature  refraction  occurs  in  the 


1894]  THE   SCIENTIFIC   WORK   OF   TYNDALL.  101 

atmosphere.  For  example,  sound  travels  more  quickly  in  warm  than  in  cold 
air.  If,  as  often  happens,  it  is  colder  above,  the  upper  part  of  the  sound 
wave  tends  to  lag  behind,  and  the  wave  is  liable  to  be  tilted  upwards  and 
so  to  be  carried  over  the  head  of  the  would-be  observer  on  the  surface  of  the 
ground.  This  explanation  of  acoustic  refraction  by  variation  of  temperature 
was  given  by  Prof.  Osborne  Reynolds.  As  Sir  G.  Stokes  showed,  refraction 
is  also  caused  by  wind.  The  difference  between  refraction  by  wind  and  by 
temperature  variations  is  that  in  one  case  everything  turns  upon  the 
direction  in  which  the  sound  is  going,  while  in  the  second  case  this  con- 
sideration is  immaterial.  The  sound  is  heard  by  an  observer  down  wind, 
and  not  so  well  by  an  observer  up  wind.  The  explanation  by  refraction  of 
the  frequent  failure  of  sound  signals  was  that  adopted  by  Prof.  Henry  in 
America,  a  distinguished  worker  upon  this  subject.  Tyndall's  investigations, 
however,  led  him  to  favour  another  explanation.  His  view  was  that  sound 
was  actually  reflected  by  atmospheric  irregularities.  He  observed,  what 
appears  to  be  amply  sufficient  to  establish  his  case,  that  prolonged  signals 
from  fog  sirens  give  rise  to  echoes  audible  after  the  signal  has  stopped. 
This  echo  was  heard  from  the  air  over  the  sea,  and  lasted  in  many  cases 
a  long  time,  up  to  15  seconds.  There  seems  here  no  alternative  but  to 
suppose  that  reflection  must  have  occurred  internally  in  the  atmosphere. 
In  some  cases  the  explanation  of  the  occasional  diminished  penetration  of 
sound  seems  to  be  rather  by  refraction,  and  in  others  by  reflection. 

Tyndall  proved  that  a  single  layer  of  hot  air  is  sufficient  to  cause 
reflection,  and  I  propose  to  repeat  his  experiment.  The  source  of  sound, 
a  toy  reed,  is  placed  at  one  end  of  one  metallic  tube,  and  a  sensitive  flame 
at  one  end  of  a  second.  The  opposite  ends  of  these  tubes  are  placed  near 
each  other,  but  in  a  position  which  does  not  permit  the  sound  waves  issuing 
from  the  one  to  enter  the  other  directly.  Accordingly  the  flame  shows  no 
response.  If,  however,  a  pane  of  glass  be  held  suitably,  the  waves  are 
reflected  back  and  the  flame  is  excited.  Tyndall's  experiment  consists  in 
the  demonstration  that  a  flat  gas  flame  is  competent  to  act  the  part  of  a 
reflector.  When  I  hold  the  gas  flame  in  the  proper  position,  the  percipient 
flame  flares ;  when  the  flat  flame  is  removed  or  held  at  an  unsuitable  angle, 
there  is  almost  complete  recovery. 

It  is  true  that  in  the  atmosphere  no  such  violent  transitions  of  density 
can  occur  as  are  met  with  in  a  flame ;  but,  on  the  other  hand,  the  inter- 
ruptions may  be  very  numerous,  as  is  indeed  rendered  probable  by  the 
phenomena  of  stellar  scintillation. 

The  third  portion  of  my  subject  must  be  treated  very  briefly.  The 
guiding  idea  of  much  of  Tyndall's  work  on  atmospheric  particles  was  the 
application  of  an  intense  illumination  to  render  them  evident.  Fine  particles 


102  THE   SCIENTIFIC   WORK   OF   TYNDALL.  [209 

of  mastic,  precipitated  on  admixture  of  varnish  with  a  large  quantity  of 
water,  had  already  been  examined  by  Briicke.  Chemically  precipitated 
sulphur  is  convenient,  and  allows  the  influence  of  size  to  be  watched  as  the 
particles  grow.  But  the  most  interesting  observations  of  Tyndall  relate  to 
precipitates  in  gases  caused  by  the  chemical  action  of  the  light  itself.  This 
may  be  illustrated  by  causing  the  concentrated  rays  of  the  electric  lamp  to 
pass  through  a  flask  containing  vapour  of  peroxide  of  chlorine.  Within  a 
few  seconds  dense  clouds  are  produced. 

When  the  particles  are  very  small  in  comparison  with  the  wave-length, 
the  laws  governing  the  dispersion  of  the  light  are  simple.  Tyndall  pursued 
the  investigation  to  the  case  where  the  particles  have  grown  beyond  the 
limit  above  indicated,  and  found  that  the  polarisation  of  the  dispersed  light 
was  affected  in  a  peculiar  and  interesting  manner. 

Atmospheric  dust,  especially  in  London,  is  largely  organic.  If,  following 
Tyndall,  we  hold  a  spirit  lamp  under  the  track  of  the  light  from  the  electric 
lamp,  the  dark  spaces,  resulting  from  the  combustion  of  the  dust,  have  all 
the  appearance  of  smoke. 

In  confined  and  undisturbed  spaces  the  dust  settles  out.  I  have  here  a 
large  flask  which  has  been  closed  for  some  days.  If  I  hold  it  to  the  lamp, 
the  track  of  the  light,  plainly  visible  before  entering  and  after  leaving  the 
flask,  is  there  interrupted.  This,  it  will  be  evident,  is  a  matter  of  consider- 
able importance  in  connexion  with  organic  germs. 

The  question  of  the  spontaneous  generation  of  life  occupied  Tyndall  for 
several  years.  He  brought  to  bear  upon  it  untiring  perseverance  and 
refined  experimental  skill,  and  his  results  are  those  now  generally  accepted. 
Guarding  himself  from  too  absolute  statements  as  to  other  times  and  other 
conditions,  he  concluded  that  under  the  circumstances  of  our  experiments 
life  is  always  founded  upon  life.  The  putrefaction  of  vegetable  and  animal 
infusions,  even  when  initially  sterilised,  is  to  be  attributed  to  the  intrusion 
of  organic  germs  from  the  atmosphere. 

The  universal  presence  of  such  germs  is  often  regarded  as  a  hypothesis 
difficult  of  acceptance.  It  may  be  illustrated  by  an  experiment  from  the 
inorganic  world.  I  have  here,  and  can  project  upon  the  screen,  glass  pots, 
each  containing  a  shallow  layer  of  a  supersaturated  solution  of  sulphate  of 
soda.  Protected  by  glass  covers,  they  have  stood  without  crystallising  for 
forty-eight  hours.  But  if  I  remove  the  cover,  a  few  seconds  or  minutes 
will  see  the  crystallisation  commence.  It  has  begun,  and  long  needles  are 
invading  the  field  of  view.  Here  it  must  be  understood  that,  with  a  few 
exceptions,  the  crystalline  germ  required  to  start  the  action  must  be  of  the 
same  nature  as  the  dissolved  salt ;  and  the  conclusion  is  that  small  crystals 
of  sulphate  of  soda  are  universally  present  in  the  atmosphere. 


1894]  THE   SCIENTIFIC   WORK   OF   TYNDALL.  103 

I  have  now  completed  my  task.  With  more  or  less  success  I  have  laid 
before  you  the  substance  of  some  of  Tyndall's  contributions  to  knowledge. 
What  I  could  not  hope  to  recall  was  the  brilliant  and  often  poetic  exposition 
by  which  his  vivid  imagination  illumined  the  dry  facts  of  science.  Some 
reminiscence  of  this  may  still  be  recovered  by  the  reader  of  his  treatises 
and  memoirs;  but  much  survives  only  as  an  influence  exerted  upon  the 
minds  of  his  contemporaries,  and  manifested  in  subsequent  advances  due 
to  his  inspiration. 


210. 


ON  AN  ANOMALY   ENCOUNTERED  IN   DETERMINATIONS 
OF  THE   DENSITY   OF  NITROGEN   GAS. 

[Proceedings  of  the  Royal  Society,  i>v.  pp.  340—344,  April,  1894.] 

IN  a  former  communication*  I  have  described  how  nitrogen,  prepared  by 
Lupton'sf  method,  proved  to  be  lighter  by  about  1/1000  part  than  that 
derived  from  air  in  the  usual  manner.  In  both  cases  a  red-hot  tube  contain- 
ing copper  is  employed,  but  with  this  difference.  In  the  latter  method  the 
atmospheric  oxygen  is  removed  by  oxidation  of  the  copper  itself,  while  in 
[Harcourt's]  method  it  combines  writh  the  hydrogen  of  ammonia,  through 
which  the  air  is  caused  to  pass  on  its  way  to  the  furnace,  the  copper  remain- 
ing unaltered.  In  order  to  exaggerate  the  effect,  the  air  was  subsequently 
replaced  by  oxygen.  Under  these  conditions  the  whole,  instead  of  only  about 
one-seventh  part  of  the  nitrogen  is  derived  from  ammonia,  and  the  dis- 
crepancy was  found  to  be  exalted  to  about  one-half  per  cent. 

Upon  the  assumption  that  similar  gas  should  be  obtained  by  both 
methods,  we  may  explain  the  discrepancy  by  supposing  either  that  the  atmo- 
spheric nitrogen  was  too  heavy  on  account  of  imperfect  removal  of  oxygen, 
or  that  the  ammonia  nitrogen  was  too  light  on  account  of  contamination  with 
gases  lighter  than  pure  nitrogen.  Independently  of  the  fact  that  the  action 
ofi  the  copper  in  the  first  case  was  pushed  to  great  lengths,  there  are  two 
arguments  which  appeared  to  exclude  the  supposition  that  oxygen  was  still 
present  in  the  prepared  gas.  One  of  these  depends  upon  the  large  quantity 
of  oxygen  that  would  be  required,  in  view  of  the  small  difference  between  the 
weights  of  the  two  gases.  As  much  as  l/30th  part  of  oxygen  would  be 
necessary  to  raise  the  density  by  1/200,  or  about  one-sixth  of  all  the  oxygen 
originally  present.  This  seemed  to  be  out  of  the  question.  But  even  if  so 
high  a  degree  of  imperfection  in  the  action  of  the  copper  could  be  admitted, 

*  "  On  the  Densities  of  the  Principal  Gases,"  Roy.  Soc.  Proc.  Vol.  LIII.  p.  146,  1893.  [Vol.  iv. 
p.  39.  See  also  p.  1.] 

t  [1902.    The  use  of  ammonia  to  burn  atmospheric  oxygen  is  due  to  Mr  Vernon  Harcourt.] 


1894]  DENSITY  OF  NITROGEN  GAS.  105 

the  large  alteration  caused  by  the  substitution  of  oxygen  for  air  in  [Harcourt's] 
process  would  remain  unexplained.  Moreover,  as  has  been  described  in  the 
former  paper,  the  introduction  of  hydrogen  into  the  gas  made  no  difference, 
such  hydrogen  being  removed  by  the  hot  oxide  of  copper  subsequently 
traversed.  It  is  surely  impossible  that  the  supposed  residual  oxygen  could 
have  survived  such  treatment. 

Another  argument  may  be  founded  upon  more  recent  results,  presently  to 
be  given,  from  which  it  appears  that  almost  exactly  the  same  density  is  found 
when  the  oxygen  of  air  is  removed  by  hot  iron  reduced  with  hydrogen, 
instead  of  by  copper,  or  in  the  cold  by  ferrous  hydrate. 

But  the  difficulties  in  the  way  of  accepting  the  second  alternative  are 
hardly  less  formidable.  For  the  question  at  once  arises,  of  what  gas,  lighter 
than  nitrogen,  does  the  contamination  consist  ?  In  order  that  the  reader  may 
the  better  judge,  it  may  be  well  to  specify  more  fully  what  were  the  arrange- 
ments adopted.  The  gas,  whether  air  or  oxygen,  after  passing  through  potash 
was  charged  with  ammonia  as  it  traversed  a  small  wash-bottle,  and  thence 
proceeded  to  the  furnace.  The  first  passage  through  the  furnace  was  in  a 
tube  packed  with  metallic  copper,  in  the  form  of  fine  wire.  Then  followed  a 
wash-bottle  of  sulphuric  acid  by  which  the  greater  part  of  the  excess  of 
ammonia  would  be  arrested,  and  a  second  passage  through  the  furnace  in  a 
tube  containing  copper  oxide.  The  gas  then  traversed  a  long  length  of  pumice 
charged  with  sulphuric  acid,  and  a  small  wash-bottle  containing  Nessler 
solution.  On  the  other  side  of  the  regulating  tap  the  arrangements  were 
always  as  formerly  described,  and  included  tubes  of  finely  divided  potash  and 
of  phosphoric  anhydride.  The  rate  of  passage  was  usually  about  half  a  litre 
per  hour. 

Of  the  possible  impurities,  lighter  than  nitrogen,  those  most  demanding 
consideration  are  hydrogen,  ammonia,  and  water  vapour.  The  last  may  be 
dismissed  at  once,  and  the  absence  of  ammonia  is  almost  equally  certain. 
The  question  of  hydrogen  appears  the  most  important.  But  this  gas,  and 
hydrocarbons,  such  as  CH4,  could  they  be  present,  should  be  burnt  by  the 
copper  oxide;  and  the  experiments  already  referred  to,  in  which  hydrogen 
was  purposely  introduced  into  atmospheric  nitrogen,  seem  to  prove  conclu- 
sively that  the  burning  would  really  take  place.  Some  further  experiments 
of  the  same  kind  will  presently  be  given. 

The  gas  from  ammonia  and  oxygen  was  sometimes  odourless,  but  at  other 
times  smelt  strongly  of  nitrous  fumes,  and,  after  mixture  with  moist  air, 
reddened  litmus  paper.  On  one  occasion  the  oxidation  of  the  nitrogen  went 
so  far  that  the  gas  showed  colour  in  the  blow-off  tube  of  the  Toppler,  although 
the  thickness  of  the  layer  was  only  about  half  an  inch.  But  the  presence 
of  nitric  oxide  is,  of  course,  no  explanation  of  the  abnormal  lightness.  The 


106  ON   AN   ANOMALY    ENCOUNTERED   IN    DETERMINATIONS  [210 

conditions  under  which  the  oxidation  takes  place  proved  to  be  difficult  of 
control,  and  it  was  thought  desirable  to  examine  nitrogen  derived  by  reduc- 
tion from  nitric  and  nitrous  oxides. 

The  former  source  was  the  first  experimented  upon.  The  gas  was  evolved 
from  copper  and  diluted  nitric  acid  in  the  usual  way,  and,  after  passing 
through  potash,  was  reduced  by  iron,  copper  not  being  sufficiently  active,  at 
least  without  a  very  high  temperature.  The  iron  was  prepared  from  black- 
smith's scale.  In  order  to  get  quit  of  carbon,  it  was  first  treated  with  a 
current  of  oxygen  at  a  red  heat,  and  afterwards  reduced  by  hydrogen,  the 
reduction  being  repeated  after  each  employment.  The  greater  part  of  the 
work  of  reducing  the  gas  was  performed  outside  the  furnace,  in  a  tube  heated 
locally  with  a  Bunsen  flame.  In  the  passage  through  the  furnace  in  a  tube 
containing  similar  iron  the  work  would  be  completed,  if  necessary.  Next 
followed  washing  with  sulphuric  acid  (as  required  in  the  ammonia  process),  a 
second  passage  through  the  furnace  over  copper  oxide,  and  further  washing 
with  sulphuric  acid.  In  order  to  obtain  an  indication  of  any  unreduced  nitric 
oxide,  a  wash-bottle  containing  ferrous  sulphate  was  introduced,  after  which 
followed  the  Nessler  test  and  drying  tubes,  as  already  described.  As  thus 
arranged,  the  apparatus  could  be  employed  without  alteration,  whether  the 
nitrogen  to  be  collected  was  derived  from  air,  from  ammonia,  from  nitric 
oxide,  from  nitrous  oxide,  or  from  ammonium  nitrite. 

The  numbers  which  follow  are  the  weights  of  the  gas  contained  by  the 
globe  at  zero,  at  the  pressure  defined  by  the  manometer  when  the  tempera- 
ture is  15°.  They  are  corrected  for  the  errors  in  the  weights,  but  not  for  the 
shrinkage  of  the  globe  when  exhausted,  and  thus  correspond  to  the  number 
2*31026,  as  formerly  given  for  nitrogen. 

Nitrogen  from  NO  by  Hot  Iron. 

November  29,  1893 2-30143  \ 

December    2,1893 2-29890      __ 

December    5,1893 2'29816      Mean'  ™™S 

December    6,  1893 2'30182  J 

Nitrogen  from  N20  by  Hot  Iron*. 

December  26,  1893 2'29869  )  .. 

December  28,  1893 2*29940  }  Mean'  2'2"°4 

Nitrogen  from  Ammonium  Nitrite  passed  over  Hot  Iron. 

January    9,  1894 2*29849  } 

January  13,  1894 2'29889  }  Mean>  2  29S 

*  The  N20  was  prepared  from  zinc  and  very  dilute  nitric  acid. 


1894]  OF  THE  DENSITY  OF  NITROGEN  GAS.  107 

With  these  are  to  be  compared  the  weights  of  nitrogen  derived  from  the 
atmosphere. 

Nitrogen  from  Air  by  Hot  Iron. 

December  12,  1893 2'31017          \ 

December  14,  1893 2'30986  (H)  I 

December  19,  1893 2*31010  (H)  f       an'  23 

December  22,  1893 2-31001          J 

Nitrogen  from  Air  by  Ferrous  Hydrate. 

January  27,  1894  2'31024  j 

January  30,  1894  2-31010  [  Mean,  2*31020 

February  1,  1894  2'31028  ) 

In  the  last  case  a  large  volume  of  air  was  confined  for  several  hours  in  a 
glass  reservoir  with  a  mixture  of  slaked  lime  and  ferrous  sulphate.  The  gas 
was  displaced  by  deoxygenated  water,  and  further  purified  by  passage  through 
a  tube  packed  with  a  similar  mixture.  The  hot  tubes  were  not  used. 

If  we  bring  together  the  means  for  atmospheric  nitrogen  obtained  by 
various  methods,  the  agreement  is  seen  to  be  good,  and  may  be  regarded  as 
inconsistent  with  the  supposition  of  residual  oxygen  in  quantity  sufficient  to 
influence  the  weights. 

Atmospheric  Nitrogen. 

By  hot  copper,  1892 2'31026 

By  hot  iron,  1893 2'31003 

By  ferrous  hydrate,  1894 2'31020 

Two  of  the  results  relating  to  hot  iron,  those  of  December  14  and  Decem- 
ber 19,  were  obtained  from  nitrogen,  into  which  hydrogen  had  been  purposely 
introduced.  An  electrolytic  generator  was  inserted  between  the  two  tubes 
containing  hot  iron,  as  formerly  described.  The  generator  worked  under  its 
own  electromotiVe  force,  and  the  current  was  measured  by  a  tangent  galvano- 
meter. Thus,  on  December  19,  the  deflection  throughout  the  time  of  filling 
was  3°,  representing  about  1/15  ampere.  In  two  hours  and  a  half  the  hydro- 
gen introduced  into  the  gas  would  be  about  70  c.c.,  sufficient,  if  retained,  to 
reduce  the  weight  by  about  4  per  cent.  The  fact  that  there  was  no  sensible 
reduction  proves  that  the  hydrogen  was  effectively  removed  by  the  copper 
oxide. 

The  nitrogen,  obtained  altogether  in  four  ways  from  chemical  compounds, 
is  materially  lighter  than  the  above,  the  difference  amounting  to  about 
11  mg.,  or  about  1/200  part  of  the  whole.  It  is  also  to  be  observed  that  the 
agreement  of  individual  results  is  less  close  in  the  case  of  chemical  nitrogen 
than  of  atmospheric  nitrogen. 


108  DENSITY    OF    NITROGEN    GAS.  [210 

I  have  made  some  experiments  to  try  whether  the  densities  were  influ- 
enced by  exposing  the  gas  to  the  silent  electric  discharge.  A  Siemens  tube, 
as  used  for  generating  ozone,  was  inserted  in  the  path  of  the  gas  after  desic- 
cation with  phosphoric  anhydride.  The  following  were  the  results : — 

Nitrogen  from  Air  by  Hot  Iron,  Electrified. 

January  1,  1894 2'31163  )  _ 

„  >  Mean,  2'310o9 
January  4,  1894 2*30956  j 

Nitrogen  from  N,O  by  Hot  Iron,  Electrified. 

January  2, 1894 2'30074  | 

January  5,  1894 2'30054  }  Mean'  2'3°°64 

The  somewhat  anomalous  result  of  January  1  is  partly  explained  by  the 
failure  to  obtain  a  subsequent  weighing  of  the  globe  empty,  and  there  is  no 
indication  that  any  effect  was  produced  by  the  electrification. 

One  more  observation  I  will  bring  forward  in  conclusion.  Nitrogen  pre- 
pared from  oxygen  and  ammonia,  and  about  one-half  per  cent,  lighter  than 
ordinary  atmospheric  nitrogen,  was  stored  in  the  globe  for  eight  months. 
The  globe  was  then  connected  to  the  apparatus,  and  the  pressure  was  re- 
adjusted in  the  usual  manner  to  the  standard  conditions.  On  re-weighing 
no  change  was  observed,  so  that  the  abnormally  light  nitrogen  did  not  become 
dense  by  keeping. 

[1902.  For  the  explanation  of  the  discrepancy  here  set  forth,  as  due  to  a 
previously  unrecognised  constituent  of  the  atmosphere,  see  the  memoir  by 
Rayleigh  and  Ramsay,  Art.  214  below.] 


211. 

ON  THE  MINIMUM  CURRENT  AUDIBLE  IN  THE  TELEPHONE. 

[Philosophical  Magazine,  xxxvm.  pp.  285—295,  1894*.] 

THE  estimates  which  have  been  put  forward  of  the  minimum  current 
perceptible  in  the  Bell  telephone  vary  largely.  Mr  Preece  gives  6  x  10~1S 
ampere  f  ;  Prof.  Tait,  for  a  current  reversed  500  times  per  second,  2  x  10~12 
ampere*.  De  la  Rue  gives  1  x  10~8  ampere,  and  the  same  figure  is  recorded 
by  Brough§  as  applicable  to  the  strongest  current  with  which  the  instrument 
is  worked.  Various  methods,  more  or  less  worthy  of  confidence,  have  been 
employed,  but  the  only  experimenter  who  has  described  his  procedure  with 
detail  sufficient  to  allow  of  criticism  is  Prof.  Ferraris  ||,  whose  results  may  be 
thus  expressed  :  — 


Fluency 

Do3  ...............  264  23x10- 

Fa3  ...............  352  17x10- 

La3  ...............  440  10x10- 

Do4  ...............  528  7x10- 

Re4  ...............  594  5x10- 

The  currents  were  from  a  make-and-break  apparatus,  and  in  each  case  are 
reckoned  as  if  only  the  first  periodic  term  of  the  Fourier  series,  representative 
of  the  actual  current,  were  effective.  On  this  account  the  quantities  in  the 
third  column  should  probably  be  increased,  for  the  presence  of  overtones 
could  hardly  fail  to  favour  audibility. 

Although  a  considerable  margin  must  be  allowed  for  varying  pitch,  vary- 
ing acuteness  of  audition,  and  varying  construction  of  the  instruments,  it  is 
scarcely  possible  to  suppose  that  all  the  results  above  mentioned  can  be 

*  Bead  at  the  Oxford  Meeting  of  the  British  Association. 
t  Brit.  Assoc.  Report,  Manchester,  1887,  p.  611. 

J  Edin.  Proc.  Vol.  ix.  p.  551  (1878).    Prof.  Tait  speaks  of  a  billion  B.A.  units,  and,  as  he 
kindly  informs  me,  a  billion  here  means  1012. 

§  Proceedings  of  the  Asiatic  Society  of  Bengal,  1877,  p.  255. 

||  Atti  delta  R.  Accad.  d.  Sci.  di  Torino,  Vol.  xm.  p.  1024  (1877). 


110  ON   THE   MINIMUM   CURRENT  AUDIBLE   IN   THE   TELEPHONE.  [211 

correct,  even  in  the  roughest  sense.  The  question  is  of  considerable  interest 
in  connexion  with  the  theory  of  the  telephone.  For  it  appears  that  a,  priori 
calculations  of  the  possible  efficiency  of  the  instrument  are  difficult  to  reconcile 
with  numbers  such  as  those  of  Tait  and  of  Preece,  at  least  without  attributing 
to  the  ear  a  degree  of  sensitiveness  to  aerial  vibration  far  surpassing  even  the 
marvellous  estimates  that  have  hitherto  been  given*. 

Under  these  circumstances  it  appeared  to  be  desirable  to  undertake  fresh 
observations,  in  which  regard  should  be  paid  to  various  sources  of  error  that 
may  have  escaped  attention  in  the  earlier  days  of  telephony.  The  importance 
of  denning  the  resistance  of  the  instruments  and  of  employing  pure  tones  of 
various  pitch  need  not  be  insisted  upon. 

As  regards  resistance,  a  low-resistance  telephone,  although  suitable  in 
certain  cases,  must  not  be  expected  to  show  the  same  sensitiveness  to  current 
as  an  instrument  of  higher  resistance.  If  we  suppose  that  the  total  space 
available  for  the  windings  is  given,  and  that  the  proportion  of  it  occupied  by 
the  copper  is  also  given,  a  simple  relation  obtains  between  the  resistance  and 
the  minimum  current.  For  if  7  be  the  current,  n  be  the  number  of  convolu- 
tions, and  r  the  resistance,  we  have,  as  in  the  theory  of  galvanometers, 
ny  =  const.,  n~-r  =  const.,  so  that  y^r  —  const.,  or  the  minimum  current  is 
inversely  as  the  square  root  of  the  resistance. 

The  telephones  employed  in  the  experiments  about  to  be  narrated  were 
two,  of  which  one  (Tj)  is  a  very  efficient  instrument  of  70-ohms  resistance. 
The  other  (T2),  of  less  finished  workmanship,  was  rewound  in  the  laboratory 
with  comparatively  thick  wire.  The  interior  diameter  of  the  windings  is 

9  mm.,  and  the  exterior  diameter  is  26  mm.     The  width  of  the  groove,  or  the 
axial  dimension  of  the  coil,  is  8  mm.,  the  number  of  windings  is  160,  and  the 
resistance  is  '8  ohm.     Since  the  dimensions  of  the  coils  are  about  the  same 
in  the  two  cases,  we  should  expect,  according  to  the  above  law,  that  about 

10  times  as  much  current  would  be  required  in  T2  as  in  Tt.     Both  instru- 
ments are  of  the  Bell  (unipolar)  type,  and  comparison  with  other  specimens 
shows  that  there  is  nothing  exceptional  in  their  sensibility. 

In  view  of  the  immense  discrepancies  above  recorded,  it  is  evident  that 
what  is  required  is  not  so  much  accuracy  of  measurement  as  assured  sound- 
ness in  method.  It  appeared  to  me  that  electromotive  forces  of  the  necessary 
harmonic  type  would  be  best  secured  by  the  employment  of  a  revolving 
magnet  in  the  proximity  of  an  inductor-coil  of  known  construction.  The 
electromotive  force  thus  generated  operates  in  a  circuit  of  known  resistance ; 
and,  if  the  self-induction  can  be  neglected,  the  calculation  of  the  current 
presents  no  difficulty.  The  sound  as  heard  in  the  telephone  may  be  reduced 

*  Proc.  Roy.  Soc.  Vol.  xxvi.  p.  248  (1877).  [Vol.  i.  p.  328 ;  sec  also  Art.  213  below.]  Also 
Wien,  Wied.  Ann.  Vol.  xxxvi.  p.  834  (1889). 


1894]        ON   THE   MINIMUM   CURRENT   AUDIBLE   IN  THE  TELEPHONE.  Ill 

to  the  required  point  either  by  varying  the  distance  (B)  between  the  magnet 
and  the  inductor,  or  by  increasing  the  resistance  (R)  of  the  circuit.  In  fact 
both  these  quantities  may  be  varied ;  and  the  agreement  of  results  obtained 
with  widely  different  values  of  R  constitutes  an  effective  test  of  the  legitimacy 
of  neglecting  self-induction.  When  R  is  too  much  reduced,  the  time-constant 
of  the  circuit  becomes  comparable  with  the  period  of  vibration,  and  the  current 
is  no  longer  increased  in  proportion  to  the  reduction  of  R.  This  complication 
is  most  likely  to  occur  when  the  pitch  is  high. 

In  order  to  keep  as  clear  as  possible  of  the  complication  due  to  self-induc- 
tion, I  employed  in  the  earlier  experiments  a  resistance-coil  of  100,000  ohms, 
constructed  as  usual  of  wire  doubled  upon  itself.  But  it  soon  appeared  that 
in  avoiding  Scylla  I  had  fallen  upon  Charybdis.  The  first  suspicion  of  some- 
thing wrong  arose  from  the  observation  that  the  sound  was  nearly  as  loud 
when  the  100,000  ohms  was  included  as  when  a  10,000-ohm  coil  was  substi- 
tuted for  it.  The  first  explanation  that  suggested  itself  was  that  the  sound 
was  being  conveyed  mechanically  instead  of  electrically,  as  is  indeed  quite 
possible  under  certain  conditions  of  experiment.  But  a  careful  observation 
of  the  effect  of  breaking  the  continuity  of  the  leads,  one  at  a  time,  proved 
that  the  propagation  was  really  electrical.  Subsequent  inquiry  showed  that 
the  anomaly  was  due  to  a  condenser,  or  leyden,  like  action  of  the  doubled 
wire  of  the  100,000-ohm  coil.  When  the  junction  at  the  middle  was  un- 
soldered, so  as  to  interrupt  the  metallic  continuity,  the  sounds  heard  in  the 
telephone  were  nearly  as  loud  as  before.  In  this  condition  the  resistance 
should  have  been  enormous,  and  was  in  fact  about  12  megohms*  as  indicated 
by  a  galvanometer.  It  was  evident  that  the  coil  was  acting  principally  as  a 
leyden  rather  than  as  a  resistance,  and  that  any  calculation  founded  upon 
results  obtained  with  it  would  be  entirely  fallacious. 

It  is  easy  to  form  an  estimate  of  the  point  at  which  the  complication  due 
to  capacity  would  begin  to  manifest  itself.  Consider  the  case  of  a  simple 
resistance  R  in  parallel  with  a  leyden  of  capacity  C,  and  let  the  currents  in 
the  two  branches  be  x  and  y  respectively.  If  V  be  the  difference  of  potential 
at  the  common  terminals,  proportional  to  eipt,  we  have 

as  =  V/R,  y  =  CdV/dt  =  ipVC; 

so  that 

x + y  =  1 + JpRG 

V  R 

The  amplitude  of  the  total  current  is  increased  by  the  leyden  in  the  ratio 
V(l  +  p^R^C1)  :  1 ;  and  the  action  of  the  leyden  becomes  important  when 
pRG=  1.  '  With  a  frequency  of  640,  p  =  4020 ;  so  that,  if  R  =  1014  C.G.S.,  the 
critical  value  of  C  is  -fa  x  10~15  C.G.S.,  or  about  ^  of  a  microfarad. 

*  Doubtless  the  insulation  between  the  wires  should  have  been  much  higher. 


112  ON  THE   MINIMUM   CURRENT   AUDIBLE    IN   THE   TELEPHONE.  [211 

It  will  be  seen  that  even  if  the  capacity  remained  unaltered,  a  reduction 
of  resistance  in  the  ratio  say  of  10  to  1  would  greatly  dimmish  the  complica- 
tion due  to  condenser-like  action  ;  but  perhaps  the  best  evidence  that  the 
results  obtained  are  not  prejudiced  in  this  manner  is  afforded  by  the  experi- 
ments in  which  the  principal  resistance  was  a  column  of  plumbago. 

The  revolving  magnet  was  of  clock-spring,  about  2|  cm.  long,  and  so  bent 
as  to  be  driven  directly,  windmill  fashion,  from  an  organ  bellows.  It  was 
mounted  transversely  upon  a  portion  of  a  sewing-needle,  the  terminals  of 
which  were  carried  in  slight  indentations  at  the  ends  of  a  U-shaped  piece  of 
brass.  As  fitted  to  the  wind-trunk,  the  axis  of  rotation  was  horizontal. 

The  inductor-coil,  with  its  plane  horizontal,  was  situated  so  that  its  centre 
was  vertically  below  that  of  the  magnet  at  distance  B.  Thus,  if  A  be  the 
mean  radius  of  the  coil,  n  the  number  of  convolutions,  the  galvanometer- 
constant  G  of  the  coil  at  the  place  occupied  by  the  magnet  is  given  by 


a) 


where  C"-  =  A--\-  B2;  and  if  m  be  the  magnetic  moment  of  the  magnet,  and 
the  angle  of  rotation,  the  mutual  potential  M  may  be  represented  by* 


(2) 

If  the  frequency  of  revolution  be  p/27r,     <f>  =  pt;  and  then 

dM/dt=Gmpcospt (3) 

The  expression  (3)  represents  the  electromotive  force  operative  in  the  circuit. 
If  the  inductance  can  be  neglected,  the  corresponding  current  is  obtained  on 
division  of  (3)  by  R,  the  total  resistance  of  the  circuit. 

The  moment  in  is  deduced  by  observation  of  the  deflection  of  a  magneto- 
meter-needle from  the  position  which  it  assumes  under  the  operation  of  the 
earth's  horizontal  force  H.  If  the  magnet  be  situated  to  the  east  at  distance 
r,  and  be  itself  directed  east  and  west,  the  angular  deflection  0  from  equili- 
brium is  given  by 

a     2»i/rs 
tan  u  =  — jj    . 

The  relation  between  the  angle  0  and  the  double  deflection  d  in  scale- 
divisions,  obtained  on  revel-sal  of  TO,  is  approximately  0  =  d/4>D,  where  D  is 
the  distance  between  mirror  and  scale ;  so  that  we  may  take 

_Hr*d 

*  Maxwell,  Electricity  and  Magnetism,  Vol.  n.  §  700. 


1894]        ON  THE   MINIMUM   CURRENT  AUDIBLE   IN   THE  TELEPHONE.  113 

The  amplitude  of  the  oscillatory  current,  generated  under  these  conditions,  is 
accordingly 


If  C.G.S.  units  are  employed,  H='l8.  A  must  of  course  be  measured  in 
centimetres  ;  but  any  units  that  are  convenient  may  be  used  for  r  and  C,  and 
for  d  and  D.  The  current  will  then  be  given  in  terms  of  the  C.G.S.  unit, 
which  is  equal  to  10  amperes. 

The  inductor-coil  used  in  most  of  the  experiments  is  wound  upon  an 
ebonite  ring,  and  is  the  one  that  was  employed  as  the  "  suspended  coil  "  in 
the  determination  of  the  electro-chemical  equivalent  of  silver*.  The  number 
of  convolutions  (n)  is  242.  The  axial  dimension  of  the  section  is  1'4  cm. 
and  the  radial  dimension  is  '97  cm.  The  mean  radius  A  is  10'25  cm.,  and 
the  resistance  is  about  10£  ohms. 

In  making  the  observations  the  current  from  the  inductor-coil  was  led  to 
a  distant  part  of  the  house  by  leads  of  doubled  wire,  and  was  there  connected 
to  the  telephone  and  resistances.  Among  the  latter  was  a  plumbago  resist- 
ance on  Prof.  F.  J.  Smith's  planf  of  about  84,000  ohms;  but  in  .most  of  the 
experiments  a  resistance-box  going  up  to  10,000  ohms  was  employed,  with 
the  advantage  of  allowing  the  adjustment  of  sound  to  be  made  by  the  observer 
at  the  telephone.  The  attempt  to  hit  off  the  least  possible  sound  was  found 
to  be  very  fatiguing  and  unsatisfactory  ;  and  in  all  the  results  here  recorded 
the  sounds  were  adjusted  so  as  to  be  easily  audible  after  attention  for  a  few 
seconds.  Experiment  showed  that  the  resistances  could  then  be  doubled 
without  losing  the  sound,  although  perhaps  it  would  not  be  caught  at  once 
by  an  unprepared  ear.  But  it  must  not  be  supposed  that  the  observation 
admits  of  precision,  at  least  without  greater  precautions  than  could  well  be 
taken.  Much  depends  upon  the  state  of  the  ear  as  regards  fatigue,  and  upon 
freedom  from  external  disturbance. 

The  pitch  was  determined  before  and  after  an  observation  by  removing 
the  added  resistance  and  comparing  the  loud  sound  then  heard  with  a  harmo- 
nium. The  octave  thus  estimated  might  be  a  little  uncertain.  It  was  verified 
by  listening  to  the  beats  of  the  sound  from  the  telephone  and  from  a  nearly 
unisonant  tuning-fork,  both  sounds  being  nearly  pure  tones. 

When  the  magnet  was  driven  at  full  speed  the  frequency  was  found  to  be 
307,  and  at  this  pitch  a  series  of  observations  was  made  with  various  values 
of  G  and  of  R.  Thus  when  5  =  7'75  inches,  or  (7=  8'7  inches,  the  resistance 
from  the  box  required  to  produce  the  standard  sound  in  telephone  T^  was 

*  Phil.  Trans.  Part  n.  1884,  p.  421.     [Vol.  n.  p.  290.] 
t  Phil.  Mag.  Vol.  xxxv.  p.  210  (1893). 


114  ON  THE   MINIMUM   CURRENT  AUDIBLE   IN  THE  TELEPHONE.  [211 

8000  ohms,  so  that  ,R  =  8100xl09.     The  quantities  required  for  the  calcu- 
lation of  (5)  are  as  follows  : — 


A  =  10-25, 
(7=8-7, 


=  2?rx307, 
=  8-25, 
=  81x10", 


=  •18, 
Z  =  140, 
>  =  1370, 


r  and  C  being  reckoned  in  inches,  d  and  Z)  in  scale-divisions  of  about  -^  inch. 
From  these  data  the  current  required  to  produce  the  standard  sound  is  found 
to  be  7'4  x  10~8  C.G.S.,  or  7*4  x  10~7  amperes,  for  telephone  T^. 

The  results  obtained  by  the  method  of  the  revolving  magnet  are  collected 
into  the  accompanying  table.  The  "  wooden  coil "  is  of  smaller  dimensions 
than  the  "  ebonite  coil,"  the  mean  radius  being  only  3'5  cm.  The  number 
of  convolutions  is  370. 


Telephone 


Frequency  =  307.     Ebonite  coil. 
R  in  ohms  Current  in  amperes  Sound 


84100  Plumbago 
8100  Box 
4100  Box 

3-8xlO-7 
7-4xlO-7 
5-2  x  10~7 

Below  standard 
Standard 

500  Box 

l'2x  10~6 

200  Box 

1-OxlO-5 

Frequency  =  307.     Wooden  coil. 


84100  Plumbago 
10100  Box 
1600  Box 
350  Box 


3-6xlO~r 
3-7x10-7 
5'4xlO-y 
1-lxlO"5 


Standard 


Frequency  =  192.     Ebonite  coil. 
7\ |          3100  Box  |        2-oxlO-6        |       Standard 

The  method  of  the  revolving  magnet  seemed  to  be  quite  satisfactory  so 
far  as  it  went,  but  it  was  desirable  to  extend  the  determinations  to  frequencies 
higher  than  could  well  be  reached  in  this  manner.  For  this  purpose  recourse 
was  had  to  magnetized  tuning-forks,  vibrating  with  known  amplitudes.  If, 
for  the  moment,  we  suppose  the  magnetic  poles  to  be  concentrated  at  the 
extremities  of  the  prongs,  a  vibrating-fork  may  be  regarded  as  a  simple 
magnet,  fixed  in  position  and  direction,  but  of  moment  proportional  to  the 
instantaneous  distance  between  the  poles.  Thus,  if  the  magnetic  axis  pass 
perpendicularly  through  the  centre  of  the  mean  plane  of  the  inductor-coil, 
the  situation  is  very  similar  to  that  obtaining  in  the  case  of  the  revolving 
magnet.  The  angle  $  in  (2)  is  no  longer  variable,  but  such  that  sin  <f>  =  1 
throughout.  On  the  other  hand  m  varies  harmonically.  If  I  be  the  mean 
distance  between  the  poles,  2#  the  extreme  arc  from  rest  to  rest  traversed  by 


1894]        ON  THE   MINIMUM   CURRENT   AUDIBLE   IN  THE  TELEPHONE.  115 

each  pole  during  the  vibration,  w0  the  mean  magnetic  moment, 

M/WO  =  1  +  2/3/1 .  sinpt, 
and 

dMJdt=Gm0p.'2fi/l.cospt ,, (6) 

The  formula  corresponding  to  (5)  is  thus  derived  from  it  by  simple  introduc- 
tion of  the  factor  2/3/1. 

The  forks  were  excited  by  bowing,  and  the  observation  of  amplitude  was 
effected  by  comparison  with  a  finely  divided  scale  under  a  magnifying-glass. 
It  was  convenient  to  observe  the  extreme  end  of  a  prong  where  the  motion  is 
greatest,  but  the  double  amplitude  thus  measured  must  be  distinguished  from 
2/3.  In  order  to  allow  for  the  distance  between  the  resultant  poles  and  the 
extremities  of  the  prongs,  the  measured  amplitude  was  reduced  in  the  ratio 
of  2  to  3.  The  observation  of  the  magnetic  moment  at  the  magnetometer  is 
not  embarrassed  by  the  diffusion  of  the  free  polarity. 

In  order  to  explain  the  determination  more  completely,  I  will  give  full 
details  of  an  observation  with  a  fork  c'  of  frequency  256.  The  distance  I 
between  the  middles  of  the  prongs  was  '875  inch,  and  the  double  amplitude 
of  the  vibration  at  the  end  of  one  of  the  prongs  was  '09  inch.  Thus  2/3  is 
reckoned  as  "06  inch.  The  inductor-coil  was  the  ebonite  coil  already  described, 
and  the  sound  was  judged  to  be  of  the  standard  distinctness  when,  for 
example,  5  =  15  inches,  or  (7=15*5  inches,  and  the  added  resistance  was 
1000  ohms,  so  that  R  =  1100  x  10°.  The  quantities  required  for  the  compu- 
tation of  (5)  as  extended  are 

n  =  242,  p  =  2-7T  x  256,  H  =  18, 

4=10-25,  r  =  15,  d  =  410, 

(7=15-5,  #  =  11x10",  D  =  1370, 

2/3  =  -06,  £  =  -875; 

and  they  give  for  the  current  corresponding  to  the  standard  sound  9'8  x  10~8 
C.G.S.,  or  9'8  x  10~7  amperes. 

A  summary  of  the  results  obtained  with  forks  of  pitch  c,  c',  e,  g',  c",  e",  g" 
is  annexed.  As  the  pitch  rose,  the  difficulties  of  observation  increased,  both 

Telephone  R  in  ohms  Current  in  amperes 

C  =  128. 
71! |  1100  |         2-8  xlO~6 

c  =  256. 


8100  Box 

1100  

500  ... 


6-8x10-7 
9-8x10-7 


1-1x10- 


116 


ON   THE   MINIMUM   CURRENT   AUDIBLE   IN   THE  TELEPHONE. 


[211 


Telephone 


2* 


R  in  ohms 


e  =  320. 


84000  Plumbago 
6100  Box 
1600  ... 


Current  in  amperes 


3-8xlO~7 
2-6x10-7 
3-1x10-' 


g'  =  384. 


T^  

84000  Plumbago 

1-4x10-' 

r,  

9500  Box 

l-6xlO~7 

7\  

2100  

1-4x10-' 

7*!  

900  

1-7  xlO-7 

T0 

600  

1-9  xlO~6 

J  2  

T7*  ... 

300  ... 

2-2  xlO~6 

c"=512. 


84000  Plumbago 

8-9x10-8 

9000  Box 

4-8x10-8 

3600  

5-2xlO-8 

700  

8-2x10-8 

11? 

5-2xlO-c? 

100  Box 



l-9xlO-« 

300  

l-4x!0-fi 

500 

2-5x10-° 

900  ... 

2-4xlO-6 

e"  =  640. 


84000  Plumbago 
5100  Box 
1100  ... 


"  =  768. 


84000  Plumbago 
7100  Box 
2100  ... 


3-8xlO-8 
3-8xlO-8 
5-5x10-8 


1-1x10-' 

•9  x!0~7 

1-1  xlO"7 


on  account  of  the  less  duration  of  the  sound  and  of  the  smaller  amplitudes 
available  for  measurement.  In  one  observation  with  telephone  T.2  at  pitch  c", 
the  resistance,  estimated  at  11  ohms,  was  that  of  the  coil,  telephone,  and 
leads  only.  No  trustworthy  result  was  to  be  expected  under  such  conditions, 
but  the  number  is  included  in  order  to  show  how  small  was  the  influence  of 
self-induction,  even  where  it  had  every  opportunity  of  manifesting  itself.  If 
we  bring  together  the  numbers*  derived  with  the  revolving  magnet  and  with 
the  forks,  we  obtain  in  the  case  of  Tl : — 


*  The  observations  recorded  were  made  with  my  own  ears.     Mr  Gordon  obtained  very  similar 
numbers  when  he  took  my  place. 


1894]         ON   THE    MINIMUM   CURRENT   AUDIBLE   IN   THE   TELEPHONE.  117 


Pitch 
128  

Source 
Fork  

Current  in  10~8  amperes 
2800 

192  
256  
307  
320  

Revolving  magnet  
Fork  
Revolving  magnet  
Fork     

250 
83 
49 
32 

384  

15 

512  

7 

640  

4-4 

768... 

10 

It  would  appear  that  the  maximum  sensitiveness  to  current  occurs  in  the 
region  of  frequency  640 ;  but  observations  at  still  higher  frequencies  would 
be  needed  to  establish  this  conclusion  beyond  doubt.  Attention  must  be  paid 
to  the  fact  that  the  sounds  were  not  the  least  that  could  be  heard,  and  that 
before  a  comparison  is  made  with  the  numbers  given  by  other  experimenters 
there  should  be  a  division  by  2,  if  not  by  3.  But  this  consideration  does  not 
fully  explain  the  difference  between  the  above  table  and  that  of  Ferraris 
already  quoted,  from  which  it  appears  that  in  his  experiments  a  current  of 
5  x  10~9  amperes  was  audible. 

It  is  interesting  to  note  that  the  sensitiveness  of  the  telephone  to  periodic 
currents  is  of  the  same  order  as  that  of  the  galvanometer  of  equal  resistance 
to  steady  currents*,  viz.  that  the  currents  (at  pitch  512)  just  audible  in  the 
telephone  would,  on  commutation,  be  just  easily  visible  by  a  deflection  in  the 
latter  instrument.  But  there  is  probably  more  room  for  further  refinements 
in  the  galvanometer  than  in  the  telephone. 

If  we  compare  the  performances  of  the  two  telephones  T^  and  T2,  we  find 
ratios  of  sensitiveness  to  current  ranging  from  13  to  30 ;  so  that  T2  shows 
itself  inferior  in  a  degree  beyond  what  may  be  accounted  for  by  the  resist- 
ances. It  is  singular  that  an  experiment  of  another  kind  led  to  the  opposite 
conclusion.  The  circuit  of  a  Daniell  cell  A  was  permanently  closed  through 
resistance-coils  of  5  ohms  and  of  1000  ohms.  The  two  telephones  in  series 
with  one  another  and  with  a  resistance-box  C  were  placed  in  a  derived  circuit 
where  was  also  a  scraping  contact-apparatus  B,  as  indicated  in  the  figure. 


The  adjustment  was  made  by  varying  the  resistance  in  C  until  the  sound  was 
just  easily  audible  in  the  telephone  under  trial.     Experiments  conducted 

*  See,  for  example,  Ayrton,  Mather,  and  Sumpnef,  Phil.  Mag.  Vol.  xxx.  p.  90,  1890,  "  On 
Galvanometers." 


118  ON   THE   MINIMUM   CURRENT   AUDIBLE   IN   THE  TELEPHONE.  [211 

upon  this  plan  showed  that  T,  was  only  about  five  times  as  sensitive  to  cur- 
rent as  T2.  It  was  noticed,  however,  that  the  sounds,  though  as  equal  as 
could  be  estimated,  were  not  of  the  same  quality,  and  in  this  probably  lies 
the  explanation  of  the  discrepancy  between  the  two  methods  of  experiment- 
ing. In  the  latter  the  original  sound  is  composite,  and  the  telephone  selects 
the  most  favourable  elements — that  is,  those  nearly  in  agreement  with  the 
natural  pitch  of  its  own  plate.  In  this  way  the  loudness  of  the  selected  sound 
becomes  a  question  of  the  freedom  of  vibration  of  the  plate,  an  element  which 
is  almost  without  influence  when  the  sound  is  of  pitch  far  removed  from  that 
of  the  proper  tone  of  the  telephone.  There  was  independent  reason  for  the 
suspicion  that  Tl  had  not  so  well  defined  a  proper  pitch  as  was  met  with  in 
the  case  of  some  other  telephones. 

P.  8. — Measurements  with  the  electro-dynamometer  have  been  made  by 
Cross  and  Page*  of  the  currents  used  in  practical  telephony.  The  experi- 
ments were  varied  by  the  employment  of  several  transmitters,  and  various 
vowel  sounds  were  investigated.  The  currents  found  were  of  the  order 
2  x  10~4  amperes. 

*  Electrical  Review,  Nov.  14,  1885.     I  owe  this  reference  to  Mr  Swinburne. 


212. 

AN  ATTEMPT  AT  A  QUANTITATIVE  THEORY  OF  THE 
TELEPHONE. 

[Philosophical  Magazine,  xxxvni.  pp.  295 — 301,  1894*.] 

THE  theory  of  the  telephone  cannot  be  said  to  be  understood,  in  any  but 
the  most  general  manner,  until  it  is  possible  to  estimate  from  the  data  of 
construction  what  its  sensitiveness  should  be,  at  least  so  far  as  to  connect  the 
magnitude  of  the  vibratory  current  with  the  resulting  condensations  and 
rarefactions  in  the  external  ear-passage.  Unfortunately  such  an  estimate  is 
a  matter  of  extreme  difficulty,  partly  on  account  of  imperfection  in  our  know- 
ledge of  the .  magnetic  properties  of  iron,  and  partly  from  mathematical  diffi- 
culties arising  from  the  particular  forms  employed  in  actual  construction ;  and 
indeed  the  problem  does  not  appear  to  have  been  attacked  hitherto.  In  view, 
however,  of  the  doubts  that  have  been  expressed  as  to  theory,  and  of  the 
highly  discrepant  estimates  of  actual  sensitiveness  which  have  been  put 
forward,  it  appears  desirable  to  make  the  attempt.  It  will  be  understood 
that  at  present  the  question  is  as  to  the  order  of  magnitude  only,  and  that 
the  result  will  not  be  without  value  should  it  prove  to  be  10  or  even  100 
times  in  error. 

One  of  the  elements  required  to  be  known,  the  number  (n)  of  convolutions, 
cannot  be  directly  observed  in  the  case  of  a  finished  instrument ;  but  it  may 
be  inferred  with  sufficient  accuracy  for  the  present  purpose  from  the  dimen- 
sions and  the  resistance  of  the  coil.  Denote  the  axial  dimension  by  £,  the 
inner  and  outer  radii  by  ^  and  tj.2,  the  section  of  the  wire  by  a-  and  its  total 
length  by  I,  so  that  l<r  is  the  total  volume  of  copper.  The  area  of  section  of 
the  coil  by  an  axial  plane  is  f  (i)2  —  ^a),  and  of  this  the  area  nor  is  occupied  by 

*  Read  at  the  Oxford  Meeting  of  the  British  Association. 


120  ON   A   QUANTITATIVE  THEORY   OF  THE  TELEPHONE  [212 

copper.     If  we  suppose  the  latter  to  be  half  the  former,  we  shall  not  be  far 
from  the  mark.     Thus 

H«--if(lfe-1h)  ...............................  (1) 

On  the  same  assumption, 

V)  ............................  (2) 


Accordingly,  if  R  be  the  whole  resistance  of  the  coil,  and  r  the  specific  resist- 
ance of  copper, 


As  applicable  to  actual  telephones  we  may  take  f  =  1  centim.,  rj2  =  S^  ;  and 
then  R  =  4firrn?.     In  C.G.S.  measure  r  =  1600,  and  thus 

w'=  477x1600  ...............................  (4) 

If  the  resistance  be  100  ohms,  .R  =  10",  and  n  =  2230. 

When  the  resistance  varies,  other  circumstances.  remaining  the  same, 


We  have  now  to  connect  the  periodic  force  upon  the  telephone-plate  with 
the  periodic  current  in  the  coil.  As  has  already  been  stated,  only  a  very 
rough  estimate  is  possible  a,  priori.  We  will  commence  by  considering  the 
case  of  an  unlimited  cylindrical  core,  divided  by  a  transverse  fracture  into  two 
parts,  and  encompassed  by  an  infinite  cylindrical  magnetizing  coil  containing 
n  turns  to  the  centimetre.  If  7  be  the  current,  the  magnetizing  force  BH 
due  to  it  is 

(5) 


If  we  regard  the  core  as  composed  of  soft  iron,  magnetized  strongly  by  a 
constant  force  H,  the  mechanical  force  with  which  the  two  parts  attract  one 
another  per  unit  of  area  is  in  the  usual  notation 


and  what  we  require  is  the  variation  of  this  quantity,  when  H  becomes 
H  +  8H.     This  may  be  written 


(6) 


The  value  of  dl/dH  to  be  here  employed  is  that  appropriate  to  small 
cyclical  changes.  It  is  greatest  when  7  is  small,  and  then*  amounts  to  about 
100/47T.  As  7  increases,  dl/dH  diminishes,  and  finally  approaches  to  zero  in 
the  state  of  saturation.  In  order  to  increase  (6)  it  is  thus  advisable  to  aug- 
ment 7  up  to  a  certain  point,  but  not  to  approach  saturation  so  nearly  as  to 

*  Phil.  Mag.  XXIH.  p.  225  (1887).     [Vol.  n.  p.  579.] 


1894]  ON   A   QUANTITATIVE   THEORY    OF   THE   TELEPHONE.  121 

bring  about  a  great  diminution  in  the  value  of  dl/dH.  In  the  absence  of 
precise  information  we  may  estimate  that  the  maximum  of  (6)  will  be  reached 
when  /  is  about  half  the  saturation  value,  or  equal  to  800*;  and  that  dl/dH 
also  has  half  its  maximum  value,  or  50/4-Tr.  At  this  rate  the  force  due  to  SH 
is  about  40,000  BH,  reckoned  per  unit  of  area  of  the  divided  core,  or  by  (5) 

40,000  x  4-7TW7 (7) 

But  before  (7)  can  be  applied  to  the  core  of  a  telephone  electromagnet  it 
must  be  subjected  to  large  deductions.  For  in  the  telephone  the  total  number 
of  Avindings  n  is  limited  to  about  one  centimetre  measured  parallel  to  the 
axis,  whereas  in  (7)  the  electromagnet  is  supposed  to  be  infinitely  long,  and 
n  denotes  the  number  of  windings  per  centimetre.  If  we  are  to  suppose  in 
(7)  that  the  windings  are  really  limited  to  one  centimetre,  lying  immediately 
on  one  side  of  the  division,  there  must  be  a  loss  of  effect  which  I  estimate  at 
5  times.  We  have  now  further  to  imagine  the  second  part  of  the  divided 
cylinder  to  be  replaced  by  the  plate  of  the  telephone,  and  that  not  in  actual 
contact  with  the  remaining  cylindrical  part.  The  reduction  of  effect  on  this 
account  I  estimate  at  4  times  "f*.  The  force  on  the  telephone-plate  per  unit 
area  of  core  is  thus 

2000  x  477-717;     (8) 

or  if,  as  for  the  telephone  of  100-ohms  resistance,  n  =  2200,  and  area  of  section 
=  '31  sq.  cm., 

force  =1-7  x  1077 (9) 

In  (9)  the  force  is  in  dynes,  and  the  current  7  is  in  c.G.8.  measure.  If  F 
denote  the  current  reckoned  in  amperes, 

force  =  1-7  xlOT, (10) 

and  this  must  be  supposed  to  be  operative  at  the  centre  of  the  plate. 

We  shall  presently  consider  what  effect  such  a  force  may  be  expected  to 
produce ;  but  before  proceeding  to  this  I  may  record  the  result  of  some  ex- 
periments directed  to  check  the  applicability  of  (10),  and  made  subsequently 
to  the  theoretical  estimates.  A  Bell  telephone,  similar  to  T1}  was  mounted 
vertically,  mouth  downwards,  having  attached  to  the  centre  of  its  plate  a 
slender  strip  of  glass.  This  strip  was  also  vertical  and  carried  at  its  lower 
end  a  small  scale-pan.  The  whole  weight  of  the  attachments  was  only  '44 
gram.  The  movement  of  the  glass  strip  in  the  direction  of  its  length  was 
observed  through  a  reading-microscope  focused  upon  accidental  markings. 
The  telephone,  itself  of  70-ohms  resistance,  was  connected  through  a  revers- 
ing-key  with  a  Daniell  cell  and  with  an  external  resistance  varied  from  time 
to  time.  In  taking  an  observation  the  current  was  first  sent  in  such  a  direc- 
tion as  to  depress  the  plate,  and  the  web  was  adjusted  upon  the  mark.  The 

*  Ewing,  Magnetic  Induction,  1891,  p.  136. 

t  I  should  say  that  these  estimates  were  all  made  in  ignorance  of  the  result  to  which 
they  would  lead. 


122  ON   A   QUANTITATIVE  THEORY   OF  THE   TELEPHONE.  [212 

current  was  then  reversed,  by  which  the  plate  was  drawn  up,  but  by  addition 
of  weights  in  the  pan  it  was  brought  back  again  to  the  same  position  as 
before.  The  force  due  to  the  current  is  thus  measured  by  the  half  of  the 
weight  applied. 

The  results  were  as  follows : — 

External  resistance  in  ohms 100     200     500 

Weight  in  grams  842 

When  1000  ohms  were  included,  the  displacement  on  reversal  was  still  just 
visible.  We  may  conclude  that  a  force  of  1  gram  weight  corresponds  to  a 
current  of  about  g^  of  an  ampere.  Now,  1  gram  weight  is  equal  to  981  dynes, 
so  that  for  comparison  with  (10) 

force  =  -6xlOT (11) 

The  force  observed  is  thus  about  the  third  part  of  that  which  had  been 
estimated,  and  the  agreement  is  sufficient. 

Although  not  needed  for  the  above  comparison,  we  shall  presently  require 
to  know  the  linear  displacement  of  the  centre  of  the  telephone-plate  due  to  a 
given  force.  Observations  with  the  aid  of  a  micrometer-eyepiece  showed  that 
a  force  of  5  grams  weight  gave  a  displacement  of  10~4  x  6'62  centim.,  or 
10~4  x  1'32  for  each  gram,  viz.  10~7  x  T34  centim.  per  dyne.  Thus  by  (11) 
the  displacement  x  due  to  a  current  T  expressed  in  amperes  is 

#=-080r (12) 

We  have  now  to  estimate  what  motion  of  the  telephone-plate  may  be 
expected  to  result  from  a  given  periodic  force  operating  at  its  centre.  The 
effect  depends  largely  upon  the  relation  between  the  frequency  of  the  imposed 
vibration  and  those  natural  to  the  plate  regarded  as  a  freely  vibrating  body. 
If  we  attempt  to  calculate  the  natural  frequencies  d  priori,  we  are  met  by 
uncertainty  as  to  the  precise  mechanical  conditions.  From  the  manner  in 
which  a  telephone-plate  is  supported  we  should  naturally  regard  the  ideal 
condition  as  one  in  which  the  whole  of  the  circular  boundary  is  clamped.  On 
this  basis  a  calculation  may  be  made,  and  it  appears*  that  the  frequency  of 
the  gravest  symmetrical  mode  should  be  about  991  in  the  case  of  the  tele- 
phone in  question.  But  it  may  well  be  doubted  whether  we  are  justified  in 
assuming  that  the  clamping  is  complete,  and  any  relaxation  tells  in  the 
direction  of  a  lowered  frequency.  A  more  trustworthy  conclusion  may  per- 
haps be  founded  upon  the  observed  connexion  between  displacement  and 
force  of  restitution,  coupled  with  an  estimate  of  the  inertia  of  the  moving 
parts.  The  total  weight  of  the  plate  is  3'4  grams;  the  outside  diameter 
is  5'7  centim.,  and  the  inside  diameter,  corresponding  to  the  free  portion  of 

*  Theory  of  Sound,  2nd  ed.  §  221  a. 


1894]  ON   A  QUANTITATIVE  THEORY   OF  THE   TELEPHONE.  123 

the  plate,  is  4'5.  The  effective  mass,  supposed  to  be  situated  at  the  centre,  I 
estimate  to  be  that  corresponding  to  a  diameter  of  2*5  centim.,  viz.  '65  gram. 
A  force  of  restitution  per  unit  displacement  equal  to  (10~7  x  1'34)~1,  or 
106  x  7*5,  is  supposed  to  urge  the  above  mass  to  its  position  of  equilibrium. 
The  frequency  of  the  resulting  vibration  is 


±     /QO'x7-5) 
27rV  I      "65      I 


With  the  aid  of  a  special  electric  maintenance  the  plate  may  be  made  to 
speak  on  its  own  account.  The  frequency  so  found,  viz.  896,  corresponds 
undoubtedly  to  a  free  vibration,  but  it  does  not  follow  that  the  vibration  is 
the  gravest  of  which  the  plate  is  capable ;  and  there  were  indications  pointing 
to  the  opposite  conclusion. 

As  it  is  almost  impossible  to  form  an  d  priori  estimate  of  the  amplitude 
of  vibration  (#)  when  the  frequency  of  the  force  is  in  the  neighbourhood  of 
any  of  the  free  frequencies,  I  will  take  for  calculation  the  case  of  frequency 
256,  which  is  presumably  much  lower  than  any  of  them.  Under  these 
circumstances  an  "  equilibrium  theory  "  may  be  employed,  the  displacement 
coexisting  with  any  applied  force  being  the  same  as  if  the  force  were  perma- 
nent. At  this  pitch  the  minimum  current  recorded  in  the  table*  is  8'3  x  10~7 
amperes;  so  that  by  (12)  the  maximum  excursion  corresponding  thereto  is 
given  by  x  =  '080  x  8'3  x  10~7=  6'8  x  10~8  centim. 

The  excursion  thus  found  must  not  be  compared  with  that  calculated 
formerly -f-  for  free  progressive  waves.  The  proper  comparison  is  rather 
between  the  condensations  s  in  the  two  cases.  In  a  progressive  wave  the 
connexion  between  s  and  v,  the  maximum  velocity,  is  v  =  as,  where  a  is  the 
velocity  of  propagation.  But  in  the  present  case  the  excursion  x  takes  effect 
upon  a  very  small  volume.  If  A  be  the  effective  area  of  the  plate,  and  8  the 
whole  volume  included  between  the  plate  and  the  tympanum  of  the  ear,  we 
may  take  s  =  AxjS.  This  relation  assumes  that  the  condensations  and  rare- 
factions are  uniform  throughout  the  space  in  question,  an  assumption  justified 
by  the  smallness  of  its  dimensions  in  comparison  with  the  wave-length,  and 
further  that  the  behaviour  is  the  same  as  if  the  space  were  closed  air-tight. 
It  would  seem  that  a  slight  deficiency  in  the  latter  respect  would  not  be 
material. 

For  the  numerical  application  I  estimate  that  A  =  4  sq.  cm.,  S  =  20  cub 
cm. ;  so  that  with  the  above  value  of  as 

s  =  l-4x!0-8, (13) 

s  being  reckoned  in  atmospheres. 

*  Supra,  p.  294.     [Vol.  iv.  p.  117.] 

t  Proc.  Roy.  Soc.  Vol.  xxvi.  p.  248  (1877).    [Vol.  i.  p.  328.] 


124  ON   A   QUANTITATIVE   THEORY   OF   THE   TELEPHONE.  [212 

The  value  of  s  corresponding  to  but  just  audible  progressive  waves  of 
frequency  256  was  found  to  be  5'9  x  10~9,  in  sufficiently  good  agreement  with 
(13)* 

But  if  the  equilibrium  theory  be  applied  to  the  notes  of  higher  pitch,  such 
as  512,  we  find  the  actual  sensitiveness  of  the  telephone  greater  than  accord- 
ing to  the  calculation.  In  this  caset  T  =  7  x  10~8;  so  that  by  (12) 

x  =  5-6  x  10-9, 
and 

l-l  x  10-9,     ........................  (14) 


decidedly  smaller  than  that  (4'5  x  10~9)  deduced  from  the  observations  upon 
progressive  waves.  The  conclusion  seems  to  be  that  for  these  frequencies  the 
equilibrium  theory  of  the  telephone-plate  fails,  and  that  in  virtue  of  resonance 
the  sensitiveness  of  the  instrument  is  specially  exalted. 

I  will  not  dwell  further  upon  these  calculations,  which  involve  too  much 
guesswork  to  be  very  satisfactory.  They  suffice,  however,  to  show  that  the 
"push  and  pull"  theory  is  capable  of  giving  an  adequate  account  of  the 
action  of  the  telephone,  so  far  at  least  as  my  own  observations  are  concerned. 
But  it  is  doubtful,  to  say  the  least,  whether  it  could  be  reconciled  with 
estimates  of  sensitiveness  such  as  those  of  Tait  and  of  Preece. 

*  I  hope  shortly  to  publish  an  account  of  the  observations  upon  which  this  statement  is 
founded.     [See  following  Art.  213.] 
t  Supra,  p.  294.     [Vol.  iv.  p.  117.] 


213. 


ON  THE  AMPLITUDE  OF  AERIAL  WAVES    WHICH    ARE   BUT 
JUST  AUDIBLE.  - 

[Philosophical  Magazine,  XXXVIH.  pp.  365—370,  1894*.] 

THE  problem  of  determining  the  .absolute  value  of  the  amplitude,  or 
particle  velocity,  of  a  sound  which  is  but  just  audible  to  the  ear,  is  one  of 
considerable  difficulty.  In  a  short  paper  published  seventeen  years  agof  I 
explained  a  method  by  which  it  was  easy  to  demonstrate  a  superior  limit. 
A  whistle,  blown  under  given  conditions,  consumes  a  known  amount  of 
energy  per  second.  Upon  the  assumption  that  the  whole  of  this  energy 
is  converted  into  sound,  that  the  sound  is  conveyed  without  loss,  and  that 
it  is  uniformly  distributed  over  the  surface  of  a  hemisphere,  it  is  easy  to 
calculate  the  amplitude  at  any  distance;  and  the  result  is  necessarily  a 
superior  limit  to  the  actual  amplitude.  In  the  case  of  the  whistle  experi- 
mented on,  of  frequency  2730,  the  superior  limit  so  arrived  at  for  a  sound 
just  easily  audible  was  8'1  x  10~8  cm.  The  maximum  particle  velocity  v  and 
the  maximum  condensation  s  are  the  quantities  more  immediately  determined 
by  the  observations,  and  they  are  related  by  the  well-known  equation  v  =  as, 
in  which  a  denotes  the  velocity  of  propagation.  In  the  experiment  above 
referred  to  the  superior  limit  for  v  was  '0014  cm.  per  second,  and  that  for  s 
was  4*1  x  10~8.  I  estimated  that  on  a  still  night  an  amplitude,  or  velocity, 
one-tenth  of  the  above  would  probably  be  audible.  A  very  similar  number 
has  been  arrived  at  by  Wien|,  who  used  an  entirely  different  method§. 

In  connexion  with  calculations  respecting  the  sensitiveness  of  telephones, 
I  was  desirous  of  checking  the  above  estimates,  and  made  some  attempts 
to  do  so  by  the  former  method.  In  order  to  avoid  possible  complications  of 

*  Bead  at  the  Oxford  Meeting  of  the  British  Association. 

+  Proc.  Roy.  Soc.  Vol.  xxvi.  p.  248  (1878).     [Vol.  i.  p.  328.] 

J  Wied.  Ann.  xxxvi.  p.  834  (1889). 

§  The  first  estimate  of  the  amplitude  of  but  just  audible  sounds,  with  which  I  have  only 
recently  become  acquainted,  is  that  of  Topler  and  Boltzmann  (Pogg.  Ann.  CXLI.  p.  321  (1870)). 
It  depends  upon  an  ingenious  application  of  v.  Helmholtz's  theory  of  the  open  organ-pipe  to  data 
relating  to  the  maximum  condensation  within  the  pipe  as  obtained  by  the  authors  experimentally. 
The  value  of  s  was  found  to  be  6-5  x  10~8  for  a  pitch  of  181.— August  21. 


126  ON  THE  AMPLITUDE   OF   AERIAL   WAVES  [213 

atmospheric  refraction  which  may  occur  when  large  distances  are  in  question, 
I  sought  to  construct  pipes  which  should  generate  sound  of  given  pitch  upon 
a  much  smaller  scale,  but  with  the  usual  economy  of  wind.  In  this  I  did  not 
succeed,  and  it  seems  as  if  there  is  some  obstacle  to  the  desired  reduction  of 
scale. 

The  experiments  here  to  be  recorded  were  conducted  with  tuning-forks. 
A  fork  of  known  dimensions,  vibrating  with  a  known  amplitude,  may  be 
regarded  as  a  store  of  energy  of  which  the  amount  may  readily  be  calculated. 
This  energy  is  gradually  consumed  by  internal  friction  and  by  generation 
of  sound.  When  a  resonator  is  employed  the  latter  element  is  the  more 
important,  and  in  some  cases  we  may  regard  the  dying  down  of  the  amplitude 
as  sufficiently  accounted  for  by  the  emission  of  sound.  Adopting  this  view 
for  the  present,  we  may  deduce  the  rate  of  emission  of  sonorous  energy  from 
the  observed  amplitude  of  the  fork  at  the  moment  in  question  and  from  the 
rate  at  which  the  amplitude  decreases.  Thus  if  the  law  of  decrease  be  erW* 
for  the  amplitude  of  the  fork,  or  e~kt  for  the  energy,  and  if  E  be  the  total 
energy  at  time  t,  the  rate  at  which  energy  is  emitted  at  that  time  is  —dE/dt, 
or  kE.  The  value  of  k  is  deducible  from  observations  of  the  rate  of  decay, 
e.g.  of  the  time  during  which  the  amplitude  is  halved.  With  these  arrange- 
ments there  is  no  difficulty  in  converting  energy  into  sound  upon  a  small 
scale,  and  thus  in  reducing  the  distance  of  audibility  to  such  a  figure  as 
30  metres.  Under  these  circumstances  the  observations  are  much  more 
manageable  than  when  the  operators  are  separated  by  half  a  mile,  and  there 
is  no  reason  to  fear  disturbance  from  atmospheric  refraction. 

The  fork  is  mounted  upon  a  stand  to  which  is  also  firmly  attached  the 
observing-microscope.  Suitable  points  of  light  are  obtained  from  starch 
grains,  and  the  line  of  light  into  which  each  point  is  extended  by  the 
vibration  is  determined  with  the  aid  of  an  eyepiece-micrometer.  Each 
division  of  the  micrometer-scale  represents  '001  centim.  The  resonator, 
when  in  use,  is  situated  in  the  position  of  maximum  effect,  with  its  mouth 
under  the  free  ends  of  the  vibrating  prongs. 

The  course  of  an  experiment  was  as  follows : — In  the  first  place  the  rates 
of  dying  down  were  observed,  with  and  without  the  resonator,  the  stand  being 
situated  upon  the  ground  in  the  middle  of  a  lawn.  The  fork  was  set  in 
vibration  with  a  bow,  and  the  time  required  for  the  double  amplitude  to  fall 
to  half  its  original  value  was  determined.  Thus  in  the  case  of  a  fork  of 
frequency  256,  the  time  during  which  the  vibration  fell  from  20  micrometer- 
divisions  to  10  micrometer-divisions  was  16s  without  the  resonator,  and  9s 
when  the  resonator  was  in  position.  These  times  of  halving  were,  as  far  as 
could  be  observed,  independent  of  the  initial  amplitude.  To  determine  the 
minimum  audible,  one  observer  (myself)  took  up  a  position  30  yards  (27'4 
metres)  from  the  fork,  and  a  second  (Mr  Gordon)  communicated  a  large 


1894]  WHICH   ARE   BUT   JUST   AUDIBLE.  127 

vibration  to  the  fork.  At  the  moment  when  the  double  amplitude  measured 
20  micrometer-divisions  the  second  observer  gave  a  signal,  and  immediately 
afterwards  withdrew  to  a  distance.  The  business  of  the  first  observer  was 
to  estimate  for  how  many  seconds  after  the  signal  the  sound  still  remained 
audible.  In  the  case  referred  to  the  time  was  12s.  When  the  distance  was 
reduced  to  15  yards  (13*7  metres),  an  initial  double  amplitude  of  10  micro- 
meter-divisions was  audible  for  almost  exactly  the  same  time. 

These  estimates  of  audibility  are  not  made  without  some  difficulty.  There 
are  usually  2  or  3  seconds  during  which  the  observer  is  in  doubt  whether 
he  hears  or  only  imagines,  and  different  individuals  decide  the  question  in 
opposite  ways.  There  is  also  of  course  room  for  a  real  difference  of  hearing, 
but  this  has  not  obtruded  itself  much.  A  given  observer  on  a  given  day  will 
often  agree  with  himself  surprisingly  well,  but  the  accuracy  thus  suggested 
is,  I  think,  illusory.  Much  depends  upon  freedom  from  disturbing  noises. 
The  wind  in  the  trees  or  the  twittering  of  birds  embarrasses  the  observer, 
and  interferes  more  or  less  with  the  accuracy  of  results. 

The  equality  of  emission  of  sound  in  various  horizontal  directions  was 
tested,  but  no  difference  could  be  found.  The  sound  issues  almost  entirely 
from  the  resonator,  and  this  may  be  expected  to  act  as  a  simple  source. 

When  the  time  of  audibility  is  regarded  as  known,  it  is  easy  to  deduce 
•the  amplitude  of  the  vibration  of  the  fork  at  the  moment  when  the  sound 
ceases  to  impress  the  observer.  From  this  the  rate  of  emission  of  sonorous 
energy  and  the  amplitude  of  the  aerial  vibration  as  it  reaches  the  observer 
are  to  be  calculated. 

The  first  step  in  the  calculation  is  the  expression  of  the  total  energy 
of  the  fork  as  a  function  of  the  amplitude  of  vibration  measured  at  the 
extremity  of  one  of  the  prongs.  This  problem  is  considered  in  §  164  of 
my  Theory  of  Sound.  If  I  be  the  length,  p  the  density,  and  »  the  sectional 
area  of  a  rod  clamped  at  one  end  and  free  at  the  other,  the  kinetic  energy  T 
is  connected  with  the  displacement  17  at  the  free  end  by  the  equation  (10) 


At  the  moment  of  passage  through  the  position  of  equilibrium  77  =  0  and 
drjjdt  has  its  maximum  value,  the  whole  energy  being  then  kinetic.  The 
maximum  value  of  drjfdt  is  connected  with  the  maximum  value  of  77  by  the 
equation 


so  that  if  we  now  denote  the  double  amplitude  by  2?;,  the  whole  energy  of 
the  vibrating  bar  is 


or  for  the  two  bars  composing  the  fork 

E=%pa>l7r*/T*.(2r))'2,  ...........................  (A) 

where  pwl  is  the  mass  of  each  prong. 


128  ON   THE   AMPLITUDE   OF   AERIAL   WAVES  [213 

The  application  of  (A)  to  the  256-fork,  vibrating  with  a  double  amplitude 
of  20  micrometer-divisions,  is  as  follows.  We  have 

1  =  14-0  cm.,         «0  =  '6xl-l=-66sq.  cm., 

l/r  =  256,        p  =  7'8,         277  =  -050  cm.; 
and  thus 

E  =  4-06  x  103  ergs. 

This  is  the  whole  energy  of  the  fork  when  the  actual  double  amplitude  at 
the  ends  of  the  prongs  is  '050  centim. 

As  has  already  been  shown,  the  energy  lost  per  second  is  kE,  if  the 
amplitude  vary  as  e~*kt.  For  the  present  purpose  k  must  be  regarded  as 
made  up  of  two  parts,  one  kt  representing  the  dissipation  which  occurs  in 
the  absence  of  the  resonator,  the  other  k2  due  to  the  resonator.  It  is  the 
latter  part  only  which  is  effective  towards  the  production  of  sound.  For 
when  the  resonator  is  out  of  use  the  fork  is  practically  silent;  and,  indeed, 
even  if  it  were  worth  while  to  make  a  correction  on  account  of  the  residual 
sound,  its  phase  would  only  accidentally  agree  with  that  of  the  sound  issuing 
from  the  resonator. 

The  values  of  jfc,  and  k  are  conveniently  derived  from  the  times,  ^  and  t, 
during  which  the  amplitude  falls  to  one-half.  Thus 


so  that 

&2  =  2  log,2  .  (l/t  -  I/*,)  =  1-386  (lit  -  I/tJ. 

And  the  energy  converted  into  sound  per  second  is  kzE. 

We  may  now  apply  these  formulae  to  the  case,  already  quoted,  of  the 
256-fork,  for  which  £  =  9,  ^  =  16.  Thus  t.2,  the  time  which  would  be  occupied 
in  halving  the  amplitude  were  the  dissipation  due  entirely  to  the  resonator, 
is  20-6;  and  k,  =  '0674.  Accordingly, 

k*E  =  267  ergs  per  second, 

corresponding  to  a  double  amplitude  represented  by  20  micrometer-divisions. 
In  the  experiment  quoted  the  duration  of  audibility  was  12  seconds,  during 
which  the  amplitude  would  fall  in  the  ratio  212/9  :  1,  and  the  energy  in  the 
ratio  412/9  :  1.  Hence  at  the  moment  when  the  sound  was  just  becoming 
inaudible  the  energy  emitted  as  sound  was  42'1  ergs  per  second*. 

*  It  is  of  interest  to  compare  with  the  energy-emission  of  a  source  of  light.  An  incandescent 
electric-lainp  of  200  candles  absorbs  about  a  horse-power,  or  say  1010  ergs  per  second.  Of  the 
total  radiation  only  about  TJff  part  acts  effectively  upon  the  eye  ;  so  that  radiation  of  suitable 
quality  consuming  5  x  103  ergs  per  second  corresponds  to  a  candle-power.  This  is  about  104  times 
that  emitted  as  sound  by  the  fork  in  the  experiment  described  above.  At  a  distance  of  102  x  30, 
or  3000  metres  the  stream  of  energy  from  the  ideal  candle  would  be  about  equal  to  the  stream  of 
energy  just  audible  to  the  ear.  It  appears  that  the  streams  of  energy  required  to  influence  the 
eye  and  the  ear  are  of  the  same  order  of  magnitude,  a  conclusion  already  drawn  by  Topler  and 
Boltzmann.  —  August  21. 


1894]  WHICH    ARE   BUT  JUST   AUDIBLE.  129 

The  question  now  remains,  What  is  the  corresponding  amplitude  or 
condensation  in  the  progressive  aerial  waves  at  27 '4  metres  from  the  source  ? 
If  we  suppose,  as  in  my  former  calculations,  that  the  ground  reflects  well, 
we  are  to  treat  the  waves  as  hemispherical.  On  the  whole  this  seems  to  be 
the  best  supposition  to  make,  although  the  reflexion  is  doubtless  imperfect. 
The  area  S  covered  at  the  distance  of  the  observer  is  thus  2?r  x  27402  sq. 
centim.,  and  since* 

S .  %apv*  =  8 .  ^ptfs*  =  421, 

2_ 421 

~7rx27402x  -00125  x341003' 

and  s  =  6'0  x  10~*. 

The  condensation  s  is  here  reckoned  in  atmospheres;  and  the  result  shows 
that  the  ear  is  able  to  recognize  the  addition  and  subtraction  of  densities 
far  less  than  those  to  be  found  in  our  highest  vacua. 

The  amplitude  of  aerial  vibration  is  given  by  asrf^Tr,  where  l/r  =  256, 
and  is  thus  equal  to  T27  x  10~7  cm. 

It  is  to  be  observed  that  the  numbers  thus  obtained  are  still  somewhat 
of  the  nature  of  superior  limits,  for  they  depend  upon  the  assumption  that 
all  the  dissipation  due  to  the  resonator  represents  production  of  sound.  This 
may  not  be  strictly  the  case  even  with  the  moderate  amplitudes  here  in 
question,  but  the  uncertainty  under  this  head  is  far  less  than  in  the  case 
of  resonators  or  organ-pipes  caused  to  speak  by  wind.  From  the  nature  of 
the  calculation  by  which  the  amplitude  or  condensation  in  the  aerial  waves 
is  deduced,  a  considerable  loss  of  energy  does  not  largely  influence  the  final 
numbers. 

Similar  experiments  have  been  tried  at  various  times  with  forks  of  pitch 
384  and  512.  The  results  were  not  quite  so  accordant  as  was  at  first  hoped 
might  be  the  case,  but  they  suffice  to  fix  with  some  approximation  the  con- 
densation necessary  for  audibility.  The  mean  results  are  as  follows : — 

c',          frequency  =  256,         s  =  6'0  x  10~9, 
gf,  „        =  384,        s  =  4-6  x  10~9, 

c",  „         =512,        s  =  4-6  x  10-9, 

no  reliable  distinction  appearing  between  the  two  last  numbers.     Even  the 
distinction  between  6'0  and  4'6  should  be  accepted  with  reserve ;   so  that  the 
comparison  must  not  be  taken  to  prove  much  more  than  that  the  condensation 
necessary  for  audibility  varies  but  slowly  in  the  singly  dashed  octave. 
*  Theory  of  Sound,  §  245. 


214. 

ARGON,  A   NEW  CONSTITUENT   OF  THE  ATMOSPHERE* 

BY  LORD  RAYLEIGH,  SEC.  R.S.,  AND  PROFESSOR  WILLIAM  RAMSAY,  F.R.S. 

[Philosophical  Transactions,  186  (A),  pp.  187—241,  1895.] 

"  Modern  discoveries  have  not  been  made  by  large  collections  of  facts,  with  subsequent 
discussion,  separation,  and  resulting  deduction  of  a  truth  thus  rendered  perceptible.  A  few 
facts  have  suggested  an  hypothesis,  which  means  a  supposition,  proper  to  explain  them. 
The  necessary  results  of  this  supposition  are  worked  out,  and  then,  and  not  till  then,  other 
facts  are  examined  to  see  if  their  ulterior  results  are  found  in  Nature." — De  Morgan, 
A  Budget  of  Paradoxes,  Ed.  1872,  p.  55. 

1.     Density  of  Nitrogen  from  Various  Sources. 

IN- a  former  paper-f-  it  has  been  shown  that  nitrogen  extracted  from 
chemical  compounds  is  about  one-half  per  cent,  lighter  than  "  atmospheric 
nitrogen." 

The  mean  numbers  for  the  weights  of  gas  contained  in  the  globe  used 

were  as  follows : — 

grams. 

From  nitric  oxide 2*3001 

From  nitrous  oxide 2'2990 

From  ammonium  nitrite    ....  2'2987 

while  for  "  atmospheric  "  nitrogen  there  was  found — 

By  hot  copper,  1892 2'3103 

By  hot  iron,  1893 2*3100 

By  ferrous  hydrate,  1894  ....  2'3102 

At  the  suggestion  of  Professor  Thorpe,  experiments  were  subsequently 
tried  with  nitrogen  liberated  from  urea  by  the  action  of  sodium  hypobromite. 

*  This  memoir  is  included  in  the  present  collection  by  kind  permission  of  Prof.  Ramsay, 
t  Rayleigh, "  On  an  Anomaly  encountered  in  Determinations  of  the  Density  of  Nitrogen  Gas," 
Proc.  Roy.  Soc.  Vol.  LV.  p.  340,  1894.     [Vol.  iv.  p.  104.] 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  131 

The  carbon  and  hydrogen  of  the  urea  are  supposed  to  be  oxidized  by  the 
reaction  to  CO2  and  H2O,  the  former  of  which  would  be  retained  by  the 
large  excess  of  alkali  employed.  It  was  accordingly  hoped  that  the  gas 
would  require  no  further  purification  than  drying.  If  it  proved  to  be  light, 
it  would  at  any  rate  be  free  from  the  suspicion  of  containing  hydrogen. 

The  hypobromite  was  prepared  from  commercial  materials  in  the  pro- 
portions recommended  for  the  analysis  of  urea — 100  grams,  caustic  soda, 
250  cub.  centims.  water,  and  25  cub.  centims.  of  bromine.  For  our  purpose 
about  one  and  a  half  times  the  above  quantities  were  required.  The  gas 
was  liberated  in  a  bottle  of  about  900  cub.  centims.  capacity,  in  which  a 
vacuum  was  first  established.  The  full  quantity  of  hypobromite  solution 
was  allowed  to  run  in  slowly,  so  that  any  dissolved  gas  might  be  at  once 
disengaged.  The  urea  was  then  fed  in,  at  first  in  a  dilute  condition,  but, 
as  the  pressure  rose,  in  a  10  per  cent,  solution.  The  washing  out  of  the 
apparatus,  being  effected  with  gas  in  a  highly  rarefied  state,  made  but  a  slight 
demand  upon  the  materials.  The  reaction  was  well  under  control,  and  the 
gas  could  be  liberated  as  slowly  as  desired. 

In  the  first  experiment,  the  gas  was  submitted  to  no  other  treatment 
than  slow  passage  through  potash  and  phosphoric  anhydride,  but  it  soon 
became  apparent  that  the  nitrogen  was  contaminated.  The  "  inert  and 
inodorous  "  gas  attacked  vigorously  the  mercury  of  the  Topler  pump,  and  was 
described  as  smelling  like  a  dead  rat.  As  to  the  weight,  it  proved  to  be  in 
excess  even  of  the  weight  of  atmospheric  nitrogen. 

The  corrosion  of  the  mercury  and  the  evil  smell  were  in  great  degree 
obviated  by  passing  the  gas  over  hot  metals.  For  the  fillings  of  June  6, 
9,  13,  the  gas  passed  through  a  short  length  of  tube  containing  copper  in 
the  form  of  fine  wire,  heated  by  a  flat  Bunsen  burner,  then  through  the 
furnace  over  red-hot  iron,  and  back  over  copper  oxide.  On  June  19  the 
furnace  tubes  were  omitted,  the  gas  being  treated  with  the  red-hot  copper 
only.  The  results,  reduced  so  as  to  correspond  with  those  above  quoted, 
were — 

June    6 2-2978 

„       9 2-2987 

„     13 '.     .  2-2982 

„     19 2-2994 

Mean     .     .     .     .  2'2985 

Without  using  heat  it  has  not  been  found  possible  to  prevent  the  cor- 
rosion of  the  mercury.  Even  when  no  urea  is  employed,  and  air  simply 
bubbled  through  the  hypobromite  solution  is  allowed  to  pass  with  constant 
shaking  over  mercury  contained  in  a  U-tube,  the  surface  of  the  metal  was 
soon  fouled.  When  hypochlorite  was  substituted  for  hypobromite  in  the  last 

9-2 


132  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

experiment  there  was  a  decided  improvement,  and  it  was  thought  desirable 
to  try  whether  the  gas  prepared  from  hypochlorite  and  urea  would  be  pure 
on  simple  desiccation.  A  filling  on  June  25  gave  as  the  weight  2-3343, 
showing  an  excess  of  36  mgs.,  as  compared  with  other  chemical  nitrogen, 
and  of  about  25  mgs.  as  compared  with  atmospheric  nitrogen.  A  test  with 
alkaline  pyrogallate  appeared  to  prove  the  absence  from  this  gas  of  free 
oxygen,  and  only  a  trace  of  carbon  could  be  detected  when  a  considerable 
quantity  of  the  gas  was  passed  over  red-hot  cupric  oxide  into  solution  of 
baryta. 

Although  the  results  relating  to  urea  nitrogen  are  interesting  for  com- 
parison with  that  obtained  from  other  nitrogen  compounds,  the  original 
object  was  not  attained  on  account  of  the  necessity  of  retaining  the  treatment 
with  hot  metals.  We  have  found,  however,  that  nitrogen  from  ammonium 
nitrite  may  be  prepared  without  the  employment  of  hot  tubes,  whose  weight 
agrees  with  that  above  quoted.  It  is  true  that  the  gas  smells  slightly  of 
ammonia,  easily  removable  by  sulphuric  acid,  and  apparently  also  of  oxides 
of  nitrogen.  The  solution  of  potassium  nitrite  and  ammonium  chloride  was 
heated  in  a  water-bath,  of  which  the  temperature  rose  to  the  boiling-point 
only  towards  the  close  of  operations.  In  the  earlier  stages  the  temperature 
required  careful  watching  in  order  to  prevent  the  decomposition  taking  place 
too  rapidly.  The  gas  was  washed  with  sulphuric  acid,  and  after  passing  a 
Nessler  test,  was  finally  treated  with  potash  and  phosphoric  anhydride  in  the 
usual  way.  The  following  results  have  been  obtained : — 

July    4 2-2983 

„      9     .......  2-2989 

13  .  2-2990 


Mean     ....  2'2987 

It  will  be  seen  that  in  spite  of  the  slight  nitrous  smell  there  is  no  appreciable 
difference  in  the  densities  of  gas  prepared  from  ammonium  nitrite  with  and 
without  the  treatment  by  hot  metals.  The  result  is  interesting,  as  showing 
that  the  agreement  of  numbers  obtained  for  chemical  nitrogen  does  not 
depend  upon  the  use  of  a  red  heat  in  the  process  of  purification. 

The  five  results  obtained  in  more  or  less  distinct   ways   for   chemical 
nitrogen  stand  thus: — 

From  nitric  oxide 2'3001 

From  nitrous  oxide 2-2990 

From  ammonium  nitrite  purified  at  a  red  heat  .     .     .  2'2987 

From  urea 2'2985 

From  ammonium  nitrite  purified  in  the  cold      .     .     .  2'2987 
Mean     .     ,  .  2'2990 


1895]  ARGON,   A   NEW   CONSTITUENT  OF   THE   ATMOSPHERE.  133 

These  numbers,  as  well  as  those  above  quoted  for  "  atmospheric  nitrogen," 
are  subject  to  a  correction  (additive)*  of  '0006  for  the  shrinkage  of  the  globe 
when  exhausted -f-.  If  they  are  then  multiplied  in  the  ratio  of  2'3108  :  1-2572, 
they  will  express  the  weights  of  the  gas  in  grams,  per  litre.  Thus,  as  regards 
.the  mean  numbers,  we  find  as  the  weight  per  litre  under  standard  conditions 
of  chemical  nitrogen  1-2511,  that  of  atmospheric  nitrogen  being  1'2572. 

It  is  of  interest  to  compare  the  density  of  nitrogen  obtained  from  chemical 
compounds  with  that  of  oxygen.  We  have  N2 :  02  =  2'2996  :  2'6276  =  O87517  ; 
so  that  if  02  =  16,  N2  =  14'003.  Thus,  when  the  comparison  is  with  chemical 
nitrogen,  the  ratio  is  very  nearly  that  of  16  : 14.  But  if  "  atmospheric  nitro- 
gen "  be  substituted,  the  ratio  of  small  integers  is  widely  departed  from. 

The  determination  by  Stas  of  the  atomic  weight  of  nitrogen  from  synthesis 
of  silver  nitrate  is  probably  the  most  trustworthy,  inasmuch  as  the  atomic 
weight  of  silver  was  determined  with  reference  to  oxygen  with  the  greatest 
care,  and  oxygen  is  assumed  to  have  the  atomic  weight  16.  If,  as  found  by 
Stas,  AgN03 :  Ag  =  1-57490  : 1,  and  Ag  :  0  =  107'930  :  16,  then 

N  :  O  =  14-049  : 16. 

To  the  above  list  may  be  added  nitrogen,  prepared  in  yet  another  manner, 
whose  weight  has  been  determined  subsequently  to  the  isolation  of  the  new 
dense  constituent  of  the  atmosphere.  In  this  case  nitrogen  was  actually 
extracted  from  air  by  means  of  magnesium.  The  nitrogen  thus  separated 
was  then  converted  into  ammonia  by  action  of  water  upon  the  magnesium 
nitride,  and  afterwards  liberated  in  the  free  state  by  means  of  calcium  hypo- 
chlorite.  The  purification  was  conducted  in  the  usual  way,  and  included 
passage  over  red-hot  copper  and  copper  oxide.  The  following  was  the 
result : — 

Globe  empty,  October  30,  November  5      .     .  2-82313 
Globe  full,  October  31 "52395 

Weight  of  gas 2-29918 

It  differs  inappreciably  from  the  mean  of  other  results,  viz.,  2'2990,  and  is 
of  special  interest  as  relating  to  gas  which,  at  one  stage  of  its  history,  formed 
part  of  the  atmosphere. 

Another  determination  with  a  different  apparatus  of  the  density  of 
"  chemical "  nitrogen  from  the  same  source,  magnesium  nitride,  which  had 
been  prepared  by  passing  "  atmospheric  "  nitrogen  over  ignited  magnesium, 
may  here  be  recorded.  The  sample  differed  from  that  previously  mentioned, 
inasmuch  as  it  had  not  been  subjected  to  treatment  with  red-hot  copper. 

[*  In  the  Abstract  of  this  paper  (Proc.  Roy.  Soc.  Vol.  LVII.  p.  265)  the  correction  of  -0006  was 
erroneously  treated  as  a  deduction. — April,  1895.] 

t  Rayleigh,  "  On  the  Densities  of  the  Principal  Gases,"  Proc.  Roy.  Soc.  Vol.  Lin.  p.  134,  1893. 
[Vol.  iv.  p.  39.] 


134  ARGON,   A    NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

After  treating  the  nitride  with  water,  the  resulting  ammonia  was  distilled 
off,  and  collected  in  hydrochloric  acid ;  the  solution  was  evaporated  to  dry- 
ness  ;  the  dry  ammonium  chloride  was  dissolved  in  water,  and  its  concentrated 
solution  added  to  a  freshly  prepared  solution  of  sodium  hypobromite.  The 
nitrogen  was  collected  in  a  gas-holder  over  water  which  had  previously  been 
boiled,  so  as  at  all  events  partially  to  expel  air.  The  nitrogen  passed  into 
the  vacuous  globe  through  a  solution  of  potassium  hydroxide,  and  through 
two  drying-tubes,  one  containing  soda-lime,  and  the  other  phosphoric  an- 
hydride. 

At  18'38°  C.  and  754'4  mgs.  pressure,  162'843  cub.  centims.  of  this  nitrogen 
weighed  0-18963  gram.  Hence:— 

Weight  of  1  litre  at  0°C.  and  760  millims.  pressure  ...  T2521  gram. 

The  mean  result  of  the  weight  of  1  litre  of  "chemical"  nitrogen  has 
been  found  to  equal  1'2511.  It  is  therefore  seen  that  "chemical"  nitrogen, 
derived  from  "atmospheric"  nitrogen,  without  any  exposure  to  red-hot 
copper,  possesses  the  usual  density. 

Experiments  were  also  made,  which  had  for  their  object  to  prove  that  the 
ammonia,  produced  from  the  magnesium  nitride,  is  identical  with  ordinary 
ammonia,  and  contains  no  other  compound  of  a  basic  character.  For  this 
purpose,  the  ammonia  was  converted  into  ammonium  chloride,  and  the 
percentage  of  chloride  determined  by  titration  with  a  solution  of  silver 
nitrate  which  had  been  standardized  by  titrating  a  specimen  of  pure 
sublimed  ammonium  chloride.  The  silver  solution  was  of  such  a  strength 
that  1  cub.  centim.  precipitated  the  chlorine  from  O'OOITOI  gram,  of  am- 
monium chloride. 

1.  Ammonium   chloride    from   orange-coloured   sample    of    magnesium 
nitride. 

0'1106  gram,  required  43'10  cub.  centims.  of  silver  nitrate  =  66'35  per 
cent,  of  chlorine. 

2.  Ammonium  chloride  from  blackish  magnesium  nitride. 

O'lllS  gram,  required  43'6  cub.  centims.  of  silver  nitrate  =  66'35  per 
cent,  of  chlorine. 

3.  Ammonium    chloride    from    nitride   containing   a   large   amount   of 
unattacked  magnesium. 

0'0630  gram,  required  24'55  cub.  centims.  of  silver  nitrate  =  66'30  per 
cent,  of  chlorine. 

Taking  for  the  atomic  weights  of  hydrogen,  H  =  T0032,  of  nitrogen, 
N  =  14-04,  and  of  chlorine,  Cl  =  35'46,  the  theoretical  amount  of  chlorine 
in  ammonium  chloride  is  66 '27  per  cent. 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  135 

From  these  results — that  nitrogen  prepared  from  magnesium  nitride 
obtained  by  passing  "  atmospheric "  nitrogen  over  red-hot  magnesium  has 
the  density  of  "  chemical "  nitrogen,  and  that  ammonium  chloride  prepared 
from  magnesium  nitride  contains  practically  the  same  percentage  of  chlorine 
as  pure  ammonium  chloride — it  may  be  concluded  that  red-hot  magnesium 
withdraws  from  "  atmospheric "  nitrogen  no  substance  other  than  nitrogen 
capable  of  forming  a  basic  compound  with  hydrogen. 

In  a  subsequent  part  of  this  paper,  attention  will  again  be  called  to  this 
statement.  (See  addendum,  p.  240.) 

2.     Reasons  for  Suspecting  a  hitherto  Undiscovered  Constituent  in  Air. 

When  the  discrepancy  of  weights  was  first  encountered,  attempts  were 
naturally  made  to  explain  it  by  contamination  with  known  impurities.  Of 
these  the  most  likely  appeared  to  be  hydrogen,  present  in  the  lighter  gas, 
in  spite  of  the  passage  over  red-hot  cupric  oxide.  But,  inasmuch  as  the 
intentional  introduction  of  hydrogen  into  the  heavier  gas,  afterwards  treated 
in  the  same  way  with  cupric  oxide,  had  no  effect  upon  its  weight,  this 
explanation  had  to  be  abandoned;  and,  finally,  it  became  clear  that  the 
difference  could  not  be  accounted  for  by  the  presence  of  any  known  impurity. 
At  this  stage  it  seemed  not  improbable  that  the  lightness  of  the  gas  extracted 
from  chemical  compounds  Avas  to  be  explained  by  partial  dissociation  of 
nitrogen  molecules  N2  into  detached  atoms.  In  order  to  test  this  suggestion, 
both  kinds  of  gas  were  submitted  to  the  action  of  the  silent  electric  discharge, 
with  the  result  that  both  retained  their  weights  unaltered.  This  was 
discouraging,  and  a  further  experiment  pointed  still  more  markedly  in  the 
negative  direction.  The  chemical  behaviour  of  nitrogen  is  such  as  to  suggest 
that  dissociated  atoms  would  possess  a  higher  degree  of  activity,  and  that, 
even  though  they  might  be  formed  in  the  first  instance,  their  life  would 
probably  be  short.  On  standing,  they  might  be  expected  to  disappear,  in 
partial  analogy  with  the  known  behaviour  of  ozone.  With  this  idea  in  view, 
a  sample  of  chemically-prepared  nitrogen  was  stored  for  eight  months.  But, 
at  the  end  of  this  time,  the  density  showed  no  sign  of  increase,  remaining 
exactly  as  at  first*. 

Regarding  it  as  established  that  one  or  other  of  the  gases  must  be  a 
mixture,  containing,  as  the  case  might  be,  an  ingredient  much  heavier  or 
much  lighter  than  ordinary  nitrogen,  we  had  to  consider  the  relative  pro- 
babilities of  the  various  possible  interpretations.  Except  upon  the  already 
discredited  hypothesis  of  dissociation,  it  was  difficult  to  see  how  the  gas  of 
chemical  origin  could  be  a  mixture.  To  suppose  this  would  be  to  admit  two 
kinds  of  nitric  acid,  hardly  reconcilable  with  the  work  of  Stas  and  others 

*  Rayleigh,  Proc.  Bay.  Soc.  Vol.  LV.  p.  344,  1894.    [Vol.  iv.  p.  108.] 


136         ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE.        [214 

upon  the  atomic  weight  of  that  substance.  The  simplest  explanation  in 
many  respects  was  to  admit  the  existence  of  a  second  ingredient  in  air  from 
which  oxygen,  moisture,  and  carbonic  anhydride  had  already  been  removed. 
The  proportional  amount  required  was  not  great.  If  the  density  of  the  sup- 
posed gas  were  double  that  of  nitrogen,  one-half  per  cent,  only  by  volume 
would  be  needed;  or,  if  the  density  were  but  half  as  much  again  as  that 
of  nitrogen,  then  one  per  cent,  would  still  suffice.  But  in  accepting  this 
explanation,  even  provisionally,  we  had  to  face  the  improbability  that  a 
gas  surrounding  us  on  all  sides,  and  present  in  enormous  quantities,  could 
have  remained  so  long  unsuspected. 

The  method  of  most  universal  application  by  which  to  test  whether  a  gas 
is  pure  or  a  mixture  of  components  of  different  densities  is  that  of  diffusion. 
By  this  means  Graham  succeeded  in  effecting  a  partial  separation  of  the 
nitrogen  and  oxygen  of  the  air,  in  spite  of  the  comparatively  small  difference 
of  densities.  If  the  atmosphere  contain  an  unknown  gas  of  anything  like 
the  density  supposed,  it  should  be  possible  to  prove  the  fact  by  operations 
conducted  upon  air  which  had  undergone  atmolysis.  If,  for  example,  the 
parts  least  disposed  to  penetrate  porous  walls  were  retained,  the  "  nitrogen  " 
derived  from  it  by  the  usual  processes  should  be  heavier  than  that  derived 
in  like  manner  from  unprepared  air.  This  experiment,  although  in  view 
from  the  first,  was  not  executed  until  a  later  stage  of  the  inquiry  (§  6),  when 
results  were  obtained  sufficient  of  themselves  to  prove  that  the  atmosphere 
contains  a  previously  unknown  gas. 

But  although  the  method  of  diffusion  was  capable  of  deciding  the  main, 
or  at  any  rate  the  first  question,  it  held  out  no  prospect  of  isolating  the  new 
constituent  of  the  atmosphere,  and  we  therefore  turned  our  attention  in  the 
first  instance  to  the  consideration  of  methods  more  strictly  chemical.  And 
here  the  question  forced  itself  upon  us  as  to  what  really  was  the  evidence 
in  favour  of  the  prevalent  doctrine  that  the  inert  residue  from  air  after 
withdrawal  of  oxygen,  water,  and  carbonic  anhydride,  is  all  of  one  kind. 

The  identification  of  "  phlogisticated  air  "  with  the  constituent  of  nitric 
acid  is  due  to  Cavendish,  whose  method  consisted  in  operating  with  electric 
sparks  upon  a  short  column  of  gas  confined  with  potash  over  mercury  at 
the  upper  end  of  an  inverted  U-tube*.  This  tube  (M)  was  only  about 
^  inch  in  diameter,  and  the  column  of  gas  was  usually  about  1  inch  in 
length.  After  describing  some  preliminary  trials,  Cavendish  proceeds : — 
"  I  introduced  into  the  tube  a  little  soap-lees  (potash),  and  then  let  up  some 
dephlogisticatedf  and  common  air,  mixed  in  the  above-mentioned  proportions 

*  "Experiments  on  Air,"  Phil.  Trans.  Vol.  LXXV.  p.  372,  1785. 

[t  The  explanation  of  combustion  in  Cavendish's  day  was  still  vague.  It  was  generally 
imagined  that  substances  capable  of  burning  contained  an  unknown  principle,  to  which  the  name 
"  phlogiston  "  was  applied,  and  which  escaped  during  combustion.  Thus,  metals  and  hydrogen 
and  other  gases  were  said  to  be  "  phlogisticated  "  if  they  were  capable  of  burning  in  air.  Oxygen 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  137 

which  rising  to  the  top  of  the  tube  M,  divided  the  soap-lees  into  its  two 
legs.  As  fast  as  the  air  was  diminished  by  the  electric  spark,  I  continued 
adding  more  of  the  same  kind,  till  no  further  diminution  took  place :  after 
which  a  little  pure  dephlogisticated  air,  and  after  that  a  little  common  air, 
were  added,  in  order  to  see  whether  the  cessation  of  diminution  was  not 
owing  to  some  imperfection  in  the  proportion  of  the  two  kinds  of  air  to 
each  other;  but  without  effect.  The  soap-lees  being  then  poured  out  of 
the  tube,  and  separated  from  the  quicksilver,  seemed  to  be  perfectly  neutra- 
lised, and  they  did  not  at  all  discolour  paper  tinged  with  the  juice  of  blue 
flowers.  Being  evaporated  to  dryness,  they  left  a  small  quantity  of  salt, 
which  was  evidently  nitre,  as  appeared  by  the  manner  in  which  paper, 
impregnated  with  a  solution  of  it,  burned." 

Attempts  to  repeat  Cavendish's  experiment  in  Cavendish's  manner  have 
only  increased  the  admiration  with  which  we  regard  this  wonderful  investi- 
gation. Working  on  almost  microscopical  quantities  of  material,  and  by 
operations  extending  over  days  and  weeks,  he  thus  established  one  of  the 
most  important  facts  in  chemistry.  And  what  is  still  more  to  the  purpose, 
he  raises  as  distinctly  as  we  could  do,  and  to  a  certain  extent  resolves,  the 
question  above  suggested.  The  passage  is  so  important  that  it  will  be 
desirable  to  quote  .it  at  full  length. 

"  As  far  as  the  experiments  hitherto  published  extend,  we  scarcely  know 
more  of  the  phlogisticated  part  of  our  atmosphere  than  that  it  is  not 
diminished  by  lime-water,  caustic  alkalies,  or  nitrous  air;  that  it  is  unfit 
to  support  fire  or  maintain  life  in  animals ;  and  that  its  specific  gravity  is 
not  much  less  than  that  of  common  air;  so  that,  though  the  nitrous  acid, 
by  being  united  to  phlogiston,  is  converted  into  air  possessed  of  these 
properties,  and  consequently,  though  it  was  reasonable  to  suppose,  that  part 
at  least  of  the  phlogisticated  air  of  the  atmosphere  consists  of  this  acid 
united  to  phlogiston,  yet  it  was  fairly  to  be  doubted  whether  the  whole 
is  of  this  kind,  or  whether  there  are  not  in  reality  many  different  substances 
confounded  together  by  us  under  the  name  of  phlogisticated  air.  .  I  therefore 
made  an  experiment  to  determine  whether  the  whole  of  a  given  portion  of 
the  phlogisticated  air  of  the  atmosphere  could  be  reduced  to  nitrous  acid,  or 
whether  there  was  not  a  part  of  a  different  nature  to  the  rest  which  would 
refuse  to  undergo  that  change.  The  foregoing  experiments  indeed  in  some 
measure  decided  this  point,  as  much  the  greatest  part  of  the  air  let  up  into 
the  tube  lost  its  elasticity;  yet  as  some  remained  unabsorbed  it  did  not 
appear  for  certain  whether  that  was  of  the  same  nature  as  the  rest  or  not. 

being  non-inflammable  was  named  "  dephlogisticated  air,"  and  nitrogen,  because  it  was  incapable 
of  supporting  combustion  or  life  was  named  by  Priestley  "  phlogisticated  air,"  although  up  till 
Cavendish's  time  it  had  not  been  made  to  unite  with  oxygen. 

The  term  used  for  oxygen  by  Cavendish  is  "  dephlogisticated  air,"  and  for  nitrogen,  "  phlogis- 
ticated air."— April,  1895.] 


138  ARGON,   A  NEW   CONSTITUENT  OF  THE   ATMOSPHERE.  [214 

For  this  purpose  I  diminished  a  similar  mixture  of  dephlogisticated  and 
common  air,  in  the  same  manner  as  before,  till  it  was  reduced  to  a  small 
part  of  its  original  bulk.  I  then,  in  order  to  decompound  as  much  as  I  could 
of  the  phlogisticated  air  which  remained  in  the  tube,  added  some  dephlo- 
gisticated air  to  it  and  continued  the  spark  until  no  further  diminution 
took  place.  Having  by  these  means  condensed  as  much  as  I  could  of  the 
phlogisticated  air,  I  let  up  some  solution  of  liver  of  sulphur  to  absorb  the 
dephlogisticated  air ;  after  which  only  a  small  bubble  of  air  remained 
unabsorbed,  which  certainly  was  not  more  than  T|7  of  the  bulk  of  the 
phlogisticated  air  let  up  into  the  tube ;  so  that,  if  there  is  any  part  of  the 
phlogisticated  air  of  our  atmosphere  which  differs  from  the  rest,  and  cannot 
be  reduced  to  nitrous  acid,  we  may  safely  conclude  that  it  is  not  more  than' 
T27  Part  °f  tne  whole." 

Although  Cavendish  was  satisfied  with  his  result,  and  does  not  decide 
whether  the  small  residue  was  genuine,  our  experiments  about  to  be  related 
render  it  not  improbable  that  his  residue  was  really  of  a  different  kind  from 
the  main  bulk  of  the  "  phlogisticated  air,"  and  contained  the  gas  now  called 
argon. 

Cavendish  gives  data*  from  which  it  is  possible  to  determine  the  rate  of 
absorption  of  the  mixed  gases  in  his  experiment.  The  electrical  machine 
used  "  was  one  of  Mr  Nairne's  patent  machines,  the  cylinder  of  which  is 
12£  inches  long  and  7  in  diameter.  A  conductor,  5  feet  long  and  6  inches 
in  diameter,  was  adapted  to  it,  and  the  ball  which  received  the  spark  was 
placed  two  or  three  inches  from  another  ball,  fixed  to  the  end  of  the 
conductor.  Now,  when  the  machine  worked  well,  Mr  Gilpin  supposes  he 
got  about  two  or  three  hundred  sparks  a  minute,  and  the  diminution  of  the 
air  during  the  half  hour  which  he  continued  working  at  a  time  varied  in 
general  from  40  to  120  measures,  but  was  usually  greatest  when  there  was 
most  air  in  the  tube,  provided  the  quantity  was  not  so  great  as  to  prevent 
the  spark  from  passing  readily."  The  "  measure  "  spoken  of  represents  the 
volume  of  one  grain  of  quicksilver,  or  '0048  cub.  centim.,  so  that  an  absorp- 
tion of  one  cub.  centim.  of  mixed  gas  per  hour  was  about  the  most  favourable 
rate.  Of  the  mixed  gas  about  two-fifths  would  be  nitrogen. 

3.     Methods  of  Causing  Free  Nitrogen  to  Combine. 

The  concord  between  the  determinations  of  density  of  nitrogen  obtained 
from  sources  other  than  the  atmosphere,  having  made  it  at  least  probable 
that  some  heavier  gas  exists  in  the  atmosphere,  hitherto  undetected,  it 
became  necessary  to  submit  atmospheric  nitrogen  to  examination,  with  a 
view  of  isolating,  if  possible,  the  unknown  and  overlooked  constituent,  or  it 
might  be  constituents. 

*  Phil.  Trans.  Vol.  LXXVHI.  p.  271,  1788. 


1895]  ARGON,   A   NEW   CONSTITUENT  OF   THE   ATMOSPHERE.  139 

Nitrogen,  however,  is  an  element  which  does  not  easily  enter  into  direct 
combination  with  other  elements;  but  with  certain  elements,  and  under 
certain  conditions,  combination  may  be  induced.  The  elements  which  have 
been  directly  united  to  nitrogen  are  (a)  boron,  (6)  silicon,  (c)  titanium, 
(d)  lithium,  (e)  strontium  and  barium,  (/)  magnesium,  (g)  aluminium, 
(h)  mercury,  (i)  manganese,  (j)  hydrogen,  and  (k)  oxygen,  the  last  two  by 
help  of  an  electrical  discharge. 

(a)  Nitride  of  boron  was  prepared  by  Wohler  and  Deville*  by  heating 
amorphous  boron  to  a  white  heat  in  a  current  of  nitrogen.  Experiments 
were  made  to  test  whether  the  reaction  would  take  place  in  a  tube  of 
difficultly  fusible  glass;  but  it  was  found  that  the  combination  took  place 
at  a  bright  red  heat  to  only  a  small  extent,  and  that  the  boron,  which  had 
been  prepared  by  heating  powdered  boron  oxide  with  magnesium  dust,  was 
only  superficially  attacked.  Boron  is,  therefore,  not  a  convenient  absorbent 
for  nitrogen.  [M.  Moissan  informs  us  that  the  reputation  it  possesses  is 
due  to  the  fact  that  early  experiments  were  made  with  boron  which  had 
been  obtained  by  means  of  sodium,  and  which  probably  contained  a  boride 
of  that  metal— April,  1895.] 

(6)  Nitride  of  silicon^  also  requires  for  its  formation  a  white  heat,  and 
complete  union  is  difficult  to  bring  about.  Moreover,  it  is  not  easy  to  obtain 
large  quantities  of  silicon.  This  method  was  therefore  not  attempted. 

(c)  Nitride  of  titanium  is  said  to  have  been  formed  by  Deville  and 
CaronJ,  by  heating  titanium  to  whiteness  in  a  current  of  nitrogen.  This 
process  was  not  tried  by  us.  As  titanium  has  an  unusual  tendency  to  unite 
with  nitrogen,  it  might,  perhaps,  be  worth  while  to  set  the  element  free  in 
presence  of  atmospheric  nitrogen,  with  a  view  to  the  absorption  of  the 
nitrogen.  This  has,  in  effect,  been  already  done  by  Wohler  and  Deville§; 
they  passed  a  mixture  of  the  vapour  of  titanium  chloride  and  nitrogen  over 
red-hot  aluminium,  and  obtained  a  large  yield  of  nitride.  It  is  possible  that 
a  mixture  of  the  precipitated  oxide  of  titanium  with  magnesium  dust  might 
be  an  effective  absorbing  agent  at  a  comparatively  low  temperature.  [Since 
writing  the  above  we  have  been  informed  by  M.  Moissan  that  titanium, 
heated  to  800°,  burns  brilliantly  in  a  current  of  nitrogen.  It  might  there- 
fore be  used  with  advantage  to  remove  nitrogen  from  air,  inasmuch  as  we 
have  found  that  it  does  not  combine  with  argon. — April,  1895.] 

(d),  (e)  Lithium  at  a  dull  red  heat  absorbs  nitrogen ||,  but  the  difficulty 
of  obtaining  the  metal  in  quantity  precludes  its  application.  On  the  other 

*  Annales  de  Chimie,  (3),  LIT.  p.  82. 
t  Schutzenberger,  Comptes  Eendus,  LXXXIX.  644. 
J  Annalen  der  Chemie  u.  Pharmacie,  ci.  360. 
§  Annalen  der  Chemie  w.  Pharmacie,  LXXIII.  34. 
||   Ouvrard,  Comptes  Eendus,  cxiv.  120. 


140  ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE.  [214 

hand,  strontium  and  barium,  prepared  by  electrolysing  solutions  of  their 
chlorides  in  contact  with  mercury,  and  subsequently  removing  the  mercury 
by  distillation,  are  said  by  Maquenne*  to  absorb  nitrogen  with  readiness. 
Although  we  have  not  tried  these  metals  for  removing  nitrogen,  still  our 
experience  with  their  amalgams  has  led  us  to  doubt  their  efficacy,  for  it  is 
extremely  difficult  to  free  them  from  mercury  by  distillation,  and  the  product 
is  a  fused  ingot,  exposing  very  little  surface  to  the  action  of  the  gas.  The 
process  might,  however,  be  worth  a  trial. 

Barium  is  the  efficient  absorbent  for  nitrogen  when  a  mixture  of  barium 
carbonate  and  carbon  is  ignited  in  a  current  of  nitrogen,  yielding  cyanide. 
Experiments  have  shown,  however,  that  the  formation  of  cyanides  takes 
place  much  more  readily  and  abundantly  at  a  high  temperature,  a  tempe- 
rature not  easily  reached  with  laboratory  appliances.  Should  the  process 
ever  come  to  be  worked  on  a  large  scale,  the  gas  rejected  by  the  barium  will 
undoubtedly  prove  a  most  convenient  source  of  argon. 

(/)  Nitride  of  magnesium  was  prepared  by  Deville  and  Caron  (loc.  tit.) 
during  the  distillation  of  impure  magnesium.  It  has  been  more  carefully 
investigated  by  Briegleb  and  Geuther-f,  who  obtained  it  by  igniting  metallic 
magnesium  in  a  current  of  nitrogen.  It  forms  an  orange-brown,  friable 
substance,  very  porous,  and  it  is  easily  produced  at  a  bright  red  heat.  When 
magnesium,  preferably  in  the  form  of  thin  turnings,  is  heated  in  a  combustion 
tube  in  a  current  of  nitrogen,  the  tube  is  attacked  superficially,  a  coating 
of  magnesium  silicide  being  formed.  As  the  temperature  rises  to  bright 
redness,  the  magnesium  begins  to  glow  brightly,  and  combustion  takes  place, 
beginning  at  that  end  of  the  tube  through  which  the  gas  is  introduced. 
The  combustion  proceeds  regularly,  the  glow  extending  down  the  tube,  until 
all  the  metal  has  united  with  nitrogen.  The  heat  developed  by  the  combi- 
nation is  considerable,  and  the  glass  softens;  but  by  careful  attention  and 
regulation  of  the  rate  of  the  current,  the  tube  lasts  out  an  operation.  A 
piece  of  combustion  tubing  of  the  usual  length  for  organic  analysis  packed 
tightly  with  magnesium  turnings,  and  containing  about  30  grams.,  absorbs 
between  seven  and  eight  litres  of  nitrogen.  It  is  essential  that  oxygen  be 
excluded  from  the  tube,  otherwise  a  fusible  substance  is  produced,  possibly 
nitrate,  which  blocks  the  tube.  With  the  precaution  of  excluding  oxygen, 
the  nitride  is  loose  and  porous,  and  can  easily  be  removed  from  the  tube  with 
a  rod ;  but  it  is  not  possible  to  use  a  tube  twice,  for  the  glass  is  generally 
softened  and  deformed. 

(<7)  Nitride  of  aluminium  has  been  investigated  by  Mallet  J.  He  ob- 
tained it  in  crystals  by  heating  the  metal  to  whiteness  in  a  carbon  crucible. 

*  Ouvrard,  Comptes  Bendus,  cxiv.  25,  and  220. 
t  Annalen  der  Chemie  u.  Pharmacie,  cxxm.  228. 
$  Journ.  Chem.  Soc.  1876,  Vol.  u.  p.  349. 


1895]  ARGON,   A    NEW   CONSTITUENT  OF   THE   ATMOSPHERE.  141 

But  aluminium  shows  no  tendency  to  unite  with  nitrogen  at  a   red   heat, 
and  cannot  be  used  as  an  absorbent  for  the  gas. 

(k)  Gerresheim  *  states  that  he  has  induced  combination  between  nitrogen 
and  mercury;  but  the  affinity  between  these  elements  is  of  the  slightest,  for 
the  compound  is  explosive. 

(i)  In  addition  to  these,  metallic  manganese  in  a  finely  divided  state  has 
been  shown  to  absorb  nitrogen  at  a  not  very  elevated  temperature,  forming 
a  nitride  of  the  formula  Mn5N2f. 

(j)  [A  mixture  of  nitrogen  with  hydrogen,  standing  over  acid,  is  absorbed 
at  a  fair  rate  under  the  influence  of  electric  sparks.  But  with  an  apparatus 
such  as  that  shown  in  Fig.  1,  the  efficiency  is  but  a  fraction  (perhaps  ^)  of 
that  obtainable  when  oxygen  is  substituted  for  hydrogen  and  alkali  for  acid. 
—April,  1895.] 


4.     Early  Experiments  on  sparking  Nitrogen  with  Oxygen  in  presence  of 

Alkali.  ' 

In  our  earliest  attempts  to  isolate  the  suspected  gas  by  the  method  of 
Cavendish,  we  used  a  Ruhmkorff  coil  of  medium  size  actuated  by  a  battery 
of  five  Grove  cells.  The  gases  were  contained  in  a  test-tube  A,  Fig.  1, 
standing  over  a  large  quantity  of  weak  alkali  B,  and  the  current  was  con- 
veyed in  wires  insulated  by  U-shaped  glass  tubes  CC  passing  through  the 
liquid  round  the  mouth  of  the  test-tube.  The  inner  platinum  ends  DD  of 
the  wires  were  sealed  into  the  glass  insulating  tubes,  but  reliance  was  not 
placed  upon  these  sealings.  In  order  to  secure  tightness  in  spite  of  cracks, 
mercury  was  placed  in  the  bends.  This  disposition  of  the  electrodes  compli- 
cates the  apparatus  somewhat  and  entails  the  use  of  a  large  depth  of  liquid 
in  order  to  render  possible  the  withdrawal  of  the  tubes,  but  it  has  the  great 
advantage  of  dispensing  with  sealing  electrodes  of  platinum  into  the  prin- 
cipal vessel,  which  might  give  way  and  cause  the  loss  of  the  experiment  at 
the  most  inconvenient  moment.  With  the  given  battery  and  coil  a  some- 
what short  spark,  or  arc,  of  about  5  millims.  was  found  to  be  more  favourable 
than  a  longer  one.  When  the  mixed  gases  were  in  the  right  proportion,  the 
rate  of  absorption  was  about  30  cub.  centims.  per  hour,  or  30  times  as  fast 
as  Cavendish  could  work  with  the  electrical  machine  of  his  day. 

To  take  an  example,  one  experiment  of  this  kind  started  with  50  cub. 
centims.  of  air.  To  this,  oxygen  was  gradually  added  until,  oxygen  being  in 
excess,  there  was  no  perceptible  contraction  during  an  hour's  sparking.  The 
remaining  gas  was  then  transferred  at  the  pneumatic  trough  to  a  small 

*  Annalen  der  Chemie  u.  Pharmacie,  cxcv.  373. 
t  0.  Prehlinger,  Monatsh.f.  Chemie,  xv.  391. 


142 


ARGON,   A   NEW   CONSTITUENT   OF   THE    ATMOSPHERE. 


[214 


measuring  vessel,  sealed  by  mercury,  in  which  the  volume  was  found  to  be 
TO  cub.  centim.  On  treatment  with  alkaline  pyrogallate,  the  gas  shrank 
to  "32  cub.  centim.  That  this  small  residue  could  not  be  nitrogen  was 
argued  from  the  fact  that  it  had  withstood  the  prolonged  action  of  the 
spark,  although  mixed  with  oxygen  in  nearly  the  most  favourable  proportion. 

Fig.  1. 


The  residue  was  then  transferred  to  the  test-tube  with  an  addition  of 
another  50  cub.  centims.  of  air,  and  the  whole  worked  up  with  oxygen  as 
before.  The  residue  was  now  2'2  cub.  centims.,  and,  after  removal  of  oxygen, 
'76  cub.  centim. 


1895]       ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE.         143 

Although  it  seemed  almost  impossible  that  these  residues  could  be  either 
nitrogen  or  hydrogen,  some  anxiety  was  not  unnatural,  seeing  that  the  final 
sparking  took  place  under  somewhat  abnormal  conditions.  The  space  was 
very  restricted,  and  the  temperature  (and  with  it  the  proportion  of  aqueous 
vapour)  was  unduly  high.  But  any  doubts  that  were  felt  upon  this  score 
were  removed  by  comparison  experiments  in  which  the  whole  quantity  of 
air  operated  on  was  very  small.  Thus,  when  a  mixture  of  5  cub.  centims.  of 
air  with  7  cub.  centims.  of  oxygen  was  sparked  for  one  hour  and  a  quarter, 
the  residue  was  '47  cub.  centim.,  and,  after  removal  of  oxygen,  '06  cub.  centim. 
Several  repetitions  having  given  similar  results,  it  became  clear  that  the  final 
residue  did  not  depend  upon  anything  that  might  happen  when  sparks  passed 
through  a  greatly  reduced  volume,  but  was  in  proportion  to  the  amount  of  air 
operated  upon. 

No  satisfactory  examination  of  the  residue  which  refused  to  be  oxidised 
could  be  made  without  the  accumulation  of  a  larger  quantity.  This,  however, 
was  difficult  of  attainment  at  the  time  in  question.  The  gas  seemed  to  rebel 
against  the  law  of  addition.  It  was  thought  that  the  cause  probably  lay  in 
the  solubility  of  the  gas  in  water,  a  suspicion  since  confirmed.  At  length, 
however,  a  sufficiency  was  collected  to  allow  of  sparking  in  a  specially  con- 
structed tube,  when  a  comparison  with  the  air  spectrum  taken  under  similar 
conditions  proved  that,  at  any  rate,  the  gas  was  not  nitrogen.  At  first 
scarcely  a  trace  of  the  principal  nitrogen  lines  could  be  seen,  but  after 
standing  over  water  for  an  hour  or  two  these  lines  became  apparent. 

[The  apparatus  shown  in  Fig.  1  has  proved  to  be  convenient  for  the  puri- 
fication of  small  quantities  of  argon,  and  for  determinations  of  the  amount  of 
argon  present  in  various  samples  of  gas,  e.g.,  in  the  gases  expelled  from 
solution  in  water.  To  set  it  in  action  an  alternating  current  is  much  to  be 
preferred  to  a  battery  and  break.  At  the  Royal  Institution  the  primary 
of  a  small  RuhmkorfF  was  fed  from  the  100- volt  alternating  current  supply, 
controlled  by  two  large  incandescent  lamps  in  series  with  the  coil.  With  this 
arrangement  the  voltage  at  the  terminals  of  the  secondary,  available  for 
starting  the  sparks,  was  about  2000,  and  could  be  raised  to  4000  by  plugging 
out  one  of  the  lamps.  With  both  lamps  in  use  the  rate  of  absorption  of 
mixed  gases  was  80  cub.  centims.  per  hour,  and  this  was  about  as  much  as 
could  well  be  carried  out  in  a  test-tube.  Even  with  this  amount  of  power  it 
was  found  better  to  abandon  the  sealings  at  D.  No  inconvenience  arises  from 
the  open  ends,  if  the  tubes  are  wide  enough  to  ensure  the  liberation  of  any 
gas  included  over  the  mercury  when  they  are  sunk  below  the  liquid. 

The  power  actually  expended  upon  the  coil  is  very  small.  When  the 
apparatus  is  at  work  the  current  taken  is  only  2 "4  amperes.  As  regards 
the  voltage,  by  far  the  greater  part  is  consumed  in  the  lamps.  The  efficient 
voltage  at  the  terminals  of  the  primary  coil  is  best  found  indirectly.  Thus,  if 


144 


ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE. 


[214 


A  be  the  current  in  amperes,  V  the  total  voltage,  V1  the  voltage  at  the 
terminals  of  the  coil,  F2  that  at  the  terminals  of  the  lamps,  the  watts 
used  are* 


In  the  present  case  a  Cardew  voltmeter  gave  F=  90£,  F2  =  88;  and 
in  the  formula  may  be  neglected.     Thus, 


=  2*4  x  2'5  =  6'0  approximately. 

The  work  consumed  by  the  coil  when  the  sparks  are  passing  is,  thus,  less 
than  y^  of  a  horse-power  ;  but,  in  designing  an  apparatus,  it  must  further  be 
remembered  that  in  order  to  maintain  the  arc,  a  pretty  high  voltage  is 
required  at  the  terminals  of  the  secondary  when  no  current  is  passing  in  it.  — 
April,  1895.] 


5.     Early  Experiments  on  Withdrawal  of  Nitrogen  from  Air  by 
means  of  Red-hot  Magnesium. 

It  having  been  proved  that  nitrogen,  at  a  bright  red  heat,  was  easily 
absorbed  by  magnesium,  best  in  the  form  of  turnings,  an  attempt  was  success- 
fully made  to  remove  that  gas  from  the  residue  left  after  eliminating  oxygen 
from  air  by  means  of  red-hot  copper. 

Fig.  2. 


The  preliminary  experiment  was  made  in  the  following  manner: — 
A  combustion  tube,  A,  was  filled  with  magnesium  turnings,  packed  tightly 
by  pushing  them  in  with  a  rod.  This  tube  was  connected  with  a  second 
piece  of  combustion  tubing,  B,  by  means  of  thick-walled  india-rubber  tubing, 

*  Ayrton  and  Sumpner,  Proc.  Roy.  Soc.  Vol.  XLIX.  p.  427,  1891. 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  145 

carefully  wired ;  B  contained  copper  oxide,  and,  in  its  turn,  was  connected 
with  the  tube  CD,  one-half  of  which  contained  soda-lime  previously  ignited  to 
expel  moisture,  while  the  other  half  was  filled  with  phosphoric  anhydride. 
E  is  a  measuring  vessel,  and  F  is  a  gas-holder  containing  "atmospheric 
nitrogen." 

In  beginning  an  experiment,  the  tubes  were  heated  with  long-flame 
burners,  and  pumped  empty;  a  little  hydrogen  was  formed  by  the  action  of 
the  moisture  on  the  metallic  magnesium ;  it  was  oxidised  by  the  copper  oxide 
and  absorbed  by  the  phosphoric  pentoxide.  A  gauge  attached  to  the 
Sprengel's  pump,  connected  with  the  apparatus,  showed  when  a  vacuum 
had  been  reached.  A  quantity  of  nitrogen  was  then  measured  in  E,  and 
admitted  into  contact  with  the  red-hot  magnesium.  Absorption  took  place, 
rapidly  at  first  and  then  slowly,  as  shown  by  the  gauge  on  the  Sprengel's 
pump.  A  fresh  quantity  was  then  measured  and  admitted,  and  these 
operations  were  repeated  until  no  more  could  be  absorbed.  The  system  of 
tubes  was  then  pumped  empty  by  means'  of  the  Sprengel's  pump,  and  the 
gas  was  collected.  The  magnesium  tube  was  then  detached  and  replaced 
by  another.  The  unabsorbed  gas  was  returned  to  the  measuring-tube  by  a 
device  shown  in  the  figure  (G)  and  the  absorption  recommenced.  After  1094 
cub.  centims.  of  gas  had  been  thus  treated,  there  was  left  about  50  cub. 
centims.  of  gas,  which  resisted  rapid  absorption.  It  still  contained  nitrogen, 
however,  judging  by  the  diminution  of  volume  which  it  experienced  when 
allowed  to  stand  in  contact  with  red-hot  magnesium.  Its  density  was, 
nevertheless,  determined  by  weighing  a  small  bulb  of  about  40  cub.  centims. 
capacity,  first  with  air,  and  afterwards  with  the  gas.  The  data  are  these  :  — 

grm. 

(a)    Weight  of  bulb  and  air  —  that  of  glass  counterpoise      .     .     0'8094 
„  „      alone  —  that  of  glass  counterpoise     .     .     .     07588 


0-0506 


(6)    Weight  of  bulb  and  gas  —  that  of  glass  counterpoise      .     .     0'8108 
„  „      alone  —  that  of  glass  counterpoise     .     .     .     0'7588 

gas 0-0520 

Taking  as  the  weight  of  a  litre  of  air,  1'29347  grms.,  the  mean  of  the 
latest  results,  and  of  oxygen  (=16)  1'42961  grms.*,  the  density  of  the 
residual  gas  is  14'88. 

This  result  was  encouraging,  although  weighted  with  the  unavoidable 
error  attaching  to  the  weighing  of  a  very  small  amount.  Still  the  fact 
remains  that  the  supposed  nitrogen  was  heavier  than  air.  It  would  hardly 
have  been  possible  to  make  a  mistake  of  2'7  milligrams. 

*  For  note  see  foot  of  p.  146. 
R.     IV.  10 


146 


ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE. 


[214 


It  is  right  here  to  place  on  record  the  fact  that  this  first  experiment 
was  to  a  great  extent  carried  out  by  Mr  Percy  Williams,  to  whose  skill 
in  manipulation  and  great  care  its  success  is  due,  and  to  whom  we  desire 
here  to  express  our  thanks. 

Experiments  were  now  begun  on  a  larger  scale,  the  apparatus  employed 
being  shown  in  Figs.  3  and  4. 

Fig.  3. 


A  and  B  are  large  glass  gas-holders  of  about  10  litres  capacity.  C  is  an 
arrangement  by  which  gas  could  be  introduced  at  will  into  the  gas-holder  A, 
either  by  means  of  an  india-rubber  tube  slipped  over  the  open  end  of  the 
U-tube,  or,  as  shown  in  the  figure,  from  a  test-tube.  The  tube  D  was  half 

*  The  results  on  which  this  and  the  subsequent  calculations  are  based  are  as  follows  (the 
weights  are  those  of  1  litre) : — 


Air 

Oxygen 

Nitrogen 

Hydrogen 

Regnault  

1-29349 

1-43011 

1-25647 

0-08988 

Von  Jolly  

1-29383 

1-42971 

1-25819 

Leduc  

1-29330 

1-42910 

1-25709 

0-08985 

Rayleigh  

1-29327 

1-42952 

1-25718 

0-09001 

Regnault's  numbers  have  an  approximate  correction  applied  to  them  by  Crafts.  The  mean  of 
these  numbers  is  taken,  that  of  Regnault  for  nitrogen  being  omitted,  as  there  is  reason  to  believe 
that  his  specimen  was  contaminated  with  hydrogen. 


Air 

Oxygen 

Nitrogen 

Hydrogen 

1-29347 

1-42961 

1-25749 

0-08991 

This  ratio  gives  for  air  the  composition  by  volume — 

Oxygen 20-91  per  cent. 

Nitrogen 79-09 

a  result  verified  by  experiment. 

It  is,  of  course,  to  be  understood  that  these  densities  of  nitrogen  refer  to  atmospheric  nitrogen, 
that  is,  to  air  from  which  oxygen,  water  vapour  carbon  dioxide,  and  ammonia  have  been  removed. 


1895]  AEGON,   A    NEW   CONSTITUENT  OF   THE    ATMOSPHERE.  147 

filled  with  soda-lime  (a),  half  with  phosphoric  anhydride  (6).  Similarly,  the 
tube  E,  which  was  kept  at  a  red  heat  by  means  of  the  long-flame  burner,  was 
filled  half  with  very  porous  copper  (a),  reduced  from  dusty  oxide  by  heating 
in  hydrogen,  half  with  copper  oxide  in  a  granular  form  (6).  The  next  tube, 
F,  contained  granular  soda-lime,  while  G  contained  magnesium  turnings,  also 
heated  to  bright  redness  by  means  of  a  long-flame  burner.  H  contained 
phosphoric  anhydride,  and  /  soda-lime.  All  joints  were  sealed,  excepting 
those  connecting  the  hard-glass  tubes  E  and  G  to  the  tubes  next  them. 

The  gas-holder  A  having  been  filled  with  nitrogen,  prepared  by  passing  air 
over  red-hot  copper,  and  introduced  at  0,  the  gas  was  slowly  passed  through 
the  system  of  tubes  into  the  gas-holder  B,  and  back  again.  The  magnesium 
in  the  tube  G  having  then  ceased  to  absorb  was  quickly  removed  and 
replaced  by  a  fresh  tube.  This  tube  was  of  course  full  of  air,  and  before  the 
tube  G  was  heated,  the  air  was  carried  back  from  B  towards  A  by  passing  a 
little  nitrogen  from  right  to  left.  The  oxygen  in  the  air  was  removed  by  the 
metallic  copper,  and  the  nitrogen  passed  into  the  gas-holder  A,  to  be  returned 
in  the  opposite  direction  to  B. 

Fig.  4. 


In  the  course  of  about  ten  days  most  of  the  nitrogen  had  been  absorbed. 
The  magnesium  was  not  always  completely  exhausted;  usually  the  nitride 
presented  the  appearance  of  a  blackish-yellow  mass,  easily  shaken  out  of  the 
tube.  It  is  needless  to  say  that  the  tube  was  always  somewhat  attacked, 
becoming  black  with  a  coating  of  magnesium  silicide.  The  nitride  of  mag- 
nesium, whether  blackish  or  orange,  if  left  for  a  few  hours  exposed  to  moist 
air,  was  completely  converted  into  white,  dusty  hydroxide,  and  during 
exposure  it  gave  off  a  strong  odour  of  ammonia.  If  kept  in  a  stoppered 
bottle,  however,  it  was  quite  stable. 

It  was  then  necessary,  in  order  to  continue  the  absorption,  to  carry  on 
operations  on  a  smaller  scale,  with  precautions  to  exclude  atmospheric  air  as 
completely  as  possible.  There  was  at  this  stage  a  residue  of  1500  cub. 

centims. 

10—2 


148  ARGON,   A   NEW   CONSTITUENT   OF  THE   ATMOSPHERE.  [214 

The  apparatus  was  therefore  altered  to  that  shown  in  Fig.  4,  so  as  to  make 
it  possible  to  withdraw  all  the  gas  out  of  the  gas-holder  A. 

The  left-hand  exit  led  to  the  Sprengel's  pump ;  the  compartment  (a)  of 
the  drying-tube  B  was  filled  with  soda-lime,  and  (b)  with  phosphoric  anhydride. 
G  is  a  tube  into  which  the  gas  could  be  drawn  from  the  gas-holder  A.  The 
stop-cock,  as  shown,  allows  gas  to  pass  through  the  horizontal  tubes,  and  does 
not  communicate  with  A  ;  but  a  vertical  groove  allows  it  to  be  placed  in  com- 
munication either  with  the  gas-holder,  or  with  the  apparatus  to  the  right. 
The  compartment  (a)  of  the  second  drying-tube  D  contained  soda-lime,  and 
(b)  phosphoric  anhydride.  The  tube  D  communicated  with  a  hard-glass  tube 
E,  heated  over  a  long-flame  burner ;  it  was  partly  filled  with  metallic  copper, 
and  partly  with  copper  oxide.  This  tube,  as  well  as  the  tube  F  filled  with 
magnesium  turnings,  was  connected  to  the  drying-tube  with  india-rubber. 
The  gas  then  entered  G,  a  graduated  reservoir,  and  the  arrangement  H 
permitted  the  removal  or  introduction  of  gas  from  or  into  the  apparatus.  The 
gas  was  gradually  transferred  from  the  gas-holder  to  the  tube  C,  and  passed 
backwards  and  forwards  over  the  red-hot  magnesium  until  only  about  200  cub. 
centims.  were  left.  It  was  necessary  to  change  the  magnesium  tube,  which 
was  made  of  smaller  size  than  formerly,  several  times  during  the  operation. 
This  was  done  by  turning  out  the  long-flame  burners  and  pumping  off  all  gas 
in  the  horizontal  tubes  by  means  of  the  Sprengel's  pump.  This  gas  was 
carefully  collected.  The  magnesium  tube  was  then  exchanged  for  a  fresh 
one,  and  after  air  had  been  exhausted  from  the  apparatus,  nitrogen  was  intro- 
duced from  the  reservoir.  Any  gas  evolved  from  the  magnesium  (and 
apparently  there  was  always  a  trace  of  hydrogen,  either  occluded  by  the 
magnesium,  or  produced  by  the  action  of  aqueous  vapour  on  the  metal)  was 
oxidised  by  the  copper  oxide.  Had  oxygen  been  present,  it  would  have  been 
absorbed  by  the  metallic  copper,  but  the  copper  preserved  its  red  appearance 
without  alteration,  whereas  a  little  copper  oxide  was  reduced  during  the 
series  of  operations.  The  gas,  which  had  been  removed  by  pumping,  was 
reintroduced  at  H,  and  the  absorption  continued. 

The  volume  of  the  gas  was  thus,  as  has  been  said,  reduced  to  about  200 
cub.  centims.  It  would  have  been  advisable  to  take  exact  measurements,  but, 
unfortunately,  some  of  the  original  nitrogen  had  been  lost  through  leakage ; 
and  a  natural  anxiety  to  see  if  there  was  any  unknown  gas  led  to  pushing  on 
operations  as  quickly  as  possible. 

The  density  of  the  gas  was  next  determined.  The  bulb  or  globe  in  which 
the  gas  was  weighed  was  sealed  to  a  two-way  stop-cock,  and  the  weight  of 
distilled  and  air-free  water  filling  it  at  17'15°  was  162'654  grms.,  correspond- 
ing to  a  capacity  of  162'843  cub.  centims.  The  shrinkage  on  removing  air 
completely  was  0'0212  cub.  centim.  Its  weight,  when  empty,  should  therefore 
be  increased  by  the  weight  of  that  volume  of  air,  which  may  be  taken  as 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  149 

O000026  grm.     This  correction,  however,  is  perhaps  hardly  worth  applying  in 
the  present  case. 

The  counterpoise  was  an  exactly  similar  bulb  of  equal  capacity,  and 
weighing  about  0*2  grm.  heavier  than  the  empty  globe.  The  balance  was  a 
very  sensitive  one  by  Oertling,  which  easily  registered  one-tenth  of  a  milligrm. 
By  the  process  of  swinging,  one-hundredth  of  a  milligrm.  could  be  determined 
with  fair  accuracy. 

In  weighing  the  empty  globe,  0'2  grm.  was  placed  on  the  same  pan  as  that 
which  hung  from  the  end  of  the  beam  to  which  it  was  suspended,  and  the  final 
weight  was  adjusted  by  means  of  a  rider,  or  by  small  weights  on  the  other 
pan.  This  process  practically  leads  to  weighing  by  substitution  of  gas  for 
weights.  The  bulb  was  always  handled  with  gloves,  to  avoid  moisture  or 
grease  from  the  fingers. 

Three  experiments,  of  which  it  is  unnecessary  to  give  details,  were  made 
to  test  the  degree  of  accuracy  with  which  a  gas  could  be.  weighed,  the  gas 
being  dried  air,  freed  from  carbon  dioxide.  The  mean  result  gave  for  the 
weight  of  one  litre  of  air  at  0°  and  760  millims.  pressure,  1*2935  grm. 
Regnault  found  1 '29340,  a  correction  having  been  applied  by  Crafts  to  allow 
for  the  estimated  alteration  of  volume  caused  by  the  contraction  of  his 
vacuous  bulb.  The  mean  result  of  determinations  by  several  observers  is 
1-29347 ;  while  one  of  us  found  1  "29327. 

The  globe  was  then  filled  with  the  carefully  dried  gas. 

Temperature,  18'80°.     Pressure,  759*3  millims. 

Weight  of  162-843  cub.  centims.  of  gas 0'21897  grm. 

Weight  of  1  litre  gas  at  0°  and  760  millims 1-4386 

Density,  that  of  air  compared  with  O,  =  16,  being  14'476  16'100     grms. 

It  is  evident  from  these  numbers  that  the  dense  constituent  of  the  air  was 
being  concentrated.  As  a  check,  the  bulb  was  pumped  empty  and  again 
weighed ;  its  weight  was  0'21903  grm.  This  makes  the  density  16105. 

It  appeared  advisable  to  continue  to  absorb  nitrogen  from  this  gas.  The 
first  tube  of  magnesium  removed  a  considerable  quantity  of  gas ;  the  nitride 
was  converted  into  ammonium  chloride,  and  the  sample  contained  6  6 '30  per 
cent,  of  chlorine,  showing,  as  has  before  been  remarked,  that  if  any  of  the 
heavier  constituent  of  the  atmosphere  had  been  absorbed,  it  formed  no  basic 
compound  with  hydrogen.  The  second  tube  of  magnesium  was  hardly 
attacked;  most  of  the  magnesium  had  melted,  and  formed  a  layer  at  the 
lower  part  of  the  tube.  That  which  was  still  left  in  the  body  of  the  tube  was 
black  on  the  surface,  but  had  evidently  not  been  much  attacked.  The 
ammonium  chloride  which  it  yielded  weighed  only  0'0035  grm. 


150  ARGON,   A    NEW   CONSTITUENT   OF   THE    ATMOSPHERE.  [214 

The  density  of  the  remaining  gas  was  then  determined.  But  as  its 
volume  was  only  a  little  over  100  cub.  centims.,  the  bulb,  the  capacity  of 
which  was  162  cub.  centims.,  had  to  be  filled  at  reduced  pressure.  This  was 
easily  done  by  replacing  the  pear-shaped  reservoir  of  the  mercury  gas-holder 
by  a  straight  tube,  and  noting  the  level  of  the  mercury  in  the  gas-holder  and 
in  the  tube  which  served  as  a  mercury  reservoir  against  a  graduated  mirror- 
scale  by  help  of  a  cathetometer  at  the  moment  of  closing  the  stop-cock  of  the 
density  bulb. 

The  details  of  the  experiment  are  these  : — 

Temperature,  19'12°  C.     Barometric  pressure,  749'8  millims.  (corr.). 
Difference  read  on  gas-holder  and  tube,  225'25  millims.  (corr.). 
Actual  pressure,  524'55  millims. 

Weight  of  162-843  cub.  centims.  of  gas      ....       017913  grm. 
Weight  of  1  litre  at  0°  and  760  millims.  pressure   .       T7054 
Density 19'086      grms. 

This  gas  is  accordingly  at  least  19  times  as  heavy  as  hydrogen. 

A  portion  of  the  gas  was  then  mixed  Avith  oxygen,  and  submitted  to  a 
rapid  discharge  of  sparks  for  four  hours  in  presence  of  caustic  potash.  It 
contracted,  and  on  absorbing  the  excess  of  oxygen  with  pyrogallate  of 
potassium  the  contraction  amounted  to  15'4  per  cent,  of  the  original  volume. 
The  question  then  arises,  if  the  gas  contain  15'4  per  cent,  of  nitrogen,  of 
density  14'014,  and  84'6  per  cent,  of  other  gas,  and  if  the  density  of  the 
mixture  were  19'086,  what  would  be  the  density  of  the  other  gas  ?  Calcula- 
tion leads  to  the  number  20'0. 

A  vacuum-tube  was  filled  with  a  specimen  of  the  gas  of  density  19'086, 
and  it  could  not  be  doubted  that  it  contained  nitrogen,  the  bands  of  which 
were  distinctly  visible.  It  was  probable,  therefore,  that  the  true  density  of 
the  pure  gas  lay  not  far  from  20  times  that  of  hydrogen.  At  the  same  time 
many  lines  were  seen  which  could  not  be  recognised  as  belonging  to  the 
spectrum  of  any  known  substance. 

Such  were  the  preliminary  experiments  made  with  the  aid  of  magnesium 
to  separate  from  atmospheric  nitrogen  its  dense  constituent.  The  methods 
adopted  in  preparing  large  quantities  will  be  subsequently  described. 

6.     Proof  of  the  Presence  of  Argon  in  Air,  by  means  of  Atmolysis. 

It  has  already  (§  2)  been  suggested  that  if  "  atmospheric  nitrogen " 
contains  two  gases  of  different  densities,  it  should  be  possible  to  obtain  direct 
evidence  of  the  fact  by  the  method  of  atmolysis.  The  present  section  contains 
an  account  of  carefully  conducted  experiments  directed  to  this  end. 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  151 

The  atmolyser  was  prepared  (after  Graham)  by  combining  a  number  of 
"  churchwarden "  tobacco  pipes.  At  first  twelve  pipes  were  used  in  three 
groups,  each  group  including  four  pipes  connected  in  series.  The  three 
groups  were  then  connected  in  parallel,  and  placed  in  a  large  glass  tube  closed 
in  such  a  way  that  a  partial  vacuum  could  be  maintained  in  the  space  outside 
the  pipes  by  a  water-pump.  One  end  of  the  combination  of  pipes  was  open 
to  the  atmosphere,  or  rather  was  connected  with  the  interior  of  an  open  bottle 
containing  sticks  of  caustic  alkali,  the  object  being  mainly  to  dry  the  air. 
The  other  end  of  the  combination  was  connected  to  a  bottle  aspirator, 
initially  full  of  water,  and  so  arranged  as  to  draw  about  two  per  cent,  of  the 
air  which  entered  the  other  end  of  the  pipes.  The  gas  collected  was  thus  a 
very  small  proportion  of  that  which  leaked  through  the  pores  of  the  pipes, 
and  should  be  relatively  rich  in  the  heavier  constituents  of  the  atmosphere. 
The  flow  of  water  from  the  aspirator  could  not  be  maintained  very  constant, 
but  the  rate  of  two  per  cent,  was  never  much  exceeded.  The  necessary  four 
litres  took  about  sixteen  hours  to  collect. 

The  air  thus  obtained  was  treated  exactly  as  ordinary  air  had  been  treated 
in  determinations  of  the  density  of  atmospheric  nitrogen.  Oxygen  was  re- 
moved by  red-hot  copper  followed  by  cupric  oxide,  ammonia  by  sulphuric 
acid,  carbonic  anhydride  and  moisture  by  potash  and  phosphoric  anhydride. 

The  following  are  the  results : — 

Globe  empty  July  10,  14 2'81789 

Globe  full  September  15  (twelve  pipes)   .     .        "50286 

Weight  of  gas 2-31503 

Ordinary  atmospheric  nitrogen 2'31016 

Difference +    '00487 

Globe  empty  September  17 2'81345 

Globe  full  September  18  (twelve  pipes)   .     .        '50191 

Weight  of  gas 2-31154 

Ordinary  atmospheric  nitrogen 2'31016 

Difference +    '00138 

Globe  empty  September  21 2'82320 

Globe  full  September  20  (twelve  pipes)   .     .        '51031 

Weight  of  gas 2-31289 

Ordinary  atmospheric  nitrogen 2'31016 

Difference  .     .  +    '00273 


152  ARGON,   A    NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

Globe  empty  September  21,  October  30  .     .      2*82306 
Globe  full  September  22  (twelve  pipes)   .     .        '51140 

Weight  of  gas 2'31166 

Ordinary  atmospheric  nitrogen 2'31016 

Difference     . +    '00150 

The  mean  excess  of  the  four  determinations  is  '00262  gram.,  or  if  we  omit 
the  first,  which  depended  upon  a  vacuum  weighing  of  two  months  old, 
•00187  gram. 

The  gas  from  prepared  air  was  thus  in  every  case  denser  than  from 
unprepared  air,  and  to  an  extent  much  beyond  the  possible  errors  of  experi- 
ment. The  excess  was,  however,  less  than  had  been  expected,  and  it  was 
thought  that  the  arrangement  of  the  pipes  could  be  improved.  The  final 
delivery  of  gas  from  each  of  the  groups  in  parallel  being  so  small  in  comparison 
with  the  whole  streams  concerned,  it  seemed  possible  that  each  group  was  not 
contributing  its  proper  share,  and  even  that  there  might  be  a  flow  in  the 
wrong  direction  at  the  delivery  end  of  one  or  two  of  them.  To  meet  this 
objection,  the  arrangement  in  parallel  had  to  be  abandoned,  and  for  the 
remaining  experiments  eight  pipes  were  connected  in  simple  series.  The 
porous  surface  in  operation  was  thus  reduced,  but  this  was  partly  compensated 
for  by  an  improved  vacuum.  Two  experiments  were  made  under  the  new 
conditions : — 

Globe  empty,  October  30,  November  5     .     .      2'82313 
Globe  full,  November  3  (eight  pipes)  .     .     .        '50930 

Weight  of  gas 2'31383 

Ordinary  atmospheric  nitrogen 2'31016 

Difference +    '00367 

Globe  empty,  November  5,  8 2'82355 

Globe  full,  November  6  (eight  pipes)  .     .     .        '51011 

Weight  of  gas 2'31344 

Ordinary  atmospheric  nitrogen 2'31016 

Difference +    '00328 

The  excess  being  larger  than  before  is  doubtless  due  to  the  greater 
efficiency  of  the  atmolysing  apparatus.  It  should  be  mentioned  that  the 
above  recorded  experiments  include  all  that  have  been  tried,  and  the  con- 
clusion seems  inevitable  that  "  atmospheric  nitrogen  "  is  a  mixture  and  not  a 
simple  body. 

It  was  hoped  that  the  concentration  of  the  heavier  constituent  would  be 
sufficient  to  facilitate  its  preparation  in  a  pure  state  by  the  use  of  prepared 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  153 

air  in  substitution  for  ordinary  air  in  the  oxygen  apparatus.  The  advance  of 
3£  mg.  on  the  11  mg.,  by  which  atmospheric  nitrogen  is  heavier  than  chemical 
nitrogen,  is  indeed  not  to  be  despised,  and  the  use  of  prepared  air  would  be 
convenient  if  the  diffusion  apparatus  could  be  set  up  on  a  large  scale  and 
be  made  thoroughly  self-acting. 

7.     Negative  Experiments  to  prove  that  Argon  is  not  derived  from  Nitrogen  or 
from  Chemical  Sources. 

Although  the  evidence  of  the  existence  of  argon  in  the  atmosphere, 
derived  from  the  comparison  of  densities  of  atmospheric  and  chemical  nitrogen 
and  from  the  diffusion  experiments  (§  6),  appeared  overwhelming,  we  have 
thought  it  undesirable  to  shrink  from  any  labour  that  would  tend  to  complete 
the  verification.  With  this  object  in  view,  an  experiment  was  undertaken 
and  carried  to  a  conclusion  on  November  13,  in  which  3  litres  of  chemical 
nitrogen,  prepared  from  ammonium  nitrite,  were  treated  with  oxygen  in 
precisely  the  manner  in  which  atmospheric  nitrogen  had  been  found  to  yield 
a  residue  of  argon.  In  the  course  of  operations  an  accident  occurred,  by  which 
no  gas  could  have  been  lost,  but  of  such  a  nature  that  from  100  to  200  cub. 
centims.  of  air  must  have  entered  the  working  vessel.  The  gas  remaining  at 
the  close  of  the  large  scale  operations  was  worked  up  as  usual  with  battery 
and  coil  until  the  spectrum  showed  only  slight  traces  of  the  nitrogen  lines. 
When  cold,  the  residue  measured  4  cub.  centims.  This  was  transferred,  and 
after  treatment  with  alkaline  pyrogallate  to  remove  oxygen,  measured  3*3  cub. 
centims.  If  atmospheric  nitrogen  had  been  employed,  the  final  residue  should 
have  been  about  30  cub.  centims.  Of  the  3'3  cub.  centims.  actually  left,  a 
part  is  accounted  for  by  the  accident  alluded  to,  and  the  result  of  the 
experiment  is  to  show  that  argon  is  not  formed  by  sparking  a  mixture  of 
oxygen  and  chemical  nitrogen. 

In  a  second  experiment  of  the  same  kind  5660  cub.  centims.  of  nitrogen 
from  ammonium  nitrite  were  treated  with  oxygen  in  the  large  apparatus 
(Fig.  7,§  8).  The  final  residue  was  3'5  cub.  centims.;  and,  as  evidenced  by  the 
spectrum,  it  consisted  mainly  of  argon. 

The  source  of  the  residual  argon  is  to  be  found  in  the  water  used  for  the 
manipulation  of  the  large  quantities  of  gas  (6  litres  of  nitrogen  and  11  litres 
of  oxygen)  employed.  Unfortunately  the  gases  had  been  collected  by  allowing 
them  to  bubble  up  into  aspirators  charged  with  ordinary  water,  and  they  were 
displaced  by  ordinary  water.  In  order  to  obtain  information  with  respect  to 
the  contamination  that  may  be  acquired  in  this  way,  a  parallel  experiment 
was  tried  with  carbonic  anhydride.  Eleven  litres  of  the  gas,  prepared  from 
marble  and  hydrochloric  acid  with  ordinary  precautions  for  the  exclusion  of 
air,  were  collected  exactly  as  oxygen  was  commonly  collected.  It  was  then 


154  ARGON,   A    NEW   CONSTITUENT  OF   THE   ATMOSPHERE.  [214 

transferred  by  displacement  with  water  to  a  gas  pipette  charged  with  a 
solution  containing  100  grins,  of  caustic  soda.  The  residue  which  refused 
absorption  measured  as  much  as  110  cub.  centims.  In  another  experiment 
where  the  water  employed  had  been  partially  de-aerated,  the  residue  left 
amounted  to  71  cub.  centims.,  of  which  26  cub.  centims.  were  oxygen.  The 
quantities  of  dissolved  gases  thus  extracted  from  water  during  the  collection 
of  oxygen  and  nitrogen  suffice  to  explain  the  residual  argon  of  the  negative 
experiments. 

It  may  perhaps  be  objected  that  the  impurity  was  contained  in  the 
carbonic  anhydride  itself  as  it  issued  from  the  generating  vessel,  and  was 
not  derived  from  the  water  in  the  gas-holder ;  and  indeed  there  seems  to  be 
a  general  impression  that  it  is  difficult  to  obtain  carbonic  anhydride  in  a  state 
of  purity.  To  test  this  question,  18  litres  of  the  gas,  made  in  the  same 
generator  and  from  the  same  materials,  were  passed  directly  into  the  absorp- 
tion pipette.  Under  these  conditions,  the  residue  was  only  6£  cub.  centims., 
corresponding  to  4  cub.  centims.  from  11  litres.  The  quantity  of  gas  employed 
was  determined  by  decomposing  the  resulting  sodium  carbonate  with  hydro- 
chloric acid,  allowance  being  made  for  a  little  carbonic  anhydride  contained 
in  the  soda  as  taken  from  the  stock  bottle.  It  will  be  seen  that  there  is  no 
difficulty  in  reducing  the  impurity  to  ^^th,  even  when  india-rubber  connec- 
tions are  freely  used,  and  no  extraordinary  precautions  are  taken.  The  large 
amount  of  impurity  found  in  the  gas  when  collected  over  water  must  therefore 
have  been  extracted  from  the  water. 


A  similar  set  of  experiments  was  carried  out  with  magnesium.  The 
nitrogen,  of  which  three  litres  were  used,  was  prepared  by  the  action  of 
bleaching-powder  on  ammonium  chloride.  It  was  circulated  in  the  usual 
apparatus  over  red-hot  magnesium,  until  its  volume  had  been  reduced  to 
about  100  cub.  centims.  An  equal  volume  of  hydrogen  was  then  added,  owing 
to  the  impossibility  of  circulating  a  vacuum.  The  circulation  then  proceeded 
until  all  absorption  had  apparently  stopped.  The  remaining  gas  was  then 
passed  over  red-hot  copper  oxide  into  the  Sprengel's  pump,  and  collected.  As 
it  appeared  still  to  contain  hydrogen,  which  had  escaped  oxidation,  owing  to 
its  great  rarefaction,  it  was  passed  over  copper  oxide  for  a  second  and  a  third 
time.  As  there  was  still  a  residue,  measuring  12'5  cub.  centims.,  the  gas  was 
left  in  contact  with  red-hot  magnesium  for  several  hours,  and  then  pumped 
out;  its  volume  was  then  4'5  cub.  centims.  Absorption  was,  however,  still 
proceeding,  when  the  experiment  terminated,  for  at  a  low  pressure,  the  rate  is 
exceedingly  slow.  This  gas,  after  being  sparked  with  oxygen  contracted  to 
3'0  cub.  centims.,  and  on  examination  was  seen  to  consist  mainly  of  argon. 
The  amount  of  residue  obtainable  from  three  litres  of  atmospheric  nitrogen 
should  have  amounted  to  a  large  multiple  of  this  quantity. 


1895]  ARGON,    A    NEW   CONSTITUENT  OF   THE   ATMOSPHERE.  155 

In  another  experiment,  15  litres  of  nitrogen  prepared  from  a  mixture  of 
ammonium  chloride  and  sodium  nitrite  by  warming  in  a  flask  (some  nitrogen 
having  first  been  drawn  off  by  a  vacuum-pump,  in  order  to  expel  all  air  from 
the  flask  and  from  the  contained  liquid)  were  collected  over  water  in  a  large 
gas-holder.  The  nitrogen  was  not  bubbled  through  the  water,  but  was 
admitted  from  above,  while  the  water  escaped  below.  This  nitrogen  was 
absorbed  by  red-hot  magnesium,  contained  in  tubes  heated  in  a  combustion- 
furnace.  The  unabsorbed  gas  was  circulated  over  red-hot  magnesium  in  a 
special  small  apparatus,  by  which  its  volume  was  reduced  to  15  cub.  centims. 
As  it  was  impracticable  further  to  reduce  the  volume  by  means  of  magnesium, 
the  residual  15  cub.  centims.  were  transferred  to  a  tube,  mixed  with  oxygen, 
and  submitted  to  sparking  over  caustic  soda.  The  residue  after  absorption  of 
oxygen,  which  undoubtedly  consisted  of  pure  argon,  amounted  to  3'5  cub. 
centims.  This  is  one-fortieth  of  the  quantity  which  would  have  been  obtained 
from  atmospheric  nitrogen,  and  its  presence  can  be  accounted  for,  we  venture 
to  think,  first  from  the  water  in  the  gas-holder,  which  had  not  been  freed  from 
dissolved  gas  by  boiling  in  vacuo  (it  has  already  been  shown  that  a  consider- 
able gain  may  ensue  from  this  source),  and  second,  from  leakage  of  air 
which  accidentally  took  place,  owing  to  the  breaking  of  a  tube.  The  leakage 
may  have  amounted  to  200  cub.  centims.,  but  it  could  not  be  accurately 
ascertained.  Quantitative  negative  experiments  of  this  nature  are  exceedingly 
difficult,  and  require  a  long  time  to  carry  them  to  a  successful  conclusion. 


8.     Reparation  of  Argon  on  a  Large  Scale. 

To  separate  nitrogen  from  "atmospheric  nitrogen"  on  a  large  scale,  by 
help  of  magnesium,  several  devices  were  tried.  It  is  not  necessary  to  describe 
them  all  in  detail.  Suffice  it  to  say  that  an  attempt  was  made  to  cause  a 
store  of  "  atmospheric  nitrogen  "  to  circulate  by  means  of  a  fan,  driven  by  a 
water-motor.  The  difficulty  encountered  here  was  leakage  at  the  bearing  of 
the  fan,  and  the  introduced  air  produced  a  cake  which  blocked  the  tube  on 
coming  into  contact  with  the  magnesium.  It  might  have  been  possible  to 
remove  oxygen  by  metallic  copper;  but  instead  of  thus  complicating  the 
apparatus,  a  water-injector  was  made  use  of  to  induce  circulation.  Here  also 
it  is  unnecessary  to  enter  into  details.  For,  though  the  plan  worked  well, 
and  although  about  120  litres  of  "  atmospheric  nitrogen"  were  absorbed,  the 
yield  of  argon  was  not  large,  about  GOO  cub.  centims.  having  been  collected. 
This  loss  was  subsequently  discovered  to  be  due  partially,  at  least,  to  the  rela- 
tively high  solubility  of  argon  in  water.  In  order  to  propel  the  gas  over 
magnesium,  through  a  long  combustion-tube  packed  with  turnings,  a  consider- 
able water-pressure,  involving  a  large  flow  of  water,  was  necessary.  The  gas 
was  brought  into  intimate  contact  with  this  water,  and  presuming  that  several 
thousand  litres  of  water  ran  through  the  injector,  it  is  obvious  that  a  not 


156  ARGON,    A    NEW   CONSTITUENT   OF  THE   ATMOSPHERE.  [214 

inconsiderable  amount  of  argon  must  have  been  dissolved.  Its  proportion 
was  increasing  at  each  circulation,  and  consequently  its  partial  pressure  also 
increased.  Hence,  towards  the  end  of  the  operation,  at  least,  there  is  every 
reason  to  believe  that  a  serious  loss  had  occurred. 

It  was  next  attempted  to  pass  "  atmospheric  nitrogen  "  from  a  gas-holder 
first  through  a  combustion  tube  of  the  usual  length  packed  with  metallic 
copper  reduced  from  the  oxide;  then  through  a  small  U-tube  containing  a 
little  water,  which  was  intended  as  an  index  of  the  rate  of  flow;  the  gas  was 
then  dried  by  passage  through  tubes  filled  with  soda-lime  and  phosphoric 
anhydride ;  and  it  next  passed  through  a  long  iron  tube  (gas-pipe)  packed 
with  magnesium  turnings,  and  heated  to  bright  redness  in  a  second  com- 
bustion-furnace. 

After  the  iron  tube  followed  a  second  small  U-tube  containing  water, 
intended  to  indicate  the  rate  at  which  the  argon  escaped  into  a  small  gas- 
holder placed  to  receive  it.  The  nitrogen  was  absorbed  rapidly,  and  argon 
entered  the  small  gas-holder.  But  there  was  reason  to  suspect  that  the  iron 
tube  is  permeable  by  argon  at  a  red  heat.  The  first  tube-full  allowed  very 
little  argon  to  pass.  After  it  had  been  removed  and  replaced  by  a  second,  the 
same  thing  was  noticed.  The  first  tube  was  difficult  to  clean ;  the  nitride  of 
magnesium  forms  a  cake  on  the  interior  of  the  tube,  and  it  was  very  difficult 
to  remove  it ;  moreover  this  rendered  the  filling  of  the  tube  very  troublesome, 
inasmuch  as  its  interior  was  so  rough  that  the  magnesium  turnings  could  only 
with  difficulty  be  forced  down.  However,  the  permeability  to  argon,  if  such 
be  the  case,  appeared  to  have  decreased.  The  iron  tube  was  coated  internally 
with  a  skin  of  magnesium  nitride,  which  appeared  to  diminish  its  permeability 
to  argon.  After  all  the  magnesium  in  the  tube  had  been  converted  into 
nitride  (and  this  was  easily  known,  because  a  bright  glow  proceeded  gradually 
from  one  end  of  the  tube  to  the  other)  the  argon  remaining  in  the  iron  tube 
was  "  washed  "  out  by  a  current  of  nitrogen ;  so  that,  after  a  number  of  opera- 
tions, the  small  gas-holder  contained  a  mixture  of  argon  with  a  considerable 
quantity  of  nitrogen. 

On  the  whole,  the  use  of  iron  tubes  is  not  to  be  recommended,  owing  to 
the  difficulty  in  cleaning  them,  and  the  possible  loss  through  their  permeability 
to  argon.  There  is  no  such  risk  of  loss  with  glass  tubes,  but  each  operation 
requires  a  new  tube,  and  the  cost  of  the  glass  is  considerable  if  much  nitrogen 
is  to  be  absorbed.  Tubes  of  porcelain  were  tried;  but  the  glaze  in  the 
interior  is  destroyed  by  the  action  of  the  red-hot  magnesium,  and  the  tubes 
crack  on  cooling. 

By  these  processes  157  litres  of  "  atmospheric  nitrogen  "  were  reduced  in 
volume  to  about  245  litres  in  all  of  a  mixture  of  nitrogen  and  argon.  This 


1895]  ARGON,   A   NEW   CONSTITUENT  OF   THE   ATMOSPHERE. 


157 


mixture  was  afterwards  circulated  over  red-hot  magnesium,  in  order  to  remove 
the  last  portion  of  nitrogen. 

Fig.  5. 


As  the  apparatus  employed  for  this  purpose  proved  very  convenient,  a  full 
description  of  its  construction  is  here  given.  A  diagram  is  shown  in  Fig.  5, 
which  sufficiently  explains  the  arrangement  of  the  apparatus.  A  is  the 
circulator.  It  consists  of  a  sort  of  Sprengel's  pump  (a)  to  which  a  supply  of 
mercury  is  admitted  from  a  small  reservoir  (6).  This  mercury  is  delivered  into 
a  gas-separator  (c),  and  the  mercury  overflows  into  the  reservoir  (d).  When 
its  level  rises,  so  that  it  blocks  the  tube  (/),  it  ascends  in  pellets  or  pistons 
into  (e),  a  reservoir  which  is  connected  through  (g)  with  a  water-pump.  The 
mercury  falls  into  (6),  and  again  passes  down  the  Sprengel  tube  (a).  No 
attention  is,  therefore,  required,  for  the  apparatus  works  quite  automatically. 
This  form  of  apparatus  was  employed  several  years  ago  by  Dr  Collie. 

The  gas  is  drawn  from  the  gas-holder  B,  and  passes  through  a  tube  C, 
which  is  heated  to  redness  by  a  long-flame  burner,  and  which  contains  in  one 
half  metallic  copper,  and  in  the  other  half  copper  oxide.  This  precaution  is 
taken  in  order  to  remove  any  oxygen  which  may  possibly  be  present,  and  also 
any  hydrogen  or  hydrocarbon.  In  practice,  it  was  never  found  that  the 
copper  became  oxidised,  or  the  oxide  reduced.  It  is,  however,  useful  to  guard 
against  any  possible  contamination.  The  gas  next  traversed  a  drying-tube  D, 
the  anterior  portion  containing  ignited  soda-lime,  and  the  posterior  portion 
phosphoric  anhydride.  From  this  it  passed  a  reservoir,  Z)',  from  which  it 
could  be  transferred,  when  all  absorption  had  ceased,  into  the  small  gas-holder. 


158  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

It  then  passed  through  E,  a  piece  of  combustion-tube,  drawn  out  at  both  ends, 
filled  with  magnesium  turnings,  and  heated  by  a  long-flame  burner  to  redness. 
Passing  through  a  small  bulb,  provided  with  electrodes,  it  again  entered  the 
fall-tube. 

After  the  magnesium  tube  E  had  done  its  work,  the  stop-cocks  were  all 
closed,  and  the  gas  was  turned  down,  so  that  the  burners  might  cool.  The 
mixture  of  argon  and  nitrogen  remaining  in  the  system  of  tubes  was  pumped 
out  by  a  Sprengel's  pump  through  F,  collected  in  a  large  test-tube,  and 
reintroduced  into  the  gas-holder  B  through  the  side-tube  G,  which  requires 
no  description.  The  magnesium  tube  was  then  replaced  by  a  fresh  one ;  the 
system  of  tubes  was  exhausted  of  air ;  argon  and  nitrogen  were  admitted  from 
the  gas-holder  B ;  the  copper-oxide  tube  and  the  magnesium  tube  were  again 
heated  ;  and  the  operation  was  repeated  until  absorption  ceased.  It  was  easy 
to  decide  when  this  point  had  been  reached,  by  making  use  of  the  graduated 
cylinder  H,  from  which  water  entered  the  gas-holder  B.  It  was  found 
advisable  to  keep  all  the  water  employed  in  these  operations,  for  it  had  become 
saturated  with  argon.  If  gas  was  withdrawn  from  the  gas-holder,  its  place 
was  taken  by  this  saturated  water. 

The  absorption  of  nitrogen  proceeds  very  slowly  towards  the  end  of  the 
operation,  and  the  diminution  in  volume  of  the  gas  is  not  greater  than  4  or  5 
cub.  centims.  per  hour.  It  is,  therefore,  somewhat  difficult  to  judge  of  the 
end-point,  as  will  be  seen  when  experiments  on  the  density  of  this  gas  are 
described.  The  magnesium  tube,  towards  the  end  of  the  operations,  was 
made  so  hot  that  the  metal  was  melted  in  the  lower  part  of  the  tube,  and 
sublimed  in  the  upper  part.  The  argon  and  residual  nitrogen  had,  therefore, 
been  thoroughly  mixed  with  gaseous  magnesium  during  its  passage  through 
the  tube  E. 

To  avoid  possible  contamination  with  air  in  the  Sprengel's  pump,  the  last 
portion  of  gas  collected  from  the  system  of  tubes  was  not  re-admitted  to  the 
gas-holder  B,  but  was  separately  stored. 

The  crude  argon  was  collected  in  two  operations.  First,  the  quantity 
made  by  absorption  by  magnesium  in  glass  tubes  with  the  water-pump 
circulator  was  purified.  Later,  after  a  second  supply  had  been  prepared  by 
absorption  in  iron  tubes,  the  mixture  of  argon  and  nitrogen  was  united  with 
the  first  quantity  and  circulated  by  means  of  the  mercury  circulator,  in  the 
gas-holder  J5.  Attention  will  be  drawn  to  the  particular  sample  of  gas 
employed  in  describing  further  experiments  made  with  the  argon. 

By  means  of  magnesium,  about  7  litres  of  nitrogen  can  be  absorbed  in  an 
hour.  The  changing  of  the  tubes  of  magnesium,  however,  takes  some  time ; 
consequently,  the  largest  amount  absorbed  in  one  day  was  nearly  30  litres. 


1895]  ARGON,   A    NEW   CONSTITUENT   OF   THE    ATMOSPHERE.  159 

At  a  later  date  a  quantitative  experiment  was  carried  out  on  a  large  scale, 
the  amount  of  argon  from  100  litres  of  "  atmospheric  "  nitrogen,  measured  at 
20°,  having  been  absorbed  by  magnesium,  and  the  resulting  argon  measured 
at  12°.  During  the  process  of  absorbing  nitrogen  in  the  combustion-furnace, 
however,  one  tube  cracked,  and  it  is  estimated  that  about  4  litres  of  nitrogen 
escaped  before  the  crack  was  noticed.  With  this  deduction,  and  assuming 
that  the  nitrogen  had  been  measured  at  12°,  93'4  litres  of  atmospheric 
nitrogen  were  taken.  The  magnesium  required  for  absorption  weighed 
409  grms.  The  amount  required  by  theory  should  have  been  285  grms.;  but 
it  must  be  remembered  that  in  many  cases  the  magnesium  was  by  no  means 
wholly  converted  into  nitride.  The  first  operation  yielded  about  3  litres  of  a 
mixture  of  nitrogen  and  argon,  which  was  purified  in  the  circulating  apparatus. 
The  total  residue,  after  absorption  of  the  nitrogen,  amounted  to  921  cub. 
centims.  The  yield  is  therefore  0'986  per  cent. 

At  first  no  doubt  the  nitrogen  gains  a  little  argon  from  the  water  over 
which  it  stands.  But,  later,  when  the  argon  forms  the  greater  portion  of  the 
gaseous  mixture,  its  solubility  in  water  must  materially  decrease  its  volume. 
It  is  difficult  to  estimate  the  loss  from  this  cause.  The  gas-holder,  from 
which  the  final  circulation  took  place,  held  three  litres  of  water.  Taking  the 
solubility  of  argon  as  4  per  cent.,  this  would  mean  a  loss  of  about  120  cub. 
centims.  If  this  is  not  an  over-estimate,  the  yield  of  argon  would  be 
increased  to  1040  cub.  centims.,  or  I'll  per  cent.  The  truth  probably  lies 
between  these  two  estimates. 

It  may  be  concluded,  with  probability,  that  the  argon  forms  approximately 
1  per  cent,  of  the  "  atmospheric  "  nitrogen. 


The  principal  objection  to  the  oxygen  method  of  isolating  argon,  as 
hitherto  described,  is  the  extreme  slowness  of  the  operation.  An  absorption 
of  30  cub.  centims.  of  mixed  gas  means  the  removal  of  but  12  cub.  centims.  of 
nitrogen.  At  this  rate  8  hours  are  required  for  the  isolation  of  1  cub.  centim. 
of  argon,  supposed  to  be  present  in  the  proportion  of  1  per  cent. 

In  extending  the  scale  of  operations  we  had  the  great  advantage  of  the 
advice  of  Mr  Crookes,  who  a  short  time  ago  called  attention  to  the  flame 
rising  from  platinum  terminals,  which  convey  a  high  tension  alternating 
electric  discharge,  and  pointed  out  its  dependence  upon  combustion  of  the 
nitrogen  and  oxygen  of  the  air*.  Mr  Crookes  was  kind  enough  to  arrange  an 
impromptu  demonstration  at  his  own  house  with  a  small  alternating  current 
plant,  in  which  it  appeared  that  the  absorption  of  mixed  gas  was  at  the  rate 
of  500  cub.  centims.  per  hour,  or  nearly  20  times  as  fast  as  with  the  battery. 

*   Chemical  News,  Vol.  LXV.  p.  301,  1892. 


160  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

The  arrangement  is  similar  to  that  first  described  by  Spottiswoode*.  The 
primary  of  a  Ruhmkorff  coil  is  connected  directly  with  the  alternator,  no 
break  or  condenser  being  required ;  so  that,  in  fact,  the  coil  acts  simply 
as  a  high  potential  transformer.  When  the  arc  is  established  the  platinum 
terminals  may  be  separated  much  beyond  the  initial  striking  distance. 

The  plant  with  which  the  large  scale  operations  have  been  made  consists 
of  a  De  Meritens  alternator,  kindly  lent  by  Professor  J.  J.  Thomson,  and  a  gas 
engine.  As  transformer,  one  of  Swinburne's  hedgehog  pattern  has  been 
employed  with  success,  but  the  ratio  of  transformation  (24  :  1)  is  scarcely 
sufficient.  A  higher  potential,  although,  perhaps,  not  more  efficient,  is  more 
convenient.  The  striking  distance  is  greater,  and  the  arc  is  not  so  liable  to 
go  out.  Accordingly  most  of  the  work  to  be  described  has  been  performed 
with  transformers  of  the  Ruhmkorff  type. 

The  apparatus  has  been  varied  greatly,  and  it  cannot  be  regarded  as 
having  even  yet  assumed  a  final  form.  But  it  will  give  a  sufficient  idea  of 
the  method  if  we  describe  an  experiment  in  which  a  tolerably  good  account 
was  kept  of  the  air  and  oxygen  employed.  The  working  vessel  was  a  glass 
flask,  A  (Fig.  6),  of  about  1500  cub.  centims.  capacity,  and  stood,  neck  down- 
wards, over  a  large  jar  of  alkali,  B.  As  in  the  small  scale  experiments,  the 
leading-in  wires  were  insulated  by  glass  tubes,  DD,  suitably  bent  and  carried 
through  the  liquid  up  the  neck.  For  the  greater  part  of  the  length  iron  wires 
were  employed,  but  the  internal  extremities,  EE,  were  of  platinum,  doubled 
upon  itself  at  the  terminals  from  which  the  discharge  escaped.  The  glass 
protecting  tubes  must  be  carried  up  for  some  distance  above  the  internal  level 
of  the  liquid,  but  it  is  desirable  that  the  arc  itself  should  not  be  much  raised 
above  that  level.  A  general  idea  of  the  disposition  of  the  electrodes  will  be 
obtained  from  Fig.  6.  To  ensure  gas  tightness  the  bends  were  occupied  by 
mercury.  A  tube,  C,  for  the  supply  or  withdrawal  of  gas  was  carried  in  the 
same  way  through  the  neck. 

The  Ruhmkorff  employed  in  this  operation  was  one  of  medium  size. 
When  the  mixture  was  rightly  proportioned  and  the  arc  of  full  length,  the 
rate  of  absorption  was  about  700  cub.  centims.  per  hour.  A  good  deal  of  time 
is  lost  in  starting,  for,  especially  when  there  is  soda  on  the  platinums,  the  arc 
is  liable  to  go  out  if  lengthened  prematurely.  After  seven  days  the  total 
quantity  of  air  let  in  amounted  to  7925  cub.  centims.,  and  of  oxygen  (prepared 
from  chlorate  of  potash)  9137  cub.  centims.  On  the  eighth  and  ninth  days 
oxygen  alone  was  added,  of  which  about  500  cub.  centims.  was  consumed, 
while  there  remained  about  700  cub.  centims.  in  the  flask.  Hence  the  pro- 
portion in  which  the  air  and  oxygen  combined  was  as  70  :  96.  On  the  eighth 
day  there  was  about  three  hours'  work,  and  the  absorption  slackened  off  to 

*  "  A  Mode  of  Exciting  an  Induction-coil,"  Phil.  Mag.  Vol.  vm.  p.  390,  1879. 


1895] 


ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE. 


161 


about  one  quarter  of  the  previous  rate.  On  the  ninth  day  (September  8)  the 
rate  fell  off  still  more,  and  after  three  hours'  work  became  very  slow.  The 
progress  towards  removal  of  nitrogen  was  examined  from  time  to  time  with 
the  spectroscope,  the  points  being  approximated  and  connected  with  a  small 

Fig.  6. 


Leyden  jar.  At  this  stage  the  yellow  nitrogen  line  was  faint,  but  plainly 
visible.  After  about  four  hours'  more  work,  the  yellow  line  had  disappeared, 
and  for  two  hours  there  had  been  no  visible  contraction.  It  will  be  seen  that 
the  removal  of  the  last  part  of  the  nitrogen  was  very  slow,  mainly  on  account 
of  the  large  excess  of  oxygen  present. 


11 


162  ARGON,   A   NEW   CONSTITUENT   OF  THE   ATMOSPHERE.  [214 

The  final  treatment  of  the  residual  700  cub.  centims.  of  gas  was  on  the 
model  of  the  small  scale  operations  already  described  (§  4).  By  means  of  a 
pipette  the  gas  was  gradually  transferred  to  a  large  test-tube  standing  over 
alkali.  Under  the  influence  of  sparks  (from  battery  and  coil)  passing  all  the 
while,  the  superfluous  oxygen  was  consumed  with  hydrogen  fed  in  slowly 
from  a  voltameter.  If  the  nitrogen  had  been  completely  removed,  and  if 
there  were  no  unknown  ingredient  in  the  atmosphere,  the  volume  under  this 
treatment  should  have  diminished  without  limit.  But  the  contraction  stopped 
at  a  volume  of  65  cub.  centims.,  and  the  volume  was  taken  backwards  and 
forwards  through  this  as  a  minimum  by  alternate  treatment  with  oxygen  and 
hydrogen  added  in  small  quantities,  with  prolonged  intervals  of  sparking. 
Whether  the  oxygen  or  the  hydrogen  were  in  excess  could  be  determined  at 
any  moment  by  a  glance  at  the  spectrum.  At  the  minimum  volume  the  gas 
was  certainly  not  hydrogen  or  oxygen.  Was  it  nitrogen  ?  On  this  point  the 
testimony  of  the  spectroscope  was  equally  decisive.  No  trace  of  the  yellow 
nitrogen  line  could  be  seen  even  with  a  wide  slit  and  under  the  most  favour- 
able conditions. 

When  the  gas  stood  for  some  days  over  water,  the  nitrogen  line  again 
asserted  itself,  and  many  hours  of  sparking  with  a  little  oxygen  were  required 
again  to  get  rid  of  it.  As  it  was  important  to  know  what  proportions  of 
nitrogen  could  be  made  visible  in  this  way,  a  little  air  was  added  to  gas  that 
had  been  sparked  for  some  time  subsequently  to  the  disappearance  of  nitrogen 
in  its  spectrum.  It  was  found  that  about  1^  per  cent,  was  clearly,  and  about 
3  per  cent,  was  conspicuously,  visible.  About  the  same  numbers  apply  to  the 
visibility  of  nitrogen  in  oxygen  when  sparked  under  these  conditions,  that  is, 
at  atmospheric  pressure,  and  with  a  jar  in  connection  with  the  secondary 
terminals. 

When  we  attempt  to  increase  the  rate  of  absorption  by  the  use  of  a 
more  powerful  electric  arc,  further  experimental  difficulties  present  them- 
selves. In  the  arrangement  already  described,  giving  an  absorption  of  700 
cub.  centims.  per  hour,  the  upper  part  of  the  flask  becomes  very  hot.  With  a 
more  powerful  arc  the  heat  rises  to  such  a  point  that  the  flask  is  filled  with 
steam  and  the  operation  comes  to  a  standstill. 

It  is  necessary  to  keep  the  vessel  cool  by  either  the  external  or  internal 
application  of  liquid  to  the  upper  surface  upon  which  the  hot  gases  from  the 
arc  impinge.  One  way  of  effecting  this  is  to  cause  a  small  fountain  of  alkali 
to  impinge  on  the  top  of  the  flask,  so  as  to  wash  the  whole  of  the  upper 
surface.  This  plan  is  very  effective,  but  it  is  open  to  the  objection  that  a  break- 
down would  be  disastrous,  and  it  would  involve  special  arrangements  to  avoid 
losing  the  argon  by  solution  in  the  large  quantity  of  alkali  required.  It  is 
simpler  in  many  respects  to  keep  the  vessel  cool  by  immersing  it  in  a  large 
body  of  water,  and  the  inverted  flask  arrangement  (Fig.  6)  has  been  applied  in 


1895] 


ARGON,   A    NEW   CONSTITUENT   OF   THE    ATMOSPHERE. 


163 


this  manner.  But,  on  the  whole,  it  appears  to  be  preferable  to  limit  the 
application  of  the  cooling  water  to  the  upper  part  of  the  external  surface, 
building  up  for  this  purpose  a  suitable  wall  of  sheet  lead  cemented  round  the 
glass.  The  most  convenient  apparatus  for  large-scale  operations  that  has 
hitherto  been  tried  is  shown  in  the  accompanying  figure  (Fig.  7). 


Fig.  7. 


Scale 


The  vessel  A  is  a  large  globe  of  about  6  litres  capacity,  intended  for 
demonstrating  the  combustion  of  phosphorus  in  oxygen  gas,  and  stands  in  an 
inclined  position.  It  is  about  half  filled  with  a  solution  of  caustic  soda.  The 
neck  is  fitted  with  a  rubber  stopper,  B,  provided  with  four  perforations.  Two 
of  these  are  fitted  with  tubes,  (7,  D,  suitable  for  the  supply  or  withdrawal  of 
gas  or  liquid.  The  other  two  allow  the  passage  of  the  stout  glass  tubes,  E,  F, 
which  contain  the  electrodes.  For  greater  security  against  leakage,  the 
interior  of  these  tubes  is  charged  with  water,  held  in  place  by  small  corks, 
and  the  outer  ends  are  cemented  up.  The  electrodes  are  formed  of  stout  iron 
wires  terminated  by  thick  platinums,  G,  H,  triply  folded  together,  and  welded 
at  the  ends.  The  lead  walls  required  to  enclose  the  cooling  water  are  partially 
shown  at  I.  For  greater  security  the  india-rubber  cork  is  also  drowned  in 
water,  held  in  place  with  the  aid  of  sheet-lead.  The  lower  part  of  the  globe 
is  occupied  by  about  3  litres  of  a  5  per  cent,  solution  of  caustic  soda,  the 
solution  rising  to  within  about  half-an-inch  of  the  platinum  terminals.  With 
this  apparatus  an  absorption  of  3  litres  of  mixed  gas  per  hour  can  be 
attained, — about  3000  times  the  rate  at  which  Cavendish  could  work. 

11—2 


164  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

When  it  is  desired  to  stop  operations,  the  feed  of  air  (or  of  chemical 
nitrogen  in  blank  experiments)  is  cut  off,  oxygen  alone  being  supplied  as  long 
as  any  visible  absorption  occurs.  Thus  at  the  close  the  gas  space  is  occupied 
by  argon  and  oxygen  with  such  nitrogen  as  cannot  readily  be  taken  up  in  a 
condition  of  so  great  dilution.  The  oxygen,  being  too  much  for  convenient 
treatment  with  hydrogen,  was  usually  absorbed  with  copper  and  ammonia, 
and  the  residual  gas  was  then  worked  over  again  as  already  described  in 
an  apparatus  constructed  upon  a  smaller  scale. 

It  is  worthy  of  notice  that  with  the  removal  of  the  nitrogen,  the  arc- 
discharge  from  the  dynamo  changes  greatly  in  appearance,  bridging  over  more 
directly  and  in  a  narrower  band  from  one  platinum  to  the  other,  and  assuming 
a  beautiful  sky-blue  colour,  instead  of  the  greenish  hue  apparent  so  long  as 
oxidation  of  nitrogen  is  in  progress. 

In  all  the  large-scale  experiments,  an  attempt  was  made  to  keep  a  reckon- 
ing of  the  air  and  oxygen  employed,  in  the  hope  of  obtaining  data  as  to  the 
proportional  volume  of  argon  in  air,  but  various  accidents  too  often  interfered. 
In  one  successful  experiment  (January,  1895),  specially  undertaken  for  the 
sake  of  measurement,  the  total  air  employed  was  9250  cub.  centims.,  and  the 
oxygen  consumed,  manipulated  with  the  aid  of  partially  de-aerated  water, 
amounted  to  10,820  cub.  centims.  The  oxygen  contained  in  the  air  would  be 
1942  cub.  centims. ;  so  that  the  quantities  of  "  atmospheric  nitrogen  "  and  of 
total  oxygen  which  enter  into  combination  would  be  7308  cub.  centims.,  and 
12,762  cub.  centims.  respectively.  This  corresponds  to  N  +  T75  0 — the  oxygen 
being  decidedly  in  excess  of  the  proportion  required  to  form  nitrous  acid — 
2HN02,  or  H2O  +  N2  +  3  O.  The  argon  ultimately  found  on  absorption  of  the 
excess  of  oxygen  was  75'0  cub.  centims.,  reduced  to  conditions  similar  to  those 
under  which  the  air  was  measured,  or  a  little  more  than  1  per  cent,  of  the 
"atmospheric  nitrogen"  used.  It  is  probable,  however,  that  some  of  the 
argon  was  lost  by  solution  during  the  protracted  operations  required  in  order 
to  get  quit  of  the  last  traces  of  nitrogen. 

[In  recent  operations  at  the  Royal  Institution,  where  a  public  supply  of 
alternating  current  at  100  volts  is  available,  the  scale  of  the  apparatus  has 
been  still  further  increased. 

The  capacity  of  the  working  vessel  is  20  litres,  of  which  about  one  half  is 
occupied  by  a  strong  solution  of  caustic  soda.  The  platinum  terminals  are 
very  massive,  and  the  flame  rising  from  them  is  prevented  from  impinging 
directly  upon  the  glass  by  a  plate  of  platinum  held  over  it  and  supported  by 
a  wire  which  passes  through  the  rubber  cork.  In  the  electrical  arrangements 
we  have  had  the  advantage  of  Mr  Swinburne's  advice.  The  transformers  are 
two  of  the  "  hedgehog  "  pattern,  the  thick  wires  being  connected  in  parallel 
and  the  thin  wires  in  series.  In  order  to  control  the  current  taken  when  the 


1895]  ARGON,   A   NEW   CONSTITUENT  OF  THE   ATMOSPHERE.  165 

arc  is  short  or  the  platinums  actually  in  contact,  a  choking-coil,  provided  with 
a  movable  core  of  fine  iron  wires,  is  inserted  in  the  thick  wire  circuit.  In 
normal  working  the  current  taken  from  the  mains  is  about  22  amperes,  so 
that  some  2£  h.  p.  is  consumed.  At  the  same  time  the  actual  voltage  at  the 
platinum  terminals  is  1500.  When  the  discharge  ceases,  the  voltage  at  the 
platinum  rises  to  3000*,  which  is  the  force  actually  available  for  re-starting 
the  discharge  if  momentarily  stopped. 

With  this  discharge,  the  rate  of  absorption  of  mixed  gases  is  about  7  litres 
per  hour.  When  the  argon  has  accumulated  to  a  considerable  extent,  the 
rate  falls  off,  and  after  several  days'  work,  about  6  litres  per  hour  becomes  the 
maximum.  In  commencing  operations  it  is  advisable  to  introduce,  first,  the 
oxygen  necessary  to  combine  with  the  already  included  air,  after  which  the 
feed  of  mixed  gases  should  consist  of  about  11  parts  of  oxygen  to  9  parts  of 
air.  The  mixed  gases  may  be  contained  in  a  large  gas-holder,  and  then,  the 
feed  being  automatic,  very  little  attention  is  required.  When  it  is  desired  to 
determine  the  rate  of  absorption,  auxiliary  gas-holders  of  glass,  graduated  into 
litres,  are  called  into  play.  If  the  rate  is  unsatisfactory,  a  determination  may 
be  made  of  the  proportion  of  oxygen  in  the  working  vessel,  and  the  necessary 
gas,  air,  or  oxygen,  as  the  case  may  be,  introduced  directly. 

In  re-starting  the  arc  after  a  period  of  intermission,  it  is  desirable  to  cut 
off  the  connection  with  the  principal  gas-holder.  The  gas  (about  two  litres  in 
amount)  ejected  from  the  working  vessel  by  the  expansion  is  then  retained  in 
the  auxiliary  holder,  and  no  argon  finds  its  way  further  back.  The  connection 
between  the  working  vessel  and  the  auxiliary  holder  should  be  made  without 
india-rubber,  which  is  liable  to  be  attacked  by  the  ozonized  gases. 

The  apparatus  has  been  kept  in  operation  lor  fourteen  hours  continuously, 
and  there  should  be  no  difficulty  in  working  day  and  night.  An  electric 
signal  could  easily  be  arranged  to  give  notice  of  the  extinction  of  the  arc, 
which  sometimes  occurs  unexpectedly;  or  an  automatic  device  for  re-striking 
the  arc  could  be  contrived. — April,  1895.] 


9.     Density  of  Argon  prepared  by  means  of  Oxygen. 

A  first  estimate  of  the  density  of  argon  prepared  by  the  oxygen  method 
was  founded  upon  the  data  recorded  already  respecting  the  volume  present  in 
air,  on  the  assumption  that  the  accurately  known  densities  of  "  atmospheric  " 
and  of  chemical  nitrogen  differ  on  account  of  the  presence  of  argon  in  the 
former,  and  that  during  the  treatment  with  oxygen  nothing  is  oxidised  except 
nitrogen.  Thus,  if 

*  A  still  higher  voltage  on  open  circuit  would  be  preferable. 


166  ARGON,   A   NEW   CONSTITUENT  OF  THE   ATMOSPHERE.  [214 

D  =  density  of  chemical  nitrogen, 

jy=      „  atmospheric  nitrogen, 

d  =      „  argon, 

a  =  proportional  volume  of  argon  in  atmospheric  nitrogen, 

the  law  of  mixtures  gives 


d  =  D+(D'-D)/a. 

In  this  formula  D'  —  D  and  a  are  both  small,  but  they  are  known  with  fair 
accuracy.     From  the  data  already  given  for  the  experiment  of  September  8th 

65 


0-79  x  7925 


=  0-0104; 


whence,  if  on  an  arbitrary  scale  of  reckoning  D  =  2*2990,  IX  =  2'3102,  we  find 
d  =  3-378.     Thus  if  N2  be  14,  or  O2  be  16,  the  density  of  argon  is  20'6. 

Again,  from  the  January  experiment, 


whence,  if  N  =  14,  the  density  of  argon  is  20'6,  as  before.  There  can  be  little 
doubt,  however,  that  these  numbers  are  too  high,  the  true  value  of  a  being 
greater  than  is  supposed  in  the  above  calculations. 

A  direct  determination  by  weighing  is  desirable,  but  hitherto  it  has  not 
been  feasible  to  collect  by  this  means  sufficient  to  fill  the  large  globe  (§1) 
employed  for  other  gases.  A  mixture  of  about  400  cub.  centims.  of  argon  with 
pure  oxygen,  however,  gave  the  weight  2'7315,  0-1045  in  excess  of  the  weight 
of  oxygen,  viz.,  2'6270.  Thus,  if  a.  be  the  ratio  of  the  volume  of  argon  to  the 
whole  volume,  the  number  for  argon  will  be 

2-6270  +  0-1045/a. 

The  value  of  a,  being  involved  only  in  the  excess  of  weight  above  that  of 
oxygen,  does  not  require  to  be  known  very  accurately.  Sufficiently  concordant 
analyses  by  two  methods  gave  a  =  0'1845  ;  whence,  for  the  weight  of  the  gas 
we  get  3*193  ;  so  that  if  O  =  16,  the  density  of  the  gas  would  be  19'45.  An 
allowance  for  residual  nitrogen,  still  visible  in  the  gas  before  admixture  of 
oxygen,  raises  this  number  to  197,  which  may  be  taken  as  the  density  of  pure 
argon  resulting  from  this  determination*. 

*  [The  proportion  of  nitrogen  (4  or  5  per  cent,  of  the  volume)  was  estimated  from  the 
appearance  of  the  nitrogen  lines  in  the  spectrum,  these  being  somewhat  more  easily  visible  than 
when  3  per  cent,  of  nitrogen  was  introduced  into  pure  argon  (§  8).  —  April,  1895.] 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  167 

10.     Density  of  Argon  prepared  by  means  of  Magnesium,*. 

It  has  already  been  stated  that  the  density  of  the  residual  gas  from  the 
first  and  preliminary  attempt  to  separate  oxygen  and  nitrogen  from  air  by 
means  of  magnesium  was  19'086,  and  allowing  for  contraction  on  sparking 
with  oxygen  the  density  is  calculable  as  20'01.  The  following  determinations 
of  density  were  also  made : — 

(a)  After  absorption  in  glass  tubes,  the  water  circulator  having  been  used, 
and  subsequent  circulation  by  means  of  mercury  circulator  until  rate  of  con- 
traction had  become  slow,  162-843  cub.  centims.,  measured  at  757*7  millims. 
(corr.)  pressure,  and  16-81°  C.,  weighed  0'2683  grm.  Hence, 

Weight  of  1  litre  at  0°  and  760  millims 1-7543  grms. 

Density  compared  with  hydrogen  (O  =  16)     .     .     .  19*63         „ 

This  gas  was  again  circulated  over  red-hot  magnesium  for  two  days. 
Before  circulation  it  contained  nitrogen  as  was  evident  from  its  spectrum; 
after  circulating,  nitrogen  appeared  to  be  absent,  and  absorption  had  com- 
pletely stopped.  The  density  was  again  determined. 

(6)  162-843  cub.  centims.,  measured  at  745'4  millims.  (corr.)  pressure,  and 
17-25°  C.,  weighed  0'2735  grm.  Hence, 

Weight  of  1  litre  at  0°  and  760  millims 1-8206  grms. 

Density  compared  with  hydrogen  (O  =  16)    .     .     .  20*38         „ 

Several  portions  of  this  gas,  having  been  withdrawn  for  various  purposes, 
were  somewhat  contaminated  with  air,  owing  to  leakage,  passage  through  the 
pump,  &c.  All  these  portions  were  united  in  the  gas-holder  with  the  main 
stock,  and  circulated  for  eight  hours,  during  the  last  three  of  which  no 
contraction  occurred.  The  gas  removed  from  the  system  of  tubes  by  the 
mercury-pump  was  not  restored  to  the  gas-holder,  but  kept  separate. 

(c)  162-843  cub.  centims.,  measured  at  758'1  millims.  (corr.)  pressure,  and 
17-09°  C.,  weighed  0'27705  grm.     Hence, 

Weight  of  1  litre  at  0°  and  760  millims 1-8124  grms. 

Density  compared  with  hydrogen  (O  =  16)    .     .     .  20'28 

The  contents  of  the  gas-holder  were  subsequently  increased  by  a  mixture 
of  nitrogen  and  argon  from  37  litres  of  atmospheric  nitrogen,  and  after 
circulating,  density  was  determined.  The  absorption  was  however  not  com- 
plete. 

(d)  162'843  cub.  centims.,  measured  at  767'6  millims.  (corr.)  pressure,  and 
16-31°  C.,  weighed  0-2703  grm.     Hence, 

*  See  Addendum,  p.  184. 


168         ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE.        [214 

Weight  of  1  litre  at  0°  and  760  millims 1-742  grms. 

Density  compared  with  hydrogen  (0  =  16)     .     .     .  19'49        „ 

The  gas  was  further  circulated,  until  all  absorption  had  ceased.  This  took 
about  six  hours.  Density  was  again  determined. 

(e)  162-843  cub.  centims.,  measured  at  767'7  millims.  (corr.)  pressure,  and 
15-00°  C.,  weighed  0'2773  grm.  Hence, 

Weight  of  1  litre  at  0°  and  760  millims 1-7784  grms. 

Density  compared  with  hydrogen  (O  =  16)    .     .     .  19*90         „ 

(/)  A  second  determination  was  carried  out,  without  further  circulation. 
162*843  cub.  centims.,  measured  at  769*0  millims.  (corr.)  pressure,  and 
16-00°  C.,  weighed  0'2757  grm.  Hence, 

Weight  of  1  litre  at  0°  and  760  millims 1*7713  grms. 

Density  compared  with  hydrogen  (O  =  16)    .     .     .  19*82         „ 

(g)     After  various  experiments  had  been  made  with  the  same  sample  of 
gas,  it  was  again  circulated  until  all  absorption  ceased.     A  vacuum-tube  was 
filled  with  it,  and  showed  no  trace  of  nitrogen. 
The  density  was  again  determined : — 

162*843  cub.  centims.,  measured  at  750  millims.  (corr.)  pressure,  and  at 
15*62°  C.,  weighed  0*26915  grm. 

Weight  of  1  litre  at  0°  and  760  millims 1*7707  grms. 

Density  compared  with  hydrogen  (O  =  16)    .     .     .  19*82         „ 

These  comprise  all  the  determinations  of  density  made.  It  should  be 
stated  that  there  was  some  uncertainty  discovered  later  about  the  weight  of 
the  vacuous  globe  in  (6)  and  (c).  Rejecting  these  weighings,  the  mean  of  (e), 
(/),  and  (g)  is  19*88.  The  density  may  be  taken  as  19*9,  with  approximate 
accuracy. 

It  is  better  to  leave  these  results  without  comment  at  this  point,  and  to 
return  to  them  later. 

11.     Spectrum  of  Argon. 

Vacuum  tubes  were  filled  with  argon  prepared  by  means  of  magnesium  at 
various  stages  in  this  work,  and  an  examination  of  these  tubes  has  been 
undertaken  by  Mr  Crookes,  to  whom  we  wish  to  express  our  cordial  thanks 
for  his  kindness  in  affording  us  helpful  information  with  regard  to  its 
spectrum.  The  first  tube  was  filled  with  the  early  preparation  of  density  19'09, 
which  obviously  contained  some  nitrogen.  A  photograph  of  the  spectrum  was 
taken,  and  compared  with  a  photograph  of  the  spectrum  of  nitrogen,  and  it 
was  at  once  evident  that  a  spectrum  different  from  that  of  nitrogen  had 
been  registered. 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  169 

Since  that  time  many  other  samples  have  been  examined. 

The  spectrum  of  argon,  seen  in  a  vacuum  tube  of  about  3  millims.  pressure, 
consists  of  a  great  number  of  lines,  distributed  over  almost  the  whole  visible 
field.  Two  lines  are  specially  characteristic;  they  are  less  refrangible  than 
the  red  lines  of  hydrogen  or  lithium,  and  serve  well  to  identify  the  gas  when 
examined  in  this  way.  Mr  Crookes,  who  gives  a  full  account  of  the  spectrum 
in  a  separate  communication,  has  kindly  furnished  us  with  the  accurate 
wave-lengths  of  these  lines  as  well  as  of  some  others  next  to  be  described ; 
they  are  respectively  696'56  and  705*64  x  lO"8  millim. 

Besides  these  red  lines,  a  bright  yellow  line,  more  refrangible  than  the 
sodium  line,  occurs  at  603*84.  A  group  of  five  bright  green  lines  occurs  next, 
besides  a  number  of  less  intensity.  Of  this  group  of  five,  the  second,  which  is 
perhaps  the  most  brilliant,  has  the  wave-length  561*00.  There  is  next  a  blue, 
or  blue- violet,  line  of  wave-length  470'2  and  last,  in  the  less  easily  visible  part 
of  the  spectrum,  there  are  five  strong  violet  lines,  of  which  the  fourth,  which 
is  the  most  brilliant,  has  the  wave-length  420-0. 

Unfortunately,  the  red  lines,  which  are  not  to  be  mistaken  for  those  of 
any  other  substance,  are  only  to  be  seen  at  atmospheric  pressure  when  a  very 
powerful  jar-discharge  is  passed  through  argon.  The  spectrum,  seen  under 
these  conditions,  has  been  examined  by  Professor  Schuster.  The  most 
characteristic  lines  are  perhaps  those  in  the  neighbourhood  of  F,  and  are  very 
easily  seen  if  there  be  not  too  much  nitrogen,  in  spite  of  the  presence  of  some 
oxygen  and  water- vapour.  The  approximate  wave-lengths  are : — 

487-91  ....  Strong. 

(486-07)  .     .     .     .  F. 

484*71  ....  Not  quite  so  strong. 

480-52  ....  Strong. 
476-50] 

473-53  >  .     .     .     .  Fairly  strong  characteristic  triplet. 
472-56  j 

It  is  necessary  to  anticipate  Mr  Crookes's  communication,  and  to  state 
that  when  the  current  is  passed  from  the  induction-coil  in  one  direction, 
that  end  of  the  capillary  tube  next  the  positive  pole  appears  of  a  redder,  and 
that  next  the  negative  of  a  bluer  hue.  There  are,  in  effect,  two  spectra, 
which  Mr  Crookes  has  succeeded  in  separating  to  a  considerable  extent. 
Mr  E.  C.  C.  Baly  *,  who  has  noticed  a  similar  phenomenon,  attributes  it  to 
the  presence  of  two  gases.  The  conclusion  would  follow  that  what  we  have 
termed  "  argon  "  is  in  reality  a  mixture  of  two  gases  which  have  as  yet  not 
been  separated.  This  conclusion,  if  true,  is  of  great  importance,  and  experi- 

*  Proc.  Phys.  Soc.  1893,  p.  147.  He  says:  "When  an  electric  current  is  passed  through  a 
mixture  of  two  gases,  one  is  separated  from  the  other,  and  appears  in  the  negative  glow." 


170  ARGON,    A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

merits  are  now  in  progress  to  test  it  by  the  use  of  other  physical  methods. 
The  full  bearing  of  this  possibility  will  appear  later. 

A  comparison  was  made  of  the  spectrum  seen  in  a  vacuum  tube  with  the 
spectrum  in  a  "plenum"  tube,  i.e.,  one  filled  at  atmospheric  pressure.  Both 
spectra  were  thrown  into  a  field  at  the  same  time.  It  was  evident  that  they 
were  identical,  although  the  relative  strengths  of  the  lines  were  not  always 
the  same.  The  seventeen  most  striking  lines  were  absolutely  coincident. 

The  presence  of  a  small  quantity  of  nitrogen  interferes  greatly  with  the 
argon  spectrum.  But  we  have  found  that  in  a  tube  with  platinum  electrodes, 
after  the  discharge  has  been  passed  for  four  hours,  the  spectrum  of  nitrogen 
disappears,  and  the  argon  spectrum  manifests  itself  in  full  purity.  A  specially 
constructed  tube,  with  magnesium  electrodes,  which  we  hoped  would  yield 
good  results,  removed  all  traces  of  nitrogen  it  is  true,  but  hydrogen  was 
evolved  from  the  magnesium,  and  showed  its  characteristic  lines  very 
strongly.  However,  these  are  easily  identified.  The  gas  evolved  on  heating 
magnesium  in  vacuo,  as  proved  by  a  separate  experiment,  consists  entirely  of 
hydrogen. 

Mr  Crookes  has  proved  the  identity  of  the  chief  lines  of  the  spectrum  of 
gas  separated  from  air-nitrogen  by  aid  of  magnesium  with  that  remaining 
after  sparking  air-nitrogen  with  oxygen,  in  presence  of  caustic  soda  solution. 

Professor  Schuster  has  also  found  the  principal  lines  identical  in  the 
spectra  of  the  two  gases,  when  taken  from  the  jar-discharge  at  atmospheric 
pressure. 

12.     Solubility  of  Argon  in   Water. 

The  tendency  of  the  gas  to  disappear  when  manipulated  over  water  in 
small  quantities  having  suggested  that  it  might  be  more  than  usually  soluble 
in  that  liquid,  special  experiments  were  tried  to  determine  the  degree  of 
solubility. 

The  most  satisfactory  measures  relating  to  the  gas  isolated  by  means  of 
oxygen  were  those  of  September  28.  The  sample  contained  a  trace  of 
oxygen,  and  (as  judged  by  the  spectrum)  a  residue  of  about  2  per  cent,  of 
nitrogen.  The  procedure  and  the  calculations  followed  pretty  closely  the 
course  marked  out  by  Bunsen*,  and  it  is  scarcely  necessary  to  record  the 
details.  The  quantity  of  gas  operated  upon  was  about  4  cub.  centims.,  of 
which  about  1£  cub.  centims.  were  absorbed.  The  final  result  for  the 
solubility  was  3'94  per  100  of  water  at  12°  C.,  about  2£  times  that  of  nitrogen. 
Similar  results  have  been  obtained  with  argon  prepared  by  means  of  mag- 
nesium. At  a  temperature  of  13'9°,  131  arbitrary  measures  of  water  absorbed 

*  Gasometry,  p.  141. 


1895] 


ARGON,   A    NEW   CONSTITUENT   OF   THE   ATMOSPHERE. 


171 


5*3  of  argon.  This  corresponds  to  a  solubility  in  distilled  water,  previously 
freed  from  dissolved  gas  by  boiling  in  vacuo  for  a  quarter  of  an  hour,  and 
admitted  to  the  tube  containing  argon  without  contact  with  air,  of  4'05  cub. 
centims.  of  argon  per  100  of  water. 

The  fact  that  the  gas  is  more  soluble  than  nitrogen  would  lead  us  to 
expect  it  in  increased  proportion  in  the  dissolved  gases  of  rain  water. 
Experiment  has  confirmed  this  anticipation.  Some  difficulty  was  at  first 
experienced  in  collecting  a  sufficiency  for  the  weighings  in  the  large  globe  of 
nearly  2  litres  capacity.  Attempts  at  extraction  by  means  of  a  Topler  pump 
without  heat  were  not  very  successful.  It  was  necessary  to  operate  upon 
large  quantities  of  water,  and  then  the  pressure  of  the  liquid  itself  acted  as  an 
obstacle  to  the  liberation  of  gas  from  all  except  the  upper  layers.  Tapping 
the  vessel  with  a  stick  of  wood  promotes  the  liberation  of  gas  in  a  remarkable 
manner,  but  to  make  this  method  effective,  some  means  of  circulating  the 
water  would  have  to  be  introduced. 

Fig.  8. 


The  extraction  of  the  gases  by  heat  proved  to  be  more  manageable. 
Although  a  large  quantity  of  water  has  to  be  brought  to  or  near  100°  C.,  a 
prolonged  boiling  is  not  necessary,  as  it  is  not  a  question  of  collecting  the 
whole  of  the  gas  contained  in  the  water.  The  apparatus  employed,  which 
worked  very  well  after  a  little  experience,  will  be  understood  from  the 
accompanying  figure.  The  boiler  A  was  constructed  from  an  old  oil-can,  and 
was  heated  by  an  ordinary  ring  Bunsen  burner.  For  the  supply  and  removal 
of  water,  two  co-axial  tubes  of  thin  brass,  and  more  than  four  feet  in  length, 


172         ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE.        [214 

were  applied  upon  the  regenerative  principle.  The  outgoing  water  flowed  in 
the  inner  tube  BG,  continued  from  C  to  D  by  a  prolongation  of  composition 
tubing.  The  inflowing  water  from  a  rain-water  cistern  was  delivered  into  a 
glass  tube  at  E,  and  passed  through  a  brass  connecting  tube  FG  into  the 
narrow  annular  space  between  the  two  principal  tubes  GH.  The  neck  of  the 
can  was  fitted  with  an  india-rubber  cork  and  delivery-tube,  by  means  of  which 
the  gases  were  collected  in  the  ordinary  way.  Any  carbonic  anhydride  was 
removed  by  alkali  before  passage  into  the  glass  aspirating  bottles  used  as 
gas-holders. 

The  convenient  working  of  this  apparatus  depends  very  much  upon  the 
maintenance  of  a  suitable  relation  between  the  heat  and  the  supply  of  water. 
It  is  desirable  that  the  water  in  the  can  should  actually  boil,  but  without  a 
great  development  of  steam ;  otherwise  not  only  is  there  a  waste  of  heat,  and 
thus  a  smaller  yield  of  gas,  but  the  inverted  flask  used  for  the  collection  of  the 
gas  becomes  inconveniently  hot  and  charged  with  steam.  It  was  found 
desirable  to  guard  against  this  by  the  application  of  a  slow  stream  of  water  to 
the  external  surface  of  the  flask.  When  the  supply  of  water  is  once  adjusted, 
nearly  half  a  litre  of  gas  per  hour  can  be  collected  with  very  little  attention. 

The  gas,  of  which  about  four  litres  are  required  for  each  operation,  was 
treated  with  red-hot  copper,  cupric  oxide,  sulphuric  acid,  potash,  and  finally 
phosphoric  anhydride,  exactly  as  atmospheric  nitrogen  was  treated  in  former 
weighings.  The  weights  found,  corresponding  to  those  recorded  in  §  1,  were 
on  two  occasions  2'3221  and  2'3227,  showing  an  excess  of  24  milligrms.  above 
the  weight  of  true  nitrogen.  Since  the  corresponding  excess  for  atmospheric 
nitrogen  is  11  milligrms.,  we  conclude  that  the  water- nitrogen  is  relatively 
twice  as  rich  in  argon. 

Unless  some  still  better  process  can  be  found,  it  may  be  desirable  to 
collect  the  gases  ejected  from  boilers,  or  from  large  supply  pipes  which  run 
over  an  elevation,  with  a  view  to  the  preparation  of  argon  upon  a  large  scale. 

The  above  experiments  relate  to  rain  water.  As  regards  spring  water,  it 
is  known  that  many  thermal  springs  emit  considerable  quantities  of  gas, 
hitherto  regarded  as  nitrogen.  The  question  early  occurred  to  us  as  to  what 
proportion,  if  any,  of  the  new  gas  was  contained  therein.  A  notable  example 
of  a  nitrogen  spring  is  that  at  Bath,  examined  by  Daubeny  in  1833.  With 
the  permission  of  the  authorities  of  Bath,  Dr  Arthur  Richardson  was  kind 
enough  to  collect  for  us  about  10  litres  of  the  gases  discharged  from  the 
King's  Spring.  A  rough  analysis  on  reception  showed  that  it  contained 
scarcely  any  oxygen  and  but  little  carbonic  anhydride.  Two  determinations 
of  density  were  made,  the  gas  being  treated  in  all  respects  as  air,  prepared 
by  diffusion  and  unprepared,  were  treated  for  the  isolation  of  atmospheric 
nitrogen.  The  results  were : — 


1895] 


ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE. 

October  29 2-30513 

November  7          .  2*30532 


173 


Mean 2-30522 

The  weight  of  the  "  nitrogen  "  from  the  Bath  gas  is  thus  about  half-way 
between  that  of  chemical  and  "  atmospheric  "  nitrogen,  suggesting  that  the 
proportion  of  argon  is  less  than  in  air,  instead  of  greater,  as  had  been 
expected. 


13.     Behaviour  at  Low  Temperatures. 

A  single  experiment  was  made  with  an  early  sample  of  gas,  of  density 
19'1,  which  certainly  contained  a  considerable  amount  of  nitrogen.  On 
compressing  it  in  a  pressure  apparatus  to  between  80  and  100  atmospheres 
pressure,  and  cooling  to  —  90°  by  means  of  boiling  nitrous  oxide,  no  appear- 
ance of  liquefaction  could  be  observed.  As  the  critical  pressure  was  not 
likely  to  be  so  high  as  the  pressure  to  which  it  had  been  exposed,  the 
non-liquefaction  was  ascribed  to  insufficient  cooling. 

VAPOUR-PRESSURES. 


Temperature 

Pressure 

Temperature 

Pressure 

Temperature 

Pressure 

-  186-9 

740-5  millims. 

-  136°'2 

27'3  atms. 

-  129°4 

35  '8  atms. 

-  139-1 

23-7  atms. 

-135-1 

29-0      „ 

-  128-6 

38-0     „ 

-  138-3 

25-3     „ 

-  134-4 

29-8      „ 

-121-0 

50-6      „ 

Density 

Gas 

Critical 
tempera- 
ture 

Critical 
pressure 

Boiling- 
point 

Freezing- 
point 

Freezing 
pressure 

Density 
of  gas 

of  liquid 
at 
boiling- 

Colour 
of 
liquid 

point 

atms. 

millims. 

- 

Hydrogen,  H2 

Below 

20-0 

?° 

?° 

? 

1 

? 

Colour- 

- 220-0° 

less 

Nitrogen,  N2  . 

-  146-0 

35-0 

-  194-4 

-214-0 

60 

14 

0-885 

5) 

Carbon  mon-  ) 
oxide,  CO...  ] 

-  139-5 

35-5 

-  190-0 

-  207-0 

100 

14 

? 

n 

Argon,  Aj    ... 

-121-0 

50-6 

-  186-9 

-  189-6 

? 

19-9 

About 

„ 

1-5 

Oxygen,  02  ... 

-118-8 

50-8 

-182-7 

? 

1 

16 

1-124 

Bluish 

Nitric  oxide,  ) 
NO                ( 

-   93-5 

71-2 

-  153-6 

-  167-0 

138 

15 

? 

Colour- 

Methane, CH4 

-    81-8 

54-9 

-  164-0 

-  185-8 

80 

8 

0-415 

less 
n 

174  ARGON,   A   NEW   CONSTITUENT  OF  THE   ATMOSPHERE.  [214 

This  supposition  turned  out  to  be  correct.  For,  on  sending  a  sample  to 
Professor  Olszewski,  the  author  of  most  of  the  accurate  measurements  of  the 
constants  of  gases  at  low  temperatures,  he  was  kind  enough  to  submit  it  to 
examination.  His  results  are  published  elsewhere;  but,  for  convenience  of 
reference,  his  tables,  showing  vapour-pressures,  and  giving  a  comparison 
between  the  constants  of  argon  and  those  of  other  gases,  are  here  reproduced. 


14.     The  ratio  of  the  Specific  Heats  of  Argon*. 

In  order  to  decide  regarding  the  elementary  or  compound  nature  of  argon, 
experiments  were  made  on  the  velocity  of  sound  in  it.  It  will  be  remem- 
bered that  from  the  velocity  of  sound,  the  ratio  of  the  specific  heat  at 
constant  pressure  to  that  at  constant  volume  can  be  deduced  by  means  of 
the  equation 


where  n  is  the  frequency,  \  is  the  wave-length  of  sound,  v  its  velocity,  e  the 
isothermal  elasticity,  d  the  density,  (1  +  at)  the  temperature-correction,  Gp 
the  specific  heat  at  constant  pressure,  and  Gv  that  at  constant  volume.  In 
comparing  two  gases  at  the  same  temperature,  each  of  which  obeys  Boyle's 
law  with  sufficient  approximation  and  in  using  the  same  sound,  many  of  these 
factors  disappear,  and  the  ratio  of  specific  heats  of  one  gas  may  be  deduced 
from  that  of  the  other,  if  known,  by  the  simple  proportion 

\*d  :  \'*d'  :  :  1'408  :  x, 

where  for  example  \  and  d  refer  to  air,  of  which  the  ratio  is  1*408,  according 
to  the  mean  of  observations  by  Rb'ntgen  (T4053),  Wiillner  (T4053),  Kayser 
(1-4106),  and  Jamin  and  Richard  (T41). 

The  apparatus  employed,  although  in  principle  the  same  as  that  usually 
employed,  differed  somewhat  from  the  ordinary  pattern,  inasmuch  as  the  tube 
was  a  narrow  one,  of  2  millims.  bore,  and  the  vibrator  consisted  of  a  glass  rod, 
sealed  into  one  end  of  the  tube,  so  that  about  15  centims.  projected  outside 
the  tube,  while  15  centims.  was  contained  in  the  tube.  By  rubbing  the 
projecting  part  longitudinally  with  a  rag  wet  with  alcohol,  vibrations  of 
exceedingly  high  pitch  of  the  gas  contained  in  the  tube  took  place,  causing 
waves  which  registered  their  nodes  by  the  usual  device  of  lycopodium  powder. 
The  temperature  was  that  of  the  atmosphere  and  varied  little  from  17'5°  ; 
the  pressure  was  also  atmospheric,  and  varied  only  one  millim.  during  the 
experiments.  Much  of  the  success  of  these  experiments  depends  on  so 
adjusting  the  length  of  the  tube  as  to  secure  a  good  echo,  else  the  wave- 
heaps  are  indistinct.  But  this  is  easily  secured  by  attaching  to  its  open  end 

*  See  Addendum,  p.  185. 


1895] 


AEGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE. 


175 


a  piece  of  thick-walled  india-rubber  tubing,  which  can  be  closed  by  a  clip  at 
a  spot  which  is  found  experimentally  to  produce  good  heaps  at  the  nodes. 

The  accuracy  of  this  instrument  has  frequently  been  tested ;  but  fresh 
experiments  were  made  with  air,  carbon  dioxide,  and  hydrogen,  so  as  to  make 
certain  that  reasonably  reliable  results  were  obtainable.  Of  these  an  account 
is  here  given. 


Number  of  observations 

Half-wave-length 

Gas  in  tube 

Eatio^ 

I. 

II. 

I. 

II. 

Air  

3 

2 

19-60 

19-59 

1-408        Assumed 

C02  

3 

15-11 

1-276       Found 

H2  

3 

... 

73-6 

1-376       Found 

To  compare  these  results  with  those  of  previous  observers,  the  following 
numbers  were  obtained  for  carbon  dioxide : — Cazin,  1*291 ;  Rontgen,  1*305 ; 
De  Lucchi,  1*292;  Miiller,  1*265;  Wiillner,  1-311;  Dulong,  1'339 ;  Masson, 
1-274;  Regnault,  1*268;  Amagat,  1'299;  and  Jamin  and  Richard,  1*29.  It 
appears  just  to  reject  Dulong's  number,  which  deviates  so  markedly  from  the 
rest ;  the  mean  of  those  remaining  is  1*288,  which  is  in  sufficient  agreement 
with  that  given  above.  For  the  ratio  of  the  specific  heats  of  hydrogen,  we 
have:— Cazin,  1*410;  Rontgen,  1'385 ;  Dulong,  1'407 ;  Masson,  1'401 ;  Reg- 
nault, 1'400 ;  and  Jamin  and  Richard,  1'410.  The  mean  of  these  numbers 
is  1'402.  This  number  appears  to  differ  considerably  from  the  one  given 
above.  But  it  must  be  noted,  first,  that  the  wave-length  which  should  have 
been  found  is  74*5,  a  number  differing  but  little  from  that  actually  found ; 
second,  that  the  waves  were  long  and  that  the  nodes  were  somewhat  difficult 
to  place  exactly;  and  third,  that  the  atomic  weight  of  hydrogen  has  been 
taken  as  unity,  whereas  it  is  more  likely  to  be  I'Ol,  if  oxygen,  as  was  done, 
be  taken  as  16.  The  atomic  weight  1*01  raises  the  found  value  of  the  ratio 
to  1*399,  a  number  differing  but  little  from  the  mean  value  found  by  other 
observers. 

Having  thus  established  the  trustworthiness  of  the  method,  we  proceed  to 
describe  our  experiments  with  argon. 

Five  series  of  measurements  were  made  with  the  sample  of  gas  of  density 
19*82.  It  will  be  remembered  that  a  previous  determination  with  the  same 
gas  gave  as  its  density  19*90.  The  mean  of  these  two  numbers  was  therefore 
taken  as  correct,  viz.,  19*86. 

The  individual  measurements  are  : 


176 


ARGON,   A   NEW   CONSTITUENT   OF  THE   ATMOSPHERE. 


[214 


I. 

II. 

III. 

IV. 

V. 

Mean 

18-16 

18-14 

18-02 

18-04 

18-03 

millims. 
18-08 

for  the  half-wave-length.     Calculating  the  ratio  of  the  specific  heats,  the 
number  1*644  is  obtained. 

The  narrowness  of  the  tube  employed  in  these  experiments  might  per- 
haps raise  a  doubt  regarding  the  accuracy  of  the  measurements,  for  it  is 
conceivable  that  in  so  narrow  a  tube  the  viscosity  of  the  gas  might  affect  the 
results.  We  therefore  repeated  the  experiments,  using  a  tube  of  8  millims. 
internal  diameter. 

The  mean  of  eleven  readings  with  air,  at  18°,  gave  a  half- wave-length  of 
34'62  millims.  With  argon  in  the  same  tube,  and  at  the  same  temperature, 
the  half-wave-length  was,  as  a  mean  of  six  concordant  readings,  31'64  millims. 
The  density  of  this  sample  of  argon,  which  had  been  transferred  from  a  water 
gas-holder  to  a  mercury  gas-holder,  was  19'82 ;  and  there  is  some  reason  to 
suspect  the  presence  of  a  trace  of  air,  for  it  had  been  standing  for  some  time. 

The  result,  however,  substantially  proves  that  the  ratio  previously  found 
was  correct.  In  the  wide  tube,  Gp  :  Cv  ::  T61  :  1.  Hence  the  conclusion 
must  be  accepted  that  the  ratio  of  specific  heats  is  practically  T66  :  1. 

It  will  be  noticed  that  this  is  the  theoretical  ratio  for  a  monatomic  gas, 
that  is,  a  gas  in  which  all  energy  imparted  to  it  at  constant  volume  is  ex- 
pended in  effecting  translational  motion.  The  only  other  gas  of  which  the 
ratio  of  specific  heats  has  been  found  to  fulfil  this  condition  is  mercury  at  a 
high  temperature*.  The  extreme  importance  of  these  observations  will  be 
discussed  later. 


15.     Attempts  to  induce  Chemical  Combination. 

A  great  number  of  attempts  were  made  to  induce  chemical  combination 
with  the  argon  obtained  by  use  of  magnesium,  but  without  any  positive 
result.  In  such  a  case  as  this,  however,  it  is  necessary  to  chronicle  negative 
results,  if  for  no  other  reason  but  that  of  justifying  its  name,  "argon."  These 
will  be  detailed  in  order. 

(a)  Oxygen  in  Presence  of  Caustic  Alkali. — This  need  not  be  further 
discussed  here ;  the  method  of  preparing  argon  is  based  on  its  inactivity 
under  such  conditions. 

*  Kundt  and  Warburg,  Pogg.  Ann.  157,  p.  353,  1876. 


1895]  ARGON,   A   NEW   CONSTITUENT   OF  THE   ATMOSPHERE.  177 

(6)  Hydrogen. — It  has  been  mentioned  that,  in  order  to  free  argon  from 
excess  of  oxygen,  hydrogen  was  admitted,  and  sparks  passed  to  cause  combi- 
nation of  hydrogen  and  oxygen.  Here  again  caustic  alkali  was  present,  and 
argon  appeared  to  be  unaffected. 

A  separate  experiment  was,  however,  made  in  absence  of  water,  though 
no  special  pains  was  taken  to  dry  the  mixture  of  gases.  The  argon  was 
admitted  up  to  half  an  atmosphere  pressure  into  a  bulb,  through  whose  sides 
passed  platinum  wires,  carrying  pointed  poles  of  gas-carbon.  Hydrogen  was 
then  admitted  until  atmospheric  pressure  had  been  attained.  Sparks  were 
then  passed  for  four  hours  by  means  of  a  large  induction  coil,  actuated  by 
four  storage  cells.  The  gas  was  confined  in  a  bulb  closed  by  two  stop-cocks, 
and  a  small  V-tube  with  bulbs  was  interposed,  to  act  as  a  gauge,  so  that  if 
expansion  or  contraction  had  taken  place,  the  escape  or  entry  of  gas  would  be 
observable.  The  apparatus,  after  the  passage  of  sparks,  was  allowed  to  cool 
to  the  temperature  of  the  atmosphere,  and,  on  opening  the  stop-cock,  the 
level  of  water  in  the  V-tube  remained  unaltered.  It  may  therefore  be  con- 
cluded that,  in  all  probability,  no  combination  has  occurred ;  or,  that  if  it  has, 
it  was  attended  with  no  change  of  volume. 

(c)  Chlorine. — Exactly  similar   experiments   were  performed  with  dry, 
and  afterwards  with  moist,  chlorine.    The  chlorine  had  been  stored  over  strong 
sulphuric  acid  for  the  first  experiment,  and  came  in  contact  with  dry  argon. 
Three  hours  sparking  produced  no  change  of  volume.     A  drop  of  water  was 
admitted  into  the  bulb.     After  four  hours  sparking,  the  volume  of  the  gas, 
after  cooling,  was  diminished  by  about  ^  cub.  centim.,  due  probably  to  the 
solution  of  a  little  chlorine  in  the  small  quantity  of  water  present. 

(d)  Phosphorus. — A  piece  of  combustion-tubing,  closed  at  one  end,  con- 
taining at  the  closed  end  a  small  piece  of  phosphorus,  was  sealed  to  the 
mercury  reservoir  containing  argon ;  connected  to  the  same  reservoir  was  a 
mercury  gauge  and  a  Sprengel's  pump.     After  removing  all  air  from  the 
tubes,  argon  was  admitted  to  a  pressure  of  600  millims.     The  middle  portion 
of  the  combustion-tube  was  then  heated  to  bright  redness,  and  the  phosphorus 
was  distilled  slowly  from  back  to  front,  so  that  its  vapour  should  come  into 
contact  with  argon  at  a  red  heat.     When  the  gas  was  hot,  the  level  of  the 
gauge  altered ;  but,  on  cooling,  it  returned  to  its  original  level,  showing  that 
no  contraction  had  taken  place.     The  experiment  was  repeated  several  times, 
the  phosphorus  being  distilled  through  the  red-hot  tube  from  open  to  closed 
end,  and  vice  versa.     In  each  case,  on  cooling,  no  change  of  pressure  was 
remarked.     Hence  it  may  be  concluded  that  phosphorus  at  a  red  heat  is 
without  action  on  argon.    It  may  be  remarked  parenthetically  that  no  gaseous 
compound  of  phosphorus  is  known,  which  does  not  possess  a  volume  different 
from  the  sum  of  those  of  its  constituents.     That  no  solid  compound  was 

R.   TV.  12 


178  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

formed  is  sufficiently  proved  by  the  absence  of  contraction.     The  phosphorus 
was  largely  converted  into  the  red  modification  during  the  experiment. 

(e)  Sulphur. — An  exactly  similar  experiment  was  performed  with  sulphur, 
again  with  negative  results.     It  may  therefore  be  concluded  that  sulphur  and 
argon  are  without  action  on  each  other  at  a  red  heat.     And  again,  no  gaseous 
compound  of  sulphur  is  known  in  which  the  volume  of  the  compound  is  equal 
to  the  sum  of  those  of  its  constituents. 

(f)  Tellurium. — As  this  element  has  a  great  tendency  to  unite  with 
heavy  metals,  it  was  thought  worth  while  to  try  its  action.     In  this,  and  in 
the  experiments  to  be  described,  a  different  form  was  given  to  the  apparatus. 
The  gas  was  circulated  over  the  reagent  employed,  a  tube  containing  it  being 
placed  in  the  circuit.     The  gas  was  dried  by  passage  over  soda-lime  and 
phosphoric  anhydride ;  it  then  passed  over  the  tellurium  or  other  reagent, 
then  through  drying  tubes,  and  then  back  to  the  gas-holder.     That  combina- 
tion did  not  occur  was  shown  by  the  unchanged  volume  of  gas  in  the  gas- 
holder ;  and  it  was  possible,  by  means  of  the  graduated  cylinder  which  ad- 
mitted water  to  the  gas-holder,  to  judge  of  as  small  an  absorption  as  half 
a  cubic  centimetre.     The  tellurium  distilled  readily  in  the  gas,  giving  the 
usual  yellow  vapours ;  and  it  condensed,  quite  unchanged,  as  a  black  subli- 
mate.    The  volume  of  the  gas,  when  all  was  cold,  was  unaltered. 

(g)  Sodium. — A  piece  of  sodium,  weighing  about  half  a  gramme,  was 
heated  in  argon.     It  attacked  the  glass  of  the  combustion  tube,  which  it 
blackened,  owing  to  liberation  of  silicon ;  but  it  distilled  over  in  drops  into 
the  cold  part  of  the  tube.     Again  no  change  of  volume  occurred,  nor  was  the 
surface  of  the  distilled  sodium  tarnished;  it  was  brilliant,  as  it  is  when  sodium 
is  distilled  in  vacuo.    It  may  probably  also  be  concluded  from  this  experiment 
that  silicon,  even  while  being  liberated,  is  without  action  on  argon. 

The  action  of  compounds  was  then  tried ;  those  chosen  were  such  as  lead 
to  oxides  or  sulphides.  Inasmuch  as  the  platinum-metals,  which  are  among 
the  most  inert  of  elements,  are  attacked  by  fused  caustic  soda,  its  action  was 
investigated. 

(h)  Fused  and  Red-hot  Caustic  Soda. — The  soda  was  prepared  from 
sodium,  in  an  iron  boat,  by  adding  drops  of  water  cautiously  to  a  lump  of  the 
metal.  When  action  had  ceased,  the  soda  was  melted,  and  the  boat  intro- 
duced into  a  piece  of  combustion-tube  placed  in  the  circuit.  After  three 
hours  circulation  no  contraction  had  occurred.  Hence  caustic  soda  has  no 
action  on  argon. 

(i)  Soda-lime  at  a  red  heat. — Thinking  that  the  want  of  porosity  of  fused 
caustic  soda  might  have  hindered  absorption,  a  precisely  similar  experiment 
was  carried  out  with  soda-lime,  a  mixture  which  can  be  heated  to  bright 
redness  without  fusion.  Again  no  result  took  place  after  three  hours  heating. 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE    ATMOSPHERE.  179 

(j)  Fused  Potassium  Nitrate  was  tried  under  the  impression  that  oxygen 
plus  a  base  might  act  where  oxygen  alone  failed.  The  nitrate  was  fused,  and 
kept  at  a  bright  red  heat  for  two  hours,  but  again  without  any  diminution  in 
volume  of  the  argon. 

(k)  Sodium  Peroxide. — Yet  another  attempt  was  made  to  induce  combi- 
nation with  oxygen  and  a  base,  by  heating  sodium  peroxide  to  redness  in  a 
current  of  argon  for  over  an  hour,  but  also  without  effect.  It  is  to  be  noticed 
that  metals  of  the  platinum  group  would  have  entered  into  combination 
under  such  treatment. 

(1)  Persulphides  of  Sodium  and  Calcium. — Soda-lime  was  heated  to 
redness  in  an  open  crucible,  and  some  sulphur  was  added  to  the  red-hot  mass, 
the  lid  of  the  crucible  being  then  put  on.  Combination  ensued,  with  forma- 
tion of  polysulphides  of  sodium  and  calcium.  This  product  was  heated  to 
redness  for  three  hours  in  a  brisk  current  of  argon,  again  with  negative  result. 
Again,  metals  of  the  platinum  group  would  have  combined  under  such  treat- 
ment. 

(ni)  Some  argon  was  shaken  in  a  tube  with  nitro-hydrochloric  acid. 
On  addition  of  potash,  so  as  to  neutralise  the  acid,  and  to  absorb  the  free 
chlorine  and  nitrosyl  chloride,  the  volume  of  the  gas  was  barely  altered.  The 
slight  alteration  was  evidently  due  to  solubility  in  the  aqueous  liquid,  and  it 
may  be  concluded  that  no  chemical  action  took  place. 

(n)  Bromine-water  was  also  without  effect.  The  bromine  vapour  was 
removed  with  potash. 

(o)  A  mixture  of  potassium  permanganate  and  hydrochloric  acid,  involv- 
ing the  presence  of  nascent  chlorine,  had  no  action,  for  on  absorbing  chlorine 
by  means  of  potash,  no  alteration  in  volume  had  occurred. 

(p)  Argon  is  not  absorbed  by  platinum  black.  A  current  was  passed 
over  a  pure  specimen  of  this  substance;  as  usual,  however,  it  contained 
occluded  oxygen.  There  was  no  absorption  in  the  cold.  At  100°  no  action 
took  place ;  and  on  heating  to  redness,  by  which  the  black  was  changed  to 
sponge,  still  no  evidence  of  absorption  was  noticed.  In  all  these  experiments, 
absorption  of  half  a  cubic  centimetre  of  argon  could  have  at  once  been 
detected. 

We  do  not  claim  to  have  exhausted  the  possible  reagents.  But  this  much 
is  certain,  that  the  gas  deserves  the  name  "  argon,"  for  it  is  a  most  astonish- 
ingly indifferent  body,  inasmuch  as  it  is  unattacked  by  elements  of  very 
opposite  character,  ranging  from  sodium  and  magnesium  on  the  one  hand,  to 
oxygen,  chlorine,  and  sulphur  on  the  other.  It  will  be  interesting  to  see  if 
fluorine  also  is  without  action,  but  for  the  present  that  experiment  must  be 
postponed,  on  account  of  difficulties  of  manipulation. 

12—2 


180  ARGON,   A    NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  [214 

It  will  also  be  necessary  to  try  whether  the  inability  of  argon  to  combine 
at  ordinary  or  at  high  temperatures  is  due  to  the  instability  of  its  possible 
compounds,  except  when  cold.  Mercury  vapour  at  800°  would  present  a 
similar  instance  of  passive  behaviour. 


16.     General  Conclusions. 

It  remains,  finally,  to  discuss  the  probable  nature  of  the  gas  or  gases 
which  we  have  succeeded  in  separating  from  atmospheric  air,  and  which  has 
been  provisionally  named  argon. 

That  argon  is  present  in  the  atmosphere,  and  is  not  manufactured  during 
the  process  of  separation  is  amply  proved  by  many  lines  of  evidence.  First, 
atmospheric  nitrogen  has  a  high  density,  while  chemical  nitrogen  is  lighter. 
That  chemical  nitrogen  is  a  uniform  substance  is  proved  by  the  identity  of 
properties  of  samples  prepared  by  several  different  processes,  and  from  several 
different  compounds.  It  follows,  therefore,  that  the  cause  of  the  high  density 
of  atmospheric  nitrogen  is  due  to  the  admixture  with  heavier  gas.  If  that 
gas  possesses  the  density  of  20  compared  with  hydrogen  as  unity,  atmospheric 
nitrogen  should  contain  of  it  approximately  1  per  cent.  This  is  found  to  be 
the  case,  for  on  causing  the  nitrogen  of  the  atmosphere  to  combine  with 
oxygen  in  presence  of  alkali,  the  residue  amounted  to  about  1  per  cent. ;  and 
on  removing  nitrogen  with  magnesium  the  result  is  similar. 

Second :  This  gas  has  been  concentrated  in  the  atmosphere  by  diffusion. 
It  is  true  that  it  cannot  be  freed  from  oxygen  and  nitrogen  by  diffusion,  but 
the  process  of  diffusion  increases  relatively  to  nitrogen  the  amount  of  argon 
in  that  portion  which  does  not  pass  through  the  porous  walls.  That  this  is 
the  case  is  proved  by  the  increase  of  density  of  that  mixture  of  argon  and 
nitrogen. 

Third :  On  removing  nitrogen  from  "  atmospheric  nitrogen  "  by  means  of 
magnesium,  the  density  of  the  residue  increases  proportionately  to  the  concen- 
tration of  the  heavier  constituent. 

Fourth :  As  the  solubility  of  argon  in  water  is  relatively  high,  it  is  to  be 
expected  that  the  density  of  the  mixture  of  argon  and  nitrogen,  pumped  out 
of  water  along  with  oxygen  should,  after  removal  of  the  oxygen,  exceed  that 
of  "  atmospheric  nitrogen."  Experiment  has  shown  that  the  density  is  con- 
siderably increased. 

Fifth:  It  is  in  the  highest  degree  improbable  that  two  processes,  so 
different  from  each  other,  should  each  manufacture  the  same  product.  The 
explanation  is  simple  if  it  be  granted  that  these  processes  merely  eliminate 
nitrogen  from  "  atmospheric  nitrogen." 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  181 

Sixth :  If  the  newly  discovered  gas  were  not  in  the  atmosphere,  the  dis- 
crepancies in  the  density  of  "  chemical "  and  "  atmospheric  "  nitrogen  would 
remain  unexplained. 

Seventh :  It  has  been  shown  that  pure  nitrogen,  prepared  from  its  com- 
pounds, leaves  a  negligible  residue  when  caused  to  enter  into  combination 
with  oxygen  or  with  magnesium. 

There  are  other  lines  of  argument  which  suggest  themselves;  but  we 
think  that  it  will  be  acknowledged  that  those  given  above  are  sufficient  to 
establish  the  existence  of  argon  in  the  atmosphere. 

It  is  practically  certain  that  the  argon  prepared  by  means  of  electric 
sparking  with  oxygen  is  identical  with  argon  prepared  by  means  of  magne- 
sium. The  samples  have  in  common : — 

First :  Spectra  which  have  been  found  by  Mr  Crookes,  Professor  Schuster, 
and  ourselves  to  be  practically  identical. 

Second :  They  have  approximately  the  same  density.  The  density  of 
argon,  prepared  by  means  of  magnesium,  was  19'9  ;  that  of  argon,  from  spark- 
ing with  oxygen,  about  197  ;  these  numbers  are  practically  identical. 

Third :  Their  solubility  in  water  is  the  same. 

That  argon  is  an  element,  or  a  mixture  of  elements,  may  be  inferred  from 
the  observations  of  §  14.  For  Clausius  has  shown  that  if  K  be  the  energy  of 
translatory  motion  of  the  molecules  of  a  gas,  and  H  their  whole  kinetic  energy, 

then 

K     3(CP-CV) 
H~  ~^CV       ' 

Cp  and  Cv  denoting  as  usual  the  specific  heat  at  constant  pressure  and  at 
constant  volume  respectively.  Hence,  if,  as  for  mercury  vapour  and  for  argon 
(§  14),  the  ratio  of  specific  heats  Gp  :  Cv  be  If,  it  follows  that  K  =  H,  or  that 
the  whole  kinetic  energy  of  the  gas  is  accounted  for  by  the  translatory  motion 
of  its  molecules.  In  the  case  of  mercury  the  absence  of  interatomic  energy 
is  regarded  as  proof  of  the  monatomic  character  of  the  vapour,  and  the 
conclusion  holds  equally  good  for  argon. 

The  only  alternative  is  to  suppose  that  if  argon  molecules  are  di-  or  poly- 
atomic, the  atoms  acquire  no  relative  motion,  even  of  rotation,  a  conclusion 
improbable  in  itself  and  one  postulating  the  sphericity  of  such  complex  groups 
of  atoms. 

Now  a  monatomic  gas  can  be  only  an  element,  or  a  mixture  of  elements ; 
and  hence  it  follows  that  argon  is  not  of  a  compound  nature. 

According  to  Avogadro,  equal  volumes  of  gases  at  the  same  temperature 
and  pressure  contain  equal  numbers  of  molecules.  The  molecule  of  hydrogen 


182         ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE.        [214 

gas,  the  density  of  which  is  taken  as  unity,  is  supposed  to  consist  of  two 
atoms.  Its  molecular  weight  is  therefore  2.  Argon  is  approximately  20 
times  as  heavy  as  hydrogen,  that  is,  its  molecular  weight  is  20  times  as  great 
as  that  of  hydrogen,  or  40.  But  its  molecule  is  monatomic,  hence  its  atomic 
weight,  or,  if  it  be  a  mixture,  the  mean  of  the  atomic  weights  of  the  elements 
in  that  mixture,  taken  for  the  proportion  in  which  they  are  present,  must 
be  40. 

This  conclusion  rests  on  the  assumption  that  all  the  molecules  of  argon 
are  monatomic.  The  result  of  the  first  experiment  is,  however,  so  nearly  that 
required  by  theory,  that  there  is  room  for  only  a  small  number  of  molecules 
of  a  different  character.  A  study  of  the  expansion  of  argon  by  heat  is  pro- 
posed, and  would  doubtless  throw  light  upon  this  question. 

There  is  evidence  both  for  and  against  the  hypothesis  that  argon  is  a 
mixture :  for,  owing  to  Mr  Crookes'  observations  of  the  dual  character  of  its 
spectrum ;  against,  because  of  Professor  Olszewski's  statement  that  it  has  a 
definite  melting-point,  a  definite  boiling-point,  and  a  definite  critical  tem- 
perature and  pressure ;  and  because  on  compressing  the  gas  in  presence  of  its 
liquid,  pressure  remains  sensibly  constant  until  all  gas  has  condensed  to 
liquid.  The  latter  experiments  are  the  well-known  criteria  of  a  pure  sub- 
stance; the  former  is  not  known  with  certainty  to  be  characteristic  of  a 
mixture.  The  conclusions  which  follow  are,  however,  so  startling,  that  in  our 
future  experimental  work  we  shall  endeavour  to  decide  the  question  by  other 
means. 

For  the  present,  however,  the  balance  of  evidence  seems  to  point  to  sim- 
plicity. We  have,  therefore,  to  discuss  the  relations  to  other  elements  of  an 
element  of  atomic  weight  40.  We  inclined  for  long  to  the  view  that  argon 
was  possibly  one,  or  more  than  one,  of  the  elements  which  might  be  expected 
to  follow  fluorine  in  the  periodic  classification  of  the  elements — elements 
which  should  have  an  atomic  weight  between  19,  that  of  fluorine,  and  23, 
that  of  sodium.  But  this  view  is  apparently  put  out  of  court  by  the  discovery 
of  the  monatomic  nature  of  its  molecules. 

The  series  of  elements  possessing  atomic  weights  near  40  are : — 

Chlorine 35'5 

Potassium 39'1 

Calcium 40'0 

Scandium 44'0 

There  can  be  no  doubt  that  potassium,  calcium,  and  scandium  follow 
legitimately  their  predecessors  in  the  vertical  columns,  lithium,  beryllium,  and 
boron,  and  that  they  are  in  almost  certain  relation  with  rubidium,  strontium, 
and  (but  not  so  certainly)  yttrium.  If  argon  be  a  single  element,  then  there 


1895]  ARGON,   A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE.  183 

is  reason  to  doubt  whether  the  periodic  classification  of  the  elements  is  com- 
plete ;  whether,  in  fact,  elements  may  not  exist  which  cannot  be  fitted  among 
those  of  which  it  is  composed.  On  the  other  hand,  if  argon  be  a  mixture  of 
two  elements,  they  might  find  place  in  the  eighth  group,  one  after  chlorine 
and  one  after  bromine.  Assuming  37  (the  approximate  mean  between  the 
atomic  weights  of  chlorine  and  potassium)  to  be  the  atomic  weight  of  the 
lighter  element,  and  40  the  mean  atomic  weight  found,  and  supposing  that 
the  second  element  has  an  atomic  weight  between  those  of  bromine,  80,  and 
rubidium,  85*5,  viz.  82,  the  mixture  should  consist  of  93'3  per  cent,  of  the 
lighter,  and  6'7  per  cent,  of  the  heavier  element.  But  it  appears  improbable 
that  such  a  high  percentage  as  6'7  of  a  heavier  element  should  have  escaped 
detection  during  liquefaction. 

If  the  atomic  weight  of  the  lighter  element  were  38,  instead  of  37,  how- 
ever, the  proportion  of  heavier  element  would  be  considerably  reduced.  Still, 
it  is  difficult  to  account  for  its  not  having  been  detected,  if  present. 

If  it  be  supposed  that  argon  belongs  to  the  eighth  group,  then  its  proper- 
ties would  fit  fairly  well  with  what  might  be  anticipated.  For  the  series, 
which  contains 

SiIV,     Pni*ndv      S™VI,    and    ClItovn 


might  be  expected  to  end  with  an  element  of  monatomic  molecules,  of  no 
valency,  i.e.  incapable  of  forming  a  compound,  or  if  forming  one,  being  an 
octad  ;  and  it  would  form  a  possible  transition  to  potassium,  with  its  mono- 
valence,  on  the  other  hand.  Such  conceptions  are,  however,  of  a  speculative 
nature  ;  yet  they  may  be  perhaps  excused,  if  they  in  any  way  lead  to  experi- 
ments which  tend  to  throw  more  light  on  the  anomalies  of  this  curious 
element. 

In  conclusion,  it  need  excite  no  astonishment  that  argon  is  so  indifferent 
to  reagents.  For  mercury,  although  a  monatomic  element,  forms  compounds 
which  are  by  no  means  stable  at  a  high  temperature  in  the  gaseous  state  ; 
and  attempts  to  produce  compounds  of  argon  may  be  likened  to  attempts  to 
cause  combination  between  mercury  gas  at  800°  and  other  elements.  As  for 
the  physical  condition  of  argon,  that  of  a  gas,  we  possess  no  knowledge  why 
carbon,  with  its  low  atomic  weight,  should  be  a  solid,  while  nitrogen  is  a  gas, 
except  in  so  far  as  we  ascribe  molecular  complexity  to  the  former  and  com- 
parative molecular  simplicity  to  the  latter.  Argon,  with  its  comparatively 
low  density  and  its  molecular  simplicity,  might  well  be  expected  to  rank 
among  the  gases.  And  its  inertness,  which  has  suggested  its  name,  suffi- 
ciently explains  why  it  has  not  previously  been  discovered  as  a  constituent  of 
compound  bodies. 

We  would  suggest  for  this  element,  assuming  provisionally  that  it  is  not 
a  mixture,  the  symbol  A. 


184 


ARGON,    A   NEW   CONSTITUENT   OF   THE   ATMOSPHERE. 


[214 


We  have  to  record  our  thanks  to  Messrs  Gordon,  Kellas,  and  Matthews, 
and  especially  to  Mr  Percy  Williams,  for  their  assistance  in  the  prosecution 
of  this  research. 

ADDENDUM  (by  Professor  W.  Ramsay). 
March  20,  1895. 

Further  determinations  of  the  density  of  argon  prepared  by  means  of 
magnesium  have  been  made.  In  each  case  the  argon  was  circulated  over 
magnesium  for  at  least  two  hours  after  all  absorption  of  nitrogen  had  stopped, 
as  well  as  over  red-hot  copper,  copper  oxide,  soda-lime,  and  phosphoric  anhy- 
dride. The  gas  also  passed  out  of  the  mercury  gas-holder  through  phosphoric 
anhydride  into  the  weighing  globe.  The  results  are  in  complete  accordance 
with  previous  determinations  of  density ;  and  for  convenience  of  reference  the 
former  numbers  are  included  in  the  table  which  follows. 

DENSITY  OF  ARGON. 


Date 

Volume 

Tempera- 
ture 

Pressure 

Weight 

Weight  of 
1  litre  at 
0°  and  760 
millims. 

Density 
(0  =  16) 

(1)  Nov.  26  

cub.  centims. 
162-843 

15°00 

millims. 
767-7 

grm. 
0-2773 

1-7784 

19-904 

(2)      „     27  

162-843 

16-00 

769-0 

0-2757 

1-7717 

19-823 

(3)  Dec.  22  

162-843 

15-62 

750-1 

0-26915 

1-7704 

19-816 

(4)  Feb.  16  

162-843 

13-45 

771-1 

0-2818 

1-7834 

19-959 

(5)      „     19  

162-843 

14-47 

768-2 

0-2789 

1-7842 

19-969 

(6)      „     24  

162-843 

17-85 

764-4 

0-2738 

1-7810 

19-932 

The  general  mean  is  19*900;  or  if  Nos.  (2)  and  (3)  be  rejected  as  sus- 
piciously low,  the  mean  of  the  remaining  four  determinations  is  19'941.  The 
molecular  weight  may  therefore  be  taken  as  39*9  without  appreciable  error. 

The  value  of  R  in  the  gas-equation  R=pv/T  has  also  been  determined 
between  —  89°  and  +  248°.  For  this  purpose,  a  gas-thermometer  was  filled 
with  argon,  and  a  direct  comparison  was  made  with  a  similar  thermometer 
filled  with  hydrogen. 

The  method  of  using  such  a  hydrogen-thermometer  has  already  been 
described  by  Ramsay  and  Shields*.  For  the  lowest  temperature,  the  ther- 
mometer bulbs  were  immersed  in  boiling  nitrous  oxide;  for  atmospheric 
temperature,  in  running  water ;  for  temperatures  near  100°  in  steam,  and  for 
the  remaining  temperatures,  in  the  vapours  of  chlorobenzene,  aniline,  and 
quinolene. 

*  Trans.  Chem.  Soc.  Vol.  63,  pp.  835,  836.  It  is  to  be  noticed  that  the  value  of  R  is  not 
involved  in  using  the  hydrogen-thermometer ;  its  constancy  alone  is  postulated. 


1895]       ARGON,  A  NEW  CONSTITUENT  OF  THE  ATMOSPHERE. 

The  results  are  collected  in  the  following  tables  : — 
HYDROGEN  THERMOMETER. 


185 


Temperature 

Pressure 

Volume  (corr.) 

E 

13-04 

millims. 
763-6 

1-00036 

2-6705 

99-84 

992-6 

1-00280 

2-6697 

130-62 

1073-8 

1-00364 

2-6701 

185-46 

1218-5 

1-00518 

2-6716 

248-66 

1385-1 

1-00703 

2-6737 

-87-92 

497-3 

0-99756 

2-6804 

The  value  of  R  is  thus  practically  constant,  and  this  affords  a  proof  that- 
the  four  last  temperatures  have  been  estimated  with  considerable  accuracy. 

ARGON  THERMOMETER. 


Temperature 

Pressure 

Volume  (corr.) 

R 

Series  I  

14-15 

millims. 
701-7 

1-000396 

2-4446 

14-27 

699-7 

1-000401 

2-4366 

14-40 

702-6 

1-000404 

2-4462 

19-96 

906-5 

1-00280 

2-4379 

100-06 

904-8 

1-00280 

2-4322 

-87-92 

455-6 

0-99756 

2-4556 

By  mischance,  air  leaked  into  the  bulb  ;  it  was  therefore  refilled. 

Series  II.  ... 

130-58 

1060-0 

1-0037 

2-6363 

185-46 

1200-3 

1-0052 

2-6317 

A  bubble  of  argon  leaked  into  the  bulb,  and  the  value  of  R  increased. 

Series  III.... 

12-05 

760-9 

1-00034 

2-6698 

12-61 

761-3 

1-00034 

2-6728 

248-66 

1384-0 

1-0070 

2-6717 

248-66 

1376-9 

1-0070 

2-6580 

-  87-92 

495-7 

0-99756 

2-6718 

It  may  be  concluded  from  these  numbers,  that  argon  undergoes  no  mole- 
cular change  between  —  88°  and  +  250°. 

Further  determinations  of  the  wave-length  of  sound  in  argon  have  been 
made,  the  wider  tube  having  been  used.     In  every  case  the  argon  was  as 


186 


ARGON,   A   NEW   CONSTITUENT   OF   THE    ATMOSPHERE. 


[214 


carefully  purified  as  possible.  In  experiment  (3)  too  much  lycopodium  dust 
was  present  in  the  tube ;  that  is  perhaps  the  cause  of  the  low  result.  For 
completeness'  sake,  the  original  result  in  the  narrow  tube  has  also  been  given. 


Date 

Density 

Half-  wave-length 

Temperature 

Ratio 

In  air 

In  argon 

Air 

Argon 

Dec   6 

19-92 
19-96 
19-97 
19-94 

19-59 
33-73 
34-10 
34-23 

18-08 
31-00 
31-31 
31-68 

17°5 
67 
7-22 
11-20 

17°5 
6-5 
8-64 
11-49 

1-644 
1-641 
1-629 
1-659 

Feb    15  

„     20  

Mar  19 

The  general  mean  of  these  numbers  is  1*643;  if  (3)  be  rejected,  it  is  1*648. 
In  the  last  experiment  every  precaution  was  taken.  The  half-wave-length 
in  air  is  the  mean  of  11  readings,  the  highest  of  which  was  34*67  and  the 
lowest  34-00.  They  run  :— 

34-67  ;  34-06  ;  34-27  ;  34'39 ;  34*00 ;  34*00 ;  34*13 ;  34-20 ;  34*20 ;  34*33 ;  34-33. 
11*25°;  11-00°;  10*80°;  10'8°;  10*0°  ;  11*0°  ;  11*3° ;  11*4°  ;  11-4°;  11*6°  ;  11*6°. 

With  argon  the  mean  is  also  that  of  11  readings,  of  which  the  highest  is 
31-83,  and  the  lowest,  31*5.  They  are  :— 

31*5  ;31-5  ;  31*66  ;  31*55  ;  31*83  ;  31*77  ;  31-81  ;  31*83  ;  31*83 ;  31-50;  31'66. 
11*8° ;  11-8° ;  11-20° ;  11*40° ;  11*60° ;  11*40° ;  11-40°;  11-4°  ;  11*5°  ;  11*5°  ;  11*4°. 

If  the  atomic  weight  of  argon  is  identical  with  its  molecular  weight,  it 
must  closely  approximate  to  39*9.  But  if  there  were  some  molecules  of  A2 
present,  mixed  with  a  much  larger  number  of  molecules  of  A];  then  the 
atomic  weight  would  be  correspondingly  reduced.  Taking  an  imaginary  case, 
the  question  may  be  put: — What  percentage  of  molecules  of  A2  would  raise 
the  density  of  A1  from  19*0  to  19*9  ?  A  density  of  19*0  would  imply  an 
atomic  weight  of  38*0,  and  argon  would  fall  into  the  gap  between  chlorine 
and  potassium.  Calculation  shows  that  in  10,000  molecules,  474  molecules 
of  A.J  would  have  this  result,  the  remaining  9526  molecules  being  those  of  Ax. 

Now  if  molecules  of  A^  be  present,  it  is  reasonable  to  suppose  that  their 
number  would  be  increased  by  lowering  the  temperature,  and  diminished  by 
heating  the  gas.  A  larger  change  of  density  should  ensue  on  lowering  than 
on  raising  the  temperature,  however,  as  on  the  above  supposition,  there  is 
not  a  large  proportion  of  molecules  of  A2  present. 

But  it  must  be  acknowledged  that  the  constancy  of  the  found  value  of  R 
is  not  favourable  to  this  supposition. 


1895]  ARGON,   A    NEW   CONSTITUENT   OF   THE    ATMOSPHERE.  187 

A  similar  calculation  is  possible  for  the  ratio  of  specific  heats.  Assuming 
the  gas  to  contain  5  per  cent,  of  molecules  of  A2,  and  95  per  cent,  of  mole- 
cules of  AI  the  value  of  7,  the  ratio  of  specific  heats,  would  be  1*648.  All 
that  can  be  said  on  this  point  is,  that  the  found  ratio  approximates  to  this 
number ;  but  whether  the  results  are  to  be  trusted  to  indicate  a  unit  in  the 
second  decimal  appears  to  me  doubtful. 

The  question  must  therefore  for  the  present  remain  open. 


ADDENDUM. 
April  9. 

It  appears  worth  while  to  chronicle  an  experiment  of  which  an  accident 
prevented  the  completion.  It  may  be  legitimately  asked,  Does  magnesium 
not  absorb  any  argon,  or  any  part  of  what  we  term  argon  ?  To  decide  this 
question,  about  500  grms.  of  magnesium  nitride,  mixed  with  metallic  mag- 
nesium which  had  remained  unacted  on,  during  extraction  of  nitrogen  from 
"  air-nitrogen,"  was  placed  in  a  flask,  to  which  a  reservoir  full  of  dilute  hydro- 
chloric acid  was  connected.  The  flask  was  coupled  with  a  tube  full  of  red-hot 
copper  oxide,  intended  to  oxidise  the  hydrogen  which  would  be  evolved  by 
the  action  of  the  hydrochloric  acid  on  the  metallic  magnesium.  To  the  end 
of  the  copper  oxide  tube  a  gas-holder  was  attached,  so  as  to  collect  any 
evolved  gas ;  and  the  system  was  attached  to  a  vacuum-pump,  in  order  to 
exhaust  the  apparatus  before  commencing  the  experiment,  as  well  as  to 
collect  all  gas  which  should  be  evolved,  and  remain  in  the  flask. 

On  admitting  hydrochloric  acid  to  the  flask  of  magnesium  nitride  a  violent 
reaction  took  place,  and  fumes  of  ammonium  chloride  passed  into  the  tube  of 
copper  oxide.  These  gave,  of  course,  free  nitrogen.  This  had  not  been  fore- 
seen ;  it  would  have  been  well  to  retain  these  fumes  by  plugs  of  glass-wool. 
The  result  of  the  experiment  was  that  about  200  cub.  centims.  of  gas  were 
collected.  After  sparking  with  oxygen  in  presence  of  caustic  soda,  the  volume 
was  reduced  to  3  cub.  centims.  of  a  gas  which  appeared  to  be  argon. 


215. 

ARGON. 

[Royal  Institution  Proceedings,  xiv.  pp.  524—538,  Ap.  1895.] 

IT  is  some  three  or  four  years  since  I  had  the  honour  of  lecturing  here  one 
Friday  evening  upon  the  densities  of  oxygen  and  hydrogen  gases,  and  upon 
the  conclusions  that  might  be  drawn  from  the  results.  It  is  not  necessary, 
therefore,  that  I  should  trouble  you  to-night  with  any  detail  as  to  the  method 
by  which  gases  can  be  accurately  weighed.  I  must  take  that  as  known, 
merely  mentioning  that  it  is  substantially  the  same  as  is  used  by  all  investi- 
gators nowadays,  and  introduced  more  than  fifty  years  ago  by  Regnault.  It 
was  not  until  after  that  lecture  that  I  turned  my  attention  to  nitrogen ;  and 
in  the  first  instance  I  employed  a  method  of  preparing  the  gas  which  originated 
with  Mr  Vernon  Harcourt,  of  Oxford.  In  this  method  the  oxygen  of  ordinary 
atmospheric  air  is  got  rid  of  with  the  aid  of  ammonia.  Air  is  bubbled  through 
liquid  ammonia,  and  then  passed  through  a  red-hot  tube.  In  its  passage  the 
oxygen  of  the  air  combines  with  the  hydrogen  of  the  ammonia,  all  the  oxygen 
being  in  that  way  burnt  up  and  converted  into  water.  The  excess  of  ammonia 
is  subsequently  absorbed  with  acid,  and  the  water  by  ordinary  desiccating 
agents.  That  method  is  very  convenient ;  and,  when  I  had  obtained  a  few 
concordant  results  by  means  of  it,  I  thought  that  the  work  was  complete,  and 
that  the  weight  of  nitrogen  was  satisfactorily  determined.  But  then  I 
reflected  that  it  is  always  advisable  to  employ  more  than  one  method,  and 
that  the  method  that  I  had  used — Mr  Vernon  Harcourt's  method — was  not 
that  which  had  been  used  by  any  of  those  who  had  preceded  me  in  weighing 
nitrogen.  The  usual  method  consists  in  absorbing  the  oxygen  of  air  by  means 
of  red-hot  copper ;  and  I  thought  that  I  ought  at  least  to  give  that  method  a 
trial,  fully  expecting  to  obtain  forthwith  a  value  in  harmony  with  that  already 
afforded  by  the  ammonia  method.  The  result,  however,  proved  otherwise. 
The  gas  obtained  by  the  copper  method,  as  I  may  call  it,  proved  to  be  one- 


1895]  ARGON.  189 

thousandth  part  heavier  than  that  obtained  by  the  ammonia  method ;  and,  on 
repetition,  that  difference  was  only  brought  out  more  clearly.  This  was  about 
three  years  ago.  In  order,  if  possible,  to  get  further  light  upon  a  discrepancy 
which  puzzled  me  very  much,  and  which,  at  that  time,  I  regarded  only  with 
disgust  and  impatience,  I  published  a  letter  in  Nature*  inviting  criticisms 
from  chemists  who  might  be  interested  in  such  questions.  I  obtained  various 
useful  suggestions,  but  none  going  to  the  root  of  the  matter.  Several  persons 
who  wrote  to  me  privately  were  inclined  to  think  that  the  explanation  was  to 
be  sought  in  a  partial  dissociation  of  the  nitrogen  derived  from  ammonia. 
For,  before  going  further,  I  ought  to  explain  that,  in  the  nitrogen  obtained  by 
the  ammonia  method,  some — about  a  seventh  part — is  derived  from  the 
ammonia,  the  larger  part,  however,  being  derived  as  usual  from  the  atmosphere. 
If  the  chemically  derived  nitrogen  were  partly  dissociated  into  its  component 
atoms,  then  the  lightness  of  the  gas  so  prepared  would  be  explained. 

The  next  step  in  the  enquiry  was,  if  possible,  to  exaggerate  the  discrepancy. 
One's  instinct  at  first  is  to  try  to  get  rid  of  a  discrepancy,  but  I  believe  that 
experience  shows  such  an  endeavour  to  be  a  mistake.  What  one  ought  to  do 
is  to  magnify  a  small  discrepancy  with  a  view  to  finding  out  the  explanation ; 
and,  as  it  appeared  in  the  present  case  that  the  root  of  the  discrepancy  lay  in 
the  fact  that  part  of  the  nitrogen  prepared  by  the  ammonia  method  was 
nitrogen  out  of  ammonia,  although  the  greater  part  remained  of  common 
origin  in  both  cases,  the  application  of  the  principle  suggested  a  trial  of  the 
weight  of  nitrogen  obtained  wholly  from  ammonia.  This  could  easily  be  done 
by  substituting  pure  oxygen  for  atmospheric  air  in  the  ammonia  method,  so 
that  the  whole,  instead  of  only  a  part,  of  the  nitrogen  collected  should  be 
derived  from  the  ammonia  itself.  The  discrepancy  was  at  once  magnified 
some  five  times.  The  nitrogen  so  obtained  from  ammonia  proved  to  be  about 
one-half  per  cent,  lighter  than  nitrogen  obtained  in  the  ordinary  way  from  the 
atmosphere,  and  which  I  may  call  for  brevity  "  atmospheric  "  nitrogen. 

That  result  stood  out  pretty  sharply  from  the  first;  but  it  was  necessary 
to  confirm  it  by  comparison  with  nitrogen  chemically  derived  in  other  ways. 
The  table  before  you  gives  a  summary  of  such  results,  the  numbers  being  the 
weights  in  grams  actually  contained  under  standard  conditions  in  the  globe 
employed. 

ATMOSPHERIC  NITROGEN. 

By  hot  copper  (1892) 2'3103 

By  hot  iron  (1893) 2'3100 

By  ferrous  hydrate  (1894) 2'3102 


Mean  2'3102 
[Vol.  iv.  p.  1.] 


190  ARGON.  [215 

CHEMICAL  NITROGEN. 

From  nitric  oxide 2'3001 

From  nitrous  oxide 2'2990 

From  ammonium  nitrite  purified  at  a  red  heat    .     .     .     2'2987 

From  urea 2'2985 

From  ammonium  nitrite  purified  in  the  cold  ....     2'2987 

Mean  2'2990 

The  difference  is  about  11  milligrams,  or  about  one-half  per  cent.;  and  it 
was  sufficient  to  prove  conclusively  that  the  two  kinds  of  nitrogen — the 
chemically  derived  nitrogen  and  the  atmospheric  nitrogen — differed  in  weight, 
and  therefore,  of  course,  in  quality,  for  some  reason  hitherto  unknown. 

I  need  not  spend  time  in  explaining  the  various  precautions  that  were 
necessary  in  order  to  establish  surely  that  conclusion.  One  had  to  be  on  one's 
guard  against  impurities,  especially  against  the  presence  of  hydrogen,  which 
might  seriously  lighten  any  gas  in  which  it  was  contained.  I  believe,  however, 
that  the  precautions  taken  were  sufficient  to  exclude  all  questions  of  that 
sort,  and  the  result,  which  I  published  about  this  time  last  year*,  stood 
sharply  out,  that  the  nitrogen  obtained  from  chemical  sources  was  different 
from  the  nitrogen  obtained  from  the  air. 

Well,  that  difference,  admitting  it  to  be  established,  was  sufficient  to  show 
that  some  hitherto  unknown  gas  is  involved  in  the  matter.  It  might  be  that 
the  new  gas  was  dissociated  nitrogen,  contained  in  that  which  was  too  light, 
the  chemical  nitrogen — and  at  first  that  was  the  explanation  to  which  I 
leaned ;  but  certain  experiments  went  a  long  way  to  discourage  such  a  suppo- 
sition. In  the  first  place,  chemical  evidence — and  in  this  matter  I  am  greatly 
dependent  upon  the  kindness  of  chemical  friends — tends  to  show  that,  even  if 
ordinary  nitrogen  could  be  dissociated  at  all  into  its  component  atoms,  such 
atoms  would  not  be  likely  to  enjoy  any  very  long  continued  existence.  Even 
ozone  goes  slowly  back  to  the  more  normal  state  of  oxygen;  and  it  was 
thought  that  dissociated  nitrogen  would  have  even  a  greater  tendency  to 
revert  to  the  normal  condition.  The  experiment  suggested  by  that  remark 
was  as  follows — to  keep  chemical  nitrogen — the  too  light  nitrogen  which 
might  be  supposed  to  contain  dissociated  molecules — for  a  good  while,  and  to 
examine  whether  it  changed  in  density.  Of  course  it  would  be  useless  to 
shut  up  gas  in  a  globe  and  weigh  it,  and  then,  after  an  interval,  to  weigh  it 
again,  for  there  would  be  no  opportunity  for  any  change  of  weight  to  occur, 
even  although  the  gas  within  the  globe  had  undergone  some  chemical 
alteration.  It  is  necessary  to  re-establish  the  standard  conditions  of  tempera- 
ture and  pressure  which  are  always  understood  when  we  speak  of  filling  a 

*  [Vol.  iv.  p.  104.] 


1895]  ARGON.  191 

globe  with  gas,  for  I  need  hardly  say  that  filling  a  globe  with  gas  is  but  a 
figure  of  speech.  Everything  depends  upon  the  temperature  and  pressure  at 
which  you  work.  However,  that  obvious  point  being  borne  in  mind,  it  was 
proved  by  experiment  that  the  gas  did  not  change  in  weight  by  standing  for 
eight  months — a  result  tending  to  show  that  the  abnormal  lightness  was  not 
the  consequence  of  dissociation. 

Further  experiments  were  tried  upon  the  action  of  the  silent  electric  dis- 
charge— both  upon  the  atmospheric  nitrogen  and  upon  the  chemically  derived 
nitrogen — but  neither  of  them  seemed  to  be  sensibly  affected  by  such 
treatment;  so  that,  altogether,  the  balance  of  evidence  seemed  to  incline 
against  the  hypothesis  of  abnormal  lightness  in  the  chemically  derived 
nitrogen  being  due  to  dissociation,  and  to  suggest  strongly,  as  almost  the  only 
possible  alternative,  that  there  must  be  in  atmospheric  nitrogen  some  con- 
stituent heavier  than  true  nitrogen. 

At  that  point  the  question  arose,  What  was  the  evidence  that  all  the  so- 
called  nitrogen  of  the  atmosphere  was  of  one  quality  ?  And  I  remember — I 
think  it  was  about  this  time  last  year,  or  a  little  earlier — putting  the  question 
to  my  colleague,  Professor  Dewar.  His  answer  was  that  he  doubted  whether 
anything  material  had  been  done  upon  the  matter  since  the  time  of  Cavendish, 
and  that  I  had  better  refer  to  Cavendish's  original  paper.  That  advice  I 
quickly  followed,  and  I  was  rather  surprised  to  find  that  Cavendish  had  him- 
self put  this  question  quite  as  sharply  as  I  could  put  it.  Translated  from  the 
old-fashioned  phraseology  connected  with  the  theory  of  phlogiston,  his  question 
was  whether  the  inert  ingredient  of  the  air  is  really  all  of  one  kind ;  whether 
all  the  nitrogen  of  the  air  is  really  the  same  as  the  nitrogen  of  nitre. 
Cavendish  not  only  asked  himself  this  question,  but  he  endeavoured  to  answer 
it  by  an  appeal  to  experiment. 

I  should  like  to  show  you  Cavendish's  experiment  in  something  like  its 
original  form.  He  inverted  a  U-tube  filled  with  mercury,  the  legs  standing  in 
two  separate  mercury  cups.  He  then  passed  up,  so  as  to  stand  above  the 
mercury,  a  mixture  of  nitrogen,  or  of  air,  and  oxygen;  and  he  caused  an 
electric  current  from  a  frictional  electrical  machine  like  the  one  I  have  before 
me  to  pass  from  the  mercury  in  the  one  leg  to  the  mercury  in  the  other, 
giving  sparks  across  the  intervening  column  of  air.  •  I  do  not  propose  to  use  a 
frictional  machine  to-night,  but  I  will  substitute  for  it  one  giving  electricity 
of  the  same  quality  of  the  construction  introduced  by  Mr  Wimshurst,  of  which 
we  have  a  fine  specimen  in  the  Institution.  It  stands  just  outside  the  door 
of  the  theatre,  and  will  supply  an  electric  current  along  insulated  wires,  lead- 
ing to  the  mercury  cups ;  and,  if  we  are  successful,  we  shall  cause  sparks  to 
pass  through  the  small  length  of  air  included  above  the  columns  of  mercury. 
There  they  are ;  and  after  a  little  time  you  will  notice  that  the  mercury  rises, 
indicating  that  the  gas  is  sensibly  absorbed  under  the  influence  of  the  sparks 


192  ARGON.  [215 

and  of  a  piece  of  potash  floating  on  the  mercury.  It  was  by  that  means  that 
Cavendish  established  his  great  discovery  of  the  nature  of  the  inert  ingredient 
in  the  atmosphere,  which  we  now  call  nitrogen;  and,  as  I  have  said,  Cavendish 
himself  proposed  the  question,  as  distinctly  as  we  can  do,  Is  this  inert 
ingredient  all  of  one  kind  ?  and  he  proceeded  to  test  that  question.  He 
found,  after  days  and  weeks  of  protracted  experiment,  that,  for  the  most  part, 
the  nitrogen  of  the  atmosphere  was  absorbed  in  this  manner,  and  converted 
into  nitrous  acid ;  but  that  there  was  a  small  residue  remaining  after  pro- 
longed treatment  with  sparks,  and  a  final  absorption  of  the  residual  oxygen. 
That  residue  amounted  to  about  T^  part  of  the  nitrogen  taken;  and  Cavendish 
draws  the  conclusion  that,  if  there  be  more  than  one  inert  ingredient  in  the 
atmosphere,  at  any  rate  the  second  ingredient  is  not  contained  to  a  greater 
extent  than  T^y  part. 

I  must  not  wait  too  long  over  the  experiment.  Mr  Gordon  tells  me  that 
a  certain  amount  of  contraction  has  already  occurred ;  and  if  we  project  the  U 
upon  the  screen,  we  shall  be  able  to  verify  the  fact.  It  is  only  a  question  of 
time  for  the  greater  part  of  the  gas  to  be  taken  up,  as  we  have  proved  by 
preliminary  experiments. 

In  what  I  have  to  say  from  this  point  onwards,  I  must  be  understood  as 
speaking  as  much  on  behalf  of  Professor  Ramsay  as  for  myself.  At  the  first, 
the  work  which  we  did  was  to  a  certain  extent  independent.  Afterwards  we 
worked  in  concert,  and  all  that  we  have  published  in  our  joint  names  must  be 
regarded  as  being  equally  the  work  of  both  of  us.  But,  of  course,  Professor 
Ramsay  must  not  be  held  responsible  for  any  chemical  blunder  into  which  I 
may  stumble  to-night. 

By  his  work  and  by  mine  the  heavier  ingredient  in  atmospheric  nitrogen 
which  was  the  origin  of  the  discrepancy  in  the  densities  has  been  isolated,  and 
we  have  given  it  the  name  of  "  argon."  For  this  purpose  we  may  use  the 
original  method  of  Cavendish,  with  the  advantages  of  modern  appliances.  We 
can  procure  more  powerful  electric  sparks  than  any  which  Cavendish  could 
command  by  the  use  of  the  ordinary  Ruhmkorff  coil  stimulated  by  a  battery 
of  Grove  cells;  and  it  is  possible  so  to  obtain  evidence  of  the  existence  of 
argon.  The  oxidation  of  nitrogen  by  that  method  goes  on  pretty  quickly.  If 
you  put  some  ordinary  air,  or,  better  still,  a  mixture  of  air  and  oxygen,  in  a  tube 
in  which  electric  sparks  are  made  to  pass  for  a  certain  time,  then  in  looking 
through  the  tube,  you  observe  the  well-known  reddish-orange  fumes  of  the 
oxides  of  nitrogen.  I  will  not  take  up  time  in  going  through  the  experiment, 
but  will  merely  exhibit  a  tube  already  prepared  (image  on  screen). 

One  can  work  more  efficiently  by  employing  the  alternate  currents  from 
dynamo  machines  which  are  now  at  our  command.  In  this  Institution  we 
have  the  advantage  of  a  public  supply;  and  if  I  pass  alternate  currents 


1895]  AKGON.  193 

originating  in  Deptford  through  this  Ruhmkorff  coil,  which  acts  as  what  is 
now  called  a  "  high  potential  transformer,"  and  allow  sparks  from  the  secondary 
to  pass  in  an  inverted  test  tube  between  platinum  points,  we  shall  be  able  to 
show  in  a  comparatively  short  time  a  pretty  rapid  absorption  of  the  gases. 
The  electric  current  is  led  into  the  working  chamber  through  bent  glass  tubes 
containing  mercury,  and  provided  at  their  inner  extremities  with  platinum 
points.  In  this  arrangement  we  avoid  the  risk,  which  would  otherwise  be 
serious,  of  a  fracture  just  when  we  least  desired  it.  I  now  start  the  sparks  by 
switching  on  the  Ruhmkorff  to  the  alternate  current  supply ;  and,  if  you  will 
take  note  of  the  level  of  the  liquid  representing  the  quantity  of  mixed  gases 
included,  I  think  you  will  see  after,  perhaps,  a  quarter  of  an  hour  that  the 
liquid  has  very  appreciably  risen,  owing  to  the  union  of  the  nitrogen  and  the 
oxygen  gases  under  the  influence  of  the  electrical  discharge,  and  subsequent 
absorption  of  the  resulting  compound  by  the  alkaline  liquid  with  which  the  gas 
space  is  enclosed. 

By  means  of  this  little  apparatus,  which  is  very  convenient  for  operations 
upon  a  moderate  scale,  such  as  analyses  of  "  nitrogen  "  for  the  amount  of  argon 
that  it  may  contain,  we  are  able  to  get  an  absorption  of  about  80  cubic  centi- 
metres per  hour,  or  about  4  inches  along  this  test  tube,  when  all  is  going  well. 
In  order,  however,  to  effect  the  isolation  of  argon  on  any  considerable 'scale 
by  means  of  the  oxygen  method,  we  must  employ  an  apparatus  still  more 
enlarged.  The  isolation  of  argon  requires  the  removal  of  nitrogen,  and,  indeed, 
of  very  large  quantities  of  nitrogen,  for,  as  it  appears,  the  proportion  of  argon 
contained  in  atmospheric  nitrogen  is  only  about  1  per  cent.,  so  that  for  every 
litre  of  argon  that  you  wish  to  get  you  must  eat  up  some  hundred  litres  of 
nitrogen.  That,  however,  can  be  done  upon  an  adequate  scale  by  calling  to 
our  aid  the  powerful  electric  discharge  now  obtainable  by  means  of  the 
alternate  current  supply  and  high  potential  transformers. 

In  what  I  have  done  upon  this  subject  I  have  had  the  advantage  of  the 
advice  of  Mr  Crookes,  who  some  years  ago  drew  special  attention  to  the 
electric  discharge  or  flame,  and  showed  that  many  of  its  properties  depended 
upon  the  fact  that  it  had  the  power  of  causing,  upon  a  very  considerable  scale, 
a  combination  of  the  nitrogen  and  the  oxygen  of  the  air  in  which  it  was 
made. 

I  had  first  thought  of  showing  in  the  lecture  room  the  actual  apparatus 
which  I  have  employed  for  the  concentration  of  argon ;  but  the  difficulty  is 
that,  as  the  apparatus  has  to  be  used,  the  working  parts  are  almost  invisible, 
and  I  came  to  the  conclusion  that  it  would  really  be  more  instructive  as  well 
as  more  convenient  to  show  the  parts  isolated,  a  very  little  effort  of  imagina- 
tion being  then  all  that  is  required  in  order  to  reconstruct  in  the  mind  the 
actual  arrangements  employed. 

R.    iv.  13 


194  ARGON.  [215 

First,  as  to  the  electric  arc  or  flame  itself.  We  have  here  a  transformer 
made  by  Pike  and  Harris.  It  is  not  the  one  that  I  have  used  in  practice ; 
but  it  is  convenient  for  certain  purposes,  and  it  can  be  connected  by  means  of 
a  switch  with  the  alternate  currents  of  100  volts  furnished  by  the  Supply 
Company.  The  platinum  terminals  that  you  see  here  are  modelled  exactly 
upon  the  plan  of  those  which  have  been  employed  in  practice.  I  may  say  a 
word  or  two  on  the  question  of  mounting.  The  terminals  require  to  be  very 
massive  on  account  of  the  heat  evolved.  In  this  case  they  consist  of  platinum 
wire  doubled  upon  itself  six  times.  The  platinums  are  continued  by  iron 
wires  going  through  glass  tubes,  and  attached  at  the  ends  to  the  copper  leads. 
For  better  security,  the  tubes  themselves  are  stopped  at  the  lower  ends  with 
corks  and  charged  with  water,  the  advantage  being  that,  when  the  whole 
arrangement  is  fitted  by  means  of  an  indiarubber  stopper  into  a  closed  vessel, 
you  have  a  witness  that,  as  long  as  the  water  remains  in  position,  no  leak  can 
have  occurred  through  the  insulating  tubes  conveying  the  electrodes. 

Now,  if  we  switch  on  the  current  and  approximate  the  points  sufficiently, 
we  get  the  electric  flame.  There  you  have  it.  It  is,  at  present,  showing  a 
certain  amount  of  soda.  That  in  time  would  burn  off.  After  the  arc  has  once 
been  struck,  the  platinums  can  be  separated ;  and  then  you  have  two  tongues 
of  fire  ascending  almost  independently  of  one  another,  but  meeting  above. 
Under  the  influence  of  such  a  flame,  the  oxygen  and  the  nitrogen  of  the  air 
combine  at  a  reasonable  rate,  and  in  this  way  the  nitrogen  is  got  rid  of.  It  is 
now  only  a  question  of  boxing  up  the  gas  in  a  closed  space,  where  the  argon 
concentrated  by  the  combustion  of  the  nitrogen  can  be  collected.  But  there 
are  difficulties  to  be  encountered  here.  One  cannot  well  use  anything  but  a 
glass  vessel.  There  is  hardly  any  metal  available  that  will  withstand  the 
action  of  strong  caustic  alkali  and  of  the  nitrous  fumes  resulting  from  the 
flame.  One  is  practically  limited  to  glass.  The  glass  vessel  employed  is  a 
large  flask  with  a  single  neck,  about  half  full  of  caustic  alkali.  The  electrodes 
are  carried  through  the  neck  by  means  of  an  indiarubber  bung  provided  also 
with  tubes  for  leading  in  the  gas.  The  electric  flame  is  situated  at  a  distance 
of  only  about  half  an  inch  above  the  caustic  alkali.  In  that  way  an  efficient 
circulation  is  established ;  the  hot  gases  as  they  rise  from  the  flame  strike  the 
top,  and  then  as  they  come  round  again  in  the  course  of  the  circulation  they 
•pass  sufficiently  close  to  the  caustic  alkali  to  ensure  an  adequate  removal  of 
the  nitrous  fumes. 

There  is  another  point  to  be  mentioned.  It  is  necessary  to  keep  the 
vessel  cool;  otherwise  the  heat  would  soon  rise  to  such  a  point  that  there 
would  be  excessive  generation  of  steam,  and  then  the  operation  would  come  to 
a  standstill.  In  order  to  meet  this  difficulty  the  upper  part  of  the  vessel  is 
provided  with  a  water-jacket,  in  which  a  circulation  can  be  established.  No 
doubt  the  glass  is  severely  treated,  but  it  seems  to  stand  it  in  a  fairly  amiable 


1895]  ARGON.  195 

By  means  of  an  arrangement  of  this  kind,  taking  nearly  three  horse-power 
from  the  electric  supply,  it  is  possible  to  consume  nitrogen  at  a  reasonable 
rate.  The  transformers  actually  used  are  the  "Hedgehog"  transformers  of 
Mr  Swinburne,  intended  to  transform  from  100  volts  to  2400  volts.  By 
Mr  Swinburne's  advice  I  have  used  two  such,  the  fine  wires  being  in  series  so 
as  to  accumulate  the  electrical  potential  and  the  thick  wires  in  parallel.  The 
rate  at  which  the  mixed  gases  are  absorbed  is  about  seven  litres  per  hour ;  and 
the  apparatus,  when  once  fairly  started,  works  very  well  as  a  rule,  going  for 
many  hours  without  attention.  At  times  the  arc  has  a  trick  of  going  out,  and 
it  then  requires  to  be  restarted  *by  approximating  the  platinums.  We  have 
already  worked  14  hours  on  end,  and  by  the  aid  of  one  or  two  automatic 
appliances  it  would,  I  think,  be  possible  to  continue  operations  day  and 
night. 

The  gases,  air  and  oxygen  in  about  equal  proportions,  are  mixed  in  a  large 
gas-holder,  and  are  fed  in  automatically  as  required.  The  argon  gradually 
accumulates ;  and  when  it  is  desired  to  stop  operations  the  supply  of  nitrogen 
is  cut  off,  and  only  pure  oxygen  allowed  admittance.  In  this  way  the  remain- 
ing nitrogen  is  consumed,  so  that,  finally,  the  working  vessel  is  charged  with 
a  mixture  of  argon  and  oxygen  only,  from  which  the  oxygen  is  removed  by 
ordinary  well-known  chemical  methods.  I  may  mention  that  at  the  close  of 
the  operation,  when  the  nitrogen  is  all  gone,  the  arc  changes  its  appearance, 
and  becomes  of  a  brilliant  blue  colour. 

I  have  said  enough  about  this  method,  and  I  must  now  pass  on  to  the 
alternative  method  which  has  been  very  successful  in  Professor  Ramsay's 
hands — that  of  absorbing  nitrogen  by  means  of  red-hot  magnesium.  By  the 
kindness  of  Professor  Ramsay  and  Mr  Matthews,  his  assistant,  we  have  here 
the  full  scale  apparatus  before  us  almost  exactly  as  they  use  it.  On  the 
left  there  is  a  reservoir  of  nitrogen  derived  from  air  by  the  simple  removal 
of  oxygen.  The  gas  is  then  dried.  Here  it  is  bubbled  through  sulphuric 
acid.  It  then  passes  through  a  long  tube  made  of  hard  glass  and  charged 
with  magnesium  in  the  form  of  thin  turnings.  During  the  passage  of  the  gas 
over  the  magnesium  at  a  bright  red  heat,  the  nitrogen  is  absorbed  in  a  great 
degree,  and  the  gas  which  finally  passes  through  is  immensely  richer  in  argon 
than  that  which  first  enters  the  hot  tube.  At  the  present  time  you  see  a 
tolerably  rapid  bubbling  on  the  left,  indicative  of  the  flow  of  atmospheric 
nitrogen  into  the  combustion  furnace ;  whereas,  on  the  right,  the  outflow  is 
very  much  slower.  Care  must  be  taken  to  prevent  the  heat  rising  to  such  a 
point  as  to  soften  the  glass.  The  concentrated  argon  is  collected  in  a  second 
gas-holder,  and  afterwards  submitted  to  further  treatment.  The  apparatus 
employed  by  Professor  Ramsay  in  the  subsequent  treatment  is  exhibited 
in  the  diagram,  and  is  very  effective  for  its  purpose ;  but  I  am  afraid  that 
the  details  of  it  would  not  readily  be  followed  from  any  explanation  that 

13—2 


196  ARGON.  [215 

I  could  give  in  the  time  at  my  disposal.  The  principle  consists  in  the 
circulation  of  the  mixture  of  nitrogen  and  argon  over  hot  magnesium,  the  gas 
being  made  to  pass  round  and  round  until  the  nitrogen  is  effectively  removed 
from  it.  At  the  end  that  operation,  as  in  the  case  of  the  oxygen  method, 
proceeds  somewhat  slowly.  When  the  greater  part  of  the  nitrogen  is  gone, 
the  remainder  seems  to  be  unwilling  to  follow,  and  it  requires  somewhat  pro- 
tracted treatment  in  order  to  be  sure  that  the  nitrogen  has  wholly  disappeared. 
When  I  say  "  wholly  disappeared,"  that,  perhaps,  would  be  too  much  to  say  in 
any  case.  What  we  can  say  is  that  the  spectrum  test  is  adequate  to  show  the 
presence,  or  at  any  rate  to  show  the  addition,'  of  about  one-and-a-half  per  cent, 
of  nitrogen  to  argon  as  pure  as  we  can  get  it ;  so  that  it  is  fair  to  argue  that 
any  nitrogen  at  that  stage  remaining  in  the  argon  is  only  a  small  fraction  of 
one-and-a-half  per  cent. 

I  should  have  liked  at  this  point  to  be  able  to  give  advice  as  to  which  of 
the  two  methods — the  oxygen  method  or  the  magnesium  method — is  the 
easier  and  the  more  to  be  recommended ;  but  I  confess  that  I  am  quite  at  a 
loss  to  do  so.  One  difficulty  in  the  comparison  arises  from  the  fact  that  they 
have  been  in  different  hands.  As  far  as  I  can  estimate,  the  quantities  of 
nitrogen  eaten  up  in  a  given  time  are  not  very  different.  In  that  respect, 
perhaps,  the  magnesium  method  has  some  advantage ;  but,  on  the  other  hand, 
it  may  be  said  that  the  magnesium  process  requires  a  much  closer  supervision, 
so  that,  perhaps,  fourteen  hours  of  the  oxygen  method  may  not  unfairly 
compare  with  eight  hours  or  so  of  the  magnesium  method.  In  practice  a 
great  deal  would  depend  upon  whether  in  any  particular  laboratory  alternate 
currents  are  available  from  a  public  supply.  If  the  alternate  currents  are  at 
hand,  I  think  it  may  probably  be  the  case  that  the  oxygen  method  is  the 
easier ;  but,  otherwise,  the  magnesium  method  would,  probably,  be  preferred, 
especially  by  chemists  who  are  familiar  with  operations  conducted  in  red-hot 
tubes. 

I  have  here  another  experiment  illustrative  of  the  reaction  between 
magnesium  and  nitrogen.  Two-  rods  of  that  metal  are  suitably  mounted  in 
an  atmosphere  of  nitrogen,  so  arranged  that  we  can  bring  them  into  contact 
and  cause  an  electric  arc  to  form  between  them.  Under  the  action  of  the 
heat  of  the  electric  arc  the  nitrogen  will  combine  with  the  magnesium ;  and 
if  we  had  time  to  carry  out  the  experiment  we  could  demonstrate  a  rapid 
absorption  of  nitrogen  by  this  method.  When  the  experiment  was  first  tried, 
I  had  hoped  that  it  might  be  possible,  by  the  aid  of  electricity,  to  start  the 
action  so  effectively  that  the  magnesium  would  continue  to  burn  independently 
under  its  own  developed  heat  in  the  atmosphere  of  nitrogen.  Possibly,  on  a 
larger  scale,  something  of  this  sort  might  succeed,  but  I  bring  it  forward  here 
only  as  an  illustration.  We  turn  on  the  electric  current,  and  bring  the 
magnesiums  together.  You  see  a  brilliant  green  light,  indicating  the  vaporisa- 


1895]  ARGON.  197 

tion  of  the  magnesium.  Under  the  influence  of  the  heat  the  magnesium 
burns,  and  there  is  collected  in  the  glass  vessel  a  certain  amount  of  brownish- 
looking  powder  which  consists  mainly  of  the  nitride  of  magnesium.  Of  course, 
if  there  is  any  oxygen  present  it  has  the  preference,  and  the  ordinary  white 
oxide  of  magnesium  is  formed. 

The  gas  thus  isolated  is  proved  to  be  inert  by  the  very  fact  of  its 
isolation.  It  refuses  to  combine  under  circumstances  in  which  nitrogen,  itself 
always  considered  very  inert,  does  combine — both  in  the  case  of  the  oxygen 
treatment  and  in  the  case  of  the  magnesium  treatment ;  and  these  facts  are, 
perhaps,  almost  enough  to  justify  the  name  which  we  have  suggested  for  it. 
But,  in  addition  to  this,  it  has  been  proved  to  be  inert  under  a  considerable 
variety  of  other  conditions  such  as  might  have  been  expected  to  tempt  it  into 
combination.  I  will  not  recapitulate  all  the  experiments  which  have  been 
tried,  almost  entirely  by  Professor  Ramsay,  to  induce  the  gas  to  combine. 
Hitherto,  in  our  hands,  it  has  not  done  so ;  and  I  may  mention  that  recently, 
since  the  publication  of  the  abstract  of  our  paper  read  before  the  Royal 
Society,  argon  has  been  submitted  to  the  action  of  titanium  at  a  red  heat, 
titanium  being  a  metal  having  a  great  affinity  for  nitrogen,  and  that  argon 
has  resisted  the  temptation  to  which  nitrogen  succumbs.  We  never  have 
asserted,  and  we  do  not  now  assert,  that  argon  can  under  no  circumstances  be 
got  to  combine.  That  would,  indeed,  be  a  rash  assertion  for  any  one  to 
venture  upon ;  and  only  within  the  last  few  weeks  there  has  been  a  most 
interesting  announcement  by  M.  Berthelot,  of  Paris,  that,  under  the  action  of 
the  silent  electric  discharge,  argon  can  be  absorbed  when  treated  in  contact 
with  the  vapour  of  benzine.  Such  a  statement,  coming  from  so  great  an 
authority,  commands  our  attention;  and  if  we  accept  the  conclusion,  as  I 
suppose  we  must  do,  it  will  follow  that  argon  has,  under  those  circumstances, 
combined. 

Argon  is  rather  freely  soluble  in  water.  That  is  a  thing  that  troubled  us 
at  first  in  trying  to  isolate  the  gas  ;  because,  when  one  was  dealing  with  very 
small  quantities,  it  seemed  to  be  always  disappearing.  In  trying  to  accumulate 
it  we  made  no  progress.  After  a  sufficient  quantity  had  been  prepared,  special 
experiments  were  made  on  the  solubility  of  argon  in  water.  It  has  been 
found  that  argon,  prepared  both  by  the  magnesium  method  and  by  the  oxygen 
method,  has  about  the  same  solubility  in  water  as  oxygen—  some  two-and-a- 
half  times  the  solubility  of  nitrogen.  This  suggests,  what  has  been  verified  by 
experiment,  that  the  dissolved  gases  of  water  should  contain  a  larger  propor- 
tion of  argon  than  does  atmospheric  nitrogen.  I  have  here  an  apparatus  of  a 
somewhat  rough  description,  which  I  have  employed  in  experiments  of  this 
kind.  The  boiler  employed  consists  of  an  old  oil-can.  The  water  is  supplied  to 
it  and  drawn  from  it  by  coaxial  tubes  of  metal.  The  incoming  cold  water  flows 
through  the  outer  annulus  between  the  two  tubes.  The  outgoing  hot  water 


198  ARGON.  [215 

passes  through  the  inner  tube,  which  ends  in  the  interior  of  the  vessel  at  a 
higher  level.  By  means  of  this  arrangement  the  heat  of  the  water  which  has 
done  its  work  is  passed  on  to  the  incoming  water  not  yet  in  operation,  and  in 
that  way  a  limited  amount  of  heat  is  made  to  bring  up  to  the  boil  a  very 
much  larger  quantity  of  water  than  would  otherwise  be  possible,  the  greater 
part  of  the  dissolved  gases  being  liberated  at  the  same  time.  These  are 
collected  in  the  ordinary  way.  What  you  see  in  this  flask  is  dissolved  air 
collected  out  of  water  in  the  course  of  the  last  three  or  four  hours.  Such  gas, 
when  treated  as  if  it  were  atmospheric  nitrogen,  that  is  to  say  after  removal 
of  the  oxygen  and  minor  impurities,  is  found  to  be  decidedly  heavier  than 
atmospheric  nitrogen  to  such  an  extent  as  to  indicate  that  the  proportion  of 
argon  contained  is  about  double.  It  is  obvious,  therefore,  that  the  dissolved 
gases  of  water  form  a  convenient  source  of  argon,  by  which  some  of  the  labour 
of  separation  from  air  is  obviated.  During  the  last  few  weeks  I  have  been 
supplied  from  Manchester  by  Mr  Macdougall,  who  has  interested  himself  in 
this  matter,  with  a  quantity  of  dissolved  gases  obtained  from  the  condensing 
water  of  his  steam  engine. 

As  to  the  spectrum,  we  have  been  indebted  from  the  first  to  Mr  Crookes, 
and  he  has  been  good  enough  to-night  to  bring  some  tubes  which  he  will 
operate,  and  which  will  show  you  at  all  events  the  light  of  the  electric 
discharge  in  argon.  I  cannot  show  you  the  spectrum  of  argon,  for  unfortunately 
the  amount  of  light  from  a  vacuum  tube  is  not  sufficient  for  the  projection  of 
its  spectrum.  Under  some  circumstances  the  light  is  red,  and  under  other 
circumstances  it  is  blue.  Of  course  when  these  lights  are  examined  with  the 
spectroscope — and  they  have  been  examined  by  Mr  Crookes  with  great  care — 
the  differences  in  the  colour  of  the  light  translate  themselves  into  different 
groups  of  spectrum  lines.  We  have  before  us  Mr  Crookes'  map,  showing  the 
two  spectra  upon  a  very  large  scale.  The  upper  is  the  spectrum  of  the  blue 
light ;  the  lower  is  the  spectrum  of  the  red  light ;  and  it  will  be  seen  that 
they  differ  very  greatly.  Some  lines  are  common  to  both ;  but  a  great  many 
lines  are  seen  only  in  the  red,  and  others  are  seen  only  in  the  blue.  It  is 
astonishing  to  notice  what  trifling  changes  in  the  conditions  of  the  discharge 
bring  about  such  extensive  alterations  in  the  spectrum. 

One  question  of  great  importance  upon  which  the  spectrum  throws  light 
is,  Is  the  argon  derived  by  the  oxygen  method  really  the  same  as  the  argon 
derived  by  the  magnesium  method  ?  By  Mr  Crookes'  kindness  I  have  had  an 
opportunity  of  examining  the  spectra  of  the  two  gases  side  by  side,  and  such 
examination  as  I  could  make  revealed  no  difference  whatever  in  the  two 
spectra,  from  which,  I  suppose,  we  may  conclude  either  that  the  gases  are 
absolutely  the  same,  or,  if  they  are  not  the  same,  that  at  any  rate  the 
ingredients  by  which  they  differ  cannot  be  present  in  more  than  a  small 
proportion  in  either  of  them. 


1895] 


ARGON. 


199 


My  own  observations  upon  the  spectrum  have  been  made  principally  at 
atmospheric  pressure.  In  the  ordinary  process  of  sparking,  the  pressure  is 
atmospheric ;  and,  if  we  wish  to  look  at  the  spectrum,  we  have  nothing  more 
to  do  than  to  include  a  jar  in  the  circuit,  and  to  put  a  direct-vision  prism  to 
the  eye.  At  my  request,  Professor  Schuster  examined  some  tubes  containing 
argon  at  atmospheric  pressure  prepared  by  the  oxygen  method,  .and  I  have 
here  a  diagram  of  a  characteristic  group.  He  also  placed  upon  the  sketch 
some  of  the  lines  of  zinc,  which  were  very  convenient  as  directing  one  exactly 
where  to  look.  See  figure. 


43 


44 


45 


46 


47 


48 


5000 


Argon 

-^Red 

Zinc 

1 

Hydrogen 

Within  the  last  few  days,  Mr  Crookes  has  charged  a  radiometer  with 
argon.  When  held  in  the  light  from  the  electric  lamp,  the  vanes  revolve 
rapidly.  Argon  is  anomalous  in  many  respects,  but  not,  you  see,  in  this. 

Next,  as  to  the  density  of  argon.  Professor  Ramsay  has  made  numerous 
and  careful  observations  upon  the  density  of  the  gas  prepared  by  the  mag- 
nesium method,  and  he  finds  a  density  of  about  19'9  as  compared  with 
hydrogen.  Equally  satisfactory  observations  upon  the  gas  derived  by  the 
oxygen  method  have  not  yet  been  made*,  but  there  is  no  reason  to  suppose 
that  the  density  is  different,  such  numbers  as  19'7  having  been  obtained. 

One  of  the  most  interesting  matters  in  connection  with  argon,  however,  is 
what  is  known  as  the  ratio  of  the  specific  heats.  I  must  not  stay  to  elaborate 
the  questions  involved,  but  it  will  be  known  to  many  who  hear  me  that  the 
velocity  of  sound  in  a  gas  depends  upon  the  ratio  of  two  specific  heats — the 
specific  heat  of  the  gas  measured  at  constant  pressure,  and  the  specific  heat 
measured  at  constant  volume.  If  we  know  the  density  of  a  gas,  and  also  the 
velocity  of  sound  in  it,  we  are  in  a  position  to  infer  this  ratio  of  specific  heats ; 
and  by  means  of  this  method,  Professor  Ramsay  has  determined  the  ratio  in 
the  case  of  argon,  arriving  at  the  very  remarkable  result  that  the  ratio  of 


*  [See  Proc.  Roy.  Soc.  Vol.  LIX.  p.  198,  1896.] 


200  ARGON.  [215 

specific  heats  is  represented  by  the  number  T65,  approaching  very  closely  to 
the  theoretical  limit,  1'67.  The  number  1/67  would  indicate  that  the  gas  has 
no  energy  except  energy  of  translation  of  its  molecules.  If  there  is  any  other 
energy  than  that,  it  would  show  itself  by  this  number  dropping  below  T67. 
Ordinary  gases,  oxygen,  nitrogen,  hydrogen,  &c.,  do  drop  below,  giving  the 
number  1*4.  Other  gases  drop  lower  still.  If  the  ratio  of  specific  heats  is 
1*65,  practically  T67,  we  may  infer  that  the  whole  energy  of  motion  is  trans- 
lational ;  and  from  that  it  would  seem  to  follow  by  arguments  which,  however, 
I  must  not  stop  to  elaborate,  that  the  gas  must  be  of  the  kind  called  by 
chemists  monatomic. 

I  had  intended  to  say  something  of  the  operation  of  determining  the  ratio 
of  specific  heats,  but  time  will  not  allow.  The  result  is,  no  doubt,  very 
awkward.  Indeed,  I  have  seen  some  indications  that  the  anomalous  properties 
of  argon  are  brought  as  a  kind  of  accusation  against  us.  But  we  had  the  very 
best  intentions  in  the  matter.  The  facts  were  too  much  for  us ;  and  all  that 
we  can  do  now  is  to  apologise  for  ourselves  and  for  the  gas. 

Several  questions  may  be  asked,  upon  which  I  should  like  to  say  a  word  or 
two,  if  you  will  allow  me  to  detain  you  a  little  longer.  The  first  question  (I  do 
not  know  whether  I  need  ask  it)  is,  Have  we  got  hold  of  a  new  gas  at  all  ? 
I  had  thought  that  that  might  be  passed  over,  but  only  this  morning  I  read  in 
a  technical  journal  the  suggestion  that  argon  was  our  old  friend  nitrous  oxide. 
Nitrous  oxide  has  roughly  the  density  of  argon ;  but  that,  so  far  as  I  can  see, 
is  the  only  point  of  resemblance  between  them. 

Well,  supposing  that  there  is  a  new  gas,  which  I  will  not  stop  to  discuss, 
because  I  think  that  the  spectrum  alone  would  be  enough  to  prove  it,  the 
next  question  that  may  be  asked  is,  Is  it  in  the  atmosphere  ?  This  matter 
naturally  engaged  our  earnest  attention  at  an  early  stage  of  the  enquiry.  I 
will  only  indicate  in  a  few  words  the  arguments  which  seem  to  us  to  show 
that  the  answer  must  be  in  the  affirmative. 

In  the  first  place,  if  argon  be  not  in  the  atmosphere,  the  original 
discrepancy  of  densities  which  formed  the  starting-point  of  the  investigation 
remains  unexplained,  and  the  discovery  of  the  new  gas  has  been  made  upon  a 
false  clue.  Passing  over  that,  we  have  the  evidence  from  the  blank  experi- 
ments, in  which  nitrogen  originally  derived  from  chemical  sources  is  treated 
either  with  oxygen  or  with  magnesium,  exactly  as  atmospheric  nitrogen  is 
treated.  If  we  use  atmospheric  nitrogen,  we  get  a  certain  proportion  of  argon, 
about  1  per  cent.  If  we  treat  chemical  nitrogen  in  the  same  way  we  get,  I 
will  not  say  absolutely  nothing,  but  a  mere  fraction  of  what  we  should  get  had 
atmospheric  nitrogen  been  the  subject.  You  may  ask,  Why  do  we  get  any 
fraction  at  all  from  chemical  nitrogen  ?  It  is  not  difficult  to  explain  the  small 
residue,  because  in  the  manipulation  of  the  gases  large  quantities  of  water  are 


1895]  ARGON.  201 

used  ;  and,  as  I  have  already  explained,  water  dissolves  argon  somewhat  freely. 
In  the  processes  of  manipulation  some  of  the  argon  will  come  out  of  solution, 
and  it  remains  after  all  the  nitrogen  has  been  consumed. 

Another  wholly  distinct  argument  is  founded  upon  the  method  of  diffusion 
introduced  by  Graham.  Graham  showed  that  if  you  pass  gas  along  porous 
tubes  you  alter  the  composition,  if  the  gas  is  a  mixture.  The  lighter  con- 
stituents go  more  readily  through  the  pores  than  do  the  heavier  ones.  The 
experiment  takes  this  form.  A  number  of  tobacco  pipes — eight  in  the  actual 
arrangement — are  joined  together  in  series  with  indiarubber  junctions,  and 
they  are  put  in  a  space  in  which  a  vacuum  can  be  made,  so  that  the  space 
outside  the  porous  pipes  is  vacuous  or  approximately  so.  Through  the  pipes 
ordinary  air  is  led.  One  end  may  be  regarded  as  open  to  the  atmosphere. 
The  other  end  is  connected  with  an  aspirator  so  arranged  that  the  gas  collected 
is  only  some  2  per  cent,  of  that  which  leaks  through  the  porosities.  The  case 
is  like  that  of  an  Australian  river  drying  up  almost  to  nothing  in  the  course 
of  its  flow.  Well,  if  we  treat  air  in  that  way,  collecting  only  the  small  residue 
which  is  less  willing  than  the  remainder  to  penetrate  the  porous  walls,  and 
then  prepare  "  nitrogen  "  from  it  by  removal  of  oxygen  and  moisture,  we 
obtain  a  gas  heavier  than  atmospheric  nitrogen,  a  result  which  proves  that  the 
ordinary  nitrogen  of  the  atmosphere  is  not  a  simple  body,  but  is  capable  of 
being  divided  into  parts  by  so  simple  an  agent  as  the  tobacco  pipe. 

If  it  be  admitted  that  the  gas  is  in  the  atmosphere,  the  further  question 
arises  as  to  its  nature. 

At  this  point  I  would  wish  to  say  a  word  of  explanation.  Neither  in  our 
original  announcement  at  Oxford,  nor  at  any  time  since,  until  the  31st  of 
January,  did  we  utter  a  word  suggesting  that  argon  was  an  element ;  and  it 
was  only  after  the  experiments  upon  the  specific  heats  that  we  thought  that 
we  had  sufficient  to  go  upon  in  order  to  make  any  such  suggestion  in  public. 
I  will  not  insist  that  that  observation  is  absolutely  conclusive.  It  is  certainly 
strong  evidence.  But  the  subject  is  difficult,  and  one  that  has  given  rise  to 
some  difference  of  opinion  among  physicists.  At  any  rate  this  property  dis- 
tinguishes argon  very  sharply  from  all  the  ordinary  gases. 

One  question  which  occurred  to  us  at  the  earliest  stage  of  the  enquiry,  as 
soon  as  we  knew  that  the  density  was  not  very  different  from  21,  was  the 
question  of  whether,  possibly,  argon  could  be  a  more  condensed  form  of 
nitrogen,  denoted  chemically  by  the  symbol  N3.  There  seem  to  be  several 
difficulties  in  the  way  of  this  supposition.  Would  such  a  constitution  be 
consistent  with  the  ratio  of  specific  heats  (1'65)  ?  That  seems  extremely 
doubtful.  Another  question  is,  Can  the  density  be  really  as  high  as  21,  the 
number  required  on  the  supposition  of  N3  ?  As  to  this  matter,  Professor 
Ramsay  has  repeated  his  measurements  of  density,  and  he  finds  that  he  cannot 


202  ARGON.  [215 

get  even  so  high  as  20.  To  suppose  that  the  density  of  argon  is  really  21, 
and  that  it  appears  to  be  20  in  consequence  of  nitrogen  still  mixed  with  it, 
would  be  to  suppose  a  contamination  with  nitrogen  oiit  of  all  proportion  to 
what  is  probable.  It  would  mean  some  14  per  cent,  of  nitrogen,  whereas  it 
seems  that  from  one-and-a-half  to  two  per  cent,  is  easily  enough  detected  by 
the  spectroscope.  Another  question  that  may  be  asked  is,  Would  N8  require 
so  much  cooling  to  condense  it  as  argon  requires  ? 

There  is  one  other  matter  on  which  I  would  like  to  say  a  word — the 
question  as  to  what  N3  would  be  like  if  we  had  it.  There  seems  to  be  a 
great  discrepancy  of  opinions.  Some  high  authorities,  among  whom  must  be 
included,  I  see,  the  celebrated  Mendeleef,  consider  that  N3  would  be  an 
exceptionally  stable  body;  but  most  of  the  chemists  with  whom  I  have 
consulted  are  of  opinion  that  N3  would  be  explosive,  or,  at  any  rate,  absolutely 
unstable.  That  is  a  question  which  may  be  left  for  the  future  to  decide.  We 
must  not  attempt  to  put  these  matters  too  positively.  The  balance  of  evidence 
still  seems  to  be  against  the  supposition  that  argon  is  N3,  but  for  my  part  I 
do  not  wish  to  dogmatise. 

A  few  weeks  ago  we  had  an  eloquent  lecture  from  Professor  Riicker  on  the 
life  and  work  of  the  illustrious  Helmholtz.  It  will  be  known  to  many  that 
during  the  last  few  months  of  his  life  Helmholtz  lay  prostrate  in  a  semi- 
paralysed  condition,  forgetful  of  many  things,  but  still  retaining  a  keen 
interest  in  science.  Some  little  while  after  his  death  we  had  a  letter  from 
his  widow,  in  which  she  described  how  interested  he  had  been  in  our 
preliminary  announcement  at  Oxford  upon  this  subject,  and  how  he  desired 
the  account  of  it  to  be  read  to  him  over  again.  He  added  the  remark,  "  I 
always  thought  that  there  must  be  something  more  in  the  atmosphere." 


216. 


ON  THE   STABILITY   OR  INSTABILITY   OF  CERTAIN 
FLUID   MOTIONS.    III.* 


[Proceedings  of  the  London  Mathematical  Society,  xxvu.  pp.  5 — 12,  1895.] 


THE  steady  motions  in  question  are  those  in  which  the  velocity  is  parallel 
to  a  fixed  line  (#),  and  such  that  U  is  a  function  of  y  only.  In  the  disturbed 
motion  U  +  u,  v,  the  infinitely  small  quantities  u,  v  are  supposed  to  be  periodic 
functions  of  x,  proportional  to  eikx,  and,  as  dependent  upon  the  time,  to  be 
proportional  to  eint,  where  n  is  a  constant,  real  or  imaginary.  Under  these 
circumstances  the  equation  determining  v  is 


The  vorticity  (Z)  of  the  steady  motion  is  ^dU/dy.  If  throughout  any  layer  Z 
be  constant,  d*U/dy*  vanishes,  and,  whenever  n  +  kU  does  not  also  vanish, 

d*v/dy*-k*v  =  Q,  (2) 

or  v  =  Aeky  +  Be~ky (3) 

If  there  are  several  layers  in  each  of  which  Z  is  constant,  the  various  solutions 
of  the  form  (3)  are  to  be  fitted  together,  the  arbitrary  constants  being  so 
chosen  as  to  satisfy  certain  boundary  conditions.  The  first  of  these  conditions 
is  evidently 

A«  =  0 (4)f 

The  second  may  be  obtained  by  integrating  (1)  across  the  boundary.     Thus 

(-      U\    &(dv}-&(—}       -0  C5) 

\k+     )'      \dy)~      \dy)'1 

*  The  two  earlier  papers  upon  this  subject  are  to  be  found  in  Proc.  Lond.  Math.  Soc.  Vol.  xi. 
p.  57,  1880  [Vol.  i.  p.  474];  Vol.  xix.  p.  67,  1887  [Vol.  in.  p.  17].  The  fluid  is  supposed  to  be 
destitute  of  viscosity. 

t  [A  being  the  symbol  of  finite  differences.] 


204  ON   THE   STABILITY   OR   INSTABILITY  [216 

At  a  fixed  wall  v  =  0. 

Equation  (2)  secures  that  the  vorticity  shall  remain  constant  in  each  layer, 
and  equation  (3)  that  there  shall  be  no  slipping  at  the  surface  of  transition. 
Equations  (2)  and  (3)  together  may  be  regarded  as  expressing  the  continuity 
of  the  motion  at  the  surface  between  the  layers. 

In  the  first  of  the  papers  above  referred  to,  I  have  applied  equation  (1)  to 
prove  that,  if  d?Ujdy2  be  of  one  sign  throughout  the  whole  interval  between 
two  fixed  walls,  n  can  have  no  imaginary  part.  It  is  true  that,  if  n+kU 
vanishes  anywhere,  the  expression  for  d^vjdy^  —  J^v  in  (1)  becomes  infinite, 
unless  indeed  v  =  0  at  the  place  in  question;  and  Lord  Kelvin*  considers  that 
the  "disturbing  infinity"  thus  introduced  vitiates  the  proof  of  stability.  To 
this  criticism  it  may  be  replied  f  that,  "  if  n  be  complex,  there  is  no  disturbing 
infinity,  and  that,  therefore,  the  argument  does  not  fail,  regarded  as  one  for 
excluding  complex  values  of  n.  What  happens  when  n  has  a  real  value, 
such  that  n  +  k U  vanishes  at  an  interior  point,  is  a  subject  for  further 
consideration." 

In  embarking  upon  this  it  will  be  convenient  to  take  first  the  case  of  (2), 
(3),  (4),  (5),  where  the  vorticity  of  the  steady  motion  is  uniform  through 
layers  of  finite  thickness.  Any  general  conclusions  arrived  at  in  this  way 
should  at  least  throw  light  upon  the  extreme  case  where  the  number  of  the 
layers  is  infinitely  great,  and  their  thickness  is  infinitely  small. 

Starting  from  the  first  wall  at  y  =  0,  let  the  surfaces  between  the  layers 
occur  at  y  =  yi,  y  =  y*,  &c.,  and  let  the  values  of  U  at  these  places  be  Ul, 
Uz,  &c.  In  conformity  with  (4)  and  with  the  condition  that  v  =  0,  when 
y  =  0,  we  may  take  in  the  first  layer 

y  =  vt  =  MI  sinh  ky ;     (6) 

in  the  second  layer 

v  =  v2  =  vl  +  Mz sinh k(y- y,};  (7) 

in  the  third  layer 

v  =  v3  =  v2  +  M3  sinh  k  (y  —  y2) ;   (8) 

and  so  onj. 

If  the  second  fixed  wall  be  in  the  rth  layer  at  y  =  y',  then 

M1  sinh  ky'-+  3fs-sinh  k  (y'  —  y^)  +  . . .  +  Mr  sinh  k  (y  —  yr-^)  =  0. . .  .(9) 

We  have  still  to  express  the  conditions  (5)  at  the  various  surfaces  of  transition. 
At  the  first  surface 

v  =  Ml  sirih  % ,  A  (dv  /  dy)  =  kM2; 

*  Phil.  Mag.  Vol.  xxiv.  p.  275,  1887. 
t  PMl.  Mag.  Vol.  xxxiv.  p.  66,  1892.     [Vol.  in.  p.  580.] 

J  This  is  the  process  followed  iu  the  second  of  the  papers  cited,  with  a  slight  difference  of 
notation. 


1895]  OF  CERTAIN  FLUID  MOTIONS.  205 

at  the  second  surface 

v  =  3/!  sink  ki/2  +  Mz  sinh  k  (y2  —  y^),  A  (dv  /  dy)  =  kM3 ; 

and  so  on.     If  we  denote  the  values  of  A  (dll/dy)  at  the  various  surfaces  by 
Ax,  A2,  &c.,  the  conditions  may  be  written 

(n  +  k  tfj)  M 2  -  Aj .  Ml  sinh  kyl  =  0 

(n  +  k  U2)  M3  -  A2 .  [Ml  sinh  ky.2  +  M,2  sinh  k  (yz  -  yt}}  =  0  [ .  . .  .(10) 


The  r  —  1  equations  (10)  together  with  (9)  suffice  to  determine  n,  and  the 
r  —  l  ratios  Ml:  Mz :  M3:  ...  :  Mr.  The  determinantal  equation  in  n  is  of 
degree  r  —  1 ,  the  number  of  the  surfaces  of  transition ;  and  corresponding  to 
each  root  there  is  an  expression  for  v,  definite  except  as  regards  a  constant 
multiplier. 

It  is  important  to  note  that  the  disturbances  thus  expressed  are  such  as 
leave  the  vorticity  unaltered  in  the  interior  of  every  layer ;  that  they  relate,  in 
fact,  merely  to  waves  upon  the  surfaces  of  transition.  The  additional  vorticity 
due  to  the  disturbance  is  proportional  to  d*v/dy*  —  k2v,  and  is  equated  to  zero 
in  (2).  If  we  wish  to  consider  the  most  general  disturbance  possible,  we  must 
provide  for  an  arbitrary  vorticity  at  every  point. 

The  nature  of  the  normal  modes  of  disturbance  not  yet  considered  will  be 
apparent  from  a  comparison  between  (1)  and  (2).  Even  though  d-U [dy*  =  0, 
the  latter  does  not  follow  from  the  former,  unless  it  be  assumed  that  n  +  k  U 
is  finite.  Wherever  n  +  kU  vanishes,  that  is,  at  the  places  where  the  wave 
velocity  is  equal  to  the  stream  velocity,  (1)  is  satisfied,  even  though  (2)  be 
violated.  Thus  any  value  of  —  kU  to  be  found  anywhere  in  the  fluid  is  an 
admissible  value  of  n,  and  the  corresponding  normal  function  (v)  is  obtained 
by  allowing  the  arbitrary  constants  in  (3)  to  be  discontinuous  at  this  place  as 
well  as  at  the  surfaces  of  transition,  subject  of  course  to  the  condition  that  v 
itself  shall  be  continuous.  The  new  arbitrary  constant  thus  disposable  allows 
all  the  conditions  to  be  satisfied  with  the  value  of  n  already  prescribed. 

The  equations  (9),  (10)  already  found  suffice  for  the  present  purpose  if  we 
introduce  a  fictitious  surface  of  transition  at  the  place  in  question.  Suppose, 
for  example,  that  A3  =  0  in  the  third  of  equations  (10).  It  will  follow  either 
that  MI  =  0,  or  that  n  +  kU3  =  0.  In  the  first  alternative  the  constants  A  and 
B  are  continuous,  and  all  local  peculiarity  disappears.  The  second  alternative 
is  the  one  with  which  we  are  now  concerned.  The  equations  suffice,  as  usual, 
to  determine  n  (equal  to  —kU,),  as  well  as  the  ratios  of  the  M's  which  give 
the  form  of  the  normal  function.  The  mode  of  disturbance  is  such  that  a  new 
vorticity  is  introduced  at  the  place,  or  rather  at  the  plane  in  question.  In 
one  sense  this  is  the  only  new  vorticity ;  but  the  waves  upon  the  surfaces  of 
transition  involve  changes  of  vorticity  as  regards  given  positions  in  space, 
though  not  as  regards  given  portions  of  fluid. 


206  ON   THE  STABILITY   OR   INSTABILITY  [216 

We  have  now  to  consider  what  occurs  at  a  second  place  in  the  fluid  where 
the  velocity  happens  to  be  the  same  as  at  the  first  place.  The  second  place 
may  be  either  within  a  layer  of  originally  uniform  vorticity  or  upon  a  surface 
of  transition.  In  the  first  case  nothing  very  special  presents  itself.  If  there 
be  no  new  vorticity  at  the  second  place,  the  value  of  v  is  definite  as  usual,  save 
as  to  an  arbitrary  multiplier.  But,  consistently  with  the  given  value  of  n,  there 
may  be  new  vorticity  at  the  second  as  well  as  at  the  first  place,  and  then  the 
complete  value  of  v  for  the  given  n  may  be  regarded  as  composed  of  two  parts, 
each  proportional  to  one  of  the  new  vorticities,  and  each  affected  by  an 
arbitrary  multiplier. 

If  the  second  place  lie  upon  a  surface  of  transition,  we  have  a  state  of 
things  corresponding  to  the  "  disturbing  infinity  "  in  (1).  In  the  above 
example,  where  A3  =  0,  n  +  k  U3  =  0,  we  have  now  further  to  suppose  that  Ul  , 
the  velocity  at  the  first  surface  of  transition,  coincides  with  U3.  From  the 
first  of  equations  (10),  since  n  +  kUl  =  0,  while  Ax  and  sinh  ky^  are  finite,  we 
see  that  Ml  must  vanish.  Hence  v  =  0  throughout  the  entire  layer  from  the 
wall  y  =  0  to  y  =  yr.  The  remainder  of  the  motion  from  y  =  yt  to  y  —  y1  is  to 
be  determined  as  usual. 

From  the  fact  that  v  =  0,  we  might  be  tempted  to  infer  that  the  surface  in 
question  behaves  like  a  fixed  wall.  But  a  closer  examination  shows  that  the 
inference  would  be  unwarranted.  In  order  to  understand  this  it  may  be  well 
to  investigate  the  relation  between  v  and  the  displacement  of  the  surface, 
supposed  also  to  be  proportional  to  eint  .  e***.  Thus,  if  the  equation  of  the 

surface  be 

F  =  y_heint+**  =  0!  ...........................  (11) 

the  condition  to  be  satisfied  is* 


so  that  -ih(n  +  kU1)+v  =  Q   ...........................  (13) 


is  the  required  relation.     Using  this,  we  see  from  the  first  of  equations  (10) 
that  h  does  not  vanish,  but  is  given  by 


The  propagation  of  a  wave  at  the  same  velocity  as  that  at  which  the  fluid 
moves  does  not  entail  the  existence  of  a  finite  velocity  v. 

That  v  vanishes  at  a  surface  of  transition  where  n  +  kU=0  follows  in 
general  from  (5),  seeing  that  the  value  of  A  (dU/dy)  is  finite.     That  region  of 

*  Lamb's  Hydrodynamics,  §  10. 


1895]  OF  CERTAIN  FLUID  MOTIONS.  207 

the  fluid,  bounded  by  this  surface  and  one  of  the  fixed  walls,  which  does  not 
include  the  added  vorticity,  will  in.  general  remain  undisturbed,  but  there  may 
be  exceptions  when  one  of  the  values  of  n  proper  to  this  region  (regarded  as 
bounded  by  fixed  walls)  happens  to  coincide  with  that  prescribed.  It  does 
not  appear  that  the  infinity  which  enters  when  n  +  kU=0  disturbs  any 
general  conclusions  as  to  the  conditions  of  stability,  or  even  seriously  modifies 
the  character  of  the  solutions  themselves. 

When  d?U/dy*  is  finite,  we  must  fall  back  upon  equation  (1).  The 
character  of  the  disturbing  infinity  at  a  place  (say,  y  =  0)  where  n  +  kU 
vanishes  would  be  most  satisfactorily  investigated  by  means  of  the  complete 
solution  of  some  particular  case.  It  is,  however,  sufficient  to  examine  the 
form  of  solution  in  the  neighbourhood  of  y  =  0,  and  for  this  purpose  the 
differential  equation  may  be  simplified.  Thus,  when  y  is  small,  n  +  kU  may 
be  regarded  as  proportional  to  y,  and  d2U/dy2  as  approximately  constant.  In 
comparison  with  the  large  term,  kzv  may  be  neglected,  and  it  suffices  to 
consider 

=  0,     ...........................  (15) 


a  known  constant  multiplying  y  being  omitted  for  brevity.     This  falls  under 
the  head  of  Riccati's  equation 

d*v/dy2  +  y*v  =  Q,     ..........................  (16) 

of  which  the  solution*  is  in  general  (m  fractional) 

v  =  Jy{AJm(Z)  +  BJ_m(Z)}>    .....................  (17) 

where  m  =  I  f(p  +  2),  £=2ra#1/!™  ...................  (18) 


When,  as  in  the  present  case,  m  is  integral,  J-m(%)  is  to  be  replaced  by 
the  function  of  the  second  kind  Ym(j;).  The  general  solution  of  (15)  is 
accordingly 

(19) 


In  passing  through  zero  y  changes  sign,  and  with  it  the  character  of  the 
functions.  If  we  regard  (19)  as  applicable  on  the  positive  side,  then  on  the 
negative  side  we  may  write 

v  =  </y{CJl(2Jy)  +  DY1(Wy)},     ..................  (20) 

the  arguments  of  the  functions  in  (20)  being  pure  imaginaries. 
The  functions  Ji(z),  Fi(*)  are  given  by 


*  Lommel,  Studien  tiber  die  BesscVschen  Functionen,  §  31,  Leipzig,  1868;  Gray  and  Mathews' 
Bessel  s  Functions,  p.  233,  1895. 


208  ON   THE   STABILITY   OR    INSTABILITY  [216 


where  Sm=  1  +£  +  £+  ...  +  1/m  ......................  (23) 

When  y  is  small,  (19)  gives 

»  =  4  (y  -  iy2}  +  B  {i  (l  -  y  +  iy)  -  log  (2  Vy)  (y  -  *y)  +  y$  -  iytf,}  ;•  •  -(24) 

so  that  ultimately 

v  =  ^B,        dv/dy  =  A-±B\ogy,        d2v/dya-  =  -  A  -%By-\  ...(25) 
v  remaining  finite  in  any  case. 

We  will  now  show  that  any  value  of  —  kU  is  an  admissible  value  of  n  in 
(1).  The  place  where  n+kU=Q  is  taken  as  origin  of  y;  and  in  the  first 
instance  we  will  suppose  that  n  +  k  U  vanishes  nowhere  else.  In  the  immediate 
neighbourhood  of  y  =  0,  the  solutions  applicable  on  the  two  sides  are  (19),  (20), 
and  they  are  subject  to  the  condition  that  v  shall  be  continuous.  Hence,  by 
(25),  B  —  D,  leaving  three  constants  arbitrary.  The  manner  in  which  the 
functions  start  from  y  =  0  being  thus  ascertained,  their  further  progress  is 
subject  to  the  original  equation  (1),  which  completely  defines  them  when  the 
three  arbitraries  are  known.  In  the  present  case  two  relations  are  given  by 
the  conditions  to  be  satisfied  at  the  fixed  walls  or  other  boundaries  of  the 
fluid,  and  thus  is  determined  the  entire  form  of  v,  save  as  to  a  constant 
multiplier.  If,  as  must  usually  be  the  case,  B  and  D  are  finite,  there  is 
infinite  vorticity  at  the  origin,  but  this  is  no  more  than  occurs  even  when 
d*U/dif  is  zero  throughout  the  region  surrounding  the  origin. 

Any  other  places  at  which  n  +  k  U  =  0  may  be  treated  in  a  similar  manner, 
and  the  most  general  solution  will  contain  as  many  arbitrary  constants  as 
there  are  places  of  infinite  vorticity.  But  the  vorticity  need  not  be  infinite 
merely  because  n  +  k  U  =  0  ;  and,  in  fact,  a  particular  solution  may  be 
obtained  with  only  one  infinite  vorticity.  At  any  other  of  the  critical  places, 
such,  for  example,  as  we  may  now  suppose  the  origin  to  be,  B  and  D  may 
vanish,  so  that 

0  =  0. 


From  this  discussion  it  would  seem  that  the  infinities  which  present  them- 
selves when  n  +  kU=-Q  do  not  seriously  interfere  with  the  application  of  the 
general  theory,  so  long  as  the  square  of  the  disturbance  from  steady  motion 
is  neglected.  The  value  of  conclusions  relating  only  to  infinitely  small 
disturbances  is  another  question. 


1895]  OF  CERTAIN  FLUID  MOTIONS.  209 

When  regard  is  paid  to  viscosity,  the  difficulties  are  of  course  much 
increased.  In  the  particular  case  where  the  original  vorticity  is  uniform,  the 
problem  of  small  disturbances  has  been  solved  by  Lord  Kelvin*,  who  shows 
that  the  motion  is  stable  by  the  aid  of  a  special  solution  not  proportional  to  a 
simple  exponential  function  of  the  time.  If  we  retain  the  supposition  of  the 
present  paper  that  the  disturbance  as  a  function  of  the  time  is  proportional  to 
eint,  we  obtain  an  equation  [(52)  in  Lord  Kelvin's  paper]  which  has  been 
discussed  by  Stokes  f.  From  his  results  it  appears  that  it  is  not  possible  to 
find  a  solution  applicable  to  an  unlimited  fluid  which  shall  be  periodic  with 
respect  to  x,  and  remain  finite  when  y  =  +  oo  ,  and  this  whether  n  be  real  or 
complex.  The  cause  of  the  failure  would  appear  to  lie  in  the  fact,  indicated 
by  Lord  Kelvin's  solution,  that  the  stability  is  ultimately  of  a  higher  order 
than  can  be  expressed  by  any  simple  exponential  function  of  the  time. 

[Addendum,  January,  1896.— It  may  be  well  to  emphasise  more  fully  that 
the  solutions  of  this  paper  only  profess  to  apply  in  the  limit,  when  the  dis- 
turbances are  infinitely  small  The  constant  factor  which  represents  the  scale 
of  the  disturbance  must  be  imagined  to  be  so  small  that  the  actual  disturbance 
nowhere  rises  to  such  a  magnitude  as  to  interfere  with  the  approximations  upon 
which  (1)  is  founded.  For  example,  in  (25),  although  dv/dy  is  infinite  at 
y  =  0  relatively  to  its  value  at  other  places,  it  must  still  be  regarded  as 
infinitely  small  throughout  in  comparison  with  the  quantities  which  define  the 
steady  motion.] 

*  Phil.  Mag.  Vol.  xxiv.  p.  191,  1887. 

t  Camb.  Phil.  Trans.  Vol.  x.  p.  105,  1857. 


217. 


ON  THE  PROPAGATION  OF  WAVES  UPON  THE  PLANE 
SURFACE  SEPARATING  TWO  PORTIONS  OF  FLUID  OF 
DIFFERENT  VORTICITIES. 

[Proceedings  of  the  London  Mathematical  Society,  xxvu.  pp.  13 — 18,  1895.] 

IN  former  papers*  I  have  considered  the  problem  of  the  motion  in  two 
dimensions  of  in  viscid  incompressible  fluid  between  two  parallel  walls.  In  the 
case  where  the  steady  motion  is  such  that  in  each  half  of  the  layer  included 
between  the  walls  the  vorticity  is  constant,  it  appeared  that  the  motion  is 
stable,  small  displacements  of  the  surface  separating  the  two  vorticities  being 
propagated  as  waves  of  constant  amplitude.  More  particularly,  if  the  velocity 
of  the  steady  motion  increase  uniformly  from  zero  at  the  walls  to  the  value  U 
in  the  middle  stratum,  a  disturbance  proportional  to  ei(nt+kx)  requires  that 

n  +  kU=  U/b.teohkb,  (1) 

where  26  is  the  distance  between  the  walls.     The  wave-length  is  2ir/k,  and 
the  fact  that  n  is  real  indicates  that  the  disturbance  is  stable. 

Discussions  upon  the  difficult  question  of  the  nature  of  the  instability 
manifested  by  fluids  in  their  flow  through  pipes  of  moderate  bore  seemed  to 
make  it  desirable  to  push  the  investigation  of  the  disturbance  from  some 
simple  case  of  steady  motion  so  far  at  least  as  to  include  the  squares  of  the 
small  quantities. 

In  the  present  paper  the  problem  chosen  for  the  purpose  is  that  above 
referred  to,  simplified  by  excluding  the  fixed  walls,  or,  what  comes  to  the 
same  thing,  by  supposing  them  removed  to  a  distance  very  great  in  comparison 
with  the  wave-length  of  the  disturbance.  We  suppose,  then,  that  in  the 
steady  motion  the  surface  of  separation  coincides  with  y  =  0,  that  when  y  is 
positive  the  vorticity  is  +  to,  and  that  when  y  is  negative  the  vorticity  is  —  CD. 

*  "  On  the  Stability  or  Instability  of  certain  Fluid  Motions,"  Proc.  Lond.  Math.  Soc.  Vol.  xi. 
p.  57,  1880  [Vol.  i.  p.  474];  Vol.  xix.  p.  67,  1887  [Vol.  in.  p.  17]. 


1895]    ON  THE  PROPAGATION  OF  WAVES  UPON  A  PLANE  SURFACE.     211 

In  the  disturbed  motion  the  surface  separating  the  two  vorticities  is  displaced, 
so  that  its  equation  becomes  y  =  h  cos  x,  k  being  put  equal  to  unity  for  the 
sake  of  brevity. 

In  virtue  of  the  incompressibility,  the  component  velocities,  denoted  as 
usual  by  u  and  v,  are  connected  with  a  stream-function  -^  by  the  relations 


(2) 

The  vorticity  is  represented  by  £V2i/r,  which  is  accordingly  equal  to  +  «. 
During  the  steady  motion  of  the  upper  fluid,  we  have 

^  =  a  +  (3y  +  nf  ...............................  (3) 

In  consequence  of  the  disturbance  ^  deviates  from  the  value  given  by  (3)  ; 
but,  since,  by  a  known  theorem,  the  vorticity  remains  throughout  equal  to  to, 
the  addition  to  i/r  must  satisfy  V2i|r  =  0.  The  additional  terms  must  also 
satisfy  the  condition  of  being  periodic  in  period  2?r;  and  thus  we  obtain 
altogether  as  the  expression  for  \jr  during  the  disturbed  motion 

i/r  =  a  +  fty  +  my2  +  e~v  (A1  cos  x  +  B^  sin  x) 

+  e-2^,,  cos  2#  +  £2  sin  2#)  +  ...,     ..................  (4) 

positive  exponents  being  excluded  by  the  condition  to  be  satisfied  when 
y  =  +  oo  .  Similarly  in  the  lower  fluid 

-»/r'  =  a  +  fly  —  my*  +  ey  (-4/  cos  x  +  B±  sin  x) 

+  e*y  (A3f  cos  2x  +  B2'  sin  2a?)  +  ......................  (5) 

From  these  values  of  ^r,  i/r'  the  velocities  u,  v  at  any  point  are  deducible 

by  (2). 

We  have  still  to  satisfy  the  conditions  at  the  surface  of  separation 

y  =  h  cosx  ..................................  (6) 

It  is  necessary  that  u  and  v,  as  given  by  i/r  and  ty',  should  there  be  continuous, 
any  sliding  of  the  one  body  of  fluid  upon  the  other  being  equivalent  to  a 
vortex-sheet,  and  therefore  excluded  by  the  conditions  of  the  problem.  Thus 
at  the  surface  we  must  have 

d(^-^')/dx  =  0,        d(^-^')/dy  =  0  ................  (7) 

For  the  purposes  of  the  first  approximation,  where  only  the  first  power  of  h  is 
retained,  y  may  be  put  equal  to  zero  in  the  exponential  terms  so  soon  as  the 
differentiations  have  been  performed.  Equations  (7)  give  accordingly 

—  sin  x  (Al  —  A/)  +  cos  x  (B1  —  B^) 

-  2  sin  2x(A2-A,')+  2  cos  2x(B2-B*)-  ............  =0, 

/3  -  £'  +  4>a>h  cos  x  -  cos  x  (A^  +  A^}  -  sin  x  (B^  +  1?,') 

2')  -2  sin  2x(B2  +  B2')-  ............  =  0; 

14—2 


212    ON  THE  PROPAGATION  OF  WAVES  UPON  THE  PLANE  SURFACE    [217 

from  which  it  appears  that  to  this  approximation  all  the  coefficients  with 
suffixes  higher  than  unity  must  vanish.     Also 


Thus  ty  =  a  +  fiy  +  (oy2  +  2(0he-ycosa;,  .....................  (8) 

^r'  —  a'  +  fiy  —  ay*  +  2wA  ev  cos  x,  .....................  (9) 

are  the  values  of  ^  determined  in  accordance  with  (6)  and  the  other  prescribed 
conditions.     From  (8)  or  (9),  we  find  as  the  values  of  u  and  v  at  the  surface 
u  =  ft,  v=2a)hsina;,   ...........................  (10) 

applicable  when  the  form  of  the  surface  is  that  given  by  (6),  at  the  moment, 
we  may  suppose,  when  t  =  0. 

By  means  of  (10)  it  is  possible  to  determine  the  form  and  position  of  the 
surface  of  separation  at  time  dt,  and  thus  to  trace  out  its  transformation.     In 
the  present  case  it  will  be  simplest  merely  to  verify  that  the  propagation  of 
the  form  (6)  with  a  certain  velocity  (  V)  satisfies  all  the  conditions.     If 

F(x,y,t)  =  y-hcoa(x-  Vt)  =  0    ..................  (11) 

be  the  equation  of  the  surface,  the  condition  to  be  satisfied*  is 


Here,  when  t  =  0, 


dF  dF  dF 

-j-  =  —  Vh  sin  x,          -j-  =  ft  sin  #,          =- 
dt  dx  dy 


so  that  (12)  becomes,  with  use  of  (10), 


showing  that  (11)  continues  to  represent  the  surface  of  separation  at  time  dt, 
provided  that 

(13) 


Accordingly,  if  (13)  be  satisfied,  equation  (11)  suffices  to  represent  the 
changes  in  the  surface  of  separation  for  any  length  of  time,  or,  in  other  words, 
the  disturbance  is  propagated  as  a  simple  wave. 

From  (8)  it  appears  that  /3  represents  the  velocity  in  the  steady  motion 
when  y  =  0,  and  the  result  is  in  accordance  with  (1),  where  tanh  kb  =  l.  The 
disturbance  may  be  supposed  to  be  got  rid  of  by  the  introduction  of  a  flexible 
lamina  at  the  surface  of  separation.  If,  by  forces  applied  to  it,  the  lamina  be 
straightened  out  so  as  to  coincide  with  y  =  0,  and  be  held  there  at  rest,  the 
steady  motion  is  recovered. 

*  Lamb's  Hydrodynamics,  §  10. 


1895]     SEPARATING  TWO  PORTIONS  OF  FLUID  OF  DIFFERENT  VORTlClTlES.     213 

In  proceeding  to  further  approximations,  in  which  higher  powers  of  h  are 
retained,  it  appears  either  from  the  equations,  or  immediately  from  the 
symmetries  involved,  that  all  the  B's  vanish,  so  that  cosines  only  occur  in  (4) 
and  (5),  that 

AI   =  AI,      A3  =  A3,      A5  =  Aa,  6LC.  ', 

A2'  =  —  A2,    A4'  =  -  A4,  &c.; 

and  further  that  ft'  =  (3.     Equations  (4)  and  (5)  may  thus  be  written 
•^r  =  a  +  @y  4-  a>?/2  +  Ate~v  cos  x  +  A2e~^  cos  2a?  +  A,er^  cos  3#+  .  .  .,  .  .  .(14) 

..........  (15) 


A!  is  of  order  h,  A2  of  order  A2,  A3  of  order  h3,  and  so  on.  If  we  are  content  to 
neglect  h6,  we  may  stop  at  A5;  and  we  find  as  the  equations  necessary  in  order 
to  secure  the  continuity  of  u  and  v  at  the  surface  (6) 

^  (2  +  T  +  n)  -  «  +  ^  (2i  +  T)  -  34>  r  - 

2^2  (2  +  2A2)  = 


From  these  equations  the  values  of  the  constants  may  be  determined  by 
successive  approximations.  Thus,  if  we  retain  terms  of  the  order  A,2,  A,  ,  A4  ,  &c., 
vanish  and 


This  is  the  second  approximation.     The  fifth  approximation  gives 


(18,19,20) 


,  l  =  ,  . 

which  values  are  to  be  substituted  in  (14),  (15). 

The  next  step  is  the  investigation  of  the  values  of  u,  v  at  the  surface  (6). 
They  are  most  conveniently  expressed  as 


214  ON   THE   PROPAGATION   OF   WAVES   UPON   A   PLANE   SURFACE.  [217 

We  get,  correct  as  far  as  A5, 


(21) 


(22) 


the  terms  containing  cos  4#  in  (21),  and  sin  5x  in  (22),  vanishing  to  this  order. 
If  we  substitute  these  values  in  (12),  we  obtain 


h  sin  x  {-  V  +  £  +  2o>  +  £o>A2  -  ^&>A4}  +  ^o>A5  sin  3x  =  0.    .  .  .(23) 

So  far,  then,  as  terms  in  h*,  the  surface  of  separation  (6)  is  propagated  as  a 
simple  wave  with  velocity  given  by 

V  =  j3  +  2co  +  $coh2;  ...........................  (24) 

but,  if  terms  in  A6  are  retained,  a  change  of  form  manifests  itself,  corresponding 
to  the  term  in  wh5  sin  3#  outstanding  in  (23). 

Hitherto  the  wave-length  has  been  supposed  to  be  2?r,  but,  if  we  now  take 
it  to  be  27T/&,  (24)  becomes 

V  =  /3  +  2(y  /  k  .  (1  +  £W),     ....................  (25) 

as  is  evident  by  "  dimensions."     The  velocity  of  propagation  is  that  of  the 
flow  of  the  fluid  in  the  steady  motion  at  the  place  where 

ky  =  l  +  ±kW  ...............................  (26) 

So  far  as  the  present  investigation  can  reach,  there  is  no  sign  of  the 
amplitude  of  a  wave  tending  spontaneously  to  increase. 


218. 

ON  SOME   PHYSICAL  PROPERTIES  OF  ARGON  AND  HELIUM*. 

[Proceedings  of  the  Royal  Society,  LIX.  pp.  198—208,  Jan.  1896.] 

Density  of  Argon. 

IN  our  original  paper-f-  are  described  determinations  by  Prof.  Ramsay,  of 
the  density  of  argon  prepared  with  the  aid  of  magnesium.  The  volume 
actually  weighed  was  163c.c.,and  the  adopted  mean  result  was  19'941,  referred 
to  O2=  16.  At  that  time  a  satisfactory  conclusion  as  to  the  density  of  argon 
prepared  by  the  oxygen  method  of  Cavendish  had  not  been  reached,  although 
a  preliminary  result  (197)  obtained  from  a  mixture  of  argon  and  oxygen  J 
went  far  to  show  that  the  densities  of  the  gases  prepared  by  the  two  methods 
were  the  same.  In  order  further  to  test  the  identity  of  the  gases,  it  was 
thought  desirable  to  pursue  the  question  of  density ;  and  I  determined,  as 
the  event  proved,  somewhat  rashly,  to  attempt  large  scale  weighings  of  pure 
argon  with  the  globe  of  1800  c.c.  capacity  employed  in  former  weighings 
of  gases  1 1  which  could  be  obtained  in  quantity. 

The  accumulation  of  the  3  litres  of  argon,  required  for  convenient  working, 
involved  the  absorption  of  some  300  litres  of  nitrogen,  or  about  800  litres  of 
the  mixture  with  oxygen.  This  was  effected  at  the  Royal  Institution  with 
the  apparatus  already  described  §,  and  which  is  capable  of  absorbing  the 
mixture  at  the  rate  of  about  7  litres  per  hour.  The  operations  extended 
themselves  over  nearly  three  weeks,  after  which  the  residual  gases  amounting 
to  about  10  litres,  still  containing  oxygen  with  a  considerable  quantity  of 

*  [Some  of  the  results  here  given  were  announced  before  the  British  Association  at  the  Ipswich 
meeting.  See  Report,  Sept.  13,  1895.] 

t  Eayleigh  and  Bamsay,  Phil.  Trans.  A,  Vol.  CLXXXVI.  pp.  221,  238,  1895.     [Vol.  iv.  p.  130.] 

J  Loc.  cit.  p.  221.     [Vol.  iv.  p.  165.] 

I!  Roy.  Soc.  Proc.  February,  1888  [Vol.  m.  p.  37] ;  February,  1892  [Vol.  in.  p.  534] ;  March, 
1893  [Vol.  iv.  p.  39]. 

§  Phil.  Trans,  loc.  cit.  p.  219.     [Vol.  iv.  p.  162.] 


216  ON   SOME   PHYSICAL   PROPERTIES   OF   ARGON    AND   HELIUM.  [218 

nitrogen,  were  removed  to  the  country  and  transferred  to  a  special  apparatus 
where  it  could  be  prepared  for  weighing. 

For  this  purpose  the  purifying  vessel  had  to  be  arranged  somewhat 
differently  from  that  employed  in  the  preliminary  absorption  of  nitrogen. 
When  the  gas  is  withdrawn  for  weighing,  the  space  left  vacant  must  be  filled 
up  with  liquid,  and  afterwards,  when  the  gas  is  brought  back  for  repurification, 
the  liquid  must  be  removed.  In  order  to  effect  this,  the  working  vessel 
(Fig.  7)*  communicates  by  means  of  a  siphon  with  a  10-litre  "aspirating 
bottle,"  the  ends  of  the  siphon  being  situated  in  both  cases  near  the  bottom 
of  the  liquid.  In  this  way  the  alkaline  solution  may  be  made  to  pass  back- 
wards and  forwards,  in  correspondence  with  the  desired  displacements  of  gas. 

There  is,  however,  one  objection  to  this  arrangement  which  requires  to  be 
met.  If  the  reserve  alkali  in  the  aspirating  bottle  were  allowed  to  come  into 
contact  with  air,  it  would  inevitably  dissolve  nitrogen,  and  this  nitrogen  would 
be  partially  liberated  again  in  the  working  vessel,  and  so  render  impossible 
a  complete  elimination  of  that  gas  from  the  mixture  of  argon  and  oxygen. 
By  means  of  two  more  aspirating  bottles  an  atmosphere  of  oxygen  was  main- 
tained in  the  first  bottle,  and  the  outermost  bottle,  connected  with  the  second 
by  a  rubber  hose,  gave  the  necessary  control  over  the  pressure. 

Five  glass  tubes  in  all  were  carried  through  the  large  rubber  cork  by 
which  the  neck  of  the  working  vessel  was  closed.  Two  of  these  convey  the 
electrodes:  one  is  the  siphon  for  the  supply  of  alkali,  while  the  fourth  and 
fifth  are  for  the  withdrawal  and  introduction  of  the  gas,  the  former  being 
bent  up  internally,  so  as  to  allow  almost  the  whole  of  the  gaseous  contents 
to  be  removed.  The  fifth  tube,  by  which  the  gas  is  returned,  communicates 
with  the  fall-tube  of  the  Topler  pump,  provision  being  made  for  the  overflow 
of  mercury.  In  this  way  the  gas,  after  weighing,  could  be  returned  to  the 
working  vessel  at  the  same  time  that  the  globe  was  exhausted.  It  would  be 
tedious  to  describe  in  detail  the  minor  arrangements.  Advantage  was  fre- 
quently taken  of  the  fact  that  oxygen  could  always  be  added  with  impunity, 
its  presence  in  the  working  vessel  being  a  necessity  in  any  case. 

When  the  nitrogen  had  been  so  far  removed  that  it  was  thought  desirable 
to  execute  a  weighing,  the  gas  on  its  way  to  the  globe  had  to  be  freed  from 
oxygen  and  moisture.  The  purifying  tubes  contained  copper  and  copper 
oxide  maintained  at  a  red  heat,  caustic  soda,  and  phosphoric  anhydride. 
In  all  other  respects  the  arrangements  were  as  described  in  the  memoir  on 
the  densities  of  the  principal  gases f,  the  weighing  globe  being  filled  at  0°, 
and  at  the  pressure  of  the  manometer  gauge. 

The  process  of  purification  with  the  means  at  my  command  proved  to  be 

*  Phil.  Trans,  loc.  cit.  p.  218.     [Vol.  iv.  p.  163.] 

t  Roy.  Soc.  Proc.  Vol.  LIII.  p.  134,  1893.     [Vol.  iv.  p.  39.] 


1896]  ON   SOME   PHYSICAL   PROPERTIES   OF    ARGON    AND   HELIUM.  217 

extremely  slow.  The  gas  contained  more  nitrogen  than  had  been  expected, 
and  the  contraction  went  on  from  day  to  day  until  I  almost  despaired  of 
reaching  a  conclusion.  But  at  last  the  visible  contraction  ceased,  and  soon 
afterwards  the  yellow  line  of  nitrogen  disappeared  from  the  spectrum  of  the 
jar  discharge*.  After  a  little  more  sparking,  a  satisfactory  weighing  was 
obtained  on  May  22,  1895 ;  but,  in  attempting  to  repeat,  a  breakage  occurred, 
by  which  a  litre  of  air  entered,  and  the  whole  process  of  purification  had  to 
be  re-commenced.  The  object  in  view  was  to  effect,  if  possible,  a  series  of 
weighings  with  intermediate  sparkings,  so  as  to  obtain  evidence  that  the 
purification  had  really  reached  a  limit.  The  second  attempt  was  scarcely 
more  successful,  another  accident  occurring  when  two  weighings  only  had 
been  completed.  Ultimately  a  series  of  four  weighings  were  successfully 
executed,  from  which  a  satisfactory  conclusion  can  be  arrived  at. 

May   22 3-2710 

June    4 3-2617 

June    7  .  3-2727 


June  13 3-2652 

June  18  ......  3-2750  ] 

June  25 3*2748  I  3'2746 

July     2 3-2741 ) 

The  results  here  recorded  are  derived  from  the  comparison  of  the  weighings 
of  the  globe  "  full "  with  the  mean  of  the  preceding  and  following  weighings 
"  empty,"  and  they  are  corrected  for  the  errors  of  the  weights  and  for  the 
shrinkage  of  the  globe  when  exhausted,  as  explained  in  former  papers.  In 
the  last  series,  the  experiment  of  June  13  gave  a  result  already  known  to  be 
too  low.  The  gas  was  accordingly  sparked  for  fourteen  hours  more.  Between 
the  weighings  of  June  18  and  June  25  there  was  nine  hours'  sparking,  and 
between  those  of  June  25  and  July  2  about  eight  hours'  sparking.  The  mean 
of  the  last  three,  viz.  3'2746,  is  taken  as  the  definitive  result,  and  it  is 
immediately  comparable  with  2'6276,  the  weight  under  similar  circumstances 
of  oxygen -f-.  If  we  take  O,2  =  16,  we  obtain  for  argon 

19-940, 

in  very  close  agreement  with  Professor  Ramsay's  result. 

The  conclusion  from  the  spectroscopic  evidence  that  the  gases  isolated 
from  the  atmosphere  by  magnesium  and  by  oxygen  are  essentially  the  same 
is  thus  confirmed. 

*  Jan.  29. — When  the  argon  is  nearly  pure,  the  arc  discharge  (no  jar  connected)  assumes 
a  peculiar  purplish  colour,  quite  distinct  from  the  greenish  hue  apparent  while  the  oxidation  of 
nitrogen  is  in  progress  and  from  the  sky-blue  observed  when  the  residue  consists  mainly  of 
oxygen. 

t  Rmj.  Soc.  Proc.  Vol.  LIII.  p.  144,  1893.     [Vol.  iv.  p.  48.] 


218  ON   SOME   PHYSICAL   PROPERTIES   OF   ARGON   AND   HELIUM.  [218 


The  Refractivity  of  Argon  and  Helium. 

The  refractivity  of  argon  was  next  investigated,  in  the  hope  that  it  might 
throw  some  light  upon  the  character  of  the  gas.  For  this  purpose  absolute 
measurements  were  not  required.  It  sufficed  to  compare  the  pressures 
necessary  in  two  columns  of  air  and  argon  of  equal  lengths,  in  order  to 
balance  the  retardations  undergone  by  light  in  traversing  them. 

The  arrangement  was  a  modification  of  one  investigated  by  Fraunhofer, 
depending  upon  the  interference  of  light  transmitted  through  two  parallel 
vertical  slits  placed  in  front  of  the  object-glass  of  a  telescope.  If  there  be 
only  one  slit,  and  if  the  original  source,  either  a  distant  point  or  a  vertical 
line  of  light,  be  in  focus,  the  field  is  of  a  certain  width,  due  to  "  diffraction," 
and  inversely  as  the  width  of  the  slit.  If  there  be  two  equal  parallel  slits 
whose  distance  apart  is  a  considerable  multiple  of  the  width  of  either,  the 
field  is  traversed  by  bands  of  width  inversely  as  the  distance  between  the 
slits.  If  from  any  cause  one  of  the  portions  of  light  be  retarded  relatively 
to  the  other,  the  bands  are  displaced  in  the  usual  manner,  and  can  be  brought 
back  to  the  original  position  only  by  abolishing  the  relative  retardation. 

When  the  object  is  merely  to  see  the  interference  bands  in  full  perfection, 
the  use  of  a  telescope  is  not  required.  The  function  of  the  telescope  is  really 
to  magnify  the  slit  system*,  and  this  is  necessary  when,  as  here,  it  is  desired 
to  operate  separately  upon  the  two  portions  of  light.  The  apparatus  is, 
however,  extremely  simple,  the  principal  objection  to  it  being  the  high 
magnifying  power  required,  leading  under  ordinary  arrangements  to  a  great 
attenuation  of  light.  I  have  found  that  this  objection  may  be  almost  entirely 
overcome  by  the  substitution  of  cylindrical  lenses,  magnifying  in  the  hori- 
zontal direction  only,  for  the  spherical  lenses  of  ordinary  eye-pieces.  For 
many  purposes  a  single  lens  suffices,  but  it  must  be  of  high  power.  In  the 
measurements  about  to  be  described  most  of  the  magnifying  was  done 
by  a  lens  of  home  manufacture.  It  consisted  simply  of  a  round  rod, 
about  £in.  (4  mm.)  in  diameter,  cut  by  Mr  Gordon  from  a  piece  of  plate 
glass  f.  This  could  be  used  alone ;  but  as  at  first  it  was  thought  necessary 
to  have  a  web,  serving  as  a  fixed  mark  to  which  the  bands  could  be  referred, 
the  rod  was  treated  as  the  object-glass  of  a  compound  cylindrical  microscope, 
the  eye-piece  being  a  commercial  cylindrical  lens  of  1^  in.  (31  mm.)  focus. 
Both  lenses  were  mounted  on  adjustable  stands,  so  that  the  cylindrical  axes 
could  be  made  accurately  vertical,  or,  rather,  accurately  parallel  to  the  length 
of  the  original  slit.  The  light  from  an  ordinary  paraffin  lamp  now  sufficed, 
although  the  magnification  was  such  as  to  allow  the  error  of  setting  to  be 

*  Brit.  Assoc.  Report,  1893,  p.  703.    [Vol.  iv.  p.  76.] 

t  Preliminary  experiments  had  been  made  with  ordinary  glass  cane  and  with  tubes  charged 
with  water. 


1896]  ON   SOME   PHYSICAL   PROPERTIES   OF   ARGON    AND   HELIUM.  219 

less  than  1/20  of  a  band  interval.  It  is  to  be  remembered  that  with  this 
arrangement  the  various  parts  of  the  length  of  a  band  correspond,  not  to  the 
various  parts  of  the  original  slit,  but  rather  to  the  various  parts  of  the  object- 
glass.  This  departure  from  the  operation  of  a  spherical  eye-piece  is  an 
advantage,  inasmuch  as  optical  defects  show  themselves  by  deformation  of 
the  bands  instead  of  by  a  more  injurious  encroachment  upon  the  distinction 
between  the  dark  and  bright  parts. 

Fig.  i. 
B 


The  collimating  lens  A  (Fig.  1)  is  situated  23  ft.  (7  metres)  from  the  source 
of  light.  B,  C  are  the  tubes,  one  containing  dry  air,  the  other  the  gas  to  be 
experimented  upon.  They  are  1  ft.  (30'5  cm.)  long,  and  of  \  in.  (1'3  cm.)  bore, 
and  they  are  closed  at  the  ends  with  small  plates  of  parallel  glass  cut  from 
the  same  strip.  E  is  the  object-glass  of  the  telescope,  about  3  in.  (7'6  cm.) 
in  diameter.  It  is  fitted  with  a  cap,  D,  perforated  by  two  parallel  slits.  Each 
slit  is  ^  in.  (6  mm.)  wide,  and  the  distance  between  the  middle  lines  of  the 
slits  is  1^  in.  (38  mm.). 

The  arrangements  for  charging  the  tubes  and  varying  the  pressures  of 
the  gases  are  sketched  in  Fig.  2.  A  gas  pipette,  DE,  communicates  with  the 
tube  C,  so  that  by  motion  of  the  reservoir  E  and  consequent  flow  of  mercury 
through  the  connecting  hose,  part  of  the  gas  may  be  transferred.  The 
pressure  was  measured  by  a  U-shaped  manometer  F,  containing  mercury. 
This  was  fitted  below  with  a  short  length  of  stout  rubber  tubing  G,  to  which 
was  applied  a  squeezer  H.  The  object  of  this  attachment  was  to  cause 
a  rise  of  mercury  in  both  limbs  immediately  before  a  reading,  and  thus  to 
avoid  the  capillary  errors  that  would  otherwise  have  entered.  A  similar- 
pipette  and  manometer  were  connected  with  the  air-tube  B.  In  order  to  be 
able,  if  desired,  to  follow  with  the  eye  a  particular  band  during  the  changes 
of  pressure  (effected  by  small  steps  and  alternately  in  the  two  tubes),  diminu- 
tive windlasses  were  provided  by  which  the  motions  of  the  reservoirs  (E) 
could  be  made  smooth  and  slow.  In  this  way  all  doubt  was  obviated  as  to 
the  identity  of  a  band ;  but  after  a  little  experience  the  precaution  was  found 
to  be  unnecessary*. 

The  manner  of  experimenting  will  now  be  evident.  By  adjustment  of 
pressures  the  centre  of  the  middle  band  was  brought  to  a  definite  position, 

*  [For  a  description  of  a  modified  apparatus  capable  of  working  with  an  extremely  small 
quantity  of  gas,  see  Proc.  Roy.  Soc.  Vol.  LXIV.  p.  97,  1898.] 


220  ON   SOME    PHYSICAL   PROPERTIES    OF    ARGON   AND   HELIUM.  [218 

determined  by  the  web  or  otherwise,  and  the  pressures  were  measured.  Both 
pressures  were  then  altered  and  adjusted  until  the  band  was  brought  back 
precisely  to  its  original  position.  The  ratio  of  the  changes  of  pressure  is  the 
inverse  ratio  of  the  refractivities  (u.  —  1)  of  the  gases.  The  process  may  be 
repeated  backwards  and  forwards  any  number  of  times,  so  as  to  eliminate  in 
great  degree  errors  of  the  settings  -and  of  the  pressure  readings. 

Fig.  2. 


To  pump. 


Scale 


During  these  observations  a  curious  effect  was  noticed,  made  possible 
by  the  independent  action  of  the  parts  of  the  object-glass  situated  at  various 
levels,  as  already  referred  to.  When  the  bands  were  stationary,  they  appeared 
straight,  or  nearly  so,  but  when  in  motion,  owing  to  changes  of  pressure,  they 
became  curved,  even  in  passing  the  fiducial  position,  and  always  in  such 
a  manner  that  the  ends  led.  The  explanation  is  readily  seen  to  depend  upon 
the  temporary  changes  of  temperature  which  accompany  compression  or 
rarefaction.  The  full  effect  of  a  compression,  for  example,  would  not  be 
attained  until  the  gas  had  cooled  back  to  its  normal  temperature,  and  this 
recovery  of  temperature  would  occur  more  quickly  at  the  top  and  bottom, 
where  the  gas  is  in  proximity  to  the  metal,  than  in  the  central  part  of 
the  tube. 

The  success  of  the  measures  evidently  requires  that  there  should  be  no 
apparent  movement  of  the  bands  apart  from  real  retardations  in  the  tubes. 


1896]  ON   SOME   PHYSICAL   PROPERTIES   OF   ARGON    AND    HELIUM.  221 

As  the  apparatus  -was  at  first  arranged,  this  condition  was  insufficiently 
satisfied.  Although  all  the  parts  were  carried  upon  the  walls  of  the  room, 
frequent  and  somewhat  sudden  displacements  of  the  bands  relatively  to  the 
web  were  seen  to  occur,  probably  in  consequence  of  the  use  of  wood  in  some 
of  the  supports.  The  observations  could  easily  be  arranged  in  such  a  manner 
that  no  systematic  error  could  thence  enter,  but  the  agreement  of  individual 
measures  was  impaired.  Subsequently  a  remedy  was  found  in  the  use  of 
a  second  system  of  bands,  formed  by  light  which  passed  just  above  the  tubes, 
to  which,  instead  of  to  the  web,  the  moveable  bands  were  referred.  The 
coincidence  of  the  two  systems  could  be  observed  with  accuracy,  and  was 
found  to  be  maintained  in  spite  of  movements  of  both  relatively  to  the  web. 

In  the  comparisons  of  argon  and  air  (with  nearly  the  same  refractivities) 
the  changes  of  pressure  employed  were  about  8  in.  (20  cm.),  being  deductions 
from  the  atmospheric  pressure.  In  one  observation  of  July  26,  the  numbers, 
representing  suctions  in  inches  of  mercury,  stood 

Argon  Air 

8-54  9-96 

0-01  177 


8-53  819 

Ratio  =  0-961, 

signifying  that  8'53  in.  of  argon  balanced  819  in.  of  dry  air.  Four  sets, 
during  which  the  air  and  argon  (from  the  globe  as  last  filled  for  weighing) 
were  changed,  taken  on  July  17,  18,  19,  26,  gave  respectively  for  the  final 
ratio  0-962,  0*961,  0'961,  0*960,  or  as  the  mean 

Refractivity  of  argon  _ 
Refractivity  of  air 

The  evidence  from  the  refractivities,  as  well  as  from  the  weights,  is  very 
unfavourable  to  the  view  that  argon  is  an  allotropic  form  of  nitrogen  such  as 
would  be  denoted  by  N3. 

The  above  measurements,  having  been  made  with  lamp-light,  refer  to  the 
most  luminous  region  of  the  spectrum,  say  in  the  neighbourhood  of  D.  But 
since  no  change  in  the  appearance  of  the  bands  at  the  two  settings  could 
be  detected,  the  inference  is  that  the  dispersions  of  the  two  gases  are 
approximately  the  same,  so  that  the  above  ratio  would  not  be  much  changed, 
even  if  another  part  of  the  spectrum  were  chosen.  It  may  be  remarked  that 
the  displacement  actually  compensated  in  the  above  experiments  amounted 
to  about  forty  bands,  each  band  corresponding  to  about  ^  in.  (5  mm.)  pressure 
of  mercury. 

Similar  comparisons  have  been  made  between  air  and  helium.  The 
latter  gas,  prepared  by  Professor  Ramsay,  was  brought  from  London  by 


222  ON   SOME   PHYSICAL   PROPERTIES   OF   ARGON   AND   HELIUM.  [218 

Mr  W.  Randall,  who  farther  gave  valuable  assistance  in  the  manipulations. 
It  appeared  at  once  that  the  refractivity  of  helium  was  remarkably  low,  13  in. 
pressure  of  the  gas  being  balanced  by  less  than  2  in.  pressure  of  air.  The 
ratios  given  by  single  comparisons  on  July  29  were  0147,  0'146,  0145,  0146, 
mean  0146 ;  and  on  July  30  0147,  0147,  0145,  0145,  mean  0146.  The 
observations  were  not  made  under  ideal  conditions,  on  account  of  the  smallness 
of  the  changes  of  air  pressure ;  but  we  may  conclude  that  with  considerable 
approximation 

Refractivity  of  helium  _  ¥ 

Refractivity  of  air 

The  lowest  refractivity  previously  known  is  that  of  hydrogen,  nearly  0'5 
of  that  of  air. 


Viscosity  of  Argon  and  Helium. 

The  viscosity  was  investigated  by  the  method  of  passage  through  capillary 
tubes.  The  approximate  formula  has  been  investigated  by  O.  Meyer^,  on 
the  basis  of  Stokes'  theory  for  incompressible  fluids.  If  the  driving  pressure 
(pi~Pa)  i&  no*  too  great,  the  volume  F2  delivered  in  time  t  through  a  tube 
of  radius  R  and  length  \  is  given  by 


the  volume  being  measured  at  the  lower  pressure  p2}  and  rj  denoting  the 
viscosity  of  the  gas.  In  the  comparison  of  different  gases  F2,  plt  p2,  R,  X 
may  be  the  same,  and  then  97  is  proportional  to  t. 

In  the  apparatus  employed  two  gas  pipettes  and  manometers,  somewhat 
similar  to  those  shown  in  Fig.  2,  were  connected  by  a  capillary  tube  of  very 
small  bore  and  about  1  metre  long.  The  volume  F2  was  about  100  c.c.,  and 
was  caused  to  pass  by  a  pressure  of  a  few  centimetres  of  mercury,  maintained 
as  uniform  as  possible  by  means  of  the  pipettes.  There  was  a  difficulty, 
almost  inherent  in  the  use  of  mercury,  in  securing  the  right  pressures  during 
the  first  few  seconds  of  an  experiment  ;  but  this  was  not  of  much  importance 
as  the  whole  time  t  amounted  to  several  minutes.  The  apparatus  was  tested 
upon  hydrogen,  and  was  found  to  give  'the  received  numbers  with  sufficient 
accuracy.  The  results,  referred  to  dry  air,  were  for  helium  0'96;  and  for 
argon  T21,  somewhat  higher  than  for  oxygen  which  at  present  stands  at 
the  head  of  the  list  of  the  principal  gases  J. 

*  [1902.  The  sample  must  have  contained  impurity  —  probably  hydrogen.  Prof.  Ramsay's 
latest  result  for  the  refractivity  of  helium  referred  to  air  is  -1238  (Proc.  Roy.  Soc.  LXVII.  p.  331, 
1900).] 

t  Pogg.  Ann.  Vol.  cxxvii.  p.  270,  1866. 

t  [1902.     Schultze  (Drude  Ann.  vi.  p.  310,  1901)  finds  for  helium  1-086  in  place  of  0'96.] 


1896]  ON   SOME   PHYSICAL   PROPERTIES   OF   ARGON    AND   HELIUM.  223 

Gas  from  the  Bath  Springs. 

In  the  original  memoir  upon  argon*  results  were  given  of  weighings  of 
the  residue  from  the  Bath  gas  after  removal  of  oxygen,  carbonic  anhydride, 
and  moisture,  from  which  it  appeared  that  the  proportion  of  argon  was  only 
one-half  of  that  contained  in  the  residue,  after  similar  treatment,  from  the 
atmosphere.  After  the  discovery  of  helium  by  Professor  Ramsay,  the  question 
presented  itself  as  to  whether  this  conclusion  might  not  be  disturbed  by  the 
presence  in  the  Bath  gas  of  helium,  whose  lightness  would  tend  to  compensate 
the  extra  density  of  argon. 

An  examination  of  the  gas  which  had  stood  in  my  laboratory  more  than 
a  year  having  shown  that  it  still  contained  no  oxygen,  it  was  thought  worth 
while  to  remove  the  nitrogen  so  as  to  determine  the  proportion  that  would 
refuse  oxidation.  For  this  purpose  200  c.c.  were  worked  up  with  oxygen  until 
the  volume,  free  from  nitrogen,  was  reduced  to  8  c.c.  On  treatment  with 
pyrogallol  and  alkali  the  residue  measured  3'3  c.c.,  representing  argon,  and 
helium,  if  present.  On  sparking  the  residue  at  atmospheric  pressure  and 
examining  the  spectrum,  it  was  seen  to  be  mainly  that  of  argon,  but  with  an 
unmistakable  exhibition  of  D3.  At  atmospheric  pressure  this  line  appears 
very  diffuse  in  a  spectroscope  of  rather  high  power,  but  the  place  was  correct. 

From  another  sample  of  residue  from  the  Bath  gas,  vacuum  tubes  were 
charged  by  my  son,  Mr  R.  J.  Strutt,  and  some  of  them  showed  D3  sharply 
denned  and  precisely  coincident  with  the  line  of  helium  in  a  vacuum  tube 
prepared  by  Professor  Ramsay. 

Although  the  presence  of  helium  in  the  Bath  gas  is  not  doubtful,  the 
quantity  seems  insufficient  to  explain  the  low  density  found  in  October, 
1894.  In  order  to  reconcile  that  density  with  the  proportion  of  residue 
(3'3/200  =  0'016)  found  in  the  experiment  just  described,  it  would  be  necessary 
to  suppose  that  the  helium  amounted  to  25  per  cent,  of  the  whole  residue  of 
argon  and  helium.  Experiment,  however,  proved  that  a  mixture  of  argon 
and  helium  containing  10  per  cent,  of  the  latter  gas  showed  D3  more  plainly 
than  did  the  Bath  residue.  It  is  just  possible  that  some  of  the  helium  was 
lost  by  diffusion  during  the  long  interval  between  the  experiments  whose 
results  are  combined  in  the  above  estimate. 


Buxton  Gas. 

Gas  from  the  Buxton  springs,  kindly  collected  for  me  by  Mr  A.  McDougall, 
was  found  to  contain  no  appreciable  oxygen.     The  argon  amounted  to  about 

*  Rayleigh  and  Ramsay,  Phil.  Trans.  A,  Vol.  CLXXXVI.  p.  227,  1895.     [Vol.  iv.  p.  172.] 


224  ON   SOME   PHYSICAL  PROPERTIES   OF   ARGON   AND   HELIUM.  [218 

2  per  cent,  of  the  volume.  When  its  spectrum  was  examined,  the  presence 
of  D3  was  suspected,  but  the  appearance  was  too  feeble  to  allow  of  a  definite 
statement  being  made.  The  proportion  of  helium  is  in  any  case  very  much 
lower  than  in  the  Bath  gas. 


Is  Helium  contained  in  the  Atmosphere? 

Apart  from  its  independent  interest,  this  question  is  important  in  con- 
nection with  the  density  of  atmospheric  argon.  Since  the  spectrum  of  this 
gas  does  not  show  the  line  D3,  we  may  probably  conclude  that  the  proportion 
of  helium  is  less  than  3  per  cent. ;  so  that  there  would  be  less  than  3  x  10~4 
of  helium  in  the  atmosphere.  The  experiment  about  to  be  described  was 
an  attempt  to  carry  the  matter  further,  and  is  founded  upon  the  observation 
by  Professor  Ramsay,  that  the  solubility  of  helium  in  water  is  only  O007,  less 
than  one-fifth  of  that  which  we  found  for  argon*. 

It  is  evident  that  if  a  mixture  of  helium  and  argon  be  dissolved  in  water 
until  there  is  only  a  small  fraction  remaining  over,  the  proportion  of  helium 
will  be  much  increased  in  the  residue.  Two  experiments  have  been  made, 
of  which  that  on  October  6,  1895,  was  the  more  elaborate.  About  60  c.c. 
of  argon  were  shaken  for  a  long  time  with  well-boiled  water  contained  in 
a  large  flask.  When  the  absorption  had  ceased,  the  residue  of  30  c.c.  was 
sparked  with  a  little  oxygen  until  no  nitrogen  could  be  seen  in  the  spectrum. 
It  was  then  treated  a  second  time  with  boiled  water  until  its  volume  was 
reduced  to  1£  c.c.  With  this  vacuum  tubes  were  charged  by  my  son  at  two 
different  pressures.  In  none  of  them  could  D3  be  detected;  nor  was  there 
any  marked  difference  to  be  seen  between  the  spectra  of  the  washed  and  the 
unwashed  argon.  If  helium  be  present  in  the  atmosphere,  it  must  be  in  very 
small  quantity,  probably  much  less  than  a  ten-thousandth  part^*. 

*  Phil.  Trans.  A,  Vol.  CLXXXVI.  p.  225,  1895.     [VoL  iv.  p.  170.] 

t  [1902.     The  presence  of  traces  of  helium  in  the  atmosphere  is  not  doubtful.] 


219. 


ON  THE  AMOUNT   OF  ARGON   AND   HELIUM   CONTAINED  IN 
THE   GAS   FROM   THE   BATH   SPRINGS* 

[Proceedings  of  the  Royal  Society,  LX.  pp.  56,  57,  1896.] 

THE  presence  of  helium  in  the  residue  after  removal  of  nitrogen  from  this 
gas  was  proved  in  a  former  paperf,  but  there  was  some  doubt  as  to  the 
relative  proportions  of  argon  and  helium.  A  fresh  sample,  kindly  collected 
by  Dr  Richardson,  has  therefore  been  examined.  Of  this  2,500  c.c.,  submitted 
to  electric  sparks  in  presence  of  oxygen,  gave  a  final  residue  of  37  c.c.,  after 
removal  of  all  gases  known  until  recently.  The  spectrum  of  the  residue, 
observed  at  atmospheric  pressure,  showed  argon,  and  the  D3  line  of  helium 
very  plainly. 

The  easy  visibility  of  D3  suggested  the  presence  of  helium  in  some  such 
proportion  as  10  per  cent.,  and  this  conjecture  has  been  confirmed  by  a 
determination  of  the  refractivity  of  the  mixture.  It  may  be  remembered 
that  while  the  refractivity  of  argon  approaches  closely  that  of  air,  the  relative 
number  being  0'961,  the  refractivity  of  helium  (as  supplied  to  me  by  Pro- 
fessor Ramsay)  is  very  low,  being  only  0'146  on  the  same  scale.  If  we  assume 
that  any  sample  of  gas  is  a  mixture  of  these  two,  its  refractivity  will  deter- 
mine the  proportions  in  which  the  components  are  present. 

The  observations  were  made  by  an  apparatus  similar  in  character  to  that 
already  described,  but  designed  to  work  with  smaller  quantities  of  gas.  The 
space  to  be  filled  is  only  about  12  c.c.,  and  if  the  gas  be  at  atmospheric 
pressure  its  refractivity  may  be  fixed  to  about  1/1000  part.  By  working  at 
pressures  below  atmosphere  very  fair  results  could  be  arrived  at  with  quan- 
tities of  gas  ordinarily  reckoned  at  only  3  or  4  c.c. 

The  refractivity  found  for  the  Bath  residue  after  desiccation  was  0'896 
referred  to  air,  so  that  the  proportional  amount  of  helium  is  8  per  cent. 
Referred  to  the  original  volume,  the  proportion  of  helium  is  1'2  parts  per 
thousand. 

*  I  am  reminded  by  Mr  Whitaker  that  helium  is  appropriately  associated  with  the  Bath 
waters,  which,  according  to  some  antiquaries,  were  called  by  the  Romans  Aqua  Solis. 
t  Boy.  Soc.  Proc.  Vol.  LIX.  p.  206,  1896.     [Vol.  iv.  p.  223.] 

R.     IV.  15 


220. 


THE   REPRODUCTION   OF  DIFFRACTION   GRATINGS. 


[Nature,  LIV.  pp.  332,  333,  1896*.] 

I  HAVE  first  to  apologise  for  the  very  informal  character  of  the  communi- 
cation which  I  am  about  to  make  to  the  Club ;  I  have  not  been  able  to  put 
anything  down  upon  paper,  but  I  thought  it  might  be  interesting  to  some  to 
hear  an  account  of  experiments  that  have  now  been  carried  on  at  intervals 
for  a  considerable  series  of  years  in  the  reproduction — mainly  the  photographic 
reproduction — of  diffraction  gratings.  Probably  most  of  you  know  that  these 
consist  of  straight  lines  ruled  very  closely,  very  accurately,  and  parallel  to 
one  another,  upon  a  piece  of  glass  or  speculum  metal.  Usually  they  are 
ruled  with  a  diamond  by  the  aid  of  a  dividing  machine ;  and  in  late  years, 
particularly  in  the  hands  of  Rutherfurd  and  Rowland,  an  extraordinary 
degree  of  perfection  has  been  attained.  It  was  many  years  ago — nearly 
25  years,  I  am  afraid — that  I  first  began  experiments  upon  the  photographic 
reproduction  of  these  divided  gratings,  each  in  itself  the  work  of  great  time 
and  trouble,  and  costing  a  good  deal  of  money.  At  that  time  the  only 
gratings  available  were  made  by  Nobert,  in  Germany,  of  which  I  had  two, 
each  containing  about  a  square  inch  of  ruled  surface,  one  of  about  3,000 
lines  to  the  inch,  and  the  other  of  about  6,000.  It  happened,  by  an  accident, 
that  the  grating  with  3,000  lines  was  the  better  of  the  two,  in  that  it 
was  more  accurately  ruled,  and  gave  much  finer  definition  upon  the  solar 
spectrum;  the  6,000  line  grating  was  brighter,  but  its  definition  was 
decidedly  inferior,  so  that  both  had  certain  advantages  according  to  the 
particular  object  in  view. 

If  it  comes  to  the  question  of  how  to  make  a  grating  by  photography, 
probably  the  first  idea  to  occur  to  one  would  be  that  it  might  be  a  com- 
paratively simple  matter  to  make  a  grating  upon  a  large  scale,  and  then 

*  [From  a  report  of]  an  address  delivered  at  the  eighth  annual  conference  of  the  Camera  Club. 


1896]  THE   REPRODUCTION   OF   DIFFRACTION   GRATINGS.  227 

reduce  it  by  photography,  but  if  one  goes  into  the  figures  the  project  is 
not  found  so  promising.  Take,  for  instance,  a  grating  with  10,000  lines  to 
the  inch ;  if  you  magnified  that,  say  100  times,  your  lines  would  then  be  100 
to  the  inch;  if  you  magnified  it  1,000  times,  they  would  still  be  10  to  the 
inch,  and  that  would  be  a  convenient  size  so  far  as  interval  between  the  lines 
was  concerned ;  but  think  what  would  be  the  area  required  to  hold  a  grating 
magnified  to  that  extent.  By  the  time  you  have  magnified  the  inch  by  100 
or  1,000,  you  would  want  a  wall  of  a  house  or  of  a  cathedral  to  hold  the 
grating.  If  the  problem  were  proposed  of  ruling  a  grating  with  6,000  lines, 
with  a  high  degree  of  accuracy,  it  would  be  easier  to  do  it  on  a  microscopic 
scale  than  upon  a  large  scale,  leaving  out  of  consideration  the  difficulty  of 
reproducing  it.  And  those  difficulties  would  be  insuperable,  because,  al- 
though with  a  good  microscopic  object-glass  it  would  be  easy  to  photograph 
lines  which  are  much  closer  together  than  3,000  or  6,000  to  the  inch, 
yet  that  could  only  be  achieved  over  a  very  small  area  of  surface — nothing 
like  a  square  inch ;  and  if  it  were  required  to  cover  a  square  inch  with  lines 
6,000  to  the  inch,  it  would  be  beyond  the  power,  not  only,  I  believe,  of  any 
microscope,  but  of  any  lens  that  was  ever  made.  So  that  that  line  of 
investigation  does  not  fulfil  the  promise  which  at  first  it  might  appear  to 
give ;  and,  in  fact,  there  is  nothing  simpler  or  better  than  to  copy  the  original 
ruled  by  a  dividing  engine,  by  the  simple  process  of  contact  printing. 

For  this  purpose  some  precautions  are  required.  You  must  use  very  flat 
glass,  by  preference  it  should  be  optically  worked  glass,  although  very  good 
results  may  be  obtained  on  selected  pieces  of  ordinary  plate.  Of  course,  no 
one  would  think  of  making  such  a  print  by  diffused  daylight,  but  the  sun 
itself,  or  a  point  of  light  from  any  suitable  source,  according  to  the  nature 
of  the  photographic  process  which  is  adopted,  permits  quite  well  of  the 
reproduction  of  any  grating  of  a  moderate  degree  of  fineness.  I  have  used 
almost  all  varieties  of  photographic  processes  in  my  time.  In  the  days 
when  I  first  worked,  the  various  dry  collodion  processes  were  better  under- 
stood than  they  are  now;  the  old  albumen  process  was  extremely  suitable 
for  such  work  as  this,  on  account  of  the  almost  complete  absence  of  structure 
in  the  film,  and  the  very  convenient  hardness  of  the  surface,  which  made  the 
result  comparatively  little  liable  to  injury.  I  used  with  success  the  dry 
collodion  processes,  the  tannin  process  among  others,  and  also  some  of  the 
direct  printing  methods,  such  as  the  collodio-chloride.  The  latter  method, 
worked  upon  glass,  gave  excellent  results,  particularly  if  the  finished  print 
was  treated  with  mercury  in  the  way  commonly  used  for  intensification, 
except  that,  in  the  treatment  of  a  grating  with  mercury,  it  is  desirable  to 
stop  at  the  mercury  and  not  to  go  on  to  the  blackening  process  used  in  the 
intensification  of  negatives.  From  the  visual  point  of  view,  the  grating, 
after  intensification — if  one  may  use  the  term — with  mercury,  looks  much 
less  intense  than  before,  but,  nevertheless,  the  spectra  seen  when  a  point  or 

15—2 


228  THE   REPRODUCTION    OF    DIFFRACTION   GRATINGS.  [220 

slit   of  light  is  looked  at  through   the  grating  becomes  very  much    more 
brilliant. 

I  used  another  process  at  that  time,  more  than  twenty  years  ago,  which 
gave  excellent  results,  but  had  not  the  degree  of  certainty  that  I  aimed  at, 
namely,  a  bichromated  gelatine  process,  similar  to  carbon  printing,  except 
that  no  pigment  was  employed.  A  glass  plate  was  simply  coated  with 
bichromated  gelatine  of  a  suitable  thickness — and  a  good  deal  depended 
upon  hitting  that  off  correctly;  if  the  coating  was  too  thin  the  grating 
showed  a  deficiency  of  brightness,  whereas,  if  it  was  too  thick,  there  might 
be  a  difficulty  in  getting  it  sufficiently  uniform  and  smooth  on  the  surface. 
However,  I  obtained  excellent  gratings  by  that  process,  most  of  them  capable 
of  showing  the  nickel  line  between  the  two  well-known  sodium  or  D  lines 
in  the  solar  spectrum,  when  suitably  examined.  The  collodio-chloride  process 
was  comparatively  slow,  and  bichromated  gelatine  required  two  or  three 
minutes  exposure  to  sunlight  to  produce  a  proper  effect;  but  for  the  more 
sensitive  developed  negative  processes  a  very  much  less  powerful  light  or 
a  reduced  exposure  was  needed. 

The  performance  of  the  copies  was  quite  good,  and,  except  where  there 
was  some  obvious  defect,  I  never  could  see  that  they  were  worse  than  the 
originals;  in  fact,  in  respect  of  brightness  it  not  unfrequently  happened 
that  the  copies  were  far  superior  to  the  originals,  so  that  in  many  cases 
they  would  be  more  useful.  I  do  not  mean  by  that,  however,  that  I  would 
rather  have  a  copy  than  an  original  if  anyone  wanted  to  make  me  a 
present.  There  seems  to  be  some  falling  off  in  copies ;  so  that  they  cannot 
well  be  copied  again,  and  if  you  want  to  work  upon  spectra  of  an  extremely 
high  order,  dispersed  to  a  great  extent  laterally  from  the  straight  line, 
a  copy  would  not  be  satisfactory.  The  reproduction  of  gratings  on  bi- 
chromated gelatine  is  easily  and  quickly  accomplished;  there  is  only  the 
coating  of  the  glass  over-night,  rapid  drying  to  avoid  crystallisation  in  the 
film,  exposure,  washing,  and  drying.  In  order  to  get  the  best  effect  it  is 
usually  desirable  to  treat  the  bichromated  copies  with  hot  water.  It  is 
a  little  difficult  to  understand  what  precisely  happens.  All  photographers 
know  that  the  action  of  light  upon  bichromated  gelatine  is  to  produce 
a  comparative  insolubility  of  the  gelatine.  In  the  carbon  process,  and 
many  others  in  which  gelatine  is  used,  the  gelatine  which  remains  soluble, 
not  having  been  sufficiently  exposed  to  light,  is  fairly  washed  away  in 
the  subsequent  treatment  with  warm  water,  but  for  that  effect  it  is  generally 
necessary  to  get  at  the  back  of  the  gelatine  film,  because  on  its  face  there 
is  usually  a  layer  which  is  so  insoluble  as  not  to  allow  of  the  washing  away 
of  any  of  the  gelatine  situated  behind.  But  in  the  present  case  there  is 
no  question  of  transferring  the  film,  which  remains  fixed  to  the  glass,  and 
therefore  it  is  difficult  to  see  how  any  gelatine  could  be  dissolved  out. 
However,  under  the  action  of  water,  the  less  exposed  gelatine  no  doubt 


1896]  THE   REPRODUCTION   OF   DIFFRACTION   GRATINGS.  229 

swells  more  than  that  which  has  received  more  exposure  and  has  thus 
lost  its  affinity  for  water;  and  while  the  gelatine  is  wet  it  is  reasonable 
that  a  rib-like  structure  should  ensue,  which  is  what  would  be  required 
in  order  to  make  a  grating,  but  when  the  gelatine  dries,  one  would  suppose 
that  all  would  again  become  flat,  and  indeed  that  happens  to  a  certain 
extent.  The  gratings  lost  a  great  deal  of  intensity  in  drying,  but,  if  properly 
treated  with  warm  water,  the  reduction  does  not  go  too  far,  and  a  considerable 
degree  of  intensity  is  left  when  the  photograph  is  dry. 

Although  it  belongs  to  another  branch  of  the  subject,  a  word  may  not 
be  out  of  place  as  to  the  accuracy  with  which  the  gratings  must  be  made. 
It  seems  a  wonderful  thing  at  first  sight,  to  rule  6,000  lines  to  an  inch 
at  all,  if  you  think  of  the  smallest  interval  that  you  can  readily  see  with 
the  eye,  perhaps  one-hundredth  of  an  inch,  and  remember  that  in  these 
gratings  there  are  sixty  lines  in  the  space  of  one-hundredth  of  an  inch, 
and  all  disposed  at  rigorously  equal  intervals.  Those  familiar  with  optics 
will  understand  the  importance  of  extreme  accuracy  if  I  give  an  illustration. 
Take  the  case  of  the  two  sodium  lines  in  the  spectrum,  the  D  lines ;  they 
differ  in  wave-length  by  about  a  thousandth  part;  the  dispersion — the 
extent  to  which  the  light  is  separated  from  the  direct  line — is  in  proportion 
to  the  wave-length  of  the  light,  and  inversely  as  the  interval  between 
the  consecutive  lines  on  the  grating ;  so  that,  if  we  had  a  grating  in  which 
the  first  half  was  ruled  at  the  rate  of  1,000  to  the  inch,  and  the  second 
half  at  the  rate  of  1,001  to  the  inch,  the  one  half  would  evidently  do  the 
same  thing  for  one  soda  line  as  the  other  half  of  the  grating  was  doing 
for  the  other  soda  line,  and  the  two  lines  would  be  mixed  together 
and  confused.  In  order,  therefore,  to  do  anything  like  good  work,  it  is 
necessary,  not  only  to  have  a  very  great  number  of  lines,  but  to  have 
them  spaced  with  most  extraordinary  precision ;  and  it  is  wonderful  what 
success  has  been  reached  by  the  beautiful  dividing  machines  of  Rutherfurd 
and  Rowland.  I  have  seen  Rowland's  machine  at  Baltimore,  and  have 
heard  him  speak  of  the  great  precautions  required  to  get  good  results. 
The  whole  operation  of  the  machine  is  automatic;  the  ruling  goes  on 
continuously  day  and  night,  and  it  is  necessary  to  pay  the  most  careful 
regard  to  uniformity  of  temperature,  for  the  slightest  expansion  or  con- 
traction due  to  change  of  temperature  of  the  different  parts  of  the  machine 
would  bring  utter  confusion  into  the  grating  and  its  resulting  spectrum. 

The  contact  in  printing  has  to  be  pretty  close  and  the  finer  the 
grating  the  closer  must  the  contact  be.  I  experimented  upon  that  point : 
one  can  get  some  kind  of  result,  theoretically,  by  preparing  a  photographic 
film  with  a  slightly  convex  surface  and  using  that  for  the  print;  then, 
where  the  contact  was  closest,  the  original  of  course  was  very  well  im- 
pressed, and  round  that,  one  got  different  degrees  of  increasingly  imperfect 
contact,  and  one  could  trace  in  the  result  what  the  effect  of  imperfect 


230  THE   REPRODUCTION   OF   DIFFRACTION    GRATINGS.  [220 

contact  is.  I  found  that,  both  with  gratings  of  3,000  and  6,000  lines  to 
the  inch,  good  enough  contact  was  obtained  with  ordinary  flat  glass ;  but 
when  you  come  to  gratings  of  17,000  or  20,000  lines  to  the  inch  the  contact 
requires  to  be  extremely  close,  and  in  order  to  get  a  good  copy  of  a  grating 
with  20,000  lines  per  inch  it  is  necessary  that  there  should  nowhere  be 
one  ten-thousandth  of  an  inch  between  the  original  and  the  printing 
surface — a  degree  of  closeness  not  easily  secured  over  the  entire  area.  It 
is  rather  singular  that  though  I  published  full  accounts  of  this  work  a  long 
time  ago,  and  distributed  a  large  number  of  copies,  the  process  of  repro- 
ducing gratings  by  photography  did  not  become  universally  known,  and 
was  re-discovered  in  France,  by  Izarn,  only  two  or  three  years  since. 

One  reason  why  photographic  reproduction  is  not  practised  to  a  very 
great  extent,  is,  that  the  modern  gratings — such  as  Rowland's — are  ruled 
almost  universally  upon  speculum  metal.  A  grating  upon  speculum  metal 
is  very  excellent  for  use,  but  does  not  well  lend  itself  to  the  process  of 
photographic  copying,  although  I  have  succeeded  to  a  certain  extent  in 
copying  a  grating  ruled  upon  speculum  metal.  For  this  purpose  the  light 
had  to  pass  first  through  the  photographic  film,  then  be  reflected  from  the 
speculum  metal,  and  so  pass  back  again  through  the  film.  Gratings,  such 
as  could  easily  be  made  by  copying  from  a  glass  original,  are  not  readily 
produced  from  one  on  speculum  metal,  and  I  think  that  is  the  reason  why 
the  process  has  not  come  into  more  regular  use.  Glass  is  much  more 
trying  than  speculum  metal  to  the  diamond,  and  that  accounts  for  the 
latter  being  generally  preferred  for  gratings ;  it  is  very  hard,  but  has  not 
ruinous  effects  upon  the  diamond;  indeed  the  principal  difficulty  consists 
in  getting  a  good  diamond  point,  and  maintaining  it  in  a  shape  suitable  for 
making  the  very  fine  cut  which  is  required. 

I  may  now  allude  to  another  method  of  photographic  reproduction  which 
I  tried  only  last  summer.  It  happened  that  I  then  went  with  Professor 
Meldola  over  Waterlow's  large  photo-mechanical  printing  establishment, 
and  I  was  much  interested,  among  other  very  interesting  things,  to  see 
the  use  of  the  old  bitumen  process — the  first  photographic  process  known. 
It  is  used  for  the  reproduction  of  cuts  in  black  and  white.  A  carefully 
cleansed  zinc  plate  is  coated  with  a  varnish  of  bitumen  dissolved  in  benzole, 
and  exposed  to  sunlight  for  about  two  hours  under  a  negative  giving  great 
contrast.  Where  the  light  penetrates  the  negative  the  bitumen  becomes 
comparatively  insoluble,  and  where  it  has  been  protected  from  the  action 
of  light  it  retains  its  original  degree  of  solubility.  When  the  exposed  plate 
is  treated  with  a  solvent,  turpentine  or  some  milder  solvent  than  benzole, 
the  protected  parts  are  dissolved  away,  leaving  the  bare  metal;  whereas 
the  parts  that  have  received  the  sunlight,  being  rendered  insoluble,  remain 
upon  the  metal  and  protect  it  in  the  subsequent  etching  process.  I  did 
not  propose  to  etch  metal,  and,  therefore,  I  simply  used  the  bitumen  varnish 


1896]  THE  REPRODUCTION   OF   DIFFRACTION  GRATINGS.  231 

spread  upon  glass  plates,  and  exposed  the  plates  so  prepared  to  sunshine 
for  about  two  hours  in  contact  with  the  grating.  They  were  then  developed, 
if  one  may  use  the  phrase,  with  turpentine;  and  this  is  the  part  of  the 
process  which  is  the  most  difficult  to  manage.  If  you  stop  development 
early  you  get  [without  difficulty]  a  grating  which  gives  fair  spectra,  but 
it  may  be  deficient  in  intensity  and  brightness;  if  you  push  development 
the  brightness  increases  up  to  a  point  at  which  the  film  disintegrates 
altogether.  In  this  way  one  is  tempted  to  pursue  the  process  to  the  very 
last  point,  and,  although  one  may  succeed  so  far  as  to  have  a  film  which 
is  quite  intact  so  long  as  the  turpentine  is  upon  it,  I  have  not  succeeded 
in  finding  any  method  of  getting  rid  of  the  turpentine  without  causing 
the  disintegration  of  the  film.  In  the  commercial  application  of  the 
process  the  bitumen  is  treated  somewhat  brutally — the  turpentine  is  rinsed 
off  with  a  jet  of  water;  I  have  tried  that,  and  many  of  my  results  have 
been  very  good.  I  have  also  tried  to  sling  off  the  turpentine  with  the  aid 
of  a  kind  of  centrifugal  machine,  but  by  either  plan  the  [too  tender]  film 
is  liable  not  to  survive  the  treatment  required  for  getting  rid  of  the 
turpentine.  If  the  solvent  is  allowed  to  remain  we  are  in  another  difficulty, 
because  then  the  developing  action  is  continued  and  the  result  is  lost. 
But  if  the  process  is  properly  managed,  and  development  stopped  at  the 
right  point,  and  if  the  film  be  of  the  right  degree  of  thickness,  you  get 
an  excellent  copy.  I  have  one  here,  6,000  lines  to  the  inch,  which  I  think 
is  about  the  very  best  copy  I  have  ever  made.  The  method  gives  results 
somewhat  superior  to  the  best  that  can  be  got  with  gelatine ;  but  I  would 
not  recommend  it  in  preference  to  the  latter,  because  it  is  much  more  difficult 
to  work  unless  some  one  can  hit  upon  an  improved  manipulation. 

I  will  not  enlarge  upon  the  importance  of  gratings;  those  acquainted 
with  optics  know  how  very  important  is  the  part  played  by  diffraction 
gratings  in  optical  research,  and  how  the  most  delicate  work  upon  spectra, 
requiring  the  highest  degree  of  optical  power,  is  made  by  means  of  gratings, 
ruled  on  speculum  metal  by  Rowland.  I  suppose  the  reason  why  no  pro- 
fessional photographer  has  taken  up  the  production  of  photographic  gratings, 
is  the  difficulty  of  getting  the  glass  originals;  they  are  very  expensive, 
and  indeed  I  do  not  know  where  they  are  now  to  be  obtained.  It  seems 
a  pity  that  photographic  copies  should  not  be  more  generally  available. 
I  have  given  a  great  many  away  myself;  but  educational  establishments 
are  increasing  all  over  the  country,  and  for  the  purpose  of  instructing 
students  it  is  desirable  that  reasonably  good  gratings  should  be  placed  in 
their  hands,  to  make  them  familiar  with  the  measurements  by  which  the 
wave-length  of  light  is  determined. 

[1902.     For  earlier  papers  upon  this  subject  see  Vol.  I.  pp.  160,  199,  504.] 


221. 


THE   ELECTRICAL  RESISTANCE   OF  ALLOYS. 


[Nature,  LIV.  pp.  154,  155,  1896.] 

THE  recent  researches  of  Profs.  Dewar  and  Fleming  upon  the  electrical 
resistance  of  metals  at  low  temperatures  have  brought  into  strong  relief 
the  difference  between  the  behaviour  of  pure  metals  and  of  alloys.  In  the 
former  case  the  resistance  shows  every  sign  of  tending  to  disappear  altogether 
as  the  absolute  zero  of  temperature  is  approached,  but  in  the  case  of  alloys 
this  condition  of  things  is  widely  departed  from,  even  when  the  admixture 
consists  only  of  a  slight  impurity. 

Some  years  ago  it  occurred  to  me  that  the  apparent  resistance  of  an 
alloy  might  be  partly  made  up  of  thermo-electric  effects,  and  as  a  rough 
illustration  I  calculated  the  case  of  a  conductor  composed  of  two  metals 
arranged  in  alternate  laminae  perpendicular  to  the  direction  of  the  current. 
Although  a  good  many  difficulties  remain  untouched,  I  think  that  the  calcu- 
lation may  perhaps  suggest  something  to  those  engaged  upon  the  subject. 
At  any  rate  it  affords  a  priori  ground  for  the  supposition  that  an  important 
distinction  may  exist  between  the  resistances  of  pure  and  alloyed  metals. 

The  general  character  of  the  effect  is  easily  explained.  According  to  the 
discovery  of  Peltier,  when  an  electric  current  flows  from  one  metal  to  another 
there  is  development  or  absorption  of  heat  at  the  junction.  The  temperature 
disturbance  thus  arising  increases  until  the  conduction  of  heat  through  the 
laminae  balances  the  Peltier  effects  at  the  junctions,  and  it  gives  rise  to  a 
thermo-electromotive  force  opposing  the  passage  of  the  current.  Inasmuch  as 
the  difference  of  temperature  at  the  alternate  j  unctions  is  itself  proportional 
to  the  current,  so  is  also  the  reverse  electromotive  force  thereby  called  into 
play.  Now  a  reverse  electromotive  force  proportional  to  current  is  indistin- 
guishable experimentally  from  a  resistance;  so  that  the  combination  of 


1896]  THE    ELECTRICAL   RESISTANCE   OF   ALLOYS.  233 

laminated  conductors  exhibits  a  false  resistance,  having  (so  far  as  is  known) 
nothing  in  common  with  the  real  resistance  of  the  metals. 

If  e  be  the  thermo-electric  force  of  the  couple  for  one  degree  difference 
of  temperature  of  the  junctions;  t,  t'  the  actual  temperatures;  then  the 
electromotive  force  for  one  couple  is  e  (t  —  t'}.  If  we  suppose  that  there  are 
n  similar  couples  per  unit  of  length  perpendicular  to  the  lamination,  the 
whole  reverse  electromotive  force  per  unit  of  length  is  ne  (t  —  t').  Again,  if 
C  be  the  current  corresponding  to  unit  of  cross-section,  the  development  of 
heat  per  second  at  each  alternate  junction  is  per  unit  of  area  273  x  e  x  C, 
the  actual  temperature  being  in  the  neighbourhood  of  zero  Cent.  This  is 
measured  in  ergs,  and  is  to  be  equated  to  the  heat  conducted  per  second 
towards  the  cold  junctions  on  the  two  sides.  If  k,  k'  be  the  conductivities 
for  heat  of  the  two  metals,  I  and  I'  the  corresponding  thicknesses,  the  heat 
conducted  per  second  is 


or  if  lf(l  +  l')=p} 

the  conducted  heat  is 

n(t-t')\k/p  +  k'/q}. 

In  this  expression  p  +  q  =  1  ,  the  symbols  p  and  q  denoting  the  proportional 
amounts  by  volume  in  which  the  two  metals  are  associated.  Thus  when  a 
stationary  state  is  reached 

273  x  e  x  C  =  n  (t  -  t')  [k/p  +  k'/q}. 

This  determines  (t  —  t')  when  C  is  given  ;  and  the  whole  back  electromotive 
force  per  unit  of  thickness  is  rC,  where 

273  x  e2 


k/p  +  k'/q' 

This  is  the  expression  for  the  false  resistance  per  unit  of  thickness,  which, 
it  should  specially  be  noted,  is  independent  of  n,  the  number  of  couples. 
The  number  of  couples  which  co-operate  is  indeed  increased  by  finer  lamina- 
tion, but  the  efficiency  of  each  is  decreased  in  the  same  proportion  by  the 
readier  conduction  of  heat  between  the  junctions.  It  is  scarcely  necessary 
to  point  out  that  the  false  resistance  is  called  into  play  only  by  currents 
which  flow  across  the  laminae. 

In  my  original  calculation  the  metals  chosen  for  illustration  were  iron 
and  copper.  In  this  case  (Everett's  C.G.8.  System  of  Units,  p.  192)  c  =  1600. 
The  conductivities  are  to  be  measured  in  ergs.  For  iron,  k  =  '164  x  4'2  x  107; 


234  THE   ELECTRICAL   RESISTANCE   OF   ALLOYS.  [221 

for  copper,  k'  =  I'll  x  4'2  x  107.     Thus,  if  the  metals  are  in  equal  volumes 

_2x  273x1600* 
4-2  x  107  x  1-27 

This  is  the  thermo-electric  addition  to  the  true  specific  resistance,  and  is 
about  1£  per  cent,  of  that  of  copper.  Such  an  addition  may  seem  small ; 
but  it  should  be  remembered  that  for  the  more  distinctively  thermo-electric 
metals  e  is  much  larger,  and  that  it  enters  by  its  square.  In  any  case  it 
seems  desirable  that  this  complication  should  be  borne  in  mind.  The 
consequences  which  follow  from  recognised  laws  for  laminated  structures, 
however  fine,  must  surely  have  some  bearing  upon  the  properties  of  alloys, 
although  in  this  case  the  fineness  is  molecular. 


222. 

ON   THE  THEORY   OF   OPTICAL  IMAGES,  WITH   SPECIAL 
REFERENCE  TO   THE  MICROSCOPE. 

[Philosophical  Magazine,  XLII.  pp.  167—195,  1896.] 

THE  special  subject  of  this  paper  has  been  treated  from  two  distinct 
points  of  view.  In  the  work  of  Helmholtz*  the  method  followed  is  analogous 
to  that  which  had  long  been  used  in  the  theory  of  the  telescope.  It  consists 
in  tracing  the  image  representative  of  a  mathematical  point  in  the  object, 
the  point  being  regarded  as  self-luminous.  The  limit  to  definition  depends 
upon  the  fact  that  owing  to  diffraction  the  image  thrown  even  by  a  perfect 
lens  is  not  confined  to  a  point,  but  distends  itself  over  a  patch  or  disk  of 
light  of  finite  diameter.  Two  points  in  the  object  can  appear  fully  separated 
only  when  the  representative  disks  are  nearly  clear  of  one  another.  The 
application  to  the  microscope  was  traced  by  means  of  a  somewhat  extended 
form  of  Lagrange's  general  optical  theorem,  and  the  conclusion  was  reached 
that  the  smallest  resolvable  distance  e  is  given  by 

e  =  iX./sina,   .................................  (1) 

X  being  the  wave-length  in  the  medium  where  the  object  is  situated,  and 
a  the  divergence  -angle  of  the  extreme  ray  (the  semi-angular  aperture)  in 
the  same  medium.  If  X0  be  the  wave-length  in  vacuum, 


/i  being  the  refractive  index  of  the  medium;  and  thus 

€  =  ^X0/yu,sin  a  ...............................  (3) 

The  denominator  /z  sin  a  is  the  quantity  now  well  known  (after  Abbe)  as 
the  "  numerical  aperture." 

The  extreme  value  possible  for  a  is  a  right  angle,  so  that  for  the  micro- 
scopic limit  we  have 


Pogg.  Ann.  Jubelband,  1874. 


236  ON  THE  THEORY  OF  OPTICAL  IMAGES,  [222 

The  limit  can  be  depressed  only  by  a  diminution  in  X0,  such  as  photography 
makes  possible,  or  by  an  increase  in  //,,  the  refractive  index  of  the  medium 
in  which  the  object  is  situated. 

This  method,  in  which  the  object  is  considered  point  by  point,  seems 
the  most  straightforward,  and  to  a  great  extent  it  solves  the  problem 
without  more  ado.  When  the  representative  disks  are  thoroughly  clear 
of  one  another,  the  two  points  in  which  they  originate  are  resolved,  and 
on  the  other  hand,  when  the  disks  overlap  the  points  are  not  distinctly 
separated.  Open  questions  can  relate  only  to  intermediate  cases  of  partial 
overlapping  and  various  degrees  of  resolution.  In  these  cases  (as  has  been 
insisted  upon  by  Dr  Stoney)  we  have  to  consider  the  relative  phases  of  the 
overlapping  lights  before  we  can  arrive  at  a  complete  conclusion. 

If  the  various  points  of  the  object  are  self-luminous,  there  is  no  per- 
manent phase-relation  between  the  lights  of  the  overlapping  disks,  and 
the  resultant  illumination  is  arrived  at  by  simple  addition  of  separate 
intensities.  This  is  the  situation  of  affairs  in  the  ordinary  use  of  a  telescope, 
whether  the  object  be  a  double  star,  the  disk  of  the  sun,  the  disk  of  the 
moon,  or  a  terrestrial  body.  The  distribution  of  light  in  the  image  of 
a  double  point,  or  of  a  double  line,  was  especially  considered  in  a  former 
paper*,  and  we  shall  return  to  the  subject  later. 

When,  as  sometimes  happens  in  the  use  of  the  telescope,  and  more 
frequently  in  the  use  of  the  microscope,  the  overlapping  lights  have  per- 
manent phase-relations,  these  intermediate  cases  require  a  further  treatment ; 
and  this  is  a  matter  of  some  importance  as  involving  the  behaviour  of  the 
instrument  in  respect  to  the  finest  detail  which  it  is  capable  of  rendering. 
We  shall  see  that  the  image  of  a  double  point  under  various  conditions 
can  be  delineated  without  difficulty. 

In  the  earliest  paper  by  Prof.  Abbef,  which  somewhat  preceded  that 
of  Helmholtz,  similar  conclusions  were  reached;  but  the  demonstrations 
were  deferred,  and,  indeed,  they  do  not  appear  ever  to  have  been  set  forth 
in  a  systematic  manner.  Although  some  of  the  positions  then  taken  up, 
as  for  example  that  the  larger  features  and  the  finer  structure  of  a  micro- 
scopic object  are  delineated  by  different  processes,  have  since  had  to  be 
abandoned^,  the  publication  of  this  paper  marks  a  great  advance,  and  has 
contributed  powerfully  to  the  modern  development  of  the  microscope§.  In 

*  "  Investigations  in  Optics,  "with  special  reference  to  the  Spectroscope,"  Phil.  Mag.  Vol.  vin. 
p.  266  (1879).  [Vol.  i.  p.  415.] 

t  Archivf.  Mikr.  Anat.  Vol.  ix.  p.  413  (1873). 

J  Dallenger's  edition  of  Carpenter's  Microscope,  p.  64,  1891. 

§  It  would  seem  that  the  present  subject,  like  many  others,  has  suffered  from  over-specializa- 
tion, much  that  is  familiar  to  the  microscopist  being  almost  unknown  to  physicists,  and  vice  versa. 
For  myself  I  must  confess  that  it  is  only  recently,  in  consequence  of  a  discussion  between 


1896] 


WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE. 


237 


Prof.  Abbe's  method  of  treating  the  matter  the  typical  object  is  not  a 
luminous  point,  but  a  grating  illuminated  by  plane  waves.  Thence  arise 
the  well-known  diffraction  spectra,  which  are  focused  near  the  back  of 
the  object-glass  in  its  principal  focal  plane.  If  the  light  be  homogeneous 
the  spectra  are  reduced  to  points,  and  the  final  image  may  be  regarded 
as  due  to  the  simultaneous  action  of  these  points  acting  as  secondary  centres 
of  light.  It  is  argued  that  the  complete  representation  of  the  object 
requires  the  co-operation  of  all  the  spectra.  When  only  a  few  are  present, 
the  representation  is  imperfect ;  and  when  there  is  only  one — for  this  purpose 
the  central  image  counts  as  a  spectrum — the  representation  wholly  fails. 

That  this  point  of  view  offers  great  advantages,  at  least  when  the  object 
under  consideration  is  really  a  grating,  is  at  once  evident.  More  especially 
is  this  the  case  in  respect  of  the  question  of  the  limit  of  resolution.  It 
is  certain  that  if  one  spectrum  only  be  operative,  the  image  must  consist 
of  a  uniform  field  of  light,  and  that  no  sign  can  appear  of  the  real  periodic 
structure  of  the  object.  From  this  consideration  the  resolving-power  is 
readily  deduced,  and  it  may  be  convenient  to  recapitulate  the  argument 
for  the  case  of  perpendicular  incidence.  In  Fig.  1  AB  represents  the  axis, 

Fig.  1. 


A  being  in  the  plane  of  the  object  (grating)  and  B  in  the  plane  of  the 
image.  The  various  diffraction  spectra  are  focused  by  the  lens  LL'  in 
the  principal  focal  plane,  S0  representing  the  central  image  due  to  rays 
which  issue  normally  from  the  grating.  After  passing  S0  the  rays  diverge 
in  a  cone  corresponding  to  the  aperture  of  the  lens  and  illuminate  a  circle 
CD  in  the  plane  of  the  image,  whose  centre  is  B.  The  first  lateral 
spectrum  /S\  is  formed  by  rays  diffracted  from  the  grating  at  a  certain 
angle;  and  in  the  critical  case  the  region  of  the  image  illuminated  by  the 
rays  diverging  from  &  just  includes  B.  The  extreme  ray  8^  evidently 

Mr  L.  Wright  and  Dr  G.  3.  Stoney  in  the  English  Mechanic  (Sept.,  Oct.,  Nov.,  1894;  Nov.  8, 
Dec.  13,  1895;  Jan.  17,  1896),  that  I  have  become  acquainted  with  the  distinguishing  features  of 
Prof.  Abbe's  work,  and  have  learned  that  it  was  conducted  upon  different  lines  to  that  of 
Helmholtz.  I  am  also  indebted  to  Dr  Stoney  for  a  demonstration  of  some  of  Abbe's  experiments. 


238  ON  THE  THEORY  OF  OPTICAL  IMAGES,  [222 

proceeds  from  A,  which  is  the  image  of  B.  The  condition  for  the  co- 
operation at  B  of  the  first  lateral  spectrum  is  thus  that  the  angle  of  diffraction 
do  not  exceed  the  semi-angular  aperture  a.  By  elementary  theory  we 
know  that  the  sine  of  the  angle  of  diffraction  is  X/e,  so  that  the  action  of 
the  lateral  spectrum  requires  that  e  exceed  X/sin  a.  If  we  allow  the  incidence 
upon  the  grating  to  be  oblique,  the  limit  becomes  ^ X/sin  a,  as  in  (1). 

We  have  seen  that  if  one  spectrum  only  illuminate  B,  the  field  shows 
no  structure.  If  two  spectra  illuminate  it  with  equal  intensities,  the  field 
is  occupied  by  ordinary  interference  bands,  exactly  as  in  the  well-known 
experiments  of  Fresnel.  And  it  is  important  to  remark  that  the  character 
of  these  bands  is  always  the  same,  both  as  respects  the  graduation  of  light 
and  shade,  and  in  the  fact  that  they  have  no  focus.  When  more  than  two 
spectra  co-operate,  the  resulting  interference  phenomena  are  more  com- 
plicated, and  there  is  opportunity  for  a  completer  representation  of  the  special 
features  of  the  original  grating*. 

While  it  is  certain  that  the  image  ultimately  formed  may  be  considered 
to  be  due  to  the  spectra  focused  at  $0,  S^..,  the  degree  of  conformity  of 
the  image  to  the  original  object  is  another  question.  From  some  of  the 
expositions  that  have  been  given  it  might  be  inferred  that  if  all  the  spectra 
emitted  from  the  grating  were  utilized,  the  image  would  be  a  complete 
representation  of  the  original.  By  considering  the  case  of  a  very  fine 
grating,  which  might  afford  no  lateral  spectra  at  all,  it  is  easy  to  see  that 
this  conclusion  is  incorrect,  but  the  matter  stands  in  need  of  further  eluci- 
dation. Again,  it  is  not  quite  clear  at  what  point  the  utilization  of  a 
spectrum  really  begins.  All  the  spectra  which  the  grating  is  competent 
to  furnish  are  focused  in  the  plane  S^;  and  some  of  them  might  be 
supposed  to  operate  partially  even  although  the  part  of  the  image  under 
examination  is  outside  the  geometrical  cone  defined  by  the  aperture  of 
the  object-glass.  For  these  and  other  reasons  it  will  be  seen  that  the 


*  These  effects  were  strikingly  illustrated  in  some  observations  upon  gratings  with  6,000  lines 
to  the  inch,  set  up  vertically  in  a  dark  room  and  illuminated  by  sunlight  from  a  distant  vertical 
slit.  The  object-glass  of  the  microscope  was  a  quarter-inch.  When  the  original  grating,  divided 
upon  glass  (by  Nobert),  was  examined  in  this  way,  the  lines  were  well  seen  if  the  instrument  was 
in  focus,  but,  as  usual,  a  comparatively  slight  disturbance  of  focus  caused  all  structure  to  disappear. 
When,  however,  a  photographic  copy  of  the  same  glass  original,  made  with  bitumen  [p.  231],  was 
substituted  for  it,  very  different  effects  ensued.  The  structure  could  be  seen  even  although  the 
object-glass  were  drawn  back  through  1£  inch  from  its  focused  position ;  and  the  visible  lines 
were  twice  as  close,  as  if  at  the  rate  of  12,000  to  the  inch.  The  difference  between  the  two  cases 
is  easily  explained  upon  Abbe's  theory.  A  soda  flame  viewed  through  the  original  showed  a  strong 
central  image  (spectrum  of  zero  order)  and  comparatively  faint  spectra  of  the  first  and  higher 
orders.  A  similar  examination  of  the  copy  revealed  very  brilliant  spectra  of  the  first  order  on  both 
sides,  and  a  relatively  feeble  central  image.  The  case  is  thus  approximately  the  same  as  when  in 
Abbe's  experiment  all  spectra  except  the  first  (on  the  two  sides)  are  blocked  out. 


1896]  WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE.  239 

spectrum  theory*,  valuable  as  it  is,  needs  a  good  deal  of  supplementing, 
even  when  the  representation  of  a  grating  under  parallel  light  is  in 
question. 

When  the  object  under  examination  is  not  a  grating  or  a  structure  in 
which  the  pattern  is  repeated  an  indefinite  number  of  times,  but  for 
example  a  double  point,  or  when  the  incident  light  is  not  parallel,  the 
spectrum  theory,  as  hitherto  developed,  is  inapplicable.  As  an  extreme 
example  of  the  latter  case  we  may  imagine  the  grating  to  be  self-luminous. 
It  is  obvious  that  the  problem  thus  presented  must  be  within  the  scope 
of  any  complete  theory,  and  equally  so  that  here  there  are  no  spectra 
formed,  as  these  require  the  radiations  from  the  different  elements  of  the 
grating  to  possess  permanent  phase-relations.  It  appears,  therefore,  to  be 
a  desideratum  that  the  matter  should  be  reconsidered  from  the  older  point 
of  view,  according  to  which  the  typical  object  is  a  point  and  not  a  grating. 
Such  a  treatment  illustrates  the  important  principle  that  the  theory  of 
resolving-power  is  essentially  the  same  for  all  instruments.  The  peculiarities 
of  the  microscope  arise  from  the  fact  that  the  divergence-angles  are  not 
limited  to  be  small,  and  from  the  different  character  of  the  illumination 
usually  employed;  but,  theoretically  considered,  these  are  differences  of 
detail.  The  investigation  can,  without  much  difficulty,  be  extended  to 
gratings,  and  the  results  so  obtained  confirm  for  the  most  part  the  conclusions 
of  the  spectrum  theory. 

It  will  be  convenient  to  Commence  our  discussion  by  a  simple  investiga- 
tion of  the  resolving-power  of  an  optical  instrument  for  a  self-luminous 
double  point,  such  as  will  be  applicable  equally  to  the  telescope  and  to 
the  microscope.  In  Fig.  2  AB  represents  the  axis,  A  being  a  point  of  the 

Fig.  2. 


object  and  B  a  point  of  the  image.  By  the  operation  of  the  object-glass  LL' 
all  the  rays  issuing  from  A  arrive  in  the  same  phase  at  B.  Thus  if  A  be 
self-luminous,  the  illumination  is  a  maximum  at  B,  where  all  the  secondary 
waves  agree  in  phase.  B  is  in  fact  the  centre  of  the  diffraction  disk  which 
constitutes  the  image  of  A.  At  neighbouring  points  the  illumination  is 

*  The  special  theory  initiated  by  Prof.  Abbe  is  usually  called  the  "diffraction  theory,"  a 
nomenclature  against  which  it  is  necessary  to  protest.  Whatever  may  be  the  view  taken,  any 
theory  of  resolving  power  of  optical  instruments  must  be  a  diffraction  theory  in  a  certain  sense, 
so  that  the  name  is  not  distinctive.  Diffraction  is  more  naturally  regarded  as  the  obstacle  to  fine 
definition,  and  not,  as  with  some  exponents  of  Prof.  Abbe's  theory,  the  machinery  by  which  good 
definition  is  brought  about. 


240  ON   THE   THEORY   OF   OPTICAL   IMAGES,  [222 

less,  in  consequence  of  the  discrepancies  of  phase  which  there  enter.  In 
like  manner,  if  we  take  a  neighbouring  point  P  in  the  plane  of  the  object, 
the  waves  which  issue  from  it  will  arrive  at  B  with  phases  no  longer 
absolutely  accordant,  and  the  discrepancy  of  phase  will  increase  as  the 
interval  AP  increases.  When  the  interval  is  very  small,  the  discrepancy 
of  phase,  though  mathematically  existent,  produces  no  practical  effect,  and 
the  illumination  at  B  due  to  P  is  as  important  as  that  due  to  A,  the 
intensities  of  the  two  luminous  centres  being  supposed  equal.  Under  these 
conditions  it  is  clear  that  A  and  P  are  not  separated  in  the  image.  The 
question  is,  to  what  amount  must  the  distance  AP  be  increased  in  order 
that  the  difference  of  situation  may  make  itself  felt  in  the  image.  This 
is  necessarily  a  question  of  degree;  but  it  does  not  require  detailed  calcu- 
lations in  order  to  show  that  the  discrepancy  first  becomes  conspicuous 
when  the  phases  corresponding  to  the  various  secondary  waves  which  travel 
from  P  to  B  range  over  about  a  complete  period.  The  illumination  at  B 
due  to  P  then  becomes  comparatively  small,  indeed  for  some  forms  of 
aperture  evanescent.  The  extreme  discrepancy  is  that  between  the  waves 
which  travel  through  the  outermost  parts  of  the  object-glass  at  L  and  L'; 
so  that,  if  we  adopt  the  above  standard  of  resolution,  the  question  is,  where 
must  P  be  situated  in  order  that  the  relative  retardation  of  the  rays  PL 
and  PL'  may  on  their  arrival  at  B  amount  to  a  wave-length  (X).  In 
virtue  of  the  general  law  that  the  reduced  optical  path  is  stationary  in 
value,  this  retardation  may  be  calculated  without  allowance  for  the  different 
paths  pursued  on  the  further  side  of  L,  L',  so  that  its  value  is  simply 
PL  —  PL.  Now  since  AP  is  very  small,  AL'  —  PL'  is  equal  to  ^LP.sina, 
where  a  is  the  semi-angular  aperture  L  AB.  In  like  manner  PL  -  AL  has 
the  same  value,  so  that 


According  to  the  standard  adopted,  the  condition  of  resolution  is  therefore 
that  AP,  or  e,  should  exceed  ^X/sina,  as  in  (1).  If  e  be'  less  than  this, 
the  images  overlap  too  much  ;  while  if  e  greatly  exceed  the  above  value 
the  images  become  unnecessarily  separated. 

In  the  above  argument  the  whole  space  between  the  object  and  the 
lens  is  supposed  to  be  occupied  by  matter  of  one  refractive  index,  and 
X  represents  the  wave-length  in  this  medium  of  the  kind  of  light  employed. 
If  the  restriction  as  to  uniformity  be  violated,  what  we  have  ultimately  to 
do  with  is  the  wave-length  in  the  medium  immediately  surrounding 
the  object. 

The  statement  of  the  law  of  resolving-power  has  been  made  in  a  form 
appropriate  to  the  microscope,  but  it  admits  also  of  immediate  application 
to  the  telescope.  If  2R  be  the  diameter  of  the  object-glass,  and  D  the 


1896]  WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE.  241 

distance  of  the  object,  the  angle  subtended  by  AP  is  e/D,  and  the  angular 
resolving-power  is  given  by 

X 


the  well-known  formula. 

This  method  of  derivation  makes  it  obvious  that  there  is  no  essential 
difference  of  principle  between  the  two  cases,  although  the  results  are 
conveniently  stated  in  different  forms.  In  the  case  of  the  telescope  we  have 
to  do  with  a  linear  measure  of  aperture  and  an  angular  limit  of  resolution, 
whereas  in  the  case  of  the  microscope  the  limit  of  resolution  is  linear  and 
it  is  expressed  in  terms  of  angular  aperture. 

In  the  above  discussion  it  has  been  supposed  for  the  sake  of  simplicity 
that  the  points  to  be  discriminated  are  self-luminous,  or  at  least  behave 
as  if  they  were  such.  It  is  of  interest  to  enquire  how  far  this  condition 
can  be  satisfied  when  the  object  is  seen  by  borrowed  light.  We  may  imagine 
that  the  object  takes  the  form  of  an  opaque  screen,  perforated  at  two  points, 
and  illuminated  by  distant  sources  situated  behind. 

If  the  source  of  light  be  reduced  to  a  point,  so  that  a  single  train  of 
plane  waves  falls  upon  the  screen,  there  is  a  permanent  phase-relation 
between  the  waves  incident  at  the  two  points,  and  therefore  also  between 
the  waves  scattered  from  them.  In  this  case  the  two  points  are  as  far  as 
possible  from  behaving  as  if  they  were  self-luminous.  If  the  incidence 
be  perpendicular,  the  secondary  waves  issue  in  the  same  phase  ;  but  in 
the  case  of  obliquity  there  is  a  permanent  phase-difference.  This  difference, 
measured  in  wave-lengths,  increases  up  to  e,  the  distance  between  the 
points,  the  limit  being  attained  as  the  incidence  becomes  grazing. 

When  the  light  originates  in  distant  independent  sources,  not  limited 
to  a  point,  there  is  no  longer  an  absolutely  definite  phase-relationship 
between  the  secondary  radiations  from  the  two  apertures  ;  but  this  condition 
of  things  may  be  practically  maintained,  if  the  angular  magnitude  of  the 
source  be  not  too  large.  For  example,  if  the  source  be  limited  to  an  angle  6 
round  the  normal  to  the  screen,  the  maximum  phase-difference  measured 
in  wave-lengths  is  esin#,  so  that  if  sin#  be  a  small  fraction  of  X/e,  the 
finiteness  of  6  has  but  little  effect.  When,  however,  sin  6  is  so  great  that 
e  sin  9  becomes  a  considerable  multiple  of  X,  the  secondary  radiations 
become  approximately  independent,  and  the  apertures  behave  like  self- 
luminous  points.  It  is  evident  that  even  with  a  complete  hemispherical 
illumination  this  condition  can  scarcely  be  attained  when  e  is  less 
than  X. 

The  use  of  a  condenser  allows  the  widely-extended  source  to  be  dispensed 
with.      By  this  means  an  image  of  a  distant  source  composed  of  indepen- 
E.   iv.  16 


242 


ON   THE   THEORY   OF   OPTICAL   IMAGES, 


[222 


dently  radiating  parts,  such  as  a  lamp-flame,  may  be  thrown  upon  the 
object,  and  it  might  at  first  sight  be  supposed  that  the  problem  under 
consideration  was  thus  completely  solved  in  all  cases,  inasmuch  as  the  two 
apertures  correspond  to  different  parts  of  the  flame.  But  we  have  to 
remember  here  and  everywhere  that  optical  images  are  not  perfect,  and 
that  to  a  point  of  the  flame  corresponds  in  the  image,  not  a  point,  but 
a  disk  of  finite  magnitude.  When  this  consideration  is  taken  into  account, 
the  same  limitation  as  before  is  encountered. 

For  what  is  the  smallest  disk  into  which  the  condenser  is  capable  of 
concentrating  the  light  received  from  a  distant  point  ?  Fig.  2  and  the 
former  argument  apply  almost  without  modification,  and  they  show  that 
the  radius  AP  of  the  disk  has  the  value  ^X/sina,  where  a  is  the  semi- 
angular  aperture  of  the  condenser.  Accordingly  the  diameter  of  the  disk 
cannot  be  reduced  below  X ;  and  if  e  be  less  than  X  the  radiations  from  the 
two  apertures  are  only  partially  independent  of  one  another. 

It  seems  fair  to  conclude  that  the  function  of  the  condenser  in  micro- 
scopic practice  is  to  cause  the  object  to  behave,  at  any  rate  in  some  degree, 
as  if  it  were  self-luminous,  and  thus  to  obviate  the  sharply-marked  inter- 
ference-bands which  arise  when  permanent  and  definite  phase-relations  are 
permitted  to  exist  between  the  radiations  which  issue  from  various  points 
of  the  object. 

As  we  shall  have  occasion  later  to  employ  Lagrange's  theorem,  it  may 
be  well  to  point  out  how  an  instantaneous  proof  of  it  may  be  given  upon 
the  principles  [especially  that  the  optical  distance  measured  along  a'  ray 
is  a  minimum]  already  applied.  As  before,  AB  (Fig.  3)  represents  the 


Fig.  3. 


axis  of  the  instrument,  A  and  B  being  conjugate  points.  P  is  a  point 
near  A  in  the  plane  through  A  perpendicular  to  the  axis,  and  Q  is  its 
image*  in  the  perpendicular  plane  through  B.  Since  A  and  B  are  conjugate, 
the  optical  distance  between  them  is  the  same  for  all  [ray-]  paths,  e.g.  for 


*  [1902.     In  the   original  diagram  Q  was  shown  upon  the  wrong  side  of  B.     I  owe  the 
correction  to  a  correspondence  with  Prof,  Everett.] 


1896]  WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE.  243 

AR8B  and  ALMB.  [For  the  same  reason  the  optical  distance  from  P 
to  Q  is  the  same  along  the  various  rays,  one  of  which  lies  infinitely  near 
to  PRSQ  and  another  to  PLMQ.]  And,  since  AP,  BQ  are  perpendicular 
to  the  axis,  the  optical  distance  from  P  to  Q  is  the  same  (to  the  first  order 
of  small  quantities  [such  as  AP])  as  from  A  to  B.  Consequently  the  optical 
distance  PRSQ  is  the  same  as  ARSB.  Thus,  if  /*,  p!  be  the  refractive 
indices  in  the  neighbourhood  of  A  and  B  respectively,  a  and  ft  the  divergence- 
angles  RAL,  SBM  for  a  given  ray,  we  have 

fi.AP.ama=p'.BQ.sm/3,  ........................  (6) 

where  AP,  BQ  denote  the  corresponding  linear  magnitudes  of  the  two 
images.  This  is  the  theorem  of  Lagrange,  extended  by  Helmholtz  so  as  to 
apply  to  finite  divergence-angles*. 

We  now  pass  on  to  the  actual  calculation  of  the  images  to  be  expected 
upon  Fresnel's  principles  in  the  various  cases  that  may  arise.  The  origin 
of  coordinates  (£  =  0,  rj  =  0)  in  the  focal  plane  is  the  geometrical  image  of 
the  radiant  point.  If  the  vibration  incident  upon  the  lens  be  represented 
by  cos  (27rVt/\),  where  V  is  the  velocity-  of  light,  the  vibration  at  any 
point  £,  i]  in  the  focal  plane  isf 


in  which  /  denotes  the  focal  length,  and  the  integration  with  respect  to  as 
and  y  is  to  be  extended  over  the  aperture  of  the  lens.  If  for  brevity 
we  write 

,    .......................  (8) 


(7)  may  be  put  into  the  form 


where 

8  =  //sin  (pas  +  qy)  dxdy,         C  =  //  cos  (px  +  qy)  dxdy.  .  .  .(10,  11) 

It  will  suffice  for  our  present  purpose  to  limit  ourselves  to  the  case  where 
the  aperture  is  symmetrical  with  respect  to  as  and  y.  We  have  then 
8  =  Q,  -and 

C  =  ffcospx  cosqy  dxdy,  ........................  (12) 

the  phase  of  the  vibration  being  the  same  at  all  points  of  the  diffraction 
pattern. 

*  I  learn  from  Czapski's  excellent  Theorie  der  Optischen  Instrumente  that  a  similar  derivation 
of  Lagrange's  theorem  from  the  principle  of  minimum  path  had  already  been  given  many  years 
ago  by  Hockin  (Micros.  Soc.  Journ.  Vol.  iv.  p.  337,  1884). 

t  See  for  example  Enc.  Brit.  "  Wave  Theory,"  p.  430  (1878).     [Vol.  in.  p.  80.] 

16—2 


244 


ON  THE  THEORY  OF  OPTICAL  IMAGES, 


[222 


When  the  aperture  is  rectangular,  of  width  a  parallel  to  x,  and  of 
width  b  parallel  to  y,  the  limits  of  integration  are  from  —  \a  to  +^a  for  x, 
and  from  —  16  to  +  16  for  y.  Thus 

sin(7ny6/X/) 


ab- 


.(13) 


and  by  (9)  the  amplitude  of  vibration  (irrespective  of  sign)  is  Cj\f.  This 
expression  gives  the  diffraction  pattern  due  to  a  single  point  of  the  object 
whose  geometrical  image  is  at  ff  =  0,  77  =  0.  Sometimes,  as  in  the  application 
to  a  grating,  we  wish  to  consider  the  image  due  to  a  uniformly  luminous 
line,  parallel  to  17,  and  this  can  always  be  derived  by  integration  from  the 
expression  applicable  to  a  point.  But  there  is  a  distinction  to  be  observed 
according  as  the  radiations  from  the  various  parts  of  the  line  are  independent 
or  are  subject  to  a  fixed  phase-relation.  In  the  former  case  we  have  to 
deal  only  with  the  intensity,  represented  by  /2  or  C2/xys;  and  we  get 


by  means  of  the  known  integral 


•dx 


dx  = 


.(15) 


This   gives,  as  a  function  of  (•,  the  intensity  due  to   a   self-luminous  line 
whose  geometrical  image  coincides  with  g  —  0. 

Under  the  second  head  of  a  fixed  phase-relation  we  need  only  consider 
the  case  where  the  radiations  from  the  various  parts  of  the  line  start  in 
the  same  phase.  We  get,  almost  as  before, 


for  the  expression  of  the  resultant  amplitude  corresponding  to  £. 

In  order  to  make  use  of  these  results  we  require  a  table  of  the  values 
of  smu/u,  and  of  sirfu/u?.     The  following  will  suffice  for  our  purposes:  — 

TABLE  I. 


4u 

IT 

sin  u 
u 

sin2  u 
u2 

4u 

IT 

sinu 
u 

sin2w 
u2 

4u 

IT 

sin  u 
u 

sin2u 
-tf~ 

0 

+1-0000 

1-0000 

6 

-•2122 

•0450 

12 

•oooo 

•oooo 

1 

•9003 

•8105 

7 

-  -1286 

•0165 

13 

-•0692 

•0048 

2 

•6366 

•4053 

8 

•oooo 

•oooo 

14 

-  -0909 

•0083 

3 

•3001 

•0901 

9 

+  •1000 

•0100 

15 

-•0600 

•0036 

4 

•0000 

•oooo 

10 

•1273 

•0162 

16 

•oooo 

•oooo 

5 

-  -1801 

•0324 

11 

•0818 

•0067 

1896] 


WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE. 


245 


When  we  have  to  deal  with  a  single  point  or  a  single  line  only,  this 
table  gives  directly  the  distribution  of  light  in  the  image,  u  being  equated 
to  Trga/Xf.  The  illumination  first  vanishes  when  u  =  ir,  or  ^/f=\/a. 

On  a  former  occasion*  it  has  been  shown  that  a  self-luminous  point 
or  line  at  u  =  —  IT  is  barely  separated  from  one  at  u  =  0.  It  will  be  of 
interest  to  consider  this  case  under  three  different  conditions  as  to  phase- 
relationship  :  (i)  when  the  phases  are  the  same,  as  will  happen  when  the 
illumination  is  by  plane  waves  incident  perpendicularly;  (ii)  when  the 
phases  are  opposite ;  and  (iii)  when  the  phase -difference  is  a  quarter  period, 
which  gives  the  same  result  for  the  intensity  as  if  the  apertures  were  self- 
luminous.  The  annexed  table  gives  the  numerical  values  required.  In 

TABLE  II. 


4w 

sinu      sin(M  +  7r) 

sin  u      sin  (u  +  IT) 

/  Isin2«      sin2(«  +  ir)j 

V  n?~+  («+-)2  I 

7T 

M                 M  +  7T 

u             11  +  ir 

-4... 

+  1-0000 

-1-0000 

+  1-000 

-3... 

+  1-2004 

-    -6002 

+   -949 

-2... 

+  1-2732 

•oooo 

+   -900 

-1... 

+  1-2004 

+   '6002 

+   -949 

0... 

+  1-0000 

+  1-0000 

+  1-000 

1... 

+    -7202 

+  1-0804 

+   '918 

2... 

+    -4244 

+   -8488 

+   -671 

3... 

+   -1715 

+   -4287 

+   -326 

4... 

•0000 

•oooo 

•000 

5... 

-    -0800 

-    -2801 

-    -206 

6... 

-   -0849 

-    -3395 

-    -247 

7... 

-    -0468 

-    -2105 

-    -152 

8... 

•0000 

•oooo 

•000 

9... 

+   -0308 

+   -1693 

+   -122 

10... 

+   -0364 

+   -2183 

+   -156 

11... 

+   -0218 

+   -1419 

+   -101 

12... 

•0000 

•oooo 

•000 

cases  (i)  and  (iii)  the  resultant  amplitude  is  symmetrical  with  respect  to 
the  point  U  =  —  ^TT  midway  between  the  two  geometrical  images;  in  case  (ii) 
the  sign  is  reversed,  but  this  of  course  has  no  effect  upon  the  intensity. 
Graphs  of  the  three  functions  are  given  in  Fig.  4,  the  geometrical  images 
being  at  the  points  marked  —  TT  and  0.  It  will  be  seen  that  while  in  case  (iii), 
relating  to  self-luminous  points  or  lines,  there  is  an  approach  to  separation, 

*  Phil.  Mag.  Vol.  vin.  p.  266,  1879.     [Vol.  i.  p.  420.] 


246 


ON  THE  THEORY  OF  OPTICAL  IMAGES, 


[222 


nothing  but  an  accurate  comparison  with  the  curve  due  to  a  single  source 
would  reveal  the  duplicity  in  case  (i).  On  the  other  hand,  in  case  (ii), 
where  there  is  a  phase-difference  of  half  a  period  between  the  radiations,  the 
separation  may  be  regarded  as  complete. 

Fig.  4. 


Wii/y 


In  a  certain  sense  the  last  conclusion  remains  undisturbed  even  when 
the  double  point  is  still  closer,  and  also  when  the  aperture  is  of  any  other 
symmetrical  form,  e.g.  circular.  For  at  the  point  of  symmetry  in  the  image, 
midway  between  the  two  geometrical  images  of  the  radiant  points,  the 
component  amplitudes  are  necessarily  equal  in  numerical  value  and  opposite 
in  sign,  so  that  the  resultant  amplitude  or  illumination  vanishes.  For 
example,  suppose  that  the  aperture  is  rectangular  and  that  the  points  or 
lines  are  twice  as  close  as  before,  the  geometrical  images  being  situated  at 
T,  u  =  0.  The  resultant  amplitude  is  represented  by  f(u),  where 


..  ,  __ 


.(17) 


The  values  of  f(u)  are  given  in  Table  III.  They  show  that  the  resultant 
vanishes  at  the  place  of  symmetry  u  =  —  \  IT,  and  rises  to  a  maximum  at 
a  point  near  u  =  %ir,  considerably  beyond  the  geometrical  image  at  u  =  0. 
Moreover,  the  value  of  the  maximum  itself  is  much  less  than  before,  a 
feature  which  would  become  more  and  more  pronounced  as  the  points  were 


1896] 


WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE. 


247 


taken  closer.  At  this  stage  the  image  becomes  only  a  very  incomplete 
representation  of  the  object ;  but  if  the  formation  of  a  black  line  in  the 
centre  of  the  pattern  be  supposed  to  constitute  resolution,  then  resolution 
occurs  at  all  degrees  of  closeness*.  We  shall  see  later,  from  calculations 
conducted  by  the  same  method,  that  a  grating  of  an  equal  degree  of  closeness 
would  show  no  structure  at  all  but  would  present  a  uniformly  illuminated 
field. 

TABLE  III. 


4u 

4« 

4W 

4u 

TT 

/(«) 

TT 

/(«) 

TT 

/(«) 

7T 

/(») 

-1  

+  •00 

2  

+  •64 

5  

-•05 

8  

-•13 

0  

+  •36 

3  

+  •48 

6  

-•21 

9  

+  •02 

1  

+  •60 

4  

+  •21 

7  

-•23 

But  before  proceeding  to  such  calculations  we  may  deduce  by  Lagrange's 
theorem  the  interval  e  in  the  original  object  corresponding  to  that  between 
u  =  0  and  u  =  TT  in  the  image,  and  thence  effect  a  comparison  with  a  grating 
by  means  of  Abbe's  theory.  The  linear  dimension  (£)  of  the  image  cor- 
responding to  u  =  TT  is  given  by  £  =  \fla\  and  from  Lagrange's  theorem 

e/ £  =  sin  /3 /  sin  a,  (If  a) 

in  which  a  is  the  "  semi-angular  aperture,"  and  /3  =  a/2/.   Thus,  corresponding 
to  U  =  TT, 


The  case  of  a  double  point  or  line  represented  in  Fig.  4  lies  therefore 
at  the  extreme  limit  of  resolution  for  a  grating  in  which  the  period  is  the 

*  These  results  are  easily  illustrated  experimentally.  I  have  used  two  parallel  slits,  formed  in 
films  of  tin-foil  or  of  chemically  deposited  silver,  of  which  one  is  conveniently  made  longer  than 
the  other.  These  slits  are  held  vertically  and  are  viewed  through  a  small  telescope,  provided  with 
a  high-power  eye-piece,  whose  horizontal  aperture  is  restricted  to  a  small  width.  The  distance 
may  first  be  so  chosen  that  when  backed  by  a  neighbouring  flame  the  double  part  of  the  slit  just 
manifests  its  character  by  a  faint  shadow  along  the  centre.  If  the  flame  is  replaced  by  sunlight 
shining  through  a  distant  vertical  slit,  the  effect  depends  upon  the  precise  adjustment.  When 
everything  is  in  line  the  image  is  at  its  brightest,  but  there  is  now  no  sign  of  resolution  of 
the  double  part  of  the  slit.  A  very  slight  sideways  displacement,  in  my  case  effected  most 
conveniently  by  moving  the  telescope,  brings  in  the  half-period  retardation,  showing  itself  by 
a  black  bar  down  the  centre.  An  increased  displacement,  leading  to  a  relative  retardation  of 
three  halves  of  a  period,  gives  much  the  same  result,  complicated,  however,  by  chromatic  effects. 

In  conformity  with  theory  the  black  bar  down  the  image  of  the  double  slit  may  still  be 
observed  when  the  distance  is  increased  much  beyond  that  at  which  duplicity  disappears  under 
flame  illumination. 

For  these  experiments  I  chose  the  telescope,  not  only  on  account  -of  the  greater  facility  of 
manipulation  which  it  allows,  but  also  in  order  to  make  it  clear  that  the  theory  is  general, 
and  that  such  effects  are  not  limited,  as  is  sometimes  supposed,  to  the  case  of  the  microscope. 


248  ON  THE  THEORY  OF  OPTICAL  IMAGES,  [222 

interval  between  the  double  points.  And  if  the  incidence  of  the  light  upon 
the  grating  were  limited  to  be  perpendicular,  the  period  would  have  to  be 
doubled  before  the  grating  could  show  any  structure. 

When  the  aperture  is  circular,  of  radius  R,  the  diffraction  pattern  is 
symmetrical  about  the  geometrical  image  (p  =  0,  q  =  0),  and  it  suffices  to 
consider  points  situated  upon  the  axis  of  £  for  which  77  (and  q)  vanish.  Thus 
from  (12) 

rr  r  +  R 

C=  llcospxdxdy  =  2  I       cos  px  V(^2  -  a?)  doc (18) 

JJ  J  -R 

This  integral  is  the  Bessel  function  of  order  unity,  definable  by 

Ji(z)  =  -  pcos^cos^sin2^ (19) 

7T  Jo 

Thus,  if  x  =  R  cos  $, 

°-»**$P (20) 

or,  if  we  write  u  =  TT|  . 


This  notation  agrees  with  that  employed  for  the  rectangular  aperture  if  we 
consider  that  2R  corresponds  with  a. 

The  illumination  at  various  parts  of  the  image  of  a  double  point  may  be 
investigated  as  before,  especially  if  we  limit  ourselves  to  points  which  lie 
upon  the  line  joining  the  two  geometrical  images.  The  only  difference  in 
the  calculations  is  that  represented  by  the  substitution  of  2/j  for  sine.  We 
shall  not,  however,  occupy  space  by  tables  and  drawings  such  as  have  been 
given  for  a  rectangular  aperture.  It  may  suffice  to  consider  the  three  prin- 
cipal points  in  the  image  due  to.  a  double  source  whose  geometrical  images 
are  situated  at  u  =  0  and  u  =  —  TT,  these  being  the  points  just  mentioned, 
and  that  midway  between  them  at  u  =  —  ^TT.  The  values  of  the  functions 
required  are 

2/x  (0)/0        =  I'OOOO  =  V{  I'OOOO). 

2J1(7r)/7r       =    -1812  =  V{'03283}. 

2«/i7r=    -7217  = 


In  the  case  (corresponding  to  i.  Fig.  4)  where  there  is  similarity  of  phase, 
we  have  at  the  geometrical  images  amplitudes  1-1812  as  against  1*4434  at 
the  point  midway  between.  When  there  is  opposition  of  phase,  the  first 
becomes  +  '8188,  and  the  last  zerof.  When  the  phases  differ  by  a  quarter 

*  Enc.  Brit.  "  Wave  Theory,"  p.  432.     [Vol.  in.  p.  87.] 

t  The  zero  illumination  extends  to  all  points  upon  the  line  of  symmetry. 


1896]  WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE.  249 

period,  or  when  the  sources  are  self-luminous  (iii.  Fig.  4),  the  amplitudes  at 
the  geometrical  images  are  V{1'0328}  or  T0163,  and  at  the  middle  point 
V{1'0418}  or  1'0207.  The  partial  separation,  indicated  by  the  central  de- 
pression in  curve  iii.  Fig.  4,  is  thus  lost  when  the  rectangular  aperture  is 
exchanged  for  a  circular  one  of  equal  width.  It  should  be  borne  in  mind 
that  these  results  do  not  apply  to  a  double  line,  which  in  the  case  of  a 
circular  aperture  behaves  differently  from  a  double  point. 


There  is  one  respect  in  which  the  theory  is  deficient,  and  the  deficiency 
is  the  more  important  the  larger  the  angular  aperture.  The  formula  (7) 
from  which  we  start  assumes  that  a  radiant  point  radiates  equally  in  all 
directions,  or  at  least  that  the  radiation  from  it  after  leaving  the  object- 
glass  is  equally  dense  over  the  whole  area  of  the  section.  In  the  case  of 
telescopes,  and  microscopes  of  moderate  angular  aperture,  this  assumption 
can  lead  to  no  appreciable  error ;  but  it  may  be  otherwise  when  the  angular 
aperture  is  very  large.  The  radiation  from  an  ideal  centre  of  transverse 
vibrations  is  certainly  not  uniform  in  various  directions,  and  indeed  vanishes 
in  that  of  primary  vibration.  If  we  suppose  such  an  ideal  source  to  be 
situated  upon  the  axis  of  a  wide-angled  object-glass,  we  might  expect  the 
diffraction  pattern  to  be  less  closely  limited  in  that  axial  plane  which  includes 
the  direction  of  primary  vibration  than  in  that  which  is  perpendicular  to  it. 
The  result  for  a  double  point  illuminated  by  borrowed  light  would  be  a 
better  degree  of  separation  when  the  primary  vibrations  are  perpendicular 
to  the  line  of  junction  than  when  they  are  parallel  to  it. 

Although  it  is  true  that  complications  and  uncertainties  under  this  head 
are  not  without  influence  upon  the  theory  of  the  microscopic  limit,  it  is  not 
to  be  supposed  that  any  considerable  variation  from  that  laid  down  by  Abbe 
and  Helmholtz  is  admissible.  Indeed,  in  the  case  of  a  grating  the  theory  of 
Abbe  is  still  adequate,  so  far  as  the  limit  of  resolution  is  concerned ;  for,  as 
Dr  Stoney  has  remarked,  the  irregularity  of  radiation  in  different  directions 
tells  only  upon  the  relative  brightness  and  not  upon  the  angular  position  of 
the  spectra.  And  it  will  remain  true  that  there  can  be  no  resolution  without 
the  cooperation  of  two  spectra  at  least. 

In  Table  II.  and  Fig.  4  we  have  considered  the  image  of  a  double  point 
or  line  as  formed  by  a  lens  of  rectangular  aperture.  It  is  now  proposed  to 
extend  the  calculation  to  the  case  where  the  series  of  points  or  lines  is 
infinite,  constituting  a  row  of  points  or  a  grating.  The  intervals  are  sup- 
posed to  be  strictly  equal,  and  also  the  luminous  intensities.  When  the 
aperture  is  rectangular,  the  calculation  is  the  same  whether  we  are  dealing" 
with  a  row  of  points  or  with  a  grating,  but  we  have  to  distinguish  according 
as  the  various  centres  radiate  independently,  viz.,  as  if  they  were  self-luminous, 


250  ON  THE  THEORY  OF  OPTICAL  IMAGES,  [222 

or  are  connected  by  phase-relations.      We  will  commence  with  the  former 
case. 

If  the  geometrical  images  of  the  various  luminous  points  are  situated 
at  u  =  0,  u  =  ±  v,  u  =  ±  2v,  «Scc.,  the  expressions  for  the  intensity  at  any  point 
u  of  the  field  may  be  written  as  an  infinite  series, 

sin2M     sin2  (u  +  t>)     sin2  (u  —  v) 

~vT"    (u  +  v)2    "    (u-v? 


Being  an  even  function  of  u   and  periodic  in  period   v,  (22)   may  be 
expanded  by  Fourier's  theorem  in  a  series  of  cosines.     Thus 


-r  f  \       T       T  T  ,nn. 

J'(u)  •/,  +  /!  008  --  +  ...  +  /rcos  -  •+  ......  ;     .........  (23) 

and  the  character  of  the  field  of  light  will  be  determined  when  the  values  of 
the  constants  /„,  /,,  &c.,  are  known.  For  these  we  have  as  usual 

7-      1  [v  T  /  N  T  T-       2  [v  T  ,  .         lirru  7 

/„  =  -     I(u)du,        Ir  =  -\   /(w)cos  --  dw;     .........  (24) 

v  J0  v  Jo  ^ 

and  it  only  remains  to  effect  the  integrations.  To  this  end  we  may  observe 
that  each  term  in  the  series  (22)  must  in  reality  make  an  equal  contribution 
to  Ir.  It  will  come  to  the  same  thing  whether,  as  indicated  in  (24),  we 
integrate  the  sum  of  the  series  from  0  to  w,  or  integrate  a  single  term  of  it, 
e.g.  the  first,  from  —  oo  to  +  oo  .  We  may  therefore  take 


,                                  TT  T                 snM                   ,           /OK  0/?N 

/»  =  -              —  du=  —  :  Ir=-              —cos  -  du.    ...(25,26) 

Vj_oo        U2                    V'  Wj-oo        W2                       W 

To  evaluate  (26)  we  have 

+oc  sin2wcossw  7  r+°°  1  cZ   ,  .  „              .  , 

-  -  -  du  =  I       -  j-  (sm2  u  cos  SM)  rfw, 

-oo          w2  )  _<»  udu^ 


and 


.7  „  2-4-s  2  _  s 

-T-  (sin2w  cos  su)  =  —  -=  sin  su  -\  --  -  —  sin  (2  +  s)  u  -\  --  -  —  sin  (2  —  s)  u  ; 


so  that  by  (15)  (s  being  positive) 


the  minus  sign  being  taken  when  2  —  s  is  negative. 
Hence 


according  as  w  exceeds  or  falls  short  of  rir. 


1896]  WITH   SPECIAL  EEFERENCE   TO   THE   MICROSCOPE.  251 

We  may  now  trace  the  effect  of  altering  the  value  of  v.  When  v  is  large, 
a  considerable  number  of  terms  in  the  Fourier  expansion  (23)  are  of  import- 
ance, and  the  discontinuous  character  of  the  luminous  grating  or  row  of  points 
is  fairly  well  represented  in  the  image.  As  v  diminishes,  the  higher  terms 
drop  out  in  succession,  until  when  v  falls  below  2ir  only  /0  and  7t  remain. 
From  this  point  onwards  7j  continues  to  diminish  until  it  also  finally  dis- 
appears when  v  drops  below  TT.  The  field  is  then  uniformly  illuminated, 
showing  no  trace  of  the  original  structure.  The  case  v  =  TT  is  that  of  Fig.  4, 
and  curve  iii.  shows  that  at  a  stage  when  an  infinite  series  shows  no  struc- 
ture, a  pair  of  luminous  points  or  lines  of  the  same  closeness  are  still  in 
some  degree  separated.  It  will  be  remembered  that  v  =  TT  corresponds  to 
e  =  l\/sin  a,  e  being  the  linear  period  of  the  original  object  and  a  the  semi- 
angular  aperture. 

We  will  now  pass  on  to  consider  the  case  of  a  grating  or  row  of  points 
perforated  in  an  opaque  screen  and  illuminated  by  plane  waves  of  light.  If 
the  incidence  be  oblique,  the  phase  of  the  radiation  emitted  varies  by  equal 
steps  as  we  pass  from  one  element  to  the  next.  But  for  the  sake  of 
simplicity  we  will  commence  with  the  case  of  perpendicular  incidence,  where 
the  radiations  from  the  various  elements  all  start  in  the  same  phase.  We 
have  now  to  superpose  amplitudes,  and  not  as  before  intensities.  If  A  be 
the  resultant  amplitude,  we  may  write 

,  _  sin  u     sin  (u  +  v)     sin  (u  —  v) 
u  u  +  v  u  —  v 


2?rw 
=  A0+A1coa  -  +  .  .  .  +  Ar  cos  -  +  ................  (28) 

When  v  is  very  small,  the  infinite  series  identifies  itself  more  and  more 
nearly  with  the  integral 

1  r+°°  sin  u  -.        .     TT 
—  du,  viz.  —  . 

V  J  _oo        U  V 

In  general  we  have,  as  in  the  last  problem, 

1  f+cc  sin  u  -.  2  f+co  sin  u        2-rrru  . 

A9  —  -l       -  du;      Ar=-  —cos  --  du:   ......  (29) 

Wj-oo       U  V  J  -x,       U  V 

so  that  A0  =  TT/V.     As  regards  Ar,  writing  s  for  Z-rrr/v,  we  have 
lprin(H.,)«  +  rin(l-.).  ; 

V  J  _«,  U  V  ^      ' 

the  lower  sign  applying  when  (1  —  s)  is  negative.     Accordingly, 

4(«)-£{i+2oM  —  +2ooB  —  *...}.,   ............  (30) 

V    [  V  V  ) 

the  series  being  continued  so  long  as  2?rr  <  v. 


252  ON  THE  THEORY  OF  OPTICAL  IMAGES,  [222 

If  the  series  (30)  were  continued  ad  infinitum,  it  would  represent  a 
discontinuous  distribution,  limited  to  the  points  (or  lines)  u  =  0,  u  =  +  v, 
u  =  ±  2v,  &c.,  so  that  the  image  formed  would  accurately  correspond  to  the 
original  object.  This  condition  of  things  is  most  nearly  realised  when  v  is 
very  great,  for  then  (30)  includes  a  large  number  of  terms.  As  v  diminishes 
the  higher  terms  drop  out  in  succession,  retaining  however  (in  contrast  with 
(27))  their  full  value  up  to  the  moment  of  disappearance.  When  v  is  less 
than  2?r,  the  series  is  reduced  to  its  constant  term,  so  that  the  field  becomes 
uniform.  Under  this  kind  of  illumination,  the  resolving-power  is  only  half 
as  great  as  when  the  object  is  self-luminous. 

These  conclusions  are  in  entire  accordance  with  Abbe's  theory.  The  first 
term  of  (30)  represents  the  central  image,  the  second  term  the  two  spectra 
of  the  first  order,  the  third  term  the  two  spectra  of  the  second  order,  and 
so  on.  Resolution  fails  at  the  moment  when  the  spectra  of  the  first  order 
cease  to  cooperate,  and  we  have  already  seen  that  this  happens  for  the  case 
of  perpendicular  incidence  when  v  =  2?r.  The  two  spectra  of  any  given  order 
fail  at  the  same  moment. 

If  the  series  stops  after  the  lateral  spectra  of  the  first  order, 

,    .....................  (31) 


showing  a  maximum  intensity  when  u  =  0,  or  \v,  and  zero  intensity  when 
u  =  %v,  or  ft;.  These  bands  are  not  the  simplest  kind  of  interference  bands. 
The  latter  require  the  operation  of  two  spectra  only  ;  whereas  in  the  present 
case  there  are  three  —  the  central  image  and  the  two  spectra  of  the  first 
order. 


We  may  now  proceed  to  consider  the  case  when  the  incident  plane  waves 
are  inclined  to  the  grating.  The  only  difference  is  that  we  require  now  to 
introduce  a  change  of  phase  between  the  image  due  to  each  element  and  its 
neighbour.  The  series  representing  the  resultant  amplitude  at  any  point  u 
may  still  be  written 


u  +  v 


For  perpendicular  incidence  m  =  0.     If  7  be  the  obliquity,  e  the  grating- 
interval,  \  the  wave-length, 

(33) 


The  series  (32),  as  it  stands,  is  not  periodic  with  respect  to  u  in  period  v, 
but  evidently  it  can  differ  from  such  a  periodic  series  only  by  the  factor  eimu. 


1896]  WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE.  253 

The  series 

e-imu  sjn  u       e-im  (u+v)  8jn  (M  +  3,) 


u  u  +  v 

e~im  <»-*>  sin  (u  -  v)     e~im(u+2v)  sin  (u  +  2v) 


—  V 

is  truly  periodic,  and  may  therefore  be  expanded  by  Fourier's  theorem  in 
periodic  terms : 

(34)  =  A0  +  iBQ  +  (A,  +  iB,)  cos  (2-n-u/v)  +  (C,  +  iD,)  sin  (2-rru/v)  +  . .. 
+  (Ar  +  iBr}  cos  (2r7ru/v)  +  (Cr  +  iDr)  sin  (2nru/v)  + (35) 

As  before,  if  s  =  2r7r/v, 

f +°°  e~imu  sin  u  cos  su  , 
$v(Ar  +  iBr)=\  —  du', 

so  that  Br  =  Q,  while 

f  +  °°  cos  mu  sin  u  cos  su  7  /0 «, 

%v.Ar=l  —  du (36) 

In  like  manner  Cr  =  0,  while 

sinm^sin^sin^ (37) 

In  the  case  of  the  zero  suffix 

•°°  cos  mu  sin  u 


n  /       cos  mi*  sn  u,  /oox 

0  =  0,        vA0=  -  —  du  ................  (38) 


When  the  products  of  sines  and  cosines  which  occur  in  (36)  &c.  are 
transformed  in  a  well-known  manner,  the  integration  may  be  effected  by 
(15).  Thus 

cos  mu  sin  u  cos  su  =  |  {sin  (1  +  m  +  s)  u  +  sin  (1  —  m  —  s)  u 

+  sin  (1  +  m  —  s)  u  +  sin  (1  —  m  +  s)  u}  ; 
so  that 

...(39) 


where  each  symbol  such  as  [1  +  m  +  s]  is  to  be  replaced  by  +  1,  the  sign 
being  that  of  (1  +  m  +  s).     In  like  manner 


[I  +  m  +  s]-[l-m-s]}.   ...(40) 
The  rth  terms  of  (35)  are  accordingly 

|L|ei«u([i  +  m  +  s]  +  [i  _m_s])+  er*»([l  +m-s]  +  [1  -m  +  s])}; 
or  for  the  original  Series  (32), 


.     ...(41) 


254  ON  THE  THEOEY  OF  OPTICAL  IMAGES,  [222 

For  the  term  of  zero  order, 

A0e™  =      ei™([I+m]  +  [I-m])  ................  (42) 


From  (41)  we  see  that  the  term  in  ei(m+g)u  vanishes  unless  (ra  +  s)  lies 
between  +  1,  and  that  then  it  is  equal  to  TT/V  ,ei(m+g}u;  also  that  the  term  in 
ei(m-s)u  vanishes  unless  (m-s)  lies  between  ±  1,  and  that  it  is  then  equal  to 
TT/V  .  el(m~s]u.  In  like  manner  the  term  in  eimu  vanishes  unless  m  lies  between 
±1,  and  when  it  does  not  vanish  it  is  equal  to  Tr/v.eimu.  This  particular 
case  is  included  in  the  general  statement  by  putting  s  =  0. 

The  image  of  the  grating,  or  row  of  points,  expressed  by  (32),  is  thus 
capable  of  representation  by  the  sum  of  terms 

TT/V  .  [eimu  +  ei(m+s^u  +  ei(m-gi)u  +  e^m+s^}u  +  ...}  ............  (43) 

where  sl  =  2'rr/v,  s2  =  4nr/v,  &c.,  every  term  being  included  for  which  the 
coefficient  of  u  lies  between  ±  1.  Each  of  these  terms  corresponds  to  a 
spectrum  of  Abbe's  theory,  and  represents  plane  progressive  waves  inclined 
at  a  certain  angle  to  the  plane  of  the  image.  Each  spectrum  when  it  occurs 
at  all  contributes  equally,  and  it  goes  out  of  operation  suddenly.  If  but  one 
spectrum  operates,  the  field  is  of  uniform  brightness.  If  two  spectra  operate, 
we  have  the  ordinary  interference  bands  due  to  two  sets  of  plane  waves 
crossing  one  another  at  a  small  angle  of  obliquity*. 

Any  consecutive  pair  of  spectra  give  the  same  interference  bands,  so  far 
as  illumination  is  concerned.  For 

ZT  }gtM[m+2r7r/»]  -J-  ei«[m+2(r+l)jr/o]J  _  ^7r  cog  ^f  gi»[i»+2  (r+i)  JT/W] 
V   l  V  V 

of  which  the  exponential  factor  influences  only  the  phase. 

In  (43)  the  critical  value  of  v  for  which  the  rth  spectrum  disappears  is 
given  by,  when  we  introduce  the  value  of  m  from  (33), 


or,  since  (as  we  have  seen) 


e  (sin  7  +  sin  a)  =  +  r\  .........................  (45) 

This  is  the  condition,  according  to  elementary  theory,  in  order  that  the 
rays  forming  the  spectrum  of  the  rth  order  should  be  inclined  at  the  angle 
a,  and  so  (Fig.  2)  be  adjusted  to  travel  from  A  to  B,  through  the  edge  of 
the  lens  L. 

*  Enc.  Brit.  "Wave  Theory,"  p.  425.     [Vol.  m.  p.  59.] 


1896]  WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE.  255 

The  discussion  of  the  theory  of  a  rectangular  aperture  may  here  close. 
This  case  has  the  advantage  that  the  calculation  is  the  same  whether  the 
object  be  a  row  of  points  or  a  grating.  A  parallel  treatment  of  other  forms 
of  aperture,  e.g.  the  circular  form,  is  not  only  limited  to  the  first  alternative, 
but  applies  there  only  to  those  points  of  the  field  which  lie  upon  the 
line  joining  the  geometrical  images  of  the  luminous  points.  Although  the 
advantage  lies  with  a  more  general  method  of  investigation  to  be  given 
presently,  it  may  be  well  to  consider  the  theory  of  a  circular  aperture  as 
specially  deduced  from  the  formula  (21)  which  gives  the  image  of  a  single 
luminous  centre. 

If  we  limit  ourselves  to  the  case  of  parallel  waves  and  perpendicular 
incidence,  the  infinite  series  to  be  discussed  is 


+  ......  (46) 

u  u  +  v  u  —  v  u  +  2v 

where  w  =  7rf.2R/\/.  ..............................  (47) 

Since  -A  is  necessarily  periodic  in  period  v,  we  may  assume 

A  (u)  =  A0  +  A  j  cos  (Ztru/v)  +  .  .  .  +  Ar  cos  (Zrirujv)  +  ...;  ......  (48) 

and,  as  in  the  case  of  the  rectangular  aperture, 

1    r  +  xJ1(M)    ,  2    [  +  XJ1(U)  2T7TU    , 

A0  =  -l       -=±**du,       Ar  =  -          -LA-^  cos  --  du  .......  (49) 

V  J  -«>        U  V  J  -x        U  V 

These  integrals  may  be  evaluated.    If  a  and  b  be  real,  and  a  be  positive  *, 


(50) 


Multiplying  by  bdb  and  integrating  from  0  to  b,  we  find 

V(*  +  »)-. 


o 
In  this  we  write  6  =  1,  a  =  is,  where  s  is  real.     Thus 


If  s2  >  1,  we  must  write  i'V(s2  —  1)  for  V(l  —  s2).     Hence,  if  s  <  1, 


=  v(1_  rJ-Wsn-j,  ...  (62>68) 

J0  x 

while,  if  s  >  1, 

rjj  (x)  cos  sx  1        _  f00  J1(x)sinsx  . 

-^-^  -  dx  =  0,  -£-*  -  dx=-M-I)  +  s.   ...(54,55) 

&  J  Q  X 

*  Gray  and  Mathews,  BesseVs  Functions,  1895,  p.  72. 


256  ON  THE  THEORY  OF  OPTICAL  IMAGES,  [222 

We  are  here  concerned  only  with  (52),  (54),  and  we  conclude  that  A0  =  2/v, 
and  that 

**,  or  0,    (56) 


according  as  s  is  less  or.  greater  than  1,  viz.  according  as  Ir-rr  is  less  or  greater 
than  v. 

If  we  compare  this  result  with  the  corresponding  one  (30)  for  a  rect- 
angular aperture  of  equal  width  (2R  =  a),  we  see  that  the  various  terms 
representing  the  several  spectra  enter  or  disappear  at  the  same  time; 
but  there  is  one  important  difference  to  be  noted.  In  the  case  of  the 
rectangular  aperture  the  spectra  enter  suddenly  and  with  their  full  effect, 
whereas  in  the  present  case  there  is  no  such  discontinuity,  the  effect  of  a 
spectrum  which  has  just  entered  being  infinitely  small.  As  will  appear 
more  clearly  by  another  method  of  investigation,  the  discontinuity  has  its 
origin  in  the  sudden  rise  of  the  ordinate  of  the  rectangular  aperture  from 
zero  to  its  full  value. 

In  the  method  referred  to  the  form  of  the  aperture  is  supposed  to  remain 
symmetrical  with  respect  to  both  axes,  but  otherwise  is  kept  open,  the 
integration  with  respect  to  x  being  postponed.  Starting  from  (12)  and 
considering  only  those  points  of  the  image  for  which  77  and  q  in  equation 
(8)  vanish,  we  have  as  applicable  to  the  image  of  a  single  luminous  source 

C  =  ffcospxdxdy  =  2fy  cospxdx  (57) 

in  which  *2y  denotes  the  whole  height  of  the  aperture  at  the  point  x.  This 
gives  the  amplitude  as  a  function  of  p.  If  there  be  a  row  of  luminous  points, 
from  which  start  radiations  in  the  same  phase,  we  have  an  infinite  series  of 
terms,  similar  to  (57)  and  derived  from  it  by  the  addition  to  p  of  positive 
and  negative  integral  multiples  of  a  constant  (p^)  representing  the  period. 
The  sum  of  the  series  A  (p)  is  necessarily  periodic,  so  that  we  may  write 

and,  as  in  previous  investigations,  we  may  take 

Ar=  I   ^Ccosspdp, (59) 

s  (not  quite  the  same  as  before)  standing  for  2irjr/pli  and  a  constant  factor 
being  omitted.  To  ensure  convergency  we  will  treat  this  as  the  limit  of 


(60) 


the  sign  of  the  exponent  being  taken  negative,  and  h  being  ultimately  made 
to  vanish.     Taking  first  the  integration  with  respect  to  p,  we  have 


1896]  WITH   SPECIAL   REFERENCE   TO   THE   MICROSCOPE.  257 


=  hydx 

' 


and  thus 


in  which  h  is  to  be  made  to  vanish.     In  the   limit   the   integrals   receive 
sensible  contributions  only  from  the  neighbourhoods  of  x  =  +  s  ;  and  since 

~ 


we  get  Ar  =  7r(yx=^  +  yx=+g)  =  27ryx=s  ...................  (62) 

From  (62)  we  see  that  the  occurrence  of  the  term  in  Ar,  i.e.  the  appear- 
ance of  the  spectrum  of  the  rth  order,  is  associated  with  the  value  of  a 
particular  ordinate  of  the  object-glass.  If  the  ordinate  be  zero,  i.e.  if  the 
abscissa  exceed  numerically  the  half-  width  of  the  object-glass,  the  term  in 
question  vanishes.  The  first  appearance  of  it  corresponds  to 


in  which  a  is  the  entire  width  of  the  object-glass  and  £  the  linear  period  in 
the  image.     By  (17  a), 


!        e  sn  a       e  sn  a 

so  that  the  condition  is,  as  before, 

e  sin  a  =  r\. 

When  Ar  has  appeared,  its  value  is  proportional  to  the  ordinate  at  x  =  s. 
Thus  in  the  case  of  a  circular  aperture  (a  =  2R)  we  have 

}  ......................  (63) 


The  above  investigation  relates  to  a  row  of  luminous  points  emitting 
light  of  the  same  intensity  and  phase,  and  it  is  limited  to  those  points  of 
the  image  for  which  17  (and  q)  vanish.  If  the  object  be  a  grating  radiating 
under  similar  conditions,  we  have  to  retain  cosqy  in  (12)  and  to  make  an 
integration  with  respect  to  q.  Taking  this  first,  and  introducing  a  factor 
e±kq,  we  have 


(64) 


This  is  now  to  be  integrated  with  respect  to  y  between  the  limits  —  y 
and  +  y.     If  this  range  be  finite,  we  have 


.(65) 
17 


258  ON  THE  THEORY  OF  OPTICAL  IMAGES,  [222 

independent  of  the  length  of  the  particular  ordinate.     Thus 

(66) 


=  l 

J 


the  integration  with  respect  to  x  extending  over  the  range  for  which  y  is 
finite,  that  is,  over  the  width  of  the  object-glass.     If  this  be  2R,  we  have 


(67) 


From  (67)  we  see  that  the  image  of  a  luminous  line,  all  parts  of  which 
radiate  in  the  same  phase,  is  independent  of  the  form  of  the  aperture  of  the 
object-glass,  being,  for  example,  the  same  for  a  circular  aperture  as  for  a 
rectangular  aperture  of  equal  width.  This  case  differs  from  that  of  a  self- 
luminous  line,  the  images  of  which  thrown  by  circular  and  rectangular 
apertures  are  of  different  types*. 

The  comparison  of  (67)  with  (20),  applicable  to  a  circular  aperture,  leads 
to  a  theorem  in  Bessel's  functions.  For,  when  q  is  finite, 


so  that,  setting  22  =  1,  we  get 


The  application  to  a  grating,  of  which  all  parts  radiate  in  the  same  phase, 
proceeds  as  before.     If,  as  in  (58),  we  suppose 


.,  (70) 

we  have  Ar=  I      CiCosspdp', (71) 

from  which  we  find  that  Ar  is  4?r2  or  0,  according  as  the  ordinate  is  finite  or 
not  finite  at  x  =  s.  The  various  spectra  enter  and  disappear  under  the  same 
conditions  as  prevailed  when  the  object  was  a  row  of  points ;  but  now  they 
enter  discontinuously  and  retain  constant  values,  instead  of  varying  with  the 
particular  ordinate  of  the  object-glass  which  corresponds  to  x  =  s. 

We  will  now  consider  the  corresponding  problems  when  the  illumination 
is  such  that  each  point  of  the  row  of  points  or  of  the  grating  radiates  in- 
dependently. The  integration  then  relates  to  the  intensity  of  the  field  as 
due  to  a  single  source. 

*  Enc.  Brit.  "  Wave  Theory,"  p.  434.    [Vol.  in.  p.  92.] 

t  This  may  be  verified  by  means  of  Neumann's  formula  (Gray  and  Mathews,  BesseVs  Functions 
(70),  p.  27). 


1896]  WITH   SPECIAL  REFERENCE   TO   THE   MICROSCOPE.  259 

By  (9),  (10),  (11),  the  intensity  72  at  the  point  (p,  q)  of  the  field,  due  to 
a  single  source  whose  geometrical  image  is  situated  at  (0,  0)  is  given  by 

Xy2  72  =  ijrjeos  (px  +  qy)  dx<lyY  +  {//sin  (px  +  qy)  dxdy}* 
=  //cos  (px  +  qy')  dx'dy'  x  //cos  (px  +  qy)  dxdy 
+  //sin  (px'  +  qy')  dxdy'  x  //sin  (px  +  qy)  dxdy 
=  ffflcos{p(x'  -x)  +  q(y'-y)}dxdydx'dy',     ...............  (72) 

the  integrations  with  respect  to  x',  y',  as  well  as  those  with  respect  to  x,  y 
being  over  the  area  of  the  aperture. 

In  the  present  application  to  sources  which  are  periodically  repeated, 
the  term  in  cos  sp  of  the  Fourier  expansion  representing  the  intensity 
at  various  points  of  the  image  has  a  coefficient  found  by  multiplying  (72) 
by  cos  sp  and  integrating  with  respect  to  p  from  p  =  —  cc  to  p  =  +  oo  .  If 
the  object  be  a  row  of  points,  we  may  take  q  =  0  ;  if  it  be  a  grating,  we 
have  to  integrate  with  respect  also  to  q  from  q  =  —  oo  to  q  =  +  <x>  . 

Considering  the  latter  case,  and  taking  first  the  integrations  with  respect 
to  p,  q,  we  introduce  the  factors  e*hp'fkq,  the  plus  or  minus  being  so  chosen  as 
to  make  the  elements  of  the  integral  vanish  at  infinity.  After  the  operations 
have  been  performed,  h  and  k  are  to  be  supposed  to  vanish*.  The  integra- 
tions are  performed  as  for  (60),  (64),  and  we  get  the  sum  of  the  two  terms 
denoted  by 


We  have  still  to  integrate  with  respect  to  dxdy  dx'dy'.     As  in  (65),  since  the 
range  for  y'  always  includes  y, 


and  we  are  left  with 


[([     2-rrhdxdydx'  7 

JJJ  K+(x'-x±sY 


If  s  were  zero,  the  integration  with  respect  to  x  would  be  precisely 
similar;  but  with  s  finite  it  will  be  only  for  certain  values  of  x  that 
(x  —  x  ±  s)  vanishes  within  the  range  of  integration.  Unless  this  evan- 
escence takes  place,  the  limit  when  h  vanishes  becomes  zero.  The  effect 
of  the  integration  with  respect  to  x'  is  thus  to  limit  the  range  of  the 
subsequent  integration  with  respect  to  x.  The  result  may  be  written 

(75) 


*  The  process  is  that  employed  by  Stokes  in  his  evaluation  of  the  integral  intensity,  Edin. 
Trans,  xx.  p.  317  (1853).     See  also  Enc,  frit,  "  Wave  Theory,"  p.  431.     [Vol.  ni.  p.  86.] 

17—2 


260  ON  THE  THEORY  OF  OPTICAL  IMAGES.  [222 

upon  the  understanding  that,  while  the  integration  for  y  ranges  over  the 
whole  vertical  aperture,  that  for  x  is  limited  to  such  values  of  x  as  bring 
x  +  s  (as  well  as  x  itself)  within  the  range  of  the  horizontal  aperture.  The 
coefficient  of  the  Fourier  component  of  the  intensity  involving  cos  sp,  or 
cos(2r7rp/p1),  is  thus  proportional  to  a  certain  part  of  the  area  of  the  aper- 
ture. Other  parts  of  the  area  are  inefficient,  and  might  be  stopped  off 
without  influencing  the  result. 

The  limit  to  resolution,  corresponding  to  r  =  1,  depends  only  on  the  width 
of  the  aperture,  and  is  therefore  for  all  forms  of  aperture  the  same  as  for 
the  case  of  the  rectangular  aperture  already  fully  investigated. 

If  the  object  be  a  row  of  points  instead  of  a  row  of  lines,  q  =  0,  and  there 
is  no  integration  with  respect  to  it.  The  process  is  nearly  the  same  as  above, 
and  the  result  for  the  coefficient  of  the  rth  term  in  the  Fourier  expansion  is 
proportional  to  fy*dx,  instead  of  fydx,  the  integration  with  respect  to  x  being 
over  the  same  parts  of  the  aperture  as  when  the  object  was  a  grating.  The 
application  to  a  circular  aperture  would  lead  to  an  evaluation  of 

'  J^  (u)  cos  su  , 
^ du. 


223. 


THEORETICAL  CONSIDERATIONS   RESPECTING  THE   SEPARA- 
TION OF  GASES  BY  DIFFUSION  AND  SIMILAR  PROCESSES. 

[Philosophical  Magazine,  XLII.  pp.  493  —  498,  1896.] 

THE  larger  part  of  the  calculations  which  follow  were  made  in  connexion 
with  experiments  upon  the  concentration  of  argon  from  the  atmosphere  by 
the  method  of  atmolysis*.  When  the  supply  of  gas  is  limited,  or  when  it 
is  desired  to  concentrate  the  lighter  ingredient,  the  conditions  of  the  question 
are  materially  altered;  but  it  will  be  convenient  to  take  first  the  problem 
which  then  presented  itself  of  the  simple  diffusion  of  a  gaseous  mixture 
into  a  vacuum,  with  special  regard  to  the  composition  of  the  residue.  The 
diffusion  tends  to  alter  this  composition  in  the  first  instance  only  in  the 
neighbourhood  of  the  porous  walls  ;  but  it  will  be  assumed  that  the  forces 
promoting  mixture  are  powerful  enough  to  allow  of  our  considering  the 
composition  to  be  uniform  throughout  the  whole  volume  of  the  residue, 
and  variable  only  with  time,  on  account  of  the  unequal  escape  of  the 
constituent  gases. 

Let  x,  y  denote  the  quantities  of  the  two  constituents  of  the  residue  at 
any  time,  so  that  —  dx,  —  dy  are  the  quantities  diffused  out  in  time  dt.  The 
values  of  dxjdt,  dy/dt  will  depend  upon  the  character  of  the  porous  partition 
and  upon  the  actual  pressure  ;  but  for  our  present  purpose  it  will  suffice  to 
express  dyjdx,  and  this  clearly  involves  only  the  ratios  of  the  constituents 
and  of  their  diffusion  rates.  Calling  the  diffusion  rates  //,,  v,  we  have 


In  this  equation  x,  y  may  be  measured  on  any  consistent  system  that 
may  be  convenient.  The  simplest  case  would  be  that  in  which  the  residue 
is  maintained  at  a  constant  volume,  when  x,  y  might  be  taken  to  represent 

*  Kayleigh  and  Ramsay,  Phil.  Trans.  CLXXXVI.  p.  206  (1895).    [Vol.  iv.  p.  130.] 


262  ON  THE   SEPARATION   OF   GASES  [223 

the  partial  pressures  of  the  two  gases.  But  the  equation  applies  equally  well 
when  the  volume  changes,  for  example  in  such  a  way  as  to  maintain  the  total 
pressure  constant. 

The  integral  of  (1)  is 

yil»=Co£l»,   (2) 

where  G  is  an  arbitrary  constant,  or 

If  X,  Y  be  simultaneous  values  of  x,  y,  regarded  as  initial, 

#-(*)""*• w 


so  that  x  =  X(fj^J  (5) 


In  like  manner  ^=F(z7FJ  (6) 

If  we  write  ^=  =  r,    (7) 


r  represents  the  enrichment  of  the  residue  as  regards  the  second  constituent, 
and  we  have  from  (5),  (6), 


an  equation  which   exhibits  the  relation  between  the  enrichment  and  the 
ratio  of  the  initial  and  final  total  quantities  of  the  mixture. 

From  (8),  or  more  simply  from  (4),  we  see  that  as  x  diminishes  with  time 
the  enrichment  tends  to  zero  or  infinity,  indicating  that  the  residue  becomes 
purer  without  limit,  and  this  whatever  may  be  the  original  proportions.  Thus 
if  the  first  gas  (x)  be  the  more  diffusive  (//,  >  v),  the  exponent  on  the  right 
of  (4)  is  negative;  and  this  indicates  that  r  becomes  infinite,  or  that  the 
first  gas  is  ultimately  eliminated  from  the  residue.  When  the  degree  of 
enrichment  required  is  specified,  an  easy  calculation  from  (8)  gives  the  degree 
to  which  the  diffusion  must  be  carried. 

In  Graham's  atmolyser  the  gaseous  mixture  is  caused  to  travel  along  a 
tobacco-pipe  on  the  outside  of  which  a  vacuum  is  maintained.  If  the 
passage  be  sufficiently  rapid  to  preclude  sensible  diffusion  along  the  length 
of  the  pipe,  the  circumstances  correspond  to  the  above  calculation  ;  but  the 
agreement  with  Graham's  numbers  is  not  good.  Thus  in  one  case  given  by 
him*  of  the  atmolysis  of  a  mixture  containing  equal  volumes  of  oxygen 
and  hydrogen,  we  have 

Y/X  =  l,        y\x  =  92-78/7-22, 

*  Phil.  Trans.  Vol.  CLIII.  p.  403  (1863). 


1896]  BY   DIFFUSION   AND  SIMILAR  PROCESSES.  263 

so  that  r  =  13  nearly.  Thus,  if  in  accordance  with  the  view  usually  held 
fjb/v  =  4,  we  should  have  from  (8) 

i  x  13-  *  +  *  x  13-*  =  -229  ; 

so  that  a  reduction  of  the  residue  to  '229  of  the  initial  quantity  should  have 
effected  the  observed  enrichment.  The  initial  and  final  volumes  given  by 
Graham  are,  however,  7'5  litres  and  '45  litre,  whose  ratio  is  '06.  The  inferior 
efficiency  of  the  apparatus  may  have  been  due  to  imperfections  in  the 
walls  or  joints  of  the  pipes.  Such  an  explanation  appears  to  be  more 
probable  than  a  failure  of  the  law  of  independent  diffusion  of  the  component 
gases  upon  which  the  theoretical  investigation  is  founded. 

In  the  concentration  of  argon  from  a  mixture  of  argon  and  nitrogen  we 
have  conditions  much  less  favourable.     In  this  case 


If  an  enrichment  of  2  :  1   is  required  and  if  the  original  mixture  is 
derived  from  the  atmosphere  by  removal  of  oxygen,  the  equation  is 

=  -99x2-6'13  +  -01  x  -2-8-13  =  -0142  +  "0029  =  "0171, 


y 

A.  + 

expressing  the  reduction  needed.     The  results  obtained  experimentally  (loc. 
cit.)  were  inferior  in  this  case  also. 


When  the  object  is  the  most  effective  separation  of  the  components  of  a 
mixture,  it  is  best,  as  supposed  in  the  above  theory,  to  maintain  a  vacuum 
on  the  further  side  of  the  porous  wall.  But  we  have  sometimes  to  consider 
cases  where  the  vacuum  is  replaced  by  an  atmosphere  of  fixed  composition, 
as  in  the  well-known  experiment  of  the  diffusion  of  hydrogen  into  air  through 
a  porous  plug.  We  will  suppose  that  there  are  only  two  gases  concerned 
and  that  the  volume  inside  is  given.  The  symbols  x,  y  will  then  denote  the 
partial  pressures  within  the  given  volume,  the  constant  partial  pressures 
outside  being  a,  /3.  Our  equations  may  be  written 


(9) 
or  on  integration 


,        y  =  /3  +  De-vt,   ..................  (10) 

C,  D  being  arbitrary  constants. 

After  a  sufficient  time  x,  y  reduce  themselves  respectively  to  a,  ft,  as  was 
to  be  expected. 

The  constants  /i,  v  are  not  known  beforehand,  depending  as  they  do  upon 


264  ON   THE   SEPARATION    OF   GASES  [223 

the  specialities  of  the  apparatus  as  well  as  upon  the  quality  of  the  gases.     If 
we  eliminate  t,  we  get 

y-ft  =  E(x-*yi*,  ...........................  (11) 

in  which  only  the  ratio  v/p  is  involved. 

As  a  particular  case  suppose  that  initially  the  inside  volume  is  occupied 
by  one  pure  gas  and  the  outside  by  another,  the  initial  pressures  being  unity. 
Then  in  (10) 

«  =  0,        /8  =  1,         (7=1,         D  =  -l; 

we  have  x  =  e-*t,        y=\-e~vt,  ........................  (12) 

and  a;  +  y  =1  +  6-^-6-*     ........................  (13) 

gives  the  total  internal  pressure.     When  this  is  a  maximum  or  minimum, 
e(iL-*)t  —  piv>  anci  the  corresponding  value  is 

a*) 


Thus  in  the  case  of  hydrogen  escaping  into  oxygen,  p/v  =  4,  and 

#  +  2/=l-3x4-J  =  -528, 
the  minimum  being  about  half  the  initial  pressure*. 

Returning  now  to  the  separation  of  gases  by  diffusion  into  a  vacuum, 
let  us  suppose  that  the  difference  between  the  gases  is  small,  so  that 
(v  —  /*)/(*,  =  K,  a  small  quantity,  and  that  at  each  operation  one-half  the  total 
volume  of  the  mixture  is  allowed  to  pass.  In  this  case  (8)  becomes 

X       -         Y       — 

=         T~K  +         T  "  = 


so  that 


This  gives  the  effect  of  the  operation  in  question  upon  the  composition  of 
the  residual  gas.  If  s  denote  the  corresponding  symbol  for  the  transmitted 
gas,  we  have 


_  -_ 

(X-x)/X     \-xlX~\-x\X~  '    1-asjX 

approximately,  since  r  is  nearly  equal  to  unity.  Accordingly 
1         1 


2- 


=  r  nearly, 


so  that  approximately  s  and  r  are  reciprocal  operations.      For  example,  if 

*  The  most  striking  effects  of  this  kind  are  when  nitrous  oxide,  or  dry  ammonia  gas,  diffuse 
into  the  air  through  indiarubber.  I  have  observed  suctions  amounting  respectively  to  53  und  64 
centimetres  of  mercury. 


1896] 


BY   DIFFUSION   AND   SIMILAR   PROCESSES. 


265 


starting  with  any  proportions  we  collect  the  transmitted  half,  and  submit  it 
to  another  operation  of  the  same  sort,  retaining  the  half  not  transmitted, 
the  final  composition  corresponding  to  the  operations  sr  is  the  same  (ap- 
proximately) as  the  composition  with  which  we  started,  and  the  same  also 
as  would  be  obtained  by  operations  taken  in  the  reverse  order,  represented 
by  rs.  A  complete  scheme*  on  these  lines  is  indicated  in  the  diagram. 


Representing  the  initial  condition  by  unity,  we  may  represent  the  result  01 
the  first  operation  by 

\r  +  $8,    or    |  (r  +  s), 

in  which  the  numerical  coefficient  gives  the  quantity  of  gas  whose  character 
is  specified  by  the  literal  symbols.  The  second  set  of  operations  gives  in  the 
first  instance 


or,  after  admixture  of  the  second  and  third  terms  (which  are  of  the  same 
quality), 


In  like  manner  the  result  of  the  third  set  of  operations  may  be  represented 
by  (j  ,  and  (as  may  be  formally  proved  by  "induction")  of  n  sets  of 
operations  by 

C-fT <ie> 

When  we  take  account  of  the  reciprocal  character  of  r  and  s,  this  may  be 
written 


L  Ln  +  n 


rn-2  +  - 


(17) 


the  number  of  parts  into  which  the  original  quantity  of  gas  is  divided  being 

*  It  differs,  however,  from  that  followed  by  Prof.  Ramsay  iu  his  recent  researches  (Proc.  Roy. 
Soc.  Vol.  LX.  p.  216,  1896). 


266  ON  THE  SEPARATION   OF  GASES   BY   DIFFUSION.  [223 

n  +  1.  If  n  is  even,  the  largest  part,  corresponding  to  the  middle  term,  has 
the  original  composition*. 

It  is  to  be  observed,  however,  that  so  far  as  the  extreme  concentration  of 
the  less  diffusive  constituent  is  concerned  these  complex  operations  are 
entirely  unnecessary.  The  same  result,  represented  by  (£)nrn  will  be  reached 
at  a  single  operation  by  continuing  the  diffusion  until  the  residue  is  reduced 
to  (£)n  of  the  original  quantity,  when  its  composition  will  be  that  denoted  by 
rn.  And  even  as  regards  the  extreme  member  at  the  other  end  in  which  the 
more  diffusive  constituent  preponderates,  it  will  be  evident  that  the  opera- 
tions really  required  are  comparatively  simple,  the  extreme  member  in  each 
row  being  derived  solely  from  the  extreme  member  of  the  row  preceding-f*. 

If  we  abandon  the  supposition,  adopted  for  simplicity,  that  the  gas  is 
divided  into  equal  parts  at  each  operation,  we  may  still  express  the  results 
in  a  similar  manner.  If  p,  a  be  the  fractions  retained  and  transmitted,  then 
p  +  a  =  1,  and  in  place  of  (15)  we  get 

r  =  Pk (18) 

The  relation  between  r  and  s  is 

pr+a8=l;  (19) 

and  the  various  portions  into  which  the  gas  is  divided  after  n  sets  of  operations 
are  represented  by  the  various  terms  of  the  expansion  of 

(pr  +  <rs)n,  (20) 

the  Greek  letters  and  the  numerical  coefficients  giving  the  quantity  of  each 
portion,  and  the  Roman  letters  giving  the  quality.  But  it  must  not  be  for- 
gotten that  this  theory  all  along  supposes  the  difference  of  diffusivities  to  be 
relatively  small. 

*  There  is  here  a  formal  analogy  with  the  problem  of  determining  the  probability  of  a  given 
combination  of  heads  and  tails  in  a  set  of  n  tosses  of  a  coin ;  and  the  result  of  supposing  n  infinite 
may  be  traced  as  in  the  theory  of  errors. 

t  Possibly  a  better  plan  for  the  concentration  of  the  lighter  constituent  would  be  diffusion 
along  a  column  of  easily  absorbable  gas,  e.g.  C02.  The  gas  which  arrives  first  at  the  remote  end 
is  infinitely  rich  in  this  constituent.  [1902.  See  Phil.  Mag.  i.  p.  105,  1901.] 


224. 

THE  THEORY  OF  SOLUTIONS. 
[Nature,  LV.  pp.  253,  254,  1897.] 

As  some  recent  viva  voce  remarks  of  mine  have  received  an  interpretation 
more  wide  than  I  intended,  I  shall  be  glad  to  be  allowed  to  explain  that 
when  (now  several  years  ago)  I  became  acquainted  with  the  work  of 
van  t'  Hoff  I  was  soon  convinced  of  the  great  importance  of  the  advances 
due  to  him  and  his  followers.  The  subject  has  been  prejudiced  by  a  good 
deal  of  careless  phraseology,  and  this  is  probably  the  reason  why  some  dis- 
tinguished physicists  and  chemists  have  refused  their  adhesion.  It  must  be 
admitted,  further,  that  the  arguments  of  van  t'  Hoff  are  often  insufficiently 
set  out,  and  are  accordingly  difficult  to  follow.  Perhaps  this  remark  applies 
especially  to  his  treatment  of  the  central  theorem,  viz.  the  identification  of 
the  osmotic  pressure  of  a  dissolved  gas  with  the  pressure  which  would  be 
exercised  by  the  gas  alone  if  it  occupied  the  same  total  volume  in  the  absence 
of  the  solvent.  From  this  follows  the  formal  extension  of  Avogadro's  law  to 
the  osmotic  pressure  of  dissolved  gases,  and  thence  by  a  natural  hypothesis 
to  the  osmotic  pressure  of  other  dissolved  substances,  even  although  they 
may  not  be  capable  of  existing  in  the  gaseous  condition.  If  I  suggest  a 
somewhat  modified  treatment,  it  is  not  that  I  see  any  unsoundness  in  van 
t'  Hoff's  argument,  but  because  of  the  importance  of  regarding  a  matter  of 
this  kind  from  various  points  of  view. 

Let  us  suppose  that  we  have  to  deal  with  an  involatile  liquid  solvent,  and 
that  its  volume,  at  the  constant  temperature  of  our  operations,  is  unaltered 
by  the  dissolved  gas — a  question  to  which  we  shall  return.  We  start  with 
a  volume  v  of  gas  under  pressure  p0,  and  with  a  volume  V  of  liquid  just 
sufficient  to  dissolve  the  gas  under  the  same  pressure,  and  we  propose  to  find 
what  amount  of  work  (positive  or  negative)  must  be  done  in  order  to  bring 
the  gas  into  solution  reversibly.  If  we  bring  the  gas  at  pressure  p0  into 
contact  with  the  liquid,  solution  takes  place  irreversibly,  but  this  difficulty 
may  be  overcome  by  a  method  which  I  employed  for  a  similar  purpose  many 


268  THE  THEORY  OF  SOLUTIONS.  [224 

years  ago*.  We  begin  by  expanding  the  gas  until  its  rarity  is  such  that  no 
sensible  dissipation  of  energy  occurs  when  contact  with  the  liquid  is  es- 
tablished. The  gas  is  then  compressed  and  solution  progresses  under  rising 
pressure  until  just  as  the  gas  disappears  the  pressure  rises  to  p0.  The  opera- 
tions are  to  be  conducted  at  constant  temperature,  and  so  slowly  that  the 
condition  never  deviates  sensibly  from  that  of  equilibrium.  The  process  is 
accordingly  reversible. 

In  order  to  calculate  the  amount  of  work  involved  in  accordance  with  the 
laws  of  Boyle  and  Henry,  we  may  conveniently  imagine  the  liquid  and  gas  to 
be  confined  under  a  piston  in  a  cylinder  of  unit  cross-section.  During  the 
first  stage  contact  is  prevented  by  a  partition  inserted  at  the  surface  of  the 
liquid.  If  the  distance  of  the  piston  from  this  surface  be  x,  we  have  initially 
a;  =  v.  At  any  stage  of  the  expansion  (x)  the  pressure  p  is  given  by  p  =p0vjx, 
and  the  work  gained  during  the  expansion  is  represented  by 


*"K 


p,vlog^, 


x  being  a  very  large  multiple  of  v.  During  the  condensation,  after  the 
partition  has  been  removed,  the  pressure  upon  the  piston  in  a  given  position 
x  is  less  than  before.  For  the  gas  which  was  previously  confined  to  the 
space  x  is  now  partly  in  solution.  If  s  denote  the  solubility,  the  available 
volume  is  practically  increased  in  the  ratio  x  :  x  +  s  V,  so  that  the  pressure  in 
position  x  is  now  given  by 


and  the  work  required  to  be  done  during  the  compression  is 

f*     dx  x  +  sV 

H0^TF  =  ^10S^F-- 

On  the  whole  the  work  lost  during  the  double  operation  is 

x  +  sV 


and  of  this  the  first  part  must  be  omitted,  as  a;  is  indefinitely  great.  As 
regards  the  second  part,  we  see  that  it  is  zero,  since  by  supposition  the 
quantity  of  liquid  is  such  as  to  be  just  capable  of  dissolving  the  gas,  so 
that  sV  =  v.  The  conclusion  then  is  that,  upon  the  whole,  there  is  no  gain 
or  loss  of  work  in  passing  reversibly  from  the  initial  to  the  final  state  of 
things. 

The  remainder  of  the  cycle,  in  which  the  gas  is  removed  from  solution 
and  restored  to  its  original  state,  may  now  be  effected  by  the  osmotic  process  ' 

*  "On  the  Work  that  may  be  gained  during  the  mixing  of  Gases,"  Phil.  Mag.  Vol.  XLIX. 
p.  811,  1875.     [Vol.  i.  p.  242.] 


1897]  THE   THEOEY   OF   SOLUTIONS.  269 

of  van  t'  Hoff*.  For  this  purpose  one  "semi-permeable  membrane,"  per- 
meable to  gas  but  not  to  liquid,  is  introduced  just  under  the  piston  which 
rests  at  the  surface  of  the  liquid.  A  second,  permeable  to  liquid  but  not  to 
gas,  is  substituted  as  a  piston  for  the  bottom  of  the  cylinder,  and  may  be 
backed  upon  its  lower  side  by  pure  solvent.  By  suitable  proportional  motions 
of  the  two  pistons,  the  upper  one  being  raised  through  the  space  v,  and  the 
lower  through  the  space  V,  the  gas  may  be  expelled,  the  pressure  of  the  gas 
retaining  the  constant  value  p0,  and  the  liquid  (which  has  not  yet  been 
expelled)  retaining  a  constant  strength,  and  therefore  a  constant  osmotic 
pressure  P.  When  the  expulsion  is  complete,  the  work  done  upon  the  lower 
piston  is  PV,  and  that  recovered  from  the  gas  is  p0v,  upon  the  whole 
PV— p0v.  Since  this  process,  as  well  as  the  first,  is  reversible,  and  since  the 
whole  cycle  has  been  conducted  at  constant  temperature,  it  follows  from  the 
second  law  of  thermo-dynamics  that  no  work  is  lost  or  gained  during  the 
cycle,  or  that  PV=p0v.  The  osmotic  pressure  P  is  thus  determined,  and  it 
is  evident  that  its  value  is  that  of  the  pressure  which  the  gas,  as  a  gas,  would 
exert  in  space  V. 

The  objection  may  perhaps  be  taken  that  the  assumption  of  unaltered 
volume  of  the  liquid  as  the  gas  dissolves  in  it  unduly  limits  the  application 
of  the  argument.  It  is  true  that  when  finite  pressures  are  in  question,  an 
expansion  (or  contraction)  of  the  liquid  would  complicate  the  results ;  but  we 
are  concerned  only,  or  at  any  rate  primarily,  with  the  osmotic  pressure  of 
dilute,  solutions.  In  this  case  the  complications  spoken  of  relate  only  to  the 
second  order  of  small  quantities,  and  in  our  theory  are  accordingly  to  be 
dismissed. 

*  Phil.  Mag.  Vol.  xxvi.  p.  88,  1888. 


225. 

OBSERVATIONS   ON   THE  OXIDATION   OF  NITROGEN   GAS. 
[Chemical  Society's  Journal,  71,  pp.  181—186,  1897.] 

THE  observations  here  described  were  made  in  connexion  with  the 
isolation  of  argon  by  removal  of  the  nitrogen  from  air,  but  they  may,  perhaps, 
possess  a  wider  interest  as  throwing  light  upon  the  behaviour  of  nitrogen 
itself. 

According  to  Davy*,  the  dissolved  nitrogen  of  water  is  oxidised  to  nitrous 
(or  nitric)  acid  when  the  liquid  is  submitted  to  electrolysis.  "  To  make  the 
experiment  in  as  refined  a  form  as  possible,  I  procured  two  hollow  cones  of 
pure  gold  containing  about  25  grains  of  water  each,  they  were  filled  with 
distilled  water  connected  together  by  a  moistened  piece  of  amianthus  which 
had  been  used  in  the  former  experiments,  and  exposed  to  the  action  of 
a  voltaic  battery  of  100  pairs.  .  .  .  In  10  minutes  the  water  in  the  negative 
tube  had  gained  the  power  of  giving  a  slight  blue  tint  to  litmus  paper :  and 
the  water  in  the  positive  tube  rendered  it  red.  The  process  was  continued 
for  14  hours ;  the  acid  increased  in  quantity  during  the  whole  time,  and  the 
water  became  at  last  very  sour  to  the  taste.  .  .  .  The  acid,  as  far  as  its 
properties  were  examined,  agreed  with  pure  nitrous  acid  having  an  excess 
of  nitrous  gas"  (p.  6). 

Further  (p.  10),  "  I  had  never  made  any  experiments,  in  which  acid 
matter  having  the  properties  of  nitrous  acid  was  not  produced,  and  the 
longer  the  operation  the  greater  was  the  quantity  which  appeared.  .  .  . 
It  was  natural  to  account  for  both  these  appearances,  from  the  combination 
of  nascent  oxygene  and  hydrogene  respectively;  with  the  nitrogene  of  the 
common  air  dissolved  in  the  water." 

Davy  was  confirmed  in   his  conclusion   by   experiments   in   which    the 
*  Phil.  Trans.  1807,  p.  1. 


1897]  OBSERVATIONS   ON   THE   OXIDATION   OF   NITROGEN   GAS.  271 

electrolytic  vessels  were  placed  in  a  vacuum  or  in  an  atmosphere  of  hydrogen. 
There  was  then  little  or  no  reddening  of  the  litmus,  even  after  prolonged 
action  of  the  battery. 

If  nitrogen  could  be  oxidised  in  this  way,  the  process  would  be  a  con- 
venient one  for  the  isolation  of  argon,  for  it  could  be  worked  on  a  large  scale 
and  be  made  self-acting.  But  it  did  not  appear  at  all  probable  that  nitrogen 
could  take  a  direct  part  in  the  electrolysis.  In  that  case,  its  oxidation  would 
be  a  secondary  action,  due,  perhaps,  to  the  formation  of  peroxide  of  hydrogen. 
This  consideration  led  me  to  try  the  effect  of  peroxide  of  sodium  on  dissolved 
nitrogen,  but  without  success.  The  nitrogen  dissolved  in  1250  c.c.  of  tap 
water  and  liberated  by  boiling,  was  found  to  be  19'1  c.c.,  and  it  was  not 
diminished  by  a  previous  addition  of  peroxide  of  sodium,  with  or  without 
acid.  Having  failed  in  this  direction,  I  endeavoured  to  repeat  Davy's  ex- 
periment nearly  in  its  original  form.  The  water  was  contained  in  two 
cavities  bored  in  a  block  of  paraffin,  and  connected  by  a  wick  of  asbestos 
which  had  been  previously  ignited.  By  means  of  platinum  terminals  con- 
nected with  a  secondary  battery,  a  potential  difference  of  100  volts  was 
maintained  between  the  cups.  The  whole  was  covered  by  a  glass  shade,  to 
exclude  any  saline  matter  that  might  be  introduced  from  the  atmosphere. 
But,  under  these  conditions,  no  difference  in  the  behaviour  of  litmus  when 
moistened  with  water  from  the  two  cups  could  be  detected,  even  after  14 
days'  exposure  to  the  100  volts.  When,  however,  the  cover  was  removed,  the 
litmus  responded  markedly  after  a  day  or  two. 

The  failure  of  several  attempts  of  this  kind  lead  me  to  doubt  the  correct- 
ness of  Davy's  view,  that  the  dissolved  nitrogen  of  water  is  oxidised  during 
electrolysis.  At  any  rate,  the  action  is  so  slow  that  the  process  holds  out  no 
promise  of  usefulness  on  a  large  scale. 

In  the  oxidation  of  nitrogen  by  gaseous  oxygen  under  the  action  of 
electric  discharge,  a  question  arises  as  to  the  influence  of  pressure.  If  the 
mass  absorbed  were  proportional  to  pressure,  or  the  volume  independent  of 
pressure,  the  electrical  energy  expended  being  the  same,  it  might  be  desirable 
to  work  with  highly  condensed  gases,  in  spite  of  the  serious  difficulties  that 
must  necessarily  be  encountered.  That  pressure  would  be  favourable  seems 
probable  a  priori,  and  is  suggested  by  certain  observations  of  Dr  Frankland. 
My  own  early  experiments  pointed  also  in  the  same  direction.  A  suitable 
mixture  of  nitrogen  and  oxygen,  standing  in  an  inverted  test-tube  over  alkali, 
was  sparked  from  a  Ruhmkorff  coil  actuated  by  five  Grove  cells ;  when  the 
total  pressure  was  about  three  atmospheres,  the  mass  absorbed  was  about 
three  times  that  absorbed  in  the  same  time  at  the  ordinary  pressure. 

This  result  made  it  necessary  to  proceed  to  operations  upon  a  larger 
scale  with  the  alternate  current  discharge.  Experiments  were  first  tried  in 


272  OBSERVATIONS   ON   THE   OXIDATION   OF   NITROGEN   GAS.  [225 

a  small  vessel  (of  250  c.c.),  which  would  be  more  easily  capable  of  withstand- 
ing internal  pressure  than  a  larger  one.  In  order  to  protect  the  glass,  which 
at  the  top  was  almost  in  contact  with  the  electric  flame,  and  to  promote 
absorption  of  the  combined  nitrogen,  the  alkali  was  used  in  the  form  of 
a  fountain,  which  struck  the  glass  immediately  over  the  flame,  and  washed 
the  whole  of  the  internal  surface*.  But,  to  my  surprise,  preliminary  trials, 
conducted  at  atmospheric  pressure,  showed  that  this  apparatus  was  not 
effective.  The  rates  of  absorption  were  about  1600  c.c.  per  hour,  the  runs 
themselves  being  for  half-an-hour.  About  double  this  rate  had  already  been 
obtained  with  the  same  electrical  appliances  and  with  stationary  alkali. 
Care  having  been  taken  that  the  quality  of  the  mixture  within  the  working 
vessel  was  maintained  throughout  the  run,  the  smaller  efficiency  could  only 
be  connected  with  the  confined  space. 

As  to  the  reason  why  a  confined  space  should  be  unfavourable,  it  is 
difficult  to  give  a  decided  opinion.  Other  things  being  the  same,  the  surface 
presented  by  the  alkali  will  be  diminished  in  a  smaller  vessel,  and  the  ab- 
sorption of  the  combined  nitrogen  may  consequently  be  less  rapid.  But  it  is 
difficult  to  accept  this  explanation,  in  view  of  the  favourable  conditions 
secured  by  the  use  of  a  fountain.  The  gases,  as  they  rise  from  the  flame, 
impinge  directly  upon  the  alkali,  which  is  itself  in  rapid  motion  over  the 
whole  internal  surface.  It  would  almost  seem  as  if  the  combined  nitrogen, 
as  it  leaves  the  flame,  is  not  yet  ready  for  absorption,  and  only  becomes  so 
after  the  lapse  of  a  certain  time.  However  this  may  be,  the  efficiency  is  in 
practice  improved  by  largely  increasing  the  capacity  of  the  working  vessel. 
A  larger  bottle,  of  370  c.c.  capacity,  allowed  a  rate  of  2000  c.c.  per  hour. 
A  flask  of  still  greater  capacity  gave  3300  c.c.  per  hour,  whilst  with  a  larger 
globe  capable  of  holding  4^  litres,  a  rate  of  6800  c.c.  per  hour  was  obtained. 
These  experiments  were  all  made  at  atmospheric  pressure  with  a  fountain  of 
alkali  and  with  the  electric  flame  in  as  nearly  as  possible  a  constant  condition. 
In  the  case  of  the  smallest  vessel,  it  was  thought  that  the  separation  of  the 
platinum  terminals  may  have  been  insufficient  for  the  best  effect,  but  the 
loss  due  to  this  cause  must  have  been  relatively  small.  Electrical  instruments 
connected  with  the  primary  circuit  of  the  Ruhmkorff  gave  readings  of  10 
amperes  and  41  volts. 

When  the  comparatively  small  vessel  of  370  c.c.  was  used  at  a  pressure 
of  about  one  additional  atmosphere,  the  volume  absorbed  was  about  the  same 
as  in  the  experiments  with  the  same  vessel  at  atmospheric  pressure,  thus 
indicating  a  double  efficiency.  This  increased  efficiency  is,  however,  of  no 
practical  importance,  inasmuch  as  a  higher  efficiency  still  can  be  obtained  at 
atmospheric  pressure  by  use  of  a  larger  vessel.  In  order  to  clear  up  the 
question,  it  was  necessary  to  compare  the  efficiencies  in  a  large  vessel  at 

*  Rayleigh  and  Ramsay,  Phil.  Trans.  1895,  p.  217.    [Vol.  iv.  p.  162.] 


1897]  OBSERVATIONS   ON   THE   OXIDATION   OF   NITROGEN    GAS.  273 

different  pressures,  an  operation  involving  considerable  difficulty  and  even 
danger. 

For  this  purpose,  a  glass  globe,  nearly  spherical  in  form,  and  having 
a  capacity  of  about  7  litres,  was  employed.  The  extra  pressure  was  nearly 
an  atmosphere  a,nd  was  obtained  by  gravity,  the  feed  and  return  pipes  for 
the  alkaline  fountain,  as  well  as  the  pipe  for  the  supply  of  water  to  the  gas- 
holder, being  carried  to  a  higher  level  than  that  at  which  the  rest  of  the 
apparatus  stood.  The  rate  of  absorption  (reduced  to  atmospheric  pressure) 
was  6880  c.c.  per  hour.  Experiments  conducted  at  atmospheric  pressure 
gave  as  a  mean  6600  c.c. 

In  order  to  examine  still  further  the  influence  of  pressure,  two  ex- 
periments were  tried  under  a  total  pressure  of  half  an  atmosphere.  The 
reduced  numbers  were  5600,  5700  c.c.  per  hour.  From  these  results,  it 
would  appear  that  the  influence  of  pressure  is  slightly  favourable.  But,  in 
comparing  the  results  for  one  atmosphere  and  for  half  an  atmosphere,  it 
should  be  remembered  that,  in  the  latter  case,  aqueous  vapour  is  responsible 
for  a  sensible  part  of  the  total  pressure.  At  any  rate,  the  results  are  much 
more  nearly  independent  of  pressure  than  proportional  to  pressure ;  so  that 
the  cases  of  large  and  small  vessels  are  sharply  distinguished,  pressure  ap- 
pearing to  be  advantageous  only  where  the  space  is  too  confined  to  admit 
of  the  best  efficiency  at  a  given  pressure  being  reached. 

Not  sorry  to  be  relieved  from  the  obligation  of  designing  a  large  scale 
apparatus  to  be  worked  at  a  high  pressure,  such  as  20  or  100  atmospheres, 
I  reverted  to  the  ordinary  pressure,  and  sought  to  obtain  a  high  rate  of 
absorption  by  employing  a  powerful  electric  flame  contained  in  a  large  vessel 
whose  walls  were  washed  internally  by  an  alkaline  fountain.  The  electrical 
arrangements  have  been  the  subject  of  much  consideration,  and  require  to  be 
different  from  what  would  naturally  be  expected.  Since  the  voltage  on  the 
final  platinums  during  discharge  is  only  from  1600  to  2000,  as  measured  by 
one  of  Lord  Kelvin's  instruments,  it  might  be  supposed  that  a  commercial 
transformer,  transforming  from  100  volts  to  2400  volts,  would  suffice  for  the 
purpose.  When,  however,  the  attempt  is  made,  it  is  soon  discovered  that 
such  an  arrangement  is  quite  unmanageable.  When,  after  some  difficulty, 
the  arc  is  started,  it  is  found  that  the  electrical  conditions  are  unstable. 
Things  may  go  well  for  a  time,  but  after  perhaps  some  hours  the  current 
will  rise  and  the  platinums  will  become  overheated  and  may  melt.  Even 
when  two  transformers  were  employed,  so  connected  as  to  give  on  open 
secondary  circuit  nearly  4800  volts,  the  conditions  were  not  steady  enough 
for  convenient  practice.  The  transformer  used  in  the  experiments  about  to 
be  described  is  by  Messrs  Swinburne,  and  is  insulated  with  oil.  On  open 
secondary,  the  voltage  is  nearly  8000*,  but  it  falls  to  2000  or  less  when  the 

*  Probably  6000  would  have  sufficed. 
H.    IV.  18 


274 


OBSERVATIONS   ON   THE   OXIDATION    OF   NITROGEN   GAS. 


[225 


discharge  is  running.  Even  with  this  transformer,  it  was  necessary  to  include 
in  its  primary  (thick  wire)  circuit  a  self-induction  coil,  provided  with, a  core 
consisting  of  a  bundle  of  iron  wires,  and  adjustable  in  position.  As  finally 
used,  the  adjustment  was  such  that  the  electromotive  force  actually  operative 
on  the  primary  was  only  about  30  volts  out  of  the  100  volts  available  at  the 
mains  of  the  public  supply.  This  reduction  of  voltage  does  not,  at  any  rate 
from  a  theoretical  point  of  view,  involve  any  loss  of  economy,  and  some  such 
reduction  seems  to  be  essential  to  steadiness.  Under  these  conditions,  the 
current  taken  amounted  to  40  amperes. 

It  is  scarcely  necessary  to  say  that  the  watts  actually  delivered  to  the 
primary  circuit  of  the  transformer  are  less  than  the  number  (1200)  derived 
by  multiplication  of  volts  and  amperes.  From  some  experiments  made 
under  similar  conditions*,  I  have  found  that  the  factor  of  reduction — the 
cosine  of  the  angle  of  lag — is  about  two-thirds,  so  that  the  watts  taken  in 
the  above  arrangement  are  about  800,  representing  a  little  more  than  a 
horse-power. 

The  working  vessel,  A,  was  of  glass,  spherical  in  form,  and  of  50  litres 
capacity.  The  neck  was  placed  downwards,  and 
was  closed  by  a  large  rubber  stopper,  through 
which  five  tubes  of  glass  penetrated.  Two  tubes 
of  substantial  construction  carried  the  electrodes, 
B,  C,  arranged  much  as  in  a  former  apparatus f ; 
two  more,  F  and  E,  were  required  for  the  supply 
tube  of  the  fountain  and  for  the  drain  of  liquid, 
whilst  the  fifth,  D,  was  for  the  supply  of  gas. 
The  external  drowning  of  the  vessel,  formerly 
necessary,  was  now  dispensed  with;  but  a  suit- 
able cooling  arrangement  for  the  alkali  (some- 
thing like  the  worm  of  a  condenser)  had  to  be 
provided  to  obviate  excessive  accumulation  of 
heat. 

As  the  solution  of  alkali  circulated  entirely 
in  the  closed  apparatus,  it  could  lose  none  of  its 
dissolved  argon.     It  was  maintained  in  circula- 
tion by  a  small  centrifugal  pump  constructed  of  iron  and  driven  from  an 
electric  motor. 

The  mixed  gases  (about  11  parts  of  oxygen  to  9  parts  of  air)  were 
supplied  from  a  large  gas-holder ;  but  an  auxiliary  holder  was  also  necessary 
in  order  to  observe  the  rate  of  absorption.  When  the  rate  became  un- 

*•  I  hope  shortly  to  publish  an  account  of  the  method  employed.     [Phil.  Mag.  XLIII.  p.  343 ; 
Art.  229  below.] 

t  Bayleigh  and  Ramsay,  Phil.  Trans.  1895,  p.  218.     [Vol.  iv.  p.  163.] 


1897]  OBSERVATIONS   ON   THE   OXIDATION   OF   NITROGEN   GAS.  275 

satisfactory,  the    mixed   gas  in  the  working  vessel   was   analysed   and  the 
necessary  rectification  effected. 

In  the  earlier  stages  of  the  operation,  the  rate  of  absorption  was  about 
21  litres  per  hour,  and  this,  by  proper  attention,  could  be  maintained  without 
much  loss  until  the  accumulation  of  argon  began  to  tell.  If  we  take  20  litres 
as  corresponding  to  800  watts,  we  have  25  c.c.  per  watt-hour,  an  efficiency 
not  very  different  from  that  found  in  operations  on  a  much  smaller  scale. 

The  present  apparatus  works  about  three  times  as  fast  as  the  former  one, 
in  which  the  vessel  was  smaller  and  the  alkali  stationary.  It  is  also  more 
interesting  to  watch,  as  the  electric  flame  is  fully  exposed  to  view.  On  the 
other  hand,  it  is  more  complicated,  owing  to  the  use  of  a  circulating  pump, 
and  probably  requires  closer  attention.  A  failure  of  the  fountain  whilst  the 
flame  was  established  would  doubtless  soon  lead  to  a  disaster. 

I  have  been  efficiently  aided  throughout  by  Mr  Gordon,  who  has  not  only 
fitted  the  apparatus,  but  has  devised  many  of  the  contrivances  necessary  to 
meet  the  ever-recurring  difficulties  which  must  be  expected  in  work  of  this 
character. 


18—2 


226. 


ON   THE   PASSAGE    OF   ELECTRIC    WAVES    THROUGH   TUBES, 
OR   THE    VIBRATIONS    OF    DIELECTRIC   CYLINDERS. 


[Philosophical  Magazine,  XLIII.  pp.  125—132,  1897.] 


General  Analytical  Investigation. 

THE  problem  here  proposed  bears  affinity  to  that  of  the  vibrations  of 
a  cylindrical  solid  treated  by  Pochhammer*  and  others,  but  when  the 
bounding  conductor  is  regarded  as  perfect  it  is  so  much  simpler  in  its 
conditions  as  to  justify  a  separate  treatment.  Some  particular  cases  of  it 
have  already  been  considered  by  Prof.  J.  J.  Thomson  f.  The  cylinder  is 
supposed  to  be  infinitely  long  and  of  arbitrary  section  ;  and  the  vibrations 
to  be  investigated  are  assumed  to  be  periodic  with  regard  both  to  the 
time  (t)  and  to  the  coordinate  (2)  measured  parallel  to  the  axis  of  the 
cylinder,  i.e.,  to  be  proportional  to  ei(mz+pt). 

By  Maxwell's  Theory,  the  components  of  electromotive  intensity  in  the 
dielectric  (P,  Q,  R)  and  those  of  magnetic  induction  (a,  b,  c)  all  satisfy 
equations  such  as 

d?R     d?R     &R_I_cPR 
dtf      dy*+  dz*  "  F2  dff'  ' 

V  being  the  velocity  of  light  ;  or  since  by  supposition 


R=0,  ........................  (2) 

where  k2  =  p*/V*-m*  .................................  (3){ 


*  Crette,  Vol.  xxxi.  1876. 

t  Recent  Researches  in  Electricity  and  Magnetism,  1893,  §  300. 

J  The  fc2  of  Prof.  J.  J.  Thomson  (loc.  cit.  %  262)  is  the  negative  of  that  here  chosen  for 
convenience. 


1897]      ON   THE   PASSAGE   OF   ELECTRIC   WAVES   THROUGH   TUBES,   ETC.          277 

The  relations  between  P,  Q,  R  and  a,  b,  c  are  expressed  as  usual  by 

da_d_Q_dR 

dt  ~  dz      dy" 

and  two  similar  equations ;  while 

da     db     dc  dP     dQ     dR 

-7 — r~3 — r  T~  =  v,  j — 7 — =—  =  0 (O.  O) 

dx     dy     dz  dx      dy      dz 

The  conditions  to  be  satisfied  at  the  boundary  are  that  the  components  of 
electromotive  intensity  parallel  to  the  surface  shall  vanish.     Accordingly 


dx/ds,  dy/ds  being  the  cosines  of  the  angles  which  the  tangent  (ds)  at  any 
point  of  the  section  makes  with  the  axes  of  x  and  y. 

Equations  (2)  and  (7)  are  met  with  in  various  two-dimensional  problems 
of  mathematical  physics.  They  are  the  equations  which  determine  the  free 
transverse  vibrations  of  a  stretched  membrane  whose  fixed  boundary  coincides 
with  that  of  the  section  of  the  cylinder.  The  quantity  k'2  is  limited  to  certain 
definite  values,  k-?,  k*, ...,  and  to  each  of  these  corresponds  a  certain  normal 
function.  In  this  way  the  possible  forms  of  R  are  determined.  A  value  of  R 
which  is  zero  throughout  is  also  possible. 

With  respect  to  P  and  Q  we  may  write 

D      dd>      d^lr  ~      d<f>      d~^r  /n   _.  _ . 

Jr  =  -j — I — ;—  ,  (J  =  j j — ',   (y,  1U1 

dx      dy  dy      dx 

where  </>  and  \Jr  are  certain  functions,  of  which  the  former  is  given  by 

dP     dQ         dR 

VSA  =  -T-+  T^=--r-  =  -imfi (11) 

dx      dy          dz 

There  are  thus  two  distinct  classes  of  solutions ;  the  first  dependent  upon  <£, 
in  which  R  has  a  finite  value,  while  \|r  =  0 ;  the  second  dependent  upon  i/r,  in 
which  R  and  <f>  vanish. 

For  a  vibration  of  the  first  class  we  have 

P=d<j>ldz,          Q=d<f>/dy,  (12) 

and  (V'  +  fc2)<£  =  0 (13) 

Accordingly  by  (11)  </>  =  ~E,   (14) 

„     im  dR             ~     im  dR  n  ^ 

QXKJL  i  = U  =  —  — ,  \  •"•**/ 

k2   dx '  n    dy 

by  which  P  and  Q  are  expressed  in  terms  of  R  supposed  already  known. 


278  ON  THE   PASSAGE   OF   ELECTRIC  WAVES   THROUGH   TUBES,  [226 

The  boundary  condition  (7)  is  satisfied  by  the  value  ascribed  to  R,  and 
the  same  value  suffices  also  to  secure  the  fulfilment  of  (8),  inasmuch  as 
•       dx         dyJmdR 
ds+^ds      fc2   ds 

The  functions  P,  Q,  R  being  now  known,  we  may  express  a,  b,  c.     From  (4) 

da  dR        in?  +  k*  dR 

-rr  =  ipa  =  imQ  —  7-  =  --  rr  —  ~r~  5 
dt      r  dy  k?       dy 


so  that  a= 


^-7—  -j—  ,  .  .,  , 

ipkz    dy  ipk*     dx 

In  vibrations  of  the  second  class  R  =  0  throughout,  so  that  (2)  and  (7)  are 
satisfied,  while  k2  is  still  at  disposal.     In  this  case 

P=d+/dy,  Q  =  -d^fdx,  .....................  (17) 

and  (V2  +  £2)x/r=0  ...............................  (18) 

By  the  third  of  equations  (4) 

dc      .         dP     dQ 


so  that  T/T  =  —  ipc/k2,  and 

ipdc  ipdc 

f  —  —  yr    -j—  ,  V^J  —  yr    -j—  ,  Xt  —  U 

A;2  dy  k*  dx 

im  dc  j      im  dc  /OAX 

AlBoby(4)  «  =  ^^,  b  =  ¥dy  ................  '  ...........  (20) 

Thus  all  the  functions  are  expressed  by  means  of  c,  which  itself  satisfies 

(V2+fc2)c  =  0  ..................................  (21) 

We  have  still  to  consider  the  second  boundary  condition  (8).  This  takes  the 
form 

dc  dx     dc  dy  _ 

dy  ds     dx  ds 

requiring  that  dcfdn,  the  variation  of  c  along  the  normal  to  the  boundary  at 
any  point,  shall  vanish.  By  (21)  and  the  boundary  condition 

dc/dn  =  0,     .................................  (22) 

the  form  of  c  is  determined,  as  well  as  the  admissible  values  of  k2.  The 
problem  as  regards  c  is  thus  the  same  as  for  the  two-dimensional  vibrations  of 
gas  within  a  cylinder  which  is  bounded  by  rigid  walls  coincident  with  the 
conductor,  or  for  the  vibrations  of  a  liquid  under  gravity  in  a  vessel  of  the 
same  form*. 

All  the  values  of  k  determined  by  (2)  and  (7),  or  by  (21)  and  (22),  are  real, 
*  Phil.  Mag.  Vol.  i.  p.  272  (1876).     [Vol.  i.  p.  265.] 


1897]  OR   THE    VIBRATIONS   OF   DIELECTRIC    CYLINDERS.  279 

but  the  reality  of  k  still  leaves  it  open  whether  m  in  (3)  shall  be  real  or 
imaginary.  If  we  are  dealing  with  free  stationary  vibrations  m  is  given 
and  real,  from  which  it  follows  that  p  is  also  real.  But  if  it  be  p  that  is 
given,  ra2  may  be  either  positive  or  negative.  In  the  former  case  the  motion 
is  really  periodic  with  respect  to  z  ;  but  in  the  latter  z  enters  in  the  forms 
em'z,  e~m'z,  and  the  motion  becomes  infinite  when  z  =  +  oo  ,  or  when  z  =  —  oo  , 
or  in  both  cases.  If  the  smallest  of  the  possible  values  of  k2  exceeds  pzj  V2, 
m  is  necessarily  imaginary,  that  is  to  say  no  periodic  waves  of  the  frequency 
in  question  can  be  propagated  along  the  cylinder. 

Rectangular  Section. 

The  simplest  case  to  which  these  formulae  can  be  applied  is  when  the 
section  of  the  cylinder  is  rectangular,  bounded,  we  may  suppose,  by  the  lines 


As  for  the  vibrations  of  stretched  membranes*,  the  appropriate  value 
of  R  applicable  to  solutions  of  the  first  class  is 

R=ei  (mz+^  sin  (/«ra?/a)  sin  (vn-y/fS)  ;   .............  (23) 

from  which  the  remaining  functions  are  deduced  so  easily  by  (15),  (16)  that 
it  is  hardly  necessary  to  write  down  the  expressions.  In  (23)  &  and  v  are 
integers,  and  by  (13) 


whence  m2  =  p*/V*  -TT*  (^  +  ^j (25) 

The  lowest  frequency  which  allows  of  the  propagation  of  periodic  waves  along 
the  cylinder  is  given  by 

7T          7T  /cu~>\ 


If  the  actual  frequency  of  a  vibration  having  its  origin  at  any  part  of  the 
cylinder  be  much  less  than  the  above,  the  resulting  disturbance  is  practically 
limited  to  a  neighbouring  finite  length  of  the  cylinder. 

For  vibrations  of  the  second  class  we  have 

c=ef(wlz+^cos(^7rA-/a)cos(^7ry//3), (27) 

the  remaining  functions  being  at  once  deducible  by  means  of  (19),  (20). 
The  satisfaction  of  (22)  requires  that  here  again  //.,  v  be  integers,  and  (21) 
gives 

*--<&+»• (28) 

identical  with  (24). 

*  Theory  of  Sound,  §  195. 


280  ON   THE   PASSAGE   OF   ELECTRIC   WAVES   THROUGH   TUBES,  [226 

If  a  >  /3,  the  smallest  value  of  k  corresponds  to  p  =  1,  v  =  0.     When  v  =  0, 
we  have  k  =  pir /a,  and  if  the  factor  eiimz+pt)  be  omitted, 

a  =  -^smfc#,         6  =  0,        c  =  cos^,  ,...(29) 

k 

a  solution  independent  of  the  value  of  ft.    There  is  no  solution  derivable  from 

Circular  Section. 

For  the  vibrations  of  the  first  class  we  have  as  the  solution  of  (2)  by  means 
of  Bessel's  functions, 

R  =  Jn(kr)cosn0,    (31) 

n  being  an  integer,  and  the  factor  ei(mi!+pt}  being  dropped  for  the  sake  of 
brevity.  In  (31)  an  arbitrary  multiplier  and  an  arbitrary  addition  to  0  are 
of  course  admissible.  The  value  of  k  is  limited  to  be  one  of  those  for  which 

Jn(kr')  =  0 (32) 

at  the  boundary  where  r  =  r. 

The  expressions  for  P,  Q,  a,  b,  c  in  (15),  (16)  involve  only  dR/dx,  dR/dy. 
For  these  we  have 

7  Tt  J  f>  J  f> 

^  =  ^F  cos  0  -  ^  sin  0  =  kJn  (kr) cos n0  cos 0  +  -Jn (kr)  sin n0 sin  0 
da;       dr  ra0  r 

(          J  } 

=  &k  cos  (n  —  1)  0  \Jn  +  i— 
I          **•) 

(kr), (33) 


according  to  known  properties  of  these  functions  ;  and  in  like  manner 

dR     dR  .          dR 

-j-  =  -j—  sm  0  +  —  ra  cos  0 
dy      dr  rd0 


(34) 

These  forms  show  directly  that  dR/d.r,  dR/dy  satisfy  the  fundamental 
equation  (2).  They  apply  when  n  is  equal  to  unity  or  any  greater  integer. 
When  n  =  0,  we  have 

R  =  Jn(kr\  ....................................  (35) 


(36) 


*  For  (18)  would  then  become  v2f  =  °;  and  this.  with  the  boundary  condition  df/dn^O, 
•would  require  that  P  and  Q,  as  well  as  R,  vanish  throughout. 


1897]  OR   THE    VIBRATIONS   OF    DIELECTRIC    CYLINDERS.  281 

The    expressions   for  the  electromotive  intensity  are   somewhat  simpler 
when  the  resolution  is  circumferential  and  radial  : 

circumf.  component 

=  QCos0-Ps™eJ^^=-™^Jn(kr)Snn0,  ..........  (37) 

radial  component 

=  Pcos0  +  Qsm0  =  'l^~  =  ~Jn'(kr)cosn0  ................  (38) 

rC       CLT  rC 

If  n  =  0,  the  circumferential  component  vanishes. 

Also  for  the  magnetization 
circ.  comp.  of  magnetization 


;         /,          •    /)  f/,   . 

=  bcos0-asm0  =  —  r-=  --  5-  =  —  r-r—  J»(kr)  cos  nd,  ........  (39) 

ipk2     dr         ipk 

rad.  comp.  of  mag.  -  a  cos  6  +  b  sin  6 

m*+k2  dR      n(m>  +  k*)  r  ,.   .     .       ., 

=  --  r-y-  --  ^  =     \    >0  —  *  Jn(kr)  smn0  .....  ...  .(40) 

ipk2    rd0          ipk*r 

The  smallest  value  of  k  for  vibrations  of  this  class  belongs  to  the  series 
n  =  0,  and  is  such  that  kr  =  2'404,  r  being  the  radius  of  the  cylinder. 

For  the  vibrations  of  the  second  class  R  =  0,  and  by  (21), 

c  =  Jn(kr)cosnO,  .......  .  ....................  (41) 

k  being  subject  to  the  boundary  condition 

Jn'(kr')  =  0  ..................................  (42) 

As  in  (33),  (34), 


dc  dc  a  dc  .  a 
-y-  =  -y-  cos  6  --  77;  sin  6 
dx  dr  rdd 


dc      dc    .    „      dc 

-j-  =  -j-  sin  6  +  —j-a  cos  9 

dy     dr  rd6 


(43) 
(44) 


so  that  by  (19),  (20)  all  the  functions  are  readily  expressed. 
When  n  =  0,  we  have 

^  =  -  kJ,  (kr)  cos  6,         (jf-=-kJ1(kr)sin0  ...............  (45) 

doc  dy 

For  the  circumferential  and  radial  components  of  magnetization  we  get 


282          ON   THE    PASSAGE    OF    ELECTRIC   WAVES   THROUGH   TUBES,    ETC.        [226 

(46) 


circ.  comp.  of  mag.  =  b  cos  6  -  a  sin  6 
im  dc          imn 


rad.  comp.  of  mag.  =  a  cos  6  +  b  sin  6 

imdc      im  T  , 
=  ~          =       Jn 


corresponding  to  (37),  (38)  for  vibrations  of  the  first  class. 

In  like  manner  equations  analogous  to  (39),  (40)  now  give  the  components 
of  electromotive  intensity.     Thus 

circ.  comp.  =  Qcos  0-P  sin  d  =  ^  ^  =|?  Jn'  (kr)  cosn0,  ............  (48) 

rad.  comp.  =  Pcos  B  +  Q  sin  0  =  -|?  ^  =  ^  Jn(kr}  sin  n0  .......  (49) 

The  smallest  value  of  k  admissible  for  vibrations  of  the  second  class  is  of 
the  series  belonging  to  n  =  1,  and  is  such  that  kr'  =  1-841,  a  smaller  value  than 
is  admissible  for  any  vibration  of  the  first  class.  Accordingly  no  real  wave  of 
any  kind  can  be  propagated  along  the  cylinder  for  which  p/V  is  less  than 
1'841/r',  where  r  denotes  the  radius.  The  transition  case  is  the  two- 
dimensional  vibration  for  which 

c  =*«*,/,  (1-841  r//>  cos  0,         p=  1-841  V/r'  .............  (50,51) 


227. 


ON  THE  PASSAGE  OF   WAVES   THROUGH  APERTURES  IN 
PLANE  SCREENS,  AND  ALLIED  PROBLEMS. 

[Philosophical  Magazine,  XLIII.  pp.  259—272,  1897.] 

THE  waves  contemplated  may  be  either  aerial  waves  of  condensation  and 
rarefaction,  or  electrical  waves  propagated  in  a  dielectric.  Plane  waves  of 
simple  type  impinge  upon  a  parallel  screen.  The  screen  is  supposed  to  be 
infinitely  thin,  and  to  be  perforated  by  some  kind  of  aperture.  Ultimately 
one  or  both  dimensions  of  the  aperture  will  be  regarded  as  infinitely  small 
in  comparison  with  the  wave-length  (X);  and  the  method  of  investigation 
consists  in  adapting  to  the  present  purpose  known  solutions  regarding  the 
flow  of  incompressible  fluids. 

If  <£  be  a  velocity-potential  satisfying 

d^jdt^V^^,  ..............................  (1) 

where  V2  =  d*/dtf  +  d*/dy*  +  d2/dz2, 

the  condition  at  the  boundary  may  be  (i)  that  d<f>fdn=  0,  or  (ii)  that  <£  =  0. 
The  first  applies  directly  to  aerial  vibrations  impinging  upon  a  fixed  wall,  and 
in  this  connexion  has  already  been  considered*. 

If  we  assume  that  the  vibration  is  everywhere  proportional  to  eint,  (1) 
becomes 

(V2  +  fc2)  <£  =  0,  ................................  (2) 

where  k  =  n/V=2Tr/\  ...............................  (3) 

It  will  conduce  to  brevity  if  we  suppress  the  factor  eint.  On  this  un- 
derstanding the  equation  of  waves  travelling  parallel  to  x  in  the  positive 
direction,  and  accordingly  incident  upon  the  negative  side  of  a  screen 
situated  at  x  =  0,  is 


Theory  of  Sound,  §  292. 


284  ON   THE   PASSAGE    OF    WAVES   THROUGH   APERTURES  [227 

When  the  solution  is  complete,  the  factor  eint  is  to  be  restored,  and  the 
imaginary  part  of  the  solution  is  to  be  rejected.  The  realized  expression 
for  the  incident  waves  will  therefore  be 

kx)  ...............................  (5) 


Perforated  Screen.  —  Boundary  Condition  d<f>/dn  =  0. 

If  the  screen  be  complete,  the  reflected  waves  under  the  above  condition 
have  the  expression  <f>  =  eikx. 

Let  us  divide  the  actual  solution  into  two  parts  %  and  -fy,  the  first  the 
solution  which  would  obtain  were  the  screen  complete,  the  second  the 
alteration  required  to  take  account  of  the  aperture  ;  and  let  us  distinguish 
by  the  suffixes  m  and  p  the  values  applicable  upon  the  negative  (minus}  and 
upon  the  positive  side  of  the  screen.  In  the  present  case  we  have 

Xp  =  0  .........................  (6) 


This   ^-solution   makes   d^m/dn  —  0,  d%p/dn  =  0  over  the  whole  plane 
x  =  0,  and  over  the  same  plane  %m  =  2,  %p  =  0. 

For  the  supplementary  solution,  distinguished  in  like  manner  upon  the 
two  sides,  we  have 


(7) 


where  r  denotes  the  distance  of  the  point  at  which  ty  is  to  be  estimated 
from  the  element  dS  of  the  aperture,  and  the  integration  is  extended 
over  the  whole  of  the  area  of  aperture.  Whatever  functions  of  position 
^m,  "Vp  may  be,  these  values  on  the  two  sides  satisfy  (2),  and  (as  is  evident 
from  symmetry)  they  make  d-fymjdn,  dtypjdn  vanish  over  the  wall,  viz.  the 
unperforated  part  of  the  screen  ;  so  that  the  required  condition  over  the  wall 
for  the  complete  solution  (^  +  \Jr)  is  already  satisfied.  It  remains  to  consider 
the  further  conditions  that  <f>  and  d<f>/dx  shall  be  continuous  across  the 
aperture. 

These  conditions  require  that  on  the  aperture 

2  +  ^m  =  ^p,       d^m/dx  =  d^p/dx  ...................  (8)* 

The  second  is  satisfied  if  %  =  -  Vm  ;  so  that 

<*S,      *,  =  -¥„  6-~  dS,  .............  (9) 


making  the  values  of  ^m  and  -fy-p  equal  and  opposite  at  all  corresponding 
points,  viz.  points  which  are  images  of  one  another  in  the  plane  x  =  0.     In 

*  The  use  of  dx  implies  that  the  variation  is  in  a  fixed  direction,  while  dn  may  be  supposed 
to  be  drawn  outwards  from  the  screen  in  both  cases. 


1897]  IN    PLANE    SCREENS,   AND   ALLIED    PROBLEMS.  285 

order  further  to  satisfy  the  first  condition  it  suffices  that  over  the  area  of 
aperture 

^m  =  -l,       ^=1  .............................  (10) 

and  the  remainder  of  the  problem  consists  in  so  determining  ^m  that  this 
shall  be  the  case. 

In  this  part  of  the  problem  we  limit  ourselves  to  the  supposition  that 
all  the  dimensions  of  the  aperture  are  small  in  comparison  with  X.  For 
points  at  a  distance  from  the  aperture  e~ikr/r  may  then  be  removed  from 
under  the  sign  of  integration,  so  that  (9)  becomes 


The  significance  of  JJ  ^mdS  is  readily  understood  from  an  electrical  inter- 
pretation. For  in  its  application  to  a  point,  itself  situated  upon  the  area 
of  aperture,  e~ikr  in  (9)  may  be  identified  with  unity,  so  that  tym  is  the 
potential  of  a  distribution  of  density  *Pm  on  S.  But  by  (10)  this  potential 
must  have  the  constant  value  —  1  ;  so  that  —ff^mdS,  or  ff^pdS,  represents 
the  electrical  capacity  of  a  conducting  disk  having  the  size  and  shape  of 
the  aperture,  and  situated  at  a  distance  from  all  other  electrified  bodies. 
If  we  denote  this  by  M,  the  solution  applicable  to  points  at  a  distance  from 
the  aperture  may  be  written 


To  these  are  to  be  added  the  values  of  ^  in  (6).     The  realized  solutions 
are  accordingly 

MC^~—  },   ...............  (13) 

(14) 


The  value  of  M  may  be  expressed*  for  an  ellipse  of  semi-major  axis  a 
and  eccentricity  e.     We  have 


M- 


J(.y 

F  being  the  symbol   of  the  complete  elliptic  function  of  the   first   kind. 
When  e  =  0,  F  (e)  =  £TT  ;  so  that  for  a  circle  M  =  2a/7r. 

It  should  be  remarked  that  M^  in  (9)  is  closely  connected  with  the  normal 
velocity  at  dS.     In  general, 


*  Theory  of  Sound,  §§  292,  306,  where  is  given  a  discussion  of  the  effect  of  ellipticity  when 
area  is  given. 


286  ON   THE   PASSAGE    OF    WAVES   THROUGH    APERTURES  [227 

At  a  point  (x)  infinitely  close  to  the  surface,  only  the  neighbouring  elements 
contribute  to  the  integral,  and  the  factor  e~ikr  may  be  omitted.  Thus 

^  = 
dx 

¥  =  -,  ..............................  (17)    - 

2-rr  dn 

d-fy/dn  being  the  normal  velocity  at  the  point  of  the  surface  in  question. 

Boundary  Condition  <f>  =  0. 

We  will  now  suppose  that  the  condition  to  be  satisfied  on  the  walls  is 
<£  =  0,  although  this  case  has  no  simple  application  to  aerial  vibrations. 
Using  a  similar  notation  to  that  previously  employed,  we  have  as  the  ex- 
pression for  the  principal  solution 

Xm  =  e-**-e**,       Xp=0,  ........................  (18) 

giving   over   the   whole   plane   (x  =  0),    %m  =  0,    %p  =  0,    d^m  \  dx  =  —  2ik, 
/dx  =  0. 

The  supplementary  solutions  now  take  the  form 


These  give  on  the  walls  -\Jrm  =  -^rp  =  0,  and  so  do  not  disturb  the  condition  of 
evanescence  already  satisfied  by  %.     It  remains  to  satisfy  over  the  aperture 
^m  =  ^p>      -2ik  +  d^m/dx  =  d^p/dx  ................  (20) 

The  first  of  these  is  satisfied  if  Wm  =  —  'Wp,  so  that  ^rm  and  typ  are  equal 
at  any  pair  of  corresponding  points  upon  the  two  sides.  The  values  of 
d^fm\dx,  dfa/dx  are  then  opposite,  and  the  remaining  condition  is  also 
satisfied  if 

d^Jdx  =  ik,       d^p/dx  =  -ik  ....................  (21) 

Thus  Wm  is  to  be  such  as  to  make  d-^rmjdx  =  ik  ;  and,  as  in  the  proof  of  (17), 
it  is  easy  to  show  that  in  (19) 

¥ro  =  ^m/27T,  %  =  -^/27T,      ...................  (22) 

where  T/rm,  -fyp  are  the  (equal)  surface-  values  at  dS. 

When  all  the  dimensions  of  8  are  small  in  comparison  with  the  wave- 
length, (19)  in  its  application  to  points  at  a  sufficient  distance  from  S 
assumes  the  form 


and  it  only  remains  to  find  what  is  the  value  of  ftypdS  which  corresponds 
to  d-dx  =  -  ik. 


1897]  IN  PLANE  SCREENS,  AND  ALLIED  PROBLEMS.  287 

Now  this  correspondence  is  ultimately  the  same  as  if  we  were  dealing 
with  an  absolutely  incompressible  fluid.  If  we  imagine  a  rigid  and  infinitely 
thin  plate  (having  the  form  of  the  aperture)  to  move  normally  through 
unlimited  fluid  with  velocity  u,  the  condition  is  satisfied  that  over  the  re- 
mainder of  the  plane  the  velocity-potential  ty  vanishes.  In  this  case  the 
values  of  ^r  at  corresponding  points  upon  the  two  sides  are  opposite  ;  but 
if  we  limit  our  attention  to  the  positive  side,  the  conditions  are  the  same 
as  in  the  present  problem.  The  kinetic  energy  of  the  motion  is  proportional 
to  uz,  and  we  will  suppose  that  twice  the  energy  upon  one  side  is  hitf.  By 
Green's  theorem  this  is  equal  to  —ff^.d-^ldn.dS,  or  -uffydS;  so  that 
//•^rdS=  —  hu.  In  the  present  application  u  =  —  ik,  so  that  the  corresponding 
value  of  ffypdS  is  ihk.  Thus  (23)  becomes 


The  same  algebraic  expression  gives  -\|rm,  if  the  minus  sign  be  omitted;  for 
as  x  itself  changes  sign  in  passing  from  one  side  to  the  other,  the  values  of 
tym  and  typ  at  corresponding  points  are  then  equal. 

The  value  of  h  can  be  determined  in  certain  cases.     For  a  circle*  of 
radius  c 


(26) 


so  that  for  a  circular  aperture  the  realized  solution  is 

^  =  -~  ~cos(nt-kr),  ..............................  (27) 

<£m  =  2  sinnt  smkx  +  ~    —  cos(nt-kr)  .............  (28) 

oA,        T 

It  will  be  remarked  that  while  in  the  first  problem  the  wave  (ty)  divergent 
from  the  aperture  is  proportional  to  the  first  power  of  the  linear  dimension, 
in  the  present  case  the  amplitude  is  very  much  less,  being  proportional  to 
the  cube  of  that  quantity. 

The  solution  for  an  elliptic  aperture  is  deducible  from  the  general  theory 
of  the  motion  of  an  ellipsoid  (a,  b,  c)  through  incompressible  fluid  f,  by 
supposing  a  =  0,  while  b  and  c  remain  finite  and  unequal;  but  the  general 
expression  does  not  appear  to  have  been  worked  out.  When  the  eccentricity 
of  the  residual  ellipse  is  small,  I  find  that 

A  =  $(te)l  (!-&*).  ...........................  (29) 

showing  that  the  effect  of  moderate  ellipticity  is  very  small  when  the  area 
is  given. 

*  Lamb's  Hydrodynamics,  §  105. 
t  Loc.  ««.,  §  111. 


288  ON    THE   PASSAGE   OF   WAVES   THROUGH   APERTURES  [227 

From  the  solutions  already  obtained  it  is  possible  to  derive  others  by 
differentiation.  If,  for  example,  we  take  the  value  of  </>  in  the  first  problem 
and  differentiate  it  with  respect  to  x,  we  obtain  a  function  which  satisfies  (2), 
which  includes  plane  waves  and  their  reflexion  on  the  negative  side,  and 
which  satisfies  over  the  wall  the  condition  of  evanescence.  It  would  seem 
at  first  sight  as  if  this  could  be  no  other  than  the  solution  of  the  second 
problem,  but  the  manner  in  which  the  linear  dimension  of  the  aperture 
enters  suffices  to  show  that  it  is  not  so.  The  fact  is  that  although  the 
proposed  function  vanishes  over  the  plane  part  of  the  wall,  it  becomes  in- 
finite at  the  edge,  and  thus  includes  the  action  of  sources  there  distributed. 
A  similar  remark  applies  to  the  solutions  that  might  be  obtained  by  differen- 
tiation of  the  second  solution  with  respect  to  y  or  z,  the  coordinates  measured 
parallel  to  the  plane  of  the  screen. 


ing  Plate.  —  d<f>/dn  =  0. 

We  now  pass  to  the  consideration  of  allied  problems  in  which  the  trans- 
parent and  opaque  parts  of  the  screen  are  interchanged.  Under  the  above- 
written  boundary  condition  the  case  is  that  of  plane  aerial  waves  incident 
upon  a  parallel  infinitely  thin  plate,  whose  dimensions  are  ultimately  sup- 
posed to  be  small  in  comparison  with  \.  The  analytical  process  of  solution 
may  be  illustrated  by  the  following  argument.  Suppose  a  motion  commu- 
nicated to  the  plate  identical  with  that  which  the  air  at  that  place  would 
execute  were  the  plate  absent.  It  is  evident  that  the  propagation  of  the 
primary  wave  will  then  be  undisturbed.  The  supplementary  solution,  re- 
presenting the  disturbance  due  to  the  plate,  must  then  correspond  to  the 
reduction  of  the  plate  to  rest,  that  is  to  a  motion  of  the  plate  equal  and 
opposite  to  that  just  imagined.  The  supplementary  solution  is  accordingly 
analogous  to  that  which  occurs  in  the  second  of  the  problems  already 
treated. 

Using  a  similar  notation,  we  have  for  the  principal  solution  upon  the 
two  sides 

Xm  =  Xp  =  e-i**>  ..............................  (30) 

giving  when  x  =  0 


The  supplementary  solution  is  of  the  form  (19),  and  gives  upon  the  aperture, 
viz.  the  part  of  the  plane  x  =  0  unoccupied  by  the  plate,  -ty-m  =  ^jrp  —  0,  and 
so  does  not  disturb  the  continuity  of  </>.  But  in  order  that  the  continuity 
of  d(j>/dx  may  be  maintained  it  is  necessary  that  '<Pp  =  '(irm;  and  then  the 


1897]  IN  PLANE  SCREENS,  AND  ALLIED  PROBLEMS.  289 

values  of  ^rm  and  typ  are  opposite  at  any  pair  of  corresponding  points  upon 
the  two  sides. 

It    remains    to    satisfy    the    necessary    conditions    at    the    plate    itself. 
These  are 


, 

dx        dx  dx        dx 

or,  since  d-^rm/dx,  d-$p/dx  are  equal, 

d+m/dx  =  d+p/dx  =  ik.   ........................  (31) 

It  follows  that  typ  has  the  opposite  value  to  that  expressed  in  (25)  ;  and  the 
realized  solution  for  a  circular  plate  of  radius  c  becomes 

<j>p  =  cos  (nt  —  kx)  +  -—2    -cos(nt  —  kr),   ..............  (32) 


^cos(nt-kr),   (33) 

the  analytical  form  being  the  same  in  the  two  cases. 

It  is  important  to  notice  that  the  reflexion  from  the  plate  is  utterly 
different  from  the  transmission  by  a  corresponding  aperture  in  an  opaque 
screen,  as  given  in  (14),  the  former  varying  as  the  cube  of  the  linear 
dimension,  and  the  latter  as  the  first  power  simply. 

Reflecting  Plate.— <j>  =  0. 

For  the  sake  of  completeness  it  may  be  well  to  indicate  the  solution  of 
a  fourth  problem  defined  by  the  above  heading.  This  has  an  affinity  with 
the  first  problem,  analogous  to  that  of  the  third  with  the  second.  The  form 
of  %  is  the  same  as  in  (30),  and  those  for  -v/rTO,  typ  the  same  as  in  (7).  These 
make  d^m/dx,  d^pjdx  vanish  on  the  aperture,  and  so  do  not  disturb  the 
continuity  of  d(f>/dx.  But  in  order  that  the  continuity  of  <f>  may  also  be 
maintained,  we  must  have  ^n  =  ^p,  and  not  as  in  (9)  Vm  =  —  %.  On  the 
plate  itself  we  must  have 

Accordingly  ^m  is  the  same  as  in  (12),  while  typ  in  (12)  must  have  its 
sign  reversed.     The  realized  solution  is 


»..,«,(,.- fa) -Jf- (34) 


19 


290  ON   THE    PASSAGE   OF   WAVES   THROUGH    APERTURES  [227 


Two-dimensional   Vibrations. 

In  the  class  of  problems  before  us  the  velocity-potential  of  a  point- 
source,  viz.  e~{kr/r,  is  replaced  by  that  of  a  linear  source;  and  this  in 
general  is  much  more  complicated.  If  we  denote  it  by  D(kr),  the  ex- 
pressions are* 


2"    '  22.42 

+  ^-^^  +  ^-^£3- ,   (35) 

where  7  is  Euler's  constant  ('5772...),  and 


Of  these  the  first  is  "  semiconvergent,"  and  is  applicable  when  kr  is  large  ; 
the  second  is  fully  convergent  and  gives  the  form  of  the  function  when 
kr  is  small. 

Since  the  complete  analytical  theory  is  rather  complicated,  it  may  be 
convenient  to  give  a  comparatively  simple  derivation  of  the  extreme  forms, 
which  includes  all  that  is  required  for  our  present  purpose,  starting  from 
the  conception  of  a  linear  source  as  composed  of  distributed  point-sources. 
If  p  be  the  distance  of  any  element  dx  of  the  linear  source  from  0,  the 
point  at  which  the  potential  is  to  be  estimated,  and  r  be  the  smallest 
value  of  p,  so  that  p2  =  r2  +  x2,  we  may  take  as  the  potential,  constant 
factors  being  omitted, 


(36) 


We  have  now  to  trace  the  form  of  (36)  when  kr  is  very  great,  and  also 
when  kr  is  very  small.  For  the  former  case  we  replace  p  by  r  +  y,  thus 
obtaining 


When  kr  is  very  great,  the  approximate  value  of  the  integral  in  (37)  may  be 
obtained  by  neglecting  the  variation  of  V(2r  +  y),  since  on  account  of  the 
rapid  fluctuation  of  sign  caused  by  the  factor  er^  we  need  attend  only  to 
small  values  of  y.  Now,  as  is  known, 


rcos  x  dx  _  r°°  sin  x  dx  _      //TT\ 
V*      ~L       V«      "Y  W§ 


*  See  for  example  Theory  of  Sound,  §  341. 


1897]  IN   PLANE   SCREENS,   AND   ALLIED   PROBLEMS.  291 

so  that  in  the  limit 


in  agreement  with  (35). 

We  have  next  to  deduce  the  limiting  form  of  (36)  when  kr  is  very  small. 
For  this  purpose  we  may  write  it  in  the  form 


The  first  integral  in  (39)  is  well  known.     We  have 
=  Ci  (kr)  -  i  {fr  4-  Si  (kr)} 


-_ 


IT 
i-- 


In  the  second  integral  of  (39)  the  function  to  be  integrated  vanishes 
when  p  is  great  compared  to  r,  and  when  p  is  not  great  in  comparison 
with  r,  kp  is  small  and  e~ikf  may  be  identified  with  unity.  Thus  in  the 
limit 


and  (39)  becomes 

^r  =  ry  4.  bg  kr  +  \ITT  -  log  2  =  7  +  log  (^ikr),  (40) 

in  agreement  with  (35). 

When  kr  is  extremely  small  (40)  may  be  considered  for  some  purposes 
to  reduce  to  log  kr ;  but  the  term  |tV  is  required  in  order  to  represent  the 
equality  of  work  done  in  the  neighbourhood  of  the  linear  source  and  at 
a  great  distance  from  it. 

We  may  now  proceed  to  solve  four  problems  relative  to  narrow  slits 
and  reflecting  blades  analogous  to  the  four  already  considered  in  which  the 
aperture  or  the  reflecting  plate  was  small  in  both  its  dimensions  in  com- 
parison with  the  wave-length. 

Narrow  Slit. — Boundary  Condition  d(f)/dn  =  Q. 
As  in  the  former  problem  the  principal  solution  is 


Xp  =  Q,   (41) 

making  dxm/dn,  dxp/dn  vanish  over  the  whole  plane  #  =  0  and  over  the 

19—2 


292  ON   THE   PASSAGE   OF   WAVES   THROUGH    APERTURES  [227 

same  plane  %m  =  2,  %p  =  0.     The  supplementary  solution,  which  represents 
the  effect  of  the  slit,  may  be  written 


(42) 


Vm,  Vp  being  certain  functions  of  y  to  be  determined,  and  the  integration 
extending  over  the  width  of  the  slit  from  y  =  -b  to  y  =  +  b. 

These  additions  do  not  disturb  the  condition  to  be  satisfied  over  the 
wall.  On  the  aperture  continuity  requires,  as  in  (8),  that 

2  +  ym  =  ^f,         d^m/dx  =  d^p/dx. 

The  second  of  these  is  satisfied  by  taking  *Pp  =  —  '*Pmi  so  that  at  all  corre- 
sponding pairs  of  points  ^m  =  —  ^p.  It  remains  to  determine  "9m  so  that  on 
the  aperture  ^frm  =  —  1  ;  and  then  by  what  has  been  said  -^rv  =  +  1. 

At  a  sufficient  distance  from  the  slit,  supposed  to  be  very  narrow,  D  (kr) 
may  be  removed  from  under  the  integral  sign  and  also  be  replaced  by  its 
limiting  form  given  in  (35).  Thus 


(43> 


The  condition  by  which  "Vm  is  determined  is  that  for  all  points  upon  the 
aperture 


(44) 


where,  since  kr  is   small   throughout,  the   second   limiting   form   given  in 
(35)  may  be  introduced. 

From  the  known  solution  for  the  flow  of  incompressible  fluid  through 
a  slit  in  an  infinite  plane  we  may  infer  that  *¥m  will  be  of  the  form 
A  (b2  —  yz)~*,  where  A  is  some  constant.  Thus  (44)  becomes 


In  this  equation  the  first  integral  is  obviously  independent  of  the  position 

of  the  point  chosen,  and  if  the  form  of  Wm  has  been  rightly  taken  the 

second  integral  must  also  be  independent  of  it.     If  its  coordinate  be  77, 
lying  between  +  6, 


and  must  be  independent  of  77.     This  can  be  verified  without  much  difficulty 


1897]  IN   PLANE   SCREENS,   AND   ALLIED   PROBLEMS.  293 

by  assuming  if  =  b  sin  a,  y  =  bsin0;  but  merely  to  determine  A  in  (45)  it 
suffices  to  consider  the  particular  case  of  tj  =  0.     Here 


so  that  (43)  becomes  ?  + 

From  this  typ  is  derived  by  simply  prefixing  a  negative  sign. 

The  realized  solution  is  obtained  from  (46)  by  omitting  the  imaginary 
part  after  introduction  of  the  suppressed  factor  eint.  If  the  imaginary  part 
of  Iog(£t%6)  be  neglected,  the  result  is 


corresponding  to  %m  =  2  cos  n£  cos  kx  ............................  (48) 

The  solution  (47)  applies  directly  to  aerial  vibrations  incident  upon  a  per- 
forated wall,  and  to  an  electrical  problem  which  will  be  specified  later. 
Perhaps  the  most  remarkable  feature  of  it  is  the  very  limited  dependence 
of  the  transmitted  vibration  on  the  width  (26)  of  the  aperture. 


Narrow  Slit.  —  Boundary  Condition  <f>  =  Q. 

The  principal  solution  is  the  same  as  in  (18)  ;  and  the  conditions  for  the 
supplementary  solution,  to  be  satisfied  over  the  aperture,  are  those  expressed 
in  (21).  In  place  of  (19) 


the  values  of  ¥m  and  %  being  opposite,  and  those  of  -»/rm  and  ^  equal  at 
corresponding  points.     At  a  distance  we  have 

*-£/>.*  ..........................  (») 

i  •  i  dD     ikx  /  TT  \*     .^  /KI\ 

in  which  -J-=—(^r)  e      ...........................  (51) 

dx        r    \2ikrJ 

There  is  a  simple  relation  between  the  value  of  Vp  at  any  point  of  the 
aperture  and  that  of  ^p  at  the  same  point.     For  in  the  application  of  (49) 


294  ON  THE  PASSAGE  OF  WAVES  THROUGH   APERTURES  [227 


to  any  point  of  the  narrow  aperture,  dDldx  =  x/r2,  showing  that  only  those 
elements  of  the  integral  are  sensible  which  lie  infinitely  near  the  point 
where  ^rp  is  to  be  estimated.  The  evaluation  is  effected  by  considering 
in  the  first  instance  a  point  for  which  x  is  finite,  and  afterwards  passing 
to  the  limit.  Thus 


so  that  (50)  becomes  fa  =       jg:if       ..........................  (52) 

It  remains  only  to  express  the  connexion  between  f^pdy  and  the  constant 
value  of  d^-p/dx  on  the  area  of  the  aperture  ;  and  this  is  effected  by  the 
known  solution  for  an  incompressible  fluid  moving  under  similar  conditions. 
The  argument  is  the  same  as  in  the  corresponding  problem  where  the 
perforation  is  circular.  In  the  motion  (a)  of  a  lamina  of  width  (26)  through 
infinite  fluid,  the  whole  kinetic  energy  per  unit  of  length  may  be  denoted 
by  hu2,  and  it  appears  from  Green's  theorem  that  ftypdy  =  ihk.  The  value 
of  h*  is  ^7r&2;  so  that 


(53) 


The  same  algebraical  expression  gives  tym,  if  the  minus  sign  be  omitted. 
The  realized  solution  from  (53)  is 

)*  «•<*-*-  W  ...............  (54) 

corresponding  to  >%m  =  2  sin  nt  sin  kx  ............................  (55) 

Reflecting  Blade.  —  Boundary  Condition  d<f>/dn  =  0. 

We  have  now  to  consider  two  problems  which  differ  from  the  last  in 
that  the  opaque  and  transparent  parts  of  the  screen  are  interchanged.  As 
in  the  case  of  the  circular  aperture,  we  shall  find  that  the  correspondence 
lies  between  the  reflecting  blade  under  the  condition  d(f>/dn  =  Q  and  the 
transmitting  aperture  under  the  condition  </>  =  0,  and  reciprocally. 

The  principal  solution  remains  as  in  (30).  The  supplementary  solution 
must  satisfy  (31),  where 


and  *PP  must  be  equal  in  order  that  the  continuity  of  d(j>/dx  over 
*  Lamb's  Hydrodynamics,  §  71. 


1897]  IN   PLANE   SCREENS,   AND   ALLIED   PROBLEMS.  295 

the  aperture  may  be  maintained.     Thus  i/rm  and  ^rp  have  opposite  values 
at  any  pair  of  corresponding  points. 

If  we  compare  these  conditions  with  those  by  which  (53)  was  determined, 
we  see  that  i/rm  has  the  same  value  as  in  that  case,  but  that  the  sign  of  ^p 
must  be  reversed.  Thus  in  the  present  problem 


corresponding  to  %m  =  %p  =  cos  (nt  —  kx)  ..........................  (58) 


Reflecting  Blade.  —  Boundary  Condition  <£  =  0. 

In  this  case  %  still  remains  as  in  (30).  The  general  forms  for  tym,  -^p 
are  as  in  (42),  which  secure  that  d^m/dx,  d-fy-pjdx  shall  vanish  on  the 
aperture  (i.e.  the  part  of  the  plane  #  =  0  unoccupied  by  the  blade).  But 
in  order  that  the  continuity  of  </>  may  also  be  maintained  over  that  area 
we  must  have  ^fm  =  ^rp.  Thus  •^rm,  -*frp  have  equal  values  at  corresponding 
points.  On  the  blade  itself  •\lrm  =  ^p  =  —  1. 

A  comparison  of  these  conditions  with  those  by  which  (46)  was  deter- 
mined shows  that  in  the  present  case 


When  log  i  in  the  denominator  of  (59)  may  be  omitted,  the  realized  form  is 
that  expressed  by  (47),  and  this  corresponds  to 

X™  =  XP  -  cos  (nt  ~  &*)  ..........................  (60) 


Various  Applications. 

Of  the  eight  problems,  whose  solutions  have  now  been  given,  four  have 
an  immediate  application  to  aerial  vibrations,  viz.  those  in  which  the  con- 
dition on  the  walls  is  d$/dn  =  0.  The  symbol  <£  then  denotes  the  velocity- 
potential,  and  the  condition  expresses  simply  that  the  fluid  does  not  penetrate 
the  boundary.  The '  four  problems  relating  to  two  dimensions  have  also 
a  direct  application  to  electrical  vibrations,  if  we  suppose  that  the  thin 
material  constituting  the  screen  (or  the  blade)  is  a  perfect  conductor.  For 
if  R  denote  the  electromotive  intensity  parallel  to  z,  the  condition  at  the 
face  of  the  conductor  is  £  =  0;  so  that  if  R  be  written  for  ^  in  (53),  (59), 
we  have  the  solutions  for  a  narrow  aperture  in  an  infinite  screen,  and  for 
a  narrow  reflecting  blade  respectively,  corresponding  to  the  incident  wave 


296  ON   THE   PASSAGE   OF   WAVES   THROUGH    APERTURES,   ETC.  [227 

R  =  e-*x.  A  narrow  aperture  parallel  to  the  electric  vibrations  transmits 
very  much  less  than  is  reflected  by  a  conductor  elongated  in  the  same 
direction. 

The  two  other  solutions  relative  to  two  dimensions  find  electrical  appli- 
cation if  we  identify  <f>  with  c,  the  component  of  magnetic  intensity  parallel 
to  z.  For  when  the  other  components  a  and  b  are  zero,  the  condition  to 
be  satisfied  at  the  face  of  a  conductor  is  dc/dn  =  0.  Thus  (46),  (57)  apply 
to  incident  vibrations  represented  by  c  =  e~ikx.  In  this  case  the  slit  transmits 
much  more  than  the  blade  reflects. 

It  may  be  remarked  that  in  general  problems  of  electrical  vibration  in 
two  dimensions  have  simple  acoustical  analogues*.  As  an  example  we  may 
refer  to  the  reflexion  of  plane  electric  waves  incident  perpendicularly  upon 
a  corrugated  surface,  the  acoustical  analogue  of  which  is  treated  in  Theory 
of  Sound,  2nd  ed.  §  272  a,  and  to  the  reflexion  of  electric  waves  from  a  con- 
ducting cylinder  (§  343). 

*  The  comparison  is  not  limited  to  the  case  of  perfect  conductors,  but  applies  also  when  the 
obstacles,  being  non-conductors,  differ  from  the  surrounding  medium  in  specific  inductive  capacity, 
or  in  magnetic  permeability,  or  in  both  properties. 


228. 


THE  LIMITS   OF  AUDITION. 


[Royal  Institution  Proceedings,  xv.  pp.  417—418,  1897.] 

IN  order  to  be  audible,  sounds  must  be  restricted  to  a  certain  range  of 
pitch.  Thus  a  sound  from  a  hydrogen  flame  vibrating  in  a  large  resonator 
was  inaudible,  as  being  too  low  in  pitch.  On  the  other  side,  a  bird-call, 
giving  about  20,000  vibrations  per  second,  was  inaudible,  although  a  sensitive 
flame  readily  gave  evidence  of  the  vibrations  and  permitted  the  wave-length 
to  be  measured.  Near  the  limit  of  hearing  the  ear  is  very  rapidly  fatigued ; 
a  sound  in  the  first  instance  loud  enough  to  be  disagreeable,  disappearing 
after  a  few  seconds.  A  momentary  intermission,  due,  for  example,  to  a  rapid 
passage  of  the  hand  past  the  ear,  again  allows  the  sound  to  be  heard. 

The  magnitude  of  vibration  necessary  for  audition  at  a  favourable  pitch 
is  an  important  subject  for  investigation.  The  earliest  estimate  is  that  of 
Boltzmann.  An  easy  road  to  a  superior  limit  is  to  find  the  amount  of  energy 
required  to  blow  a  whistle  and  the  distance  to  which  the  sound  can  be  heard 
(e.g.  one-half  a  mile).  Experiments  upon  this  plan  gave  for  the  amplitude 
8  x  10~8  cm.,  a  distance  which  would  need  to  be  multiplied  100  times  in  order 
to  make  it  visible  in  any  possible  microscope.  Better  results  may  be  obtained 
by  using  a  vibrating  fork  as  a  source  of  sound.  The  energy  resident  in  the 
fork  at  any  time  may  be  deduced  from  the  amplitude  as  observed  under 
a  microscope.  From  this  the  rate  at  which  energy  is  emitted  follows  when 
we  know  the  rate  at  which  the  vibrations  of  the  fork  die  down  (say  to  one- 
half).  In  this  way  the  distance  of  audibility  may  be  reduced  to  30  metres, 
and  the  results  are  less  liable  to  be  disturbed  by  atmospheric  irregularities. 
If  s  be  the  proportional  condensation  in  the  waves  which  are  just  capable  of 
exciting  audition,  the  results  may  be  expressed : — 


frequency  =  256 


:512 


=  6'OxlO-9 
=  4-6x  10-9 


4-6x  10-9 


298  THE   LIMITS   OF   AUDITION.  [228 

showing  that  the  ear  is  capable  of  recognising  vibrations  which  involve  far 
less  changes  of  pressure  than  the  total  pressure  outstanding  in  our  highest 


In  such  experiments  the  whole  energy  emitted  is  very  small,  and  contrasts 
strangely  with  the  60  horse-power  thrown  into  the  fog-signals  of  the  Trinity 
House.  If  we  calculate  according  to  the  law  of  inverse  squares  how  far 
a  sound  absorbing  60  horse-power  should  be  audible,  the  answer  is  2700  kilo- 
metres !  The  conclusion  plainly  follows  that  there  is  some  important  source 
of  loss  beyond  the  mere  diffusion  over  a  larger  surface.  Many  years  ago 
Sir  George  Stokes  calculated  the  effect  of  radiation  upon  the  propagation 
of  sound.  His  conclusion  may  be  thus  stated.  The  amplitude  of  sound 
propagated  in  plane  waves  would  fall  to  half  its  value  in  six  times  the  interval 
of  time  occupied  by  a  mass  of  air  heated  above  its  surroundings  in  cooling 
through  half  the  excess  of  temperature.  There  appear  to  be  no  data  by 
which  the  latter  interval  can  be  fixed  with  any  approach  to  precision ;  but  if 
we  take  it  at  one  minute,  the  conclusion  is  that  sound  would  be  propagated 
for  six  minutes,  or  travel  over  about  seventy  miles,  without  very  serious  loss 
from  this  cause. 

The  real  reason  for  the  falling  off  at  great  distances  is  doubtless  to  be 
found  principally  in  atmospheric  refraction  due  to  variation  of  temperature, 
and  of  wind,  with  height.  In  a  normal  state  of  things  the  air  is  cooler  over- 
head, sound  is  propagated  more  slowly,  and  a  wave  is  tilted  up  so  as  to 
pass  over  the  head  of  an  observer  at  a  distance.  [Illustrated  by  a  model.] 
The  theory  of  these  effects  has  been  given  by  Stokes  and  Reynolds,  and  their 
application  to  the  explanation  of  the  vagaries  of  fog-signals  by  Henry. 
Progress  would  be  promoted  by  a  better  knowledge  of  what  is  passing  in 
the  atmosphere  over  our  heads. 

The  lecture  concluded  with  an  account  of  the  observations  of  Preyer  upon 
the  delicacy  of  pitch  perception,  and  of  the  results  of  Kohlrausch  upon  the 
estimation  of  pitch  when  the  total  number  of  vibrations  is  small.  In  illustra- 
tion of  the  latter  subject  an  experiment  (after  Lodge)  was  shown,  in  which 
the  sound  was  due  to  the  oscillating  discharge  of  a  Leyden  battery  through 
coils  of  insulated  wire.  Observation  of  the  spark  proved  that  the  total 
number  of  (aerial)  vibrations  was  four  or  five.  The  effect  upon  the  pitch 
of  moving  one  of  the  coils  so  as  to  vary  the  self-induction  was  very  apparent. 


229. 


ON  THE  MEASUREMENT  OF  ALTERNATE  CURRENTS  BY 
MEANS  OF  AN  OBLIQUELY  SITUATED  GALVANOMETER 
NEEDLE,  WITH  A  METHOD  OF  DETERMINING  THE  ANGLE 
OF  LAG. 

[Philosophical  Magazine,  XLIII.  pp.  343 — 349,  1897.] 

IT  is  many  years*  since,  as  the  result  of  some  experiments  upon  induction, 
I  proposed  a  soft  iron  needle  for  use  with  alternate  currents  in  place  of  the 
permanently  magnetized  steel  needle  ordinarily  employed  in  the  galvanometer 
for  the  measurement  of  steady  currents.  An  instrument  of  this  kind  designed 
for  telephonic  currents  has  since  been  constructed  by  Giltay ;  but,  so  far  as 
I  am  aware,  no  application  has  been  made  of  it  to  measurements  upon  a  large 
scale,  although  the  principle  of  alternately  reversed  magnetism  is  the  founda- 
tion of  several  successful  commercial  instruments. 

The  theory  of  the  behaviour  of  an  elongated  needle  is  sufficiently  simple, 
so  long  as  it  can  be  assumed  that  the  magnetism  is  made  up  of  two  parts, 
one  of  which  is  constant  and  the  other  proportional  to  the  magnetizing  force. 
If  internal  induced  currents  can  be  neglected,  this  assumption  may  be 
regarded  as  legitimate  so  long  as  the  forces  are  small  f.  In  the  ordinary  case 
of  alternate  currents,  where  upon  the  whole  there  is  no  transfer  of  electricity 
in  either  direction,  the  constant  part  of  the  magnetism  has  no  effect ;  while 
the  variable  part  gives  rise  to  a  deflecting  couple  proportional  on  the  one 
hand  to  the  mean  value  of  the  square  of  the  magnetizing  force  or  current, 
and  upon  the  other  to  the  sine  of  twice  the  angle  between  the  direction  of 
the  force  and  the  length  of  the  needle.  The  deflecting  couple  is  thus 
evanescent  when  the  needle  stands  either  parallel  or  perpendicular  to  the 
magnetizing  force,  and  rises  to  a  maximum  at  the  angle  of  45°.  For  practical 

*  Brit.  Assoc.  Report,  1868;  Phil.  Mag.  Vol.  in.  p.  43  (1887).     [Vol.  i.  p.  310.] 
t  Phil.  Mag.  Vol.  xxur.  p.  225  (1887).     [Vol.  n.  p.  579.] 


300 


ON  THE   MEASUREMENT   OF   ALTERNATE   CURRENTS   BY   MEANS 


[229 


purposes  the  law  of  proportionality  to  the  mean  square  of  current  would 
seem  to  be  trustworthy  so  long  as  no  great  change  occurs  in  the  frequency 
or  type  of  current ;  otherwise  eddy  currents  in  the  iron  might  lead  to  error, 
unless  the  metal  were  finely  subdivided. 

It  is  hardly  to  be  supposed  that  for  ordinary  purposes  a  suspended 
iron  needle  would  compete  in  convenience  with  the  excellent  instruments 
now  generally  available ;  but  having  found  it  suitable  for  a  special  purpose 
of  my  own,  I  think  it  may  be  worth  while  to  draw  to  it  the  attention  of  those 
interested.  In  experiments  upon  the  oxidation  of  nitrogen  by  the  electric 
arc  or  flame  it  was  desired  to  ascertain  the  relation  between  the  electric 
power  absorbed  and  the  amount  of  nitrogen  oxidized.  A  transformer  with 
an  unclosed  magnetic  circuit  was  employed  to  raise  the  potential  from  that 
of  the  supply  to  the  3000  volts  or  more  needed  at  the  platinum  terminals. 
Commercial  ampere-meters  and  volt-meters  gave  with  all  needed  precision 
the  current  and  potential  at  the  primary  of  the  transformer ;  but,  as  is  well 
known,  these  data  do  not  suffice  for  an  estimate  of  power.  The  latter  depends 
also  upon  the  angle  of  lag,  or  retardation  of  current  relatively  to  potential- 
difference.  If  this  angle  be  0,  the  power  actually  employed  is  to  be  found 
by  multiplying  the  product  of  volts  and  amperes  by  cos  0,  so  that  the  actual 
power  may  be  less  to  any  extent  than  the  apparent  power  represented  by 
the  simple  product.  Various  watt-meters  have  been  introduced  for  measuring 
the  actual  power  directly,  but  I  could  not  hear  of  one  suitable  for  the  large 
current  of  40  amperes  used  at  the  Royal  Institution.  Working  subsequently 
in  the  country  I  returned  to  the  problem,  and  succeeded  in  determining  the 
angle  of  lag  very  easily  by  means  of  the  principle  now  to  be  explained. 

The  soft  iron  needle  of  2  centim.  in  length,  suspended  by  a  fine  torsion- 
fibre  of  glass  and  carrying  a  mirror  in  the  usual  way,  is  inclined  at  45°  to 
the  direction  of  the  magnetic  force.  This  force  is  due  to  currents  in  two  coils, 
the  common  axis  of  the  coils  being  horizontal  and  passing  through  the  centre 
of  the  needle.  As  in  ordinary  galvanometers,  the  mean  plane  of  each  coil 
may  include  the  centre  of  the  needle ;  but  it  was  found  better  to  dispose  the 


coils  on  opposite  sides  and  at  distances  from  the  needle  which  could  be  varied. 
A  plan  of  the  arrangement  is  sketched  diagrammatically  in  the  woodcut, 


1897]  OF   AN   OBLIQUELY   SITUATED    GALVANOMETER   NEEDLE.  301 

where  MM,  SS  represent  the  two  coils,  the  common  axis  HK  passing  through 
the  centre  of  the  needle  37.  If  the  currents  in  the  coils  are  of  the  same 
frequency  and  of  simple  type,  the  magnetizing  forces  along  HK  may  be 
denoted  by  A  cos  nt,  B  cos  (nt  —  e),  e  being  the  phase-difference.  If  either 
force  act  alone,  the  deflecting  couple  is  represented  by  .A2  or  by  B2 ;  but  if 
the  two  forces  cooperate  the  corresponding  effect  is 

(72  =  A2  +  B2  +  2AB  cos  e,     (1) 

reducing  itself  to  (A  +  J5)2  or  (A  —  B}z  only  in  the  cases  where  e  is  zero  or  two 
right  angles.  The  method  consists  in  measuring  upon  any  common  scale  all 
the  three  quantities  A*,  B2,  and  (72,  from  which  e  can  be  deduced  by  trigono- 
metrical tables,  or  more  simply  in  many  cases  by  constructing  the  triangle 
whose  sides  are  A,  B,  and  C.  The  determination  of  the  phase-difference 
between  the  currents  is  thus  independent  of  any  measurement  of  their 
absolute  values. 

The  best  method  of  estimating  the  deflecting  couples  may  depend  upon 
the  circumstances  of  the  particular  case.  The  most  accurate  in  principle  is 
the  restoration  of  the  needle  to  the  zero  position  by  means  of  a  torsion-head. 
But  when  the  conditions  are  so  arranged  that  the  angular  deflexions  are 
moderate,  it  will  usually  suffice  merely  to  read  them,  either  objectively  by 
a  spot  of  light  thrown  upon  a  scale,  or  by  means  of  a  telescope.  In  any  case 
where  it  may  be  desired  to  push  the  deflexions  beyond  the  region  where  the 
law  of  proportionality  can  be  relied  upon,  all  risk  of  error  may  be  avoided  by 
comparison  with  another  instrument  of  trustworthy  calibration,  one  coil  only 
of  the  soft  iron  apparatus  being  employed. 

In  certain  cases  the  advantages  which  accompany  the  restoration  of  the 
zero  position  of  the  needle  may  be  secured  by  causing  the  deflexions  them- 
selves to  assume  a  constant  value,  e.g.  by  making  known  changes  of  resistance 
in  one  or  both  of  the  circuits,  or  by  motion  of  the  coils  altering  their 
efficiencies  in  a  known  ratio. 

In  the  particular  experiments  for  which  the  apparatus  was  set  up  the 
coil  MM  (see  woodcut)  was  reduced  to  a  single  turn  of  about  17  centim. 
diameter  and  conveyed  the  main  current  (about  10  amperes)  which  traversed 
the  primary  circuit  of  the  transformer.  This,  it  may  be  mentioned,  was 
a  home-made  instrument,  somewhat  of  the  Ruhmkorff  type,  and  was  placed 
at  a  sufficient  distance  from  the  measuring  apparatus.  The  shunt-coil  SS 
was  of  somewhat  less  diameter,  and  contained  32  convolutions.  The  shunt- 
circuit  included  also  two  electric  lamps,  joined  in  series,  and  its  terminals 
were  connected  with  two  points  of  the  main  circuit  outside  the  apparatus, 
where  the  difference  of  potentials  was  about  40  volts.  Provision  was  made 
for  diverting  the  main  current  at  pleasure  from  MM,  and  by  means  of  a  re- 
verser  the  direction  of  the  current  in  SS  could  be  altered,  equivalent  to 


302  ON    THE   MEASUREMENT   OF    ALTERNATE    CURRENTS    BY    MEANS         [229 

a  change  of  e  by  180°.  The  measurements  to  be  made  are  the  effects  of  MM 
and  of  88  acting  separately,  and  of  MM  and  88  acting  together  in  one  or 
both  positions  of  the  reverser. 

The  best  arrangement  of  the  details  of  observation  will  depend  somewhat 
upon  the  particular  value  of  e  to  be  dealt  with.  If  this  be  60°,  or  there- 
abouts, the  method  can  be  applied  with  peculiar  advantage.  For  by  pre- 
liminary adjustment  of  the  coils,  if  movable,  or  by  inclusion  of  (unknown) 
resistance  in  the  shunt-circuit,  the  deflexions  due  to  MM  and  SS  may  be 
made  equal  to  one  another  ;  so  that  in  the  case  supposed  the  same  deflexion 
will  ensue  from  the  simultaneous  action  of  the  two  currents  in  one  of  the 
ways  in  which  they  may  be  combined. 

This  condition  of  things  was  somewhat  approached  in  the  actual  measures 
relating  to  the  electric  flame.  Thus  in  one  trial  the  coils  were  adjusted  so 
as  to  make  the  deflexions,  due  to  each  of  the  currents  acting  singly,  equal 
to  one  another.  The  value  was  40  divisions  of  the  scale.  When  both  currents 
were  turned  on,  the  deflexion  was  26  J  divisions.  Thus 


whence  cose  =  '67,         or  e  =  48°. 

In  a  second  experiment  the  deflexion  due  to  both  currents  acting  together 
was  made  equal  to  that  of  the  main  acting  alone.     Here 


whence  cos  e  =  '665. 

The  accuracy  was  limited  by  the  unsteadiness  of  the  electric  flame  and  of  the 
primary  currents  (from  a  gas-driven  De  Meritens)  rather  than  by  want  of 
delicacy  in  the  measuring  apparatus. 

When  the  phase-difference  is  about  a  quarter  of  a  period,  cose  is  small, 
and  its  value  is  best  found  by  observing  the  effect  of  reversing  the  shunt- 
current  while  the  main  current  continues  running.  The  difference  is  4>AB  cos  e, 
from  which,  combined  with  a  knowledge  of  A  and  B,  the  value  of  cos  e  is  ad- 
vantageously derived.  If  cos  e  is  absolutely  zero,  the  reversal  does  not  alter 
the  reading. 

If  the  currents  are  in  the  same,  or  in  opposite  phases,  it  is  possible  to 
reduce  the  joint  effect  to  zero  by  suitable  adjustment  of  the  coils  or  of  the 
shunt  resistance. 

The  application  of  principal  interest  is  when  the  shunt-current  may  be 
assumed  to  have  the  same  phase  as  the  potential-difference  at  its  terminals, 
for  then  cos  e  is  the  factor  by  which  the  true  watts  may  be  derived  from  the 
apparent  watts.  We  will  presently  consider  the  question  of  the  negligibility 


1897]  OF   AN    OBLIQUELY   SITUATED   GALVANOMETER   NEEDLE.  303 

of  the  self-induction  of  the  shunt-current,  but  before  proceeding  to  this  it 
may  be  well  to  show  the  application  of  the  formulae  when  the  currents  deviate 
from  the  sine  type. 

If  a  be  the  instantaneous  current,  and  v  the  instantaneous  potential- 
difference  at  the  terminals,  the  work  done  is  fav  dt.  The  readings  of  the  soft 
iron  galvanometer  for  either  current  alone  may  be  represented  by 

A2  =  k2fa2dt,        B'*  =  k*fv*dt,    .............  ........  (2) 

where  h,  k  are  constants  depending  upon  the  disposition  of  the  apparatus. 
When  both  currents  act,  we  have  the  readings 

Cj2  or  C2z  =  j(ha±kvJ2dt  ..........  ..................  (3) 

Taking  the  first  alternative,  we  find 

C?  =  h?ja?dt  +  Zhkfavdt  +  k*Jv*dt, 


The  fraction  on  the  right  of  (4)  is  the  ratio  of  true  and  apparent  watts  ;  and 
we  see  that,  whether  the  currents  follow  the  sine  law  or  not,  the  ratio  is  given 
by  cos  e,  where,  as  before,  e  is  the  angle  of  the  triangle  constructed  with  sides 
proportional  to  the  square  roots  of  the  three  readings. 

Another  formula  for  cos  e  is 


In  the  final  formula  (4)  the  factors  of  efficiency  of  the  separate  coils  (h,  k) 
do  not  enter.     This  result  depends,  however,  upon  the  fulfilment  of  the  con- 
dition of  parallelism  between  the  two  coils.     If  the  magnetic  forces  due  to 
the  coils  be  inclined  at  different  angles  %,  %'  to  the  length  of  the  needle,  we 
have  in  place  of  (3), 

C2  =  /  (a  cos  x  +  v  cos  %')  (a  sin  %  +  v  sin  %')  dt 

=/  [|  a2  sin  2%  +  $v2  sin  2^'  +  av  sin  (^  +  x')]  dt  ;    ......  (6) 

while  ^2  =  £sin2x/a2d£,         Bt=±sm2tffiPdt  ................  (7) 

Accordingly 

/  av  dt          _  C*-A*-B*  V  {sin  2#  .  sin  2^'} 
{Jra^dtx~fv^df\^~        2AB  sin(x  +  x')       '     .........  (  ' 

in  which  the  second  fraction  on  the  right  represents  the  influence  of  the 
defect  in  parallelism.  If  ^  and  %'  are  both  nearly  equal  to  45°,  then  approxi- 
mately 

V  {sin  2  X.  sin  2^}  _ 

'  *(*     %>  ......................  ( 


304  ON   THE   MEASUREMENT   OF    ALTERNATE   CURRENTS,   ETC.  [229 

We  have  now  to  consider  under  what  conditions  the  shunt-current  may 
be  assumed  to  be  proportional  to  the  instantaneous  value  of  the  potential- 
difference  at  its  terminals.  The  obstacles  are  principally  the  self-induction  of 
the  shunt-coil  itself,  and  the  mutual  induction  between  it  and  the  coil  which 
conveys  the  main  current.  As  to  the  former,  we  know*  that  if  the  mean 
radius  of  a  coil  be  a,  and  if  the  section  be  circular  of  radius  c,  and  if  n  be  the 
number  of  convolutions, 

"*T-i} <10> 

To  take  an  example  from  the  shunt-coil  used  in  the  experiments  above 
referred  to,  where 

a  =  6  cm.,       c  =  1  cm.,       n  =  32, 

L  is  of  the  order  105cm.  The  time-constant  of  the  shunt-circuit  (T)  is  equal 
to  L/R,  where  R  is  the  resistance  in  C.G.S.  measure.  If  r  be  the  resistance 
measured  in  ohms,  R  =  r  x  109,  so  that 

1 

~  r  x  104 ' 

What  we  are  concerned  with  is  the  ratio  of  T  to  the  period  of  the  currents ; 
if  the  latter  be  y^  second,  the  ratio  is  l/100r,  so  that  if  r  be  a  good  number 
of  ohms — it  must  have  exceeded  100  in  the  particular  experiments — there  is 
nothing  to  fear  from  self-induction.  It  would  seem  to  follow  generally  that 
if  the  voltage  be  not  too  small,  say  not  falling  below  10  volts,  there  should  be 
no  difficulty  in  obtaining  sufficient  effect  from  a  shunt-coil  whose  self-induction 
may  be  neglected.  It  may  be  remarked  that  since  the  efficiency  of  the  coil 
varies  as  n,  while  L  varies  as  n2,  it  will  be  advantageous  to  keep  n  (and  r) 
down  so  long  as  the  self-induction  of  the  whole  shunt-circuit  is  mainly  that 
of  the  coil. 

If  the  main  and  the  shunt-coils  were  wound  closely  together,  the  disturb- 
ance due  to  mutual  induction  would  be  of  the  same  order  of  magnitude  as 
that  due  to  self-induction.  If  the  coils  are  separated,  as  is  otherwise  con- 
venient, the  influence  of  mutual  induction  will  be  less,  and  may  be  neglected 
under  the  conditions  above  defined. 

As  to  the  effect  of  self-induction,  if  present,  we  know  that  the  lag  (<£)  is 
given  by 

tan  </>  =  Lp  I  R,  (11) 

where  p  =  ZTT  x  frequency.  The  angle  of  lag  of  the  main  current  (6),  which 
it  is  the  object  of  the  measurements  to  determine,  is  then  given  by 

6  =  e  +  <f>,  (12) 

e  being  the  phase-difference  of  the  two  currents  as  found  directly  from  the 
observations. 

*  Maxwell's  Electricity,  §  706. 


230. 


ON  THE  INCIDENCE  OF  AERIAL  AND  ELECTRIC  WAVES 
UPON  SMALL  OBSTACLES  IN  THE  FORM  OF  ELLIPSOIDS 
OR  ELLIPTIC  CYLINDERS,  AND  ON  THE  PASSAGE  OF 
ELECTRIC  WAVES  THROUGH  A  CIRCULAR  APERTURE  IN 
A  CONDUCTING  SCREEN. 

[Philosophical  Magazine,  XLIV.  pp.  28 — 52,  1897.] 

THE  present  paper  may  be  regarded  as  a  development  of  previous 
researches  by  the  author  upon  allied  subjects.  When  the  character  of  the 
obstacle  differs  only  infinitesimally  from  that  of  the  surrounding  medium, 
a  solution  may  be  obtained  independently  of  the  size  and  the  form  which 
it  presents.  But  when  this  limitation  is  disregarded,  when,  for  example, 
in  the  case  of  aerial  vibrations  the  obstacle  is  of  arbitrary  compressibility 
and  density,  or  in  the  case  of  electric  vibrations  when  the  dielectric  constant 
and  the  permeability  are  arbitrary,  the  solutions  hitherto  given  are  confined 
to  the  case  of  small  spheres,  or  circular  cylinders.  In  the  present  investiga- 
tion extension  is  made  to  ellipsoids,  including  flat  circular  disks  and  thin 
blades. 

The  results  arrived  at  are  limiting  values,  strictly  applicable  only  when 
the  dimensions  of  the  obstacles  are  infinitesimal,  and  at  distances  outwards 
which  are  infinitely  great  in  comparison  with  the  wave-length  (X).  The 
method  proceeds  by  considering  in  the  first  instance  what  occurs  in  an  inter- 
mediate region,  where  the  distance  (r)  is  at  once  great  in  comparison  with 
the  dimensions  of  the  obstacle  and  small  in  comparison  with  X.  Throughout 
this  region  and  within  it  the  calculation  proceeds  as  if  A,  were  infinite,  and 
depends  only  upon  the  properties  of  the  common  potential.  When  this 
problem  is  solved,  extension  is  made  without  much  difficulty  to  the  exterior 
region  where  r  is  great  in  comparison  with  X,  and  where  the  common 
potential  no  longer  avails. 

R.    iv.  20 


306  ON   THE   INCIDENCE   OF   AERIAL 

At  the  close  of  the  paper  a  problem  of  some  importance  is  considered 
relative  to  the  escape  of  electric  waves  through  small  circular  apertures 
in  metallic  screens.  The  case  of  narrow  elongated  slits  has  already  been 
treated*. 


Obstacle  in  a  Uniform  Field. 

The  analytical  problem  with  which  we  commence  is  the  same  whether 
the  flow  be  thermal,  electric,  or  magnetic,  the  obstacle  differing  from  the 
surrounding  medium  in  conductivity,  specific  inductive  capacity,  or  per- 
meability respectively.  If  <f>  denote  its  potential,  the  uniform  field  is 

defined  by 

wz;     ...........................  (1) 


u,  v,  w  being  the  fluxes  in  the  direction  of  fixed,  arbitrarily  chosen,  rectangular 
axes.  If  i/r  be  the  potential  in  the  uniform  medium  due  to  the  obstacle,  so 
that  the  complete  potential  is  </>  -I-  ^r,  ^  may  be  expanded  in  the  series  of 
spherical  harmonics 


the  origin  of  r  being  within  the  obstacle.  Since  there  is  no  source,  S0 
vanishes.  Further,  at  a  great  distance  S2)S3,...  maybe  neglected,  so  that 
ir  there  reduces  to 


The  disturbance  (3)  corresponds  to  (1).     If  we  express  separately  the 
parts  corresponding  to  u,  v,  lu,  writing  A'  =  Alu  +  A2v  +  A3w,  &c.,  we  have 

r3^  =  u  (AlX  +  B,y  +  C.z)  +  v  (A2a;  +  B2y  +  C2z)  +  w  (A3x  +  B3y  +  C3z)  ; 

.........  (4) 

but  the  nine  coefficients  are  not  independent.  By  the  law  of  reciprocity  the 
coefficient  of  the  #-part  due  to  v  must  be  the  same  as  that  of  the  y-part  due 
to  u,  and  so  on-f*.  Thus  B±  =  A2,  &c.,  and  we  may  write  (4)  in  the  form 

dF       dF        dF 
^  +  =  Udx+Vdy+Wdz>  ........................  (5> 

where  F=^A1a?  +  \E^  +  $C3z2  +  B^xy  +  C2yz  +  G&x  ..........  (6) 

In  the  case  of  a  body,  like  an  ellipsoid,  symmetrical  with  respect  to  three 
planes  chosen  as  coordinate  planes, 

B,  =  C2  =  0,  =  0, 

*  Phil.  Mag.  Vol.  xmi.  p.  272.    [Vol.  iv.  p.  295.] 

t  Theory  of  Sound,  §  109.     u  and  v  may  be  supposed  to  be  due  to  point-sources  situated  at  a 
great  distance  B  along  the  axes  of  x  and  y  respectively. 


1897]  AND   ELECTRIC   WAVES    UPON   SMALL   OBSTACLES.  307 

and  (4)  reduces  to 


(7) 

It  will  now  be  shown  that  by  a  suitable  choice  of  coordinates  this  reduc- 
tion may  be  effected  in  any  case.  Let  u,  v,  w  originate  in  a  source  at  distance 
R,  whose  coordinates  are  x',  y',  /,  so  that  u  =  x'jR3,  &c.  Then  (5)  becomes 

•i  rr  -I-TI  jrr 

^^  =  X>  dx  +  y'  dy  +  Z>  dz  =  AlXX'  +  E*yy'  +  °3ZZ' 
+  B,  (x'y  +  y'x}  +  C2  (y'z  +  z'y}  +  C,  (z'x  +  x'z) 
=  F(x  +  x',  y  +  y',  z  +  /)  -  F  (x,  y,z}-F  (x',  y',  /). 

Now  by  a  suitable  transformation  of  coordinates  F(x,  y,  z),  and  therefore 
F  (x,  y',  z')  and  F  (x  +  x',  y  +  y',  z  +  /),  may  be  reduced  to  the  form 

A^  +  B2y2  +  C,z*,   &c. 

If  this  be  done, 

r^R3^  =  A^xx  +  B2yy  +  C3zz', 

or  reverting  to  u,  v,  w,  reckoned  parallel  to  the  new  axes, 

r3^  =  A^ux  +  B^vy+  C3ivz,  ........................  (8) 

as  in  (7)  for  the  ellipsoid.  It  should  be  observed  that  this  reduction  of  the 
potential  at  a  distance  from  the  obstacle  to  the  form  (8)  is  independent  of 
the  question  whether  the  material  composing  the  obstacle  is  uniform. 

For  the  case  of  the  ellipsoid  (a,  b,  c)  of  uniform  quality  the  solution  may 
be  completely  carried  out.  Thus*,  if  T  be  the  volume,  so  that 

T=%7rabc,     .................................  (9) 

we  have  AlU  =  -AT,      B,v  =  -  BT,      G3w^-GT,     ............  (10) 


(1 


where  L  =  ^abcQ   ^^--—^     ............  (12) 

with  similar  expressions  for  M  and  N. 

In  (11)  K  denotes  the  susceptibility  to  magnetization.  In  terms  of  the 
permeability  /*,  analogous  to  conductivity  in  the  allied  problems,  we  have,  if 
/jf  relate  to  the  ellipsoid  and  n  to  the  surrounding  medium, 

(13) 


so  that 


with  similar  equations  for  B  and  C. 

*  The  magnetic  problem  is  considered  in  Maxwell's  Electricity  and  Magnetism,  1873,  §  437, 
and  in  Mascart's  Lecons,  1896,  §§  52,  53,  276. 

20—2 


308  ON  THE  INCIDENCE  OF  AERIAL  [230 

Two    extreme   cases   are   worthy   of  especial    notice.     If  /////*  =  oo ,    the 
general  equation  for  ty  becomes 


r^r     ux     vy     wz 

T     L+M+N- 


On  the  other  hand,  if  /*'  ///.  =  0, 

r3-^         ux  vy  wz 


In  the  case  of  the  sphere  (a) 

L  =  M=N  =  $7r; 
so  that  (15)  becomes 

^=-^(wc  +  vy  +  wz),      .....................  (18) 

giving,  when  r  =  a,  0  +  ^  =  0.     This  is  the  case  of  the  perfect  conductor. 
In  like  manner  for  the  non-conducting  sphere  (16)  gives 

*  =  j^(ux  +  vy  +  wz)  .........................  (19) 

If  the  conductivity  of  the  sphere  be  finite  (//), 


which  includes  (18)  and  (19)  as  particular  cases. 

If  the  ellipsoid  has  two  axes  equal,  and  is  of  the  planetary  or  flattened 
form, 

6  =  c  =       a  r  =  f7rcV(l-*2);    ............  (21) 


(22) 
(23) 


In  the  extreme  case  of  a  disk,  when  e  =  1  nearly. 

L  =  47r  -  27rV(l  -  e2),     ........................  (24) 

M=N  =  -n*</(l-el)  .........................  (25) 

Thus  in  the  limit  from  (14),  (21)  TA  =  0,  unless  //  =  0  ;  and  when  /*'  =  0, 


In  like  manner  the  limiting  values  of  TB,  TO  are  zero,  unless  //  =  oo  , 
and  then 

(27) 


1897]  AND   ELECTRIC   WAVES   UPON   SMALL  OBSTACLES.  309 

In  all  cases 

t,_n*?+a>+ca (28) 

gives  the  disturbance  due  to  the  ellipsoid. 

If  the  ellipsoid  of  revolution  be  of  .the  ovary  or  elongated  form, 

tt  =  6  =  0^(1-0;    (29) 


In  the  case  of  a  very  elongated  ovoid  L  and  M  approximate  to  the  value  2?r, 
while  N  approximates  to  the  form 


(32) 
vanishing  when  e  =  1. 

/w  Two  Dimensions. 

The  case  of  an  elliptical  cylinder  in  two  dimensions  may  be  deduced  from 
(12)  by  making  c  infinite,  when  the  integration  is  readily  effected.     We  find 

T      4nrb  M      4?ra 

L  =  -  7,          M  =  ---  , 
a  +  b  a+b 

A  and  B  are  then  given  by  (14)  as  before,  and  finally 

_      ab  (a  +  6)  (  (//  -p)ux     (//  -  /*) 


corresponding  to 

(35) 


In  the  case  of  circular  section  L  =  M  =  2?r,  so  that 


When  b  =  0,  that  is  when  the  obstacle  reduces  itself  to  an  infinitely  thin 
blade,  ty  vanishes  unless  /*'  =  0  or  //  =  oo  .     In  the  first  case 

0.-.0)        +-^;     ........................  (37) 

in  the  second 

.  ,  a?ux 

(/-oo)        ^r=-  —  ......................  (38) 

*  There  are  slight  errors  in  the  values  of  L,  M,  N  recorded  for  this  case  in  both  the  works 
cited. 


310  ON   THE   INCIDENCE   OF   AERIAL  [230 

Aerial   Waves. 

We  may  now  proceed  to  investigate  the  disturbance  of  plane  aerial  waves 
by  obstacles  whose  largest  diameter  is  small  in  comparison  with  the  wave- 
length (X).  The  volume  occupied  by  the  obstacle  will  be  denoted  by  T  ;  as 
to  its  shape  we  shall  at  first  impose  ho  restriction  beyond  the  exclusion  of 
very  special  cases,  such  as  would  involve  resonance  in  spite  of  the  small 
dimensions.  The  compressibilities  and  densities  of  the  medium  and  of  the 
obstacle  are  denoted  by  m,  m'  ;  a-,  a  ;  so  that  if  V,  V  be  the  velocities  of 
propagation 

F2  =  m/o-,        V*=m'/<r'  .........................  (39) 

The  velocity-potential  of  the  undisturbed  plane  waves  is  represented  by 

$  =  eikvt.eik*,    ..............................  (40) 

in  which  k  =  27r/\.    The  time  factor  eikvt,  which  operates  throughout,  may  be 
omitted  for  the  sake  of  brevity. 

The  velocity-potential  (T/T)  of  the  disturbance  propagated  outwards  from 
T  may  be  expanded  in  spherical  harmonic  terms  * 

+  ...},     ............  (41) 


n(n  +  l)  ,  (n-l). 
where  /.  <0;r)  =  1  +  -^^  +       2    4 

+  ......  +'2.4.6...2n(tfcr)»  ................  (42) 

At  a  great  distance  from  the  obstacle  fn(ikr}  =  1;  and  the  relative  importance 
of  the  various  harmonic  terms  decreases  in  going  outwards  with  the  order  of 
the  harmonic.  For  the  present  purpose  we  shall  need  to  regard  only  the 
terms  of  order  0  and  1.  Of  these  the  term  of  order  0  depends  upon  the 
variation  of  compressibility,  and  that  of  order  1  upon  the  variation  of  density. 

The  relation  between  the  variable  part  of  the  pressure  Bp,  the  conden- 
sation s,  and  0  is 

rw-t-SE; 

at       a 

so  that  during  the  passage  of  the  undisturbed  primary  waves  the  rate  at 
which  fluid  enters  the  volume  T  (supposed  for  the  moment  to  be  of  the  same 
quality  as  the  surrounding  medium)  is 


If  the  obstacle  present  an  unyielding  surface,  its  effect  is  to  prevent  the 
entrance  of  the  fluid  (43)  ;  that  is,  to  superpose  upon  the  plane  waves  such  a 

*  Theory  of  Sound,  §§  323,  324. 


1897]  AND   ELECTRIC   WAVES   UPON   SMALL   OBSTACLES.  311 

disturbance  as  is  caused  by  the  introduction  of  (43)  into  the  medium.     Thus, 
if  the  potential  of  this  disturbance  be 

^  =  8^,     ..............................  (44) 

S0  is  to  be  determined  by  the  condition  that  when  r  —  0 


so  that  £0=  -  k*TI4nr,  and 


This  result  corresponds  with  m'  =  oo  representing  absolute  incompressibility. 
The  effect  of  finite  compressibility,  differing  from  that  of  the  surrounding 
medium,  is  readily  inferred  by  means  of  the  pressure  relation  (8p  =  ms).  The 
effect  of  the  variation  of  compressibility  at  the  obstacle  is  to  increase  the  rate 
of  introduction  of  fluid  into  T  from  what  it  would  otherwise  be  in  the  ratio 
m  :  m'  ;  and  thus  (45)  now  becomes 


or  if  we  restore  the  factor  eikvt  and  throw  away  the  imaginary  part  of  the 
solution, 

TT  L  tn  —  m        ,  ,-.r        ,  /  1)-,\ 

•f  =  --       —  —  cos  k  (  Vt  -  r)  ...................  (47) 

A/r       m 

This  is  superposed  upon  the  primary  waves 

x)  ............................  (48) 


When  m  =  0,  i.e.,  when  the  material  composing  the  obstacle  offers  no 
resistance  to  compression,  (47)  fails.  In  this  case  the  condition  to  be  satisfied 
at  the  surface  of  T  is  the  evanescence  of  Bp,  or  of  the  total  potential  (<£  +  ^). 
In  the  neighbourhood  of  the  obstacle  <£  =  1  ;  and  thus  if  M  '  denote  the 
electrical  "  capacity  "  of  a  conducting  body  of  form  T  situated  in  the  open, 
•fr  =  —M'/r,  r  being  supposed  to  be  large  in  comparison  with  the  linear 
dimension  of  T  but  small  in  comparison  with  X.  The  latter  restriction  is 
removed  by  the  insertion  of  the  factor  e~ikr  ;  and  thus,  in  place  of  (46).  we 
now  have 

t—  ^  ............................  (49) 

The  value  of  M'  may  be  expressed  when  T  is  in  the  form  of  an  ellipsoid. 
For  a  sphere  of  radius  R, 

M'=R;    .................................  (50) 

for  a  circular  plate  of  radius  R, 

M'  =  2RlTr  ...............................  (51) 


312  ON  THE  INCIDENCE  OF  AERIAL  [230 

When  the  density  of  the  obstacle  (a-')  is  the  same  as  that  of  the  sur- 
rounding medium,  (47)  constitutes  the  complete  solution.  Otherwise  the 
difference  of  densities  causes  an  interference  with  the  flow  of  fluid,  giving 
rise  to  a  disturbance  of  order  1  in  spherical  harmonics.  This  disturbance  is 
independent  of  that  already  considered,  and  the  flow  in  the  neighbourhood  of 
the  obstacle  may  be  calculated  as  if  the  fluid  were  incompressible.  We  thus 
fall  back  upon  the  problem  considered  in  the  earlier  part  of  this  paper,  and 
the  results  will  be  applicable  as  soon  as  we  have  established  the  corre- 
spondence between  density  and  conductivity. 

In  the  present  problem,  if  ^  denote  the  whole  velocity-potential,  the 
conditions  to  be  satisfied  at  any  part  of  the  surface  of  the  obstacle  are  the 
continuity  of  d%/dn  and  of  <r%,  the  latter  of  which  represents  the  pressure. 
Thus,  if  we  regard  <T%  as  the  variable,  the  conditions  are  the  continuity  of 
(o"x)  and  of  a-~l  d  (ay)  I  dn.  In  the  conductivity  problem  the  conditions  to  be 
satisfied  by  the  potential  (^')  are  the  continuity  of  ^'  and  of  ^d-^jdn. 

In  an  expression  relating  only  to  the  external  region  where  <r  is  constant, 
it  makes  no  difference  whether  we  are  dealing  with  o-%  or  with  ^;  and 
accordingly  there  is  correspondence  between  the  two  problems  provided  that 
we  suppose  the  ratio  of  //,'s  in  the  one  problem  to  be  the  reciprocal  of  the 
ratio  of  the  cr's  in  the  other. 

We  may  now  proceed  to  the  calculation  of  the  disturbance  due  to  an 
obstacle,  based  upon  the  assumption  that  there  is  a  region  over  which  r  is 
large  compared  with  the  linear  dimension  of  T,  but  small  in  comparison 
with  X.  Within  this  region  i/r  is  given  by  (8)  if  the  motion  be  referred  to 
certain  principal  axes  determined  by  the  nature  and  form  of  the  obstacle,  the 
quantities  u,  v,  w  being  the  components  of  flow  in  the  primary  waves.  By 
(41),  (42),  this  is  to  be  identified  with 

p-ikr  t  1    x 

+  =  &—  (l  +  ~},     (52) 

r    \        ikr) 


when  r  is  small  in  comparison  with  \  ;  so  that 


C3wz)  .„„. 

—  ...................  (oo) 


At  a  great  distance  from  T,  (52)  reduces  to 

^  =  ik  (A,ux  +  Bzvy  +  C,wz)  e~ikr  ( 

—a  term  of  order  1,  to  be  added  to  that  of  zero  order  given  in  (46). 

In  general,  the  axis  of  the  harmonic  in  (54)  is  inclined  to  the  direction  of 
propagation  of  the  primary  waves  ;  but  there  are  certain  cases  of  exception. 
For  example,  v  and  w  vanish  if  the  primary  propagation  be  parallel  to  #  (one 


1897]  AND   ELECTRIC   WAVES   UPON    SMALL   OBSTACLES.  313 

of  the  principal  axes).     Again,  as  for  a  sphere  or  a  cube,  A1}  Bz,  G3  may 
be  equal. 

We  will  now  limit  ourselves  to  the  case  of  the  ellipsoid,  and  for  brevity 
will  further  suppose  that  the  primary  waves  move  parallel  to  x,  so  that 
v  —  w  —  0.  The  terms  corresponding  to  u  and  v,  if  existent,  are  simply 
superposed.  If,  as  hitherto,  <J>  =  eikx,  u  =  ik;  so  that  by  (14),  a  being  sub- 
stituted for  //  and  a'  for  /A, 


In  the  intermediate   region  by  (28)  ^  =  —  TAxfr*,  and   thus   at   a   great 
distance 


(56) 

or  on  substitution  of  the  values  of  A  and  k, 


to(<r'-<r) 
X2r2       to<r'  +  (<r-<r')L'  ' 

Equations  (46),  (57)  express  the  complete  solution  in  the  case  supposed. 

For  an  obstacle  which  is  rigid  and  fixed,  we  may  deduce  the  result  by 
supposing  in  our  equations  m'  =  oo  ,  a-'  =  oo  .     Thus 


Certain  particular  cases  are  worthy  of  notice.    For  the  sphere  L  =  |TT,  and 


If  the  ellipsoid  reduce  to  an  infinitely  thin  circular  disk  of  radius  c,  T  =  0 
and  the  term  of  zero  order  vanishes.  The  term  of  the  first  order  also 
vanishes  if  the  plane  of  the  disk  be  parallel  to  x.  If  the  plane  of  the  disk  be 
perpendicular  to  x,  4nr  —  L  is  infinitesimal.  By  (21),  (24)  we  get  in  this  case 

toT    _8c3. 
4>7T-L~  3  ' 


(60) 


If  the  axis  of  the  disk  be  inclined  to  that  of  x,  -fy  retains  its  symmetry 
with  respect  to  the  former  axis,  and  is  reduced  in  magnitude  in  the  ratio  of 
the  cosine  of  the  angle  of  inclination  to  unity. 

In  the  case  of  the  sphere  the  general  solution  is 


*** 


Theory  of  Sound,  §  334.  t  L  c.  cit.  §  335. 


314  ON   THE   INCIDENCE   OF   AERIAL  [230 

Waves  in  Two  Dimensions. 

In  the  case  of  two  dimensions  (x,  y)  the  waves  diverging  from  a  cylindrical 
obstacle  have  the  expression,  analogous  to  (41), 


1(kr)+...,      ..................  (62) 

where  S0>  Sl  ...  are  the  plane  circular  functions  of  the  various  orders,  and 


3   A^r4 

+...,     ......  (63) 


d(kr)    ~  \2kr 


As  in  the  case  of  three  dimensions  already  considered,  the  term  of  zero 
order  in  -ty  depends  upon  the  variation  of  compressibility.  If  we  again  begin 
with  the  case  of  an  unyielding  boundary,  the  constant  S0  is  to  be  found  from 
the  condition  that  when  r  =  0 


T  denoting  now  the  area  of  cross-section.     When  r  is  small, 

dP0  (kr)  _  1 
dr      ~r' 

and  thus  S0  =  k2T/2-n; 


................  (65) 

when  r  is  very  great.     This  corresponds  to  (45). 

In  like  manner,  if  the  compressibility  of  the  obstacle  be  finite, 


The  factor  i~*  =  e~*iir  ;  and  thus  if  we  restore  the  time-factor  e*7*,  and  reject 
the  imaginary  part  of  the  solution,  we  have 


—  /Tr,  .  ^. 

2src08T<F'-r-*x)'  ...............  (67) 

See  Theory  of  Sound,  §  341  ;  Phil.  Mag.  April,  1897,  p.  266.   [Vol.  iv.  p.  290.] 


1897]  AND   ELECTRIC   WAVES   UPON   SMALL   OBSTACLES.  315 

corresponding  to  the  plane  waves 

............................  (68) 


In  considering  the  term  of  the  first  order  we  will  limit  ourselves  to  the 
case  of  the  cylinder  of  elliptic  section,  and  suppose  that  one  of  the  principal 
axes  of  the  ellipse  is  parallel  to  the  direction  (#)  of  primary  wave-propagation. 
Thus  in  (34),  which  gives  the  value  of  -»/r  at  a  distance  from  the  cylinder 
which  is  great  in  comparison  with  a  and  b,  but  small  in  comparison  with  X, 
we  are  to  suppose  u  =  ik,  v  =  0,  at  the  same  time  substituting  a,  a'  for  ///, 
fi  respectively.  Thus  for  the  region  in  question 

,ab.ikx    o-'-o-a  +  fr). 


and  this  is  to  be  identified  with  8lD1(kr)  when  kr  is  small,  i.e.  with  Sl/kr. 
Accordingly 

o  _x  ik2ab  (a  -  <r)  (a  +  6) 
r      2  a-'a  +  <rb 

so  that,  at  a  distance  r  great  in  comparison  with  X,  -fy  becomes 

'--  b)  x 

&      > 


T  being  written  for  trab.  The  complete  solution  for  a  great  distance  is  given 
by  addition  of  (66)  and  (70),  and  corresponds  to  <£  =  €**•*. 

In  the  case  of  circular  section  (b  =  a)  we  have  altogether  * 

+  =-k*a*e-*r  (^  K-^  +  ^  *1  ,  ...(71) 

\2ikrJ    (    2m         <r  +  cr  r) 

which  may  be  realized  as  in  (67).  If  the  material  be  unyielding,  the  corre- 
sponding result  is  obtained  by  making  m'  =  oo  ,  </  =  oo  in  (71).  The  realized 
value  is  then  f 


(72) 


In  general,  if  the  material  be  unyielding,  we  get  from  (66),  (70) 

(73) 


The  most  interesting  case  of  a  difference  between  a  and  b  is  when  one  of 
them  vanishes,  so  that  the  cylinder  reduces  to  an  infinitely  thin  blade.     If 

*  Theory  of  Sound,  §  343. 
t  Loc.  cit.  equation  (17). 


316  ON   THE    INCIDENCE   OF   AERIAL  [230 

b  =  0,  i/r  vanishes  as  to  both  its  parts ;  but  if  a  —  0,  although  the  term  of  zero 
order  vanishes,  that  of  the  first  order  remains  finite,  and  we  have 


(74) 


in  agreement  with  the  value  formerly  obtained*. 

It  remains  to  consider  the  extreme  case  which  arises  when  m'  =  0.  The 
term  of  zero  order  in  circular  harmonics,  as  given  in  (66),  then  becomes 
infinite,  and  that  of  the  first  order  (70)  is  relatively  negligible.  The  con- 
dition to  be  satisfied  at  the  surface  of  the  obstacle  is  now  the  evanescence  of 
the  total  potential  (<£  +--V/T),  in  which  <£  =  1. 

It  will  conduce  to  clearness  to  take  first  the  case  of  the  circular  cylinder 
(a).  By  (62),  (63)  the  surface  condition  is 

S0{y  +  \og($ika)}  +  l=Q  .........................  (75) 

Thus  at  a  distance  r  great  in  comparison  with  A,  we  have 

(76) 


When  the  section  of  the  obstacle  is  other  than  circular,  a  less  direct 
process  must  be  followed.  Let  us  consider  a  circle  of  radius  p  concentric 
with  the  obstacle,  where  p  is  large  in  comparison  with  the  dimensions  of  the 
obstacle  but  small  in  comparison  with  X.  Within  this  circle  the  flow  may  be 
identified  with  that  of  an  incompressible  fluid.  On  the  circle  we  have 

(77) 
(78) 

of  which  the  latter  expresses  the  flow  of  fluid  across  the  circumference.  This 
flow  in  the  region  between  the  circle  and  the  obstacle  corresponds  to  the 
potential-difference  (77).  Thus,  if  R  denote  the  electrical  resistance  between 
the  two  surfaces  (reckoned  of  course  for  unit  length  parallel  to  z), 

S«{7  +  log(W-27r.R}  =  l,  .....................  (79) 

and  ^  =  S0D0  (&r)>  as  usual. 

The  value  of  S0  in  (79)  is  of  course  independent  of  the  actual  value  of  p, 
so  long  as  it  is  large.  If  the  obstacle  be  circular, 


The  problem  of  determining  R  for  an  elliptic  section  (a,  6)  can,  as  is  well 
known,  be  solved  by  the  method  of  conjugate  functions.     If  we  take 

x  —  c  cosh  £  cos  77,          y  =  c  sinh  f  sin  77,   ...............  (80) 

*  Phil.  Mag.  April  1897,  p.  271.     [Vol.  rv.  p.  295.]     The  primary  waves  are  there  supposed  to 
travel  in  the  direction  of  +  x,  but  here  in  the  direction  of  -  x. 


1897]         -  AND   ELECTRIC   WAVES   UPON   SMALL   OBSTACLES.  317 

the  confocal  ellipses 


are  the  equipotential   curves.     One  of  these,  for  which   f  is  large,  can   be 
identified  with  the  circle  of  radius  p,  the  relation  between  p  and  f  being 


An  inner  one,  for  which   £=£„,  is  to  be  identified  with  the  ellipse  (a,  b), 
so  that 

a  =  c  cosh  f0,         b  =  c  sinh  £0, 

whence  c2  =  u?  -  62,         tanh  %0  =  bfa. 

Thus  27r£  =  £-£0  =  log;      ...................  (32) 


and  then  (79)  gives  as  applicable  at  a  great  distance 


—          ,vy 


T  ~_~  i M  »'7, /„  i  JAI  I  2ikr) '***v 


The  result  for  an  infinitely  thin  blade  is  obtained  by  merely  putting  6  =  0 
in  (83). 

For  some  purposes  the  imaginary  part  of  the  logarithmic  term  may  be 
omitted.     The  realized  solution  is  then 


/jr  \*  coBft(Ft-r-fr) 

UtoV    7  +  log  {fk  (a  +  &)}' 


7  +  log  {|fc(a  +  6)} 
corresponding,  as  usual,  to 

<f>  =  cosk(Vt  +  x) (85) 

Electrical  Applications. 

The  problems  in  two  dimensions  for  aerial  waves  incident  upon  an 
obstructing  cylinder  of  small  transverse  dimensions  are  analytically  identical 
with  certain  electric  problems  which  will  now  be  specified.  The  general 
equation  (v2  +  A;2)  =  0  is  satisfied  in  all  cases.  In  the  ordinary  electrical 
notation  V2  =  l/K/j,,  F'2  =  1  jK'p! ;  while  in  the  acoustical  problem  F2  =  ra/<r, 
V'2  =  m'/<r'.  The  boundary  conditions  are  also  of  the  same  general  form. 
Thus  if  the  primary  waves  be  denoted  by  7  =  eikx,  y  being  the  magnetic  force 
parallel  to  z,  the  conditions  to  be  satisfied  at  the  surface  of  the  cylinder  are 
the  continuity  of  7  and  of  K~l  dy/dn.  Comparing  with  the  acoustical 
conditions  we  see  that  K  replaces  or,  and  consequently  (by  the  value  of  F2) 
fj,  replaces  1/w.  These  substitutions  with  that  of  7,  or  c  (the  magnetic 
induction),  for  ^  and  </>  suffice  to  make  (66),  (70)  applicable  to  the  electrical 


318  ON   THE   INCIDENCE   OF   AERIAL  [230 

problem.     For  example,  in  the  case  of  the  circular  cylinder,  we  have  for  the 
dispersed  wave 

'  (86) 


corresponding  to  the  primary  waves 

c  =  eikx  ...................................  (87) 

An  important  particular  case  is  obtained  by  making  K'  =  oo  ,  yu/  =  0,  in 
such  a  way  that  V  remains  finite.  This  is  equivalent  to  endowing  the 
obstacle  with  the  character  of  a  perfect  conductor,  and  we  get 


which,  when  realized,  coincides  with  (72). 

The  other  two-dimensional  electrical  problem  is  that  in  which  everything 
is  expressed  by  means  of  R,  the  electromotive  intensity  parallel  to  z.  The 
conditions  at  the  surface  are  now  the  continuity  of  R  and  of  p^dR/dri. 
Thus  K  and  p  are  simply  interchanged,  /j,  replacing  a  and  K  replacing  1/ra 
in  (66),  (70),  </>  and  i/r  also  being  replaced  by  R.  In  the  case  of  the  circular 
cylinder 

'  (89) 


corresponding  to  the  primary  waves 

R  =  e**  ..................................  (90) 

If  in  order  to  obtain  the  solution  for  a  perfectly  conducting  obstacle  we 
make  K'  =  oo  ,  //  =  0,  (89)  becomes  infinite,  and  must  be  replaced  by  the 
analogue  of  (83).  Thus  for  the  perfectly  conducting  circular  obstacle 


which  may  be  realized  as  in  (84). 

The  problem  of  a  conducting  cylinder  is  treated  by  Prof.  J.  J.  Thomson  in 
his  valuable  Recent  Researches  in  Electricity  and  Magnetism,  §  364  ;  but  his 
result  differs  from  (84),  not  only  in  respect  to  the  sign  of  ^X,  but  also  in  the 
value  of  the  denominator*.  The  values  here  given  are  those  which  follow 
from  the  equations  (9),  (17)  of  §  343  Theory  of  Sound. 

Electric   Waves  in  Three  Dimensions. 

In  the  problems  which  arise  under  this  head  the  simple  acoustical 
analogue  no  longer  suffices,  and  we  must  appeal  to  the  general  electrical 

*  It  should  be  borne  in  mind  that  y  here  is  the  same  as  Prof.  Thomson's  log  y. 


1897]  AND   ELECTRIC   WAVES   UPON   SMALL   ORSTACLES.  319 

equations  of  Maxwell.  The  components  of  electric  polarization  (f,  g,  h)  and 
of  magnetic  force  (a,  /3,  7),  being  proportional  to  eikvt,  all  satisfy  the  funda- 
mental equation 

(V2  +  £2)  =  0.    ..............................  (92) 

and  they  are  connected  together  by  such  relations  as 


da      ,    Tro  fdq      dh\ 
or  —  =  4?rF  *(-/--!-  ),    .......................  (94) 

dt  *  dz      dy) 

in  which  any  differentiation  with  respect  to  t  is  equivalent  to  the  introduction 
of  the  factor  ikV.     Further 

dj,     dk  *«  +  *8  +  £  g 

as     dy     dz  dx      dy      dz 

The  electromotive  intensity  (P,  Q,  R)  and  the  magnetization  (a,  b,  c)  are 
connected  with  the  quantities  already  defined  by  the  relations 

a,  b,  c  =  /*(«,  /S,  7);  .........  (96) 


in  which  K  denotes  the  specific  inductive  capacity  and  /i  the  permeability  ; 
so  that  F~2  =  Kft. 

The  problem  before  us  is  the  investigation  of  the  disturbance  due  to  a 
small  obstacle  (K',  /*')  situated  at  the  origin,  upon  which  impinge  primary 
waves  denoted  by 

/o  =  0,         <7o  =  0,         A,  =  ««*  .....................  (97) 

or,  as  follows  from  (94), 

a0  =  0,         00  =  47rFeto,         7o  =  0  ...................  (98) 

The  method  of  solution,  analogous  to  that  already  several  times  employed, 
depends  upon  the  principle  that  in  the  neighbourhood  of  the  obstacle  and  up 
to  a  distance  from  it  great  in  comparison  with  the  dimensions  of  the  obstacle 
but  small  in  comparison  with  \,  the  condition  at  any  moment  may  be 
identified  with  a  steady  condition  such  as  is  determined  by  the  solution  of  a 
problem  in  conduction.  When  this  is  known,  the  disturbance  at  a  distance 
from  the  obstacle  may  afterwards  be  derived. 

We  will  commence  with  the  case  of  the  sphere,  and  consider  first  the 
magnetic  functions  as  disturbed  by  the  change  of  permeability  from  ^  to  /*'. 
Since  in  the  neighbourhood  of  the  sphere  the  problem  is  one  of  steady 
distribution,  ot,  /3,  7  are  derivable  from  a  potential.  By  (98),  in  which  we 
may  write  eikx  =  1,  the  primary  potential  is  4nrVy;  so  that  in  (1)  we  are  to 
take  u  =  0,  v  =  4-rrF,  w  =  0.  Hence  by  (20)  a,  ft,  7  for  the  disturbance  are 
given  by 


320  ON   THE    INCIDENCE   OF    AERIAL  [230 

where  .'  f__4,ri<      .........................  (99) 


In  like  manner  f,  g,  h  are  derivable  from  a  potential  %.  The  primary 
potential  is  z  simply,  so  that  in  (1),  u  =  0,  v  =  0,  w  =  1.  Hence  by  (20) 

K'-K  a?z 
X  =  -£T^2K^>     ........................  (1< 

from  which  /,  g,  h  for  the  disturbance  are  derived  by  simple  differentiations 
with  respect  to  x,  y,  z  respectively. 

Since  /.  g,  h,  a,  /8,  7  all  satisfy  (92),  the  values  at  a  distance  can  be 
derived  by  means  of  (41).  The  terms  resulting  from  (99),  (100)  are  of  the 
second  order  in  spherical  harmonics.  When  r  is  small, 


and  when  r  is  great 

r-i  e-*r 

so  that,  as  regards  an  harmonic  of  the  second  order,  the  value  at  a  distance 
will  be  deduced  from  that  in  the  neighbourhood  of  the  origin  by  the  intro- 
duction of  the  factor  -  ^kzr2e~ikr.  Thus,  for  example,/  in  the  neighbourhood 
of  the  origin  is 


so  that  at  a  great  distance  we  get 

f__K^K.**^  ......................  (102) 

In  this  way  the  terms  of  the  second  order  in  spherical  harmonics  are  at 
once  obtained,  but  they  do  not  constitute  the  complete  solution  of  the 
problem.  We  have  also  to  consider  the  possible  occurrence  of  terms  of  other 
orders  in  spherical  harmonics.  Terms  of  order  higher  than  the  second  are 
indeed  excluded,  because  in  the  passage  from  r  small  to  r  great  they  suffer 
more  than  do  the  terms  of  the  second  order.  But  for  a  like  reason  it  may 
happen  that  terms  of  order  zero  and  1  in  spherical  harmonics  rise  in  relative 
importance  so  as  to  be  comparable  at  a  distance  with  the  term  of  the  second 
order,  although  relatively  negligible  in  the  neighbourhood  of  the  obstacle. 
The  factor,  analogous  to  —%feir*e~ikr  for  the  second  order,  is  for  the  first  order 
ikre~i}cr,  and  for  zero  order  e~ikr.  Thus,  although  (101)  gives  the  value  of  f 
with  sufficient  completeness  for  the  neighbourhood  of  the  obstacle,  (102)  may 
need  to  be  supplemented  by  terms  of  the  first  and  zero  orders  in  spherical 
harmonics  of  the  same  importance  as  itself.  The  supplementary  terms  may 
be  obtained  without  much  difficulty  from  those  already  arrived  at  by  means 
of  the  relations  (93),  (94),  *(95)  ;  but  the  process  is  rather  cumbrous,  and 


1897]         AND  ELECTRIC  WAVES  UPON  SMALL  OBSTACLES.          321 

it  seems  better  to  avail  ourselves  of  the  forms  deduced  by  Hertz  *  for  electric 
vibrations  radiated  from  a  centre. 

If  we  write  Tl  =  Ae~ikr/r,  the  solution  corresponding  to  an  impressed 
electric  force  acting  at  the  origin  parallel  to  z  is 


(104) 


These  values  evidently  satisfy  (92)  since  H  does  so,  and  they  harmonize  with 
(93),  (94),  (95). 

In  the  neighbourhood  of  the  origin,  where  kr  is  small,  e~ikr  may  be 
identified  with  unity,  so  that  II  =  A  jr.     In  this  case  (103)  may  be  written 

/•__^!E  <M  M 

'         dxdz'        9        dydz'  dz*  ' 

and  all  that  remains  is  to  identify  —  dU/dz  with  ^  in  (100).     Accordingly 

^  =  -a°         .........................  <105> 


The  values  of  f,  g,  h  in  (103)  are  now  determined.  Those  of  a,  /3,  7  are 
relatively  negligible  in  the  neighbourhood  of  the  origin.  At  a  great  distance 
we  have 


f=-A 

J  ~          dxdz  \    r      ~      r    dxdz 

so  that  (103),  (104)  may  be  written 


K'  —  K   k*a?e~ikr  (    xz         yz      a?  +  y2\ 

f>9>h=vr-^r- 7— (-^»  ~^>    -pr-)' <106) 


a, 


y        x        \ 
r'   "r'    °J 


These  equations  give  the  values  of  the  functions  for  a  disturbance 
radiating  from  a  small  spherical  obstacle,  so  far  as  it  depends  upon  (K'  —  K). 
We  have  to  add  a  similar  solution  dependent  upon  the  change  from  /j.  to  ///. 
In  this  (103),  (104)  are  replaced  by 


_,  ___. 

2          *  T      2  '          2  '  ' 


_ 

dxdy  '       F2      dx*       rf*2  '       F2 


*  Ausbreitung  tier  electrischen  Kraft,  Leipzig,  1892,  p.  150.     It  may  be  observed  that  the 
solution  for  the  analogous  but  more  difficult  problem  relating  to  an  elastic  solid  was  given  much 
earlier  by  Stokes  (Camb.  Trans.,  Vol.  ix.  p.  1,  1849).     Compare  Theory  of  Sound,  2nd  ed.  §  378. 
R.     IV.  21 


322  ON  THE  INCIDENCE  OF  AERIAL  [230 

where  H  =  Be~ikrlr,  corresponding  to  an  impressed  magnetic  force  parallel 
to  y.     In  the  neighbourhood  of  the  origin  (108)  becomes 

a          d2H          ft  _     d-Tl          7  _     d*H 
Vz         dy~  '       V*        dzdy ' 


so  that  -f  in  (99)  is  to  be  identified  with  -  V2dU/dy.     Thus 

-.'...'.  ;•      •B=-1f?T^ <110> 

At  a  great  distance  we  have 

...(111) 


a,  ft,  7_  p' -p   t?a3e-ikr(    xy    tf  +  z*     _zy\ 
"4arV  ~  p'  +  ty         r'      \     r*  '       r*  r*  J ' 

By  addition  of  (111)  to  (106)  and  of  (112)  to  (107)  we  obtain  the  com- 
plete values  of  f,  g,  h,  a,  ft,  7  when  both  the  dielectric  constant  and  the 
permeability  undergo  variation.  The  disturbance  corresponding  to  the 
primary  waves  h  =  eikx  is  thus  determined. 

When  the  changes  in  the  electric  constants  are  small,  (106),  (111)  may 
be  written 


(113) 

\      f\.     i~       p.    'i  / 

'  _.,  i    &Kiiz\ 
9  =  ^.e 


where  T=§TTO?,  Ar=27r/X.  These  are  the  results  given  formerly*  as  applic- 
able in  this  case  to  an  obstacle  of  volume  T  and  of  arbitrary  form.  When 
the  obstacle  is  spherical  and  &KJ  K  is  not  small,  it  was  further  shown  that 
&KJK  should  be  replaced  by  3(K'  —  K)/(K'  +  2^T)f,  and  similar  reasoning 
would  have  applied  to  A/A //A. 

The  solution  for  the  case  of  a  spherical  obstacle  having  the  character  of  a 
perfect  conductor  may  be  derived  from  the  general  expressions  by  supposing 
that  K'  =  x ,  and  (in  order  that  V  may  remain  finite)  //  =  0.  We  get 
from  (106),  (111), 

*  "Electromagnetic  Theory  of  Light,"  Phil.  Mag.  Vol.  xii.  p.  90  (1881).    [Vol.  i.  p.  526.} 
t  [1902.     The  "  3  "  was  inadvertently  omitted  in  the  original  of  the  present  paper.] 


1897]  AND   ELECTRIC   WAVES   UPON   SMALL   OBSTACLES.  323 

(nfi) 


in  agreement  with  the  results  of  Prof.  J.  J.  Thomson*.  As  was  to  be 
expected,  in  every  case  the  vectors  (f,  g,  h),  (a,  ft,  7),  (x,  y,  z)  are  mutually 
perpendicular. 

Obstacle  in  the  Form  of  an  Ellipsoid. 

The  case  of  an  ellipsoidal  obstacle  of  volume  T,  whose  principal  axes  are 
parallel  to  those  of  x,  y,  z,  i.  e.  parallel  to  the  directions  of  propagation  and  of 
vibration  in  the  primary  waves,  is  scarcely  more  complicated.  The  passage 
from  the  values  of  the  disturbance  in  the  neighbourhood  of  the  obstacle  to 
that  at  a  great  distance  takes  place  exactly  as  in  .the  case  of  the  sphere. 
The  primary  magnetic  potential  in  the  neighbourhood  of  the  obstacle  is 
4?r  Vy,  and  thus,  as  before,  u  =  0,  v  =  4nrV,  w  =  Q  in  (1).  Accordingly,  by  (14), 
A  =  0,  C  =  0  ;  and  (28)  gives 

'  gy,  ..................  (119) 

3 


r 

47r/u,  +  (fjL  — 

corresponding  to  (99)  for  the  sphere.     In  like  manner  the  electric  potential  is 

—  /i  on\ 

x==~  *7rK  +  (K'-K}N  ^  ' 

These  potentials  give  by  differentiation  the  values  of  a,  /3,  7  and  f,  g,  h 
respectively  in  the  neighbourhood  of  the  ellipsoid.  Thus  at  a  great  distance 
we  obtain  for  the  part  dependent  on  (K1  —  K},  as  generalizations  of  (106), 
(107), 


y     _x 
'          '         ' 


_  __ 

4-TrK   ~±TrK  +  (K'-K)N         r        \r'       r' 

To  these  are  to  be  added  corresponding  terms  dependent  upon  (//—  /i),  viz.:  — 

'-,  0,   -?);    ......  (128) 

r'  rj 


a,  0,  7  =  ^  -n       _  k*Te~ikr  (xy      x>  +  z*     _zy 

4-TrF       4nrfji  +  (»'  -  p)  M        r       \     r*  '        r2  r* 

*  Recent  Researches,  §  377,  1893. 

21—2 


324  ON   THE   INCIDENCE   OF   AERIAL  [230 

The  sum  gives  the  disturbance  at  a  distance  due  to  the  impact  of  the 
primary  waves, 

(125) 


upon  the  ellipsoid  T  of  dielectric  capacity  K'  and  of  permeability  /*'. 

As  in  the  case  of  the  sphere,  the  result  for  an  ellipsoid  of  perfect  conduc- 
tivity is  obtained  by  making  K'  =  oo  ,  /*'  =  0.     Thus 

(T  xz          T 


,        tee-**  (T  xz          T       z\ 

~^(N^  +  4^Mr)'    

.(127) 


Next  to  the  sphere  the  case  of  greatest  interest  is  that  of  a  flat  circular 
disk  (radius  =  R).  The  volume  of  the  obstacle  then  vanishes,  but  the  effect 
remains  finite  in  certain  cases  notwithstanding.  Thus,  if  the  axis  of  the  disk 
be  parallel  to  x,  that  is  to  the  direction  of  primary  propagation,  we  have 

(21),  (25), 

T     4R3  T 


In  spite  of  its  thinness,  the  plate  being  a  perfect  conductor  disturbs  the 
electric  field  in  its  neighbourhood;  but  the  magnetic  disturbance  vanishes, 
the  zero  permeability  having  no  effect  upon  the  magnetic  flow  parallel  to  its 
face.  If  the  axis  of  the  disk  be  parallel  to  y  (see  (24)), 


and  if  the  axis  be  parallel  to  z, 

£-0  *      -0 

N  4>-rr  —  M 

so  that  in  this  case  the  obstacle  produces  no  effect  at  all. 

Circular  Aperture  in  Conducting  Screen. 

The  problem  proposed  is  the  incidence  of  plane  waves  (A0  =  e***)  upon  an 
infinitely  thin  screen  at  x  =  0  endowed  with  perfect  electric  conductivity  and 
perforated  by  a  circular  aperture.  In  the  absence  of  a  perforation  there 
would  of  course  be  no  waves  upon  the  negative  side,  and  upon  the  positive 
side  the  effect  of  the  screen  would  merely  be  to  superpose  the  reflected  waves 
denoted  by  /*0  =  -  e~ikx.  We  wish  to  calculate  the  influence  of  a  small 
circular  aperture  of  radius  R, 


1897]  AND   ELECTRIC   WAVES   UPON   SMALL   OBSTACLES.  325 

In  accordance  with  the  general  principle  the  condition  of  things  is 
determined  by  what  happens  in  the  neighbourhood  of  the  aperture,  and  this 
is  substantially  the  same  as  if  the  wave-length  were  infinite.  The  problem 
is  then  expressible  by  means  of  a  common  potential.  The  magnetic  force  at 
a  distance  from  the  aperture  on  the  positive  side  is  altogether  87rV,  and  on 
the  negative  side  zero ;  while  the  condition  to  be  satisfied  upon  the  faces  of 
the  screen  is  that  the  force  be  entirely  tangential.  The  general  character  of 
the  flow  is  indicated  in  Fig.  1. 

Fig.  1.  Fig.  2.  Fig.  3. 


The  problem  here  proposed  is  closely  connected  with  those  which  we  have 
already  considered  where  no  infinite  screen  was  present,  but  a  flat  finite 
obstacle,  which  may  be  imagined  to  coincide  with  the  proposed  aperture. 
The  primary  magnetic  field  being  /9  =  4>7rV,  and  the  disk  of  radius  R  being  of 
infinite  permeability,  the  potential  at  .  a  distance  great  compared  with  R  (but 
small  compared  with  X)  is  by  (27),  (28) 


(132) 


By  the  symmetry  the  part  of  the  plane  x  =  0  external  to  the  disk  is  not 
crossed  by  the  lines  of  flow,  and  thus  it  will  make  no  difference  in  the 
conditions  if  this  area  be  filled  up  by  a  screen  of  zero  permeability.  On  the 
other  hand,  the  part  of  the  plane  #  =  0  represented  by  the  disk  is  met 
normally  by  the  lines  of  flow.  This  state  of  things  is  indicated  in  Fig.  2. 

The  introduction  of  the  lamina  of  zero  permeability  effects  the  isolation 
of  the  positive  and  negative  sides.  We  may  therefore  now  reverse  the  flow 
upon  the  negative  side,  giving  the  state  of  things  indicated  in  Fig.  3.  But 
the  plate  of  infinite  permeability  then  loses  its  influence  and  may  be  removed, 
so  as  to  re-establish  a  communication  between  the  positive  and  negative  sides 
through  an  aperture.  The  passage  from  the  present  state  of  things  to  that 
of  Fig.  1  is  effected  by  superposition  upon  the  whole  field  of  ft  =  4-TrF,  so  as  to 
destroy  the  field  at  a  distance  from  the  aperture  upon  the  negative  side  and 
upon  the  positive  side  to  double  it. 


326  ON   THE   INCIDENCE   OF   AERIAL    AND   ELECTRIC   WAVES.  [230 

As  regards  the  solution  of  the  proposed  problem  we  have  then  on  the 
positive  side 

and  on  the  negative  side 


Thus  on  the  negative  side  at  a  distance  great  in  comparison  with  the  wave- 
length we  get,  as  in  (99),  (111),  (112), 


-  ^  -? 

On  the  positive  side  these  values  are  to  be  reversed,  and  addition  made  of 
A,  =  eite_<r«*         £0  =  47rF(e'^  +  e-to),      .........  (137) 

representing  the  plane  waves  incident  and  reflected. 

The  solution  for  h  in  (135)  may  be  compared  with  that  obtained  (27),  (28) 
in  a  former  paper*,  where,  however,  the  primary  waves  were  supposed  to 
travel  in  the  positive,  instead  of,  as  here,  in  the  negative  direction.  It  had  at 
first  been  supposed  that  the  solution  for  <£  there  given  might  be  applied 
directly  to  h,  which  satisfies  the  condition  (imposed  upon  <£)  of  vanishing 
upon  the  faces  of  the  screen.  If  this  were  admitted,  as  also  g  =  0  throughout, 
the  value  of  h  would  follow  by  (95).  The  argument  was,  however,  felt  to  be 
insufficient  on  account  of  the  discontinuities  which  occur  at  the  edge  of  the 
aperture,  and  the  value  now  obtained,  though  of  the  same  form,  is  doubly 
as  great. 

*  *'  On  the  Passage  of  Waves  through  Apertures  in  Plane  Screens,  and  Allied  Problems," 
Phil.  Mag.  Vol.  XLIII.  p.  264  (1897).  [Vol.  iv.  p.  287.] 


231. 

r- 

ON   THE   PROPAGATION   OF   ELECTRIC   WAVES   ALONG 
CYLINDRICAL  CONDUCTORS   OF  ANY   SECTION. 

[Philosophical  Magazine,  XLIV.  pp.  199—204,  1897.] 

THE  problem  of  the  propagation  of  waves  along  conductors  has  been 
considered  by  Mr  Heaviside  and  Prof.  J.  J.  Thomson,  for  the  most  part  with 
limitation  to  the  case  of  a  wire  of  circular  section  with  a  coaxal  sheath 
serving  as  a  return.  For  practical  applications  it  is  essential  to  treat  the 
conductivity  of  the  wire  as  finite;  but  for  some  scientific  purposes  the 
conductivity  may  be  supposed  perfect  without  much  loss  of  interest.  Under 
this  condition  the  problem  is  so  much  simplified  that  important  extensions 
may  be  made  in  other  directions.  For  example,  the  complete  solution  may 
be  obtained  for  the  case  of  parallel  wires,  even  although  the  distance  between 
them  be  not  great  in  comparison  with  their  diameters. 

We  may  start  from  the  general  equations  of  Maxwell  involving  the 
electromotive  intensity  (P,  Q,  R)  and  the  magnetic  induction  (a,  b,  c), 
introducing  the  supposition  that  all  the  functions  are  proportional  to  ei{pt+mz}, 
and  further  that  m=p/V,  just  as  in  the  case  of  uninterrupted  plane  waves 
propagated  parallel  to  z.  Accordingly  d*/dt2  =  V*d2/dz2,  and  any  equation 
such  as 

d*P     d*P     d*P      1    d*P 

dtf  +  djf  +  df^T*  ~dr  

fJ2P       d-P 

reduces  to  ^  +  ^  =  ° <2> 

They  may  be  summarized  in  the  form 

+ £)CP,  Q,R,a,b,c)  =  0 (3) 

dx2     dy*J  ^ 


328  ON   THE   PROPAGATION    OF   ELECTRIC   WAVES  [231 

The  case  to  be  here  treated  is  characterized  by  the  conditions  R  =  0,  c  =  0; 
but  it  would  suffice  to  assume  one  of  them,  say  the  latter.  Since  in  general 
throughout  the  dielectric 

dc/dt  =  dP/dy-dQ/da;,      ........................  (4) 

* 

it  follows  that  P  and  Q  are  derivatives  of  a  function  (<£),  also  proportional  to 
ei(pt+mz)^  which  as  a  function  of  x  and  y  may  be  regarded  as  a  potential  since 
it  satisfies  the  form  (2).  Thus  dP/dx  +  dQ/dy  =  Q,  from  which  it  follows 
that  dRjdz  and  R  vanish.  It  will  be  convenient  to  express  all  the  functions 
by  means  of  <f>.  We  have  at  once 

P  =  d(f>/dx,     Q  =  d<J)/dy,     E  =  0  ...................  (5) 

Again,  by  the  general  equation  analogous  to  (4),  since  R  =  0,  ipa  =  imQ  ; 
so  that 

a=V-ld<j>/dy,        b  =  -V-*d$ldx,        c  =  0  .............  (6) 


Thus  the  same  function  0  serves  as  a  potential  for  P,  Q  and  as  a  stream- 
function  for  a,  b. 

The  problem  is  accordingly  reduced  to  dependence  upon  a  simple  potential 
problem  in  two  dimensions.     Throughout  the  dielectric  <f>  satisfies 


(7) 


At  the  boundary  of  a  conductor,  supposed  to  be  perfect,  the  condition  is 
that  the  electromotive  intensity  be  entirely  normal.  So  far  as  regards  the 
component  parallel  to  z  this  is  satisfied  already,  since  R  =  0  throughout. 
The  remaining  condition  is  that  </>  be  constant  over  the  contour  of  any 
continuous  conductor.  This  condition  secures  also  that  the  magnetic  in- 
duction shall  be  exclusively  tangential. 

It  is  to  be  observed  that  R  is  not  equal  to  dffr/dz.  The  former  quantity 
vanishes  throughout,  while  d<j>/dz  remains  finite,  since  <j>  <x  e{  {pt+mz}  .  In- 
asmuch as  <j>  satisfies  Laplace's  equation  in  two  dimensions,  but  not  in  three, 
it  will  be  convenient  to  use  language  applicable  to  two  dimensions,  referring 
the  conductors  to  their  sections  by  the  plane  xy. 

If  a  boundary  of  a  conductor  be  in  the  form  of  a  closed  curve,  the  included 
dielectric  is  incapable  of  any  vibration  of  the  kind  now  under  consideration. 
For  a  function  satisfying  (7)  and  retaining  a  constant  value  over  a  closed 
contour  cannot  deviate  from  that  value  in  the  interior.  Thus  the  derivatives 
of  </>  vanish,  and  there  is  no  disturbance.  The  question  of  dielectric  vibrations 
within  closed  tubes,  when  ra  is  not  limited  to  equality  with  p/V,  was  con- 
sidered in  a  former  paper*. 

*  Phil.  Mag.  Vol.  XLIII.  p.  125  (1897).     [Vol.  iv.  p.  276.] 


1897]  ALONG   CYLINDRICAL   CONDUCTORS   OF   ANY   SECTION.  329 

For  the  case  of  a  dielectric  bounded  by  two  planes  perpendicular  to  x  we 
may  take 


giving  p  =  eHpt+mz)> 


(10) 


in  which,  as  usual,  m—pj  V.  Since  Q  =  0,  R  =  0  throughout,  the  dielectric 
may  be  regarded  as  limited  by  conductors  at  any  planes  (perpendicular  to  x) 
that  may  be  desired. 

If  the  dielectric  be  bounded  by  conductors  in  the  form  of  coaxal  circular 
cylinders,  we  have  the  familiar  wire  with  sheath  return,  first,  I  believe, 
considered  on  the  basis  of  these  equations  by  Mr  Heaviside.  We  may  take, 
with  omission  of  a  constant  addition  to  log  r  which  has  here  no  significance, 


(11) 


gvng  ,,      =  *,.      y-,     o)  ,   ...............  (12) 


(13) 


And  here  again  it  makes  no  difference  to  these  forms  at  what  points  (rl}  r2) 
the  dielectric  is  replaced  by  conductors. 

For  the  moment  these  simple  examples  may  suffice  to  illustrate  the 
manner  in  which  the  propagation  along  z  takes  place,  and  to  show  that  $ 
is  determined  by  conditions  completely  independent  of  p  and  its  associated  m. 
In  further  discussions  it  will  save  much  circumlocution  to  suppose  that  p  and 
m  are  zero  and  thus  to  drop  the  exponential  factor.  The  problem  is  then 
strictly  reduced  to  two  dimensions  and  relates  to  charges  and  steady  currents 
upon  cylindrical  conductors,  the  currents  being  still  entirely  superficial. 
When  <f>  is  once  determined  for  any  case  of  this  kind,  the  exponential  factor 
may  be  restored  at  pleasure  with  an  arbitrary  value  assigned  to  p  and  the 
corresponding  value,  viz.  p/V,  to  m. 

The  usual  expressions  for  electric  and  magnetic  energies  will  then  apply, 
everything  being  reckoned  per  unit  length  parallel  to  z.  It  suffices  for 
practical  purposes  to  limit  ourselves  to  the  case  of  a  single  outgoing  and  a 
single  return  conductor.  We  may  then  write 

Electric  energy    .<&"&!.,  ........................  (14) 

2  x  capacity 

Magnetic  energy  =  |-  x  self-induction  x  (current)2  ;     .........  (15) 

and  the  value  of  the  self-induction  in  the  latter  case  is  the  reciprocal  of  that 
of  the  capacity  in  the  former. 


330  ON   THE   PROPAGATION    OF   ELECTRIC   WAVES  [231 

Thus,  for  a  dielectric  bounded  by  coaxal  conductors  at  r  =  r^  and  r  =  >:„ 
we  have  $  =  log  r,  and 

self-induction  =  (capacity)-1  =  2  log  -  ................  (16) 

Among  the  cases  for  which  the  solution  can  be  completely  effected  may  be 
mentioned  that  of  a  dielectric  bounded  by  confocal  elliptical  cylinders. 

More  important  in  practice  is  the  case  of  parallel  circular  wires.  In 
Lecher's  arrangement,  which  has  been  employed  by  numerous  experimenters, 
the  wires  are  of  equal  diameter  ;  and  it  is  usually  supposed  to  be  necessary  to 
maintain  them  at  a  distance  apart  which  is  very  great  in  comparison  with 
that  diameter.  The  general  theory  above  given  shows  that  there  is  no  need 
for  any  such  restriction,  the  manner  and  velocity  of  propagation  along  the 
length  being  the  same  whatever  may  be  the  character  of  the  cross-section  of 
the  system. 

The  form  of  <f>,  and  the  self-induction  of  the  system,  may  be  determined 
in  this  case,  whatever  may  be  the  radii  (oj,  a.2)  of  the  wires  and  the  distance 
(6)  between  their  centres.  If  rl}  r2  are  the  distances  of  any  point  P  in  the 
plane  from  fixed  points  Oi,  02,  the  equipotential  curves  for  which  <£,  equal  to 
Iog(?*2/r1),  assumes  constant  values  are  a  system  of  circles,  two  of  which  can 
be  identified  with  the  boundaries  of  the  conductors.  The  details  of  the 
investigation,  consisting  mainly  of  the  geometrical  relations  between  the 
ultimate  points  0j,  02  and  the  circles  of  radii  a,,  a,,  are  here  passed  over. 
The  result  for  the  self-induction  per  unit  length  L,  or  for  the  capacity,  may 
be  written* 


As  was  to  be  expected,  L  vanishes  when  6  =  a,  +  a*,  that  is,  when  the 
conductors  are  just  in  contact. 

When  Oj,  a?  are  small  in  comparison  with  b,  the  approximate  value  is 

..................  (18) 


•or,  if  a,  =  «,  =  <«,  i  =  4log-        .........................  (19) 


The  first  term  of  (19)  is  the  value  usually  given.     The  same  expression 
represents  the  reciprocal  of  the  capacity  of  the  system  per  unit  length. 

In  the  application  of  Lecher's  arrangement  to  the  investigation  of  re- 
fractive indices,  we  have  to  consider  the  effect  of  a  variation  of  the  dielectric 

*  Compare  Mncdonald,  Camb.  Phil.  Trans.  Vol.  xv.   p.  303  (1894). 


1897]  ALONG   CYLINDRICAL   CONDUCTORS   OF   ANY   SECTION.  331 

occurring  at  planes  for  which  z  is  constant.  It  will  be  seen  that  no  new 
difficulty  arises  in  the  case  of  systems  for  which  the  appropriate  function  <£  in 
two  dimensions  can  be  assigned. 

Regarding  <£  as  a  given  function,  e.g.  log  r  for  the  case  of  a  coaxal  wire 
and  sheath  (compare  (11)),  we  may  take  as  the  solution  for  any  length  of 
uniform  dielectric 


,    ,      ,    ,     -,,     (20) 

dx       dy         ) 

j±        \ 

...(21) 

in  which  pa  =  a,  &c.,  and  provision  is  made  for  waves  travelling  in  both 
directions. 

At  a  plane  where  the  dielectric  changes,  the  conditions  to  be  satisfied  are 
the  continuity  of  P,  Q  and  of  a,  /8 ;  and  this  is  secured  if 

A  eimz  +  Be~imz,  (22) 

~(Aeimz-Be-imz),  (23) 

are  continuous.  It  will  be  seen  that  the  conditions  are  altogether  indepen- 
dent of  the  section  of  the  conductors,  being  the  same  in  fact  as  if  there  were 
no  conductors  and  we  were  dealing  with  infinite  plane  waves  represented 
by  <£  =  x. 

As  a  particular  case  we  may  suppose  that  waves  travelling  in  the  negative 
direction  in  the  dielectric  (  F,  //,)  meet  at  z  =  0  a  dielectric  of  altered  character 
(V,  //).  The  expressions  (20),  (21)  represent  the  incident  (A)  and  reflected 
(B)  waves.  For  the  second  medium  it  suffices  to  accent  V  and  /*,  writing 
also  A'  for  A  and  0  for  B.  Thus  (22),  (23)  give 

by  which  B  and  A'  are  determined.     For  the  reflected  wave 


or   if  the   difference   between   the  dielectrics  relate  only  to  the  dielectric 
constants  (K,  K'), 

'--' 


in  agreement  with  Young's  well-known  optical  formula. 

Whether  the  dielectric  consist  of  uniform  portions  with  discontinuous 
changes  of  character  at  the  boundaries,  or  whether  it  be  a  continuous  function 
of  z,  the  solution  of  the  problem  is  the  same,  whatever  be  the  character  of  the 
cylindrical  conductors.  It  is  only  the  form  of  <£  that  is  influenced  by  the 
latter  consideration. 


232. 


THE   ELECTRO-CHEMICAL   EQUIVALENT  OF  SILVER. 

[Nature,  LVI.  p.  292,  1897.] 

IN  Nature,  Vol.  LVI.  p.  259,  Mr  Griffiths  points  out  that  recent  comparisons 
of  the  values  of  the  mechanical  equivalent  of  heat,  obtained  by  mechanical 
and  electrical  methods,  suggest  that  the  adopted  value  of  the  equivalent  of 
silver  may  be  in  error  to  the  extent  of  Y^TT-  This  adopted  value  rests, 
I  believe,  almost  entirely  upon  experiments  made  by  Kohlrausch,  and  by 
myself  with  Mrs  Sidgwick  in  1882  ;  and  the  question  has  been  frequently  put 
to  me  as  to  the  limits  within  which  it  is  trustworthy.  Such  questions  are 
more  easily  asked  than  answered,  and  experience  shows  that  estimates  of 
possible  error  given  by  experimenters  themselves  are  usually  framed  in  far  too 
sanguine  a  spirit. 

When  our  work  was  undertaken  the  generally  accepted  number  was  '01136 
obtained  by  Kohlrausch  in  1873.  Mascart  had  recently  given  '01124,  sub- 
sequently corrected  to  '011156.  The  uncertainty,  therefore,  at  that  time 
amounted  to  at  least  1  per  cent.  The  experiments  of  Mrs  Sidgwick  and 
myself  were  very  carefully  conducted,  and  we  certainly  hoped  to  have  attained 
an  accuracy  of  -^wo^-  So  ^ar  as  errors  that  can  be  eliminated  by  repetition 
are  concerned,  this  was  doubtless  the  case,  as  is  proved  by  an  examination  of 
our  tabular  results.  But,  as  every  experimenter  knows,  or  ought  to  know, 
this  class  of  errors  is  not  really  the  most  dangerous.  Security  is  only  to  be 
obtained  by  coincidence  of  numbers  derived  by  different  methods  and  by 
different  individuals.  It  was,  therefore,  a  great  satisfaction  to  find  our 
number  (Phil.  Trans.  1884)  ('011179)  confirmed  by  that  of  Kohlrausch 
('011183),  resulting  from  experiments  made  at  about  the  same  time. 

It  would,  however,  in  my  opinion,  be  rash  to  exclude  the  question  of  an 
error  of  y^.  Indeed,  I  have  more  than  once  publicly  expressed  surprise  at 
the  little  attention  given  to  this  subject  in  comparison  with  that  lavished 
upon  the  ohm.  I  do  not  know  of  any  better  method  of  measuring  currents 
absolutely  than  that  followed  in  1882,  but  an  ingenious  critic  would  doubtless 
be  able  to  suggest  improvements  in  details.  The  only  thing  that  has  occurred 
to  me  is  that  perhaps  sufficient  attention  was  not  given  to  the  change  in 
dimensions  that  must  accompany  the  heating  of  the  suspended  coil  when 
conveying  the  current  of  £  ampere.  Recent  experiments  upon  the  coil  (which 
exists  intact)  show  that,  as  judged  by  resistance,  the  heating  effect  due  to 
this  current  is  2|°  C.  But  it  does  not  appear  possible  that  the  expansion  of 
mean  radius  thence  arising  could  be  comparable  with  y^j.  [See  Vol.  II.  p.  278.] 


233. 


ON  AN   OPTICAL   DEVICE   FOR  THE  INTENSIFICATION   OF 
PHOTOGRAPHIC   PICTURES. 

[Philosophical  Magazine,  XLIV.  pp.  282 — 285,  1897.] 

WHETHER  from  insufficient  exposure  or  from  other  causes,  it  not  unfre- 
quently  happens  that  a  photographic  negative  is  deficient  in  density,  the  ratio 
of  light-transmissions  for  the  transparent  and  opaque  parts  being  too  low  for 
effective  contrast.  In  many  cases  an  adequate  remedy  is  found  in  chemical 
processes  of  intensification,  but  modern  gelatine  plates  do  not  always  lend 
themselves  well  to  this  treatment. 

The  method  now  proposed  may  be  described  as  one  of  using  the  negative 
twice  over.  Many  years  ago  a  pleasing  style  of  portrait  was  current  depen- 
dent upon  a  similar  principle.  A  thin  positive  transparency  is  developed 
upon  a  collodion  plate  by  acid  pyrogallol.  Viewed  in  the  ordinary  way  by 
holding  up  to  the  light,  the  picture  is  altogether  too  faint;  but  when  the 
film  side  is  placed  in  contact  with  paper  and  the  combination  viewed 
by  reflected  light,  the  contrast  is  sufficient.  Through  the  transparent 
parts  the  paper  is  seen  with  but  little  loss  of  brilliancy,  while  the  opaque 
parts  act,  as  it  were,  twice  over,  once  before  the  light  reaches  the  paper,  and 
again  after  reflexion  on  its  way  to  the  eye.  For  this  purpose  it  is  necessary 
that  the  deposit,  constituting  the  more  opaque  parts  of  the  picture,  be  of 
such  a  nature  as  not  itself  to  reflect  light  back  to  the  eye  in  appreciable 
degree — a  condition  very  far  from  being  satisfied  by  ordinary  gelatine 
negatives.  But  by  a  modification  of  the  process  the  objection  may  be  met 
without  much  difficulty. 

To  obtain  an  intensified  copy  (positive)  of  a  feeble  negative,  a  small  source 
of  illumination,  e.g.  a  candle,  is  employed,  and  it  is  placed  just  alongside  of 
the  copying-lens.  The  white  paper  is  replaced  by  a  flat  polished  reflector, 
and  the  film  side  of  the  negative  is  brought  into  close  contact  with  it.  On 


334  ON    AN   OPTICAL   DEVICE   FOR  THE  [233 

the  other  side  of  the  negative  and  pretty  close  to  it  is  a  field,  or  condensing, 
lens  of  such  power  that  the  light  from  the  candle  is  made  parallel  by  it. 
After  reflexion  the  light  again  traverses  the  lens  and  forms  an  image  of  the 
candle  centred  upon  the  photographic  copying-lens.  The  condenser  must  be 
large  enough  to  include  the  picture  and  must  be  free  from  dirt  and  scratches ; 
otherwise  it  does  not  need  to  be  of  good  optical  quality.  If  the  positive  is  to 
preserve  the  original  scale,  the  focal  length  of  the  condenser  must  be  about 
twice  that  of  the  copying-lens. 

In  carrying  this  method  into  execution  there  are  two  points  which  require 
special  attention.  The  first  is  the  elimination  of  false  light  reflected  from  the 
optical  surfaces  employed.  As  regards  the  condensing-lens,  the  difficulty  is 
easily  met  by  giving  it  a  moderate  slope.  But  the  light  reflected  from  the 
glass  face  of  the  negative  to  be  copied  is  less  easily  dealt  with.  If  allowed 
to  remain,  it  gives  a  uniform  illumination  over  the  whole  field,  which  in  many 
cases  would  go  far  to  neutralize  the  advantages  otherwise  obtainable  by  the 
method.  The  difficulty  arises  from  the  parallelism  of  the  two  surfaces  of  the 
negative,  and  is  obviated  by  using  for  the  support  of  the  film  a  glass  whose 
faces  are  inclined.  The  false  light  can  then  be  thrown  to  one  side  and 
rendered  inoperative.  In  practice  it  suffices  to  bring  into  contact  with  the 
negative  (taken  as  usual  upon  a  parallel  plate)  a  wedge-shaped  glass  of  equal 
or  greater  area,  the  reflexion  from  the  adjoining  faces  being  almost  destroyed 
by  the  interposition  of  a  layer  of  turpentine.  By  these  devices  the  false 
light  is  practically  eliminated,  and  none  reaches  the  sensitive  film  but  what 
has  twice  traversed  the  original  negative. 

The  other  point  requiring  attention  is  to  secure  adequate  superposition  of 
the  negative  and  its  image  in  the  associated  reflector.  On  account  of  the 
slight  lateral  interval  between  the  copying-lens  and  the  source  of  light,  the 
incidence  of  the  rays  upon  the  reflector  is  not  accurately  perpendicular,  and 
thus  any  imperfection  of  contact  between  the  negative  film  and  the  reflector 
leads  to  a  displacement  prejudicial  to  definition.  The  linear  displacement  is 
evidently  2£  sin  6,  if  t  denote  the  interval  between  the  surfaces  and  0  the 
angle  of  incidence,  and  it  can  be  calculated  in  any  particular  case.  It  is  the 
necessity  for  a  small  t  that  imposes  the  use  of  a  speculum  as  a  reflector.  In 
practice  20  can  easily  be  reduced  to  ^ ;  so  that  if  t  were  ^  inch,  the  dis- 
placement would  not  exceed  -^  inch,  and  for  most  purposes  might  be 
disregarded*.  The  obliquity  6  could  be  got  rid  of  altogether  by  introducing 
the  light  with  the  aid  of  a  parallel  glass  reflector  placed  at  45°;  but  this  com- 
plication is  hardly  to  be  recommended. 

The  scale  of  the  apparatus  depends,  of  course,  upon  the  size  of  the 
negatives  to  be  copied.  In  my  own  experiments  ^-plates  (4£  in.  x  3£  in.) 

*  If  the  glass  of  the  negative  were  flat,  its  approximation  to  the  reflector  might  be  much  closer 
than  is  here  supposed. 


1897] 


INTENSIFICATION    OF    PHOTOGRAPHIC    PICTURES. 


335 


were  employed.  The  condenser  is  of  plate-glass  6  in.  diameter  and  36  in. 
focus.  The  reflector  is  of  silver  deposited  on  glass*.  The  wedge-shaped 
glass f  attached  to  the  negative  with  turpentine  is  4  x  4  ins.  and  the  angle 
between  the  faces  is  2°.  The  photographic  lens  is  of  3  inch  aperture  and 
about  18  inch  principal  focus.  It  stands  at  about  36  inches  from  the  negative 
to  be  copied.  [Inch  =  2'54  cm.] 


The  accompanying  sketch  shows  the  disposition  of  some  of  the  parts. 
It  represents  a  section  by  a  horizontal  plane.  A  is  the  condensing-lens, 
B  the  wedge,  C  the  negative  temporarily  cemented  to  B  by  fluid  turpentine, 
D  the  speculum. 

[1902.  An  almost  identical  procedure  had  been  described  about  three 
years  earlier  by  Mach  (Eder's  Jakrbuch  fur  Photographic}.  The  method  of 
double  transmission  was  employed  in  a  former  research  (Phil.  Mag.  Oct.  1892; 
Vol.  iv.  of  this  collection,  p.  10).] 

*  For  a  systematic  use  of  the  method  a  reflector  of  speculum  metal  would  probably  be 
preferable. 

f  It  is  one  of  those  employed  for  a  similar  purpose  in  the  projection  of  Newton's  rings  (Proc. 
Roy.  Iiist.  March,  1893 ;  Nature,  Vol.  XLVIII.  p.  212  [Vol.  iv.  p.  54]). 


234. 


ON  THE  VISCOSITY  OF  HYDROGEN   AS  AFFECTED  BY 
MOISTURE. 

[Proceedings  of  the  Royal  Society,  LXII.  pp.  112—116,  1897.] 

IN  Sir  W.  Crookes's  important  work  upon  the  viscosity  of  gases*  the  case 
of  hydrogen  was  found  to  present  peculiar  difficulty.  "  With  each  improve- 
ment in  purification  and  drying  I  have  obtained  a  lower  value  for  hydrogen, 
and  have  consequently  diminished  the  number  expressing  the  ratio  of  the 
viscosity  of  hydrogen  to  that  of  air.  In  1876  I  found  the  ratio  to  be  0'508. 
In  1877  I  reduced  this  ratio  to  0'462.  Last  year,  with  improved  apparatus, 
I  obtained  the  ratio  O458,  and  I  have  now  got  it  as  low  as  0'4439"  (p.  425). 
The  difficulty  was  attributed  to  moisture.  Thus  (p.  422) :  "  After  working  at 
the  subject  for  more  than  a  year,  it  was  discovered  that  the  discrepancy  arose 
from  a  trace  of  water  obstinately  held  by  the  hydrogen — an  impurity  which 
behaved  as  I  explain  farther  on  in  the  case  of  air  and  water  vapour." 

When  occupied  in  1888  with  the  density  of  hydrogen,  I  thought  that 
viscosity  might  serve  as  a  useful  test  of  purity,  and  I  set  up  an  apparatus 
somewhat  on  the  lines  of  Sir  W.  Crookes.  A  light  mirror,  18  mm.  in 
diameter,  was  hung  by  a  fine  fibre  (of  quartz  I  believe)  about  60  cm.  long. 
A  small  attached  magnet  gave  the  means  of  starting  the  vibrations  whose 
subsidence  was  to  be  observed.  The  viscosity  chamber  was  of  glass,  and 
carried  tubes  sealed  to  it  above  and  below.  The  window,  through  which  the 
light  passed  to  and  fro,  was  of  thick  plate  glass  cemented  to  a  ground  face. 
This  arrangement  has  great  optical  advantages,  and  though  unsuitable  for 
experiments  involving  high  exhaustions,  appeared  to  be  satisfactory  for  the 
purpose  in  hand,  viz.,  the  comparison  of  various  samples  of  hydrogen  at 
atmospheric  pressure.  The  Topler  pump,  as  well  as  the  gas  generating 
apparatus  and  purifying  tubes,  were  connected  by  sealing.  But  I  was  not 
able  to  establish  any  sensible  differences  among  the  various  samples  of 
hydrogen  experimented  upon  at  that  time. 

*  Phil.  Trans.  1881,  p.  387. 


1897]      ON  THE   VISCOSITY  OF   HYDROGEN  AS   AFFECTED   BY  MOISTURE.      337 

In  view  of  the  importance  of  the  question,  I  have  lately  resumed  these 
experiments.  If  hydrogen,  carefully  prepared  and  desiccated  in  the  ordinary 
way,  is  liable  to  possess  a  viscosity  of  10  per  cent,  in  excess,  a  similar  un- 
certainty in  less  degree  may  affect  the  density.  I  must  confess  that  I  was 
sceptical  as  to  the  large  effect  attributed  to  water  vapour  in  gas  which  had 
passed  over  phosphoric  anhydride.  Sir  W.  Crookes  himself  described  an 
experiment  (p.  428)  from  which  it  appeared  that  a  residue  of  water  vapour 
in  his  apparatus  indicated  the  viscosity  due  to  hydrogen,  and,  without 
deciding  between  them,  he  offered  two  alternative  explanations.  Either  the 
viscosity  of  water  vapour  is  really  the  same  as  that  of  hydrogen,  or  under 
the  action  of  the  falling  mercury  in  the  Sprengel  pump  decomposition 
occurred  with  absorption  of  oxygen,  so  that  the  residual  gas  was  actually 
hydrogen.  It  does  not  appear  that  the  latter  explanation  can  be  accepted,  at 
any  rate  as  regards  the  earlier  stages  of  the  exhaustion,  when  a  rapid  current 
of  aqueous  vapour  must  set  in  the  direction  of  the  pump ;  but  if  we  adopt 
the  former,  how  comes  it  that  small  traces  of  water  vapour  have  so  much 
effect  upon  the  viscosity  of  hydrogen  ? 

It  is  a  fact,  as  was  found  many  years  ago  by  Kundt  and  Warburg*  (and 
as  I  have  confirmed),  that  the  viscosity  of  aqueous  vapour  is  but  little  greater 
than  that  of  hydrogen.  The  numbers  (relatively  to  air)  given  by  them  are 
0-5256  and  0'488.  It  is  difficult  to  believe  that  small  traces  of  a  foreign  gas 
having  a  six  per  cent,  greater  viscosity  could  produce  an  effect  reaching  to 
10  per  cent. 

In  the  recent  experiments  the  hydrogen  was  prepared  from  amalgamated 
zinc  and  sulphuric  acid  in  a  closed  generator  constituting  in  fact  a  Smee  cell, 
and  it  could  be  liberated  at  any  desired  rate  by  closing  the  circuit  externally 
through  a  wire  resistance.  The  generating  vessel  was  so  arranged  as  to  admit 
of  exhaustion,  and  the  materials  did  not  need  to  be  renewed  during  the 
whole  course  of  the  -experiments.  The  gas  entered  the  viscosity  chamber 
from  below,  and  could  be  made  to  pass  out  above  through  the  upper  tube 
(which  served  also  to  contain  the  fibre)  into  the  pump  head  of  the  Tdpler. 
By  suitable  taps  the  viscosity  chamber  could  be  isolated,  when  observations 
were  to  be  commenced. 

The  vibrations  were  started  by  a  kind  of  galvanometer  coil  in  connexion 
(through  a  key)  with  a  Leclanche  cell.  As  a  sample  set  of  observations  the 
following  relating  to  hydrogen  at  atmospheric  pressure  and  at  58°  F.,  which 
had  been  purified  by  passage  over  fragments  of  sulphur  and  solid  soda 
(without  phosphoric  anhydride),  may  be  given: — 

*  Fogg.  Ann.  1875,  Vol.  CLV.  p.  547. 


22 


338     ON  THE  VISCOSITY  OF  HYDROGEN  AS  AFFECTED  BY  MOISTURE.     [234 
ORSERVATIONS  ON  JUNE  7,  1897. 


_ 

65-4 





423-7 

88-9 

358-3 

2-554 

— 

401-3 

110-0 

312-4 

2-495 

0-059 

381-5 

128-9 

271-5 

2-434 

0-061 

364-4 

144-1 

235-5 

2-372 

0-062 

349-7 

158-6 

205-6 

2-313 

0-059 

336-8 

169-8 

178-2 

2-251 

0-062 

325-7 

180-6 

155-9 

2-193 

0-058 

315-7 

189-8 

135-1 

2-131 

0-062 

307-2 

197-8 

117-4 

2-070 

0-061 

300-0 

204-6                     102-2 

2-009 

0-061 

293-7 

210-6 

89-1 

1-950 

0-059 

287-8 

— 

77-2 

1-888 

0-062 

Meaft  log.  dec.  =0-0604. 

The  two  first  columns  contain  the  actually  observed  elongations  upon  the 
two  sides.  They  require  no  correction,  since  the  scale  was  bent  to  a  circular  arc 
centred  at  the  mirror.  The  third  column  gives  the  actual  arcs  of  vibration, 
the  fourth  their  (common)  logarithms,  and  the  fifth  the  differences  of  these, 
which  should  be  constant.  The  mean  logarithmic  decrement  can  be  obtained 
from  the  first  and  last  arcs  only,  but  the  intermediate  values  are  useful  as  a 
check.  The  time  of  (complete)  vibration  was  determined  occasionally.  It 
was  constant,  whether  hydrogen  or  air  occupied  the  chamber,  at  26'2  seconds. 

The  observations  extended  themselves  over  two  months,  and  it  would  be 
tedious  to  give  the  results  in  any  detail.  One  of  the  points  to  which  I 
attached  importance  was  a  comparison  between  hydrogen  as  it  issued  from 
the  generator  without  any  desiccation  whatever  and  hydrogen  carefully  dried 
by  passage  through  a  long  tube  packed  with  phosphoric  anhydride.  The 
difference  proved  itself  to  be  comparatively  trifling.  For  the  wet  hydrogen 
there  were  obtained  on  May  10,  11,  such  log.  decs,  as  0'0594,  0'0590,  O0591, 
or  as  a  mean  0*0592.  The  dried  hydrogen,  on  the  other  hand,  gave  0'0588, 
0'0586,  0'0584,  0'0590  on  various  repetitions  with  renewed  supplies  of  gas, 
or  as  a  mean  0'0587,  about  1  per  cent,  smaller  than  for  the  wet  hydrogen. 
It  appeared  that  the  dry  hydrogen  might  stand  for  several  days  in  the 
viscosity  chamber  without  alteration  of  logarithmic  decrement.  It  should  be 
mentioned  that  the  apparatus  was  set  up  underground,  and  that  the  changes 
of  temperature  were  usually  small  enough  to  be  disregarded. 

In  the  next  experiments  the  phosphoric  tube  was  replaced  by  others 
containing  sulphur  (with  the  view  of  removing  mercury  vapour)  and  solid 
soda.  Numbers  were  obtained  on  different  days  such  as  0'0591,  0-0586, 
0-0588,  0-0587,  mean  0'0588,  showing  that  the  desiccation  by  soda  was  practi- 
cally as  efficient  as  that  by  phosphoric  anhydride. 


1897]      ON   THE   VISCOSITY   OF   HYDROGEN   AS   AFFECTED   BY   MOISTURE.      339 

At  this  stage  the  apparatus  was  rearranged.  As  shown  by  observations 
upon  air  (at  10  cm.  residual  pressure),  the  logarithmic  decrements  were 
increased,  probably  owing  to  a  slight  displacement  of  the  mirror  relatively  to 
the  containing  walls  of  the  chamber.  The  sulphur  and  soda  tubes  were 
retained,  but  with  the  addition  of  one  of  hard  glass  containing  turnings  of 
magnesium.  Before  the  magnesium  was  heated  the  mean  number  for 
hydrogen  (always  at  atmospheric  pressure)  was  0*0600.  The  heating  of  the 
magnesium  to  redness,  which  it  was  supposed  might  remove  residual  water, 
had  the  effect  of  increasing  the  viscosity  of  the  gas,  especially  at  first*. 
After  a  few  operations  the  logarithmic  decrement  from  gas  which  had  passed 
over  the  hot  magnesium  seemed  to  settle  itself  at  0*0606.  When  the 
magnesium  was  allowed  to  remain  cold,  fresh  fillings  gave  again  0'0602, 
0-0601,  0-0598,  mean  O'OGOO.  Dried  air  at  10cm.  residual  pressure  gave 
0-01114,  0-01122,  0-01118,  0-01126,  0-01120,  mean  0-01120. 

In  the  next  experiments  a  phosphoric  tube  was  added  about  60  cm.  long 
and  closely  packed  with  fresh  material.  The  viscosity  appeared  to  be  slightly 
increased,  but  hardly  more  than  would  be  accounted  for  by  an  accidental 
rise  of  temperature.  The  mean  unconnected  number  may  be  taken  as  0*0603. 

The  evidence  from  these  experiments  tends  to  show  that  residual  moisture 
is  without  appreciable  influence  upon  the  viscosity  of  hydrogen ;  so  much  so 
that,  were  there  no  other  evidence,  this  conclusion  would  appear  to  me  to  be 
sufficiently  established.  It  remains  barely  possible  that  the  best  desiccation 
to  which  I  could  attain  was  still  inadequate,  and  that  absolutely  dry  hydrogen 
would  exhibit  a  less  viscosity.  It  must  be  admitted  that  an  apparatus 
containing  cemented  joints  and  greased  stop-cocks  is  in  some  respects  at  a 
disadvantage.  Moreover,  it  should  be  noticed  that  the  ratio  0'0600  :  01120, 
viz.  0*536,  for  the  viscosities  of  hydrogen  and  air  is  decidedly  higher  than 
that  (0'500)  deduced  by  Sir  G.  Stokes  from  Crookes's  observations.  Accord- 
ing to  the  theory  of  the  former,  a  fair  comparison  may  be  made  by  taking,  as 
above,  the  logarithmic  decrements  for  hydrogen  at  atmospheric  pressure,  and 
for  air  at  a  pressure  of  10  cm.  of  mercury.  I  may  mention  that  moderate 
rarefactions,  down  say  to  a  residual  pressure  of  5  cm.,  had  no  influence  on  the 
logarithmic  decrement  observed  with  hydrogen. 

I  am  not  able  to  explain  the  discrepancy  in  the  ratios  thus  exhibited. 
A  viscous  quality  in  the  suspension,  leading  to  a  subsidence  of  vibrations 
independent  of  the  gaseous  atmosphere,  would  tend  to  diminish  the  apparent 
differences  between  various  kinds  of  gas,  but  I  can  hardly  regard  this  cause 
as  operative  in  my  experiments.  For  actual  comparisons  of  widely  differing 
viscosities  I  should  prefer  an  apparatus  designed  on  Maxwell's  principle,  in 
which  the  gas  subjected  to  shearing  should  form  a  comparatively  thin  layer 
bounded  on  one  side  by  a  moving  plane  and  on  the  other  by  a  fixed  plane. 

*  The  glass  was  somewhat  attacked,  and  it  is  supposed  that  silicon  compounds  may  have 
contaminated  the  hydrogen. 

22—2 


235. 


ON  THE   PROPAGATION   OF  WAVES  ALONG   CONNECTED 
SYSTEMS  OF  SIMILAR  BODIES. 

[Philosophical  Magazine,  XLIV.  pp.  356—362,  1897.] 

FOR  simplicity  of  conception  the  bodies  are  imagined  to  be  similarly 
disposed  at  equal  intervals  (a)  along  a  straight  line.  The  position  of  each 
body,  as  displaced  from  equilibrium,  is  supposed  to  be  given  by  one  coor- 
dinate, which  for  the  rth  body  is  denoted  by  -ty-r.  A  wave  propagated  in  one 
direction  is  represented  by  taking  tyr  proportional  to  ei(nt+r®.  If  we  take  an 
instantaneous  view  of  the  system,  the  disturbance  is  periodic  when  rj3 
increases  by  2?r,  or  when  ra  increases  by  2?ra/y8.  This  is  the  wave-length, 
commonly  denoted  by  X,  ;  so  that,  if  k  =  2-Tr/X,,  k  =  /3/a.  The  velocity  of 
propagation  (V)  is  given  by  V=nfk;  and  the  principal  object  of  the  investi- 
gation is  to  find  the  relation  between  n  or  V  and  X. 

The  forces  acting  upon  each  body,  which  determine  the  vibration  of  the 
system  about  its  configuration  of  equilibrium,  are  assumed  to  be  due  solely 
to  the  neighbours  situated.  within  a  limited  distance.  The  simplest  case  of 
all  is  that  in  which  there  is  no  mutual  reaction  between  the  bodies,  the  kinetic 
and  potential  energies  of  the  system  being  then  given  by 

T=\A£+*>  P  =  i<702fr»,     ..................  (1) 

similarity  requiring  that  the  coefficients  A0,  C0  be  the  same  for  all  values 
of  r.  In  this  system  each  body  vibrates  independently,  according  to  the 
equation 


and  n*  =  C0/A0  ..................................  (3) 

The  frequency  is  of  course  independent  of  the  wave-length  in  which  the 
phases  may  be  arranged  to  repeat  themselves,  so  that  n  is  independent  of  k, 
while  V  equal  to  n/k  varies  inversely  as  k,  or  directly  as  X.  The  propagation 
of  waves  along  a  system  of  this  kind  has  been  considered  by  Reynolds. 


1897]  ON   WAVES   ALONG   SYSTEMS   OF   SIMILAR   BODIES.  341 

In  the  general  problem  the  expression  for  P  will  include  also  products  of 
tyr  with  the  neighbouring  coordinates  ...^r-2>  tyr-i,  tyr+i,  tyr+z-',  and  a 
similar  statement  holds  good  for  T.  Exhibiting  only  the  terms  which  involve 
r,  we  may  write 


-  A2^r_2  -  A2^r+2  -...,    .......  (4) 

P=  ...  +  i<W-  d  Wv-i  -  £  Wr» 

-<72^rfr_2-C2TMrr+2-...,       .........  (5)      ' 

where  A1}  A2,  ...  G1}  C2)  ...  are  constants,  finite  for  a  certain  number  of  terms 
and  then  vanishing.  The  equation  for  tyr  is  accordingly 

A0$'r-Al-f>r-l  -  A^r+j.  -  A2fyr_2  -  A^r+s  -  ...... 

+  C^r  -  C^M  -  C^r+l  -  C2+r_2  -  C^r+2  -  ......  =  0  .......  (6) 

In  the  other  equations  of  the  system  r  is  changed,  but  without  entailing  any 
other  alteration  in  (6).  Since  all  the  quantities  i/r  are  proportional  to  eint, 
the  double  differentiation  is  accounted  for  by  the  introduction  of  the  factor 
—  n2.  Making  this  substitution  and  remembering  that  tyr  is  also  proportional 
to  eirP,  we  get  as  the  equivalent  of  any  one  of  the  equations  (6) 

n?  (A0  -  A^-V  -  A^V  -  A2e~^  -  A2e^  -  ...) 
=  CQ  -  C^e-*  -  C.e*  -  C2e~w  -  C,e^  -  ..., 

2  _  C0  -  2^  cos  ka  -  2G2  cos  2feg  -  .  .  .  ,,_, 

~  2o^"237cos  ka  -  1As  cos  2ka  -  .  .  .  '    ' 

in  which  ^  is  replaced  by  its  equivalent  ka.  By  (7)  n  is  determined  as  a 
function  of  k  and  of  the  fundamental  constants  of  the  system. 

In  most  of  the  examples  which  naturally  suggest  themselves  A1}  A2,  ... 
vanish,  so  that  T  has  the  same  simple  form  as  in  (1).  If  we  suppose  for 
brevity  that  A0  is  unity,  (7)  becomes 

n?=  C0  -  20!  cos  ka  -  2C2  cos  2ka  -  ....................  (8) 

When  the  waves  are  very  long,  k  approximates  to  zero.     In  the  limit 

n2=C0-2Cl-2G2-  .............................  (9) 

If  we  call  the  limiting  value  C,  we  may  write  (8)  in  the  form 


sn 


(10) 


In  an  important  class  of  cases  C  vanishes,  that  is  the  frequency  diminishes 
without  limit  as  \  increases.  If  at  the  same  time  but  one  of  the  constants 
G!,  C2,  ...  be  finite,  the  equation  simplifies.  For  example,  if  Cl  alone  be 

finite, 

(11) 


342  ON  THE  PROPAGATION  OF  WAVES   ALONG  [235 

In  any  case  when  n  is  known  V  follows  immediately.     Thus  from  (10)  with 
C  evanescent,  we  get 


A.  simple  case  included  under  (11)  is  that  of  a  stretched  string,  itself 
without  mass,  but  carrying  unit  loads  at  equal  intervals  (a)*.  The  expression 
for  the  potential  energy  is 


T!  representing  the  tension.     Thus  by  comparison  with  (5) 
ao  =  22T1/a,         Ci  =  Zya,         0,  =  0,    &c.; 

so  that  by  (8) 

2rx     2T, 

ri>  =  —  -  --  cos  tea, 
a        a 


IJL  being  introduced  to  represent  the  mass  of  each  load  with  greater  generality. 
The  value  of  V  is  obtained  by  division  of  (14)  by  k.  In  order  more  easily  to 
compare  with  a  known  formula  we  may  introduce  the  longitudinal  density  p, 
such  that  /i  =  ap.  Thus 


V=-=     /(^}    sin(^a>  (15) 

k     V  \P  /         %ka 


reducing  to  the  well-known  value  of  the  constant  velocity  of  propagation 
along  a  uniform  string  when  a  is  made  infinitesimal.  Lord  Kelvin's  wave- 
model  (Popular  Lectures  and  Addresses,  Vol.  I.  2nd  ed.  p.  164)  is  also  included 
under  the  class  of  systems  for  which  P  has  the  form  (13). 

Another  example  in  which  again  C2,  G3 . . .  vanish  is  proposed  by  Fitzgerald-f*. 
It  consists  of  a  linear  system  of  rotating  magnets  (Fig.  1)  with  their  poles 

Fig.  l. 


close  to  one  another  and  disturbed  to  an  amount  small  compared  with  the 
distance  apart  of  the  poles.  The  force  of  restitution  is  here  proportional  to 
the  sum  of  the  angular  displacements  (-^)  of  contiguous  magnets,  so  that  P 
is  proportional  to 


*  See  Theory  of  Sound,  §§  120,  148. 
t  Brit.  Assoc.  Report,  1893,  p.  689. 


1897]  CONNECTED   SYSTEMS   OF   SIMILAR   BODIES.  343 

Here  Ol  =  -  £  <70,  and  (8)  gives  ?i2  =C0(l  +  cos  ka\ 

or  n  =  n0  .  cos  (Pa),    ...........................  (16) 

if  n0  represent  the  value  of  n  appropriate  to  k  —  0,  i.e.  to  infinitely  long  waves. 
Here  n  =  0,  when  \  =  2a.     In  this  case 


Fitzgerald  considers,  further,  a  more  general  linear  system  constructed  by 
connecting  a  series  of  equidistant  wheels  by  means  of  indiarubber  bands. 
"  By  connecting  the  wheels  each  with  its  next  neighbour  we  get  the  simplest 
system.  If  to  this  be  superposed  a  system  of  connexion  of  each  with  its 
next  neighbour  but  two,  and  so  on.  complex  systems  with  very  various 
relations  between  wave-length  and  velocity  can  be  constructed  depending 
on  the  relative  strengths  of  the  bands  employed."  If  the  bands  may  be 
crossed,  the  potential  energy  takes  the  form 

P  =      7l         r  ±       r-l)"  +  i7l  (*r  ±  *r+l)* 
r  ±  ^+2)2 


which  is  only  less  general  than  (5)  by  the  limitation 

^±£±^±...  =  0  .........................  (18) 

Prof.  Fitzgerald  appears  to  limit  himself  to  the  lower  sign  in  the  alternatives, 
so  that  C  in  (10)  vanishes.  This  leads  to  (12),  from  which  his  result  differs, 
but  probably  only  by  a  slip  of  the  pen. 

If  we  take  the  upper  sign  throughout,  (8)  becomes 

-  Jn«  =  0icos'^  +  a,cos*  ^  +  C,cos'  ~  +  ..........  (19) 

£  2  .  ~Z 

It  may  be  observed  that  Prof.  Fitzgerald's  system  will  have  the  most 
general  potential  energy  possible  (5),  if  in  addition  to  the  elastic  connexions 
between  the  wheels  there  be  introduced  a  force  of  restitution  acting  upon 
each  wheel  independently. 

As  an  example  in  which  (72  is  finite  as  well  as  C,,  let  us  imagine  a  system 
of  masses  of  which  each  is  connected  to  its  immediate  neighbours  on  the  two 
sides  by  an  elastic  rod  capable  of  bending  but  without  inertia.  Here 

P  =  .  .  .  +  *c  (2^r_,  -  ^r_2  -  ^  +  i  c  (2f  r  -  tr-i  -  ^+i)2 

+  ic(2^r+1-tr-^+2)*+  .............  (20> 

A  comparison  with  (5)  gives 

00  =  6c,         ai  =  4c,         Ca  =  -c, 
so  that  0=00-201-2C'2  =  0. 


344  ON  THE  PROPAGATION   OF  WAVES   ALONG  [235 

Accordingly  by  (10), 

n2  =  16c  sin3  (%ka)  -  4c  sin2  ka  =  16c  sin4  (%ka), 
or  n  =  4c*  sin2  ($ka)  ............................  (21) 


Thus  far  we  have  considered  the  propagation  of  waves  along  an  unlimited 
series  of  bodies.  If  we  suppose  that  the  total  number  is  m  and  that  they 
form  a  closed  chain,  -^  must  be  such  that 

+r+m  =  +r,    .................................  (22) 

from  which  it  follows  that 

@  =  ka  =  2s7rlm,  ..............................  (23) 

s  being  an  integer.     Thus  (8)  becomes 

n2  =  (70  -  2  C1  cos  (  2s?r/w)  -  2(72  cos  (4s?r/m)  -  .......  ...  (24) 

When  the  chain,  composed  of  a  limited  series  of  bodies,  is  open  at  the 
ends  instead  of  closed,  the  general  problem  becomes  more  complicated.  A 
simple  example  is  that  treated  by  I^agrange,  of  a  stretched  massless  string, 
carrying  a  finite  number  of  loads  and  fixed  at  its  extremities*.  The  open 
chain  of  m  magnets,  for  which 


a  +  tm)2,  ......  (25) 

is  considered  by  Fitzgerald.     The  equations  are 

^  (1  -  n*)  +  >/r2       =  0, 
=0, 


of  which  the  first  and  last  may  be  brought  under  the  same  form  as  the  others 
if  we  introduce  T/TO  and  ^m+i,  such  that 

^o  +  ^i  =  0,         ^  +  ^1  =  0 (27) 

If  we  assume 

^rr  =  cosnt  sin (r/8 -  £/8),  (28) 

the  first  of  equations  (27)  is  satisfied.     The  second  is  also  satisfied  provided 
that 

=  0,     or    @  =  S7r/m (29) 

*  Theory  of  Sound,  §  120. 


1897]  CONNECTED   SYSTEMS   OF   SIMILAR   BODIES.  345 

The  equations  (26)  are  satisfied  if 


that  is,  if  n  =  2  cos  (s7r/2m)  ............................  (30) 

In  (29),  (30)  s  may  assume  the  ra  values  1  to  m  inclusive.     In  the  last  case 
n  =  0,  and  £  =  TT  ;  and  from  (28), 


The  equal  amplitudes  and  opposite  phases  of  consecutive  coordinates,  i.e. 
angular  displacements  of  the  magnets,  give  rise  to  no  potential  energy,  and 
therefore  to  a  zero  frequency  of  vibration.  In  the  first  case  (s  =  1)  the 
angular  deflexions  are  all  in  the  same  direction,  and  the  frequency  is  the 
highest  admissible.  If  at  the  same  time  m  be  very  great,  n  reaches  its 
maximum  value,  corresponding  to  parallel  positions  of  all  the  magnets.  If 
we  call  this  value  N,  the  generalized  form  of  (30),  applicable  to  all  masses 
and  degrees  of  magnetization,  may  be  written 


(31) 
If  m  is  great  and  s  relatively  small,  (31)  becomes  approximately 


so  that  as  s  diminishes  we  have  a  series  of  frequencies  approaching  N  as  an 
upper  limit,  and  are  reminded  (as  Fitzgerald  remarks)  of  certain  groups  of 
spectrum  lines.  A  nearer  approach  to  the  remarkable  laws  of  Balmer  for 
hydrogen*  and  of  Kayser  and  Runge  for  the  alkalies  is  arrived  at  by 
supposing  s  constant  while  m  varies.  In  this  case,  instead  of  supposing  that 
the  whole  series  of  lines  correspond  to  various  modes  of  one  highly  compound 
system,  we  attribute  each  line  to  a  different  system  vibrating  in  a  given 
special  mode.  Apart  from  the  better  agreement  of  frequencies,  this  point  of 
view  seems  the  more  advantageous  as  we  are  spared  the  necessity  of  selecting 
and  justifying  a  special  high  value  of  m.  If  we  were  to  take  s  =  2  in  (31) 
and  attribute  to  m  integral  values  3,  4,  5,  .  .  .  ,  we  should  have  a  series  of 
frequencies  of  the  same  general  character  as  the  hydrogen  series,  but  still 
differing  considerably  in  actual  values. 

There  is  one  circumstance  which  suggests  doubts  whether  the  analogue 
of  radiating  bodies  is  to  be  sought  at  all  in  ordinary  mechanical  or  acoustical 
systems  vibrating  about  equilibrium.  For  the  latter,  even  when  gyratory 
terms  are  admitted,  give  rise  to  equations  involving  the  square  of  the 
frequency;  and  it  is  only  in  certain  exceptional  cases,  e.g.  (31),  that  the 
frequency  itself  can  be  simply  expressed.  On  the  other  hand,  the  formulae 

*  Viz.  n=tf  (l-4m-2),  with  m=B,  4,  5,  &c. 


346  ON  WAVES   ALONG  SYSTEMS   OF  SIMILAR   BODIES.  [235 

and  laws  derived  from  observation  of  the  spectrum  appear  to  introduce  more 
naturally  the  first  power  of  the  frequency.  For  example,  this  is  the  case 
with  Balmer's  formula.  Again,  when  the  spectrum  of  a  body  shows  several 
doublets,  the  intervals  between  the  components  correspond  closely  to  a 
constant  difference  of  frequency,  and  could  not  be  simply  expressed  in  terms 
of  squares  of  frequency.  Further,  the  remarkable  law,  discovered  indepen- 
dently by  Rydberg  and  by  Schuster,  connecting  the  convergence  frequencies  of 
different  series  belonging  to  the  same  substance,  points  in  the  same  direction. 

What  particular  conclusion  follows  from  this  consideration,  even  if  force 
be  allowed  to  it,  may  be  difficult  to  say.  The  occurrence  of  the  first  power 
of  the  frequency  seems  suggestive  rather  of  kinematic  relations*  than  of  those 
of  dynamics. 

[1902.  See  further  on  the  subject  of  the  present  paper,  Phil.  Mag.  Dec. 
1898,  "  On  Iso-periodic  Systems,"  Art.  242,  below.] 

*  E.g.  as  in  the  phases  of  the  moon. 


236. 


ON  THE  DENSITIES   OF  CARBONIC   OXIDE,  CARBONIC 
ANHYDRIDE,  AND  NITROUS   OXIDE. 

[Proceedings  of  the  Royal  Society,  LXII.  pp.  204—209,  1897.] 

THE  observations  here  recorded  were  carried  out  by  the  method  and  with 
the  apparatus  described  in  a  former  paper*,  to  which  reference  must  be  made 
for  details.  It  must  suffice  to  say  that  the  globe  containing  the  gas  to  be 
weighed  was  filled  at  0°  C.,  and  to  a  pressure  determined  by  a  manometric 
gauge.  This  pressure,  nearly  atmospheric,  is  slightly  variable  with  tempera- 
ture on  account  of  the  expansion  of  the  mercury  and  iron  involved.  The 
actually  observed  weights  are  corrected  so  as  to  correspond  with  a  temperature 
of  15°  C.  of  the  gauge,  as  well  as  for  the  errors  in  the  platinum  and  brass 
weights  employed.  In  the  present,  as  well  as  in  the  former,  experiments  I 
have  been  ably  assisted  by  Mr  George  Gordon. 

Carbonic  Oxide. 

This  gas  was  prepared  by  three  methods.  In  the  first  method  a  flask, 
sealed  to  the  rest  of  the  apparatus,  was  charged  with  80  grams  recrystallised 
ferrocyanide  of  potassium  and  360  c.c.  strong  sulphuric  acid.  The  generation 
of  gas  could  be  started  by  the  application  of  heat,  and  with  cares  it  could  be 
checked  and  finally  stopped  by  the  removal  of  the  flame  with  subsequent 
application,  if  necessary,  of  wet  cotton-wool  to  the  exterior  of  the  flask.  In 
this  way  one  charge  could  be  utilised  with  great  advantage  for  several  fillings. 
On  leaving  the  flask  the  gas  was  passed  through  a  bubbler  containing  potash 
solution  (convenient  as  allowing  the  rate  of  production  to  be  more  easily 
estimated)  and  thence  through  tubes  charged  with  fragments  of  potash  and 
phosphoric  anhydride,  all  connected  by  sealing.  When  possible,  the  weight 

*  "On  the  Densities   of   the  Principal   Gases,"  Roy.  Soc.   Proc.  Vol.  LIII.  p.  134,  1893. 
[Vol.  iv.  p.  39.] 


348  ON   THE   DENSITIES   OF    CARBONIC   OXIDE,  [236 

of  the  globe  full  was  compared  with  the  mean  of  the  preceding  and  following 
weights  empty.  Four  experiments  were  made  with  results  agreeing  to  within 
a  few  tenths  of  a  milligram. 

In  the  second  set  of  experiments  the  flask  was  charged  with  100  grams 
of  oxalic  acid  and  500  c.c.  strong  sulphuric  acid.  To  absorb  the  large 
quantity  of  CO2  simultaneously  evolved,  a  plentiful  supply  of  alkali  was 
required.  A  wash-bottle  and  a  long  nearly  horizontal  tube  contained  strong 
alkaline  solution,  and  these  were  followed  by  the  tubes  containing  solid  potash 
and  phosphoric  anhydride  as  before. 

For  the  experiments  of  the  third  set  oxalic  acid  was  replaced  by  formic, 
which  is  more  convenient  as  not  entailing  the  absorption  of  large  volumes  of 
C02.  In  this  case  the  charge  consisted  of  50  grams  formate  of  soda,  300  c.c. 
strong  sulphuric  acid,  and  150  c.c.  distilled  water.  The  water  is  necessary  in 
order  to  prevent  action  in  the  cold,  and  the  amount  requires  to  be  somewhat 
carefully  adjusted.  As  purifiers,  the  long  horizontal  bubbler  was  retained 
and  the  tubes  charged  with  solid  potash  and  phosphoric  anhydride.  In  this 
set  there  were  four  concordant  experiments.  The  immediate  results  stand 
thus  :— 

Carbonic  Oxide. 

From  ferrocyanide 2*29843 

„      oxalic  acid 2*29852 

formate  of  soda   .  2*29854 


Mean 2*29850 

This  corresponds  to  the  number  2*62704  for  oxygen*,  and  is  subject  to  a 
correction  (additive)  of  0*00056  for  the  diminution  of  the  external  volume  of 
the  globe  when  exhausted. 

The  ratio  of  the  densities  of  carbonic  oxide  and  oxygen  is  thus 
2*29906  :  2*62760 ; 

so  that  if  the  density  of  oxygen  be  taken  as  32,  that  of  carbonic  oxide  will  be 
27*9989.  If,  as  some  preliminary  experiments  by  Dr  Scottf  indicate,  equal 
volumes  may  be  taken  as  accurately  representative  of  CO  and  of  02,  the 
atomic  weight  of  carbon  will  be  11*9989  on  the  scale  of  oxygen  =  16. 

The  very  close  agreement  between  the  weights  of  carbonic  oxide  prepared 
in  three  different  ways  is  some  guarantee  against  the  presence  of  an  impurity 
of  widely  differing  density.  On  the  other  hand,  some  careful  experiments 
led  Mr  T.  W.  Richards J  to  the  conclusion  that  carbonic  oxide  is  liable  to 

*  "On  the  Densities  of  the  Principal  Gases,"  Roy.  Soc.  Proc.  Vol.  LIII.  p.  144,  1893. 
[Vol.  rv.  p.  39.] 

t  Camb.  Phil.  Proc.  Vol.  ix.  p.  144,  1896. 
J  Amer.  Acad.  Proc.  Vol.  xvin.  p.  279,  1891. 


1897]  CARBONIC   ANHYDRIDE,   AND   NITROUS   OXIDE.  349 

contain  considerable  quantities  of  hydrogen  or  of  hydrocarbons.  From 
5£  litres  of  carbonic  oxide  passed  over  hot  cupric  oxide  he  collected  no  less 
than  25  milligrams  of  water,  and  the  evidence  appeared  to  prove  that  the 
hydrogen  was  really  derived  from  the  carbonic  oxide.  Such  a  proportion  of 
hydrogen  would  entail  a  deficiency  in  the  weight  of  the  globe  of  about  11 
milligrams,  and  seems  improbable  in  view  of  the  good  agreement  of  the 
numbers  recorded.  The  presence  of  so  much  hydrogen  in  carbonic  oxide  is 
also  difficult  to  reconcile  with  the  well-known  experiments  of  Professor  Dixon, 
who  found  that  prolonged  treatment  with  phosphoric  anhydride  was  required 
in  order  to  render  the  mixture  of  carbonic  oxide  and  oxygen  inexplosive.  In 
the  presence  of  relatively  large  quantities  of  free  hydrogen  (or  hydrocarbons) 
why  should  traces  of  water  vapour  be  so  important  ? 

In  an  experiment  by  Dr  Scott*,  4  litres  of  carbon  monoxide  gave  only 
1*3  milligrams  to  the  drying  tube  after  oxidation. 

I  have  myself  made  several  trials  of  the  same  sort  with  gas  prepared  from 
formate  of  soda  exactly  as  for  weighing.  The  results  were  not  so  concordant 
as  I  had  hoped -f,  but  the  amount  of  water  collected  was  even  less  than  that 
given  by  Dr  Scott.  Indeed,  I  do  not  regard  as  proved  the  presence  of 
hydrogen  at  all  in  the  gas  that  I  have  employed  J. 


Carbonic  A  nhydride. 

This  gas  was  prepared  from  hydrochloric  acid  and  marble,  and  after 
passing  a  bubbler  charged  with  a  solution  of  carbonate  of  soda,  was  dried  by 
phosphoric  anhydride.  Previous  to  use,  the  acid  was  caused  to  boil  for  some 
time  by  the  passage  of  hydrochloric  acid  vapour  from  a  flask  containing 
another  charge  of  the  acid.  In  a  second  set  of  experiments  the  marble  was 
replaced  by  a  solution  of  carbonate  of  soda.  There  is  no  appreciable 
difference  between  the  results  obtained  in  the  two  ways;  and  the  mean, 
corrected  for  the  errors  of  weights  and  for  the  shrinkage  of  the  globe  when 
exhausted,  is  3'6349,  corresponding  to  2'6276  for  oxygen.  The  temperature 
at  which  the  globe  was  charged  was  0°  C.,  and  the  actual  pressure  that  of  the 
manometric  gauge  at  about  20°,  reduction  being  made  to  15°  by  the  use  of 
Boyle's  law.  From  the  former  paper  it  appears  that  the  actual  height  of  the 
mercury  column  at  15°  is  762*511  mm. 

*  Chem.  Soc.  Trans.  1897,  p.  564. 

t  One  obstacle  was  the  difficulty  of  re-oxidising  the  copper  reduced  by  carbonic  oxide.  I  have 
never  encountered  this  difficulty  after  reduction  by  hydrogen. 

$  In  Mr  Richards'  work  the  gas  in  an  imperfectly  dried  condition  was  treated  with  hot 
platinum  black.  Is  it  possible  that  the  hydrogen  was  introduced  at  this  stage? 


350  ON  THE   DENSITIES  OF   CARBONIC   OXIDE,  [236 

Nitrous  Oxide. 

In  preliminary  experiments  the  gas  was  prepared  in  the  laboratory,  at  as 
low  a  temperature  as  possible,  from  nitrate  of  ammonia,  or  was  drawn  from 
the  iron  bottles  in  which  it  is  commercially  supplied.  The  purification  was 
by  passage  over  potash  and  phosphoric  anhydride.  Unless  special  precautions 
are  taken  the  gas  so  obtained  is  ten  or  more  milligrams  too  light,  presumably 
from  admixture  with  nitrogen.  In  the  case  of  the  commercial  supply,  a  better 
result  is  obtained  by  placing  the  bottles  in  an  inverted  position  so  as  to  draw 
from  the  liquid  rather  than  from  the  gaseous  portion. 

Higher  and  more  consistent  results  were  arrived  at  from  gas  which  had 
been  specially  treated.  In  consequence  of  the  high  relative  solubility  of 
nitrous  oxide  in  water,  the  gas  held  in  solution  after  prolonged  agitation,  of 
the  liquid  with  impure  gas  from  any  supply,  will  contain  a  much  diminished 
proportion  of  nitrogen.  To  carry  out  this  method  on  the  scale  required,  a 
large  (11 -litre)  flask  was  mounted  on  an  apparatus  in  connexion  with  the 
lathe  so  that  it  could  be  vigorously  shaken.  After  the  dissolved  air  had  been 
sufficiently  expelled  by  preliminary  passage  of  N2O,  the  water  was  cooled  to 
near  0°  C.  and  violently  shaken  for  a  considerable  time  while  the  gas  was 
passing  in  large  excess.  The  nitrous  oxide  thus  purified  was  expelled  from 
solution  by  heat,  and  was  used  to  fill  the  globe  in  the  usual  manner. 

For  comparison  with  the  results  so  obtained,  gas  purified  in  another 
manner  was  also  examined.  A  small  iron  bottle,  fully  charged  with  the  com- 
mercial material,  was  cooled  in  salt  and  ice  and  allowed  somewhat  suddenly 
to  blow  off  half  its  contents.  The  residue  drawn  from  the  bottle  in  one  or 
other  position  was  employed  for  the  weighings. 

Nitrous  Oxide  (1896). 

Aug.  15     Expelled  from  water 3'6359 

,,17  „  „  3-6354 

„     19     From  residue  after  blow  off,  valve  downwards  3'6364 

„     21  „  „  valve  upwards    .  3'6358 

„     22  „  „  valve  downwards  3'6360 

Mean 3'6359 

The  mean  value  may  be  taken  to  represent  the  corrected  weight  of  the  gas 
which  fills  the  globe  at  0°  C.  and  at  the  pressure  of  the  gauge  (at  15°),  corre- 
sponding to  2'6276  for  oxygen. 

One  of  the  objects  which  I  had  in  view  in  determining  the  density  of 
nitrous  oxide  was  to  obtain,  if  it  were  possible,  evidence  as  to  the  atomic 
weight  of  nitrogen.  It  may  be  remembered  that  observations  upon  the 


1897]  CARBONIC   ANHYDRIDE,   AND   NITROUS  OXIDE.  351 

density  of  pure  nitrogen,  as  distinguished  from  the  atmospheric  mixture 
containing  argon  which,  until  recently,  had  been  confounded  with  pure 
nitrogen,  led*  to  the  conclusion  that  the  densities  of  oxygen  and  nitrogen 
were  as  16  :  14'003,  thus  suggesting  that  the  atomic  weight  of  nitrogen  might 
really  be  14  in  place  of  14'05,  as  generally  received.  The  chemical  evidence 
upon  which  the  latter  number  rests  is  very  indirect,  and  it  appeared  that  a 
direct  comparison  of  the  weight  of  nitrous  oxide  and  of  its  contained  nitrogen 
might  be  of  value.  A  suitable  vessel  would  be  filled,  under  known  conditions, 
with  the  nitrous  oxide,  which  would  then  be  submitted  to  the  action  off  a 
spiral  of  copper  or  iron  wire  rendered  incandescent  by  an  electric  current. 
When  all  the  oxygen  was  removed,  the  residual  nitrogen  would  be  measured, 
from  which  the  ratio  of  equivalents  could  readily  be  deduced.  The  fact  that 
the  residual  nitrogen  would  possess  nearly  the  same  volume  as  the  nitrous 
oxide  from  which  it  was  derived  would  present  certain  experimental  advan- 
tages. If  indeed  the  atomic  weights  were  really  as  14  :  16,  the  ratio  (*•)  of 
volumes,  after  and  before  operations,  would  be  given  by 

2-2996  xx  14 


7  x  3-6359 
Whence  '  °  11  x  2-2996  -1"0061' 

3-6359  and  2*2996  being  the  relative  weights  of  nitrous  oxide  and  of 
nitrogen  which  (at  0°  C.  and  at  the  pressure  of  the  gauge)  occupy  the  same 
volume.  The  integral  numbers  for  the  atomic  weights  would  thus  correspond 
to  an  expansion,  after  chemical  reduction,  of  about  one-half  per  cent. 

But  in  practical  operation  the  method  lost  most  of  its  apparent  simplicity. 
It  was  found  that  copper  became  unmanageable  at  a  temperature  sufficiently 
high  for  the  purpose,  and  recourse  was  had  to  iron.  Coils  of  iron  suitably 
prepared  and  supported  could  be  adequately  heated  by  the  current  from  a 
dynamo  without  twisting  hopelessly  out  of  shape  ;  but  the  use  of  iron  leads 
to  fresh  difficulties.  The  emission  of  carbonic  oxide  from  the  iron  heated  in 
vacuum  continues  for  a  very  long  time,  and  the  attempt  to  get  rid  of  this  gas 
by  preliminary  treatment  had  to  be  abandoned.  By  final  addition  of  a  small 
quantity  of  oxygen  (obtained  by  heating  some  permanganate  of  potash  sealed 
up  in  one  of  the  leading  tubes)  the  CO  could  be  oxidised  to  CO2,  and  thus, 
along  with  any  H20,  be  absorbed  by  a  lump  of  potash  placed  beforehand  in 
the  working  vessel.  To  get  rid  of  superfluous  oxygen,  a  coil  of  incandescent 
copper  had  then  to  be  invoked,  and  thus  the  apparatus  became  rather 
complicated. 

It  is  believed  that  the  difficulties  thus  far  mentioned  were  overcome,  but 
nevertheless  a  satisfactory  concordance  in  the  final  numbers  was  not  attained. 

*  Bayleigh  and  Ramsay,  Phil.  Trans.  Vol.  CLXXXVI.  p.  190,  1895.     [Vol.  iv.  p.  133.] 


352  ON   THE   DENSITIES   OF   CARBONIC   OXIDE,   ETC.  [236 

In  the  present  position  of  the  question  no  results  are  of  value  which  do  not 
discriminate  with  certainty  between  14'05  and  14*00.  The  obstacle  appeared 
to  lie  in  a  tendency  of  the  nitrogen  to  pass  to  higher  degrees  of  oxidation. 
On  more  than  one  occasion  mercury  (which  formed  the  movable  boundary  of 
an  overflow  chamber)  was  observed  to  be  attacked.  Under  these  circum- 
stances I  do  not  think  it  worth  while  to  enter  into  further  detail  regarding 
the  experiments  in  question. 

The  following  summary  gives  the  densities  of  the  various  gases  relatively 
to  air,  all  obtained  by  the  same  apparatus*.  The  last  figure  is  of  little 
significance. 

Air  free  from  H20  and  C02    .    .    .     .     .  1 '00000 

Oxygen 110535 

Nitrogen  and  argon  (atmospheric)    .     .     .  0'97209 

Nitrogen 0'96737 

Argon 1-37752 

Carbonic  oxide 0'96716 

Carbonic  anhydride 1*52909 

Nitrous  oxide 1-52951 

The  value  obtained  for  hydrogen  upon  the  same  scale  was  0'06960 ;  but 
the  researches  of  M.  Leduc  and  of  Professor  Morley  appear  to  show  that  this 
number  is  a  little  too  high. 

[1902.     For  the  absolute  densities  of  air  and  oxygen,  see  Vol.  IV.  p.  51.] 

*  Boy.  Soc.  Proc.  Vol.  LHI.  p.  148,  1893 ;  Vol.  LV.  p.  340,  1894 ;   Phil.  Trans.  Vol.  CLXXXVI. 
p.  189,  1895 ;  Roy.  Soc.  Proc.  Vol.  LIX.  p.  201,  1896.     [Vol.  iv.  pp.  52,  104,  130,  215.] 


237. 

RONTGEN  RAYS  AND   ORDINARY  LIGHT. 

[Nature,  LVII.  p.  607,  1898.] 

r 

ACCORDING  to  the  theory  of  the  Rontgen  rays  suggested  by  Sir  G.  Stokes*, 
and  recently  developed  by  Prof.  J.  J.  Thomson f,  their  origin  is  to  be  sought 
in  impacts  of  the  charged  atoms  constituting  the  kathode-stream,  whereby 
pulses  of  disturbance  are  generated  in  the  ether.  This  theory  has  certainly 
much  to  recommend  it ;  but  I  cannot  see  that  it  carries  with  it  some  of  the 
consequences  which  have  been  deduced  as  to  the  distinction  between  Rontgen 
rays  and  ordinary  luminous  and  non-luminous  radiation.  The  conclusion  of 
the  authors  above  mentioned]:,  "  that  the  Rontgen  rays  are  not  waves  of  very 
short  wave-length,  but  impulses,"  surprises  me.  From  the  fact  of  their  being 
highly  condensed  impulses,  I  should  conclude  on  the  contrary  that  they  are 
waves  of  short  wave-length.  If  short  waves  are  inadmissible,  longer  waves 
are  still  more  inadmissible.  What  then  becomes  of  Fourier's  theorem  and 
its  assertion  that  any  disturbance  may  be  analysed  into  regular  waves  ? 

Is  it  contended  that  previous  to  resolution  (whether  merely  theoretical, 
or  practically  effected  by  the  spectroscope)  the  vibrations  of  ordinary 
(e.g.  white)  light  are  regular,  and  thus  distinguished  from  disturbances  made 
up  of  impulses  ?  This  view  was  certainly  supported  in  the  past  by  high 
authorities,  but  it  has  been  shown  to  be  untenable  by  Gouy§,  Schuster ||,  and 
the  present  writer  1T.  A  curve  representative  of  white  light,  if  it  were  drawn 
upon  paper,  would  show  no  sequences  of  similar  waves. 

In  the  second  of  the  papers  referred  to,  I  endeavoured  to  show  in  detail 
that  white  light  might  be  supposed  to  have  the  very  constitution  now 
ascribed  to  the  Rontgen  radiation,  except  that  of  course  the  impulses  would 
have  to  be  less  condensed.  The  peculiar  behaviour  of  the  Rontgen  radiation 
with  respect  to  diffraction  and  refraction  would  thus  be  attributable  merely 
to  the  extreme  shortness  of  the  waves  composing  it. 

[1902.  In  a  reply  to  the  above  (Nature,  LVIII.  p.  8),  Prof.  Thomson 
expresses  the  opinion  that  "the  difference  between  us  is  one  of  terminology."] 

*  Manchester  Memoirs,  Vol.  XLI.  No.  15,  1897. 
t  Phil.  Mag.  Vol.  XLV.  p.  172,  1898. 

J  See  also  Prof.  S.  P.  Thompson's  Light  Visible  and  Invisible  (London,  1897),  p.  273. 
§  Journ.  de  Physique,  1886,  p.  354. 
||  Phil.  Mag.  Vol.  xxxvn.  p.  509,  1894. 

IT  Enc.  Brit.  "  Wave  Theory,"  1888.     [Vol.  in.  p.  60.]    Phil.  Mag.  Vol.  xxvn.  p.  461,  1889. 
[Vol.  HI.  p.  270.] 

R.   iv.  23 


238. 

NOTE  ON  THE  PRESSURE  OF  RADIATION,  SHOWING  AN 
APPARENT  FAILURE  OF  THE  USUAL  ELECTROMAGNETIC 
EQUATIONS. 

[Philosophical  Magazine,  XLV.  pp.  522—525,  1898.] 

FOLLOWING  a  suggestion  of  Bartoli,  Boltzmann*  and  W.  Wienf  have 
arrived  at  the  remarkable  conclusion  that  that  part  of  the  energy  of  radiation 
from  a  black  body  at  absolute  temperature  6,  which  lies  between  wave-lengths 
X,  and  X  +  d\,  has  the  expression 

e^(6\)d\    (1) 

where  <£  is  an  arbitrary  function  of  the  single  variable  0\.  The  law  of 
Stefan,  according  to  which  the  total  radiation  is  as  0*,  is  therein  included. 
The  argument  employed  by  these  authors  is  very  ingenious,  and  I  think 
convincing  when  the  postulates  are  once  admitted.  The  most  important  of 
them  relates  to  the  pressure  of  radiation,  supposed  to  be  operative  upon  the 
walls  within  which  the  radiation  is  confined,  and  estimated  at  one-third  of 
the  density  of  the  energy  in  the  case  when  the  radiation  is  alike  in  all 
directions.  The  argument  by  which  Maxwell  originally  deduced  the  pressure 
of  radiation  not  being  clear  to  me,  I  was  led  to  look  into  the  question  a 
little  more  closely,  with  the  result  that  certain  discrepancies  have  presented 
themselves  which  I  desire  to  lay  before  those  who  have  made  a  special  study 
of  the  electric  equations.  The  criticism  which  appears  to  be  called  for  extends 
indeed  much  beyond  the  occasion  which  gave  rise  to  it. 

A  straightforward  calculation  of  the  pressure  exercised  by  plane  electric 
waves  incident  perpendicularly  upon  a  metallic  reflector  is  given  by  Prof. 
J.  J.  Thomson :[.  The  face  of  the  reflector  coincides  with  x  =  0,  and  in  the 
vibrations  under  consideration  the  magnetic  force  reduces  itself  to  the  com- 
ponent (/3)  parallel  to  y,  and  the  current  to  the  component  (w)  parallel  to  z. 
The  waves  which  penetrate  the  conducting  mass  die  out  more  or  less  quickly 
according  to  the  conductivity.  If  the  conductivity  is  great,  most  of  the 
energy  is  reflected,  and  such  part  as  is  propagated  into  the  conductor  is 
limited  to  a  thin  skin  at  x  =  0.  According  to  the  usual  equations  the 

*  Wied.  Ann.  Vol.  xxn.  pp.  31,  291  (1884). 

t  Berlin.  Sitzungsber.  Feb.  1893. 

J  Elements  of  Electricity  and  Magnetism,  Cambridge,  1895,  §  241. 


1898]  ON   THE   PRESSURE   OF   RADIATION.  355 

mechanical  force  exercised  upon  unit  of  area  of  the  slice  dx  of  the  conductor 
is  —  wbdx,  or  altogether 

I    wbdx  ..................................  (2) 

Here  b  denotes  the  magnetic  induction,  and  is  equal  to  //,/3,  if  //,  be  the  per- 
meability and  /8  the  magnetic  force.     Now 

4>7rw  =  d(3/dx, 
so  that  the  integral  becomes 


where  /30  is  the  value  of  /3  within  the  conductor  at  x  =  0,  and  ftx  =  0,  if  the 
conducting  slab  be  sufficiently  thick.  Since  there  is  no  discontinuity  of 
magnetic  force  at  x  =  0,  /30  may  be  taken  also  to  refer  to  the  value  at  x  =  0 
just  outside  the  metallic  surface. 

The  expression  (3)  gives  the  force  at  any  moment  ;  but  we  are  concerned 
only  with  the  mean  value.  Since  the  mean  value  of  ft?  is  one-half  the  maxi- 
mum value,  we  have  for  the  pressure 


It  only  remains  to  compare  with  the  density  of  the  energy  outside  the 
metal,  and  we  may  limit  ourselves  to  the  case  of  complete  reflexion.  The 
constant  energy  of  the  stationary  waves  passes  alternately  between  the  electric 
and  magnetic  forms.  If  we  estimate  it  at  the  moment  of  maximum  magnetic 
force,  we  have 

energy  =  ^jfjpdxdyde  .........................  (5) 

In  (5)  ft  is  variable  with  x.  If  j9max.  denote  the  maximum  value  which 
occurs  at  x  =  0,  the  mean  of  /32  =  l/S^x.  Thus 

density  of  energy  =  2J53B1  .  ^_  ^     ...............  (6) 

Thus,  if  the  permeability  /ju  of  the  metal  be  unity,  (4)  and  (6)  coincide  ; 
and  we  conclude  that  in  this  case  the  pressure  is  equal  to  the  density  of  the 
energy  in  the  neighbourhood  of  the  metal.  This  is  Maxwell's  result.  When 
we  consider  radiation  in  all  directions,  the  pressure  is  expressed  as  one-third 
of  the  density  of  energy. 

The  difficulty  that  I  have  to  raise  relates  to  the  case  where  //,  is  not  equal 
to  unity.  The  conclusion  in  (4)  that  the  pressure  is  proportional  to  p  would 
make  havoc  of  the  theory  of  Boltzmann  and  Wien  and  must,  I  think,  be 
rejected.  So  long  as  the  reflexion  is  complete  —  and  it  may  be  complete 
independently  of  /*  —  the  radiation  is  similarly  influenced,  and  (one  would 
suppose)  must  exercise  a  similar  force  upon  the  reflector.  But  if  the  con- 

23—2 


356  ON   THE   PRESSURE   OF   RADIATION.  [238 

elusion  is  impossible,  where  is  the  flaw  in  the  process  by  which  it  is  arrived 
at  ?  Being  unable  to  find  any  fault  with  the  deduction  above  given  (after 
Prof.  J.  J.  Thomson),  I  was  led  to  scrutinize  more  closely  the  fundamental 
equation  itself;  and  I  will  now  explain  why  it  appears  to  me  to  be  incorrect. 

For  this  purpose  let  us  apply  it  to  the  very  simple  case  of  a  wire  of 
circular  section,  parallel  to  z,  moving  in  the  direction  of  x  across  an  originally 
uniform  magnetic  field  (yS).  The  uniformity  of  the  field  is  disturbed  in  two 
ways :  (i)  by  the  operation  of  the  current  (w)  flowing  in  the  various  filaments 
of  the  wire,  and  (ii)  independently  of  a  current,  by  the  magnetic  effect  of  the 
material  composing  the  wire  whose  permeability  (fi)  is  supposed  to  be  great. 
In  estimating  as  in  (2)  the  mechanical  force  parallel  to  x  operative  upon  the 
wire,  we  should  have  to  integrate  wb  over  the  cross-section.  In  this  w  is 
supposed  to  be  constant,  and  the  local  value  is  everywhere  to  be  attribed  to  b. 
We  may  indeed,  if  we  please,  omit  from  b  the  part  due  to  the  currents  in  the 
wire,  which  will  in  the  end  contribute  nothing  to  the  result ;  but  we  are 
directed  to  use  the  actual  value  of  6  as  disturbed  by  the  presence  of  the 
magnetic  material.  In  the  particular  case  supposed,  where  fj,  is  great,  the 
value  of  b  within  the  wire  is  uniform,  and  just  twice  as  great  as  at  a  distance. 
It  follows,  when  the  integration  is  effected,  that  the  force  parallel  to  x  acting 
upon  the  wire  is  greater  (in  the  particular  case  doubly  greater)  than  it  would 
be  if  the  value  of  /*  were  unity. 

But  this  conclusion  cannot  be  accepted.  The  force  depends  upon  the 
number  of  lines  of  force  to  be  crossed  when  the  wire  makes  a  movement 
parallel  to  x.  And  it  is  clear  that  the  lines  effectively  crossed  in  such  a 
movement  are  not  the  condensed  lines  due  to  the  magnetic  quality  of  the 
wire,  but  are  to  be  reckoned  from  the  intensity  of  the  undisturbed  field.  The 
mechanical  force  cannot  really  depend  upon  p,,  and  the  formula  which  leads  to 
such  a  result  must  be  erroneous. 

As  regards  the  problem  of  the  pressure  of  radiation,  I  conclude  that  in 
this  case  also,  and  in  spite  of  the  formula,  the  permeability  of  the  reflector  is 
without  effect,  and  that  the  consequences  deduced  by  Boltzmann  and  Wien 
remain  undisturbed. 

Another  investigation  to  which  perhaps  similar  considerations  will  apply 
is  that  of  the  mechanical  force  between  parallel  slabs  conveying  rapidly 
alternating  electric  currents.  Prof.  J.  J.  Thomson's  conclusion*  is  that  the 
electromagnetic  repulsion  is  p  times  the  electrostatic  attraction,  so  that  a 
balance  will  occur  only  when  p,  =  1.  It  seems  more  probable  that  the  factor 
p,  should  be  omitted,  and  that  balance  between  the  two  kinds  of  force  is 
realized  in  every  case. 

[1902.  See  Phil.  Mag.  XLVL  p.  154,  1898,  where  Prof.  J.  J.  Thomson 
returns  to  the  consideration  of  the  question  above  raised.] 

*  Recent  Researches  in  Electricity  and  Magnetism,  1893,  §  277. 


239. 


SOME  EXPERIMENTS   WITH  THE  TELEPHONE. 

[Roy.  Inst.  Proc.  xv.  pp.  786—789,  1898;  Nature,  LVIII. 
pp.  429—430,  1898.] 

EARLY  estimates  of  the  minimum  current  of  suitable  frequency  audible 
in  the  telephone  having  led  to  results  difficult  of  reconciliation  with  the 
theory  of  the  instrument,  experiments  were  undertaken  to  clear  up  the 
question.  The  currents  were  induced  in  a  coil  of  known  construction,  either 
by  a  revolving  magnet  of  known  magnetic  moment,  or  by  a  magnetised 
tuning-fork  vibrating  through  a  measured  arc.  The  connexion  with  the 
telephone  was  completed  through  a  resistance  which  was  gradually  increased 
until  the  residual  current  was  but  just  easily  audible.  For  a  frequency  of  512 
the  current  was  found  to  be  7  x  10~8  amperes*.  This  is  a  much  less  degree 
of  sensitiveness  than  was  claimed  by  the  earlier  observers,  but  it  is  more  in 
harmony  with  what  might  be  expected  upon  theoretical  grounds. 

In  order  to  illustrate  before  an  audience  these  and  other  experiments 
requiring  the  use  of  a  telephone,  a  combination  of  that  instrument  with  a 
sensitive  flame  was  introduced.  The  gas,  at  a  pressure  less  than  that  of  the 
ordinary  supply,  issues  from  a  pin-hole  burner^  into  a  cavity  from  which  air 
is  excluded  (see  figure).  Above  the  cavity,  and  immediately  over  the  burner, 
is  mounted  a  brass  tube,  somewhat  contracted  at  the  top  where  ignition  first 
occurs J.  In  this  arrangement  the  flame  is  in  strictness  only  an  indicator, 
the  really  sensitive  organ  being  the  jet  of  gas  moving  within  the  cavity  and 
surrounded  by  a  similar  atmosphere.  When  the  pressure  is  not  too  high, 
and  the  jet  is  protected  from  sound,  the  flame  is  rather  tall  and  burns  bluish. 
Under  the  influence  of  sound  of  suitable  pitch  the  jet  is  dispersed.  At 
first  the  flame  falls,  becoming  for  a  moment  almost  invisible ;  afterwards 
it  assumes  a  more  smoky  and  luminous  appearance,  easily  distinguishable 
from  the  unexcited  flame. 

When  the  sounds  to  be  observed  come  through  the  air,  they  find  access 
by  a  diaphragm  of  tissue  paper  with  which  the  cavity  is  faced.  This 
serves  to  admit  vibration  while  sufficiently  excluding  air.  To  get  the  best 
results  the  gas  pressure  must  be  steady,  and  be  carefully  adjusted  to  the 
maximum  (about  1  inch)  at  which  the  flame  remains  undisturbed.  A  hiss 

*  The  details  are  given  in  Phil.  Mag.  Vol.  xxxvm.  p.  285  (1894).    [Vol.  iv.  p.  109.] 
f  The  diameter  of  the  pin-hole  may  be  0-03".     [inch  =  2-54  cm.] 
t  Camb.  Proc.  Vol.  iv.  p.  17,  1880.    [Vol,  i.  p.  500.] 


358 


SOME   EXPERIMENTS   WITH   THE   TELEPHONE. 


[239 


from  the  mouth  then  brings  about  the  transformation,  while  a  clap  of  the 
hands   or  the   sudden   crackling   of  a 
piece  of  paper  often  causes  extinction, 
especially   soon    after    the    flame    has 
been  lighted. 

When  the  vibrations  to  be  indicated 
are  electrical,  the  telephone  takes  the 
place  of  the  disc  of  tissue  paper,  and  it 
is  advantageous  to  lead  a  short  tube 
from  the  aperture  of  the  telephone  into 
closer  proximity  with  the  burner.  The 
earlier  trials  of  the  combination  were 
comparative  failures,  from  a  cause  that 
could  not  at  first  be  traced.  As  applied, 
for  instance,  to  a  Hughes'  induction 
balance,  the  apparatus  failed  to  indicate 
with  certainty  the  introduction  of  a 
shilling  into  one  of  the  cups,  and  the 
performance,  such  as  it  was,  seemed  to 
deteriorate  after  a  few  minutes'  experi- 
menting. At  this  stage  an  observation 
was  made  which  ultimately  afforded  a 
clue  to  the  anomalous  behaviour.  It 
was  found  that  the  telephone  became 
dewed.  At  first  it  seemed  incredible 
that  this  could  come  from  the  water  of 
combustion,  seeing  that  the  lowest  part 
of  the  flame  was  many  inches  higher. 
But  desiccation  of  the  gas  on  its  way 
to  the  nozzle  was  no  remedy,  and  it 
was  soon  afterwards  observed  that  no 
dewing  ensued  if  the  flame  were  all 
the  while  under  excitation,  either  from 
excess  of  pressure  or  from  the  action 
of  sound.  The  dewing  was  thus  con- 
nected with  the  unexcited  condition. 
Eventually  it  appeared  that  the  flame 
in  this  condition,  though  apparently 
filling  up  the  aperture  from  which  it 
issues,  was  nevertheless  surrounded  by 
a  descending  current  of  air  carrying 
with  it  part  of  the  moisture  of  combus- 
tion. The  deposition  of  dew  upon  the  nozzle  was  thus  presumably  the  source 


1898]  SOME   EXPERIMENTS   WITH   THE   TELEPHONE.  359 

of  the  trouble,  and  a  remedy  was  found  in  keeping  the  nozzle  warm  by 
means  of  a  stout  copper  wire  (not  shown)  conducting  the  heat  downwards 
from  the  hot  tube  above. 

The  existence  of  the  downward  current  could  be  made  evident  to  private 
observation  in  various  ways,  perhaps  most  easily  by  projecting  little  scraps 
of  tinder  into  the  flame,  whereupon  bright  sparks  were  seen  to  pass  rapidly 
downwards.  In  this  form  the  experiment  could  not  be  shown  to  an  audience, 
but  the  matter  was  illustrated  with  the  aid  of  a  very  delicate  ether  mano- 
meter devised  by  Professor  Dewar.  This  was  connected  with  the  upper  part 
of  the  brass  tube  by  means  of  a  small  lateral  perforation  just  below  the  root 
of  the  flame.  The  influence  of  sound  and  consequent  passage  of  the  flame 
from  the  unexcited  to  the  excited  condition  was  readily  shown  by  the  mano- 
meter, the  pressure  indicated  being  less  in  the  former  state  of  things. 

The  downward  current  is  evidently  closely  associated  with  the  change  of 
appearance  presented  by  the  flame.  In  the  excited  state  the  gas  issues 
at  the  large  aperture  above  as  from  a  reservoir  at  very  low  pressure.  The 
unexcited  flame  rises  higher,  and  must  issue  at  a  greater  speed,  carrying  with 
it  not  only  the  material  supplied  from  the  nozzle,  and  constituting  the 
original  jet,  but  also  some  of  the  gaseous  atmosphere  in  the  cavity  surround- 
ing it.  The  downward  draught  thus  appears  necessary  in  order  to  equalise 
the  total  issue  from  the  upper  aperture  in  the  two  cases. 

Although  the  flame  falls  behind  the  ear  in  delicacy,  the  combination 
is  sufficiently  sensitive  to  allow  of  the  exhibition  of  a  great  variety  of  in- 
teresting experiments.  In  the  lecture  the  introduction  of  a  threepenny 
piece  into  one  of  the  cups  of  a  Hughes'  induction  balance  was  made  evident, 
the  source  of  current  being  three  Leclanche  cells,  and  the  interrupter  being 
of  the  scraping  contact  type  actuated  by  clockwork. 

Among  other  experiments  was  shown  one  to  prove  that  in  certain  cases 
the  parts  into  which  a  rapidly  alternating  electric  current  is  divided  may 
be  greater  than  the  whole*.  The  divided  circuit  was  formed  from  the  three 
wires  with  which,  side  by  side,  a  large  flat  coil  is  wound.  One  branch  is 
formed  by  two  of  these  wires  connected  in  series,  the  other  (in  parallel  with 
the  first),  by  the  third  wire.  Steady  currents  would  traverse  all  three  wires 
in  the  same  direction.  But  the  rapidly  periodic  currents  from  the  interrupter 
distribute  themselves  so  as  to  make  the  self-induction,  and  consequently  the 
magnetic  field,  a  minimum ;  and  this  is  effected  by  the  assumption  of 
opposite  values  in  the  two  branches,  the  ratio. of  currents  being  as  2:— 1. 
On  the  same  scale  the  total  or  main  current  is  +  1.  It  was  shown  by  means 
of  the  telephone  and  flame  that  the  current  in  one  branch  was  about  the 
same  (arithmetically)  as  in  the  main,  and  that  the  current  in  the  other 
branch  was  much  greater. 

*  See  Phil.  Mag.  Vol.  xxn.  p,  496  (1886).     [Vol.  n.  p.  575.] 


240. 

LIQUID  AIR  AT  ONE   OPERATION. 

[Nature,  LVIII.  p.  199,  1898.] 

IT  is  to  be  hoped  that  personal  matters  will  not  divert  attention  from  the 
very  interesting  scientific  questions  involved.  The  liquefaction  of  air  at  one 
operation  by  Linde  and  Hampson  is  indeed  a  great  feat,  and  a  triumph  for 
the  principle  of  regeneration.  But  it  must  not  be  overlooked  that  to  allow 
the  air  to  expand  without  doing  work,  or  rather  to  allow  the  work  of  ex- 
pansion to  appear  as  heat  at  the  very  place  where  the  utmost  cooling  is 
desired,  is  very  bad  thermodynamics.  The  work  of  expansion  should  not  be 
dissipated  within,  but  be  conducted  to  the  exterior. 

I  understand  that  attempts  to  expand  the  air  under  a  piston  in  a  cylinder 
have  led  to  practical  difficulties  connected  with  the  low  temperature.  But 
surely  a  turbine  of  some  sort  might  be  made  to  work.  This  would  occupy 
little  space,  and  even  if  of  low  efficiency,  would  still  allow  a  considerable 
fraction  of  the  work  of  expansion  to  be  conveyed  away.  The  worst  turbine 
would  be  better  than  none,  and  would  probably  allow  the  pressures  to  be 
reduced.  It  should  be  understood  that  the  object  is  not  so  much  to  save  the 
work,  as  to  obviate  the  very  prejudicial  heating  arising  from  its  dissipation 
in  the  coldest  part  of  the  apparatus.  It  seems  to  me  that  the  future  may 
bring  great  developments  in  this  direction,  and  that  it  may  thus  be  possible 
to  liquefy  even  hydrogen  at  one  operation. 


241. 


ON  THE  CHARACTER  OF  THE  IMPURITY  FOUND  IN  NITRO- 
GEN GAS  DERIVED  FROM  UREA  [WITH  AN  APPENDIX 
CONTAINING  DETAILS  OF  REFRACTOMETER]. 

[Proceedings  of  the  Royal  Society,  LXIV.  pp.  95—100,  1898.] 

IT  has  already*  been  recorded  that  nitrogen,  prepared  from  urea  by 
the  action  of  sodium  hypobromite  or  hypochlorite,  is  contaminated  with 
an  impurity  heavier  than  nitrogen.  The  weight  of  pure  nitrogen  in  the 
globe  employed  being  2-299  grams,  the  gas  obtained  with  hypochlorite  was 
36  milligrams,  or  about  1^  per  cent.,  heavier.  "  A  test  with  alkaline  pyro- 
gallate  appeared  to  prove  the  absence  from  this  gas  of  free  oxygen,  and  only 
a  trace  of  carbon  could  be  detected  when  a  considerable  quantity  of  the  gas 
was  passed  over  red-hot  cupric  oxide  into  solution  of  baryta."  Most  gases 
heavier  than  nitrogen  are  excluded  from  consideration  by  the  thorough  treat- 
ment with  alkali  to  which  the  material  in  question  is  subjected.  In  view  of 
the  large  amount  of  the  impurity,  and  of  the  fact  that  it  was  removed  by 
passage  over  red-hot  iron,  I  inclined  to  identify  it  with  nitrous  oxide ;  but  it 
appeared  that  there  were  strong  chemical  objections  to  this  explanation,  and 
so  the  matter  was  left  open  at  that  time.  This  summer  I  have  returned  to 
it ;  and  although  it  is  difficult  to  establish  by  direct  evidence  the  presence  of 
nitrous  oxide,  I  think  there  can  remain  little  doubt  that  this  is  the  true 
explanation  of  the  anomaly.  I  need  scarcely  say  that  there  is  here  no 
question  of  argon  beyond  the  minute  traces  that  might  be  dissolved  in  the 
liquids  employed. 

In  the  present  experiments  hypochlorite  has  been  employed,  and  the 
procedure  has  been  the  same  as  before.  The  generating  bottle,  previously 
exhausted,  is  first  charged  with  the  full  quantity  of  hypochlorite  solution,  and 
the  urea  is  subsequently  fed  in  by  degrees.  The  gas  passes  in  succession 
over  cold  copper  turnings,  solid  caustic  soda,  and  phosphoric  anhydride.  In 
various  experiments  the  excess  of  weight  was  found  to  be  variable,  from  23 
to  36  milligrams.  In  order  to  identify  the  impurity  it  was  desirable  to  have 

*  Eayleigh  and  Ramsay,  Phil.  Trans.,  A  (1895),  p.  188.     [Vol.  iv.  p.  131.] 


362  ON   THE    CHARACTER   OF   THE    IMPURITY    FOUND   IN  [241 

as  much  of  it  as  possible,  and  experiments  were  undertaken  to  find  out  the 
conditions  of  maximum  weight.  A  change  of  procedure  to  one  in  which  the 
urea  was  first  introduced,  so  that  the  hypochlorite  would  always  be  on  the 
point  of  exhaustion,  led  in  the  wrong  direction,  giving  an  excess  of  but 
7  milligrams.  Determinations  of  refractivity  by  the  apparatus  *,  which  uses 
only  12  c.c.  of  gas,  allowed  the  substitution  of  a  miniature  generating  vessel, 
and  showed  that  the  refractivity  (and  along  with  it  the  density)  was  increased 
by  a  previous  heating  of  the  hypochlorite  to  about  140°  F.  [60°  C.].  Acting 
upon  this  information,  arrangements  were  made  for  a  preliminary  heating 
of  the  large  generating  vessel  and  its  charge,  with  the  result  that  the 
excess  of  weight  was  raised  to  55  milligrams,  or  about  2£  per  cent,  of  the 
whole.  In  any  case  heat  is  developed  during  the  reaction,  and  the  heavier 
weights  of  some  of  the  earlier  trials  probably  resulted  from  a  more  rapid 
generation  of  gas. 

In  seeking  to  obtain  evidence  as  to  the  nature  of  the  impurity,  the  most 
important  question  is  as  to  the  presence  or  the  absence  of  carbon.  The 
former  experiment  has  been  more  than  once  repeated,  with  the  result  that 
the  baryta  showed  a  slight  clouding.  Parallel  experiments,  in  which  C02  was 
purposely  introduced,  indicated  that  the  whole  carbon  in  a  charge  of  gas 
weighing  30  milligrams  in  excess  was  about  1  milligram.  It  is  possible 
(though  scarcely,  I  think,  probable)  that  this  carbon  is  not  to  be  attributed 
to  the  gas  at  all,  and  in  any  case  the  amount  appears  to  be  too  small  to  afford 
an  explanation  of  the  30  milligrams  excess  of  weight.  If  carbon  be  excluded, 
the  range  for  conjecture  is  much  narrowed.  As  to  oxygen,  only  traces  were 
found  in  most  of  the  samples  examined,  whereas  enormous  quantities  would 
be  needed  to  explain  the  excessive  weight.  It  should  be  noted,  however,  that 
the  extra  heavy  sample,  showing  55  milligrams  excess,  gave  evidence  of  con- 
taining a  more  appreciable  quantity  of  oxygen. 

It  seems  difficult  to  suggest  any  other  impurity  than  nitrous  oxide  which 
could  account  for  the  anomalous  weight.  Unfortunately  there  is  no  direct 
test  for  nitrous  oxide,  but  so  far  as  the  examination  has  been  carried,  the 
behaviour  of  the  gas  is  consistent  with  the  view  that  this  is  the  principal 
impurity.  The  gas  as  collected  has  no  smell.  The  proportion  of  nitrous 
oxide  indicated  by  the  refractometer  is  nearly  the  same  as  that  deduced  from 
the  weight.  For  example,  the  refractivity  was  observed  of  some  of  the  gas 
which  weighed  55  milligrams  in  excess.  The  proportion  by  volume  (a?)  of 
NaO  in  the  whole  required  to  explain  the  excess  of  weight  is  given  by 

22  2-299  +  0-055 

*X14  +  1-*=         2-299         ' 

whence  x  =  0'042. 

*  Roy.  Soc.  Proc.,  Vol.  LIX.  p.  201,  1896  [Vol.  iv.  p.  218];  Vol.  LX.  p.  56,  1896  [Vol.  iv. 
p.  225].  See  also  Appendix. 


1898]  NITROGEN   GAS   DERIVED   FROM   UREA.  363 

The  refractivity  (referred  to  air  as  unity)  of  the  same  gas  was  deter- 
mined by  two  independent  sets  of  observations  as  T047,  1*048;  mean, 
T0475.  If  we  assume  that  there  are  only  nitrogen  and  nitrous  oxide  present, 
the  proportion  (x)  of  the  latter  can  be  deduced  from  the  known  refrac- 
tivities  (/A  —  1)  of  nitrous  oxide,  nitrogen,  and  air,  which  are  respectively 
0-0005159,  0-0002977,  0'0002927,  the  number  for  air  being  less  than  for 
nitrogen.  Thus, 

x  x  5159  +  (l  -  a?)  x  2977  =  1-0475  x  2927, 
giving  x  =  0-0408. 

The  slight  want  of  agreement  can  be  explained  by  the  presence  of  a 
little  oxygen,  the  recognition  of  which  would  lead  to  a  rise  in  the  second 
value  of  x,  and  a  fall  in  the  first.  Examination  of  the  gas  from  the  refracto- 
meter  with  alkaline  pyrogallate  proved  that  oxygen  was  actually  present. 

Evidence  may  also  be  obtained  by  exploding  the  gas  with  excess  of 
hydrogen  for  which  purpose  oxy-hydrogen  gas  must  be  added.  But  when 
nitrous  oxide  is  in  question,  operations  over  water  are  useless,  while  for  the 
more  exact  procedure  with  mercury,  experience  and  appliances  were  somewhat 
deficient.  The  contraction  observed  was  rather  in  excess  of  the  volume  of 
nitrous  oxide  supposed  to  be  present,  but  of  this  a  good  part  is  readily  explained 
by  a  small  proportion  of  free  oxygen. 

If  the  impurity  is  really  nitrous  oxide,  it  should  admit  of  concentration 
by  solution  in  water.  To  test  this,  about  1  litre  of  water  (cooled  with  ice) 
was  shaken  with  the  contents  of  a  globe  (about  2  litres).  The  dissolved 
gases  were  then  expelled  by  boiling,  and  were  collected  over  water  rendered 
alkaline,  in  order  to  guard  against  the  introduction  of  C02.  The  quantity 
was,  of  course,  too  small  for  weighing,  but  it  could  readily  be  examined  in 
the  refractometer.  Of  one  sample,  after  desiccation,  the  refractivity  rela- 
tively to  air  was  found  to  be  as  high  as  1*207,  although  some  air  was  known 
to  have  entered  accidentally.  The  proportion  of  nitrous  oxide  in  a  mixture 
with  nitrogen  which  would  have  this  refractivity  is  0*255.  The  impurity  thus 
agrees  with  nitrous  oxide  in  being  very  much  more  soluble  in  water  than  are 
the  gases  of  the  atmosphere. 

In  the  analytical  use  of  hypobromite  for  the  determination  of  urea,  it 
has  been  noticed*  that  the  nitrogen  collected  is  deficient  by  about  8  per  cent., 
but  the  matter  does  not  appear  to  have  been  further  examined.  The 
deficiency  might  be  attributed  to  a  part  of  the  urea  remaining  undecomposed, 
but  more  probably  to  oxidation  of  nitrogen.  In  default  of  analysis  any 
nitrogen  collected  as  nitrous  oxide  would  not  appear  anomalous,  and  the 
explanation  suggested  requires  the  formation  in  addition  of  higher  oxides 
retained  by  the  alkali. 

*  Russell  and  West,  Chem.  Soc.  Journ.,  Vol.  xn.  p.  749,  1874. 


364  ON   THE    CHARACTER    OF   THE   IMPURITY   FOUND   IN  [241 

There  is  reason  to  suspect  that  nitrogen  prepared  by  the  action  of  chlorine 
upon  ammonia  is  also  contaminated  with  nitrous  oxide,  and  this  is  a  matter 
of  interest,  for  the  contamination  in  this  case  cannot  well  be  referred  to  a 
carbon  compound.  In  two  trials  with  distinct  samples  the  refractivities  were 
decidedly  in  excess  of  that  of  pure  nitrogen. 


APPENDIX. 

Details  of  Refractometer. 

Determinations  of  refractivity  have  proved  so  useful  and  can  be  made  so 
readily  and  upon  such  small  quantities  of  gas,  that  it  may  be  desirable  to 
give  further  details  of  the  apparatus  employed,  referring  for  explanation  of 
the  principles  involved  to  the  former  communication  already  cited. 

The  optical  parts,  other  than  the  tubes  containing  the  gases,  are  mounted 
independently  of  everything  else  upon  a  bar  of  T-iron  90  cm.  in  length  over 
all.  The  telescopes  are  cheap  instruments,  of  about  3  cm.  aperture  and 
30  cm.  focus,  from  which  the  eye-pieces  are  removed.  At  one  end  of  the 
T-iron  and  in  the  focus  of  the  collimating  telescope  the  original  slit  is  fixed. 
This  requires  to  be  rather  narrow,  and  was  made  by  scraping  a  fine  line 
upon  a  piece  of  silvered  glass.  At  the  further  end  the  object-glass  of  the 
observing  telescope  carries  two  slits  which  give  passage  to  the  interfering 
pencils,  and  are  situated  opposite  to  the  axes  of  the  tubes  holding  the  gases. 
The  sole  eye-piece  is  a  short  length  of  glass  rod — the  same  as  formerly 
described — of  about  4  mm.  diameter,  which  serves  as  horizontal  magnifier. 
The  gas  tubes  are  of  brass,  about  20  cm.  long  and  6  mm.  in  bore.  These  are 
soldered  together  side  by  side  and  are  closed  at  the  ends  by  plates  of  worked 
glass,  so  cemented  as  to  obstruct  as  little  as  possible  the  passage  of  light 
immediately  over  the  tubes.  There  are  two  systems  of  bands,  one  formed  by 
light  which  has  traversed  the  gases  within  the  tubes,  the  other  by  light 
which  passes  independently  above;  and  an  observation  consists  in  so  adjusting 
the  pressures  within  the  tubes  that  the  two  systems  fit  one  another.  Unless 
some  further  provision  be  made,  there  is  necessarily  a  dark  interval  between 
the  two  systems  of  bands  corresponding  to  the  thickness  of  the  walls  of  the 
tubes  and  any  projecting  cement.  It  is,  perhaps,  an  improvement  to  bring 
the  two  sets  of  bands  into  closer  juxtaposition.  The 
interval  can  be  abolished  with  the  aid  of  a  bi-plate 
[see  figure],  formed  of  worked  glass  4  or  5  mm.  thick*. 
This  is  placed  immediately  in  front  of  the  object-glass 
of  the  observing  telescope,  the  plane  of  junction  of  the 
two  glasses  being  horizontal  and  at  the  level  of  the 
obstacles  which  are  to  be  blotted  out  of  the  field  of  view. 

*  Compare  Mascart,  Traite  d'Optique,  Vol.  i.  p.  495,  1889. 


1898]  NITROGEN    GAS    DERIVED   FROM   UREA.  365 

The  objects  sought  in  the  design  of  the  remainder  of  the  apparatus 
were  (i)  the  use  of  a  minimum  of  gas,  and  (ii)  independence  of  other 
pumping  appliances.  To  this  end  the  glass  tubes  associated  with  each 
optical  tube  were  arranged  so  as  to  serve  both  as  manometer  tubes  and 
as  a  sort  of  Geissler  pump.  The  two  halves  of  the  apparatus  being  inde- 
pendent and  similar,  it  will  suffice  to  speak  of  that  which  contained  the  gas 
to  be  investigated.  The  tubes  in  which  the  levels  of  mercury  are  observed 
are  about  1  cm.  in  diameter.  The  fixed  one,  corresponding  to  the  "  pump- 
head"  of  a  Geissler  or  Topler,  is  33  cm.  in  length,  and  is  surmounted  by  a' 
three-way  tap,  allowing  it  to  be  placed  in  communication  either  with  the 
optical  tube  or  with  one  of  narrow  bore  ending  in  a  U,  drowned  in  a  deep 
mercury  trough.  The  bottom  of  the  fixed  tube,  prolonged  by  92  cm.  of 
narrower  bore,  is  connected  through  a  hose  of  black  rubber  with  the  movable 
manometer  tube.  The  latter  is  70  cm.  long  and  of  one  bore  (1  cm.)  through- 
out. It  can  either  be  held  in  the  hand  or  placed  in  a  groove  (parallel  to  the 
fixed  tube)  along  which  it  can  slide.  The  four  columns  of  mercury  stand 
side  by  side,  and  the  levels  are  referred  by  a  cathetometer  to  a  metre  scale 
which  occupies  the  central  position.  It  is  not  proposed  to  describe  the  cathe- 
tometer in  detail,  but  it  may  be  mentioned  that  it  is  of  home  construction, 
and  is  mounted  on  centres  attached  to  the  floor  and  ceiling  of  the  room.  It 
sufficed  to  record  the  levels  to  tenths  of  millimetres.  The  whole  apparatus 
was  constructed  by  Mr  Gordon. 

If  the  glasses  closing  the  optical  tubes  were  perfect,  there  would  be  coin- 
cidence of  bands  corresponding  to  complete  exhaustion  of  both  optical  tubes. 
A  correction  could  be  made  for  the  residual  error  once  for  all  determined,  but 
it  is  safer  to  make  two  independent  settings,  one  at  pressures  as  nearly  atmo- 
spheric as  the  case  admits,  and  a  second  at  minimum  pressures.  There  are 
then  in  all  eight  readings  to  be  combined.  An  example  may  be  taken  from  a 
case  already  referred  to : — 

I.  II.  III.  IV. 

9770       9371       9749       9790 
7272       2165       2469       7445 

Columns  I,  II  refer  to  the  anomalous  nitrogen,  III  and  IV  to  the  dried 
air  used  as  a  standard  of  comparison.  I  and  IV  are  the  fixed  manometer 
tubes  in  communication  with  the  optical  tubes.  The  reduction  may  be 
effected  by  subtraction  of  the  rows : 

2498  7206  7280  2345 

Thus  4708,  the  difference  between  II  and  I,  of  the  nitrogen  balances 
4935,  the  difference  between  III  and  IV,  of  air.  The  refractivity  referred  to 
air  is  accordingly  ffff,  or  T048. 


366         THE    CHARACTER   OF   THE   IMPURITY    FOUND    IN   NITROGEN    GAS.       [241 

In  this  example  the  range  of  pressures  for  the  air  is  493'5  mm.,  or  about 
two-thirds  of  an  atmosphere. 

Great  care  is  sometimes  required  to  ensure  matching  the  same  bands  in 
the  two  settings.  A  mistake  of  one  band  in  the  above  example  would  entail 
nearly  2  per  cent,  error  in  the  final  result,  inasmuch  as  the  whole  number  of 
bands  concerned  is  about  96  per  atmosphere  of  air,  or  about  62  over  the 
range  actually  used.  It  is  wise  always  to  include  a  match  with  pressures 
about  midway  between  the  extremes.  If  the  results  harmonise,  an  error  of 
a  single  band  is  excluded ;  and  it  is  hardly  possible  to  make  a  mistake  of 
two  bands. 

As  regards  accuracy,  independent  final  results  usually  agree  to  one- 
thousandth  part. 


242. 

ON  ISO-PERIODIC   SYSTEMS. 

[Philosophical  Magazine,  XLVI.  pp.  567—569,  1898.] 

IN  general  a  system  with  m  degrees  of  freedom  vibrating  about  a  con- 
figuration of  equilibrium  has  m  distinct  periods,  or  frequencies,  of  vibration, 
but  in  particular  cases  two  or  more  of  these  frequencies  may  be  equal.  The 
simple  spherical  pendulum  is  an  obvious  example  of  two  degrees  of  freedom 
whose  frequencies  are  equal.  It  is  proposed  to  point  out  the  properties 
of  vibrating  systems  of  such  a  character  that  all  the  frequencies  are  equal. 

In  the  general  case  when  a  system  is  referred  to  its  normal  coordinates 
</>!,  </>2,  ...  we  have  for  the  kinetic  and  potential  energies*, 


-*tflk... 

and  for  the  vibrations 

4>1  =  Acos(n1t-a),        </>2  =  B  cos(n2t  -/3),          &c  ..........  (2) 

where  A,  B,  ...  a,  @  ...  are  arbitrary  constants  and 

n^c,/^,         n22=c2/a2,         &c  ......................  (3) 

If  «!,  n^,  &c.,  are  all  equal,  T  and  V  are  of  the  same  form  except  as 
to  a  constant  multiplier.  By  supposing  a,  ft  .  .  .  equal,  we  see  that  any 
prescribed  ratios  may  be  assigned  to  fa,  <£2  ...,  so  that  vibrations  of  arbitrary 
type  are  normal  and  can  be  executed  without  constraint.  In  particular  any 
parts  of  the  system  may  remain  at  rest. 

If  x,  y,  z  be  the  space  coordinates  (measured  from  the  equilibrium  position) 
of  any  point  of  the  system,  the  most  general  values  are  given  by 

x  =  X±  cos  nt  +  X2  sin  nt  \ 

y=  Fjcos  nt+  Y^sinnt  L  ........................  (4) 

z  =  Z1  cos  nt  +  Z2  sin  nt  } 
*  See,  for  example,  Theory  of  Sound,  §  87. 


368  ON   ISO-PERIODIC   SYSTEMS.  [242 

where  Xl}  X2,  &c.  are  constants  for  each  point.  These  equations  indicate 
elliptic  motion  in  the  plane 

x(Y1Z2-Z1Y2)  +  y(ZlX2-X1Z2)  +  z(XlY2-Y1XJ  =  0 (5) 

Thus  every  point  of  the  system  describes  an  elliptic  orbit  in  the  same  periodic 
time. 

An  interesting  case  is  afforded  by  a  line  of  similar  bodies  of  which  each 
is  similarly  connected  to  its  neighbours*.  The  general  formula  for  w2  is 

_  C,  -  2fl  cos  ka  -  2 C2  cos  2ka  -  . . . 
~ A0-  2A,  cos  ka  -  2A2  cos  2ka  -  . . .  ' 

in  which  the  constants  CQ,  C^  ...  refer  to  the  potential,  and  Al}  A^  ...  to 
the  kinetic  energy.  Here  C1}  A^  represent  the  influence  of  immediate 
neighbours  distant  a  from  one  another,  C2,  A2  the  influence  of  neighbours 
distant  2a,  and  so  on.  Further,  k  denotes  2-Tr/A,,  X  being  the  wave-length. 
If  C-i,  C2  ...  ,  Alt  A2 ...  vanish,  each  body  is  uninfluenced  by  its  neighbours, 
and  the  case  is  one  considered  by  Reynolds  of  a  number  of  similar  and 
disconnected  pendulums  hanging  side  by  side  at  equal  distances.  It  is 
obvious  that  a  vibration  of  any  type  is  normal  and  is  executed  in  the  same 
time.  If  we  consider  a  progressive  wave,  its  velocity  is  proportional  to  A,. 
A  disturbance  communicated  to  any  region  has  no  tendency  to  propagate 
itself ;  the  "  group  velocity  "  is  zero. 

Although  the  line  of  disconnected  pendulums  is  interesting  and  throws 
light  upon  the  general  theory  of  wave  and  group  propagation,  one  can  hardly 
avoid  the  feeling  that  it  is  only  by  compliment  that  it  is  regarded  as  a  single 
system.  It  is  therefore  not  without  importance  to  notice  that  there  are  other 
cases  for  which  n  assumes  a  constant,  and  the  group-velocity  a  zero,  value. 
To  this  end  it  is  only  necessary  that 

C0:C1:Ca:...=A0:A1:A,: (7) 

If  this  condition  be  satisfied,  the  connexion  of  neighbouring  bodies  does  not 
entail  the  propagation  of  disturbance.  Any  number  of  the  bodies  may  remain 
at  rest,  and  all  vibrations  have  the  same  period. 

We  might  consider  particular  systems  for  which  C2,  C3 ...  A.2,  A3...  vanish, 
while  CJ/CQ  =  A1/A0 ;  but  it  is  perhaps  more  interesting  to  draw  an  illustra- 
tion from  the  case  of  continuous  linear  bodies.  Consider  a  wire  stretched 
with  tension  T1}  each  element  dx  of  which  is  urged  to  its  position  of  equili- 
brium (y  =  0)  by  a  force  equal  to  pydx.  The  potential  energyf  is  given  by 


(8) 


*  Phil.  Mag.  Vol.  XLIV.  p.  356,  1897.     [Vol.  iv.  p.  340.] 
t  See  Theory  of  Sound,  §§  122,  162,  188. 


1898]  ON   ISO-PERIODIC   SYSTEMS.  369 

If  the  "rotatory  inertia"  be  included,  the  corresponding  expression  for  the 
kinetic  energy  is 


in  which  p  is  the  volume  density,  to  the  area  of  cross  section,  and  K  the  radius 
of  gyration  of  the  cross  section  about  an  axis  perpendicular  to  the  plane 
of  bending.  In  waves  along  an  actual  wire  vibrating  transversely  the  second 
term  would  be  relatively  unimportant,  but  there  is  no  contradiction  in  the 
supposition  that  the  rotatory  term  is  predominant.  The  differential  equation 
derived  from  (8)  and  (9)  is 

a2S+c22/=°>    (10) 


where  a*=T1/pa>,         c2  =  /i//xo  ......................  (11) 

If  we  suppose  that  there  is  no  tension  and  no  rotatory  inertia,  a  =  0,  K  =  0, 
and  the  solution  of  (10)  may  be  written 

y  =  cos  ct  .  yl  +  sinc£  .  yZi    ..................  .  .....  (12) 

2/i  >  2/2  being  arbitrary  functions  of  x.     If  yl  =  cos  mx,  yz—  sin  mx,  (12)  becomes 
y  =  cos  (ct  —  mx),      ...........................  (13) 

and  the  velocity  of  propagation  (elm)  is  proportional  to  \,  equal  to  27r/m. 
This  is  the  case  of  the  disconnected  pendulums. 

On  the  other  hand  we  may  equally  well  suppose  that  c  is  zero  and  that 
the  rotatory  inertia  is  paramount,  so  that  (10)  reduces  to 


The  periodic  part  of  the  solution  is  again  of  the  form  (12),  and  has  the  same 
peculiar  properties  as  before. 

In  the  general  case  we  have  the  solution  for  stationary  vibrations 

y  =  sin  mx  cos  nt,  ..............................  (14) 

where  m=  ITT  /I,  i  being  an  integer,  and 


This  gives  the  frequencies  for  the  various  modes  of  vibration  of  a  wire  of 
length  I  fastened  at  the  ends. 

If  /e2  =  a2/c2,  n  becomes  independent  of  m  as  before. 

If  K2  <  a2/c2,w2  increases,  as  i  and  m  increase,  and  approaches  a  finite  upper 
limit  a2//c2.  The  series  of  frequencies  is  thus  analogous  to  those  met  with  in 
the  spectra  of  certain  bodies*. 

*  Compare  Schuster,  Nature,  Vol.  LV.  p.  200  (1890). 
R.    iv.  24 


243. 

ON  JAMES   BERNOULLI'S  THEOREM   IN    PROBABILITIES. 

[Philosophical  Magazine,  XLVII.  pp.  246 — 251,  1899.] 

IF  p  denote  the  probability  of  an  event,  then  the  probability  that  in  p, 
trials  the  event  will  happen  in  times  and  fail  n  times  is  equal  to  a  certain 
term  in  the  expansion  of  (p  +  q)*,  namely, 


m\n\r 
where  p  +  q=l,  m  +  n  =  fjt,. 

"  Now  it  is  known  from  Algebra  that  if  m  and  n  vary  subject  to  the 
condition  that  m  +  n  is  constant,  the  greatest  value  of  the  above  term  is 
when  m/n  is  as  nearly  as  possible  equal  to  p/q,  so  that  m  and  n  are  as  nearly 
as  possible  equal  to  pp  and  pq  respectively.  WTe  say  as  nearly  as  possible, 
because  p.p  is  not  necessarily  an  integer,  while  m  is.  We  may  denote  the 
value  of  m  by  up  +  z,  where  z  is  some  proper  fraction,  positive  or  negative ; 
and  then  n  =  p,q  —  z" 

The  rth  term,  counting  onwards,  in  the  expansion  of  (p  +  q)*  after  (1)  is 


—  r\ 


(2) 


The  approximate  value  of  (2)  when  in  and  n  are  large  numbers  may  be 
obtained  with  the  aid  of  Stirling's  theorem,  viz. 

(3) 


The  process  is  given  in  detail  after  Laplace  in  Todhunter's  History  of  the 
Theory  of  Probability,  p.  549,  from  which  the  above  paragraph  is  quoted. 
The  expression  for  the  rth  term  after  the  greatest  is 


n</pLprzr(n-m)_T*     ,    1*  }  . 
rmn}  (        mn          2mn         6m2  T  6w2)  ' 


1899]          ON  JAMES  BERNOULLI'S  THEOREM  IN  PROBABILITIES.  371 

and  that  for  the  rth  term  before  the  greatest  may  be  deduced  by  changing 
the  sign  of  r  in  (4). 

It  is  assumed  that  r2  does  not  surpass  p,  in  order  of  magnitude,  and 
fractions  of  the  order  I//JL  are  neglected. 

There  is  an  important  case  in  which  the  circumstances  are  simpler  than 
in  general.  It  arises  when  p  =  q  =  £ ,  and  //,  is  an  even  number,  so  that 
m  =  n=  £//,.  Here  z  disappears  ab  initio,  and  (4)  reduces  to 


representing  (2),  which  now  becomes 

(6) 


An  important  application  of  (5)  is  to  the  theory  of  random  vibrations. 
If  /A  vibrations  are  combined,  each  of  the  same  phase  but  of  amplitudes  which 
are  at  random  either  +1  or  —  1,  (5)  represents  the  probability  of  \p  +  r  of 
them  being  positive  vibrations,  and  accordingly  \^—r  being  negative.  In 
this  case,  and  in  this  case  only,  is  the  resultant  +  2r.  Hence  if  x  represent 
the  resultant,  the  chance  of  x,  which  is  necessarily  an  even  integer,  is 


The  next  greater  resultant  is  (x  +  2);  so  that  when  x  is  great  the  above 
expression  may  be  supposed  to  correspond  to  a  range  for  x  equal  to  2.  If  we 
represent  the  range  by  dx,  the  chance  of  a  resultant  lying  between  x  and 
x  +  dx  is  given  by 


Another  view  of  this  matter,  leading  to  (5)  or  (7)  without  the  aid  of 
Stirling's  theorem,  or  even  of  formula  (1),  is  given  (somewhat  imperfectly)  in 
Theory  of  Sound,  2nd  ed.  §  42  a.  It  depends  upon  a  transition  from  an 
equation  in  finite  differences  .to  the  well-known  equation  for  the  conduction 
of  heat  and  the  use  of  one  of  Fourier's  solutions  of  the  latter.  Let/(/*,  r) 
denote  the  chance  that  the  number  of  events  occurring  (in  the  special  ap- 
plication positive  vibrations)  is  \p  +  r,  so  that  the  excess  is  r.  Suppose  that 
each  random  combination  of  /*  receives  two  more  random  contributions  —  two 
in  order  that  the  whole  number  may  remain  even,  —  and  inquire  into  the 
chance  of  a  subsequent  excess  r,  denoted  by  /(ft  +  2,  r).  The  excess  after  the 
addition  can  only  be  r  if  previously  it  were  r  —  1,  r,  or  r  +  1.  In  the  first 
case  the  excess  becomes  r  by  the  occurrence  of  both  of  the  two  new  events, 

*  Phil.  Mag.  Vol.  x.  p.  75  (1880).    [Vol.  i.  p.  491.] 


372  ON  JAMES  BERNOULLI'S  THEOREM  IN  PROBABILITIES.  [243 

of  which  the  chance  is  \  .  In  the  second  case  the  excess  remains  r  in  conse- 
quence of  one  event  happening  and  the  other  failing,  of  which  the  chance  is 
£;  and  in  the  third  case  the  excess  becomes  r  in  consequence  of  the  failure 
of  both  the  new  events,  of  which  the  chance  is  \.  Thus 

/(/*  +  2,  r)  =  If  (p.,  r  -  1)  +  i/0*.  r)  +  £/(,.,  r  +  1)  .......  (8) 

According  to  the  present  method  the  limiting  form  of  f  is  to  be  derived  from 
(8).  We  know,  however,  that/  has  actually  the  value  given  in  (6),  by  means 
of  which  (8)  may  be  verified. 

Writing  (8)  in  the  form 

/(/*  +  2,  r)  -f(p.,  r)  =  i/0*.  r  -  1)  -  1/0*.  r)  +  £/(,*,  r  +  1),  ...(9) 

we  see  that  when  p.  and  r  are  infinite  the  left-hand  member  becomes  Zdf/dfj,, 
and  the  right-hand  member  becomes  ^d^f/dr2,  so  that  (9)  passes  into  the 
differential  equation 


In  (9),  (10)  r  is  the  excess  of  the  actual  occurrences  over  |/z.  If  we  take  # 
to  represent  the  difference  between  the  number  of  occurrences  and  the  number 
of  failures,  x  =  2r  and  (10)  becomes 

#-*#'  (11) 

dp,      2dx*' 

In  the  application  to  vibrations  /(/A,  #)  then  denotes  the  chance  of  a  resultant 
+  x  from  a  combination  of  p,  unit  vibrations  which  are  positive  or  negative 
at  random. 

In  the  formation  of  (10)  we  have  supposed  for  simplicity  that  the  addition 
to  p,  is  2,  the  lowest  possible  consistently  with  the  total  number  remaining 
even.  But  if  we  please  we  may  suppose  the  addition  to  be  any  even  number 
//.  The  analogue  of  (8)  is  then 

2*'  ./(/,  +  /,  r)  =  /(,*,  r  -  I,*')  -f  p.'  /(p.,  r  -  ^  +  1) 

+  /-i)/(^  r  _  ^  +  2)  +  _ 


and  when  /A  is  treated  as  very  great  the  right-hand  member  becomes 


*'  (/*'  -  2)2  +  1  .  //2    . 


1899]          ON  JAMES  BERNOULLI'S  THEOREM  IN  PROBABILITIES.  373 

The  series  which  multiplies  f  is  (1  +  \Y'>  or  2M/.  The  second  series  is 
equal  to  jjf  .  2**  ',  as  may  be  seen  by  comparison  of  coefficients  of  #2  in  the 
equivalent  forms 

(e*  +  e~x)n  =  2"  (1  +  %x*  +  .  .  .)» 


. 
The  value  of  the  left-hand  member  becomes  simultaneously 


so  that  we  arrive  at  the  same  differential  equation  (10)  as  before. 

This  is  the  well-known  equation  for  the  conduction  of  heat,  and  the 
solution  developed  by  Fourier  is  at  once  applicable.  The  symbol  /JL  corre- 
sponds to  time  and  r  to  a  linear  coordinate.  The  special  condition  is  that 
initially  —  that  is  when  /*  is  relatively  small  —  /must  vanish  for  all  values  of  r 
that  are  not  small.  We  take  therefore 


which  may  be  verified  by  differentiation. 

The  constant  A  may  be  determined  by  the  understanding  that/(/ci,  r)dr 
is  to  represent  the  chance  of  an  excess  lying  between  r  and  r  +  dr,  and  that 
accordingly 

+)rfr  =  l  ............................  (13) 


r+oo 

Since   I      e~lftdz  =  *Jir,  we  have 

£ 

and,  finally,  as  the  chance  that  the  excess  lies  between  r  and  r  +  dr, 


Another  method  by  which   A   in  (12)  might  be  determined  would  be  by 
comparison  with  (6)  in  the  case  of  r  =  0.     In  this  way  we  find 

A  til  1.3.5...0*-!) 


\     2.4.6 


J  ( 


—}  by  Wallis'  theorem. 


374  ON  JAMES  BERNOULLI'S  THEOREM  IN  PROBABILITIES.  [243 

If,  as  is  natural  in  the  problem  of  random  vibrations,  we  replace  r 
by  x,  denoting  the  difference  between  the  number  of  occurrences  and 
the  number  of  failures,  we  have  as  the  chance  that  x  lies  between  x  and 
x  +  dx 


identical  with  (7). 

In  the  general  case  when  p  and  q  are  not  limited  to  the  values  £, 
it  is  more  difficult  to  exhibit  the  argument  in  a  satisfactory  form, 
because  the  most  probable  numbers  of  occurrences  and  failures  are  no 
longer  definite,  or  at  any  rate  simple,  fractions  of  /i.  But  the  general 
idea  is  substantially  the  same.  The  excess  of  occurrences  over  the  most 
probable  number  is  still  denoted  by  r,  and  its  probability  by  /(/*,  r}.  We 
regard  r  as  continuous,  and  we  then  suppose  that  p  increases  by  unity. 
If  the  event  occurs,  of  which  the  chance  is  p,  the  total  number  of  occurrences 
is  increased  by  unity.  But  since  the  most  probable  number  of  occurrences 
is  increased  by  p,  r  undergoes  only  an  increase  measured  by  1  —  p  or  q. 
In  like  manner  if  the  event  fails,  r  undergoes  a  decrease  measured  by  p. 
Accordingly 

(17) 


On  the  right  of  (17)  we  expandy(/A,  r  —  q),  f([i,  r  +  p)  in  powers  of  p  and  q. 
Thus 


so  that  the  right-hand  member  is 


The  left-hand  member  may  be  represented  by/+  df/d/j,,  so  that  ultimately 


Accordingly  by  the  same  argument  as  before  the  chance  of  an  excess  r  lying 
between  r  and  r  +  dr  is  given  by 


(19) 


We  have  already  considered  the  case  of  p  =  q  =  |.  Another  particular  case 
of  importance  arises  when  p  is  very  small,  and  accordingly  q  is  nearly  equal 
to  unity.  The  whole  number  /*  is  supposed  to  be  so  large  that  pjj,,  or  m, 


1899]  ON  JAMES  BERNOULLI'S  THEOREM  IN  PROBABILITIES.  375 

representing  the  most  probable  number  of  occurrences,  is  also  large.     The 
general  formula  now  reduces  to 

1 

_r2/2r/iJr.  /9ft  \ 

V(2^)e 

which  gives  the  probability  that  the  number  of  occurrences  shall  lie  between 
m  +  r  and  m  +  r  +  dr.     It  is  a  function  of  m  and  r  only. 


The  probability  of  the  deviation  from  m  lying  between  +  r 


(21) 


where  r  =  r/\/(2m).  This  is  equal  to  '84  when  r  =  TO,  or  r  =  ^(2m)  ;  so  that 
the  chance  is  comparatively  small  of  a  deviation  from  m  exceeding  +  V(2w). 
For  example,  if  m  is  50,  there  is  a  rather  strong  probability  that  the  actual 
number  of  occurrences  will  lie  between  40  and  60. 

The  formula  (20)  has  a  direct  application  to  many  kinds  of  statistics. 


244. 


ON  THE  COOLING  OF  AIR  BY  RADIATION  AND  CONDUCTION, 
AND  ON   THE   PROPAGATION   OF  SOUND. 

[Philosophical  Magazine,  XLVII.  pp.  308—314,  1899.] 

ACCORDING  to  Laplace's  theory  of  the  propagation  of  Sound  the  expansions 
(and  contractions)  of  the  air  are  supposed  to  take  place  without  transfer  of 
heat.  Many  years  ago  Sir  G.  Stokes*  discussed  the  question  of  the  influence 
of  radiation  from  the  heated  air  upon  the  propagation  of  sound.  He  showed 
that  such  small  radiating  power  as  is  admissible  would  tell  rather  upon  the 
intensity  than  upon  the  velocity.  If  a;  be  measured  in  the  direction  of 
propagation,  the  factor  expressing  the  diminution  of  amplitude  is  e~mx,  where 

m  =  Tl±£m  ...(i) 

7     2a 

In  (1)  7  represents  the  ratio  of  specific  heats  (1'41),  a  is  the  velocity  of  sound, 
and  q  is  such  that  e~qt  represents  the  law  of  cooling  by  radiation  of  a  small 
mass  of  air  maintained  at  constant  volume.  If  r  denote  the  time  required  to 
traverse  the  distance  x,  r  =  x/a,  and  (1)  may  be  taken  to  assert  that  the 
amplitude  falls  to  any  fraction,  e.g.  one-half,  of  its  original  value  in  7  times 
the  interval  of  time  required  by  a  mass  of  air  to  cool  to  the  same  fraction 
of  its  original  excess  of  temperature.  "  There  appear  to  be  no  data  by  which 
the  latter  interval  can  be  fixed  with  any  approach  to  precision ;  but  if  we 
take  it  at  one  minute,  the  conclusion  is  that  sound  would  be  propagated  for 
(seven)  minutes,  or  travel  over  about  (80)  miles,  without  very  serious  loss  from 
this  cause  f."  We  shall  presently  return  to  the  consideration  of  the  probable 
value  of  q. 

Besides  radiation  there  is  also  to  be  considered  the  influence  of  conductivity 
in  causing  transfer  of  heat,  and  further  there  are  the  effects  of  viscosity. 

«  Phil.  Mag.  [4]  i.  p.  305,  1851 ;  Theory  of  Sound,  §  247. 
t  Proc.  Roy.  Inst.  April  9,  1897.     [Vol.  iv.  p.  298.] 


1899]       ON   THE   COOLING   OF   AIR   BY   RADIATION   AND   CONDUCTION.  377 

The  problems  thus  suggested  have  been  solved  by  Stokes  and  Kirchhoff*. 
If  the  law  of  propagation  be 

U  =  e-m'*co8(nt-as/a),    (2) 


then 


in  which  the  frequency  of  vibration  is  w/2-Tr,  /jf  is  the  kinematic  viscosity,  and 
v  the  thermometric  conductivity.  In  c.G.S.  measure  we  may  take  //  =  "14, 
v  =  '26,  so  that 


To  take  a  particular  case,  let  the  frequency  be  256  ;  then  since  a  =  33200, 
we  find  for  the  time  of  propagation  during  which  the  amplitude  diminishes 
in  the  ratio  of  e  :  1, 

(ma)-1  =  3560  seconds. 

Accordingly  it  is  only  very  high  sounds  whose  propagation  can  be  ap- 
preciably influenced  by  viscosity  and  conductivity. 

If  we  combine  the  effects  of  radiation  with  those  of  viscosity  and  conduction, 
we  have  as  the  factor  of  attenuation 

Q—  (m+m')x 

where  m  +  m'  =  "14<  (q  /  a)  +  !12(n9/a*)  ......................  (4) 

In  actual  observations  of  sound  we  must  expect  the  intensity  to  fall  off 
in  accordance  with  the  law  of  inverse  squares  of  distances.  A  very  little 
experience  of  moderately  distant  sounds  shows  that  in  fact  the  intensity  is  in 
a  high  degree  uncertain.  These  discrepancies  are  attributable  to  atmospheric 
refraction  and  reflexion,  and  they  are  sometimes  very  surprising.  But  the 
question  remains  whether  in  a  uniform  condition  of  the  atmosphere  the 
attenuation  is  sensibly  more  rapid  than  can  be  accounted  for  by  the  law  of 
inverse  squares.  Some  interesting  experiments  towards  the  elucidation  of 
this  matter  have  been  published  by  Mr  Wilmer  Duff  -f-,  who  compared  the 
distances  of  audibility  of  sounds  proceeding  respectively  from  two  and  from 
eight  similar  whistles.  On  an  average  the  eight  whistles  were  audible  only 
about  one-fourth  further  than  a  pair  of  whistles  ;  whereas,  if  the  sphericity  of 
the  waves  had  been  the  only  cause  of  attenuation,  the  distances  would  have 
been  as  2  to  1.  Mr  Duff  considers  that  in  the  circumstances  of  his  experi- 
ments there  was  little  opportunity  for  atmospheric  irregularities,  and  he 
attributes  the  greater  part  of  the  falling  off  to  radiation.  Calculating  from 
(1)  he  deduces  a  radiating  power  such  that  a  mass  of  air  at  any  given  excess 
of  temperature  above  its  surroundings  will  (if  its  volume  remain  constant) 
fall  by  radiation  to  one-half  of  that  excess  in  about  one-twelfth  of  a  second. 

*  Fogg.  Ann.  Vol.  cxxxiv.  p.  177,  1868  ;  Theory  of  Sound,  2nd  ed.  §  348. 
t  Phys.  Review,  Vol.  vi.  p.  129,  1898. 


378  ON   THE   COOLING   OF   AIB   BY    RADIATION    AND    CONDUCTION,  [244 

In  this  paper  I  propose  to  discuss  further  the  question  of  the  radiating 
power  of  air,  and  I  shall  contend  that  on  various  grounds  it  is  necessary  to 
restrict  it  to  a  value  hundreds  of  times  smaller  than  that  above  mentioned. 
On  this  view  Mr  Duff's  results  remain  unexplained.  For  myself  I  should 
still  be  disposed  to  attribute  them  to  atmospheric  refraction.  If  further 
experiment  should  establish  a  rate  of  attenuation  of  the  order  in  question 
as  applicable  in  uniform  air,  it  will  I  think  be  necessary  to  look  for  a  cause 
not  hitherto  taken  into  account.  We  might  imagine  a  delay  in  the  equaliza- 
tion of  the  different  sorts  of  energy  in  a  gas  undergoing  compression,  not 
wholly  insensible  in  comparison  with  the  time  of  vibration  of  the  sound.  If 
in  the  dynamical  theory  we  assimilate  the  molecules  of  a  gas  to  hard  smooth 
bodies  which  are  nearly  but  not  absolutely  spherical,  and  trace  the  effect  of  a 
rapid  compression,  we  see  that  at  the  first  moment  the  increment  of  energy  is 
wholly  translational  and  thus  produces  a  maximum  effect  in  opposing  the 
compression.  A  little  later  a  due  proportion  of  the  excess  of  energy  will 
have  passed  into  rotational  forms  which  do  not  influence  the  pressure,  and 
this  will  accordingly  fall  off.  Any  effect  of  the  kind  must  give  rise  to 
dissipation,  and  the  amount  of  it  will  increase  with  the  time  required  for  the 
transformations,  i.e.  in  the  above  mentioned  illustration  with  the  degree  of 
approximation  to  the  spherical  form.  In  the  case  of  absolute  spheres  no 
transformation  of  translatory  into  rotatory  energy,  or  vice  versa,  would 
occur  in  a  finite  time.  There  appears  to  be  nothing  in  the  behaviour  of 
gases,  as  revealed  to  us  by  experiment,  which  forbids  the  supposition  of 
a  delay  capable  of  influencing  the  propagation  of  sound. 

Returning  now  to  the  question  of  the  radiating  power  of  air,  we  may 
establish  a  sort  of  superior  limit  by  an  argument  based  upon  the  theory  of 
exchanges,  itself  firmly  established  by  the  researches  of  B.  Stewart.  Consider 
a  spherical  mass  of  radius  r,  slightly  and  uniformly  heated.  Whatever  may 
be  the  radiation  proceeding  from  a  unit  of  surface,  it  must  be  less  than  the 
radiation  from  an  ideal  black  surface  under  the  same  conditions.  Let  us, 
however,  suppose  that  the  radiation  is  the  same  in  both  cases  and  inquire 
what  would  then  be  the  rate  of  cooling.  According  to  Bottomley*  the 
emissivity  of  a  blackened  surface  moderately  heated  is  '0001.  This  is  the 
amount  of  heat  reckoned  in  water-gram-degree  units  emitted  in  one  second 
from  a  square  centimetre  of  surface  heated  1°  C.  If  the  excess  of  temperature 
be  6,  the  whole  emission  is 

0  x  47rr2  x  -0001 

On  the  other  hand,  the  capacity  for  heat  is 

fur3  x  -0013  x  -24, 

the  first  factor  being  the  volume,  the  second  the  density,  and  the  third  the 
*  Everett,  C.G.S.  Units,  1891,  p.  134. 


1899]  AND   ON   THE   PROPAGATION   OF   SOUND.  379 

specific  heat  of  air  referred,  as  usual,  to  water.     Thus  for  the  rate  of  cooling, 
d6  '0003  1 


whence  0  =  00ertlr,     .................................  (5) 

00  being   the   initial  value  of  0.     The  time  in  seconds  of  cooling   in  •  the 
ratio  of  e  :  1   is  thus  represented  numerically  by  r  expressed  in  centims. 

When  r  is  very  great,  the  suppositions  on  which  (5)  is  calculated  will 
be  approximately  correct,  and  that  equation  will  then  represent  the  actual 
law  of  cooling  of  the  sphere  of  air,  supposed  to  be  maintained  uniform  by 
mixing  if  necessary.  But  ordinary  experience,  and  more  especially  the 
observations  of  Tyndall  upon  the  diathermancy  of  air,  would  lead  us  to 
suppose  that  this  condition  of  things  would  not  be  approached  until  r 
reached  1000  or  perhaps  10,000  centims.  For  values  of  r  comparable  with 
the  half  wave-length  of  ordinary  sounds,  e.g.  30  centim.,  it  would  seem  that 
the  real  time  of  cooling  must  be  a  large  multiple  of  that  given  by  (5). 
At  this  rate  the  time  of  cooling  of  a  mass  of  air  must  exceed,  and  probably 
largely  exceed,  60  seconds.  To  suppose  that  this  time  is  one-twelfth  of  a 
second  would  require  a  sphere  of  air  2  millim.  in  diameter  to  radiate  as  much 
heat  as  if  it  were  of  blackened  copper  at  the  same  temperature. 

Although,  if  the  above  argument  is  correct,  there  seems  little  likelihood 
of  the  cooling  of  moderate  masses  of  air  being  sensibly  influenced  by  radiation, 

1  thought  it  would  be  of  interest  to  inquire  whether  the  observed  cooling  (or 
heating)  in  an  experiment  on  the  lines  of  Clement  and  Desormes  could  be 
adequately  explained  by  the  conduction  of  heat  from  the  walls  of  the  vessel 
in  accordance  with  the  known  conductivity  of  air.     A  nearly  spherical  vessel 
of  glass  of  about  35  centim.  diameter,  well  encased,  was  fitted,  air-tight,  with 
two  tubes.     One  of  these  led  to  a  manometer  charged  with  water  or  sulphuric 
acid;  the  other  was  provided  with  a  stopcock  and  connected  with  an  air- 
pump.     In  making  an  experiment  the  stopcock  was  closed  and  a  vacuum 
established  in  a  limited  volume  upon  the  further  side.     A  rapid  opening  and 
reclosing  of  the  cock  allowed  a  certain  quantity  of  air  to  escape  suddenly,  and 
thus  gave  rise  to  a  nearly  uniform  cooling  of  that  remaining  behind  in  the 
vessel.     At  the  same   moment  the  liquid  rose  in  the  manometer,  and  the 
observation  consisted  in  noting  the  times  (given  by  a  metronome  beating 
seconds)  at  which  the  liquid  in  its  descent  passed  the  divisions  of  a  scale, 
as  the  air  recovered  the  temperature  of  the  containing  vessel.     The  first 
record  would  usually  be  at  the  third  or  fourth  second  from  the  turning  of  the 
cock,  and  the  last  after  perhaps  120  seconds.     In  this  way  data  are  obtained 
for  a  plot  of  the  curve  of  pressure  ;  and  the  part  actually  observed  has  to 
be  supplemented  by  extrapolation,  so  as  to  go  back  to  the  zero  of  time  (the 
moment  of  turning  the  tap)  and  to  allow  for  the  drop  which  might  occur 


380  ON   THE   COOLING  OF   AIR  BY  RADIATION   AND   CONDUCTION,          [244 

subsequent  to  the  last  observation.  An  estimate,  which  cannot  be  much  in 
error,  is  thus  obtained  of  the  whole  rise  in  pressure  during  the  recovery  of 
temperature,  and  for  the  time,  reckoned  from  the  commencement,  at  which 
the  rise  is  equal  to  one-half  of  the  total. 

In  some  of  the  earlier  experiments  the  whole  rise  of  pressure  (fall  in  the 
manometer)  during  the  recovery  of  temperature  was  about  20  millim.  of 
water,  and  the  time  of  half  recovery  was  15  seconds.  I  was  desirous  of 
working  with  the  minimum  range,  since  only  in  this  way  could  it  be  hoped 
to  eliminate  the  effect  of  gravity,  whereby  the  interior  and  still  cool  parts 
of  the  included  air  would  be  made  to  fall  and  so  come  into  closer  proximity 
to  the  walls,  and  thus  accelerate  the  mean  cooling.  In  order  to  diminish 
the  disturbance  due  to  capillarity,  the  bore  of  the  manometer-tube,  which 
stood  in  a  large  open  cistern,  was  increased  to  about  18  millim.*,  and  suitable 
optical  arrangements  were  introduced  to  render  small  movements  easily 
visible.  By  degrees  the  range  was  diminished,  with  a  prolongation  of  the 
time  of  half  recovery  to  18,  22,  24,  and  finally  to  about  26  seconds.  The 
minimum  range  attained  was  represented  by  3  or  4  millim.  of  water,  and  at 
this  stage  there  did  not  appear  to  be  much  further  prolongation  of  cooling 
in  progress.  There  seemed  to  be  no  appreciable  difference  whether  the 
air  was  artificially  dried  or  not,  but  in  no  case  was  the  moisture  sufficient 
to  develop  fog  under  the  very  small  expansions  employed.  The  result  of  the 
experiments  may  be  taken  to  be  that  when  the  influence  of  gravity  was, 
as  far  as  practicable,  eliminated,  the  time  of  half  recovery  of  temperature  was 
about  26  seconds. 

It  may  perhaps  be  well  to  give  an  example  of  an  actual  experiment. 
Thus  in  one  trial  on  Nov.  1,  the  recorded  times  of  passage  across  the  divisions 
of  the  scale  were  3,  6,  11,  18,  26,  35,  47,  67,  114  seconds.  The  divisions 
themselves  were  millimetres,  but  the  actual  movements  of  the  meniscus  were 
less  in  the  proportion  of  about  2£ :  1.  A  plot  of  these  numbers  shows  that 
one  division  must  be  added  to  represent  the  movement  between  0s  and  3s, 
and  about  as  much  for  the  movement  to  be  expected  between  114s  and  oo . 
The  whole  range  is  thus  10  divisions  (corresponding  to  4  millim.  at  the 
meniscus),  and  the  mid-point  occurs  at  26s.  On  each  occasion  3  or  4 
sets  of  readings  were  taken  under  given  conditions  with  fairly  accordant 
results. 

It  now  remains  to  compare  with  the  time  of  heating  derived  from  theory. 
The  calculation  is  complicated  by  the  consideration  that  when  during  the 
process  any  part  becomes  heated,  it  expands  and  compresses  all  the  other 
parts,  thereby  developing  heat  in  them.  From  the  investigation  which 

*  It  must  not  be  forgotten  that  too  large  a  diameter  is  objectionable,  as  leading  to  an 
augmentation  of  volume  during  an  experiment,  as  the  liquid  falls. 


1899]  AND   ON   THE   PROPAGATION   OF   SOUND.  381 

follows  *,  we  see  that  the  time  of  half  recovery  t  is  given  by  the  formula 


in  which  a  is  the  radius  of  the  sphere,  7  the  ratio  of  specific  heats  (1'41),  and 
v  is  the  thermometric  conductivity,  found  by  dividing  the  ordinary  or  calori- 
metric  conductivity  by  the  thermal  capacity  of  unit  volume.  This  thermal 
capacity  is  to  be  taken  with  volume  constant,  and  it  will  be  less  than  the 
thermal  capacity  with  pressure  constant  in  the  ratio  of  7  :  1.  Accordingly  v/y 
in  (6)  represents  the  latter  thermal  capacity,  of  which  the  experimental  value 
is  '00128  x  '239,  the  first  factor  representing  the  density  of  air  referred  to 
water.  Thus,  if  we  take  the  calorimetric  conductivity  at  '000056,  we  have  in 
C.G.s.  measure 

i>  =  -258,         i;/7  =  183; 
and  thence 

t  =  '102a2. 

In  the  present  apparatus  a,  determined  by  the  contents,  is  16'4  centim., 
whence 

t  =  2  7  '4  seconds. 

The  agreement  of  the  observed  and  calculated  values  is  quite  as  close 
as  could  have  been  expected,  and  confirms  the  view  that  the  transfer  of  heat 
is  due  to  conduction,  and  that  the  part  played  by  radiation  is  insensible. 
From  a  comparison  of  the  experimental  and  calculated  curves,  however, 
it  seems  probable  that  the  effect  of  gravity  was  not  wholly  eliminated,  and 
that  the  later  stages  of  the  phenomenon,  at  any  rate,  may  still  have  been 
a  little  influenced  by  a  downward  movement  of  the  central  parts. 

*  See  next  paper. 


245. 


ON  THE  CONDUCTION  OF  HEAT  IN  A  SPHERICAL  MASS 
OF  AIR  CONFINED  BY  WALLS  AT  A  CONSTANT 
TEMPERATURE. 

[Philosophical  'Magazine,  XLVII.  pp.  314  —  325,  1899.] 

IT  is  proposed  to  investigate  the  subsidence  to  thermal  equilibrium  of 
a  gas  slightly  disturbed  therefrom  and  included  in  a  solid  vessel  whose 
walls  retain  a  constant  temperature.  The  problem  differs  from  those  con- 
sidered by  Fourier  in  consequence  of  the  mobility  of  the  gas,  which  may  give 
rise  to  two  kinds  of  complication.  In  the  first  place  gravity,  taking  ad- 
vantage of  the  different  densities  prevailing  in  various  parts,  tends  to  produce 
circulation.  In  many  cases  the  subsidence  to  equilibrium  must  be  greatly 
modified  thereby.  But  this  effect  diminishes  with  the  amount  of  the 
temperature  disturbance,  and  for  infinitesimal  disturbances  the  influence 
of  gravity  disappears.  On  the  other  hand,  the  second  complication  remains, 
even  though  we  limit  ourselves  to  infinitesimal  disturbances.  When  one 
part  of  the  gas  expands  in  consequence  of  reception  of  heat  by  radiation 
or  conduction,  it  compresses  the  remaining  parts,  and  these  in  their  turn 
become  heated  in  accordance  with  the  laws  of  gases.  To  take  account  of 
this  effect  a  special  investigation  is  necessary. 

But  although  the  fixity  of  the  boundary  does  not  suffice  to  prevent  local 
expansions  and  contractions  and  consequent  motions  of  the  gas,  we  may 
nevertheless  neglect  the  inertia  of  these  motions  since  they  are  very  slow 
in  comparison  with  the  free  oscillations  of  the  mass  regarded  as  a  resonator. 
Accordingly  the  pressure,  although  variable  with  time,  may  be  treated  as 
uniform  at  any  one  moment  throughout  the  mass. 

In  the  usual  notation*,  if  s  be  the  condensation  and  6  the  excess  of 
temperature,  the  pressure  p  is  given  by 

(1) 


*  Theory  of  Sound,  §  247. 


1899]      ON  THE   CONDUCTION  OF   HEAT   IN   A   SPHERICAL  MASS   OF   AIR.      383 

The  effect  of  a  small  sudden  condensation  s  is  to  produce  an  elevation  of 
temperature,  which  may  be  denoted  by  fts.  Let  dQ  be  the  quantity  of  heat 
entering  the  element  of  volume  in  the  time  dt,  measured  by  the  rise  of 
temperature  which  it  would  produce,  if  there  were  no  "  condensation." 
Then 

dO         ds     d 


and,  if  the  passage  of  dQ  be  the  result  of  radiation  and  conduction,  we  have 

f  =  vw-qe  ..............................  .(3) 

In  (3)  v  represents  the  "  therrnometric  conductivity  "  found  by  dividing  the 
conductivity  by  the  thermal  capacity  of  the  gas  (per  unit  volume),  at  constant 
volume.  Its  value  for  air  at  0°  and  atmospheric  pressure  may  be  taken  to  be 
•26  cm2.  /sec.  Also  q  represents  the  radiation,  supposed  to  depend  only  upon 
the  excess  of  temperature  of  the  gas  over  that  of  the  enclosure. 

If  dQ  =  0,  0  =  /3s,  and  in  (1) 


so  that 

l  +  «/9  =  7,   .................................  (4) 

where  7  is  the  well-known  ratio  of  specific  heats,  whose  value  for  air  and 
several  other  gases  is  very  nearly  1/41. 

In  general  from  (2)  and  (3) 


In  order  to  find  the  normal  modes  into  which  the  most  general  subsidence 
may  be  analysed,  we  are  to  assume  that  s  and  6  are  functions  of  the  time 
solely  through  the  factor  e~ht.     Since  p  is  uniform,  s  +  a.6  must  by  (1)  be  of 
the  form  He~ht,  where  H  is  some  constant  ;  so  that  if  for  brevity  the  factor 
e~ht  be  dropped, 

s  +  a0  =  H;   .................................  (6) 

while  from  (5) 

q)e  =  hps  .........................  (7) 


Eliminating  s  between  (5)  and  (7),  we  get 

V20  +  m*  (6  -  C)  =  0,    ...........................  (8) 

where 

m,  =  h-_q        0_Wff  ...................... 

v  hj  —  q 

These   equations   are   applicable  in  the   general  case,  but   when  radiation 
and  conduction  are  both  operative  the  equation  by  which  ra  is  determined 


384         ON   THE   CONDUCTION   OF   HEAT   IN    A   SPHERICAL   MASS   OF   AIR       [245 

becomes  rather  complicated.     If  there  be  no  conduction,  v  =  0.     The  solution 
is  then  very  simple,  and  may  be  worth  a  moment's  attention. 


Equations  (6)  and  (7)  give 

hftH 


.(10) 


hy-q' 

Now  the  mean  value  of  s  throughout  the  mass,  which  does  not  change  with 
the  time,  must  be  zero  ;  so  that  from  (10)  we  obtain  the  alternatives 

(i)     h  =  q,         (ii)     H  =  0. 
Corresponding  to  (i)  we  have  with  restoration  of  the  time-factor 

«=0  ......................  (11) 


In  this  solution  the  temperature  is  uniform  and  the  condensation  zero 
throughout  the  mass.  By  means  of  it  any  initial  mean  temperature  may  be 
provided  for,  so  that  in  the  remaining  solutions  the  mean  temperature  may 
be  considered  to  be  zero. 

In  the  second  alternative  H—  0,  so  that  s  =  -  aO.     Using  this  in  (7)  with 
v  evanescent,  we  get 

07-00  =  0  ...............................  (12) 

The  second  solution  is  accordingly 

.........  (13) 


where  <f>  denotes  a  function  arbitrary  throughout  the  mass,  except  for  the 
restriction  that  its  mean  value  must  be  zero. 

Thus  if  ©  denote  the  initial  value  of  0  as  a  function  of  x,  y,  z,  and  ©0  its 
mean  value,  the  complete  solution  may  be  written 

e  =  ®0e-<it  +  (®-G0)e-#iY  \ 

k     ..................  (14) 

8=  _a(e-@0)e-9'/yJ 

giving 

s  +  a0=a®Qe-#  ............................  (15) 

It  is  on  (15)  that  the  variable  part  of  the  pressure  depends. 

When  the  conductivity  v  is  finite,  the  solutions  are  less  simple  and  involve 
the  form  of  the  vessel  in  which  the  gas  is  contained.  As  a  first  example 
we  may  take  the  case  of  gas  bounded  by  two  parallel  planes  perpendicular 
to  x,  the  temperature  and  condensation  being  even  functions  of  x  measured 
from  the  mid-plane.  In  this  case  V2  =  d?/da?,  and  we  get 

6  =  C  +  A  cos  mx,        -s/a  =  D  +  Acosmx,  ............  (16) 

<*C-aD  =  H.  ........................  (17) 


1899]  CONFINED   BY  WALLS   AT   A   CONSTANT  TEMPERATURE.  385 

By  (9),  (17) 


y-q 


There  remain  two  conditions  to  be  satisfied.     The  first  is  simply  that  6  =  0 
when  x  =  ±  a,  2a  being  the  distance  between  the  walls.     This  gives 

0  +  Acosma=0  ............................  (19) 

The  remaining  condition  is  given  by  the  consideration  that  the  mean  value 
of  s,  proportional  to  jsdx,  must  vanish.     Accordingly 

ma.D  +  sinma.A=Q  .........................  (20) 

From  (18),  (19),  (20)  we  have  as  the  equation  for  the  admissible  values 
of  m, 

tan  ma  _   a@q  —  vm? 
ma     ~  z       ' 

reducing  for  the  case  of  evanescent  q  to 


ma  a/3' 

The  general  solution  may  be  expressed  in  the  series 


} 

) 


(23) 


where  h1}  h2>...  are  the  values  of  h  corresponding  according  to  (9)  with  the 
various  values  of  m,  and  0l}  02 ...  are  of  the  form 

0l  =  cos  TOI#  —  cos  TO!«.  ) 

I (24) 

It  only  remains  to  determine  the  arbitrary  constants  Alt  A2,  ...  to  suit 
prescribed  initial  conditions.  We  will  limit  ourselves  to  the  simpler  case 
of  q  =  0,  so  that  the  values  of  m  are  given  by  (22).  With  use  of  this  relation 
and  putting  for  brevity  a  =  1,  we  find  from  (24) 

r1  a/3  +  1 

J— -5 —  cos  TO!  cos  ra2, 

a/3  +  1 
s^dsc  = ^7^ —  cos  TO!  cos  m^; 

so  that 

0,    (25) 


'o  Jo 

?,,  02  being  any  (different)  functions  of  the  form  (24).     Also 

E.   jv.  25 


386         ON  THE   CONDUCTION   OF   HEAT   IN   A   SPHERICAL   MASS   OF   AIR       [245 

There  is  now  no  difficulty  in  finding  Alt  Az,  ...  to  suit  arbitrary  initial 
values  of  6  and  its  associated  s,  i.e.  so  that 

&  =  A10l  +  A,0«+...  } 

.........................  (27) 

S=AISI  +  A*SS  +  ...  J 

Thus  to  determine  Al} 

\l(%0l  +  /3/a  .  SSl)  dx  =  A,  P(0f  +  y3/a  .  O  dx 
o  Jo 


in  which  the  coefficients  of  A2,  As  ...  vanish  by  (25);  so  that  by  (26) 


An  important  particular  case  is  that  in  which  0  is  constant,  and  accordingly 
S  =  0.     Since 


f1  „ 
I    6l 

Jo 


sin  m,  1  4-  a/3 

—    --  cos  7^1  =  --  7r-  cos  ???i  , 


ap 
we  have 


For  the  pressure  we  have 


- 
a/3 

or  in  the  particular  case  of  (29), 


cos  w,  . 


a 


(30) 


If  /3  =  0,  we  fall  back  upon  a  problem  of  the  Fourier  type.     By  (22)  in 
that  case 

ma  =  |TT  (1,  3,  5,  . . . )     and     cos2  ma  =  a-fi2/ 

so  that  (30)  becomes 

or  initially 

80  n       1       !_ 

The  values  of  h  are  given  by 

...(32) 


1899]  CONFINED   BY  WALLS   AT   CONSTANT  TEMPERATURE.  387 

We  will  now  pass  on  to  the  more  important  practical  case  of  a  spherical 
envelope  of  radius  a.  The  equation  (8)  for  (6  —  C)  is  identical  with  that 
which  determines  the  vibrations  of  air*  in  a  spherical  case,  and  the  solution 
may  be  expanded  in  Laplace's  series.  The  typical  term  is 


(mr).Yn,  .....................  (33) 

Yn  being  the  surface  spherical  harmonic  of  order  n  where  n  =  0,  1,  2,  3  ...  , 
and  J  the  symbol  of  Bessel's  functions.  In  virtue  of  (6)  we  may  as  before 
equate  -  s/a  -  D,  where  D  is  another  constant,  to  the  right-hand  member  of 
(33).  The  two  conditions  yet  to  be  satisfied  are  that  6  =  0  when  r  =  a,  and 
that  the  mean  value  of  s  throughout  the  sphere  shall  vanish. 

When  the  value  of  n  is  greater  than  zero,  the  first  of  these  conditions 
gives  (7=0  and  the  second  D  —  0  ;  so  that 

0  =  -s(*  =  (mr)-Un+i(mr).Yn,   ..................  (34) 

and  s  +  ad  =  0.  Accordingly  these  terms  contribute  nothing  to  the  pressure. 
It  is  further  required  that 

Jn+l(ma)  =  0,  ..............................  (35) 

by  which  the  admissible  values  of  m  are  determined.  The  roots  of  (35) 
are  discussed  in  Theory  of  Sound,  §  206...  ;  but  it  is  not  necessary  to  go 
further  into  the  matter  here,  as  interest  centres  rather  upon  the  case  n  =  0. 

If  we  assume  symmetry  with  respect  to  the  centre  of  the  sphere,  we  may 

1  d2 
replace  V2  in  (8)  by  -  r~z  r,  thus  obtaining 


(36) 


of  which  the  general  solution  is 


But  for  the  present  purpose  the  term  r~l  cos  mr  is  excluded,  so  that  we  may 
write 


,     .........  (37) 

mr  mr 

giving 

s  +  a0  =  a(C-D)=H.  .....................  (37  bis) 

The  first  special  condition  gives 

maC  +  B  sin  ma  =  0  .........................  (38) 

The  second,  that  the  mean  value  of  s  shall  vanish,  gives  on  integration 

^m3a?D  +  B  (sin  ma  —  ma  cos  ma)  =  0  ................  (39) 

*  Theory  of  Sound,  Vol.  IT.  ch.  xvii. 

25—2 


388         ON   THE   CONDUCTION   OF   HEAT   IN    A   SPHERICAL   MASS   OF   AIR        [245 

Equations  (18),  derived  from  (9)  and  (37  bis),  giving  C  and  D  in  terms 
of  H,  hold  good  as  before.     Thus 

* 


G~  haft      aft(q+Vm*)' 
Equating  this  ratio  to  that  derived  from  (38),  (39),  we  find 

3     ma  cos  ma  —  sin  ma  _    vmz  —  aftq  .  -  . 

m2a2  sin  ma  aft  (vmz  +  q)  ' 

This  is  the  equation  from  which  m  is  to  be  found,  after  which  h  is  given 

by  (9). 

In  the  further  discussion  we  will  limit  ourselves  to  the  case  of  q  =  0, 
when  (41)  reduces  to 

l),    ........................  (42) 


in  which  a  has  been  put  equal  to  unity.     Here  by  (40) 

D  =  -C/aft. 
Thus  we  may  set,  as  in  (23), 

6  =  B1e-h>t0l  +  Bze-h*t02+  ......  ) 

k      ....'  ...........  (43) 

s=Ble-h*tsl+B9erUs2  +  ......  j 

in  which 

..      sin  ??ij?'     sin  m^a  sin  w,r      1  sin  VIM 

0i=  —      ---      —  ,          «!  =  —  «—  —     ...(44) 

m{r  m^a,  my        ft     mta 

and  by  (9)     J^—vm^/y.     Also 


The  process  for  determining  B1}  B2,  ...  follows  the  same  lines  as  before. 
By  direct  integration  from  (44)  we  find 


_  sin  (m-i  —  m^)  _  sin  (m^  +  m2)     2  sin  ml  sin  m2 
Wj  —  77^2  ml  +  m,t  3«/3 

a  being  put  equal  to  unity.  By  means  of  equation  (42)  satisfied  by  m± 
and  ra2  we  may  show  that  the  quantity  on  the  right  in  the  above  equation 
vanishes.  For  the  sum  of  the  first  two  fractions  is 

2m2  sin  ml  cos  ra2  —  2m1  sin  w2  cos  m^ 


of  which  the  denominator  by  (42)  is  equal  to 

3a/3  (nh  cot  ml  —  m2  cot  r?i2). 


1899]  CONFINED   BY   WALLS   AT   CONSTANT  TEMPERATURE.  389 

Accordingly  f  (010.,  +  ^/a.s1s.z)r2dr  =  0  ......................  (46) 

Jo 


Also 


To  determine  the  arbitrary  constants  Bl  ...  from  the  given  initial  values 
of  9  and  s,  say  ®  and  8,  we  proceed  as  usual.  We  limit  ourselves  to  the 
term  of  zero  order  in  spherical  harmonics,  i.e.  to  tne  supposition  that  6,  s 
are  functions  of  r  only.  The  terms  of  higher  order  in  spherical  harmonics,  if 
present,  are  treated  more  easily,  exactly  as  in  the  ordinary  theory  of  the 
conduction  of  heat.  By  (43) 


and  thus  I  \0  6l  +  01  a  .  SsJ  f2dr  =  B,  fW*  +  £/  a  .  6V2)  i*dr 

Jo  Jo 


z  !\0102  +  /3/a  .  8,8,}  r*dr  +  ......  , 

Jo 


in  which  the  coefficients  of  B«,  B3>  ...  vanish  by  (46).     The  coefficient  of  Bt 
is  given  by  (47).     Thus 


by  which  Bl  is  determined. 

An  important  particular  case  is  that  where  ®  is  constant  and  accordingly 
S  vanishes.     Now  with  use  of  (42) 

f1  sin  ml  —  m1  cos  m1     sin  m1  _      (1  +  a/3)  sin  ml 

Jo1  mf  ~~3m^~~ 

so  that 


sin  2m1      2sin2m!]  2m^  sin  m^  .  ®  /Kn, 

"2^"      "S^")"  3«/3 

Bl,  B.,,  ...  being  thus  known,  0  and  s  are  given  as  functions  of  the  time  and 
of  the  space  coordinates  by  (43),  (44). 

To  determine  the  pressure  in  this  case  we  have  from  (45) 

0  +  s/a.  _  I  +a/3         _  sin2  m  .  e~ht  (     , 

~~  sin  2m\  ' 


the  summation  extending  to  all  the  values  of  m  in  (42).  Since  (for  each 
term)  the  mean  value  of  s  is  zero,  the  right-hand  member  of  (51)  represents 
also  0/®,  where  0  is  the  mean  value  of  0. 

If  in  (51)  we  suppose  /3  =  0,  we  fall  back  upon  a  known  Fourier  solution, 


390       ON   THE   CONDUCTION   OF   HEAT   IN    A   SPHERICAL   MASS   OF    AIR        [245 


relative  to  the  mean  temperature  of  a  spherical  solid  which,  having  been 
initially  at  uniform  temperature  ®  throughout,  is  afterwards  maintained 
at  zero  all  over  the  surface.  From  (42)  we  see  that  in  this  case  sin  in  is 
small  and  of  order  /3.  Approximately 

sin  m  =  3a/3lm ; 
and  (51)  reduces  to 

0      6   ,e-M     e-M     e-/M 


of  which  by  a  known  formula  the  right-hand  member  identifies  itself  with 
unity  when  t  =  0.     By  (9)  with  restoration  of  a, 

h  =  (I2,  32,  52,  ...)*/7r2/a2 (53) 

In  the  general  case  we  may  obtain  from  (42)  an  approximate  value 
applicable  when  m  is  moderately  large.  The  first  approximation  is  m  =  ITT, 
i  denoting  an  integer.  Successive  operations  give 

3a£     ISa-ft2  +  9a3/33 

m  =  17T  +    — ; ; • (54) 

ITT  i  77"^ 

In  like  manner  we  find  approximately  in  (51) 

sin2  m  (1  +  qff)/a/3       =  6  (1  +  a/8)  L      15ay8  +  9a2y8-- } 

.  3a n 

sin2  m  -\ — -, 


sin 


ftH 


•  •  -(55) 


showing  that  the  coefficients  of  the  terms  of  high  order  in  (51)  differ  from  the 
corresponding  terms  in  (52)  only  by  the  factor  (1  +  a/3)  or  7. 

In  the  numerical  computation  we  take  7  =  1*41,  a/3  =  '41.  The  series  (54) 
suffices  for  finding  m  when  i  is  greater  than  2.  The  first  two  terms  are 
found  by  trial  and  error  with  trigonometrical  tables  from  (42).  In  like 
manner  the  approximate  value  of  the  left-hand  member  of  (51)  therein  given 
suffices  when  i  is  greater  than  3.  The  results  as  far  as  i  =  12  are  recorded  in 
the  annexed  table. 


i 

mjw 

Left-hand 
member 
of  (55) 

i 

m/T 

Left-hand 
member 
of  (55) 

1  

1-0994 

•4942 

7  

7-0177 

•0175 

2  

2-0581 

•1799 

8  

8-0156 

•0134 

3  

3-0401 

•0871 

9  

9-0138 

•0106 

4  

4-0305 

•0510 

10  

10-0125 

•0086 

5  

5-0246 

•0332 

11  

11-0113 

•0071 

6  

6-0206 

•0233 

12  

12-0104 

•0060 

Thus  the  solution  (51)  of  our  problem  is  represented  by 
0/0  =  •4942e-(1-°9!M>!i<'-l--l799e-(2-0581)2t'+  ... 


.(56) 


1899] 


CONFINED    BY    WALLS    AT   CONSTANT  TEMPERATURE. 


391 


where  by  (9),  with  omission  of  q  and  restoration  of  a, 

t'/t  =  Tr'vlyct?  ...............................  (57) 

The  numbers  entered  in  the  third  column  of  the  above  table  would 
add  up  to  unity  if  continued  far  enough.  The  verification  is  best  made 
by  a  comparison  with  the  simpler  series  (52).  If  with  t  zero  we  call  this 
series  2'  and  the  present  series  2,  both  2  and  2'  have  unity  for  their  sum, 
and  accordingly  7^'  —  2  =  7  —  1,  or 


Here  Qy/tr2  =  '8573,  and  the  difference  between  this  and  the  first  term  of 
S,  i.e.  '4942,  is  '3631.  The  differences  of  the  second,  third,  &c.  terms  are 
•0344,  -0082,  -0026,  '0011,  '0005,  '0000,  &c.,  making  a  total  of  '4099. 

We  are  now  in  a  position  to  compute  the  right-hand  member  of  (56) 
as  a  function  of  t'.     The  annexed  table  contains  sufficient  to  give  an  idea 


t' 

(56) 

t' 

(56) 

t' 

(56) 

•oo  

1-0000 

•40  

•3401 

•90  

•1705 

•05  

•7037 

•50  

•2926 

1-00  

•1502 

•10  

•6037 

•60  

•2538 

1-50  

•0809 

•20  

•4811 

•70  

•2215 

2-00  

•0441 

•30  

•4002 

•80  

•1940 

of  the  course  of  the  function.  It  is  plotted  in  the  figure.  The  second  entry 
(t'  =  -05)  requires  the  inclusion  of  9  terms  of  the  series.  After  t'  =  '7  two 
terms  suffice ;  and  after  t'  =  2'0  the  first  term  represents  the  series  to  four 
places  of  decimals. 


By  interpolation  we  find  that  the  series  attains  the  value  '5  when 


(58) 


246. 

TRANSPARENCY  AND   OPACITY. 

[Proc.  Roy.  Inst.  xvi.  pp.  116—119,  1899;  Nature,  LX.  pp.  64,  65,  1899.] 

ONE  kind  of  opacity  is  due  to  absorption;  but  the  lecture  dealt  rather 
with  that  deficiency  of  transparency  which  depends  upon  irregular  reflections 
and  refractions.  One  of  the  best  examples  is  that  met  with  in  Christiansen's 
experiment.  Powdered  glass,  all  from  one  piece  and  free  from  dirt,  is  placed 
in  a  bottle  with  parallel  flat  sides.  In  this  state  it  is  quite  opaque ;  but 
if  the  interstices  between  the  fragments  are  filled  up  with  a  liquid  mixture 
of  bisulphide  of  carbon  and  benzole,  carefully  adjusted  so  as  to  be  of  equal 
refractivity  with  the  glass,  the  mass  becomes  optically  homogeneous,  and 
therefore  transparent.  In  consequence,  however,  of  the  different  dispersive 
powers  of  the  two  substances,  the  adjustment  is  good  for  one  part  only  of  the 
spectrum,  other  parts  being  scattered  in  transmission  much  as  if  no  liquid 
were  employed,  though,  of  course,  in  a  less  degree.  The  consequence  is  that 
a  small  source  of  light,  backed  preferably  by  a  dark  ground,  is  seen  in  its 
natural  outlines  but  strongly  coloured.  The  colour  depends  upon  the  precise 
composition  of  the  liquid,  and  further  varies  with  the  temperature,  a  few 
degrees  of  warmth  sufficing  to  cause  a  transition  from  red  through  yellow  to 
green. 

The  lecturer  had  long  been  aware  that  the  light  regularly  transmitted 
through  a  stratum  from  15  to  20  mm.  thick  was  of  a  high  degree  of  purity, 
but  it  was  only  recently  that  he  found  to  his  astonishment,  as  the  result  of  a 
more  particular  observation,  that  the  range  of  refrangibility  included  was  but 
two  and  a  half  times  that  embraced  by  the  two  D-lines.  The  poverty  of 
general  effect,  when  the  darkness  of  the  background  is  not  attended  to,  was 
thus  explained;  for  the  highly  monochromatic  and  accordingly  attenuated 
light  from  the  special  source  is  then  overlaid  by  diffused  light  of  other 
colours. 


1899]  TRANSPARENCY   AND   OPACITY.  393 

More  precise  determinations  of  the  range  of  light  transmitted  were 
subsequently  effected  with  thinner  strata  of  glass  powder  contained  in  cells 
formed  of  parallel  glass.  The  cell  may  be  placed  between  the  prisms  of  the 
spectroscope  and  the  object-glass  of  the  collimator.  With  the  above  mentioned 
liquids  a  stratum  5  mm.  thick  transmitted,  without  appreciable  disturbance,  a 
range  of  the  spectrum  measured  by  11 '3  times  the  interval  of  the  D's.  In 
another  cell  of  the  same  thickness  an  effort  was  made  to  reduce  the  difference 
of  dispersive  powers.  To  this  end  the  powder  was  of  plate  glass  and  the 
liquid  oil  of  cedar- wood  adjusted  with  a  little  bisulphide  of  carbon.  The 
general  transparency  of  this  cell  was  the  highest  yet  observed.  When  it 
was  tested  upon  the  spectrum,  the  range  of  refrangibility  transmitted  was 
estimated  at  34  times  the  interval  of  the  D's. 

As  regards  the  substitution  of  other  transparent  solid  material  for  glass, 
the  choice  is  restricted  by  the  presumed  necessity  of  avoiding  appreciable 
double  refraction.  Common  salt  is  singly  refracting,  but  attempts  to  use 
it  were  not  successful.  Opaque  patches  always  interfered.  With  the  idea 
that  these  might  be  due  to  included  mother- liquor,  the  salt  was  heated  to 
incipient  redness,  but  with  little  advantage.  Transparent  rock-salt  artificially 
broken  may,  however,  be  used  with  good  effect,  but  there  is  some  difficulty  in 
preventing  the  approximately  rectangular  fragments  from  arranging  them- 
selves too  closely. 

The  principle  of  evanescent  refraction  may  also  be  applied  to  the  spectro- 
scope. Some  twenty  years  ago,  an  instrument  had  been  constructed  upon 
this  plan*.  Twelve  90°  prisms  of  Chance's  "dense  flint"  were  cemented  in  a 
row  upon  a  strip  of  glass  (Fig.  1),  and  the  whole  was  immersed  in  a  liquid 
mixture  of  bisulphide  of  carbon  with  a  little  benzole.  The  dispersive  power 
of  the  liquid  exceeds  that  of  the  solid,  and  the  difference  amounts  to  about 
three-quarters  of  the  dispersive  power  of  Chance's  "  extra  dense  flint."  The 

Fig.  1. 


resolving  power  of  the  latter  glass  is  measured  by  the  number  of  centimetres 
of  available  thickness,  if  we  take  the  power  required  to  resolve  the  jD-lines  as 
unity.  The  compound  spectroscope  had  an  available  thickness  of  12  inches 
or  30  cm.,  so  that  its  theoretical  resolving  power  (in  the  yellow  region  of  the 
spectrum)  would  be  about  22.  With  the  aid  of  a  reflector  the  prism  could  be 
used  twice  over,  and  then  the  resolving  power  is  doubled. 

*  [Vol.  i.  p.  456.] 


394  TRANSPARENCY   AND  OPACITY.  [246 

One  of  the  objections  to  a  spectroscope  depending  upon  bisulphide  of 
carbon  is  the  sensitiveness  to  temperature.  In  the  ordinary  arrangement  of 
prisms  the  refracting  edges  are  vertical.  If,  as  often  happens,  the  upper  part 
of  a  fluid  prism  is  warmer  than  the  lower,  the  definition  is  ruined,  one  degree 
(Centigrade)  of  temperature  making  nine  times  as  great  a  difference  of 
refraction  as  a  passage  from  Z^  to  D.2.  The  objection  is  to  a  great  extent 
obviated  by  so  mounting  the  compound  prism  that  the  refracting  edges  are 
horizontal,  which  of  course  entails  a  horizontal  slit.  The  disturbance 
due  to  a  stratified  temperature  is  then  largely  compensated  by  a  change 
of  focus. 

In  the  instrument  above  described  the  dispersive  power  is  great — the 
D-lines  are  seen  widely  separated  with  the  naked  eye — but  the  aperture  is 
inconveniently  small  (|-inch).  In  the  new  instrument  exhibited  the  prisms 
(supplied  by  Messrs  Watson)  are  larger,  so  that  a  line  of  ten  prisms  occupies 
20  inches.  Thus,  while  the  resolving  power  is  much  greater,  the  dispersion 
is  less  than  before*. 

In  the  course  of  the  lecture  the  instrument  was  applied  to  show  the 
duplicity  of  the  reversed  soda  lines.  The  interval  on  the  screen  between  the 
centres  of  the  dark  lines  was  about  half  an  inch. 

It  is  instructive  to  compare  the  action  of  the  glass  powder  with  that  of 
the  spectroscope.  In  the  latter  the  disposition  of  the  prisms  is  regular,  and 
in  passing  from  one  edge  of  the  beam  to  the  other  there  is  complete  substitu- 
tion of  liquid  for  glass  over  the  whole  length.  For  one  kind  of  light  there  is 
no  relative  retardation ;  and  the  resolving  power  depends  upon  the  question 
of  what  change  of  wave-length  is  required  in  order  that  its  relative  retardation 
may  be  altered  from  zero  to  the  quarter  wave-length.  All  kinds  of  light  for 
which  the  relative  retardation  is  less  than  this  remain  mixed.  In  the  case 
of  the  powder  we  have  similar  questions  to  consider.  For  one  kind  of  light 
the  medium  is  optically  homogeneous,  i.e.  the  retardation  is  the  same  along 
all  rays.  If  we  now  suppose  the  quality  of  the  light  slightly  varied,  the 
retardation  is  no  longer  precisely  the  same  along  all  rays ;  but  if  the  variation 
from  the  mean  falls  short  of  the  quarter  wave-length,  it  is  without  importance, 
and  the  medium  still  behaves  practically  as  if  it  were  homogeneous.  The 
difference  between  the  action  of  the  powder  and  that  of  the  regular  prisms  in 
the  spectroscope  depends  upon  this,  that  in  the  latter  there  is  complete 
substitution  of  glass  for  liquid  along  the  extreme  rays,  while  in  the  former  the 
paths  of  all  the  rays  lie  partly  through  glass  and  partly  through  liquid  in 
nearly  the  same  proportions.  The  difference  of  retardations  along  various 
rays  is  thus  a  question  of  a  deviation  from  an  average. 

*  [1902.  When  carefully  used  this  instrument  gives  about  as  good  definition  in  the  greeii 
as  a  first-rate  Rowland  grating.] 


1899]  TRANSPARENCY   AND  OPACITY.  395 

It  is  true  that  we  may  imagine  a  relative  distribution  of  glass  and  liquid 
that  would  more  nearly  assimilate  the  two  cases.  If,  for  example,  the  glass 
consisted  of  equal  spheres  resting  against  one  another  in  cubic  order,  some 
rays  might  pass  entirely  through  glass  and  others  entirely  through  liquid, 
and  then  the  quarter  wave-length  of  relative  retardation  would  enter  at  the 
same  total  thickness  in  both  cases.  But  such  an  arrangement  would  be 
highly  unstable;  and,  if  the  spheres  be  packed  in  close  order,  the  extreme 
relative  retardation  would  be  much  less.  The  latter  arrangement,  for  which 
exact  results  could  readily  be  calculated,  represents  the  glass  powder  more 
nearly  than  does  the  cubic  order. 

A  simplified  problem,  in  which  the  element  of  chance  is  retained,  may 
be  constructed  by  supposing  the  particles  of  glass  replaced  by  thin  parallel 
discs  which  are  distributed  entirely  at  random  over  a  certain  stratum.  We 
may  go  further  and  imagine  the  discs  limited  to  a  particular  plane.  Each 
disc  is  supposed  to  exercise  a  minute  retarding  influence  on  the  light  which 
traverses  it,  and  they  are  supposed  to  be  so  numerous  that  it  is  improbable 
that  a  ray  can  pass  the  plane  without  encountering  a  large  number.  A 
certain  number  (m)  of  encounters  is  more  probable  than  any  other,  but  if 
every  ray  encountered  the  same  number  of  discs,  the  retardation  would  be 
uniform  and  lead  to  no  disturbance. 

It  is  a  question  of  Probabilities  to  determine  the  chance  of  a  prescribed 
number  of  encounters,  or  of  a  prescribed  deviation  from  the  mean.  In  the 
notation  of  the  integral  calculus  the  chance  of  the  deviation  from  in  lying 
between  ±  r  is* 


where  r  =  r/\/(2w).  This  is  equal  to  '84  when  r=l*0,  or  r=\f(2m);  so 
that  the  chance  is  comparatively  small  of  a  deviation  from  m  exceeding 
±  V(2w). 

To  represent  the  glass  powder  occupying  a  stratum  of  2  cm.  thick,  we  may 
perhaps  suppose  that  m  =  72.  There  would  thus  be  a  moderate  chance  of  a 
difference  of  retardations  equal  to,  say,  one-fifth  of  the  extreme  difference 
corresponding  to  a  substitution  of  glass  for  liquid  throughout  the  whole 
thickness.  The  range  of  wave-lengths  in  the  light  regularly  transmitted  by 
the  powder  would  thus  be  about  five  times  the  range  of  wave-lengths  still 
unseparated  in  a  spectroscope  of  equal  (2cm.)  thickness.  Of  course,  no 
calculation  of  this  kind  can  give  more  than  a  rough  idea  of  the  action  of  the 
powder,  whose  disposition,  though  partly  a  matter  of  chance,  is  also  influenced 
by  mechanical  considerations  ;  but  it  appears,  at  any  rate,  that  the  character 

*  See  Phil.  Mag.  1899,  Vol.  XLVII.  p.  251.     [Vol.  zv.  p.  375.] 


396  TRANSPARENCY   AND  OPACITY.  [246 

of  the  light  regularly  transmitted  by  the  powder  is  such  as  may  reasonably 
be  explained. 

As  regards  the  size  of  the  grains  of  glass,  it  will  be  seen  that  as  great  or  a 
greater  degree  of  purity  may  be  obtained  in  a  given  thickness  from  coarse 
grains  as  from  fine  ones,  but  the  light  not  regularly  transmitted  is  dispersed 
through  smaller  angles.  Here  again  the  comparison  with  the  regularly 
disposed  prisms  of  an  actual  spectroscope  is  useful. 

At  the  close  of  the  lecture  the  failure  of  transparency  which  arises  from 
the  presence  of  particles  small  compared  to  the  wave-length  of  light  was 
discussed.  The  tints  of  the  setting  sun  were  illustrated  by  passing  the 
light  from  the  electric  lamp  through  a  liquid  in  which  a  precipitate  of 
sulphur  was  slowly  forming*.  The  lecturer  gave  reasons  for  his  opinion 
that  the  blue  of  the  sky  is  not  wholly,  or  even  principally,  due  to  particles 
of  foreign  matter.  The  molecules  of  air  themselves  are  competent  to  dis- 
perse a  light  not  greatly  inferior  in  brightness  to  that  which  we  receive 
from  the  sky. 

*  Op.  cit.  1881,  Vol.  xn.  p.  96.     [Vol.  i.  p.  531.] 


247. 


ON  THE  TRANSMISSION  OF  LIGHT  THROUGH  AN  ATMO- 
SPHERE CONTAINING  SMALL  PARTICLES  IN  SUSPENSION, 
AND  ON  THE  ORIGIN  OF  THE  BLUE  OF  THE  SKY. 

[Philosophical  Magazine,  XLVII.  pp.  375—384,  1899.] 

THIS  subject  has  been  treated  in  papers  published  many  years  ago*. 
I  resume  it  in  order  to  examine  more  closely  than  hitherto  the  attenuation 
undergone  by  the  primary  light  on  its  passage  through  a  medium  containing 
small  particles,  as  dependent  upon  the  number  and  size  of  the  particles. 
Closely  connected  with  this  is  the  interesting  question  whether  the  light 
from  the  sky  can  be  explained  by  diffraction  from  the  molecules  of  air 
themselves,  or  whether  it  is  necessary  to  appeal  to  suspended  particles 
composed  of  foreign  matter,  solid  or  liquid.  It  will  appear,  I  think,  that 
even  in  the  absence  of  foreign  particles  we  should  still  have  a  blue  skyf. 

The  calculations  of  the  present  paper  are  not  needed  in  order  to  explain 
the  general  character  of  the  effects  produced.  In  the  earliest  of  those  above 

*  Phil.  Mag.  XLI.  pp.  107,  274,  447  (1871);  xn.  p.  81  (1881).     [Vol.  i.  pp.  87,  104,  518.] 

f  My  attention  was  specially  directed  to  this  question  a  long  while  ago  by  Maxwell  in  a 
letter  which  I  may  be  pardoned  for  reproducing  here.  Under  date  Aug.  28,  1873,  he  wrote : — 

"I  have  left  your  papers  on  the  light  of  the  sky,  &c.  at  Cambridge,  and  it  would  take  me,  even 
if  I  had  them,  some  time  to  get  them  assimilated  sufficiently  to  answer  the  following  question, 
which  I  think  will  involve  less  expense  to  the  energy  of  the  race  if  you  stick  the  data  into  your 
formula  and  send  me  the  result.... 

"  Suppose  that  there  are  N  spheres  of  density  p  and  diameter  s  in  unit  of  volume  of  the 
medium.  Find  the  index  of  refraction  of  the  compound  medium  and  the  coefficient  of  extinction 
of  light  passing  through  it. 

"  The  object  of  the  enquiry  is,  of  course,  to  obtain  data  about  the  size  of  the  molecules  of  air. 
Perhaps  it  may  lead  also  to  data  involving  the  density  of  the  aether.    The  following  quantities 
are  known,  being  combinations  of  the  three  unknowns, 
M=ms.ss  of  molecule  of  hydrogen ; 

N= number  of  molecules  of  any  gas  in  a  cubic  centimetre  at  0°  C.  and  760  B. 
s  =  diameter  of  molecule  in  any  gas  :— 


398  ON   THE   TRANSMISSION   OF   LIGHT  THROUGH   AN  [247 

referred  to  I  illustrated  by  curves  the  gradual  reddening  of  the  transmitted 
light  by  which  we  see  the  sun  a  little  before  sunset.  The  same  reasoning 
proved,  of  course,  that  the  spectrum  of  even  a  vertical  sun  is  modified  by  the 
atmosphere  in  the  direction  of  favouring  the  waves  of  greater  length. 

For  such  a  purpose  as  the  present  it  makes  little  difference  whether 
we  speak  in  terms  of  the  electromagnetic  theory  or  of  the  elastic  solid 
theory  of  light  ;  but  to  facilitate  comparison  with  former  papers  on  the  light 
from  the  sky,  it  will  be  convenient  to  follow  the  latter  course.  The  small 
particle  of  volume  T  is  supposed  to  be  small  in  all  its  dimensions  in  comparison 
with  the  wave-length  (X),  and  to  be  of  optical  density  D'  differing  from  that 
(D)  of  the  surrounding  medium.  Then,  if  the  incident  vibration  be  taken 
as  unity,  the  expression  for  the  vibration  scattered  from  the  particle  in  a 
direction  making  an  angle  6  with  that  of  primary  vibration  is 

-       irT   .  fa  ,,,        ,* 

(6(-r)»,      ..................  U) 


r  being  the  distance  from  T  of  any  point  along  the  secondary  ray. 

In  order  to  find  the  whole  emission  of  energy  from  T  we  have  to  integrate 
the  square  of  (1)  over  the  surface  of  a  sphere  of  radius  r.  The  element 
of  area  being  far3  sin  Odd,  we  have 


r  *™^  far-  sin  0d0  =  47T  f  '"sin"  OdO  =  ^  ; 
Jo     r*  Jo  o 


o     r  o 

so  that  the  energy  emitted  from  T  is  represented  by 


Known  Combinations. 
M  N=  density. 
A/s2    from  diffusion  or  viscosity. 

Conjectural  Combination. 
•  —3  =  density  of  molecule. 

"  If  you  can  give  us  (i)  the  quantity  of  light  scattered  in  a  given  direction  by  a  stratum  of  a 
certain  density  and  thickness  ;  (ii)  the  quantity  cut  out  of  the  direct  ray  ;  and  (iii)  the  effect  of 
the  molecules  on  the  index  of  refraction,  which  I  think  ought  to  come  out  easily,  we  might  get 
a  little  more  information  about  these  little  bodies. 

"  You  will  see  by  Nature,  Aug.  14,  1873,  that  I  make  the  diameter  of  molecules  about  j^Vu  of 
a  wave-length. 

"  The  enquiry  into  scattering  must  begin  by  accounting  for  the  great  observed  transparency  of 
air.  I  suppose  we  have  no  numerical  data  about  its  absorption. 

"But  the  index  of  refraction  can  be  numerically  determined,  though  the  observation  is  of 
a  delicate  kind,  and  a  comparison  of  the  result  with  the  dynamical  theory  may  lead  to  some  new 
information." 

Subsequently  he  wrote,  "Your  letter  of  Nov.  17  quite  accounts  for  the  observed  transparency 
of  any  gas."  So  far  as  I  remember,  my  argument  was  of  a  general  character  only. 

*  The  factor  TT  was  inadvertently  omitted  in  the  original  memoir. 


1899]        ATMOSPHERE    CONTAINING   SMALL   PARTICLES    IN   SUSPENSION.  399 


on  such  a  scale  that  the  energy  of  the  primary  wave  is  unity  per  unit  of 
wave-front  area. 

The  above  relates  to  a  single  particle.  If  there  be  n  similar  particles  per 
unit  volume,  the  energy  emitted  from  a  stratum  of  thickness  dx  and  of  unit 
area  is  found  from  (2)  by  introduction  of  the  factor  ndx.  Since  there  is 
no  waste  of  energy  on  the  whole,  this  represents  the  loss  of  energy  in  the 
primary  wave.  Accordingly,  if  E  be  the  energy  of  the  -primary  wave, 

1  dE         87rsn(D'-D)*T* 

Edx  =      ~3  --  W~V>      .....................  (3) 


whence 
where 


E  =  Ene~hx, 

8>rr3n(D'-D)* 
- 


(4) 


If  we  had  a  sufficiently  complete  expression  for  the  scattered  light,  we 
might  investigate  (5)  somewhat  more  directly  by  considering  the  resultant 
of  the  primary  vibration  and  of  the  secondary  vibrations  which  travel  in  the 
same  direction.  If,  however,  we  apply  this  process  to  (1),  we  find  that  it 
fails  to  lead  us  to  (5),  though  it  furnishes  another  result  of  interest.  The 
combination  of  the  secondary  waves  which  travel  in  the  direction  in  question 
has  this  peculiarity,  that  the  phases  are  no  more  distributed  at  random. 
The  intensity  of  the  secondary  light  is  no  longer  to  be  arrived  at  by  addition 
of  individual  intensities,  but  must  be  calculated  with  consideration  of  the 
particular  phases  involved.  If  we  consider  a  number  of  particles  which  all 
lie  upon  a  primary  ray,  we  see  that  the  phases  of  the  secondary  vibrations 
which  issue  along  this  line  are  all  the  same. 

The  actual  calculation  follows  a  similar  course  to  that  by  which  Huygens' 
conception  of  the  resolution  of  a  wave  into  components 
corresponding  to  the  various  parts  of  the  wave-front 
is  usually  verified.  [See  for  example  Vol.  in.  p.  74.] 
Consider  the  particles  which  occupy  a  thin  stratum  dx 
perpendicular  to  the  primary  ray  x.  Let  AP  (Fig.  1)  be 
this  stratum  and  0  the  point  where  the  vibration  is  to 
be  estimated.  If  AP  =  p,  the  element  of  volume  is 
dx.^Trpdp,  and  the  number  of  particles  to  be  found  in 
it  is  deduced  by  introduction  of  the  factor  n.  Moreover, 
if  OP  =  r,  A0  =  x,  r*  =  x*  +  p\  and  pdp  =  rdr.  The 
resultant  at  0  of  all  the  secondary  vibrations  which  issue 
from  the  stratum  dx  is  by  (1),  with  sin  6  equal  to  unity, 


ndx 


or 


c^D'-DirT       Sir  ,,, 
.  I     —  j;  ----  —  cos  —  -  (bt  — 

J  x          U          7*  A*  A* 


,     D'-DirT  .    27r/r 
ndx.—j.  ---  —  sm  —  (bt  —  x) 

.L/  A*  A* 


(6) 


400  ON  THE  TRANSMISSION   OF   LIGHT  THROUGH   AN  [247 

To  this  is  to  be  added  the  expression  for  the  primary  wave  itself,  supposed 
to  advance  undisturbed,  viz.,  cos  -^  (bt  -  x\  and  the  resultant  will  then 

A. 

represent  the  whole  actual  disturbance  at  0  as  modified  by  the  particles 
in  the  stratum  da. 

It  appears,  therefore,  that  to  the  order  of  approximation  afforded  by  (1) 
the  effect  of  the  particles  in  dec  is  to  modify  the  phase,  but  not  the  intensity, 
of  the  light  which  passes  them.  If  this  be  represented  by 

cos  ^  (fa- a: -S),     (7) 

8  is  the  retardation  due  to  the  particles,  and  we  have 

If  fi  be  the  refractive  index  of  the  medium  as  modified  by  the  particles, 
that  of  the  original  medium  being  taken  as  unity,  8  =  (/u,  —  1)  dx,  and 

p.-  1  =nT(D'  —  D)/2D (9) 

If  ft    denote  the  refractive  index  of  the  material  composing  the  particles 
regarded  as  continuous,  D'/D  =  /*'*,  and 

reducing  to 

in  the  case  where  p!  —  1  can  be  regarded  as  small. 

It  is  only  in  the  latter  case  that  the  formulae  of  the  elastic-solid  theory 
are  applicable  to  light.  In  the  electric  theory,  to  be  preferred  on  every 
ground  except  that  of  easy  intelligibility,  the  results  are  more  complicated 
in  that  when  (//  —  1)  is  not  small,  the  scattered  ray  depends  upon  the  shape 
and  not  merely  upon  the  volume  of  the  small  obstacle.  In  the  case  of  spheres 
we  are  to  replace  (D'  -  D)/D  by  3  (K'  -  K)/(K'+  2K),  where  K,  K'  are 
the  dielectric  constants  proper  to  the  medium  and  to  the  obstacle  respectively*; 
so  that  instead  of  (10) 

onf  //,2  —  1  ,     . 

/*-!  =  "y^pq^ (12) 

On  the  same  suppositions  (5)  is  replaced  by 
On  either  theory 


*  Phil.  Mag.  xn.  p.  98  (1881).    [Vol.  i.  p.  533.]    For  the  corresponding  theory  in  the  case  of 
an  ellipsoidal  obstacle,  see  Phil.  Map.  Vol.  xuv.  p.  48  (1897).     [Vol.  iv.  p.  305.] 


1899]        ATMOSPHERE   CONTAINING  SMALL  PARTICLES   IN  SUSPENSION.         401 

a  formula  giving  the  coefficient  of  transmission  in  terms  of  the  refraction, 
and  of  the  number  of  particles  per  unit  volume. 

We  have  seen  that  when  we  attempt  to  find  directly  from  (1)  the  effect 
of  the  particles  upon  the  transmitted  primary  wave,  we  succeed  only  so  far 
as  regards  the  retardation.  In  order  to  determine  the  attenuation  by  this 
process  it  would  be  necessary  to  supplement  (1)  by  a  terra  involving 

sin  2?r  (6*  -  r)/\; 

but  this  is  of  higher  order  of  smallness.  We  could,  however,  reverse  the 
process  and  determine  the  small  term  in  question  a  posteriori  by  means  of 
the  value  of  the  attenuation  obtained  indirectly  from  (1),  at  least  as  far  as 
concerns  the  secondary  light  emitted  in  the  direction  of  the  primary  ray. 

The  theory  of  these  effects  may  be  illustrated  by  a  completely  worked 
out  case,  such  as  that  of  a  small  rigid  and  fixed  spherical  obstacle  (radius  c) 
upon  which  plane  waves  of  sound  impinge*.  It  would  take  too  much  space 
to  give  full  details  here,  but  a  few  indications  may  be  of  use  to  a  reader 
desirous  of  pursuing  the  matter  further. 

The  expressions  for  the  terms  of  orders  0  and  1  in  spherical  harmonics  of 
the  velocity-potential  of  the  secondary  disturbance  are  given  in  equations 
(16),  (17),  §  334.  With  introduction  of  approximate  values  of  70  and  7^  viz. 


70  +  kc  =  %k?c3,      7x  +  kc  =  \ir 
we  get 

[*.]  +  [*J  =  -  ^  (l  +  y)  cos  k  (at  -  r)  +  ^  (l  -  ^)  sin  k  (at  -  r),  .  .  .(15)f 

in  which  c  is  the  radius  of  the  sphere,  and  k  =  27T/X..  This  corresponds  to 
the  primary  wave 

[</>]  =  cos  k  (at  +  x),  ...........................  (16) 

and  includes  the  most  important  terms  from  all  sources  in  the  multipliers 
of  cos  k  (at  -  r),  sin  k  (at  —  r).  Along  the  course  of  the  primary  ray  (JM  =  —  1) 
it  reduces  to 


~  r)  .......  (17) 

We  have  now  to  calculate  by  the  method  of  Fresnel's  zones  the  effect 
of  a  distribution  of  n  spheres  per  unit  volume.  We  find,  corresponding 
to  (6),  for  the  effect  of  a  layer  of  thickness  dx, 

2-rrndx  {%kc*  sin  k  (at  +  x)  -  ^JfcV  cos  k  (at  +  x)}  ..........  (18) 

*  Theory  of  Sound,  2nd  ed.  §  334. 
t  [1902.     n  here  denotes  the  sine  of  the  latitude.] 
B.    IV.  26 


402  ON  THE  TRANSMISSION  OF  LIGHT  THROUGH   AN  [247 

To  this  is  to  be  added  the  expression  (16)  for  the  primary  wave.  The 
coefficient  of  cos  k  (at  +  x)  is  thus  altered  by  the  particles  in  the  layer  dx 
from  unity  to  (1  —  ^T^c^Trndx),  and  the  coefficient  of  sink(at  +  x)  from  0 
to  \k<?Trndx.  Thus,  if  E  be  the  energy  of  the  primary  wave, 

dEj  E  =  -  ^kWirndx  ; 

so  that  if,  as  in  (4),  E=E0e~hx, 

(19) 


The  same  result  may  be  obtained  indirectly  from  the  first  term  of  (15). 
For  the  whole  energy  emitted  from  one  sphere  may  be  reckoned  as 


(20) 


unity  representing  the  energy  of  the  primary  wave  per  unit  area  of  wave- 
front.     From  (20)  we  deduce  the  same  value  of  h  as  in  (19). 

The  first  term  of  (18)  gives  the  refractivity  of  the  medium.     If  8  be  the 
retardation  due  to  the  spheres  of  the  stratum  dx, 


or  &  =  %7rn(?dx  ...............................  (21) 

Thus,  if  jj,  be  the  refractive  index  as  modified  by  the  spheres,  that  of  the 
original  medium  being  unity, 

ip,  ...........................  (22) 


where  p  denotes  the  (small)  ratio  of  the  volume  occupied  by  the  spheres 
to  the  whole  volume.  This  result  agrees  with  equations  formerly  obtained 
for  the  refractivity  of  a  medium  containing  spherical  obstacles  disposed  in 
cubic  order*. 

Let  us  now  inquire  what  degree  of  transparency  of  air  is  admitted  by  its 
molecular  constitution,  i.e.,  in  the  absence  of  all  foreign  matter.  We  may 
take  X  =  6  x  10~8  centim.,  p  —  1  =  '0003  ;  whence  from  (14)  we  obtain  as 
the  distance  x,  equal  to  I/  h,  which  light  must  travel  in  order  to  undergo 
attenuation  in  the  ratio  e:I, 

x  =  4>-4>  x  10~13  x  n  ............................  (23) 

The  completion  of  the  calculation  requires  the  value  of  n.  Unfortunately 
this  number  —  according  to  Avogadro's  law  the  same  for  all  gases  —  can 
hardly  be  regarded  as  known.  Maxwell  f  estimates  the  number  of  molecules 
under  standard  conditions  as  19  x  1018  per  cub.  centim.  If  we  use  this  value 
of  n,  we  find 

x  =  8'3  x  10"  cm.  =  83  kilometres, 

*  Phil.  Mag.  Vol.  xxxiv.  p.  499  (1892).     [Vol.  iv.  p.  35.]     Suppose  m=o>  ,  <r=cc  . 
t  "Molecules,"  Nature,  vni.  p.  440  (1873). 


1899]      ATMOSPHERE   CONTAINING   SMALL   PARTICLES    IN   SUSPENSION.  403 

as  the  distance  through  which  light  must  pass  in  air  at  atmospheric  pres- 
sure before  its  intensity  is  reduced  in  the  ratio  of  2*7  :  1. 

Although  Mount  Everest  appears  fairly  bright  at  100  miles  distance 
as  seen  from  the  neighbourhood  of  Darjeeling,  we  cannot  suppose  that 
the  atmosphere  is  as  transparent  as  is  implied  in  the  above  numbers; 
and  of  course  this  is  not  to  be  expected,  since  there  is  certainly  suspended 
matter  to  be  reckoned  with.  Perhaps  the  best  data  for  a  comparison  are 
those  afforded  by  the  varying  brightness  of  stars  at  various  altitudes.  Bouguer 
and  others  estimate  about  '8  for  the  transmission  of  light  through  the  entire 
atmosphere  from  a  star  in  the  zenith.  This  corresponds  to  8'3  kilometres 
of  air  at  standard  pressure.  At  this  rate  the  transmission  through  83  kilo- 
metres would  be  (-8)10,  or  '11,  instead  of  l/e  or  '37.  It  appears  then  that 
the  actual  transmission  through  83  kilometres  is  only  about  3  times  less 
than  that  calculated  (with  the  above  value  of  n)  from  molecular  diffraction 
without  any  allowance  for  foreign  matter  at  all.  And  we  may  conclude 
that  the  light  scattered  from  the  molecules  would  suffice  to  give  us  a  blue 
sky,  not  so  very  greatly  darker  than  that  actually  enjoyed. 

If  n  be  regarded  as  altogether  unknown,  we  may  reverse  our  argument, 
and  we  then  arrive  at  the  conclusion  that  n  cannot  be  greatly  less  than 
was  estimated  by  Maxwell.  A  lower  limit  for  n,  say  7  x  1018  per  cubic  centi- 
metre, is  somewhat  sharply  indicated.  For  a  still  smaller  value,  or  rather 
the  increased  individual  efficacy  which  according  to  the  observed  refraction 
would  be  its  accompaniment,  must  lead  to  a  less  degree  of  transparency  than 
is  actually  found.  When  we  take  into  account  the  known  presence  of  foreign 
matter,  we  shall  probably  see  no  ground  for  any  reduction  of  Maxwell's 
number. 

The  results  which  we  have  obtained  are  based  upon  (14),  and  are  as  true 
as  the  theories  from  which  that  equation  was  derived.  In  the  electromagnetic 
theory  we  have  treated  the  molecules  as  spherical  continuous  bodies  differing 
from  the  rest  of  the  medium  merely  in  the  value  of  their  dielectric  constant. 
If  we  abandon  the  restriction  as  to  sphericity,  the  results  will  be  modified  in 
a  manner  that  cannot  be  precisely  defined  until  the  shape  is  specified.  On 
the  whole,  however,  it  does  not  appear  probable  that  this  consideration  would 
greatly  affect  the  calculation  as  to  transparency,  since  the  particles  must  be 
supposed  to  be  oriented  in  all  directions  indifferently.  But  the  theoretical 
conclusion  that  the  light  diffracted  in  a  direction  perpendicular  to  the  primary 
rays  should  be  completely  polarized  may  well  be  seriously  disturbed.  If  the 
view,  suggested  in  the  present  paper,  that  a  large  part  of  the  light  from 
the  sky  is  diffracted  from  the  molecules  themselves,  be  correct,  the  observed 
incomplete  polarization  at  90°  from  the  Sun  may  be  partly  due  to  the 
molecules  behaving  rather  as  elongated  bodies  with  indifferent  orientation 
than  as  spheres  of  homogeneous  material, 

26—2 


404  ON  THE  TRANSMISSION  OF  LIGHT  THROUGH   AN  [247 

Again,  the  suppositions  upon  which  we  have  proceeded  give  no  account 
of  dispersion.  That  the  refraction  of  gases  increases  as  the  wave-length 
diminishes  is  an  observed  fact ;  and  it  is  probable  that  the  relation  between 
refraction  and  transparency  expressed  in  (14)  holds  good  for  each  wave- 
length. If  so,  the  falling  off  of  transparency  at  the  blue  end  of  the  spectrum 
will  be  even  more  marked  than  according  to  the  inverse  fourth  power  of  the 
wave-length. 

An  interesting  question  arises  as  to  whether  (14)  can  be  applied  to 
highly  compressed  gases  and  to  liquids  or  solids.  Since  approximately 
(p  —  1)  is  proportional  to  n,  so  also  is  h  according  to  (14).  We  have  no 
reason  to  suppose  that  the  purest  water  is  any  more  transparent  than  (14) 
would  indicate;  but  it  is  more  than  doubtful  whether  the  calculations  are 
applicable  to  such  a  case,  where  the  fundamental  supposition,  that  the  phases 
are  entirely  at  random,  is  violated.  When  the  volume  occupied  by  the 
molecules  is  no  longer  very  small  compared  with  the  whole  volume,  the  fact 
that  two  molecules  cannot  occupy  the  same  space  detracts  from  the  random 
character  of  the  distribution.  And  when,  as  in  liquids  and  solids,  there  is 
some  approach  to  a  regular  spacing,  the  scattered  light  must  be  much  less 
than  upon  a  theory  of  random  distribution. 

Hitherto  we  have  considered  the  case  of  obstacles  small  compared  to  the 
wave-length.  In  conclusion  it  may  not  be  inappropriate  to  make  a  few 
remarks  upon  the  opposite  extreme  case  and  to  consider  briefly  the  obstruction 
presented,  for  example,  by  a  shower  of  rain,  where  the  diameters  of  the 
drops  are  large  multiples  of  the  wave-length  of  light. 

The  full  solution  of  the  problem  presented  by  spherical  drops  of  water 
would  include  the  theory  of  the  rainbow,  and  if  practicable  at  all  would  be 
a  very  complicated  matter.  But  so  far  as  the  direct  light  is  concerned,  it 
would  seem  to  make  little  difference  whether  we  have  to  do  with  a  spherical 
refracting  drop,  or  with  an  opaque  disk  of  the  same  diameter.  Let  us  suppose 
then  that  a  large  number  of  small  disks  are  distributed  at  random  over  a 
plane  parallel  to  a  wave-front,  and  let  us  consider  their  effect  upon  the  direct 
light  at  a  great  distance  behind.  The  plane  of  the  disks  may  be  divided 
into  a  system  of  Fresnel's  zones,  each  of  which  will  by  hypothesis  include 
a  large  number  of  disks.  If  a  be  the  area  of  each  disk,  and  v  the  number 
distributed  per  unit  of  area  of  the  plane,  the  efficiency  of  each  zone  is 
diminished  in  the  ratio  1  : 1  —  vet,  and,  so  far  as  the  direct  wave  is  concerned, 
this  is  the  only  effect.  The  amplitude  of  the  direct  wave  is  accordingly 
reduced  in  the  ratio  1  : 1  —  va,  or,  if  we  denote  the  relative  opaque  area  by  ra, 
in  the  ratio  1  : 1  —  m*.  A  second  operation  of  the  same  kind  will  reduce  the 

*  The  intensity  of  the  direct  wave  is  l-2m,  and  that  of  the   scattered  light  m,  making 
altogether  1  —  m. 


1899]      ATMOSPHERE   CONTAINING   SMALL   PARTICLES   IN   SUSPENSION.  405 

amplitude  to  (1  —  ra)2,  and  so  on.  After  x  passages  the  amplitude  is  (1  —  ra)*, 
which  if  m  be  very  small  may  be  equated  to  e~mx.  Here  mx  denotes  the 
whole  opaque  area  passed,  reckoned  per  unit  area  of  wave-front;  and  it 
would  seem  that  the  result  is  applicable  to  any  sufficiently  sparse  random 
distribution  of  obstacles. 

It  may  be  of  interest  to  give  a  numerical  example.  If  the  unit  of  length 
be  the  centimetre  and  x  the  distance  travelled,  m  will  denote  the  projected 
area  of  the  drops  situated  in  one  cubic  centimetre.  Suppose  now  that  a  is 
the  radius  of  a  drop,  and  n  the  number  of  drops  per  cubic  centimetre,  then 
m  =  WTra2.  The  distance  required  to  reduce  the  amplitude  in  the  ratio  e  :  1 
is  given  by 

a;  =  l/W7ra2. 

Suppose  that  a  =  -^  centim.,  then  the  above-named  reduction  will  occur 
in  a  distance  of  one  kilometre  (x=  105)  when  n  is  about  10~3,  i.e.  when  there 
is  about  one  drop  of  one  millimetre  diameter  per  litre. 

It  should  be  noticed  that  according  to  this  theory  a  distant  point  of  light 
seen  through  a  shower  of  rain  ultimately  becomes  invisible,  not  by  failure 
of  definition,  but  by  loss  of  intensity  either  absolutely  or  relatively  to  the 
scattered  light. 


248. 


THE  INTERFEROMETER. 

[Nature,  LIX.  p.  533,  1899.] 

THE  questions  raised  by  Mr  Preston  (Nature,  March  23)  can  only  be  fully 
answered  by  Prof.  Michelson  himself;  but  as  one  of  the  few  who  have  used 
the  interferometer  in  observations  involving  high  interference,  I  should 
like  to  make  a  remark  or  two.  My  opportunity  was  due  to  the  kindness 
of  Prof.  Michelson,  who  some  years  ago  left  in  my  hands  a  small  instrument 
of  his  model. 

I  do  not  understand  in  what  way  the  working  is  supposed  to  be  prejudiced 
by  "  diffraction."  My  experience  certainly  suggested  nothing  of  the  sort,  and 
I  do  not  see  why  it  is  to  be  expected  upon  theoretical  grounds. 

The  estimation  of  the  "visibility"  of  the  bands,  and  the  deduction  of 
the  structure  of  the  spectrum  line  from  the  visibility  curve,  are  no  doubt 
rather  delicate  matters.  I  have  remarked  upon  a  former  occasion  (Phil.  Mag. 
November,  1892)*  that,  strictly  speaking,  the  structure  cannot  be  deduced 
from  the  visibility  curve  without  an  auxiliary  assumption.  But  in  the 
application  to  radiation  in  a  magnetic  field  the  assumption  of  symmetry 
would  appear  to  be  justified. 

My  observations  were  made  with  a  modification  of  the  original  apparatus, 
which  it  may  be  worth  while  briefly  to  describe.  In  order  to  increase  the 
retardation  it  is  necessary  to  move  backwards,  parallel  to  itself,  one  of  the 
perpendicularly  reflecting  mirrors.  Unless  the  ways  upon  which  the  sliding 
piece  travels  are  extremely  true,  this  involves  a  troublesome  readjustment 
of  the  mirror  after  each  change  of  distance.  The  difficulty  is  avoided  by 
the  use  of  a  fluid  surface  as  reflector,  which  after  each  movement  automatically 
sets  itself  rigorously  horizontal.  If  mercury  be  contained  in  a  glass  dish, 
the  depth  must  be  considerable,  and  then  the  surface  is  inconveniently 
mobile.  A  better  plan  is  to  use  a  thin  layer  standing  on  a  piece  of  copper 
plate  carefully  amalgamated.  A  screw  movement  for  raising  and  lowering 
the  mercury  reflector  is  still  desirable,  though  not  absolutely  necessary. 
*  [Vol.  iv.  p.  15.] 


249. 


ON  THE  CALCULATION  OF  THE  FREQUENCY  OF  VIBRATION 
OF  A  SYSTEM  IN  ITS  GRAVEST  MODE,  WITH  AN 
EXAMPLE  FROM  HYDRODYNAMICS. 

[Philosophical  Magazine,  XLVII.  pp.  566  —  572,  1899.] 

WHEN  the  expressions  for  the  kinetic  (T)  and  potential  (F)  energy  of  a 
system  moving  about  a  configuration  of  stable  equilibrium  are  given,  the 
possible  frequencies  of  vibration  are  determined  by  an  algebraic  equation 
of  degree  (in  the  square  of  the  frequency)  equal  to  the  number  of  independent 
motions  of  which  the  system  is  capable.  Thus  in  the  case  of  a  system  whose 
position  is  defined  by  two  coordinates  q1  and  q2,  we  have 


and  if  in  a  free  vibration  the  coordinates  are   proportional  to  cos  pt,  the 
determinantal  equation  is 

A  —  V&J.          R  —  rfM 

=  0,     (2) 


C-p*N 


(3) 


And  whatever  be  the  number  of  coordinates,  the  possible  frequencies  are 
given  by  a  determinantal  equation  analogous  to  (2). 

When  the  determinantal  equation  is  fully  expressed,  the  smallest  root, 
or  indeed  any  other  root,  can  be  found  by  the  ordinary  processes  of  successive 
approximation.  In  many  of  the  most  interesting  cases,  however,  the  number 
of  coordinates  is  infinite,  and  the  inclusion  of  even  a  moderate  number  of 
them  in  the  expressions  for  T  and  V  would  lead  to  laborious  calculations. 
We  may  then  avail  ourselves  of  the  following  method  of  approximating  to 
the  value  of  the  smallest  root. 


408  ON  THE   CALCULATION   OF   THE   FREQUENCY   OF  [249 

The  method  is  founded  upon  the  principle*  that  the  introduction  of  a 
constraint  can  never  lower,  and  must  in  general  raise,  the  frequency  of  any 
mode  of  a  vibrating  system.  The  first  constraint  that  we  impose  is  the 
evanescence  of  one  coordinate,  say  the  last.  The  lowest  frequency  of  the 
system  thus  constrained  is  higher  than  the  lowest  frequency  of  the  uncon- 
strained system.  Next  impose  as  an  additional  constraint  the  evanescence 
of  the  last  coordinate  but  one.  The  lowest  frequency  is  again  raised.  If 
we  continue  this  process  until  only  one  coordinate  is  left  free  to  vary,  we 
obtain  a  series  of  continually  increasing  quantities  as  the  lowest  frequencies 
of  the  various  systems.  Or,  if  we  contemplate  the  operations  in  the  reverse 
order,  we  obtain  a  series  of  decreasing  quantities  ending  in  the  precise 
quantity  sought.  The  first  of  the  series,  resulting  from  the  sole  variation 
of  the  first  coordinate,  is  given  by  an  equation  of  the  first  degree,  viz. 
A  —  p*L  =  0.  The  second  is  the  lower  root  of  the  determinant  (2)  of  the 
second  order.  The  third  is  the  lowest  root  of  a  determinant  of  the  third 
order  formed  by  the  addition  of  one  row  and  one  column  to  (2),  and  so  on. 
This  series  of  quantities  may  accordingly  be  regarded  as  successive  approxi- 
mations to  the  value  required.  Each  is  nearer  than  its  predecessor  to  the 
truth,  and  all  (except  of  course  the  last  itself)  are  too  high. 

The  practical  success  of  the  method  must  depend  upon  the  choice  of 
coordinates  and  of  the  order  in  which  they  are  employed.  The  object  is 
so  to  arrange  matters  that  the  variation  of  the  first  two  or  three  coordinates 
shall  allow  a  good  approximation  to  the  actual  mode  of  vibration. 

The  example  by  which  I  propose  to  illustrate  the  method  is  one  already 
considered  by  Prof.  Lamb.  It  is  that  of  the  transverse  vibration  of  a  liquid 
mass,  contained  in  a  horizontal  cylindrical  vessel,  and  of  such  quantity  that 
the  free  surface  contains  the  axis  of  the  cylinder  (r  =  0).  If  we  measure  9 
vertically  downwards,  the  fluid  is  limited  by  r  =  0,  r  =  c,  and  by  0  =  —  l^rr, 
0=+  \TT.  Between  the  above  limits  of  6  and  when  r  =  c  the  motion  must 
be  exclusively  tangential. 

In  the  gravest  mode  of  vibration  the  fluid  swings  from  one  side  to  the 
other  in  such  a  manner  that  the  horizontal  motions  are  equal  and  the  vertical 
motions  opposite  at  any  two  points  which  are  images  of  one  another  in  the 
line  0  =  0.  This  relation,  which  holds  also  at  the  two  halves  of  the  free 
surface,  implies  a  stream-function  ty  which  is  symmetrical  with  respect 
to  0  =  0. 

Let  ?/,  denoting  the  elevation  of  the  surface  at  a  distance  r  from  the 
centre  on  the  side  for  which  d  =  \TT,  be  expressed  by 

/c)3-6^(r/c)5+...;   ............  (4) 


*  Theory  of  Sound,  §§  88,  89.    [See  Vol.  i.  p.  170.] 


1899]  VIBRATION   OF    A   SYSTEM    IN    ITS   GRAVEST   MODE.  409 

then  the  potential  energy  for  the  whole  mass  (supposed  to  be  of  unit  density) 
is  given  by 

°  ...)  .............  (5) 


The  more  difficult  part  of  the  problem  lies  in  determining  the  motion 
and  in  the  calculation  of  the  kinetic  energy.  It  may  be  solved  by  the 
method  of  Sir  G.  Stokes,  who  treated  a  particular  case,  corresponding  in 
fact  to  our  first  approximation  in  which  (4)  reduces  to  its  first  term.  It 
is  required  to  find  the  motion  of  an  incompressible  fluid  in  two  dimensions 
within  the  semicylinder,  the  normal  velocity  being  zero  over  the  whole  of 
the  curved  boundary  (r  =  c,  %TT  >  0  >  —  ^TT)  and  over  the  flat  boundary  having 
values  prescribed  by  (4).  If  i|r  be  the  stream-function,  satisfying 


the  conditions  are  that  ty  shall  be  symmetrical  with  respect  to  6  =  0,  that  it 
be  constant  when  r  =  c  from  6  =  0  to  6  =  |TT,  and  that  when  6  =  £TT, 


=  -2q,  (r/c)  +  4g4  (r/c)3 
or  ^/c  =  -q,(r/c)*  +  q4(r/c)*-q6(r/c)s  +  .............  (6) 

At  the  edge,  where  r  =  c, 

^/c  =  -q,  +  q4-qe-...,    ........................  (7) 

and  this  value  must  obtain  also  over  the  curved  boundary. 
The  conditions  may  be  satisfied*  by  assuming 

ijr/c  =  q2  (r/c)2  cos  26  +  qt  (r/c)4  cos  4^  +  ... 

0,     .....................  (8) 


in  which  n  =  0,  1,  2,  &c.  This  form  satisfies  Laplace's  equation  and  the 
condition  of  symmetry  since  cosines  of  6  alone  occur.  When  B  =  ^TT,  it 
reduces  to  (6).  It  remains  only  to  secure  the  reduction  to  (7)  whenr  =  c, 
and  this  can  be  effected  by  Fourier's  method.  It  is  required  that  from 
0  =  0  to  Q  =  \TT 

^Am+l  cos  (2n  +  1)  6  =  -  q2  (1  +  cos  26)  +  q4  (1  -  cos  40)  -  .......  (9) 

It  will  be  convenient  to  write 

^+1  =  q^ll  +  qiA^1  +  ...,      ..................  (10) 

so  that 

ZA^  cos  (2n  +1)0  =  (-!)»-  cos  2s0  .............  (11) 

In  (11)  s  may  have  the  values  1,  2,  3,  &c. 

*  Lamb's  Hydrodynamics,  §  72. 


410  ON   THE    CALCULATION   OF   THE    FREQUENCY   OF  [249 

The  values  of  the  constants  in  (11)  are  to  be  found  as  usual.     Since 

2  I  'cos  (2n  + 1)  0 .  cos  (2m  +  1)  6  dd 

Jo 

vanishes  when  m  and  n  are  different,  and  when  m  and  n  coincide  has  the 
value  £TT,  and  since 


2  /"**{(_  l)o  -  cos  2s0]  cos  (2»  + 1)  0  d6 


we  get 

J<2«>_/      1V+n 

in  which  s  =  1,  2,  3,  &c.,  n  =  0,  1,  2,  &c. 

The  value  of  i/r  in  (8)  is  now  completely  determined  when  <?2,  &c.  are 
known.  The  velocity-potential  </>  is  deducible  by  merely  writing  sines,  in 
place  of  cosines,  of  the  multiples  of  6. 

We  have  now  to  calculate  the  kinetic  energy  T  of  the  motion  thus 
expressed,  supposing  for  brevity  that  the  density  is  unity.  We  have  in 
general 


where  dn  is  drawn  normally  outwards  and  the  integration  extends  over  the 
whole  contour.  In  the  present  case,  however,  d<j>/dn  vanishes  over  the 
circular  boundary,  so  that  the  integration  may  be  limited  to  the  plane  part. 
Of  this  the  two  halves  contribute  equally.  Now  when  6  =  \ir, 


(14) 
(15) 


Thus 

-...,    .........  (16) 


where  Am+l  is  given  by  (10)  and  (12);  it  is  of  course  a  quadratic  function 


The  summation  with  respect  to  n  is  easily  effected  in  particular  cases 
by  decomposition  into  partial  fractions  according  to  the  general  formula 


(2n+2s+I)(2n  +  2s'  +  I)     2(s-s') 


1899]  VIBRATION   OF   A   SYSTEM   IN    ITS   GRAVEST   MODE.  411 

If  s'  =  —  s,  we  have 
1 




(2n  +  2s  +  1)  (2ra  -  2s  +  1) 


4s  \2n  -  2s  +  1      2n  +  2s  +  ij 


If  s'  =  s,  (17)  fails,  but  we  have  by  a  known  formula 


/  ~  8  32     52  (2s  -  I)2 ' 

Thus  for  the  term  in  <j22,  we  have  in  (16) 

in  which  by  (18)  2  (2n  +  3)-1  (2n  - 1)'1  =  0, 

by  (17)         2  (2ra  +  3)"1  (2n  +  I)"1  =  £2  (2w  +  I)-1  -  J2  (2n  +  3)"1 
11  \      1/1       1 


and  by  (19)  2(2n  +3)~2=  |wa-l. 

The  complete  term  (20)  in  q?  is  accordingly 

*?<*-*+> <21> 

The  first  approximation  to  p2  is  therefore  from  (5),  (21) 

-..(22) 


or  p  =  T1690  (g/c)*, (23) 

which  is  Prof.  Lamb's  result*. 

For  the  second  approximation  we  require  also  the  terms  in  (16)  which 
involve  q?  and  qzqt,  and  they  are  calculated  as  before.     The  term  in  <j42  is 


2 


_ 

TT     V9       8  /  ' 

The  term  in  q2  qt  is  made  up  of  two  parts.     Its  complete  value  is 

64c2  .  /9_, 

-9^**  ...............................  (25) 

*  Hydrodynamics,  §  238. 


412  VIBRATION   OF   A   SYSTEM   IN   ITS   GRAVEST   MODE.  [249 

Thus 

+™'  ......  (26> 


which  with  (5)  gives  materials  for  the  second  approximation.  In  proceeding 
to  this  we  may  drop  the  symbols  c  and  g,  which  can  at  any  moment  be 
restored  by  consideration  of  dimensions.  Also  the  factor  8  may  be  omitted 
from  the  expressions  for  T  and  V.  On  this  understanding  we  have  by 
comparison  with  (1), 

44        B~\.        04 

<-!-£-   *--£•   *-£-,. 

or  on  introduction  of  the  value  of  TT, 

L  =  -2439204,        M  =  -  -2829420,        N  =  -3463696. 
The  coefficients  of  the  quadratic  (3)  are  thence  found  to  be 

LN-M*=  -00443040,        AC-&=  -0304762, 

2MB  ^LC-NA=-  -0284860  ; 
whence  on  restoration  of  the  factor  (0/c)*, 

^  =  1-1644  (#/c)*,        p2  =  2-2525  fo/c)*,     ............  (2V) 

the  first  of  which  constitutes  the  second  approximation  to  the  value  of  p 
in  cos  pt,  corresponding  to  the  gravest  mode  of  vibration.  The  small  differ- 
ence between  (23)  and  (27)  shows  the  success  of  the  method  and  indicates 
that  (27)  is  but  very  little  in  excess  of  the  truth. 

If  the  result  were  of  special  importance  it  would  be  quite  practicable 
to  take  another  step  in  the  approximation,  determining  p*  as  the  lowest  root 
of  a  cubic  equation. 

A  question  naturally  suggests  itself  as  to  the  significance  of  the  value 
of'  p2  in  (27).  The  general  theory  of  constraints*  shows  that  it  may  be 
regarded  as  a  first,  but  probably  a  rather  rough,  approximation  to  the 
frequency  of  the  second  lowest  mode  of  the  complete  system.  Just  as  for 
the  gravest  mode  of  all,  the  second  lowest  roots  of  the  series  of  determinants 
(of  the  2nd,  3rd,  and  following  orders)  form  successive  approximations  to 
the  true  value,  each  value  being  lower  and  truer  than  its  predecessor.  The 
second  approximation  would  be  the  middle  root  of  the  cubic  above  mentioned. 
But  for  this  purpose  it  is  doubtful  whether  the  method  is  practical. 

*  Theory  of  Sound,  2nd  ed.  §  92  a. 


250. 


THE  THEORY   OF  ANOMALOUS   DISPERSION. 


[Philosophical  Magazine,  XLVIII.  pp.  151,  152,  1899.] 


I  HAVE  lately  discovered  that  Maxwell,  earlier  than  Sellmeier  or  any 
other  writer,  had  considered  this  question.  His  results  are  given  in  the 
Mathematical  Tripos  Examination  for  1869  (see  Cambridge  Calendar  for 
that  year).  In  the  paper  for  Jan.  21,  l|h—  4h,  Question  IX.  is  :— 

"  Show  from  dynamical  principles  that  if  the  elasticity  of  a  medium  be 
such  that  a  tangential  displacement  77  (in  the  direction  of  y)  of  one  surface 
of  a  stratum  of  thickness  a  calls  into  action  a  force  of  restitution  equal 
to  Ei)  /  a  per  unit  of  area,  then  the  equation  of  propagation  of  such  displace- 
ments is 


"Suppose  that  every  part  of  this  medium  is  connected  with  an  atom 
of  other  matter  by  an  attractive  force  varying  as  distance,  and  that  there 
is  also  a  force  of  resistance  between  the  medium  and  the  atoms  varying 
as  their  relative  velocity,  the  atoms  being  independent  of  each  other  ;  show 
that  the  equations  of  propagation  of  waves  in  this  compound  medium  are 


where  p  and  a  are  the  quantities  of  the  medium  and  of  the  atoms  respectively 
in  unit  of  volume,  y  is  the  displacement  of  the  medium,  and  77  +  f  that 
of  the  atoms,  <rp2£  is  the  attraction,  and  <rRd£/dt  is  the  resistance  to  the 
relative  motion  per  unit  of  volume. 


414  THE  THEORY   OF   ANOMALOUS   DISPERSION.  [250 

"  If  one  term  of  the  value  of  77  be  Ge~xl1  cos  n  (t  —  xjv),  show  that 

1         1    _  p  +  <r      av?         p1  —  n2 
&  +  Vn*~~E~*  W  (pi-tf 

2        <rtf  En 


"  If  <r  be  very  small,  one  of  the  values  of  tf  will  be  less  than  E/p,  and 
if  R  be  very  small  v  will  diminish  as  n  increases,  except  when  n  is  nearly 
equal  to  p,  and  in  the  last  case  I  will  have  its  lowest  values.  Assuming 
these  results,  interpret  them  in  the  language  of  the  undulatory  theory  of 
light." 

If  we  suppose  that  R  =  0, 

L  =  £+  <L      P* 
v2     E     E  p*-n?' 
and 


v2  p  p*-n2' 

if  v0  be  the  velocity  corresponding  to  a-  =  0. 


251. 


INVESTIGATIONS  IN  CAPILLARITY :— THE  SIZE  OF  DROPS.— 
THE  LIBERATION  OF  GAS  FROM  SUPERSATURATED  SOLU- 
TIONS. —  COLLIDING  JETS.  — THE  TENSION  OF  CONTA- 
MINATED WATER-SURFACES.— A  CURIOUS  OBSERVATION. 

[Philosophical  Magazine,  XLVIII.  pp.  321—337,  1899.] 

The  Size  of  Drops. 

THE  relation  between  the  diameter  of  a  tube  and  the  weight  of  the  drop 
which  it  delivers  appears  to  have  been  first  investigated  by  Tate*,  whose 
experiments  led  him  to  the  conclusion  that  "  other  things  being  the  same, 
the  weight  of  a  drop  of  liquid  is  proportional  to  the  diameter  of  the  tube 
in  which  it  is  formed."  Sufficient  time  must  of  course  be  allowed  for  the 
formation  of  the  drops ;  otherwise  no  simple  results  can  be  expected.  In 
Tate's  experiments  the  period  was  never  less  than  40  seconds. 

The  magnitude  of  a  drop  delivered  from  a  tube,  even  when  the  formation 
up  to  the  phase  of  instability  is  infinitely  slow,  cannot  be  calculated  a  priori. 
The  weight  is  sometimes  equated  to  the  product  of  the  capillary  tension  (T) 
and  the  circumference  of  the  tube  (27ra),  but  with  little  justification.  Even 
if  the  tension  at  the  circumference  of  the  tube  acted  vertically,  and  the  whole 
of  the  liquid  below  this  level  passed  into  the  drop,  the  calculation  would  still 
be  vitiated  by  the  assumption  that  the  internal  pressure  at  the  level  in 
question  is  atmospheric.  It  would  be  necessary  to  consider  the  curvatures 
of  the  fluid  surface  at  the  edge  of  attachment.  If  the  surface  could  be 
treated  as  a  cylindrical  prolongation  of  the  tube  (radius  a),  the  pressure  would 
be  T/a,  and  the  resulting  force  acting  downwards  upon  the  drop  would 
amount  to  one-half  (jraT)  of  the  direct  upward  pull  of  the  tension  along  the 
circumference.  At  this  rate  the  drop  would  be  but  one-half  of  that  above 

*  Phil.  Mag.  Vol.  xxvn.  p.  176  (1864). 


416  INVESTIGATIONS   IN   CAPILLARITY.  [251 

reckoned.  But  the  truth  is  that  a  complete  solution  of  the  statical  problem 
for  all  forms  up  to  that  at  which  instability  sets  in,  would  not  suffice  for  the 
present  purpose,  The  detachment  of  the  drop  is  a  dynamical  effect,  and 
it  is  influenced  by  collateral  circumstances.  For  example,  the  bore  of  the 
tube  is  no  longer  a  matter  of  indifference,  even  though  the  attachment  of 
the  drop  occurs  entirely  at  the  outer  edge.  It  will  appear  presently  that 
when  the  external  diameter  exceeds  a  certain  value,  the  weight  of  a  drop 
of  water  is  sensibly  different  in  the  two  extreme  cases  of  a  very  small  and  of 
a  very  large  bore. 

But  although  a  complete  solution  of  the  dynamical  problem  is  im- 
practicable, much  interesting  information  may  be  obtained  from  the  principle 
of  dynamical  similarity.  The  argument  has  already  been  applied  by  Dupre 
(Theorie  Mecanique  de  la  Chaleur,  Paris,  1869,  p.  328),  but  his  presentation 
of  it  is  rather  obscure.  We  will  assume  that  when,  as  in  most  cases,  viscosity 
may  be  neglected,  the  mass  (M)  of  a  drop  depends  only  upon  the  density  (cr), 
the  capillary  tension  (T),  the  acceleration  of  gravity  (g),  and  the  linear 
dimension  of  the  tube  (a).  In  order  to  justify  this  assumption,  the  form- 
ation of  the  drop  must  be  sufficiently  slow,  and  certain  restrictions  must  be 
imposed  upon  the  shape  of  the  tube.  For  example,  in  the  case  of  water 
delivered  from  a  glass  tube,  which  is  cut  off  square  and  held  vertically,  a  will 
be  the  external  radius;  and  it  will  be  necessary  to  suppose  that  the  ratio 
of  the  internal  radius  to  a  is  constant,  the  cases  of  a  ratio  infinitely  small,  or 
infinitely  near  unity,  being  included.  But  if  the  fluid  be  mercury,  the  flat 
end  of  the  tube  remains  unwetted,  and  the  formation  of  the  drop  depends 
upon  the  internal  diameter  only. 

The  "  dimensions  "  of  the  quantities  on  which  M  depends  are  : — 
o-  =  (Mass)1  (Length)"3, 
T  =  (Force)1  (Length)-1  =  (Mass)1  (Time)-2, 
g  =  Acceleration  =  (Length)1  (Time)"2, 
of  which  M,  a  mass,  is  to  be  expressed  as  a  function.     If  we  assume 

M  oc  Tx  .  gv  .  <TZ  .  au, 
we  have,  considering  in  turn  length,  time,  and  mass, 

y  -  32  +  u  =  0,     2#  +  2y  =  0, 
so  that  y=.  —  x,     z  =  1  —  x,     u  =  3  — 

Ta  (  T  \x~l 
Accordingly  M «  —  (  —       . 


1899]  INVESTIGATIONS   IN   CAPILLARITY.  417 

Since  as  is  undetermined,  all  that  we  can  conclude  is  that  M  is  of  the  -form 


9 
where  F  denotes  an  arbitrary  function. 

Dynamical  similarity  requires  that  T/ga-a?  be  constant  ;  or,  if  g  be  sup- 
posed to  be  so,  that  a2  varies  as  T/<r.  If  this  condition  be  satisfied,  the  mass 
(or  weight)  of  the  drop  is  proportional  to  T  and  to  a. 

If  Tate's  law  be  true,  that  cwteris  paribus  M  varies  as  a,  it  follows  from 
(1)  that  F  is  constant.  For  all  fluids  and  for  all  similar  tubes  similarly 
wetted,  the  weight  of  a  drop  would  then  be  proportional  not  only  to  the 
diameter  of  the  tube  but  also  to  the  superficial  tension,  and  it  would  be 
independent  of  the  density. 

In  order  to  examine  how  far  Tate's  law  can  be  relied  upon,  I  have  thought 
it  desirable,  with  the  assistance  of  Mr  Gordon,  to  institute  fresh  experiments 
with  water,  in  which  necessary  precautions  were  observed,  especially  against 
the  presence  of  grease.  Attention  has  been  given  principally  to  the  two 
extreme  cases,  (i)  when  the  wall  of  the  tube  is  thin,  so  that  the  external 
and  internal  diameters  of  the  tube  are  nearly  equal;  (ii)  when  the  bore 
is  small  in  comparison  with  the  external  diameter.  The  event  showed  that 
up  to  an  external  diameter  of  one  centimetre  or  more,  the  size  of  the  bore 
is  of  little  consequence,  but  that  for  larger  diameters  the  weight  of  the  drop 
in  (ii)  is  sensibly  less  than  in  (i).  It  scarcely  needs  to  be  pointed  out  that 
in  (i)  the  diameter  can  only  be  increased  up  to  a  certain  limit,  after  which 
the  tube  would  not  remain  full.  In  (ii)  the  diameter  can  be  increased  to  any 
extent,  but  the  drop  falling  from  it  reaches  a  limit.  The  experiments  of  Tate 
extended  also  to  case  (ii),  but  his  results  are,  I  believe,  erroneous.  For 
a  diameter  of  one-half  an  inch  (1'27  cm.)  he  found  for  the  two  cases  drops  in 
the  ratio  of  T56  :  2'84. 

In  my  experiments  the  thin-walled  tubes  were  of  glass,  the  ends  being 
ground  to  a  plane,  and  carefully  levelled.  Ten  drops,  following  one  another 
at  intervals  of  about  50  seconds,  were  usually  weighed  together.  As  to  the 
interval,  sufficient  time  must  be  allowed  for  the  normal  formation  of  the  drop, 
but  the  fact  that  evaporation  is  usually  in  progress  forbids  too  great  a  pro- 
longation. The  accuracy  attained  was  not  so  great  as  had  been  hoped  for. 
Successive  collections,  made  without  disturbance,  gave  indeed  closely  accord- 
ant weights  (often  to  one-thousandth  part),  but  repetitions  after  cleaning 
and  remounting  indicated  discrepancies  amounting  to  one-half  per  cent.,  or 
even  to  one  per  cent.  The  cause  of  these  minor  variations  has  not  been  fully 
traced;  but  the  results  recorded,  being  the  mean  of  several  experiments, 
must  be  free  from  serious  error.  Attention  may  be  called  to  tubes  11  and  12 
of  nearly  the  same  (external)  diameter.  Of  these  11  was  plugged  so  as  to 
R.  iv.  27 


418 


INVESTIGATIONS    IN   CAPILLARITY. 


[251 

It  will  be  seen 


leave  only  a  small  bore,  the  end  being  carefully  ground  flat, 
that  the  difference  in  the  weights  of  the  drops  was  but  small. 

Again,  No.  10  was  of  barometer- tubing,  having  a  comparatively  small 
bore,  which  accounts  for  the  slightly  diminished  weight  of  the  drop.  The 
other  tubes  were  thin-walled.  In  all  cases  care  was  taken  that  the  cylindrical 
part  of  the  tube,  though  clean,  should  remain  unwetted,  a  condition  which 
precluded  the  use  of  diameters  much  less  than  those  recorded. 


Glass 

Metal 

1 

•088       -0375 

15 

•400 

•1446 

2 

•134 

•0526 

16 

•450 

•1662 

3 

•191 

•0712 

17 

•500 

•1882 

4 

•200 

•0755 

18 

•530 

•2023 

5 

•256 

•0923 

19 

•550 

•2130 

' 

6 

•354 

•1151 

20 

•559 

•2167 

7 

•383 

•1362 

21 

•580 

•2256 

8 

•406 

•1461    ;   22 

•597 

•2295 

9 

•459 

•1703 

23 

•621 

•2389 

10 

•465 

•1698 

24 

•640 

•2454 

11 

•521 

•1969 

25 

•680 

•2510 

12 

•523 

•2023 

26 

•730 

•2531 

13 

•566 

•2210 

27 

•800 

•2509 

14 

•584 

•2339 

Fig.  1. 


The  numbers  in  the  second  column  are  the  external  diameters  measured 
in  inches  (one  inch  =  2'54  cm.),  while  the  third  column  gives  the  weight 
in  grams  of  a  single  drop,  corrected  for  temperature  to  15°  C.,  upon  the 
supposition  (corresponding  to  Tate's  law)  that  the  weight  is  proportional  to 
surface-tension. 

The  entries  under  the  heading  "  Metal "  relate  to  experiments  in  which 
the  glass  tubes  were  replaced  by  metal  disks,  bored  cen- 
trally and  turned  true  in  the  lathe.  The  water  was  supplied 
from  above  through  a  metal  tube  soldered  to  the  back 
(upper)  face  of  the  disk  (Fig.  1).  At  the  time  of  use  only 
the  lower  face  was  wetted. 

A  plot  of  both  sets  of  numbers  is  shown  in  the  Figure  (2). 
The  two  curves  practically  coincide  up  to  diameters  of 
about  '4  inch,  after  which  that  corresponding  to  the  disks 
falls  below.  The  lower  curve  shows  some  irregularities, 
especially  in  the  region  of  diameters  equal  to  "6  inch. 
These  appear  to  be  genuine ;  they  may  originate  in  a  sort  of  reflexion  from 


1899] 


INVESTIGATIONS    IN    CAPILLARITY. 


419 


the  circumference  of  the  disk  of  the  disturbance  caused  by  the  breaking  away 
of  the  drop.  It  is  possible  that  at  this  stage  the  phenomenon  is  sensibly 
influenced  by  fluid  viscosity. 

Fig.  2. 


That  the  size  of  the  bore  should  be  of  secondary  importance  is  easily 
understood.  Up  to  the  phase  of  instability,  the  phenomenon  is  merely 
a  statical  one,  and  the  element  of  the  size  of  the  bore  does  not  enter.  It  is 
only  the  rapid  motion  which  occurs  during  the  separation  of  the  drop  that 
could  be  influenced.  When  the  diameter  is  moderate,  the  most  rapid  motions 
occur  at  a  level  considerably  below  the  tube,  and  the  obstruction  presented 
by  the  flat  face  of  a  thick- walled  tube  is  unimportant. 

The  observations  give  materials  for  the  determination  of  the  function  F 
in  (1).  In  the  following  table,  applicable  to  thin-walled  tubes,  the  first 
column  gives  values  of  Tjgcr^,  and  the  second  column  those  of  gM/Ta,  all 
the  quantities  concerned  being  in  c.G.S.  measure,  or  other  consistent  system. 


Tlgoa? 

gM/Ta 

2-58 

4-13 

tie 

3-97 

•708 

3-80 

•441 

3-73 

•277 

3-78 

•220 

3-90 

•169 

4-06 

27—2 


420  INVESTIGATIONS   IN   CAPILLARITY.  [251 

From  this  the  weight  of  a  drop  of  any  liquid  of  which  the  density  and  the 
surface-tension  are  known  can  be  calculated.  For  many  purposes  it  may 
suffice  to  treat  F  as  constant,  say  3'8.  The  formula  for  the  weight  of  a  drop 
is  then  simply 

....................................  (2) 


in  which  3'8  replaces  the  2?r  of  the  faulty  theory  alluded  to  earlier. 

The  Liberation  of  Gas  from  Supersaturated  Solutions. 

The  formation  of  bubbles  upon  the  sides  of  a  vessel  containing  "  soda- 
water"  or  a  gas-free  liquid  heated  above  its  boiling-point,  is  a  subject 
upon  which  there  has  been  much  difference  of  opinion.  In  one  view,  ably 
advocated  by  Gernez,  the  nucleus  is  invariably  gaseous.  That  a  small  volume 
of  gas,  visible  or  invisible,  provided  that  its  dimensions  exceed  molecular 
distances,  must  act  in  this  way  is  certain,  and  the  activity  of  porous  solids  is 
thus  naturally  and  easily  explained.  But  Gernez  goes  much  further,  and 
holds  that  the  activity  of  glass  or  metal  rods,  immersed  in  the  liquid  without 
precaution,  is  of  the  same  nature,  and  to  be  attributed  to  the  film  of  air 
which  all  bodies  acquire  when  left  for  some  time  in  contact  with  the  atmo- 
sphere. If  a  body  is  rendered  inactive  by  prolonged  standing  in  cold  water  ; 
by  treatment  with  alcohol,  ether,  &c.,  "qui  dissolvent  les  gaz  de  1'air,  plus 
abondamment  que  1'eau  *  "  ;  or  by  heating  in  a  flame  ;  it  is  because  by  such 
processes  the  film  of  air  is  removed.  One  cannot  but  sympathise  with 
Tomlinson-f*  in  his  repugnance  to  such  an  explanation  ;  but  the  position  main- 
tained by  the  latter,  that  activity  is  due  to  contamination  with  grease,  is  also 
not  without  its  difficulties. 

The  question  whether  contact  with  air  suffices  to  restore  the  activity  of 
a  piece  of  glass  or  metal  that  has  been  rendered  inactive  by  heat  or  otherwise, 
appears  to  be  amenable  to  experiment,  and  should  not  remain  an  open  one. 
In  1892  I  had  a  number  of  glass  tubes  prepared  of  about  1  cm.  diameter 
for  experiments  in  this  direction.  After  a  thorough  heating  in  the  blowpipe- 
flame,  the  ends  of  the  tubes  were  hermetically  sealed.  At  intervals  since 
that  date  some  of  the  tubes  have  been  opened  and  compared  with  others 
which  had  undergone  no  preparation.  Short  lengths  of  rubber  provided  with 
pinch-cocks  are  fitted  to  the  upper  ends,  by  means  of  which  aerated  water 
is  easily  drawn  in  from  a  shallow  vessel.  Three  tubes  remaining  over  from 
the  batch  above  mentioned  were  tried  a  few  weeks  ago,  and  establish  the 
conclusion  that  seven  years  contact  with  air  fails  to  restore  activity.  A  similar 
experiment  may  be  made  with  iron  wires.  If  these  be  heated  and  sealed  up 
in  glass  tubes,  they  remain  inactive,  but  exposure  to  the  air  of  the  laboratory 
for  a  day  or  two  restores  activity. 

*  Annales  de  VEcole  Normak,  p.  319,  1875. 
t  Phil.  May.  Vol.  xux.  p.  305  (1875). 


1899]  INVESTIGATIONS   IN   CAPILLARITY.  421 

In  opposition  to  the  contention  that  grease  is  the  primary  cause  of 
activity,  Gernez  brings  forward  a  striking  experiment  from  which  it  appears 
that  a  drop  of  olive-oil  itself  liberates  no  gas  when  introduced  with  pre- 
caution. "  Quant  au  r61e  que  jouent  les  corps  gras,  il  est  facile  de  s'en  rendre 
compte:  lorsqu'on  frotte  un  corps  quelconque  entre  les  doigts  legerement 
graisses,  on  produit  a  sa  surface  une  se*rie  d'eminences  line'aires  se"pare"es  par 
les  sillons  qui  correspondent  aux  lignes  de  1'epiderme ;  les  cavity's  forment  un 
reseau  de  conduits  qui  contiennent  de  1'air,  sont  difficilement  mouilles  par 
1'eau  et,  par  consequent,  constituent  au  sein  du  liquide  une  atmosphere 
eminemment  favorable  au  de*gagement  des  gaz*." 

It  seems  to  me  that  Tomlinson  was  substantially  correct  in  attributing 
the  activity  of  a  non-porous  surface  to  imperfect  adhesion.  We  have  to 
consider  in  detail  the  course  of  events  when  a  surface,  e.g.  of  glass,  is  intro- 
duced into  the  liquid.  If  the  surface  be  clean,  it  is  wetted  by  the  water 
advancing  over  it,  whether  there  be  a  film  -of  air  condensed  upon  it  or  not, 
and  no  gas  is  liberated  from  the  liquid.  But  if  the  surface  be  greasy,  even 
in  a  very  slight  degree,  the  behaviour  is  different.  We  know  that  a  drop 
of  water  is  reluctant  to  spread  over  a  glass  that  is  not  scrupulously  clean. 
If  a  large  quantity  of  water  be  employed,  some  sort  of  spreading  follows 
under  the  influence  of  gravity,  but  there  is  no  proper  adhesion,  at  least  for 
a  time,  as  appears  at  once  on  pouring  the  water  off  again.  The  precise 
character  of  the  transition  from  glass  to  water  when  there  is  grease  between 
is  not  well  understood.  It  may  be  that  there  is  something  which  can  fairly 
be  called  a  film  of  air.  If  so,  its  existence  is  a  consequence  of  the  presence 
of  the  grease.  On  the  other  hand,  it  appears  at  least  equally  probable  that 
air  is  not  concerned,  and  that  the  activity  of  the  surface  is  directly  due  to  the 
thin  film  of  grease,  whose  properties,  as  in  the  case  of  greased  water  surfaces, 
are  materially  different  from  those  of  a  thick  layer. 

On  this  principle,  too,  it  is  easier  to  understand  the  retention  of  a  visible 
bubble  when  formed — a  retention  which  often  lasts  for  a  long  time.  So 
soon  as  the  gas  is  entirely  surrounded  by  liquid  of  thickness  exceeding  the 
capillary  limit,  the  bubble  is  bound  to  rise.  It  is  difficult  to  see  how  the 
hypothetical  film  of  air  explains  the  failure  of  the  liquid  to  penetrate  between 
the  bubble  and  the  solid. 

Colliding  Jets. 

In  various  papers  (Proc.  Roy.  Soc.  Feb.  1879,  May  1879,  June  1882) 
[Vol.  I.  pp.  372,  377 ;  Vol.  n.  p.  103]  I  have  examined  the  behaviour  of 
colliding  drops  and  jets.  Experiments  with  drops  are  very  simply  carried 
out  by  the  observation  of  nearly  vertical  fountains,  rising  say  to  two  feet 
from  nozzles  ^  inch  in  diameter.  The  scattering  of  the  drops,  when  the 

*  Loc.  cit.  p.  346. 


422  INVESTIGATIONS   IN   CAPILLARITY.  [251 

water  is  clean  and  not  acted  upon  by  electricity,  shows  that  collision  is 
followed  by  rebound.  If  the  water  is  milky,  or  soapy  with  unclarified  soap, 
or  if  the  jet,  though  clean,  is  under  the  influence  of  feeble  electricity,  the 
apparent  coherence  and  the  heaviness  of  the  patter  made  by  the  falling  water 
are  evidence  that  rebound  no  longer  ensues,  but  that  collision  results  in 
amalgamation.  Eye  observation,  or  photography,  with  the  instantaneous 
illumination  of  electric  sparks  renders  the  course  of  events  perfectly  clear. 
[1902.  The  annexed  illustrations  are  from  instantaneous  photographs  of  the 
same  fountain  with  and  without  electrical  influence.] 

The  form  of  the  experiment  in  which  are  employed  jets,  issuing  at 
moderate  velocities  and  meeting  at  high  obliquities,  is  the  more  instructive ; 
but  it  is  liable  to  be  troublesome  in  consequence  of  the  tendency  of  the  jets 
to  unite  spontaneously.  It  is  important  to  avoid  dust  both  in  the  water  and 
in  the  atmosphere  where  the  collision  occurs.  An  electromotive  force  of  one 
volt  suffices  to  determine  union ;  but  so  long  as  the  jets  rebound  there  is 
complete  electrical  insulation  between  them. 

As  to  the  manner  in  which  electricity  acts,  two  views  were  suggested.  It 
was  thought  probable  that  union  was  the  result  of  actual  discharge  across  the 
thin  layer  of  intervening  insulation;  but  it  was  also  pointed  out  that  the 
result  might  be  due  to  the  augmented  pressure  to  be  expected  from  the 
electrical  charges  upon  the  opposed  surfaces.  From  observations  upon  the 
colours  of  thin  plates  exhibited  at  the  region  of  contact,  which  he  found 
to  be  undisturbed  by  such  electrical  forces  as  would  not  produce  union, 
Mr  Newall*  concluded  that  the  second  of  the  above-mentioned  explanations 
must  be  discarded. 

On  the  other  hand,  as  has  been  pointed  out  by  Kaiser f,  the  progress  of 
knowledge  concerning  electrical  discharge  has  rendered  the  first  explanation 
more  difficult  of  acceptance.  It  would  appear  that  some  hundreds  of  volts 
are  needed  in  order  to  start  a  spark,  and  that  mere  diminution  of  the  interval 
to  be  crossed  would  not  compensate  for  want  of  electromotive  force. 

A  more  attentive  examination  of  the  conditions  of  the  experiment  may 
perhaps  remove  some  of  the  difficulties  which  seem  to  stand  in  the  way  of 
the  second  explanation.  As  the  liquid  masses  approach  one  another,  the 
intervening  air  has  to  be  squeezed  out.  In  the  earlier  stages  of  approxima- 
tion the  obstacle  thus  arising  may  not  be  important ;  but  when  the  thickness 
of  the  layer  of  air  is  reduced  to  the  point  at  which  the  colours  of  thin  plates 
are  visible,  the  approximation  must  be  sensibly  resisted  by  the  viscosity  of 
the  air  which  still  remains  to  be  got  rid  of.  No  change  in  the  capillary 

*  Phil.  Mag.  Vol.  xx.  p.  33  (1885). 

t  Wied.  Ann.  LIII.  p.  667  (1894).  Kaiser's  own  experiments  were  made  upon  the  modification 
of  the  phenomenon  observed  by  Boys,  where  the  contact  takes  place  between  two  soap-films. 


1S99]  INVESTIGATIONS   IN   CAPILLARITY.  423 


In  natural  condition. 


424  INVESTIGATIONS   IN   CAPILLARITY.  [251 

conditions  can  arise  until  the  interval  is  reduced  to  a  small  fraction  of  a  wave- 
length of  light;  but  such  a  reduction,  unless  extremely  local,  is  strongly 
opposed  by  the  remaining  air.  It  is  of  course  true  that  this  opposition  is 
temporary.  The  question  is  whether  the  air  can  be  anywhere  squeezed  out 
during  the  short  time  over  which  the  collision  extends. 

It  would  seem  that  the  electrical  forces  act  with  peculiar  advantage. 
If  we  suppose  that  upon  the  whole  the  air  cannot  be  removed,  so  that  the 
mean  distance  between  the  opposed  surfaces  remains  constant,  the  electric 
attractions  tend  to  produce  an  instability  whereby  the  smaller  intervals  are 
diminished  while  the  larger  are  increased.  Extremely  local  contacts  of  the 
liquids,  while  opposed  by  capillary  tension  which  tends  to  keep  the  surfaces 
flat,  are  thus  favoured  by  the  electrical  forces,  which  moreover  at  the  small 
distances  in  question  act  with  exaggerated  power. 

It  is  probably  by  promoting  local  approximations  in  opposition  to  capillary 
forces  that  dust,  finding  its  way  to  the  surfaces,  brings  about  union. 

A  question  remains  as  to  the  mode  of  action  of  milk  or  soapy  turbidity. 
The  observation,  formerly  recorded,  that  it  is  possible  for  soap  to  be  in  excess 
may  here  have  significance.  It  would  seem  that  the  surfaces,  coming  into 
collision  within  a  fraction  of  a  second  of  their  birth,  would  still  be  subject 
to  further  contamination  from  the  interior.  A  particle  of  soap  rising  acci- 
dentally to  the  surface  would  spread  itself  with  rapidity.  Now  such  an 
outward  movement  of  the  liquid  is  just  what  is  required  to  hasten  the 
removal  of  the  intervening  air.  It  is  obvious  that  this  effect  would  fail  if 
the  contamination  of  the  surface  had  proceeded  too  far  previously  to  the 
collision. 

In  order  to  illustrate  the  importance  of  the  part  played  by  the  intervening 
gas,  I  thought  that  it  would  be  interesting  to  compare  the  behaviour  of  the 
jets  when  situated  in  atmospheres  of  different  gases.  It  seemed  that  gases 
more  freely  soluble  in  water  than  the  atmospheric  gases  would  be  more  easily 
got  rid  of  in  the  later  stages  of  the  collision,  and  that  thus  union  might  more 
readily  be  brought  about.  This  expectation  has  been  confirmed  in  trials 
made  on  several  different  occasions.  It  was  found  sufficient  to  allow  a  pretty 
strong  stream  of  the  gas  under  examination  to  play  upon  the  jets  at  and 
above  the  place  of  collision.  Jets  of  air,  of  oxygen,  and  of  coal-gas  were 
found  to  be  without  effect.  On  the  other  hand,  carbonic  acid,  nitrous  oxide, 
sulphurous  anhydride,  and  steam  at  once  caused  union.  Only  in  the  case 
of  hydrogen  was  there  an  ambiguity.  On  some  occasions  the  hydrogen 
appeared  to  be  without  effect,  but  on  others  (when  perhaps  the  pressure  of 
collision  was  higher)  union  uniformly  followed.  Care  was  taken  to  verify 
that  air  blown  through  the  same  tube  as  had  supplied  the  hydrogen  was 
inactive,  so  that  the  effect  of  the  hydrogen  could  not  be  attributed  to  dust. 


1899]  INVESTIGATIONS   IN   CAPILLARITY.  425 

The  action  of  hydrogen  cannot  be  explained  by  its  solubility.  Hydrogen  is, 
however,  much  less  viscous  than  other  gases,  and  to  this  we  may  plausibly 
attribute  its  activity  in  promoting  union.  A  layer  of  hydrogen  may  be 
effectively  squeezed  out  in  a  time  that  would  be  insufficient  in  the  case  of 
air  and  oxygen. 

The  Tension  of  Contaminated  Water-Surfaces. 

In  my  experiments  upon  the  superficial  viscosity  of  water  (Proc.  Roy.  Soc. 
June  1890)  [Vol.  ill.  p.  375]  I  had  occasion  to  notice  that  the  last  traces 
of  residual  contamination  had  very  little  influence  upon  the  surface-tension, 
but  that  they  became  apparent  when  compressed  in  front  of  the  vibrating 
needle  of  Plateau's  apparatus.  Subsequently  I  showed  (Phil.  Mag.  Vol. 
xxxni.  p.  470,  1892)  [Vol.  in.  p.  572]  that  according  to  Laplace's  theory 
of  Capillarity,  in  which  matter  is  regarded  as  continuous,  the  effect  of  a  thin 
surface-film  in  diminishing  the  tension  of  pure  water  should  be  as  the 
square  of  the  thickness  of  the  film. 

The  tension  of  slightly  contaminated  surfaces  was  made  the  subject 
of  special  experiments  by  Miss  Pockels  (Nature,  Vol.  XLIII.  p.  437, 1891),  who 
concluded  that  a  water-surface  can  "  exist  in  two  sharply  contrasted  conditions; 
the  normal  condition,  in  which  the  displacement  of  the  partition  [altering 
the  density  of  the  contamination]  makes  no  impression  upon  the  tension, 
and  the  anomalous  condition,  in  which  every  increase  or  decrease  alters 
the  tension."  It  is  only  since  I  have  myself  made  experiments  upon  the 
same  lines  that  I  have  appreciated  the  full  significance  of  Miss  Pockels' 
statement.  The  conclusion  that,  judged  by  surface-tension,  the  effect  of 
contamination  comes  on  suddenly,  seems  to  be  of  considerable  importance, 
and  I  propose  to  illustrate  it  further  by  actual  curves  embodying  results 
recently  obtained. 

The  water  is  contained  in  a  trough  modelled  after  that  of  Miss  Pockels. 
It  is  of  tin-plate,  70  cm.  long,  10  cm.  broad,  and  2  cm.  deep,  and  it  is  filled 
nearly  to  the  brim.  The  partitions,  by  which  the  oil  is  confined,  are  made 
of  strips  of  glass  resting  upon  the  edge  of  the  trough  in  such  a  manner  that 
their  lower  surfaces  are  wetted  while  the  upper  surfaces  remain  dry.  The 
strips  may  be  1£  cm.  wide,  and  for  convenience  of  handling  their  length 
should  exceed  considerably  the  width  of  the  trough.  I  have  found  advantage 
in  cementing  (with  hard  cement)  slight  webs  of  glass  to  the  lower  faces.  The 
length  of  these  is  a  rough  fit  with  the  width  of  the  trough,  enabling  them  to 
serve  as  guides  preventing  motion  of  the  strips  parallel  to  their  length. 

In  order  to  observe  the  surface-tension  Miss  Pockels  used  a  small  disk 
(6  mm.  in  diameter)  in  contact  with  the  surface,  measuring  the  force  necessary 
to  detach  it.  In  my  own  experiments  I  have  employed  the  method  of 


426  INVESTIGATIONS   IN   CAPILLARITY.  [251 

Wilhelmy,  which  appears  to  be  better  adapted  to  the  purpose.  A  thin 
blade  is  mounted  in  a  balance,  its  plane  being  vertical  and  its  lower  horizontal 
edge  dipping  under  the  surface  of  the  liquid.  If  absolute  measures  are 
7-equired,  the  edge  of  the  blade  should  lie  at  the  general  level  of  the  surface 
when  the  pointer  of  the  balance  stands  at  zero.  If  m  be  the  mass  in  the 
other  pan  needed  to  compensate  the  effect  of  the  liquid,  I  the  length  of 
the  blade,  the  surface-tension  (T)  may  be  deduced  from  the  equation 

2lT=m(/ (1) 

When  only  differences  of  tension  are  concerned,  the  precise  level  of  the  strip 
is  of  no  consequence.  As  regards  material,  glass  is  to  be  preferred  and  it 
should  be  thin  in  order  not  unduly  to  diminish  the  sensitiveness  of  the 
balance  by  the  displacement  of  water.  I  have  used  a  small  frame  carrying 
three  parallel  blades,  the  total  length  being  27  cm.,  while  the  thickness  may 
be  considered  nearly  negligible.  Before  use  the  glass  is  cleaned  with  strong 
sulphuric  acid,  and  the  angle  of  contact  with  the  water  when  the  balance  is 
raised  appears  to  be  zero.  The  total  value  of  m  for  a  clean  surface  may  then 
be  calculated  from  (1),  taking  T  at  74.  We  find  m  =  4'1  gms.  The  balance 
could  be  read  without  difficulty  to  '01  gm.,  giving  abundant  accuracy. 

The  position  of  the  barrier,  giving  the  length  of  the  surface  to  which  the 
grease  is  confined,  is  measured  by  a  millimetre-scale,  but  is  subject  to  a 
correction  needed  in  order  to  take  account  of  the  additional  surface  operative 
when  the  suspended  strip  is  raised.  This  amounts  to  about  3  cm.,  and  is 
to  be  added  to  the  measured  length.  In  a  set  of  experiments  where  the 
grease  is  successfully  confined,  the  density  is  proportional  to  the  reciprocal 
of  the  above  corrected  length.  It  sometimes  happens  that  continuity  is 
lost  by  the  passage  of  grease  across  the  barrier.  This  is  of  course  most 
likely  to  happen  when  the  tensions  on  the  two  sides  differ  considerably, 
and  the  danger  may  be  mitigated  by  the  use  of  a  second  barrier,  so  manipu- 
lated that  the  densities  are  nearly  the  same  on  the  two  sides  of  the  principal 
barrier. 

In  commencing  a  set  of  observations  the  first  step  is  to  secure  the 
cleanness  of  the  surface.  To  this  end  the  surface  is  scraped,  if  the  expression 
may  be  allowed,  along  the  whole  length  by  one  of  the  movable  partitions, 
and,  if  thought  necessary,  the  accumulated  grease  at  the  far  end  may  be 
removed  with  strips  of  paper.  The  operation  should  be  repeated  two  or 
three  times  with  intermediate  insertion  of  the  balanced  strips  until  it  is 
certain  that  no  grease  remains,  competent  to  affect  the  tension  even  when 
concentrated.  The  weights  now  necessary  to  bring  the  pointer  to  zero  give 
the  standard  with  which  the  contaminated  surfaces  are  to  be  compared. 

If  it  be  desired  to  begin  with  small  contaminations,  it  is  best  to  contract 
the  area,  say  to  about  one-half  the  maximum,  and  then  to  apply  the  grease 


1899] 


INVESTIGATIONS   IN    CAPILLAEITY. 


427 


under  examination  with  a  previously  ignited  platinum  wire  until  a  small 
effect,  such  as  '02  gm.,  is  observed  at  the  balance.  If  the  surface  be  now 
extended  to  the  maximum,  the  attenuated  grease  will  have  lost  its  power, 
and  the  original  reading  for  clean  surfaces  will  be  recovered.  The  barrier 
may  now  be  advanced,  readings  being  taken  at  intervals  as  the  grease  is 
concentrated.  It  is  often  more  convenient  to  make  the  final  adjustment  by 
moving  the  barrier  rather  than  by  correcting  the  weights. 

An  example  will  make  manifest  at  once  the  character  of  the  results 
obtained.  On  May  15,  the  weight  for  the  clean  surface  being  1-65  gm., 
the  water  was  greased  with  castor-oil.  With  the  barrier  at  63  cm.  this 
grease  had  no  effect.  The  corrected  length  is  66,  and  the  reciprocal  of  this, 
viz.  152,  represents  (for  this  series  of  observations)  the  density  of  the  oil. 
With  the  barrier  at  40,  viz.  at  density  233,  there  was  no  change  of  the  order 
of  '005  gm.  At  36  cm.,  or  density  256,  the  oil  had  just  begun  to  show  itself 
distinctly,  the  weight  being  then  T64.  At  density  278  the  weight  became 
1'62.  From  this  point  onwards  increase  of  density  tells  rapidly.  At  308  the 
weight  was  I' 55,  and  at  334  the  weight  was  1-40.  A  plot  of  these  results 
is  given  in  Fig.  3,  and  brings  out  more  vividly  than  any  description  the 
striking  character  of  the  law  discovered  by  Miss  Pockels. 

The  effect  of  concentration  beyond  571,  giving  *70  gm.,  could  not  be 
examined  in  the  same  series.  It  was  necessary  to  add  more  oil,  and  then 
of  course  the  reciprocals  of  the  corrected  lengths  represent  the  densities 
on  a  different  scale  from  before.  Corresponding  to  63  cm.,  of  which  the 
reciprocal  is  159,  the  weight  was  now  T20  gm.,  falling  to  TOO  at  175,  '80 
at  204,  -70  at  233,  '60  at  351,  '55  at  488,  and  finally  '52  at  625.  These 
values  are  plotted  in  Fig.  4,  and  they  show  that  from  a  certain  density 
onwards  the  tension  falls  very  slowly.  This  curve  may  be  continued 

Figs.  3-6. 
0  133456 


-/erf* 


428  INVESTIGATIONS   IN   CAPILLARITY.  [251 

backwards  by  means  of  the  results  of  Fig.  3,  for  of  course  the  densities 
corresponding  to  any  particular  weight,  e.g.  1'20  gm.,  are  really  the  same  in 
the  two  series. 

It  is  of  interest  to  inquire  what  point  on  these  curves  corresponds  to  the 
deadening  of  the  movements  of  small  particles  of  camphor  deposited  upon  the 
surface.  On  a  former  occasion  I  have  shown  (Phil.  Mag.  Vol.  xxxm.  p.  366, 
1892)  [Vol.  in.  p.  565]  that  whatever  may  be  the  character  of  the  grease  the 
cessation  of  the  movements  indicates  that  the  tension  falls  short  of  a  particular 
value.  In  the  present  method  of  experimenting  there  is  no  difficulty  in  deter- 
mining what  for  brevity  may  be  called  the  camphor-point.  Two  precautions 
should,  however,  be  observed.  It  is  desirable  not  to  try  the  camphor  until  near 
the  close  of  a  set  of  experiments,  and  then  to  avoid  too  great  a  quantity.  It 
would  seem  that  the  addition  of  camphor  may  sometimes  lower  the  tension 
below  the  point  due  to  the  grease.  The  second  precaution  required  is  the 
raising  of  the  balanced  strip ;  otherwise  when  a  weight  is  taken  the  density 
of  the  grease  is  altered.  In  several  trials  with  castor  and  other  oils  the 
camphor-point  'was  found  to  correspond  with  a  drop  of  tension  from  that 
of  clean  water  amounting  to  '9  gm.  The  points  thus  fixed  are  marked  in 
Figs.  (3)  and  (4)  with  the  letter  G. 

At  this  stage  a  certain  discrepancy  from  former  results  should  be  remarked 
upon.  Working  by  the  method  of  ripples  I  had  concluded  that  the  camphor- 
point  corresponded  to  a  tension  '72  of  that  of  pure  water,  i.e.  to  a  drop  of  28 
per  cent.  But  the  '9  gm.  is  only  22  per  cent,  of  the  calculated  weight  for 
pure  water,  i.e.  4*1  gms.  At  this  rate  the  72  per  cent,  would  become  78  per 
cent.,  and  the  difference  seems  larger  than  can  well  be  explained  as  an 
alteration  of  standard  in  judging  when  the  fragments  are  nearly  dead. 

One  of  the  most  striking  conclusions  to  be  drawn  from  an  inspection 
of  the  curves  is  the  slowness  of  the  fall  of  tension  which  sets  in  soon  after 
passing  the  camphor-point.  On  a  rough  view  it  would  seem  as  if  a  second 
limit  were  being  approached.  But  this  idea  is  scarcely  confirmed  by  actual 
further  additions  of  oil,  for  the  tension  continues  to  fall  slightly  after  each 
addition,  even  when  large  quantities  are  already  present.  But  there  is  one 
peculiarity  in  the  behaviour  of  the  oil  which  suggests  that  the  failure  to 
reach  a  limit  may  be  due  to  want  of  homogeneity.  As  is  well  known,  the 
disk  into  which  a  drop  deposited  upon  an  already  oiled  surface  at  first  spreads, 
soon  breaks  up,  and  the  superfluous  oil  collects  itself  into  little  lenses.  After 
this  stage  is  reached  it  would  be  natural  to  suppose  that  the  affinity  of  the 
surface  for  oil  was  fully  satisfied,  and  that  no  further  alteration  in  tension 
could  occur.  And  in  fact  the  balance  usually  indicated  the  absence  of 
immediate  effect.  But  if  the  surface  were  expanded  so  as  to  spread  the 
added  oil  more  effectively  and  then  contracted  again,  a  fall  in  tension  was 
almost  always  observed.  It  would  seem  as  if  the  surface  still  retained  an 


1899] 


INVESTIGATIONS    IN    CAPILLARITY. 


affinity  for  some  minor  ingredient  capable  of  being  extracted,  though  satiated 
as  regards  the  principal  ingredient. 

The  comparison  of  the  present  with  former  results  throws  an  interesting 
light  upon  molecular  magnitudes.  It  has  been  shown  (Proc.  Roy.  Soc.  March 
1890)  [Vol.  in.  p.  347]  that  the  thickness  of  the  film  of  olive-oil,  calculated 
as  if  continuous,  which  corresponds  to  the  camphor-point,  is  about  2'0  /*/*,*; 
while  from  the  present  curves  it  follows  that  the  point  at  which  the  tension 
begins  to  fall  is  about  half  as  much,  or  TO  /A/Z.  Now  this  is  only  a  moderate 
multiple  of  the  supposed  diameter  of  a  gaseous  molecule,  and  perhaps  scarcely 
exceeds  at  all  the  diameter  to  be  attributed  to  a  molecule  of  oil.  It  is  obvious 
therefore  that  the  present  phenomena  lie  entirely  outside  the  scope  of  a  theory 
such  as  Laplace's,  in  which  matter  is  regarded  as  continuous,  and  that  an 
explanation  requires  a  direct  consideration  of  molecules. 

If  we  begin  by  supposing  the  number  of  molecules  of  oil  upon  a  water 
surface  to  be  small  enough,  not  only  will  every  molecule  be  able  to  approach 
the  water  as  closely  as  it  desires,  but  any  repulsion  between  molecules  will 
have  exhausted  itself.  Under  these  conditions  there  is  nothing  to  oppose 
the  contraction  of  the  surface — the  tension  is  the  same  as  that  of  pure  water. 


Castor  Oil. 
May  15. 
Fig.  (3) 

Castor  Oil. 
May  15. 
Fig.  (4) 

Olive  Oil. 
May  3. 
Fig.  (5) 

Cod  Liver  Oil. 
May  11. 
Fig.  (6) 

Density 

Weight 
(in  grams) 

Density 

Weight 
(in  grams) 

,    Density 

Weight 
(in  grams) 

Density 

Weight 
(in  grams) 

0 

1-65 

0  obs. 

1-65 

0 

8-45 

0 

8-28 

152 

1-65 

98  calc.            1-65 

159 

8-45 

77 

8-27 

213 

1-65 

108     „               1-64 

324 

8-30 

113 

8-25 

233 

1-65 

117     „               1-62 

350 

8-20 

125 

8-20 

256 

1-64 

122     „               1-60 

376 

8-10 

137 

8-10 

278 

1-62 

130    „               1-55 

405 

8-00 

147 

8-00 

290 

1-60 

136     „               1-50 

430 

7-90 

154 

7-90 

308 

1-55 

141     „ 

1-40 

461 

7'80 

171 

7-70 

323 
334 

1-50 
1-40 

148  calc. 
159  obs. 

1-30 
1-20 

483 
518 

7'70 
7-60 

213 
303 

7'60 
7-50 

351 

1-30 

175     „ 

1-00 

392 

7-45 

377 

1-20 

204     „ 

•80 

465 

7-40 

408 

1-10 

233     „ 

•70 

526 

7-35 

435 

1-00 

351     „ 

•60 

625 

7-30 

472 

•90 

488     „ 

•55 

510 

•80 

625  obs. 

•52 

571 

•70 

430  INVESTIGATIONS   IN   CAPILLARITY.  [251 

The  next  question  for  consideration  is — at  what  point  will  an  opposition 
to  contraction  arise  ?  The  answer  must  depend  upon  the  forces  supposed 
to  be  operative  between  the  molecules  of  oil.  If  they  behave  like  the  smooth 
rigid  spheres  of  gaseous  theory,  no  forces  will  be  called  into  play  until  they 
are  closely  packed.  According  to  this  view  the  tension  would  remain  constant 
up  to  the  point  where  a  double  layer  commences  to  form.  It  would  then 
suddenly  change,  to  remain  constant  at  the  new  value  until  the  second  layer 
is  complete.  The  actual  course  of  the  curve  of  tension  deviates  somewhat 
widely  from  the  above  description,  but  perhaps  not  more  than  could  be 
explained  by  heterogeneity  of  the  oil,  whereby  some  molecules  would  mount 
more  easily  than  others,  or  by  reference  to  the  molecular  motions  which 
cannot  be  entirely  ignored.  If  we  accept  this  view  as  substantially  true, 
we  conclude  that  the  first  drop  in  tension  corresponds  to  a  complete  layer  one 
molecule  thick,  and  that  the  diameter  of  a  molecule  of  oil  is  about  1*0  ///*. 

An  attractive  force  between  molecules  extending  to  a  distance  of  many 
diameters,  such  as  is  postulated  in  Laplace's  theory,  would  not  apparently 
interfere  with  the  above  reasoning.  An  essentially  different  result  would 
seem  to  require  a  repulsive  force  between  the  molecules,  resisting  concentra- 
tion long  before  the  first  layer  is  complete.  In  this  case  the  tension  would 
begin  to  fall  as  soon  as  the  density  is  sufficient  to  bring  the  repulsion  into 
play.  On  the  whole  this  view  appears  less  probable  than  the  former,  the 
more  as  it  involves  a  molecular  diameter  much  exceeding  TO/z/u,. 

EXPLANATION  OF  FIGURES. 

In  the  Figures  (and  in  the  tables)  there  is  no  relation  between  the  scales  of 
the  abscissae  representing  the  densities  in  the  various  cases.  As  regards  the 
ordinates,  representing  weights  or  tensions,  the  scale  is  the  same  in  all  the  cases, 
but  the  zero  point  is  arbitrary.  It  may  be  supposed  to  be  situated  on  the  line 
of  zero  densities  at  a  point  4*1  below  the  starting-point  of  the  curve. 

A  Curious  Observation. 

[1902.  The  present  paragraph  was  accidentally  omitted  in  the  original 
publication.  In  experimenting  upon  a  shallow  layer  of  mercury  contained  in 
a  glass  vessel  with  a  flat  bottom,  it  was  noticed  that  a  piece  of  iron  gauze 
pressed  under  the  mercury  upon  the  bottom  of  the  vessel  unexpectedly  re- 
mained down.  There  was  no  sticky  substance  present  to  which  the  effect 
could  be  referred,  and  on  inspection  from  below  it  was  seen  that  the  mercury 
was  out  of  contact  with  the  bottom  at  places  where  the  gauze  was  closest. 
The  phenomenon  was  thus  plainly  of  a  capillary  nature,  the  mercury  refusing 
to  fill  up  the  narrowest  chinks,  even  though  the  alternative  was  a  vacuum. 
The  experiment  may  be  repeated  in  a  simpler  form  by  substituting  for 
the  gauze  a  piece  of  plate  glass  a  few  cms.  square.  If  the  bottom  of  the 
vessel  be  also  of  plate  glass,  the  expulsion  of  the  mercury  may  be  observed 
from  the  whole  of  the  contiguous  areas.] 


252. 


THE  MUTUAL  INDUCTION  OF  COAXIAL  HELICES. 

[British  Association  Report,  pp.  241,  242,  1899.] 

PROFESSOR  J.  V.  JONES*  has  shown  that  the  coefficient  of  mutual  induction 
etween  a  circle  and  a  coaxial  helix  is  the  same  as  between  the  circle 
and  a  uniform  circular  cylindrical  current-sheet  of  the  same  radial  and  axial 
dimensions  as  the  helix,  if  the  currents  per  unit  length  in  helix  and  sheet  be 
the  same.  This  conclusion  is  arrived  at  by  comparison  of  the  integrals 
resulting  from  an  application  of  Neumann's  formula;  and  it  may  be  of 
interest  to  show  that  it  can  be  deduced  directly  from  the  general  theory 
of  lines  of  force. 

In  the  first  place,  it  may  be  well  to  remark  that  the  circuit  of  the  helix 
must  be  supposed  to  be  completed,  and  that  the  result  will  depend  upon  the 
manner  in  which  the  completion  is  arranged.  In  the  general  case  the 
return  to  the  starting-point  might  be  by  a  second  helix  lying  upon  the 
same  cylinder;  but  for  practical  purposes  it  will  suffice  to  treat  of  helices 
including  an  integral  number  of  revolutions,  so  that  the  initial  and  final 
points  lie  upon  the  same  generating  line.  The  return  will  then  naturally 
be  effected  along  this  straight  line. 

Let  us  now  suppose  that  the  helix,  consisting  of  one  revolution  or  of  any 
number  of  complete  revolutions,  is  situated  in  a  field  of  magnetic  force 
symmetrical  with  respect  to  the  axis  of  the  helix.  In  considering  the  number 
of  lines  of  force  included  in  the  complete  circuit,  it  is  convenient  to  follow  in 
imagination  a  radius-vector  drawn  perpendicularly  to  the  axis  from  any  point 
of  the  circuit.  The  number  of  lines  cut  by  this  radius,  as  the  complete 
circuit  is  described,  is  the  number  required,  and  it  is  at  once  evident  that  the 
part  of  the  circuit  corresponding  to  the  straight  return  contributes  nothing 
to  the  total  f.  As  regards  any  part  of  the  helix  corresponding  to  a  rotation 

*  Proc.  Roy.  Soc.  Vol.  LXIII.  (1897),  p.  192. 

t  This  would  be  true  so  long  as  the  return  lies  anywhere  in  the  meridianal  plane.  In  the 
general  case,  where  the  number  of  convolutions  is  incomplete,  the  return  may  be  made  along 
a  path  composed  of  the  extreme  radii  vectores  and  of  the  part  of  the  axis  intercepted  between 
them. 


432  THE   MUTUAL   INDUCTION   OF   COAXIAL   HELICES.  [252 

of  the  radius  through  an  angle  dO,  it  is  equally  evident  that  in  the  limit  the 
number  of  lines  cut  through  is  the  same  as  in  describing  an  equal  angle 
of  the  circular  section  of  the  cylinder  at  the  place  in  question,  whence 
Professor  Jones's  result  follows  immediately.  Every  circular  section  is  sampled, 
as  it  were,  by  the  helix,  and  contributes  proportionally  to  the  result,  since  at 
every  point  the  advance  of  the  vector  parallel  to  the  axis  is  in  strict  pro- 
portion to  the  rotation.  It  is  remarkable  that  the  case  of  the  helix  (with 
straight  return)  is  simpler  than  that  of  a  system  of  true  circles  in  parallel 
planes  at  intervals  equal  to  the  pitch  of  the  helix. 

The  replacement  of  the  helix  by  a  uniform  current-sheet  shows  that  the 
force  operative  upon  it  in  the  direction  of  the  axis  (dH  jdx)  depends  only  upon 
the  values  of  M  appropriate  to  the  two  terminal  circles. 

If  the  field  is  itself  due  to  a  current  flowing  in  a  helix,  the  condition  of 
symmetry  about  the  axis  is  only  approximately  satisfied.  The  question 
whether  both  helices  may  be  replaced  by  the  corresponding  current-sheets 
is  to  be  answered  in  the  negative,  as  may  be  seen  from  consideration  of  the 
case  where  there  are  two  helices  of  the  same  pitch  on  cylinders  of  nearly 
equal  diameters.  In  one  relative  position  of  the  cylinders  the  paths  are  in 
close  proximity  throughout,  and  the  value  of  M  will  be  large  ;  but  this  state 
of  things  may  be  greatly  altered  by  a  relative  rotation  through  two  right 
angles. 

But  although  in  strictness  the  helices  cannot  be  replaced  by  current- 
sheets,  the  complication  thence  arising  can  be  eliminated  in  experimental 
applications  by  a  relative  rotation.  For  instance,  if  the  helix  to  which 
the  field  is  supposed  to  be  due  be  rotated,  the  mean  field  is  strictly  sym- 
metrical, and  accordingly  the  mean  M  is  the  same  as  if  the  other  helix  were 
replaced  by  a  current-sheet.  A  further  application  of  Professor  Jones's 
theorem  now  proves  that  the  first  helix  may  also  be  so  replaced.  Under 
such  conditions  as  would  arise  in  practice,  the  mean  of  two  positions  distant 
180°,  or  at  any  rate  of  four  distant  90°,  would  suffice  to  eliminate  any  dif- 
ference between  the  helices  and  the  corresponding  current-sheets,  if  indeed 
such  difference  were  sensible  at  all. 

The  same  process  of  averaging  suffices  to  justify  the  neglect  of  spirality 
when  the  observation  relates  to  the  mutual  attraction  of  two  helices  as 
employed  in  current  determinations. 


253. 


THE   LAW   OF   PARTITION   OF   KINETIC   ENERGY. 


[Philosophical  Magazine,  XLIX.  pp.  98 — 118,  1900.] 

THE  law  of  equal  partition,  enunciated  first  by  Waterston  for  the  case 
of  point  molecules  of  varying  mass,  and  the  associated  Boltzmann-Maxwell 
doctrine  respecting  steady  distributions  have  been  the  subject  of  much 
difference  of  opinion.  Indeed,  it  would  hardly  be  too  much  to  say  that  no 
two  writers  are  fully  agreed.  The  discussion  has  turned  mainly  upon 
Maxwell's  paper  of  1879*,  to  which  objections  •[•  have  been  taken  by  Lord 
Kelvin  and  Prof.  Bryan,  and  in  a  minor  degree  by  Prof.  Boltzmann  and 
myself.  Lord  Kelvin's  objections  are  the  most  fundamental.  He  writes^ : 
"  But,  conceding  Maxwell's  fundamental  assumption,  I  do  not  see  in  the 
mathematical  workings  of  his  paper  any  proof  of  his  conclusion  '  that  the 
average  kinetic  energy  corresponding  to  any  one  of  the  variables  is  the  same 
for  every  one  of  the  variables  of  the  system.'  Indeed,  as  a  general  pro- 
position its  meaning  is  not  explained,  and  it  seems  to  me  inexplicable.  The 
reduction  of  the  kinetic  energy  to  a  sum  of  squares  leaves  the  several 
parts  of  the  whole  with  no  correspondence  to  any  defined  or  definable  set 
of  independent  variables." 

In  a  short  note  §  written  soon  afterwards  I  pointed  out  some  considera- 
tions which  appeared  to  me  to  justify  Maxwell's  argument,  and  I  suggested 
the  substitution  of  Hamilton's  principal  function  for  the  one  employed  by 
Maxwell ||.  The  views  that  I  then  expressed  still  commend  themselves  to 

*  Collected  Scientific  Papers,  Vol.  n.  p.  713. 

t  I  am  speaking  here  of  objections  to  the  dynamical  and  statistical  reasoning  of  the  paper. 
Difficulties  in  the  way  of  reconciling  the  results  with  a  kinetic  theory  of  matter  are  another 
question. 

J  Proc.  Roy.  Soc.  Vol.  L.  p.  85  (1891). 

§  Phil.  Mag.  April  1892,  p.  356.     [Vol.  m.  p.  554.] 

||  See  also  Dr  Watson's  Kinetic  Theory  of  Gases,  2nd  edit.  1893. 
R.    iv.  28 


434  THE   LAW  OF   PAKTITION   OF   KINETIC   ENERGY.  [253 

me ;  and  I  think  that  it  may  be  worth  while  to  develop  them  a  little  further, 
and  to  illustrate  Maxwell's  argument  by  applying  it  to  a  particular  case 
where  the  simplicity  of  the  circumstances  and  the  familiarity  of  the  notation 
may  help  to  fix  our  ideas. 

But  in  the  mean  time  it  may  be  well  to  consider  Lord  Kelvin's  "  Decisive 
Test-case  disproving  the  Maxwell-Boltzmann  Doctrine  regarding  Distribution 
of  Kinetic  Energy*,"  which  appeared  shortly  after  the  publication  of  my 
note.  The  following  is  the  substance  of  the  argument : — 

"Let  the  system  consist  of  three  bodies,  A,  B,  C,  all  movable  only  in  one 
straight  line,  KHL : 

"  B  being  a  simple  vibrator  controlled  by  a  spring  so  stiff  that  when, 
at  any  time,  it  has  very  nearly  the  whole  energy  of  the  system,  its  extreme 
excursions  on  each  side  of  its  position  of  equilibrium  are  small : 

"  G  and  A,  equal  masses : 

"  C,  unacted  upon  by  force  except  when  it  strikes  L,  a  fixed  barrier, 
and  when  it  strikes  or  is  struck  by  B : 

"A,  unacted  on  by  force  except  when  it  strikes  or  is  struck  by  B,  and 
when  it  is  at  less  than  a  certain  distance,  HK,  from  a  fixed  repellent  barrier, 
E,  repelling  with  a  force,  F,  varying  according  to  any  law,  or  constant,  when 
A  is  between  K  and  H,  but  becoming  infinitely  great  when  (if  at  any  time) 
A  reaches  K,  and  goes  infinitesimally  beyond  it. 

"Suppose  now  A,  B,  C  to  be  all  moving  to  and  fro.  The  collisions 
between  B  and  the  equal  bodies  A  and  C  on  its  two  sides  must  equalize, 
and  keep  equal,  the  average  kinetic  energy  of  A,  immediately  before  and 
after  these  collisions,  to  the  average  kinetic  energy  of  G.  Hence,  when 
the  times  of  A  being  in  the  space  between  H  and  K  are  included  in  the 
average,  the  average  of  the  sum  of  the  potential  and  kinetic  energies  of  A 
is  equal  to  the  average  kinetic  energy  of  C.  But  the  potential  energy  of 
A  at  every  point  in  the  space  HK  is  positive,  because,  according  to  our 
supposition,  the  velocity  of  A  is  diminished  during  every  time  of  its  motion 
from  H  towards  K,  and  increased  to  the  same  value  again  during  motion 
from  K  to  H.  Hence,  the  average  kinetic  energy  of  A  is  less  than  the 
average  kinetic  energy  of  Gl" 

The  apparent  disproof  of  the  law  of  partition  of  energy  in  this  simple 
problem  seems  to  have  shaken  the  faith  even  of  such  experts  as  Dr  Watson 
and  Mr  Burburyf*.  M.  Poincare,  however,  considering  a  special  case  of  Lord 


*  Phil.  Mag.  May  1892,  p.  466. 
t  Nature,  Vol.  XLVI.  p.  100  (1892). 


1900]  THE   LAW   OF   PARTITION   OF    KINETIC   ENERGY.  435 

Kelvin's  problem*,  arrives  at  a  conclusion  in  harmony  with  Maxwell's  law. 
Prof.  Bryan^f*  considers  that  the  test-case  "  shows  the  impossibility  of  drawing 
general  conclusions  as  to  the  distribution  of  energy  in  a  single  system  from 
the  possible  law  of  permanent  distribution  in  a  large  number  of  systems." 
It  is  indeed  true  that  Maxwell's  theorem  relates  in  the  first  instance  to 
a  large  number  of  systems;  but,  as  I  shall  show  more  fully  later,  the  ex- 
tension to  the  time-average  for  a  single  system  requires  only  the  application 
of  Maxwell's  assumption  that  all  phases,  i.e.  all  states,  defined  both  in  respect 
to  configuration  and  velocity,  which  are  consistent  with  the  energy  condition 
lie  on  the  same  path,  i.e.  are  attained  by  the  system  in  its  free  motion  sooner 
or  later.  This  fundamental  assumption,  though  certainly  untrue  in  special 
cases,  would  appear  to  apply  in  Lord  Kelvin's  problem ;  and,  if  so,  Maxwell's 
argument  requires  the  equality  of  kinetic  energies  for  A  and  C  in  the  time- 
averages  of  a  single  system. 

In  view  of  this  contradiction  we  may  infer  that  there  must  be  a  weak 
place  in  one  or  other  argument ;  and  I  think  I  can  show  that  Lord  Kelvin's 
conclusion  above  that  the  average  of  the  sum  of  the  potential  and  kinetic 
energies  of  A  is  equal  to  the  average  kinetic  energy  of  C,  is  not  generally 
true.  In  order  to  see  this  let  us  suppose  the  repulsive  force  F  to  be  limited 
to  a  very  thin  stratum  at  H,  so  that  A  after  penetrating  this  stratum  is 
subject  to  no  further  force  until  it  reaches  the  barrier  K ;  and  let  us  compare 
two  cases,  the  whole  energy  being  the  same  in  both. 

In  case  (i)  F  is  so  powerful  that  with  whatever  velocity  (within  the 
possible  limits)  A  can  approach,  it  is  reflected  at  H,  which  then  behaves 
like  a  fixed  barrier.  In  case  (ii)  F  is  still  powerful  enough  to  produce  this 
result,  except  when  A  approaches  it  with  a  kinetic  energy  nearly  equal 
to  the  whole  energy  of  the  system.  A  then  penetrates  beyond  H,  moving 
slowly  from  H  to  K  and  back  again  from  K  to  H,  thus  remaining  for  a 
relatively  long  time  beyond  H.  Lord  Kelvin's  statement  requires  that 
the  average  total  energy  of  A  should  be  the  same  in  the  two  cases  ;  but  this 
it  cannot  be.  For  during  the  occasional  penetrations  beyond  H  in  case 
(ii)  A  has  nearly  the  whole  energy  of  the  system ;  and  its  enjoyment  of 
this  is  prolonged  by  the  penetration.  Hence  in  case  (ii)  A  has  a  higher 
average  total  energy  than  in  case  (i);  and  a  margin  is  provided  which 
may  allow  the  average  kinetic  energies  to  be  equal.  I  believe  that  the 
consideration  here  advanced  goes  to  the  root  of  the  matter,  and  shows 
why  it  is  that  the  possession  of  potential  energy  may  involve  no  deduction 
from  the  full  share  of  kinetic  energy. 

Lord  Kelvin's  "  decisive  test-case "  is  entirely  covered  by  Maxwell's 
reasoning — a  reasoning  in  my  view  substantially  correct.  It  would  be 

*  Revue  generate  des  Sciences,  July  1894. 

t  "Report  on  Thermodynamics,"  Part  II.  S  26.     Brit.  Ass.  Rep.  1894. 

28—2 


436  THE   LAW  OF  PARTITION  OF   KINETIC   ENERGY.  [253 

possible,  therefore,  to  take  this  case  as  a  typical  example  in  illustration 
of  the  general  argument ;  but  I  prefer  for  this  purpose,  as  somewhat  simpler, 
another  test-case,  also  proposed  by  Lord  Kelvin.  This  is  simply  that  of 
a  particle  moving  in  two  dimensions;  and  it  may  be  symbolized  by  the 
motion  of  the  ball  upon  a  billiard-table.  If  there  is  to  be  potential  energy, 
the  table  may  be  supposed  to  be  out  of  level.  The  reconsideration  of  this 
problem  may  perhaps  be  thought  superfluous,  seeing  that  it  has  been  ably 
treated  already  by  Prof.  Boltzmann*.  But  his  method,  though  (I  believe) 
quite  satisfactory,  is  somewhat  special.  My  object  is  rather  to  follow  closely 
the  steps  of  the  general  theory.  If  objections  are  taken  to  the  argument 
of  the  particular  case,  they  should  be  easy  to  specify.  If,  on  the  other  hand, 
the  argument  of  the  particular  case  is  admitted,  the  issue  is  much  narrowed. 
I  shall  have  occasion  myself  to  make  some  comments  relating  to  one  point 
in  the  general  theory  not  raised  by  the  particular  case. 

In  the  general  theory  the  coordinates f  of  the  system  at  time  t  are  denoted 
by  qlf  q2, ...  qn,  and  the  momenta  by  pl} p2}  ...  pn.  At  an  earlier  time  1f  the 
coordinates  and  momenta  of  the  same  motion  are  represented  by  correspond- 
ing letters  accented,  and  the  first  step  is  the  establishment  of  the  theorem 
usually,  if  somewhat  enigmatically,  expressed 

dq\  dq'2 . . .  dq'ndp\dp2 . . .  dp'n  =  dq^qz . . .  dq^dftdp,  . . .  dpn (1) 

In  the  present  case  ql}  q2  are  the  ordinary  Cartesian  coordinates  (x,  y)  of 
the  particle ;  and  if  we  identify  the  mass  with  unity,  pl}  p2  are  simply  the 
corresponding  velocity-components  (u,  v) ;  so  that  (1)  becomes 

dx'  dy  du'  dv'  =  dx  dy  du  dv (2) 

For  the  sake  of  completeness  I  will  now  establish  (2)  de  novo. 

In  a  possible  motion  the  particle  passes  from  the  phase  (x,  y',  u,  v') 
at  time  t'  to  the  phase  (x,  y,  u,  v)  at  time  t.  In  the  following  discussion 
t'  and  t  are  absolutely  fixed  times,  but  the  other  quantities  are  regarded 
as  susceptible  of  variation.  These  variations  are  of  course  not  independent. 
The  whole  motion  is  determined  if  either  the  four  accented,  or  the  four 
unaccented,  symbols  be  given.  Either  set  may  therefore  be  regarded  as 
definite  functions  of  the  other  set.  Or  again,  the  four  coordinates  x ',  y ',  x,  y 
may  be  regarded  as  independent  variables,  of  which  u',  v',  u,  v  are  then 
functions. 

The  relations  which  we  require  are  readily  obtained  by  means  of  Hamilton's 
principal  function  S,  where 

S=f\T-V)dt (3) 

*  Phil.  Mag.  Vol.  xxxv.  p.  156  (1893). 

t  Generalized  coordinates  appear  to  have  been  first  applied  to  these  problems  by  Boltzmann. 


1900]  THE   LAW  OF   PARTITION  OF   KINETIC   ENERGY.  437 

In  this  V  denotes  the  potential  energy  in  any  position,  and  T  is  the  kinetic 
energy,  so  that 


S  may  here  be  regarded  as  a  function  of  the  initial  and  final  coordinates  ; 
and  we  proceed  to  form  the  expression  for  BS  in  terms  of  Sx,  By',  Bx,  By. 
By  (3) 

B8=     (ST-SV)dt,  ...........................  (5) 

J  t1 
and 


so  that 


=    £&»  +  #%     -  t(xBx  +  yBy)dt; 
SS  =  IxBx  +  y  ByY  -  I  \x  Bx  +  yBy  +  8  V)  dt. 


By  the  general  equation  of  dynamics  the  term  under  the  integral  sign 
vanishes  throughout,  and  thus  finally 

BS=uSx  +  vSy-u'Bx'-v'By (6) 

In  the  general  theory  the  corresponding  equation  is 

BS^ZpBq-Zp'Bq' (7) 

Equation  (6)  is  equivalent  to 

u=-dS/dx',       u  =  dS/dx, 


.(8) 
v  =  dS/dy.  ' 

It  is  important  to  appreciate  clearly  the  meaning  of  these  equations. 
S  is  in  general  a  function  of  x,  y,  x,  y' ;  and  (e.g.)  the  second  equation 
signifies  that  u  is  equal  to  the  rate  at  which  S  varies  with  x,  when  y,  x',  y' 
are  kept  constant,  and  so  in  the  other  cases. 

We  have  now  to  consider,  not  merely  a  single  particle,  but  an  immense 
number  of  similar  particles,  moving  independently  of  one  another  under  the 
same  law  (V),  and  distributed  at  time  t  over  all  possible  phases  (x,  y,  u,  v). 
The  most  general  expression  for  the  law  of  distribution  is 

f(x,  y,  u,  v)  dxdydudv,      (9) 

signifying   that   the   number   of  particles  to  be  found  at  time  t  within  a 
prescribed  range  of  phase  is  to  be  obtained  by  integrating  (9)  over  the  range 

*  As  is  not  unusual  in  the  integral  calculus,  we  employ  the  same  symbols  x,  &c.  to  denote  the 
current  and  the  final  values  of  the  variables.  If  desired,  the  final  values  may  be  temporarily 
distinguished  as  x",  &c. 


438  THE   LAW  OF  PARTITION   OF   KINETIC   ENERGY.  [253 

in  question.  But  such  a  distribution  would  in  general  be  unsteady.  If  it 
obtained  at  time  t,  it  would  be  departed  from  at  time  t',  and  vice  versa, 
owing  to  the  natural  motions  of  the  particles.  The  question  before  us  is 
to  ascertain  what  distributions  are  steady,  i.e.  are  maintained  unaltered 
notwithstanding  the  motions. 

It  will  be  seen  that  it  is  the  spontaneous  passage  of  a  particle  from  one 
phase  to  another  that  limits  the  generality  of  the  function  /.  If  there  be 
no  possibility  of  passage,  say,  from  the  phase  (#',  y',  u',  v)  to  the  phase 
(x,  y,  u,  v),  or,  as  it  may  be  expressed,  if  these  phases  do  not  lie  upon  the 
same  path,  then  there  is  no  relation  imposed  upon  the  corresponding  values 
of/.  An  example,  given  by  Prof.  Bryan  (1.  c.  §  17),  well  illustrates  this  point. 
Suppose  that  F=0,  so  that  every  particle  pursues  a  straight  course  with 
uniform  velocity.  The  phases  (x',  y,  u',  v)  and  (x,  y,  u,  v)  can  lie  upon  the 
same  path  only  if  u  =  u,  v'  =  v.  Accordingly  /  remains  arbitrary  so  far 
as  regards  u  and  v.  For  instance,  a  distribution 

f(u,v)dxdydudv     (10) 

is  permanent  whatever  may  be  the  form  of  f,  understood  to  be  independent 
of  x  and  y.  In  this  case  the  distribution  is  uniform  in  space,  but  uniformity 
is  not  indispensable.  Suppose,  for  example,  that  all  the  particles  move 
parallel  to  x,  so  that  f  vanishes  unless  v  =  0.  The  general  form  (9)  now 
reduces  to 

f(x,  y,  u)  dxdydu;    (11) 

and  permanency  requires  that  the  distribution  be  uniform  along  any  line 
for  which  y  is  constant.  Accordingly,  f  must  be  independent  of  x,  so 
that  permanent  distributions  are  of  the  form 

f(y,  u)  dxdydu,      (12) 

in  which  /  is  an  arbitrary  function  of  y  and  u.  If  either  y  or  u  be  varied,  we 
are  dealing  with  a  different  path  (in  the  sense  here  involved),  and  there  is  no 
connexion  between  the  corresponding  values  of  f.  But  if  while  y  and  u 
remain  constant,  x  be  varied,  the  value  of  f  must  remain  unchanged,  for  the 
different  values  of  x  relate  to  the  same  path. 

Before  taking  up  the  general  question  in  two  dimensions,  it  may  be  well 
to  consider  the  relatively  simple  case  of  motion  in  one  dimension,  which, 
however,  is  not  so  simple  but  that  it  will  introduce  us  to  some  of  the  points 
of  difficulty.  The  particles  are  supposed  to  move  independently  upon  one 
straight  line,  and  the  phase  of  any  one  of  them  is  determined  by  the  co- 
ordinate x  and  the  velocity  u.  At  time  t'  the  phase  of  a  particle  will  be 
denoted  by  (x',  u'),  and  at  time  t  the  phase  of  the  same  particle  will  be  (x,  u), 
where  u  will  in  general  differ  from  u',  since  we  no  longer  suppose  that  V  is 
constant,  but  rather  that  it  is  variable  in  a  known  manner,  i.e.  is  a  known 


1900]  THE   LAW  OF  PARTITION  OF   KINETIC   ENERGY.  439 

function  of  x.  The  number  of  particles  which  at  time  t  lie  within  the  limits 
of  phase  represented  by  dxdu  is  f(x,  u)  dxdu,  and  the  question  is  whether 
this  distribution  is  steady,  and  in  particular  whether  it  was  the  same  at 
time  t'.  In  order  to  find  the  distribution  at  time  t1,  we  regard  x,  u  as  known 
functions  of  x',  u,  and  transform  the  multiple  differential.  The  result  of  this 
transformation  is  best  seen  by  comparison  with  intermediate  transformations 
in  which  dxdu  and  dxdu  are  compared  with  dxdx'.  We  have 


(13) 


du' 
dx'du'=dxdx'  x  -r-  ............................  (14) 

In  du/dx  of  (13)  x  is  to  be  kept  constant,  and  in  du  /dx  of  (14)  x'  is  to 
be  kept  constant.  If  we  disregard  algebraic  sign,  both  are  by  (8)  equal 
to  d*S/dxdx',  and  are  therefore  equal  to  one  another.  Hence  we  may 
write 

dxdu  =  dx'du  ;   ..............................  (15) 

and  the  transformation  is  expressed  by 

f(x,  u)  dx  du  =/  (x',  u')  dx'du  ,     ..................  (16) 

where/j  (x',  u)  is  the  result  of  substituting  for  x,  u  inf(x,  u)  their  values  in 
terms  of  x,  u'.  The  right-hand  member  of  (16)  expresses  the  distribution 
at  time  t'  corresponding  to  the  distribution  at  time  t  expressed  by  the  left- 
hand  member,  as  determined  by  the  laws  of  motion  between  the  two  phases. 
If  the  distribution  is  to  be  steady,  /  (x',  u')  must  be  identical  with  /  (#',  u'); 
in  other  words  f(x,  u)  must  be  such  a  function  of  (x,  u)  that  it  remains 
unchanged  when  (x,  u)  refers  to  various  phases  of  the  motion  of  the  same 
particle.  Now,  if  E  denote  the  total  energy,  so  that 

E=$u*+V,     ..............................  (17) 

then  E  remains  constant  during  the  motion  ;  and  thus,  if  for  the  moment 
we  suppose  f  expressed  in  terms  of  E  and  x,  we  see  that  x  cannot  enter,  or 
that  /is  a  function  of  E  only.  The  only  permanent  distributions  accordingly 
are  those  included  under  the  form 

j(E}dxdu,     ..............................  (18) 

where  E  is  given  by  (17),  and  /is  an  arbitrary  function. 

It  is  especially  to  be  noticed  that  the  limitation  to  the  form  (18)  holds 
only  for  phases  lying  upon  the  same  path.  If  two  phases  have  different 
energies,  they  do  not  lie  upon  the  same  path,  but  in  this  case  the  independence 
of  the  distributions  in  the  two  phases  is  already  guaranteed  by  the  form 
of  (18).  The  question  is  whether  all  phases  of  given  energy  lie  upon  the 


440 


THE   LAW   OF   PARTITION   OF   KINETIC   ENERGY. 


[253 


same  path.  It  is  easy  to  invent  cases  for  which  the  answer  will  be  in  the 
negative.  Suppose,  for  example,  that  there  are  two  centres  of  force  0,  0' 
on  the  line  of  motion  which  attract  with  a  force  at  first  proportional  to 
distance  but  vanishing  when  the  distance  exceeds  a  certain  value  less  than 
the  interval  00'.  A  particle  may  then  vibrate  with  the  same  (small)  energy 
either  round  0  or  round  0' ;  but  the  phases  of  the  two  motions  do  not  lie 
upon  the  same  path.  Consequently  /  is  not  limited  by  the  condition  of 
steadiness  to  be  the  same  in  the  two  groups  of  phases.  In  all  cases  steadiness 
is  ensured  by  the  form  (18);  and  if  all  phases  of  equal  energy  lie  upon  the 
same  path,  this  form  is  necessary  as  well  as  sufficient. 

All  the  essential  difficulties  of  the  theory  appear  to  be  raised  by  the 
particular  case  just  discussed,  and  the  reader  to  whom  the  subject  is  new 
is  recommended  to  give  it  his  careful  attention. 

In  the  more  general  problem  of  motion  in  two  dimensions  the  discussion 
follows  a  parallel  course.  In  order  to  find  the  distribution  at  time  If  cor- 
responding to  (9)  at  time  t,  we  have  to  transform  the  multiple  differential, 
regarding  x,  y,u,v  as  known  functions  of  x',  y',  u',  v'.  Here  again  we  take 
the  initial  and  final  coordinates  x,  y,  x'.  y  as  an  intermediate  set  of  variables. 
Thus 


dad dy' du' dv'  =  dx'dy'dxdy  x 


dxdydudv  =  dxdydx'dy'  x 


dx 


_ 
dy 

du 


du 


dv' 
dx 

dtf 
dy 

dv 
dx' 

dy 


.(19) 


.(20) 


In  the  determinants  of  (19),  (20)  the  motion  is  regarded  as  a  function  of 
x,  y,  x',  y',  and  the  three  quantities  which  do  not  appear  in  the  denominator 
of  any  differential  coefficient  are  to  be  considered  constant.  This  was  also 
the  understanding  in  equations  (8),  from  which  we  infer  that  the  two  deter- 
minants are  equal,  being  each  equivalent  to 


dxdx' '    dxdy' 
d*S         d*S 


.(21) 


dx'dy'   dydy 
Hence  we  may  write 

dxdydudv  =  dx'dy'du'dv, (22) 

an  equation  analogous  to  (15).     By  the  same  reasoning  as  was  employed 


1900]  THE   LAW   OF   PARTITION   OF   KINETIC   ENERGY.  441 

for  motion  in  one  dimension  it  follows  that,  if  the  distribution  is  to  be  steady, 
f(x,  y,  u,  v)  in  (9)  must  remain  constant  for  all  phases  which  lie  upon  the 
same  path.  A  distribution  represented  by 

f(E)dxdydudv,  ..............................  (23) 

where 

*  +  V,     ...........................  (24) 


will  satisfy  the  conditions  of  steadiness  whatever  be  the  form  of/;  but  this 
form  is  only  necessary  under  the  restriction  known  as  Maxwell's  assumption 
or  postulate,  viz.  that  all  phases  of  equal  energy  lie  upon  the  same  path. 

It  is  easy  to  give  examples  in  which  Maxwell's  assumption  is  violated, 
and  in  which  accordingly  steady  distributions  are  not  limited  to  (23).  Thus, 
if  no  force  act  parallel  to  y,  so  that  V  reduces  to  a  function  of  x  only,  the 
component  velocity  v  remains  constant  for  each  particle,  and  no  phases 
for  which  v  differs  lie  upon  the  same  path.  A  distribution 

f(E,v)dxdydudv    ...........................  (25) 

is  then  steady,  whatever  function  /may  be  of  E  and  v. 

That  under  the  distribution  (23)  the  kinetic  energy  is  equally  divided 
between  the  component  velocities  u  and  v  is  evident  from  symmetry.  It 
is  to  be  observed  that  the  law  of  equal  partition  applies  not  merely  upon  the 
whole,  but  for  every  element  of  area  dxdy,  and  for  every  value  of  the  total 
energy,  and  at  every  moment  of  time.  When  x  and  y  are  prescribed  as  well 
as  E,  the  value  of  the  resultant  velocity  itself  is  determined  by  (24). 

Another  feature  worthy  of  attention  is  the  spacial  distribution  ;  and  it 
happens  that  this  is  peculiar  in  the  present  problem.  To  investigate  it 
we  must  integrate  (23)  with  respect  to  u  and  v,  x  and  y  being  constant. 
Since  x  and  y  are  constant,  V  is  constant  ;  so  that,  if  we  suppose  E  to  lie 
within  narrow  limits  E  and  E  +  dE,  the  resultant  velocity  U  will  lie  between 

limits  given  by 

UdU=dE  ...............................  (26) 

If  we  transform  from  u,  v  to  U,  6,  where 

u=Ucos0,        v=Usin0,    .....................  (27) 

dudv  becomes  UdUdO',  so  that  on  integration  with  respect  to  6  we  have, 
with  use  of  (26), 

2TrF(E)dE  .dxdy  ............................  (28) 

The  spacial  distribution  is  therefore  uniform. 

In  order  to  show  the  special  character  of  the  last  result,  it  may  be  well 
to  refer  briefly  to  the  corresponding  problem  in  three  dimensions,  where  the 


442  THE  LAW  OF  PARTITION   OF   KINETIC   ENERGY.  [253 

coordinates  of  a  particle  are  x,  y,  z  and  the  component  velocities  are  u,  v,  w. 
The  steady  distribution  corresponding  to  (23)  is 

/(E)dxdydzdudvdw,     ........................  (29) 

in  which 


.     ...............  (30) 

Here  equation  (26)  still  holds  good,  and  the  transformation  of  dudvdw  is, 
as  is  well  known,  4nrU2dU.  Accordingly  (29)  becomes 

4>TrF(E)dE.(2E-2V^da;dy,     ..................  (31) 

no  longer  uniform  in  space,  since  V  is  a  function  of  x,  y. 

In  (31)  the  density  of  distribution  decreases  as  V  increases.  For  the 
corresponding  problem  in  one  dimension  (18)  gives 

F(E)dE.(2E-2V)-ldx,      .....................  (32) 

so  that  in  this  case  the  density  increases  with  increasing  V. 

The  uniform  distribution  of  the  two-dimensional  problem  is  thus  peculiar. 
Although  an  immediate  consequence  of  Maxwell's  equation  (41),  see  (41) 
below,  I  failed  to  remark  it  in  the  note  before  referred  to,  where  I  wrote 
as  if  a  uniform  distribution  in  the  billiard-table  example  required  that  V  =  0. 
In  order  to  guard  against  a  misunderstanding  it  may  be  well  to  say  that 
the  uniform  distribution  does  not  necessarily  extend  over  the  whole  plane. 
Wherever  (E  —  V)  falls  below  zero  there  is  of  course  no  distribution. 

We  have  thus  investigated  for  a  particle  in  two  dimensions  the  law  of 
steady  distribution,  and  the  equal  partition  of  energy  which  is  its  necessary 
consequence.  And  we  see  that  "  the  only  assumption  necessary  to  the  direct 
proof  is  that  the  system,  if  left  to  itself  in  its  actual  state  of  motion,  will, 
sooner  or  later,  pass  through  every  phase  which  is  consistent  with  the 
equation  of  energy"  (Maxwell).  It  will  be  observed  that  so  far  nothing 
whatever  has  been  said  as  to  time-averages  for  a  single  particle.  The  law  of 
equal  partition,  as  hitherto  stated,  relates  to  a  large  number  of  particles  and 
to  a  single  moment  of  time. 

The  extension  to  time-averages,  the  aspect  under  which  Lord  Kelvin  has 
always  considered  the  problem,  is  important,  the  more  that  some  authors 
appear  to  doubt  the  possibility  of  such  extension.  Thus  Prof.  Bryan  (Report, 
§  11,  1894),  speaking  of  Maxwell's  assumption,  writes  :  —  "  To  discover,  if 
possible,  a  general  class  of  dynamical  systems  satisfying  the  assumption 
would  form  an  interesting  subject  for  future  investigation.  It  is,  however, 
doubtful  how  far  Maxwell's  law  would  be  applicable  to  the  time-averages 
of  the  energies  in  any  such  system.  We  shall  see,  in  what  follows,  that  the 
law  of  permanent  distribution  of  a  very  large  number  of  systems  is  in  many 
cases  not  unique.  Where  there  is  more  than  one  possible  distribution  it 


1900]  THE   LAW   OF   PARTITION   OF   KINETIC    ENERGY.  443 

would  be  difficult  to  draw  any  inference  with  regard  to  the  average  distri- 
bution (taken  with  respect  to  the  time)  for  one  system." 

The  extension  to  time-averages  appears  to  me  to  require  nothing  more 
than  Maxwell's  assumption,  without  which  the  law  of  distribution  itself 
is  only  an  artificial  arrangement,  sufficient  indeed  but  not  necessary  for 
steadiness.  We  shall  still  speak  of  the  particle  moving  in  two  dimensions, 
though  the  argument  is  general.  It  has  been  shown  that  at  any  moment 
the  w-energy  and  the  v-energy  of  the  group  of  particles  is  the  same ;  and 
it  is  evident  that  the  equality  subsists  if  we  integrate  over  any  period  of 
time.  But  if  this  period  be  sufficiently  prolonged,  and  if  Maxwell's  assumption 
be  applicable,  it  makes  no  difference  whether  we  contemplate  the  whole  group 
of  particles  or  limit  ourselves  to  a  single  member  of  it.  It  follows  that 
for  a  single  particle  the  time-averages  of  u2  and  v2  are  equal,  provided  the 
averages  be  taken  over  a  sufficient  length  of  time. 

On  the  other  hand,  if  in  any  case  Maxwell's  assumption  be  untrue,  not 
only  is  the  special  distribution  unnecessary  for  steadiness,  but  even  if  it 
be  artificially  arranged,  the  law  of  equal  time-averages  does  not  follow  as 
a  consequence. 

Having  now  considered  the  special  problem  at  full — I  hope  it  may  not 
be  thought  at  undue — length,  I  pass  on  to  some  remarks  on  the  general 
investigation.  This  proceeds  upon  precisely  parallel  lines,  and  the  additional 
difficulties  are  merely  those  entailed  by  the  use  of  generalized  coordinates. 
Thus  (1)  follows  from  (7)  by  substantially  the  same  process  (given  in  my 
former  note)  that  (22)  follows  from  (6).  Again,  if  E  denote  the  total  energy 
of  a  system,  the  distribution 

f(E)dq1...dqndp1  ...dpn,  (33) 

where  f  is  an  arbitrary  function,  satisfies  the  condition  of  permanency  ;  and, 
if  Maxwell's  assumption  be  applicable,  it  is  the  only  form  of  distribution  that 
can  be  permanent. 

As  I  hinted  before,  some  of  the  difficulties  that  have  been  felt  upon  this 
subject  may  be  met  by  a  fuller  recognition  of  the  invariantic  character  of 
the  expressions.  This  point  has  been  ably  developed  by  Prof.  Bryan,  who 
has  given  (loc.  cit.  §  14)  a  formal  verification  that  (33)  is  unaltered  by  a  change 
of  coordinates.  If  we  follow  attentively  the  process  by  which  (1)  is  established, 
we  see  that  in  (3)  there  is  no  assumption  that  the  system  of  coordinates  is 
the  same  at  times  t'  and  t,  and  that  accordingly  we  are  not  tied  to  one  system 
in  (33).  Indeed,  so  far  as  I  can  see,  there  would  be  no  meaning  in  the 
assertion  that  the  system  of  generalized  coordinates  employed  for  two  different 
configurations  was  the  same*. 

*  It  would  be  like  saying  that  two  points  lie  upon  the  same  curve,  when  the  character  of  the 
curve  is  not  denned. 


444  THE  LAW  OF   PARTITION   OF   KINETIC   ENERGY.  [253 

We  come  now  to  the  deduction  from  (33)  of  Maxwell's  law  of  partition 
of  energy.  On  this  Prof.  Bryan  (loc.  cit.  §  20)  remarks:  —  "Objections  have 
been  raised  to  this  step  in  Maxwell's  work  by  myself  ('  Report  on  Thermo- 
dynamics/ Part  I.  §  44)  on  the  ground  that  the  kinetic  energy  cannot  in 
general  be  expressed  as  the  sum  of  squares  of  generalized  momenta  corre- 
sponding to  generalized  coordinates  of  the  system,  and  by  Lord  Kelvin 
(Nature,  Aug.  13,  1891)  on  the  ground  that  the  conclusion  to  which  it  leads 
has  no  intelligible  meaning.  Boltzmann  (Phil.  Mag.  March  1893)  has  put  the 
investigation  into  a  slightly  modified  form  which  meets  the  first  objection, 
and  which  imposes  a  certain  restriction  upon  the  generality  of  the  result. 
Under  this  limitation  the  result  is  perfectly  intelligible,  and  the  second 
objection  is  therefore  also  met."  At  this  point  I  find  myself  in  disagreement 
with  all  the  above  quoted  authorities,  and  in  the  position  of  maintaining 
the  correctness  of  Maxwell's  original  deduction. 

Prof.  Boltzmann  considers  that  "  Maxwell  committed  an  error  in  assuming 
that  by  choosing  suitable  coordinates  the  expression  for  the  vis  viva  could 
always  be  made  to  contain  only  the  squares  of  the  momenta."  This  is 
precisely  the  objection  which  I  supposed  myself  to  have  already  answered 
in  1892.  I  wrote,  "  It  seems  to  be  overlooked  that  Maxwell  is  limiting  his 
attention  to  systems  in  a  given  configuration,  and  that  no  dynamics  is  founded 
upon  the  reduced  expression  for  T.  The  reduction  can  be  effected  in  an 
infinite  number  of  ways.  We  may  imagine  the  configuration  in  question 
rendered  one  of  stable  equilibrium  by  the  introduction  of  suitable  forces 
proportional  to  displacements.  The  principal  modes  of  isochronous  vibration 
thus  resulting  will  serve  the  required  purpose." 

It  is  possible,  therefore,  so  to  choose  the  coordinates  that  for  a  given 
configuration  (and  for  configurations  differing  infinitely  little  therefrom)  the 
kinetic  energy  T,.which  is  always  a  quadratic  function  of  the  velocities,  shall 
reduce  to  a  sum  of  squares  with,  if  we  please,  given  coefficients.  Thus  in 
the  given  configuration 

T  =  W  +  b&+...+W;    .....................  (34) 

and,  since  in  general  p  =  dT/dq, 


so  that  T  =  to2  +  ib>22+---+to2  ......................  (35) 

Whether  the  coordinates  required  to  effect  a  similar  reduction  for  other 
configurations  are  the  same  is  a  question  with  which  we  are  not  concerned. 

The  mean  value  of  pr2  for  all  the  systems  in  the  given  configuration  is, 
according  to  (33), 


*  Confer  Bryan,  loc.  cit. 


1900]  THE   LAW  OF   PARTITION  OF   KINETIC   ENERGY.  445 

The  limits  for  each  variable  may  be  supposed  to  be  ±00;  but  the  large 
values  do  not  really  enter  if  we  suppose  F(E)  to  be  finite  for  moderate, 
perhaps  for  nearly  definite,  values  of  E  only. 

It  is  now  evident  that  the  mean  value  is  the  same  for  all  the  momenta  p  ; 
and  accordingly  that  for  each  the  mean  value  of  £p2  is  1/n  of  the  mean 
value  of  T.  This  result  holds  good  for  every  moment  of  time,  for  every 
configuration,  for  every  value  of  E,  and  for  every  system  of  resolution  (of 
which  there  are  an  infinite  number)  which  allows  T  to  be  expressed  in  the 
form  (35). 

In  the  case  where  the  "  system  "  consists  of  a  single  particle,  (35)  is  justified 
by  any  system  of  rectangular  coordinates  ;  and  although  we  are  not  bound  to 
use  the  same  system  for  different  positions  of  the  particle,  it  would  conduce 
to  simplicity  to  do  so.  If  the  system  be  a  rigid  body,  we  may  measure  the 
velocities  of  the  centre  of  inertia  parallel  to  three  fixed  rectangular  axes, 
while  the  remaining  momenta  refer  to  rotations  about  the  principal  axes 
of  the  body.  If  Maxwell's  assumption  hold  good,  a  permanent  distribution 
is  such  that  in  one,  or  in  any  number  of  positions,  the  mean  energy  of  each 
rotation  and  of  each  translation  is  the  same.  And  under  the  same  restriction 
a  similar  assertion  may  be  made  respecting  the  time-averages  for  a  single 
rigid  body. 

There  is  much  difficulty  in  judging  of  the  applicability  of  Maxwell's 
assumption.  As  Maxwell  himself  showed.it  is  easy  to  find  cases  of  exception; 
but  in  most  of  these  the  conditions  strike  one  as  rather  special.  It  must 
be  observed,  however,  that  if  we  take  it  quite  literally,  the  assumption  is 
of  a  severely  restrictive  character;  for  it  asserts  that  the  system,  starting 
from  any  phase,  will  traverse  every  other  phase  (consistent  with  the  energy 
condition)  before  returning  to  the  initial  phase.  As  soon  as  the  initial 
phase  is  recovered,  a  cycle  is  established,  and  no  new  phases  can  be  reached, 
however  long  the  motion  may  continue. 

We  return  now  to  the  question  of  the  distribution  of  momenta  among 
the  systems  which  occupy  a  given  configuration,  still  supposing  the  coordinates 
so  chosen  as  to  reduce  T  to  a  sum  of  squares  (35).  It  will  be  convenient 
to  fix  our  attention  upon  systems  for  which  E  lies  within  narrow  limits, 
E  and  E  +  dE.  Since  E  is  given,  there  is  a  relation  between  pl}p2,  ...  pn, 
and  we  may  suppose  pn  expressed  in  terms  of  E  and  the  remaining  momenta. 
By  (35) 


since  the  configuration  is  given,  and  thus  (33)  becomes 

f(E)dE.dq1...dqn.pn-1dpl...dpn-l  ................  (37) 


446  THE  LAW  OF  PARTITION   OF  KINETIC   ENERGY.  [253 

For  the  present  purpose  the  latter  factors  alone  concern  us,  so  that  what 
we  have  to  consider  is 


in  which  T,  being  equal  to  E  —  V,  is  given.     For  the  moment  we  may  suppose 
that  2T  is  unity. 

The  whole  number  of  systems  is  to  be  found  by  integrating  (38),  the 
integral  being  so  taken  as  to  give  the  variables  all  values  consistent  with  the 
condition  that  p?  +  p2*  +  .  .  .  +  p*n-!  is  not  greater  than  unity.  Now 


(1  -pflP-idp! (39) 

J        J  V   {J-  —  Pi'  —  ••-  —p~n-i<        *-    \^'*—  2J-'  -I 

and 


(40) 


in  which  F  (|)  =  V71"-     Thus  the  whole  number  of  systems  is 


or  on  restoration  of  2T,  equal  to  2E  —  2V, 

y^{2#-2Fp-  .........................  (41) 

To  this  we  shall  return  later;  but  for  the  present  what  we  require  to 
ascertain  is  the  distribution  of  one  of  the  momenta,  say  plt  irrespectively 
of  the  values  of  the  remaining  momenta.  By  (39),  (40)  the  number  of 
systems  for  which  pl  lies  between  pi  and  p±  +  dp^  in  comparison  with  the 
whole  number  of  systems  is 

_  £  u»-i   dPl 

1  ' 


rg)r(t»-» 

This  is  substantially  Maxwell's  investigation,  and  (42)  corresponds  with  his 
equation  (51).  As  was  to  be  expected,  the  law  of  distribution  is  the  same 
for  all  the  momenta.  From  the  manner  of  its  formation,  we  note  that  the 
integral  of  (42),  taken  between  the  limits  pt  =  +  \f  (2T),  is  equal  to  unity. 

Maxwell  next  proceeds  to  the  consideration  of  the  special  form  assumed 
by  (42),  when  the  number  n  of  degrees  of  freedom  is  extremely  great*.  This 
part  of  the  work  seems  to  be  very  important  ;  but  it  has  been  much  neglected, 
probably  because  the  result  was  not  correctly  stated. 

*  The  particular  cases  where  n  =  2,  or  n=B,  are  also  worthy  of  notice. 


1900]  THE   LAW   OF   PARTITION   OF   KINETIC   ENERGY.  447 

Dropping  the  suffix  as  unnecessary,  we  have  to  consider  the  form  of 


I-"        -' 


when  n  is  very  great,  the  mean  value  of  p*  becoming  at  the  same  time  small 
in  comparison  with  2T.     If  we  write 


...........................  (43) 

we  have 


Limit    l  -  =e-*'/4jr  =  <rP*'2pl  ................  (44) 

The  limit  of  the  fraction  containing  the  F  functions  may  be  obtained 
by  the  formula 

r  (m  +  1)  =  e-mmm  V  (2m7r)  ; 


and  the  limiting  form  of  (42)  becomes 

dp 

P  ...................  (4< 


-***      dp 


It  may  be  observed  that  the  integral  of  (45)  between  the  limits  +  oo  is 
unity,  and  that  this  fact  might  have  been  used  to  determine  the  numerical 
factor. 

Maxwell's  result  is  given  in  terms  of  a  quantity  k,  analogous  to  K,  and 
defined  by 

W  =  k  ..................................  (46) 

It  is 

•;.;;   ,.,       ;  Tifez''**  ............................  (47) 

The  corresponding  form  from  (45)  is 


In  like  manner  if  we  inquire  what  proportion  of  the  whole  number  of 
systems  have  momenta  lying  within  the  limits  denoted  by  dpidp^  ...  dpr, 
where  r  is  a  number  very  small  relatively  to  n,  we  get 

...  dpr 

' 


or,  if  we  prefer  it, 

-(*»+:'4*>IU*  .dpr 


These  results  follow  from  the  general  expression  (38),  in  the  same  way  as 
.does  (45),  by  stopping  the  multiple  integration  at  an  earlier  stage.     The 


448  THE   LAW  OF  PARTITION   OF   KINETIC   ENERGY.  [253 

remaining  variables  range  over  values  which  may  be  considered  in  each  case 
to  be  unlimited.  If  the  integration  between  ±  oo  be  carried  out  completely, 
we  recover  the  value  unity. 

The  interest  of  the  case  where  n  is  very  great  lies  of  course  in  the 
application  to  a  gas  supposed  to  consist  of  an  immense  number  of  similar 
molecules*,  or  of  several  sets  of  similar  molecules;  and  the  question  arises 
whether  (45)  can  be  applied  to  deduce  the  Maxwellian  law  of  distribution 
of  velocities  among  the  molecules  of  a  single  system  at  a  given  instant  of 
time.  A  caution  may  usefully  be  interposed  here  as  to  the  sense  in  which 
the  Maxwellian  distribution  is  to  be  understood.  It  would  be  absurd  to 
attempt  to  prove  that  the  distribution  in  a  single  system  is  necessarily  such 
and  such,  for  we  have  already  assumed  that  every  phase,  including  every 
distribution  of  velocities,  is  attainable,  and  indeed  attained  if  sufficient  time 
be  allowed.  The  most  that  can  be  proved  is  that  the  distribution  will 
approximate  to  a  particular  law  for  the  greater  part  of  the  time,  and  that 
if  sensible  deviations  occur  they  will  be  transitory. 

In  applying  (45)  to  a  gas  it  will  be  convenient  to  suppose  in  the  first 
instance  that  all  the  molecules  are  similar.  Each  molecule  has  several 
degrees  of  freedom,  but  we  may  fix  our  attention  upon  one  of  them,  say 
the  ^-velocity  of  the  centre  of  inertia,  usually  denoted  by  u.  In  (45)  the 
whole  system  is  supposed  to  occupy  a  given  configuration ;  and  the  expression 
gives  us  the  distribution  of  velocity  at  a  given  time  for  a  single  molecule 
among  all  the  systems.  The  distribution  of  velocity  is  the  same  for  every 
other  molecule,  and  thus  the  expression  applies  to  the  statistics  of  all  the 
molecules  of  all  the  systems.  Does  it  also  apply  to  the  statistics  of  all  the 
molecules  of  a  single  system  ?  In  order  to  make  this  inference  we  must 
assume  that  the  statistics  are  the  same  (at  the  same  time)  for  all  the  systems, 
or,  what  comes  to  the  same  thing  (if  Maxwell's  assumption  be  allowed),  that 
they  are  the  same  for  the  same  system  at  the  various  times  when  it  passes 
through  a  given  configuration. 

Thus  far  the  argument  relates  only  to  a  single  configuration.  If  the 
configuration  be  changed,  there  will  be  in  general  a  change  of  potential 
energy  and  a  corresponding  change  in  the  kinetic  energy  to  be  distributed 
amongst  the  degrees  of  freedom.  But  in  the  case  of  a  gas,  of  which  the 
statistics  are  assumed  to  be  regular,  the  potential  energy  remains  approxi- 
mately constant  when  exclusion  is  made  of  exceptional  conditions.  The  same 
law  of  distribution  of  velocity  then  applies  to  every  configuration,  that  is, 
it  may  be  asserted  without  reference  to  the  question  of  configuration.  We 
thus  arrive  at  the  Maxwellian  law  of  velocities  in  a  single  gas,  as  well  as  the 

*  The  terms  "gas"  and  "molecule"  are  introduced  for  the  sake  of  brevity.  The  question  is 
still  purely  dynamical. 


1900]  THE    LAW   OF   PARTITION   OF   KINETIC    ENERGY.  449 

relation  between  the  velocities  in  a  mixture  of  molecules  of  different  kinds 
first  laid  down  by  Waterston. 

The  assumptions  which  we  have  made  as  to  the  practical  regularity  of 
statistics  are  those  upon  which  the  usual  theory  of  ideal  gases  is  founded; 
but  the  results  are  far  more  general.  Nothing  whatever  has  been  said  as 
to  the  character  of  the  forces  with  which  the  molecules  act  upon  one  another, 
or  are  acted  upon  by  external  agencies.  Although  for  distinctness  a  gas  has 
been  spoken  of,  the  results  apply  equally  to  a  medium  constituted  as  a  liquid 
or  a  solid  is  supposed  to  be.  A  kinetic  theory  of  matter,  as  usually  under- 
stood, appears  to  require  that  in  equilibrium  the  whole  kinetic  energy  shall 
be  equally  shared  among  all  the  degrees  of  freedom,  and  within  each  degree 
of  freedom  be  distributed  according  to  the  same  law.  It  is  included  in  this 
statement  that  temperature  is  a  matter  of  kinetic  energy  only,  e.  g.  that  when 
a  vertical  column  of  gas  is  in  equilibrium,  the  mean  velocity  of  a  molecule 
is  the  same  at  the  top  as  at  the  bottom  of  the  column. 

Reverting  to  (37),  (41),  in  order  to  consider  the  distribution  of  the 
systems  as  dependent  upon  the  coordinates  independently  of  the  velocities, 
we  have,  omitting  unnecessary  factors, 

{E-V}^dq,dq,...dqn  .........................  (51) 

If  n=  2,  e.g.  in  the  case  already  considered  of  a  single  particle  moving  in  two 
dimensions,  or  of  two  particles  moving  in  one  dimension,  or  again  whatever 
n  may  be,  provided  V  vanish,  the  first  factor  disappears,  so  that  the  distri- 
bution is  uniform  with  respect  to  the  coordinates  ql  ...  qn.  If  n  >  2  and  V 
be  finite,  the  distribution  is  such  as  to  favour  those  configurations  for  which 
V  is  least. 

"  When  the  number  of  variables  is  very  great,  and  when  the  potential 
energy  of  the  specified  configuration  is  very  small  compared  with  the  total 
energy  of  the  system,  we  may  obtain  a  useful  approximation  to  the  value  of 
{E  —  V}^~1  in  an  exponential  form  ;  for  if  we  write  (as  before)  E  =  nK, 


(52) 

nearly,  provided  n  is  very  great  and  V  is  small  compared  with  E.  The 
expression  is  no  longer  approximate  when  V  is  nearly  as  great  as  E,  and  it 
does  not  vanish,  as  it  ought  to  do,  when  V  =  E"  (Maxwell.) 

In  the  case  of  gas  composed  of  molecules  whose  mutual  influence  is 
limited  to  a  small  distance  and  which  are  not  subject  to  external  forces, 
the  distribution  expressed  by  (51)  is  uniform  in  space  except  near  the 
boundary.  For  if  q1  denote  the  ^-coordinate  of  a  particular  molecule,  and 
if  we  effect  the  integration  with  respect  to  all  the  coordinates  of  other 
molecules  as  well  as  the  other  coordinates  of  the  particular  molecule,  we 
must  arrive  at  a  result  independent  of  x,  provided  x  relate  to  a  point  well 
R.  iv.  29 


450  THE   LAW  OF   PARTITION   OF   KINETIC    ENERGY.  [253 

in  the  interior.  That  is  to  say  in  the  various  systems  contemplated  the 
particular  molecule  is  uniformly  distributed  with  respect  to  x.  The  same 
is  true  of  y  and  z,  and  thus  the  whole  spacial  distribution  is  uniform.  If 
the  single  system  constituting  the  gas  has  uniform  statistics,  it  will  follow 
that  the  distribution  in  it  of  molecules  similar  to  the  particular  molecule 
is  uniform. 

The  uniformity  of  the  distribution  is  disturbed  if  an  external  force  acts. 
In  illustration  of  this  we  may  consider  the  case  of  gravity.  From  (52)  the 
distribution  with  respect  to  the  coordinates  of  the  particular  molecule 
will  be 


and  the  same  formula  gives  the  density  of  molecules  similar  to  the  particular 
molecule  in  a  single  system. 

The  main  purpose  of  this  paper  is  now  accomplished  ;  but  I  will  take 
the  opportunity  to  make  a  few  remarks  upon  some  general  aspects  of  a 
kinetic  theory  of  matter.  Many  writers  appear  to  commit  themselves  to 
absolute  statements,  but  Kelvin*  and  Boltzmann  and  Maxwell  fully  recognize 
that  conclusions  can  never  be  more  than  probable.  The  second  law  of  thermo- 
dynamics itself  is  in  this  predicament.  Indeed  it  might  seem  at  first  sight 
as  if  the  case  were  even  worse  than  this.  Mr  Culverwell  has  emphasized 
a  difficulty,  which  must  have  been  pretty  generally  felt,  arising  out  of  the 
reversibility  of  a  d3rnamical  system.  If  during  one  motion  of  a  system  energy 
is  dissipated,  restoration  must  occur  when  the  motion  is  reversed.  How  then 
is  one  process  more  probable  than  the  other  ?  Prof.  Boltzmann  has  replied 
to  this  objection,  upon  the  whole  I  think  satisfactorily,  in  a  very  interesting 
letterf.  The  available  (internal)  energy  of  a  system  tends  to  zero,  or  rather 
to  a  small  value,  only  because  the  conditions,  or  phases  as  we  have  called 
them,  corresponding  to  small  values  are  more  probable,  i.e.  more  numerous. 
If  there  is  considerable  available  energy  at  any  moment,  it  is  because  the 
condition  is  then  exceptional  and  peculiar.  After  a  short  interval  of  time 
the  condition  may  become  more  peculiar  still,  and  the  available  energy  may 
increase,  but  this  is  improbable.  The  probability  is  that  the  available  energy 
will,  if  not  at  once,  at  any  rate  after  a  short  interval,  decrease  owing  to  the 
substitution  of  a  more  nearly  normal  state  of  things. 

There  is,  however,  another  side  to  this  question,  which  perhaps  has  been 
too  much  neglected.  Small  values  of  the  available  energy  are  indeed  more 

*  Witness  the  following  remarkable  passage  :  —  "  It  is  a  strange  but  nevertheless  a  true 
conception  of  the  old  well-known  law  of  the  conduction  of  heat  to  say  that  it  is  very  improbable 
that  in  the  course  of  1000  years  one-half  the  bar  of  iron  shall  of  itself  become  warmer  by  a  degree 
than  the  other  half;  and  that  the  probability  of  this  happening  before  1,000,000  years  pass 
is  1000  times  as  great  as  that  it  will  happen  in  the  course  of  1000  years,  and  that  it  certainly  will 
happen  in  the  course  of  some  very  long  time."—  (Nature,  Vol.  ix.  p.  443,  1874.) 

t  Nature,  Vol.  LI.  p.  413  (1895). 


1900]  THE   LAW   OF   PARTITION   OF    KINETIC    ENERGY.  451 

probable  than  large  ones,  but  there  is  a  degree  of  smallness  below  which  it  is 
improbable  that  the  value  will  lie.  If  at  any  time  the  value  lies  extremely 
low,  it  is  an  increase  and  not  a  decrease  which  is  probable.  Maxwell  showed 
long  ago  how  a  being  capable  of  dealing  with  individual  molecules  would 
be  in  a  position  to  circumvent  the  second  law.  It  is  important  to  notice 
that  for  this  end  it  is  not  necessary  to  deal  with  individual  molecules.  It 
would  suffice  to  take  advantage  of  local  reversals  of  the  second  law,  which 
will  involve,  not  very  rarely,  a  considerable  number  of  neighbouring  molecules. 
Similar  considerations  apply  to  other  departures  from  a  normal  state  of 
things,  such,  for  example,  as  unequal  mixing  of  two  kinds  of  molecules, 
or  such  a  departure  from  the  Waterston  relation  (of  equal  mean  kinetic 
energies)  as  has  been  investigated  by  Maxwell  and  by  Tait  and  Burbury. 

The  difficulties  connected  with  the  application  of  the  law  of  equal  parti- 
tion of  energy  to  actual  gases  have  long  been  felt.  In  the  case  of  argon 
and  helium  and  mercury  vapour  the  ratio  of  specific  heats  (1'67)  limits  the 
degrees  of  freedom  of  each  molecule  to  the  three  required  for  translatory 
motion.  The  value  (1'4)  applicable  to  the  principal  diatomic  gases  gives 
room  for  the  three  kinds  of  translation  and  for  two  kinds  of  rotation.  Nothing 
is  left  for  rotation  round  the  line  joining  the  atoms,  nor  for  relative  motion  of 
the  atoms  in  this  line.  Even  if  we  regard  the  atoms  as  mere  points,  whose 
rotation  means  nothing,  there  must  still  exist  energy  of  the  last-mentioned 
kind,  and  its  amount  (according  to  the  law)  should  not  be  inferior. 

We  are  here  brought  face  to  face  with  a  fundamental  difficulty,  relating 
not  to  the  theory  of  gases  merely,  but  rather  to  general  dynamics.  In  most 
questions  of  dynamics  a  condition  whose  violation  involves  a  large  amount  of 
potential  energy  may  be  treated  as  a  constraint.  It  is  on  this  principle  that 
solids  are  regarded  as  rigid,  strings  as  inextensible,  and  so  on.  And  it  is  upon 
the  recognition  of  such  constraints  that  Lagrange's  method  is  founded.  But 
the  law  of  equal  partition  disregards  potential  energy.  However  great  may 
be  the  energy  required  to  alter  the  distance  of  the  two  atoms  in  a  diatomic 
molecule,  practical  rigidity  is  never  secured,  and  the  kinetic  energy  of  the 
relative  motion  in  the  line  of  junction  is  the  same  as  if  the  tie  were  of 
the  feeblest.  The  two  atoms,  however  related,  remain  two  atoms,  and  the 
degrees  of  freedom  remain  six  in  number. 

What  would  appear  to  be  wanted  is  some  escape  from  the  destructive 
simplicity  of  the  general  conclusion  relating  to  partition  of  kinetic  energy, 
whereby  the  energy  of  motions  involving  larger  amounts  of  potential  energy 
should  be  allowed  to  be  diminished  in  consequence.  If  the  argument,  as 
above  set  forth  after  Maxwell,  be  valid,  such  escape  must  involve  a  repudiation 
of  Maxwell's  fundamental  postulate  as  practically  applicable  to  systems  with 
an  immense  number  of  degrees  of  freedom. 

29-2 


254. 


ON  THE  VISCOSITY  OF  ARGON  AS  AFFECTED  BY 
TEMPERATURE. 


[Proceedings  of  the  Royal  Society,  LXVI.  pp.  68—74,  1900.] 

ACCORDING  to  the  kinetic  theory,  as  developed  by  Maxwell,  the  viscosity 
of  a  gas  is  independent  of  its  density,  whatever  may  be  the  character  of  the 
encounters  taking  place  between  the  molecules.  In  the  typical  case  of  a 
gas  subject  to  a  uniform  shearing  motion,  we  may  suppose  that  of  the  three 
component  velocities  v  and  w  vanish,  while  u  is  a  linear  function  of  y,  indepen- 
dent of  x  and  z.  If  p  be  the  viscosity,  the  force  transmitted  tangentially 
across  unit  of  area  perpendicular  to  y  is  measured  by  pdu/dy.  This  repre- 
sents the  relative  momentum,  parallel  to  x,  which  in  unit  of  time  crosses  the 
area  in  one  direction,  the  area  being  supposed  to  move  with  the  velocity 
of  the  fluid  at  the  place  in  question.  We  may  suppose,  for  the  sake  of 
simplicity,  and  without  real  loss  of  generality,  that  u  is  zero  at  the  plane. 
The  momentum,  which  may  now  be  reckoned  absolutely,  does  not  vanish, 
as  in  the  case  of  a  gas  at  rest  throughout,  because  the  molecules  come  from 
a  greater  or  less  distance,  where  (e.g.)  the  value  of  u  is  positive.  The  distance 
from  which  (upon  the  average)  the  molecules  may  be  supposed  to  have  come 
depends  upon  circumstances.  If,  for  example,  the  molecules,  retaining  their 
number  and  velocity,  interfere  less  with  each  other's  motion,  the  distance 
in  question  will  be  increased.  The  same  effect  will  be  produced,  without 
a  change  of  quality,  by  a  simple  reduction  in  the  number  of  molecules,  i.e., 
in  the  density  of  the  gas,  and  it  is  not  difficult  to  recognize  that  the  distance 
from  which  the  molecules  may  be  supposed  to  have  come  is  inversely  as  the 
density.  On  this  account  the  passage  of  tangential  momentum  per  molecule 
is  inversely  as  the  density,  and  since  the  number  of  molecules  crossing  is 
directly  as  the  density,  the  two  effects  compensate,  and  upon  the  whole 
the  tangential  force  and  therefore  the  viscosity  remain  unaltered  by  a  change 
of  density. 


1900]     ON   THE   VISCOSITY   OF   ARGON    AS   AFFECTED   BY    TEMPERATURE.       453 

On  the  other  hand,  the  manner  in  which  this  viscosity  varies  with 
temperature  depends  upon  the  nature  of  the  encounters.  If  the  molecules 
behaved  like  Boscovich  points,  which  exercise  no  force  upon  one  another  until 
the  distance  falls  to  a  certain  value,  and  which  then  repel  one  another 
infinitely  (erroneously  called  the  theory  of  elastic  spheres),  then,  as  Maxwell 
proved,  the  viscosity  would  be  proportional  to  the  square  root  of  the  absolute 
temperature.  Or  again,  if  the  law  of  repulsion  were  as  the  inverse  fifth 
power  of  the  distance,  viscosity  would  be  as  the  absolute  temperature. 

In  the  more  general  case  where  the  repulsive  force  varies  as  r~n,  the 
dependence  of  p  upon  temperature  may  also  be  given.  If  v  be  the  velocity 
of  mean  square,  proportional  to  the  square  root  of  the  temperature,  p,  varies 

n+% 

as  vn~l,  a  formula  which  includes  the  cases  (n  =  5,  n=  oo  )  already  specified. 
If  we  assume  the  law  already  discussed  —  that  p  is  independent  of  density  — 
this  conclusion  may  be  arrived  at  very  simply  by  the  method  of  "  dimensions." 

In  order  to  see  this  we  note  that  the  only  quantities  (besides  the  density) 
on  which  //,  can  depend  are  m  the  mass  of  a  particle,  v  the  velocity  of  mean 
square,  and  k  the  repulsive  force  at  unit  distance.  The  dimensions  of  these 
quantities  are  as  follows  :  — 

p  =  (mass)1  (length)"1  (time)"1, 
TO  =  (mass)1, 
v  =  (length)1  (time)"1, 
k  =  (mass)1  (length)"*1  (time)-2. 

Thus,  if  we  assume 

pKmv.vy.k'     ..............................  (1) 

we  have         l=x  +  z,         —  l=y  +  (n  +  l)z,         —  l=  —  y  —  2z, 

n+l  n+B  2 

whence  *  =  ^>         2/  =  —  r  =  —  r 

n+l       n+8  2 

Accordingly  f*  =  a.mn-1  .vn~l  .k~n-\  ........................  (2) 

where  a  is  a  purely  numerical  coefficient.  For  a  given  kind  of  molecule, 
w  and  k  are  constant.  Thus 

n+8  n+3 

...............................  (3) 


The  case  of  sudden  impacts  (n  =  oo  )  gives,  as  already  remarked, 
Hence  k  disappears,  and  the  consideration  of  dimensions  shows  that  fj,  oc  d~2, 
where  d  is  the  diameter  of  the  particles. 


454        ON  THE  VISCOSITY  OF   ARGON   AS   AFFECTED   BY  TEMPERATURE.      [254 

The  best  experiments  on  air  show  that,  so  far  as  a  formula  of  this  kind 
can  represent  the  facts,  p  oc  00"77.  It  may  be  observed  that  n  =  8  corre- 
sponds to  /*  oc  00"79. 

When  we  remember  that  the  principal  gases,  such  as  oxygen,  hydrogen, 
and  nitrogen,  are  regarded  as  diatomic,  we  may  be  inclined  to  attribute 
the  want  of  simplicity  in  the  law  connecting  viscosity  and  temperature  to 
the  complication  introduced  by  the  want  of  symmetry  in  the  molecules  and 
consequent  diversities  of  presentation  in  an  encounter.  It  was  with  this  idea 
that  I  thought  it  would  be  interesting  to  examine  the  influence  of  tempera- 
ture upon  the  viscosity  of  argon,  which  in  the  matter  of  specific  heat  behaves 
as  if  composed  of  single  atoms.  From  the  fact  that  no  appreciable  part  of 
the  total  energy  is  rotatory,  we  may  infer  that  the  forces  called  into  play 
during  one  encounter  are  of  a  symmetrical  character.  It  seemed,  therefore, 
more  likely  that  a  simple  relation  between  viscosity  and  temperature  would 
obtain  in  the  case  of  argon  than  in  the  case  of  the  "  diatomic  "  gases. 

The  best  experimental  arrangement  for  examining  this  question  is  probably 
that  of  Holman*,  in  which  the  same  constant  stream  of  gas  passes  in  suc- 
cession through  two  capillaries  at  different  temperatures,  the  pressures  being 
determined  before  the  first  and  after  the  second  passage,  as  well  as  between 
the  two.  But  to  a  gas  like  argon,  available  in  small  quantities  only,  the 
application  of  this  method  is  difficult.  And  it  seemed  unnecessary  to  insist 
upon  the  use  of  constant  pressures,  seeing  that  it  was  not  proposed  to  in- 
vestigate experimentally  the  dependence  of  transpiration  upon  pressure. 

The  theoretical  formula  for  the  volume  of  gas  transpired,  analogous 
to  that  first  given  by  Stokes  for  an  incompressible  fluid,  was  developed  by 
O.  E.  Meyerf.  Although  not  quite  rigorous,  it  probably  suffices  for  the 
purpose  in  hand.  If  pli  Vl  denote  the  pressure  and  volume  of  the  gas  as 
it  enters  the  capillary,  p^,  F2  as  it  leaves  the  capillary,  we  have 


In  this  equation  t  denotes  the  time  of  transpiration,  R  the  radius  of  the 
tube,  I  its  length,  and  jj,  the  viscosity  measured  in  the  usual  way. 

In  order  to  understand  the  application  of  the  formula  for  our  present 
purpose,  it  will  be  simplest  to  consider  first  the  passage  of  equal  volumes 
of  different  gases  through  the  capillary,  the  initial  pressures,  and  the  constant 
temperature  being  the  same.  In  an  apparatus,  such  as  that  about  to  be 
described,  the  pressures  change  as  the  gas  flows,  but  if  the  pressures  are 
definite  functions  of  the  amount  of  gas  which  at  any  moment  has  passed  the 

*  Phil.  Mag.  Vol.  m.  p.  81  (1877). 

t  Fogg.  Ann.  Vol.  cxxvii.  p.  269  (1866). 


1900]     ON   THE    VISCOSITY   OF    ARGON    AS    AFFECTED   BY   TEMPERATURE.       455 

capillary,  this  variation  does  not  interfere  with  the  proportionality  between 
t  and  fj,.  For  example,  if  the  viscosity  be  doubled,  the  flow  takes  place 
precisely  as  before,  except  that  the  scale  of  time  is  doubled.  It  will  take 
twice  as  long  as  before  to  pass  the  same  quantity  of  gas. 

Although  different  gases  have  been  employed  in  the  present  experiments, 
there  has  been  no  attempt  to  compare  their  viscosities,  and  indeed  such 
a  comparison  would  be  difficult  to  carry  out  by  this  method.  The  question 
has  been,  how  is  the  viscosity  of  a  given  gas  affected  by  a  change  of  tempera- 
ture ?  In  one  set  of  experiments  the  capillary  is  at  the  temperature  of  the 
room;  in  a  closely  following  set  the  capillary  is  bathed  in  saturated  steam 
at  a  temperature  that  can  be  calculated  from  the  height  of  the  barometer. 

If  the  temperature  were  changed  throughout  the  whole  apparatus  from 
one  absolute  temperature  0  to  another  absolute  temperature  6'  '  ,  we  could 
make  immediate  application  of  (4)  ;  the  viscosities  (/A,  /*')  at  the  two  tempera- 
tures would  be  directly  as  the  times  of  transpiration  (t,  t').  The  matter  is 
not  quite  so  simple  when,  as  in  these  experiments,  the  change  of  temperature 
takes  place  only  in  the  capillary.  A  rise  of  temperature  in  the  capillary  now 
acts  in  two  ways.  Not  only  does  it  change  the  viscosity,  but  it  increases  the 
volume  of  gas  which  has  to  pass.  The  ratio  of  volumes  is  6',  0;  and  thus 


subject  to  a  small  correction  for  the  effect  of  temperature  upon  the  dimensions 
of  the  capillary.  It  is  assumed  that  the  temperature  of  the  reservoirs  is  the 
same  in  both  transpirations. 

The  apparatus  is  shown  in  the  figure.  The  gas  flows  to  and  fro  between 
the  bulbs  A  and  B,  the  flow  from  A  to  B  only  being  timed.  It  is  confined  by 
mercury,  which  can  pass  through  U  connexions  of  blown  glass  from  A  to  C 
and  from  B  to  D.  The  bulbs  B,  C,  D  are  supported  upon  their  seats  with 
a  little  plaster  of  Paris.  The  capillary  is  nearly  5  feet  (150  cm.)  in  length 
and  is  connected  with  the  bulbs  by  gas  tubing  of  moderate  diameter,  all 
joints  being  blown.  E  represents  the  jacket  through  which  steam  can  be 
passed  ;  its  length  exceeds  that  of  the  capillary  by  a  few  inches. 

In  order  to  charge  the  apparatus,  the  first  step  is  the  exhaustion.  This 
is  effected  through  the  tap,  F,  with  the  aid  of  a  Topler  pump,  and  it  is 
necessary  to  make  a  corresponding  exhaustion  in  C  and  D,  or  the  mercury 
would  be  drawn  over.  To  this  end  the  rubber  terminal  H  is  temporarily 
connected  with  0,  while  /  leads  to  a  common  air-pump.  When  the  exhaustion 
is  complete,  the  gas  to  be  tried  is  admitted  gradually  at  F,  the  atmosphere 
being  allowed  again  to  exert  its  pressure  in  G  and  D.  When  the  charge  is 
sufficient,  F  is  turned  off,  after  which  G  remains  open  to  the  atmosphere,  and 
H  is  connected  to  a  manometer. 


456         ON  THE   VISCOSITY   OF   ARGON   AS   AFFECTED   BY  TEMPERATURE.     [254 

When  a  measurement  is  commenced,  the  first  step  is  to  read  the  tempera- 
tures of  the  bulbs  and  of  the  capillary  ;  /  is  then  connected  to  a  force  pump, 
and  pressure  is  applied  until  so  much  of  the  gas  is  driven  over  that  the 
mercury  below  A  and  in  B  assumes  the  positions  shown  in  the  diagram. 
/  is  then  suddenly  released  so  that  the  atmospheric  pressure  asserts  itself 
in  D,  and  the  gas  begins  to  flow  back  into  B.  The  bulb  /  allows  the  flow 
a  short  time  in  which  to  establish  itself  before  the  time  measurement  begins 
as  the  mercury  passes  the  connexion  passage  K.  When  the  mercury  reaches 
L,  the  time  measurement  is  closed. 

One  of  the  points  to  be  kept  in  view  in  designing  the  apparatus  is  to 
secure  long  enough  time  of  transpiration  without  unduly  lowering  the  driving 
pressure.  At  the  beginning  of  the  measured  transpiration  the  pressure 
in  A  was  about  30  cm.  of  mercury  above  atmosphere,  and  that  in  B  about 


2  cm.  below  atmosphere.  At  the  end  the  pressure  in  A  was  20  cm.,  and 
in  B  3  cm.,  both  above  atmosphere.  Accordingly  the  driving  pressure  fell 
from  32  to  17  cm. 


1900]     ON   THE   VISCOSITY   OF    ARGON   AS    AFFECTED    BY   TEMPERATURE.       457 

Three,  or,  in  the  case  of  hydrogen,  five,  observations  of  the  time  were 
usually  taken,  and  the  agreement  was  such  as  to  indicate  that  the  mean 
would  be  correct  to  perhaps  one-tenth  of  a  second.  The  time  for  air  at  the 
temperature  of  the  room  was  about  ninety  seconds,  and  for  hydrogen  forty-four 
seconds,  but  these  numbers  are  not  strictly  comparable. 

When  the  low  temperature  observations  were  finished,  the  gas  was  lighted 
under  a  small  boiler  placed  upon  a  shelf  above  the  apparatus,  and  steam  was 
passed  through  the  jacket.  It  was  necessary  to  see  that  there  was  enough 
heat  to  maintain  a  steady  issue  of  steam,  yet  not  so  much  as  to  risk  a  sensible 
back  pressure  in  the  jacket.  The  time  of  transpiration  for  air  was  now  about 
139  seconds.  Care  was  always  taken  to  maintain  the  temperature  of  the 
bulbs  at  the  same  point  as  in  the  first  observations. 

There  are  one  or  two  matters  as  to  which  an  apparatus  on  these  lines 
is  necessarily  somewhat  imperfect.  In  the  high  temperature  measurements 
the  whole  of  the  gas  in  the  capillary  is  assumed  to  be  at  the  temperature 
of  boiling  water,  and  all  that  is  not  in  the  capillary  to  be  at  the  temperature 
of  the  room,  assumptions  not  strictly  compatible.  The  compromise  adopted 
was  to  enclose  in  the  jacket  the  whole  of  the  capillary  and  about  2  inches 
at  each  end  of  the  approaches,  and  seems  sufficient  to  exclude  sensible  error 
when  we  remember  the  rapidity  with  which  heat  is  conducted  in  small  spaces. 
A  second  weak  point  is  the  assumption  that  the  instantaneous  pressures  are 
represented  by  the  heights  of  the  moving  mercury  columns.  If  the  connecting 
U -tubes  are  too  narrow,  the  resistance  to  the  flow  of  mercury  enters  into 
the  question  in  much  the  same  way  as  the  flow  of  gas  in  the  capillary.  In 
order  to  obtain  a  check  upon  this  source  of  error  the  apparatus  has  been 
varied.  In  an  earlier  form  the  connecting  U -tubes  were  comparatively 
narrow;  but  the  result  for  the  ratio  of  viscosities  of  hot  and  cold  air  was 
substantially  the  same  as  that  subsequently  obtained  with  the  improved 
apparatus,  in  which  these  tubes  were  much  widened.  Even  if  there  be  a 
sensible  residual  error  arising  from  this  cause,  it  can  hardly  affect  the  com- 
parison of  temperature -coefficients  of  gases  whose  viscosity  is  nearly  the  same. 

I  will  now  give  an  example  in  detail  from  the  observations  of  December  21 
with  purified  argon.  The  times  of  transpiration  at  the  temperature  of  the 
room  (15°  C.)  were  in  seconds 

104f ,         104£,         104|.         Mean,  104'67. 

When  the  capillaries  were  bathed  in  steam,  the  corresponding  times  were 
167$,         167i,         167f.         Mean,  167-58. 

The  barometer  reading  (corrected)  being  767'4  mm.,  we  deduce  as  the 
temperature  of  the  jacket  100'27°  C.  Thus  6  =  287'5,  0'  =  372'8.  The  re- 
duction was  effected  by  assuming 


458        ON   THE   VISCOSITY   OF   ARGON   AS   AFFECTED   BY   TEMPERATURE.      [254 

With  the  above  values  we  get 

#=1-812. 

As  appears  from  (5),  the  integral  part  of  x  relates  merely  to  the  expansion 
of  the  gas  by  temperature.     If  we  take 


we  get  n  =  0*812. 

This  number  is,  however,  subject  to  a  small  correction  for  the  expansion 
of  the  glass  of  the  capillary.  As  appears  from  (4),  the  ratio  //,  /*  as  used 
above  requires  to  be  altered  in  the  same  ratio  as  that  in  which  the  glass 
expands  by  volume.  The  value  of  n  must  accordingly  be  increased  by  O'OIO, 
making 

n  =  0-822. 

The  following  table  embodies  the  results  obtained  in  a  somewhat  extended 
series  of  observations.  The  numbers  given  are  the  values  of  n  in  (7),  corrected 
for  the  expansion  of  the  glass. 

Air  (dry) 0754 

Oxygen 0782 

Hydrogen 0'681 

Argon  (impure)      ....   0'801 
Argon  (best) 0'815 

In  the  last  trials,  the  argon  was  probably  within  1  or  2  per  cent,  of 
absolute  purity.  The  nitrogen  lines  could  no  longer  be  seen,  and  scarcely  any 
further  contraction  could  be  effected  on  sparking  with  oxygen  or  hydrogen. 

It  will  be  seen  that  the  temperature  change  of  viscosity  in  argon  does 
not  differ  very  greatly  from  the  corresponding  change  in  air  and  oxygen. 
At  any  rate  the  simpler  conditions  under  which  we  may  suppose  the  collisions 
to  occur,  do  not  lead  to  values  of  n  such  as  0'5,  or  I'O,  discussed  by  theoretical 
writers. 

I  may  recall  that,  on  a  former  occasion*,  I  found  the  viscosity  of  argon 
to  be  1-21  relatively  to  that  of  air,  both  being  observed  at  the  temperature 
of  the  room. 

[1902.     See  further,  Vol.  iv.  p.  481.] 

*  Roy.  Soc.  Proc.  January,  1896.     [Vol.  iv.  p.  222.] 


255. 


ON  THE   PASSAGE  .OF  ARGON   THROUGH  THIN   FILMS   OF 
INDIARUBBER. 


[Philosophical  Magazine,  XLix.  pp.  220,  221,  1900.] 

SOON  after  the  discovery  of  Argon  it  was  thought  desirable  to  compare 
the  percolation  of  the  gas  through  indiarubber  with  that  of  nitrogen,  and 
Sir  W.  Roberts- Austen  kindly  gave  me  some  advice  upon  the  subject.  The 
proposal  was  simply  to  allow  atmospheric  air  to  percolate  through  the  rubber 
film  into  a  vacuum,  after  the  manner  of  Graham,  and  then  to  determine  the 
proportion  of  argon.  It  will  be  remembered  that  Graham  found  that  the 
percentage  of  oxygen  was  raised  in  this  manner  from  the  21  of  the  atmo- 
sphere to  about  40.  At  the  time  the  experiment  fell  through,  but  during 
the  last  year  I  have  carried  it  out  with  the  assistance  of  Mr  Gordon. 

The  rubber  balloon  was  first  charged  with  dry  boxwood  sawdust.  This 
rather  troublesome  operation  was  facilitated  by  so  mounting  the  balloon  that 
with  the  aid  of  an  air-pump  the  external  pressure  could  be  reduced.  When 
sufficiently  distended  the  balloon  was  connected  with  a  large  Tb'pler  pump, 
into  the  vacuous  head  of  which  the  diffused  gases  could  collect.  At  intervals 
they  were  drawn  off  in  the  usual  way. 

The  diffusion  was  not  conducted  under  ideal  conditions.  In  order  to 
make  the  most  of  the  time,  the  apparatus  was  left  at  work  during  the  night, 
so  that  by  the  morning  the  internal  pressure  had  risen  to  perhaps  three 
inches  of  mercury.  The  proportion  of  oxygen  in  the  gas  collected  was  deter- 
mined from  time  to  time.  It  varied  from  34  per  cent,  when  the  vacuum 
was  bad  to  about  39  per  cent,  when  the  vacuum  was  good.  On  an  average 
it  was  estimated  that  the  proportion  of  oxygen  would  be  about  37  per  cent, 
of  the  whole.  The  total  quantity  of  diffused  gas  reckoned  at  atmospheric 
pressure  was  about  300  c.c.  per  twenty-four  hours. 


460       PASSAGE  OF   ARGON  THROUGH  THIN   FILMS   OF   INDIARUBBER.  [255 

On  removal  from  the  pump  the  gas  was  introduced  into  an  inverted  flask 
standing  over  alkali,  and  with  addition  of  oxygen  as  required  was  treated 
with  the  electrical  discharge  from  a  transformer  in  connexion  with  the  public 
supply  of  alternating  current.  In  this  way  the  nitrogen  was  gradually 
oxidized  and  absorbed.  Towards  the  close  of  operations  the  gas  was  trans- 
ferred to  a  smaller  vessel,  where  it  was  further  sparked  until  no  further 
contraction  occurred,  and  the  lines  of  nitrogen  had  disappeared  from  the 
spectrum.  The  excess  of  oxygen  was  then  removed  by  phosphorus. 

It  remains  only  to  record  the  final  figures.  The  residue,  free  of  oxygen 
and  nitrogen,  from  3205  c.c.  of  diffused  gas  was  39  c.c.  The  most  instructive 
way  of  stating  the  result  is  perhaps  to  reckon  the  argon  as  a  percentage,  not 
of  the  whole,  but  of  the  nitrogen  and  argon  only.  Of  the  3205  c.c.  total, 
2020  c.c.  would  be  nitrogen  and  argon,  and  of  this  the  39  c.c.  argon  would 
be  1*93  per  cent.  Since,  according  to  Kellas  (Proc.  Roy.  Soc.  Vol.  LIX.  p.  67, 
1895),  100  c.c.  of  mixed  atmospheric  nitrogen  and  argon  contains  T19  per  cent, 
of  argon,  we  see  that  in  the  diffused  gas  the  proportion  of  argon  is  about  half 
as  great  again  as  in  the  atmosphere.  Argon  then  passes  the  indiarubber  film 
more  readily  than  nitrogen,  but  not  in  such  a  degree  as  to  render  the  diffusion 
process  a  useful  one  for  the  concentration  of  argon  from  the  atmosphere. 


256. 

ON  THE   WEIGHT   OF  HYDROGEN  DESICCATED  BY 
LIQUID  AIR. 

[Proceedings  of  the  Royal  Society,  LXVI.  p.  344,  1900.] 

IN  recent  experiments  by  myself  and  by  others  upon  the  density  of 
hydrogen,  the  gas  has  always  been  dried  by  means  of  phosphoric  anhydride ; 
and  a  doubt  may  remain  whether  on  the  one  hand  the  removal  of  aqueous 
vapour  is  sufficiently  complete,  and  on  the  other  whether  some  new  impurity 
may  not  be  introduced.  I  thought  that  it  would  be  interesting  to  weigh 
hydrogen  dried  in  an  entirely  different  manner,  and  this  I  have  recently  been 
able  to  effect  with  the  aid  of  liquid  air,  acting  as  a  cooling  agent,  supplied 
by  the  kindness  of  Professor  Dewar  from  the  Royal  Institution.  The  opera- 
tions of  filling  and  weighing  were  carried  out  in  the  country  as  hitherto. 
I  ought,  perhaps,  to  explain  that  the  object  was  not  so  much  to  make  a  new 
determination  of  the  highest  possible  accuracy,  as  to  test  whether  any  serious 
error  could  be  involved  in  the  use  of  phosphoric  anhydride,  such  as  might 
explain  the  departure  of  the  ratio  of  densities  of  oxygen  and  hydrogen  from 
that  of  16  : 1.  I  may  say  at  once  that  the  result  was  negative. 

Each  supply  consisted  of  about  6  litres  of  the  liquid,  contained  in  two 
large  vacuum-jacketed  vessels  of  Professor  Dewar's  design,  and  it  sufficed 
for  two  fillings  with  hydrogen  at  an  interval  of  two  days.  The  intermediate 
day  was  devoted  to  a  weighing  of  the  globe  empty.  There  were  four  fillings 
in  all,  but  one  proved  to  be  abortive  owing  to  a  discrepancy  in  the  weights 
when  the  globe  was  empty,  before  and  after  the  filling.  The  gas  was  exposed 
to  the  action  of  the  liquid  air  during  its  passage  in  a  slow  stream  of  about 
half  a  litre  per  hour  through  a  tube  of  thin  glass. 

I  have  said  that  the  result  was  negative.  In  point  of  fact  the  actual 
weights  found  were  ^  to  -^  milligrams  heavier  than  in  the  case  of  hydrogen 
dried  by  phosphoric  anhydride.  But  I  doubt  whether  the  small  excess  is  of 
any  significance.  It  seems  improbable  that  it  could  have  been  due  to  residual 
vapour,  and  it  is  perhaps  not  outside  the  error  of  experiment,  considering 
that  the  apparatus  was  not  in  the  best  condition. 


257. 

THE  MECHANICAL  PRINCIPLES   OF  FLIGHT. 

[Manchester  Memoirs,  XLIV.  pp.  1 — 26,  1900.] 

THE  subject  under  discussion  includes  both  natural  and  artificial  flight. 
Although  we  are  familiar  with  the  flight  of  birds,  there  are  many  interesting 
questions  which  arise  in  connexion  with  natural  flight,  and  some  of  them  are 
yet  very  obscure. 

In  still  air  a  bird,  being  heavier  than  the  fluid  displaced,  cannot  maintain 
his  level  for  more  than  a  short  time  without  working  his  wings.  In  this 
matter  the  vicarious  principle  holds  good.  If  the  bird  is  not  to  fall,  some- 
thing must  fall  instead  of  him,  and  this  can  only  be  air.  The  maintenance 
of  the  bird  thus  implies  the  perpetual  formation  of  a  downward  current  of 
air,  and  involves  therefore  performance  of  work.  Later  we  shall  consider 
more  particularly  how  this  work  is  applied;  but  a  preliminary  difficulty 
remains  to  be  discussed.  It  is  well  known  that  large  birds,  such  as  vultures 
and  pelicans,  are  often  observed  to  maintain  their  level  for  considerable 
periods  of  time,  without  flapping  or  visibly  working  their  wings.  On  a 
smaller  scale,  and  in  more  special  situations,  sea-gulls  in  these  latitudes 
perform  similar  feats.  This  question  of  the  soaring  or  sailing  flight  of  birds 
has  given  rise  to  much  difference  of  opinion.  Few  of  the  naturalists,  to 
whom  we  owe  the  observations,  are  familiar  with  mechanical  principles,  and 
thus  statements  are  often  put  forward  which  amount  to  mechanical  impossi- 
bilities. The  arm-chair  theorist  at  home,  on  the  other  hand,  may  be  too 
willing  to  discredit  reports  of  actual  observations,  especially  when  they  are 
made  in  other  parts  of  the  world.  On  both  sides  it  seems  to  be  admitted 
that  there  is  no  sailing  flight  in  the  absence  of  wind ;  but  observers,  un- 
trained in  dynamics  and  misled  by  the  analogy  of  the  kite,  are  apt  to  suppose 
that  the  existence  of  wind  at  once  removes  the  difficulty.  The  doctrine  of 
relative  motion  shows  however  that,  so  long  as  there  is  no  connexion  with 
the  ground,  a  uniform  horizontal  wind  is  for  this  purpose  the  same  thing  as 
absolutely  still  air. 


1900]  THE    MECHANICAL    PRINCIPLES    OF    FLIGHT.  463 

In  a  short  paper  upon  this  subject  (Nature,  xxvu.  p.  534,  1883  [Vol.  n. 
p.  194])  I  pointed  out  that,  "  Whenever  a  bird  pursues  his  course  for  some 
time  without  working  his  wings,  we  must  conclude  either  (1)  that  the  course 
is  not  horizontal,  (2)  that  the  wind  is  not  horizontal,  or  (3)  that  the  wind  is 
not  uniform.  It  is  probable  that  the  truth  is  usually  represented  by  (1)  or 
(2),  but  the  question  I  wish  to  raise  is  whether  the  cause  suggested  by  (3) 
may  not  sometimes  come  into  operation."  Case  (1)  is  that  of  a  rook  gliding 
downwards  from  a  tree  in  still  air  with  motionless  wings.  We  shall  presently 
consider  upon  what  conditions  depend  the  time  and  distance  of  travel  possible 
with  a  given  descent.  Case  (2)  is  closely  related  to  case  (1).  If  the  air  have 
an  upward  velocity  equal  to  that  at  which  the  rook  falls  through  it  in  a 
vertical  direction,  the  vertical  motion  is  compensated,  and  the  course  of  the 
rook  relatively  to  the  ground  becomes  horizontal.  It  is  not  necessary,  of 
course,  that  the  whole  motion  of  the  air  be  upwards  ;  a  horizontal  motion  of 
the  air  is  simply  superposed.  A  bird  gliding  into  a  wind  having  a  small 
upward  component  may  thus  maintain  relatively  to  the  ground  an  absolutely 
fixed  position,  or  he  may  advance  over  the  ground  to  windward  at  a  fixed 
level. 

There  can  be  no  doubt  that  the  vertical  component  of  wind  plays  a  large 
part,  not  merely  in  the  flight  of  birds,  but  in  general  atmospheric  phenomena. 
Living  at  the  bottom  of  the  atmospheric  ocean,  where  the  wind  is  necessarily 
parallel  to  the  ground,  we  are  liable  to  overlook  the  importance  of  vertical 
motions.  This  is  the  more  remarkable  when  we  consider  that  wind  is  due  to 
atmospheric  expansion  and  condensation,  so  that  the  primary  movements  are 
vertical  and  not  horizontal.  Thus  the  inhabitants  of  an  oceanic  island  are 
specially  interested  in  the  so-called  land  and  sea  breezes,  but  the  primary 
phenomenon  is  the  rise  and  fall  of  air  over  the  island  as  it  is  heated  by  the 
sun  during  the  day  and  cooled  by  radiation  at  night. 

A  recent  American  observer  (Huffaker,  Smithsonian  Report  for  1897)  has 
recorded  many  examples  of  vultures  soaring  under  circumstances  which  sug- 
gested that  they  take  advantage  of  the  upward  currents  which  rise  locally 
from  the  ground  when  it  is  strongly  heated  by  the  sun.  On  dull  days  and 
in  light  winds  the  vultures  were  not  seen  to  soar.  There  is  no  doubt  that 
under  the  influence  of  a  strong  sun  the  layers  of  air  near  the  ground  approach 
an  unstable  condition,  and  that  comparatively  slight  causes  may  determine 
local  upward  currents.  Mr  Huffaker  suggests  that  in  some  cases  the  birds 
themselves,  by  flying  round,  may  determine  the  upward  current.  Some  of 
his  observations  certainly  point  in  this  direction ;  but  it  must  be  remembered 
that  the  immediate  effect  of  flight  will  be  a  downward  and  not  an  upward 
current. 

The  more  obvious  examples  of  upward  motion  occur  when  an  otherwise 
horizontal  wind  meets  an  obstruction.  Some  years  ago  I  visited  the  north 


464  THE   MECHANICAL  PRINCIPLES   OF   FLIGHT.  [257 

side  of  Madeira,  where  cliffs,  nearly  2,000  feet  high,  rise  perpendicularly 
from  the  sea.  Being  on  the  top  of  the  cliff,  we  had  difficulty  in  finding 
a  sheltered  spot  until  we  noticed  that  close  to  the  edge  there  was  almost  com- 
plete calm.  Lying  upon  the  ground  and  moving  only  one's  arms,  it  was 
possible  to  hold  a  handkerchief  by  the  corner  so  that  a  little  behind  the 
plane  of  the  cliff  it  hung  downwards  as  in  still  air,  and  a  little  in  front  of  the 
cliff  was  carried  upwards  in  the  vertically  rising  stream.  A  ball  of  crumpled 
paper  thrown  outwards  was  carried  up  high  over  our  heads.  Of  course  gulls 
and  other  birds  found  no  difficulty  in  rising  up  the  face  of  the  cliff  without 
working  their  wings.  During  a  recent  visit  to  India,  I  frequently  watched 
the  effect  of  similar  upward  currents  deflected  by  rocky  fortresses  which  rise 
from  the  plains.  Kites  could  be  seen  to  maintain  themselves  for  minutes 
together  without  a  single  flap  of  the  wings.  When  this  occurred,  the  birds 
were  sailing  to  and  fro  over  the  windward  side  of  the  rock. 

We  now  turn  to  the  consideration  of  case  (3). 

"In  a  uniform  wind  the  available  energy  at  the  disposal  of  the  bird 
depends  upon  his  velocity  relatively  to  the  air  about  him.  With  only  a 
moderate  waste  this  energy  can  at  any  moment  be  applied  to  gain  elevation, 
the  gain  of  elevation  being  proportional  to  the  loss  of  relative  velocity 
squared.  It  will  be  convenient  for  the  moment  to  ignore  the  waste  referred 
to,  and  to  suppose  that  the  whole  energy  available  remains  constant,  so  that 
however  the  bird  may  ascend  or  descend,  the  relative  velocity  is  that  due  to 
a  fall  from  a  certain  level  to  the  actual  position,  the  certain  level  being  of 
course  that  to  which  the  bird  might  just  rise  by  the  complete  sacrifice  of 
relative  velocity." 

In  illustration  of  case  (3)  I  instanced  a  wind  blowing  everywhere  hori- 
zontally but  with  a  velocity  increasing  upwards,  taking  for  the  sake  of 
simplicity  the  imaginary  case  of  a  wind  uniform  above  and  below  a  certain 
plane  where  the  velocity  changes.  Since  a  uniform  motion  has  no  effect,  we 
may  suppose  without  further  loss  of  generality,  that  the  velocities  of  the 
wind  above  and  below  the  plane  are  +  u  and  —  u.  Let  us  consider  how  a 
bird,  sailing  somewhat  above  the  plane  of  separation  and  endowed  with  an 
initial  relative  velocity  v,  might  take  advantage  of  the  position  in  which  he 
finds  himself. 

The  first  step  is,  if  necessary,  to  turn  round  until  the  relative  motion  is 
down  wind  (in  the  upper  stratum)  and  then  to  drop  through  the  plane  of 
separation.  In  falling  down  to  the  level  of  the  plane  there  is  a  gain  of 
relative  velocity,  but  this  of  no  significance  for  the  present  purpose,  as  it  is 
purchased  by  the  loss  of  elevation ;  but  in  passing  through  the  plane  there 
is  a  really  effective  gain.  In  entering  the  lower  stratum  the  actual  velocity 
is  indeed  unaltered,  but  the  velocity  relatively  to  the  surrounding  air  (moving 
in  the  opposite  direction)  is  increased. 


1900]  THE   MECHANICAL    PRINCIPLES    OF    FLIGHT.  465 

If  h  denote  the  height  above  the  plane  of  separation  to  which  the  initial 
relative  velocity  v  is  due,  we  have  v2  =  *2gh.  Here  v  is  the  velocity,  relatively 
to  the  air  in  the  upper  stratum,  with  which  the  bird  crosses  the  plane. 
After  crossing,  the  velocity,  now  reckoned  relatively  to  the  air  in  the  lower 
stratum,  becomes  v  +  Zu,  and  the  new  value  of  h  is  given  by 

2gh'  =  (v  +  2w)2, 
so  that  2g  (h'  —  h)  =  4<uv  +  4u2  =  4<u  (u  +  v). 

Here  (h'  —  h)  is  the  gain  of  potential  elevation  and,  if  u  is  given,  it  increases 
as  v  increases. 

At  this  stage  the  bird  is  moving  against  the  direction  of  the  wind  in  the 
lower  stratum.  He  next  turns  round — it  is  supposed  without  loss  of  relative 
velocity — until  his  direction  is  reversed  so  as  to  be  with  the  wind  of  the 
lower  stratum  and  contrary  to  the  wind  of  the  upper  stratum.  A  passage 
upwards  through  the  plane  now  secures  another  gain  of  relative  velocity, 
or  of  potential  elevation,  of  nearly  the  same  value  as  before.  The  process 
may  be  repeated.  At  every  passage  through  the  plane  (whether  in  the 
upwards  or  in  the  downwards  direction)  there  is  a  gain  of  potential  elevation, 
and  if  this  gain  outweighs  the  losses  all  the  while  in  progress,  the  bird  may 
maintain  or  improve  his  position  without  doing  a  stroke  of  work. 

It  may  be  of  interest  to  consider  a  numerical  example. 
Suppose  that 

v  =  30  miles  per  hour  =  1'34  x  10s  cm.  per  second, 
and  that  h'  -  h  =  10  feet  =  305  cm. ; 

then  in  C.G.s.  measure 

(v  +  2uJ>  =  v*  +  2g  (Ji  - h)  =  T80  x  10B  +  -60  x  10"  =  2'40  x  106, 
and  w +  2w  =  1-55  x  10s; 

so  that  2w  =  '21  x  103  cm./sec.  =  47  miles/ hour. 

In  this  case  a  freshening  of  the  wind  amounting  to  4'7  miles  per  hour 
is  equivalent  to  a  gain  of  10  feet  of  potential  elevation. 

In  order  to  take  advantage  of  the  gradual  increase  of  wind  with  elevation 
usually  to  be  met  with,  a  bird  may  describe  circles  in  an  inclined  plane, 
always  descending  when  moving  to  leeward  and  ascending  when  moving  to 
windward.  Whether  the  differences  of  velocity  available  at  considerable 
elevations  in  the  atmosphere  are  sufficient  to  allow  a  bird  to  maintain  his 
position  without  working  his  wings  appears  to  be  doubtful.  Near  the  level 
of  the  ground  or  sea  these  differences  are  greater,  and  probably  suffice  to 
explain  much  of  the  sailing  flight  of  albatrosses  and  other  sea-birds. 

R.   iv.  30 


466  THE   MECHANICAL   PRINCIPLES   OF   FLIGHT.  [257 

Another  way  in  which  a  bird  may  draw  upon  the  internal  energy  of  the 
wind  has  been  specially  discussed  by  Dr  Langley  (Smithsonian  Contributions, 
1893),  who  calls  attention  to  the  fact  that  the  well-known  gustiness  of  the 
wind,  at  any  rate  near  the  earth's  surface,  is  underestimated  in  the  usual 
meteorological  records.  The  differences  of  horizontal  velocity  involved  in 
what  are  commonly  called  gusts  of  wind  imply  in  general  vertical  motions 
also,  but  near  the  ground  these  latter  may,  perhaps,  be  left  out  of  account. 
The  advantage  which  a  bird  may  take  of  the  variations  in  the  speed  of  the 
wind  is  explicable  upon  the  principles  already  applied,  the  inertia  of  the 
bird  playing  in  some  sort  the  part  of  the  string  of  a  kite. 

If  u  denote  the  speed  of  the  wind  at  any  moment,  and  v  the  speed  of  the 
bird  in  the  opposite  direction,  both  e.g.,  reckoned  relatively  to  the  ground, 
the  available  energy  is  measured  by  ^  (v  +  uf.  Suppose  now  that  the  wind 
freshens,  u  becoming  u  +  du,  while  v  remains  constant.  The  increment  of 
available  energy  is 

k(v  +  u  +  duf  -  £  0  +  uf  =  (v  +  u)  du ; 

rt 

or  in  time  t,  I   (v  +  u)du (1) 

.'o 

The  speed  of  the  wind  being  supposed  to  be  periodic,  and  the  integration 
being  taken  over  a  sufficiently  long  period  of  time,  we  have 


and  thus  the  mechanical  advantage  may  be  reckoned  as 

vdu (2) 


In  order  that  this  may  have  a  finite  value,  v  must  vary  ;  the  principle 
being  that  to  get  the  most  advantage  v  must  be  great  when  du  is  positive, 
that  is  when  the  wind  is  freshening,  and  smaller  when  the  wind  is  failing. 
The  higher  velocity  required  to  meet  the  freshening  wind  is  to  be  obtained 
by  a  previous  fall  to  a  lower  level. 

As  an  example,  let  us  suppose  that  u  and  v  are  periodic,  so  that 
«  =  M0  +  ui  sin  pt,         v  =  va  +  vl  cos  (pt  +  e)  ; 


then  I  vdu  =  pu^v^  I  cos  pt  .  cos  (pt  +  e)  dt, 

and,  when  t  is  great,  Ivdu  =  ^pttt1vlcose  ............................  (3) 

The  mechanical  advantage  obtained  in  time  t  is  greatest  when  e  vanishes, 
i.e.,  when  du  and  v  are  in  the  same  phase.     This  mechanical  advantage  is  to 


1900]  THE   MECHANICAL  PRINCIPLES  OF   FLIGHT.  467 

be  set  against  the  frictional  and  other  losses  neglected  in  our  original  suppo- 
sition. Were  there  no  such  losses,  the  value  of  v,  or  of  the  elevation,  might 
continually  increase. 

This  example  shows  that  it  is  quite  possible  for  a  bird  moving  in  a  very 
natural  manner  against  a  strong  and  variable  wind  to  maintain  himself  and 
to  advance  over  the  ground  without  working  his  wings.  Observations  of  this 
kind  are  recorded  by  Mr  Huffaker.  It  will  be  understood,  of  course,  that 
a  bird,  not  being  interested  in  simplifying  the  calculation,  will  take  any 
advantage  that  offers  itself  of  the  internal  energy  of  the  wind  and  of  upward 
currents  in  order  to  attain  his  objects. 

In  the  preceding  discussions  we  have  assumed,  for  the  sake  of  simplicity, 
that  a  bird  or  a  flying  machine  is  able  to  glide  in  still  air  without  loss  of 
energy.  It  is  needless  to  say  that  the  truth  of  such  an  assumption  can, 
at  best,  be  only  approximate.  Apart  from  frictional  losses,  the  maintenance 
of  a  given  level  implies  the  continual  formation  of  a  downward  aerial  current, 
and  consequent  expenditure  of  energy.  We  have  next  to  consider  the  mag- 
nitude of  these  losses,  taking  the  case  of  a  plane  moving  at  a  uniform  speed. 
And,  in  the  first  instance,  we  shall  neglect  the  frictional  forces,  assuming  that 
the  reaction  of  the  air  upon  the  plane  is  truly  normal. 

Before  we  can  advance  a  step  in  the  desired  direction  we  must  know  how 
the  normal  pressure  upon  an  aeroplane  is  related  to  the  size  and  shape  of  the 
plane,  to  the  velocity  of  the  motion,  and  above  all  to  the  angle  between  the 
plane  and  the  direction  of  motion.  According  to  an  erroneous  theory,  to 
some  extent  sanctioned  by  Newton,  the  mean  pressure  would  depend  only 
upon  the  area  of  the  plane  and  the  resolved  part  of  the  velocity  in  a  direction 
perpendicular  to  the  plane.  If  V  be  the  velocity,  a  the  angle  between  V  and 
the  plane,  p  the  density  of  the  air  (or  other  fluid  concerned),  the  pressure  p 
would  be  given  by 

(4) 


That  this  formula  is  quite  erroneous,  especially  when  a  is  small,  has  long 
been  known*.  At  small  angles  the  pressure  is  more  nearly  proportional  to 
sin  a  than  to  sin2  a  and,  as  was  strongly  emphasized  by  Wenham  in  an  early 
and  important  paper  on  aerial  locomotion  f,  the  question  of  shape  and  pre- 
sentation is  by  no  means  indifferent.  In  the  case  of  an  elongated  shape 
moving  with  given  velocity  V,  and  at  a  given  small  inclination  a,  the  pressure 
is  much  greater  when  the  long  dimension  of  the  plane  is  perpendicular  than 
when  it  is  (nearly)  parallel  to  V. 

*  A  further  discussion  will  be  found  in  Phil.  Mag.  Vol.  n.  p.  430,  1876  ;  Scientific  Papers, 
Vol.  i.  p.  287;  and  in  Nature,  Vol.  XLV.  p.  108,  1892.     [Vol.  in.  p.  491.] 
t  Report  of  Aeronautical  Society,  1866,  p.  10. 

30—2 


468  THE   MECHANICAL   PRINCIPLES  OF   FLIGHT.  [257 

According  to  a  theoretical  formula  developed  on  the  basis  of  Kirchhoff's 
analysis  (Phil.  Mag.  loc.  cit.)  we  should  have  for  the  mean  pressure,  instead 
of  (4), 

Trsma  

v     4  +  TT  sm  a  H 

This  applies  strictly  to  motion  in  two  dimensions,  or  practically  to  the  case 
of  a  very  elongated  blade,  whose  length  is  perpendicular  to  V. 

At  perpendicular  incidence  (a  =  90°)  the  difference  between  (4)  and  (5)  is 
not  important ;  but  when  a  is  small,  the  value  of  p  in  (5)  may  be  enormously 
greater  than  the  corresponding  value  from  (4). 

As  regards  numerical  values,  if  we  use  C.G.S.  measure,  so  that  V  is 
measured  in  centimetres  per  second,  we  have  in  the  case  of  air  under  standard 
conditions  p  =  '00128,  and  p,  at  perpendicular  incidence,  measured  in  dynes 
per  square  centimetre,  is  according  to  (4), 

^  =  •000641^ (6) 

This  does  not  differ  greatly  from  the  data  given  in  engineering  tables.  To 
compare  with  Langley's  more  recent  experiments,  we  may  express  V  in 
metres  per  second  and  p  in  grams  weight  per  square  centimetre.  Thus 

p'  =  -0065F'2;      (7) 

while  the  mean  of  Langley's  numbers  gives 

p'  =  -0087F'2,   (8) 

about  30  per  cent,  greater.  The  difference  is  accounted  for,  at  any  rate 
partly,  by  the  suction,  which  experiment  shows  to  exist  at  the  back  of  the 
plate. 

As  regards  the  law  of  obliquity,  the  early  experiments  of  Vince  (1798) 
sufficed  to  show  that  the  effect  was  more  nearly  as  sin  a  than  as  sin2  a.  In 
recent  times  this  subject  has  been  very  thoroughly  investigated  by  Langley, 
who  has  examined  not  only  the  influence  of  obliquity,  but  also  of  the  shape 
and  presentation  of  the  plane.  His  results  for  the  case  to  which  (5)  relates 
indicate  an  even  greater  relative  effect  at  small  angles,  probably  referable  to 
the  back  suction.  A  laboratory  experiment  to  demonstrate  the  reality  of 
this  suction  was  described  in  one  of  the  papers  already  referred  to  (Nature, 
loc.  cit.). 

Experiments  upon  the  law  of  obliquity,  as  executed  for  the  case  of  air,  by 
Dines*  and  Langley f,  involve  cumbrous  and  costly  whirling  machines,  and 
if  made  in  the  open  are  greatly  embarrassed  by  wind.  An  apparatus  capable 

*  Proc.  Roy.  Soc.  June,  1890. 

t  Smithsonian  Contributions  to  Knowledge,  1891. 


1900]  THE   MECHANICAL  PRINCIPLES  OF   FLIGHT.  469 

of  working  in  the  laboratory,  or  as  a  lecture  illustration,  has  long  been  a 
desideratum.  With  the  aid  of  Mr  Gordon  I  have  recently  constructed  one 
which,  while  very  simple  and  inexpensive,  performs  sufficiently  well.  It  may 
be  regarded  as  a  kind  of  adjustable  windmill.  An  axis  of  hard  steel,  finely 
pointed  at  the  ends,  is  carried  by  agate  cups.  From  a  central  boss  six  spokes 
of  round  steel  project  symmetrically,  carrying  at  their  ends  six  similar  vanes 
of  tin-plate.  The  vanes  are  provided  with  projecting  sockets  of  brass  tubing, 
which  fit  the  spokes  somewhat  tightly,  but  yet  allow  the  vanes  to  be  rotated 
when  desired.  The  vanes  are  4  inches  long  and  !•£  inches  wide,  the  distance 
of  their  inner  ends  from  the  axis  being  about  3*7  inches.  The  whole  appa- 
ratus is  as  light  as  may  be  (about  120  gm.)  consistently  with  the  necessary 
rigidity. 

If  the  vanes  are  all  inclined  at  the  same  angle,  the  apparatus  works  like 
an  ordinary  windmill,  and  may  be  set  into  rapid  rotation  by  a  motion  through 
the  air  parallel  to  the  axis.  This  motion  may  take  place  either  in  a  horizontal 
or  in  a  vertical  direction.  If  means  were  provided  for  estimating  the  couple 
needed  to  prevent  rotation,  we  should  obtain  the  efficiency  of  the  vanes  at 
the  given  obliquity  and  speed.  Observations  at  the  same  speed  and  at  other 
obliquities  would  then  give  the  means  of  determining  the  law  of  obliquity. 

Such  a  procedure  would  be  analogous  to  that  adopted  in  former  ex- 
periments with  whirling  machines.  The  essential  feature  of  the  present 
method  consists  in  setting  some  of  the  vanes  to  compensate  others  inclined 
at  different  angles.  The  balance  of  effects  is  independent  of  the  speed  of  the 
wind,  so  long  as  it  is  uniform  over  the  whole  section  in  operation.  To  guard 
against  errors  that  might  arise  from  a  deficient  fulfilment  of  this  condition, 
I  have  preferred  so  to  arrange  that  opposite  vanes  were  inclined  always  at 
the  same  angle.  For  example,  two  pairs  of  opposite  vanes  might  be  set  so 
that  their  planes  make  an  angle  of  6°  with  the  axis.  The  remaining  pair 
of  opposite  vanes  would  then  be  set  at  a  greater  angle,  and  this  would  be 
varied  until  no  tendency  remained  to  turn  in  either  direction.  The  exact 
point  of  balance  could  be  inferred  either  from  the  absence  of  observable 
effect,  or  by  interpolation  from  equal  slight  effects  in  opposite  directions. 

As  has  been  suggested,  the  motion  itself  may  be  either  horizontal  or 
vertical.  Fair  results  may  be  obtained  indoors  at  a  walking  speed,  and  my 
first  idea  was  to  determine  balances  by  holding  the  wheel  overhead  while 
travelling  in  a  dog-cart  at  10  or  12  miles  per  hour.  But  when  the  axis  is 
horizontal,  much  time  is  lost  owing  to  the  necessity  of  readjusting  the  centre 
of  gravity  after  almost  every  shifting  of  the  vanes.  With  a  nearly  vertical 
motion  the  position  of  the  centre  of  gravity  is  of  less  consequence,  and  it 
was  found  that  very  good  results  could  be  arrived  at  by  somewhat  rapidly 
lowering  the  apparatus  while  held  in  the  hands  with  axis  vertical.  It  is 
possible  that  part  of  the  delicacy  obtained  in  this  way  is  due  to  a  partial 


470 


THE   MECHANICAL   PRINCIPLES   OF    FLIGHT. 


[257 


annulment  of  gravity  during  the  downward  acceleration  and  consequent 
diminution  of  frictional  effect  at  the  bearings. 

Some  of  the  observations  presently  to  be  discussed  were  made  in  this 
way,  but  in  most  of  them  the  arrangement  was  rather  different.  The  wheel 
was  removed  from  its  bearings  and  suspended  by  a  fine  wire,  whose  torsion 
was  insufficient  to  check  the  rotation  seriously.  The  wire  was  pulled  up 
vertically  by  a  cord  running  over  a  pulley  overhead.  Although  this  arrange- 
ment offered  some  advantages,  they  were  largely  neutralised  by  disturbances 
due  to  draughts;  and  it  is  probable  that  equally  good  balances  might  be 
obtained  by  the  simpler  method. 

According  to  an  old  and  long  discredited  law,  the  normal  pressure  upon 
a  vane  moving  through  the  air  at  given  speed  would  be  proportional  to  the 
square  of  the  sine  of  the  angle  (a)  between  the  plane  of  the  vane  and  the 
direction  of  motion.  The  resolved  part  of  this  in  the  direction  of  rotation 
would  be  sin2  a  cos  a,  which  expression  would  represent  the  efficiency  of  the 
vanes  of  our  mill  as  dependent  upon  the  angle  of  setting.  When  a  is  small, 
the  second  factor  is  of  little  importance.  A  very  simple  experiment  will 
now  decide  whether  the  law  of  sin2  a  is,  or  is  not,  an  approximation  to  the 
truth.  We  find,  in  fact,  that  four  vanes  set  at  6°  markedly  overpower  two 
vanes  set  at  9°,  whereas  according  to  the  law  of  sin2  a  the  reverse  should 
happen.  In  order  to  balance  the  four  vanes  at  6°,  the  two  vanes  need  to 
be  at  about  14|°. 

By  observations  of  this  kind  materials  are  collected  for  a  complete  plotting 
out  of  the  curve  of  efficiency.  The  efficiency  necessarily  vanishes  when 
a  =  0,  and  also  on  account  of  the  resolving  factor,  when  a  =  90°.  In  order  to 
balance  four  vanes  set  at  5°,  we  may  set  the  remaining  two  vanes  either  at 
10£°  or  at  about  58°.  The  efficiency  reaches  a  maximum  in  the  neighbour- 
hood of  27°.  The  results  are  shown  in  A  (Fig.  1),  or  in  the  second  column 
of  the  accompanying  table.  The  scale  of  the  ordinates  is,  of  course,  arbitrary. 
The  efficiency  for  5°  is  assumed  to  be  10. 


a 

Rotatory  E  fficiency 

Normal  Pressure 

a 

Rotatory  Efficiency 

Normal  Pressure 

0 

o-o 

0 

40 

27-0 

88 

5 

10-0 

25 

50 

23-5 

91 

10 

19-0 

48 

60 

19-0 

94 

15 

24-8 

64 

70 

13-2 

96 

20                  28-0 

75 

80 

6-9 

99 

25                  29-2 

80 

90 

o-o 

100 

30 

29-0 

84 

: 

In  order   to  deduce   the  normal  pressure,  the  results  for  the  rotatory 
efficiency  must  be  divided  by  cos  a,  and  accuracy  is  necessarily  lost  in  the 


1900] 


THE   MECHANICAL   PRINCIPLES   OF    FLIGHT. 


471 


case  of  the  larger  angles.  The  numbers  thus  arrived  at  are  plotted  in 
curve  B,  and  are  given  in  the  third  column  of  the  table,  reduced  so  as  to 
make  the  maximum  (at  90°)  equal  to  100.  As  regards  the  relative  pressures 
at  the  smaller  angles,  the  results  appear  to  be  at  least  as  accurate  as  those 
obtained  on  a  larger  scale  with  the  whirling  machine ;  but  the  reference  to 
the  pressure  operative  at  90°  is  probably  less  accurate.  The  principal  con- 
clusion that  at  small  angles  the  pressure  is  proportional  to  sin  a,  and  by  no 
means  to  sin2  a,  is  abundantly  established. 


472  THE   MECHANICAL   PRINCIPLES   OF    FLIGHT.  [257 

In  applying  these  results,  the  first  problem  which  suggests  itself  for 
solution  is  that  of  the  gliding  motion  of  an  aeroplane.  It  was  first  success- 
fully treated  by  Penaud*,  and  it  may  be  taken  under  slightly  different  forms. 
We  may  begin  by  supposing  the  motion  to  be  strictly  horizontal,  the  velocity 
being  V  and  the  inclination  of  the  plane  to  the  horizon  being  a.  Under  these 
circumstances  a  propelling  force  F  is  required,  which  we  suppose  to  act 
horizontally.  The  mean  pressure  upon  the  plane  we  will  denote  by  /cV'2sma, 
the  assumption  of  proportionality  to  sin  a  being  amply  sufficient  for  the  case 
of  small  angles,  with  which  alone  we  are  practically  concerned.  If  S  be  the 
area  of  the  plane,  the  whole  normal  force  is  /c$F2sina.  In  view  of  the 
smallness  of  a,  we  may  equate  this  to  the  weight  (W)  supported.  Thus 

W  =  /eSF2sina,    (9) 

also  F=KSV*am*a (10) 

If  F  be  independent  of  V,  as  approximately  in  the  method  of  rocket 
propulsion,  these  equations  show  at  once  that  there  is  no  limit  to  the  weight 
that  may  be  supported  by  a  given  F.  It  is  only  necessary  to  make  a  small 
enough,  and  to  take  V  large  enough  to  satisfy  (9). 

In  other  methods  of  propulsion  we  should  have  to  do  rather  with  the 
rate  (H)  at  which  energy  is  expended  than  with  the  force  F  itself.  The 
relation  is 

H  =  FV,    (11) 

so  that  in  place  of  (10)  H=  tcSV3  sin3  a (12) 

Or,  again,  since  in  many  cases  the  power  that  might  be  expended  is  pro- 
portional to  the  weight  lifted,  we  may  conveniently  write 

H=  WU.      (13) 

From  these  equations  we  derive 

W*         W 
V-      -  -  — 
~KSH~KSU'  '• 


and  it  is  possible  so  to  determine  F  and  a  that,  with  a  given  U  and  a  given 
S,  any  weight  W  can  be  supported.  As  W  increases,  F  must  be  greater  and 
a  smaller.  The  same  is  true,  in  an  enhanced  degree,  if  it  be  H  that  is  given 
in  place  of  U. 

According  to  what  has  been  shown  (6),  (7),  (8),  Fig.  1,  we  have  in  C.G.s. 
measure 

K  sin  5°  =  -25  x  -00085, 

so  that  «  =  '0024 (16) 

*  Societe  Philomathique  de  Paris,  1876;   Report  of  Aeronautical  Society,  1876.     See  also 
W.  Froude,  Glasgow  Proceedings,  VoL  xvm.  p.  65,  1891. 


1900]  THE   MECHANICAL   PRINCIPLES   OF    FLIGHT.  473 

In  the  case  of  a  very  elongated  plane  the  value  of  K  would  be  a  little 
higher.  We  must  remember  that  V  is  reckoned  in  centimetres  per  second, 
S  in  square  centimetres,  and  the  normal  force  in  dynes. 

The  conclusion  that  a  weight,  however  great,  may  be  supported  with  a 
given  S  and  a  given  U,  or  even  a  given  H,  is  unpractical  for  more  than  one 
reason.  There  must  be  a  limit  below  which  a  cannot  be  reduced,  if  only 
because  of  the  high  degree  of  instability  that  such  an  adjustment  must  have 
to  contend  with.  Another  important  matter  is  the  tangential  force  upon  the 
plane,  although  some  distinguished  experimenters  have  expressed  the  opinion 
that  it  is  negligible.  In  order  to  take  account  of  it,  we  may  add  to  the 
right-hand  member  of  (10)  a  term  proportional  to  V2,  but  independent  of  a. 
Thus  (12)  becomes 

H  =  WU=(KSsm2a  +  n)  V3,  .....................  (17) 

(9)  remaining  unchanged.     Eliminating  V,  we  find 


W 


We  may  apply  (18)  to  find  for  what  value  of  sin  a  the  quantity  t/"2  attains  a 
minimum.  By  the  ordinary  rules, 

sin-a^^g,    (19) 

and,  of  course,  this  value  of  sin3  a  must  be  small,  if  the  investigation  is  to  be 
applicable.  If  //-  vanish,  sin  a  diminishes  without  limit.  In  general  the 
minimum  value  of  U2  is  given  by 

WV>    »    *  ...(20) 


and  the  corresponding  value  of  F2  by 

^2  =  OA  ..A  ..i  vu (21) 


These  equations  show  that  the  necessary  work  depends  entirely  upon  /*,  and 
that  without  a  knowledge  of  this  element  no  numerical  conclusions  can  be 
arrived  at. 

It  might  be  supposed  that  /JL,  so  far  as  it  depends  upon  the  aeroplane, 
would  be  proportional  to  S,  but  this  relation  is  more  than  doubtful.  In  any 
case  of  a  practical  machine  there  must  at  any  rate  be  a  part  of  p  not 
proportional  to  8. 

It  may  be  well  to  recall  that  U  represents  the  velocity  at  which  a  weight 
equal  to  W  would  have  to  be  raised  in  order  to  do  work  equal  to  that  done 
by  the  propelling  force  F.  By  (20),  caeteris  paribus,  U  varies  as  /S>~*. 

We  may  now  pass  to  the  case  of  an  aeroplane  gliding  in  still  air,  the  path 
being  slightly  inclined  downwards.  If  0  be  the  small  angle  between  the 


474  THE   MECHANICAL   PRINCIPLES   OF   FLIGHT.  [257 

path  and  the  horizontal,  we  may  regard  the  component  of  gravity  in  this 
direction,  viz.,   W  sin  6,  as  the  propelling  force  F.     Thus 


.....................  (22) 

so  that  U=VsmO  ...............................  (23) 

The  same  equations  apply  as  before,  with  the  understanding  that  a,  being 
the  inclination  of  the  plane  to  the  direction  of  motion  through  the  air,  is  no 
longer  identical  with  the  inclination  of  the  plane  to  the  horizon.  The  latter 
angle,  reckoned  positive  when  the  leading  edge  is  downwards,  will  now  be 
denoted  by  (6  -  a). 

Introducing  (23)  into  (14),  (15),  we  get 

W  «SF2sm2tf 

V*  =  —~-r—6,        sma=  ---  -Wr—  -,      ............  (24) 

/cS  sm  0 

from  which  it  appears  that  whatever  may  be  the  values  of  W  and  8,  0  may 
still  be  as  small  as  we  please.  Thus,  if  frictional  forces  can  be  neglected,  a 
high  speed  is  all  that  is  required  in  order  to  glide  without  loss  of  energy. 
This  is  the  supposition  upon  which  we  discussed  the  manner  in  which  a  bird 
may  take  advantage  of  the  internal  work  of  the  wind  ;  and  we  see  that  the 
motion  of  the  bird  must  be  of  such  a  character  that  he  always  retains  a  high 
velocity  relatively  to  the  surrounding  air.  The  advantage  that  we  showed  to 
be  obtainable  must  be  set  against  losses  due  to  friction  and  to  imperfect 
fulfilment  of  the  condition  just  specified. 

When  frictional  forces  are  included  we  may  use  equation  (18),  merely 
substituting  V  sin  0  for  U.  The  problem  already  considered  of  making  U  a 
minimum  is  still  pertinent,  since  U  denotes  the  rate  of  vertical  descent.  By 
(19),  (20),  (21) 

3/A  .  na     U2     16/i,  /ocv 

-"'«-;&  ^'-Ti-ia  ................  (25) 

so  that,  0  and  a  being  small, 

a  =  f0,         0-a  =  £a  =  i0  ......................  (26) 

This  result,  due  to  Penaud,  shows  that  when  the  rate  of  vertical  descent  is 
slowest,  or  when  the  time  of  falling  a  given  height  is  greatest,  the  slope  of 
the  plane  to  the  horizon  is  downwards  in  front,  and  equal  to  one-quarter  of 
the  slope  of  the  line  of  motion.  The  actual  minimum  rate  of  vertical  descent 
is  given  by  (20).  This  rate  is  relative  to  still  air.  If  there  be  a  wind  having 
a  vertical  component  of  the  same  amount,  the  course  of  the  plane  -may  be 
horizontal. 

Another  slightly  different  minimum  problem  is  also  treated  by  Pe'naud, 
in  which  it  is  required  to  determine  how  far  it  is  possible  to  glide  while 


1900]  THE   MECHANICAL   PRINCIPLES   OF    FLIGHT.  475 

falling  through  a  given  vertical  height.     From  (9),  (17),  (23),  we  have  in 
general 

s\n0  =  S^«  +  ^KS  .........................  (27) 

sin  a. 

When  0  is  a  minimum  by  variation  of  a, 

sin  a  =  i  sin  0  =  VO*/*S)  ......................  (28) 

In  this  case  the  plane  bisects  the  angle  between  the  horizontal  and  the 
direction  of  motion. 

In  the  flying  machines  of  Penaud,  Langley,  and  Maxim,  the  propelling 
force  is  obtained  by  a  screw,  acting  like  the  screw-propeller  of  a  ship. 
A  rough  theory  of  this  action  is  easily  given  and  is  of  interest,  not  only  in 
the  application  to  the  horizontal  propulsion  of  an  aeroplane,  but  also  because 
a  screw  rotating  about  a  vertical  axis  may  be  used  for  direct  maintenance. 
The  latter  question  may  conveniently  be  considered  first. 

The  screw  is  supposed  to  maintain  a  weight  W  at  a  fixed  position  in  still 
air.  This  it  does  by  creating  a  downward  current  of  velocity  v.  If  $'  be  the 
area  of  section  of  the  current,  equal  to  that  swept  through  by  the  screw,  the 
volume  of  air  acted  upon  per  second  is  S'v,  and  the  momentum  generated 
per  second  is  S'v.pv,  or  S'pv2.  Hence 

W  =  S'pv*  ...............................  (29) 


Again,  the  kinetic  energy  generated  per  second  is  ^S'ptf;  so  that  if  U  be  the 
velocity  at  which  W  would  have  to  be  lifted  to  do  a  corresponding  amount  of 
work,  we  may,  neglecting  frictioiial  losses,  equate  the  above  to  UW.  Thus 


(30) 
From  (29),  (30),  $v=U, 


So  far  as  these  equations  are  concerned,  any  weight  can  be  maintained  by  a 
limited  expenditure  of  work,  but  the  smaller  the  power  available  the  larger 
must  be  the  section  of  the  stream  of  air,  and  consequently  of  the  screw,  or 
other  machinery,  by  which  the  air  is  set  in  motion.  Again  from  (31) 

(32) 

so  that  if  S'  be  given,  the  whole  power  required  varies  as  TP*. 

To  obtain  numbers  applicable  to  the  case  of  a  man  supporting  himself  in 
this  way  by  his  own  muscular  power,  we  take  in  c.G.s.  measure 

W  =68000  x  981,         U=15,        p  =  sh> 
thus  finding  S'  =  G'O  x  107  sq.  cm. 


476  THE   MECHANICAL  PRINCIPLES   OF   FLIGHT.  [257 

This  represents  the  cross-section  of  the  descending  column  of  air.  If  we 
equate  S'  to  ^ird2,  d  will  be  the  diameter  of  the  screw  required,  and  we  get 
d  =  90  metres.  It  is  to  be  observed  that  this  assumed  value  of  U  corresponds 
to  the  power  which  a  man  may  exercise  when  working  for  eight  hours  a  day. 
But  even  if  he  could  do  ten  times  as  much  for  a  few  minutes,  d  would  still 
amount  to  9  metres  ;  and  in  this  estimate  nothing  has  been  allowed  for  the 
weight  of  the  mechanism,  or  for  frictional  losses.  It  seems  safe  to  conclude 
that  a  man  will  never  support  himself  in  this  manner  by  his  own  muscular 
power. 

A  screw  works  to  better  advantage  when  it  has  a  forward  motion  through 
the  fluid,  for  then  a  larger  mass  comes  under  its  influence.  Let  us  suppose 
that  a  screw,  now  rotating  about  a  horizontal  axis,  is  advancing  through  still 
air  with  horizontal  velocity  V.  Also  let  v  be  the  actual  velocity  with  which 
the  column  of  air  leaves  it.  The  volume  acted  on  per  second  is  S'(V  +  v). 
If  F  be  the  propulsive  force 

F=S'p(V  +  v)v  ............................  (33) 

Again,  the  work  per  second  required  to  generate  the  kinetic  energy  of  the 
column  is 

%S'p(V+v)v*  ...............................  (34) 

The  whole  work  expended  per  second  (H')  is  accordingly 

)  .............  (35) 


When  V  is  great  compared  with  v,  the  right-hand  member  of  (35)  reduces  to 
its  first  term.  We  conclude  that  when  a  screw  advances  at  a  sufficiently 
rapid  rate,  the  energy  left  behind  in  the  fluid  is  negligible,  so  that  the  whole 
work  done  is  available  for  propulsion.  The  distinction  between  H  '  and  H, 
as  formerly  employed,  then  disappears. 

If  U  denote  the  rate  at  which  W  would  have  to  be  lifted  in  order  to  do 
the  work  actually  performed  by  the  machine,  we  may  now  take  from  (15),  as 
applicable  to  the  rapid  flight  of  an  aeroplane, 


(36) 


In  the  case  of  direct  maintenance  by  a  screw  rotating  about  a  vertical 
axis,  (31)  gives 


It  may  be  interesting  to  compare  the  powers  required  in  the  two  methods, 
especially  as  some  high  authorities  have  favoured  direct  maintenance,  without 


1900]  THE   MECHANICAL   PRINCIPLES    OF    FLIGHT.  477 

the  use  of  an  aeroplane,  as  the  more  economical.     The  ratio  of  the  values  of 
U  in  (36),  (37)  is 


or,  in  the  case  of  air,  since  K  =  '0024,  p  =  "0012, 

V(2  sin  a  .  S'/S)  ............................  (39) 

Since  a  may  be  made  small,  and  8  the  area  of  the  plane  may  be  a  large 
multiple  of  S'  the  area  swept  over  by  the  screw,  it  would  appear  that  the 
advantage  must  lie  with  the  aeroplane,  even  if  the  object  be  mere  main- 
tenance, and  not  a  rapid  transit  from  place  to  place. 

But  although  the  flying  machine  of  the  future  will,  as  it  appears  to  me, 
be  on  the  principle  of  the  aeroplane,  it  cannot  be  denied  that  the  method 
of  direct  maintenance  by  a  vertically  rotating  screw  offers  certain  present 
advantages.  Among  the  most  important  of  these  are  a  much  better  ensured 
stability,  and  less  danger  in  alighting  owing  to  the  absence  of  rapid  horizontal 
motion.  The  first  experiments  might  well  be  made  with  screws  driven  by 
electric  motors,  the  power  being  supplied  from  the  ground  by  means  of 
vertical  wires  30  or  40  feet  long.  In  this  way  the  necessary  experience 
would  be  easily  gained,  and  most  of  the  doubtful  points  settled,  before  a 
completely  self-contained  machine  was  attempted. 

In  natural  flight  revolving  mechanism  is  not,  and  apparently  could  not 
have  been,  used.  As  we  all  know,  a  bird  flying  horizontally  through  still 
air  performs  the  necessary  work  by  flapping  his  wings.  The  effect  of  a 
reciprocating  motion  in  modifying  the  action  of  an  aeroplane  was,  I  believe, 
first  considered  in  detail  by  Professor  M.  Fitzgerald*.  It  may  be  convenient 
to  give,  as  naturally  connected  with  the  foregoing,  an  outline  of  this  theory 
in  a  modified  form,  following  Professor  Fitzgerald  in  assimilating  the  wing 
to  a  simple  aeroplane,  upon  which  is  imposed  (without  rotation)  a  vertical 
reciprocating  motion. 

We  denote  by  u  the  horizontal  velocity  of  the  plane  supposed  uniform, 
by  v  the  vertical  velocity  at  time  t,  by  6  the  inclination  of  the  plane  to  the 
horizon  at  time  t,  while  S  and  W  denote  the  area  and  weight  as  before.  If 
we  assume  the  same  formula  for  the  pressure  as  before,  although  the 
application  is  now  to  an  unsteady  motion,  and  further  suppose  that  v/u 
and  0  are  always  small,  we  get  as  in  (9)  for  the  whole  normal  pressure  upon 
the  plane  at  time  t 

icS(u?  +  tf)(e  +  v/u),      ........................  (40) 

in  which  however  vz  in  (w2  +  v2)  may  be  omitted. 

*   Proc.  Roy.  Soc.  Vol.  LXIV.  p.  420,  1899. 


478  THE   MECHANICAL   PRINCIPLES    OF    FLIGHT.  [257 

We  now  assume  that  6  and  v  are  periodic,  for  example  that 

0=0a  +  01cospt,     ...........................  (41) 

v(u  =  j3cos(pt  +  €),    ................  ...........  (42) 

where  the  periodic  time  r  is  related  to  p  according  to 


At  this  stage  the  criticism  may  present  itself  that  the  assumed  motion 
involves  a  reaction  for  which  we  have  made  no  provision.  In  practice  the 
reaction  is  supplied  by  the  inertia  of  the  body  of  the  bird  to  which  the  wings 
are  attached.  The  difficulty  would  be  got  over  by  supposing  that  there  are 
several  planes  executing  similar  movements,  but  in  different  phases  regularly 
disposed.  It  seems  hardly  worth  while  to  complicate  the  present  investigation 
by  introducing  a  vertical  movement  of  the  weight. 

By  (40)  the  whole  pressure  at  time  t,  perpendicular  to  the  plane,  is 

KSu*{00  +  01cospt  +  l3cos(pt  +  €)}  ................  (43) 

Of  this  the  mean  value  is  to  be  equated  to  the  weight  W  supported,  so  that 

W  =KSu?e<>  ...............................  (44) 

The  horizontal  component  of  the  whole  pressure  at  time  t  is 

S.Ku*.{0  +  v/u}0,    ...........................  (45) 

and  of  this  the  mean  value  is  to  be  supposed  to  be  zero,  in  order  that  the 
plane  may  move  with  uniform  horizontal  velocity.  Thus 

6>o2  +  W  +  I/30J  cos  e  =  0  ......................  (46) 

Again,  if  WU  be  the  (mean)  rate  of  expenditure  of  work, 


*).  ...(47) 
If  we  eliminate  /3  between  (46),  (47),  we  get 

**1**'),  ............  (48) 


from  which  we  see  that  if  0l  be  given  (as  well  as  S,  W,  u),  U  is  least  when 
6  =  0,  viz.,  when  the  phase  of  maximum  vertical  velocity  coincides  with  the 
phase  of  greatest  inclination.  In  this  case  by  use  of  (44)  we  have 


.(49) 


If  we  regard  W,  S,  u  as  given,  the  smallest  value  of  U  corresponds  to  01  being 
large  in  comparison  with  00  which  is  given  by  (44)*. 

*  It  must  not  be  forgotten  that  Ol  itself  has  been  assumed  to  be  small. 


1900]  THE    MECHANICAL   PRINCIPLES   OF    FLIGHT.  479 

The  smallest  value  is 


The  work  required  to  be  done  is  here  the  same  function  of  S,  W,  and  the 
horizontal  velocity  as  was  found  in  (14),  where  V  has  the  meaning  here 
assigned  to  u. 

We  see  from  (46)  that,  under  the  circumstances  supposed,  0l  +  ft  is 
numerically  small  in  comparison  with  00,  and  a  fortiori  in  comparison  with 
0j.  Accordingly  the  forward  edge  of  the  plane  is  inclined  downwards  when 
the  motion  of  the  plane  is  downwards. 

As  regards  the  pressure,  it  is  by  (43)  proportional  to 


in  which  the  second  term  is  relatively  small.  The  pressure  acts  always  upon 
the  under  side  of  the  plane,  and  the  weight  is  approximately  supported  in  all 
phases. 


258. 

ON  THE  LAW  OF  RECIPROCITY  IN  DIFFUSE  REFLEXION. 
[Philosophical  Magazine,  XLIX.  pp.  324,  325,  1900.] 

IN  the  current  number  of  the  Philosophical  Magazine  (Vol.  XLIX.  p.  199) 
Dr  Wright  discusses  the  question  of  the  amount  of  light  diffusely  reflected 
from  a  given  area  of  a  matt  surface  as  dependent  upon  the  angle  of  incidence 
(i)  and  the  angle  of  emission  (e).  According  to  Lambert's  law  the  function 
of  i  and  e  is 

cos  i  cos  e ; (1) 

and  this  law,  though  in  the  present  case  without  theoretical  foundation, 
appears  approximately  to  represent  the  facts.  The  question  may  indeed  be 
raised  whether  it  is  possible  so  to  define  an  ideally  matt  surface  that 
Lambert's  law  may  become  strictly  applicable. 

The  conclusion  drawn  by  Dr  Wright  from  his  experiments  with  com- 
pressed powders  upon  which  I  desire  to  comment  is  that  numbered  (4)  in  his 
resume  of  results,  viz.  "  A  law  for  the  intensity  of  reflected  scattered  light 
cannot  be  symmetric  in  reference  to  i  and  e."  It  appears  to  me  that  this 
statement  is  in  contradiction  to  a  fundamental  principle  of  reciprocity,  of 
such  generality  that  escape  from  it  is  difficult.  This  principle  is  discussed 
at  length  in  my  book  on  the  Theory  of  Sound,  §  109.  Its  application  to  the 
present  question  may  be  thus  stated : — Suppose  that  in  any  direction  (i)  and 
at  any  distance  r  from  a  small  surface  (S)  reflecting  in  any  manner  there  be 
situated  a  radiant  point  (A)  of  given  intensity,  and  consider  the  intensity  of 
the  reflected  vibrations  at  any  point  B  situated  in  direction  e  and  at  distance 
r'  from  S.  The  theorem  is  to  the  effect  that  the  intensity  is  the  same  as  it 
would  be  at  A  if  the  radiant  point  were  transferred  to  B*.  The  conclusion 
follows  that  whatever  may  be  its  character  in  other  respects,  the  function 
of  i  and  e  which  represents  the  intensity  of  the  reflected  scattered  light  must 
be  symmetrical  with  respect  to  these  quantities. 

The  actual  departures  from  the  reciprocal  relation  found  by  Dr  Wright 
were  not  very  large,  and  they  may  possibly  be  of  the  nature  of  experimental 
errors.  In  any  case  it  seems  desirable  that  the  theoretical  difficulty  in 
accepting  Dr  Wright's  conclusion  should  be  pointed  out. 

*  I  have  not  thought  it  necessary  to  enter  into  questions  connected  with  polarization,  but 
a  more  particular  statement  could  easily  be  made. 


259. 


ON  THE  VISCOSITY  OF  GASES  AS  AFFECTED  BY 
TEMPERATURE. 

[Proceedings  of  the  Royal  Society,  LXVII.  pp.  137 — 139,  1900.] 

A  FOEMER  paper*  describes  the  apparatus  by  which  I  examined  the  in- 
fluence of  temperature  upon  the  viscosity  of  argon  and  other  gases.  I  have 
recently  had  the  opportunity  of  testing,  in  the  same  way,  an  interesting 
sample  of  gas  prepared  by  Professor  Dewar,  being  the  residue,  uncondensed 
by  liquid  hydrogen,  from  a  large  quantity  collected  at  the  Bath  springs.  As 
was  to  be  expected -f-,  it  consists  mainly  of  helium,  as  is  evidenced  by  its 
spectrum  when  rendered  luminous  in  a  vacuum  tube.  A  line,  not  visible 
from  another  helium  tube,  approximately  in  the  position  of  D5  (Neon)  is  also 
apparent]:. 

The  result  of  the  comparison  of  viscosities  at  about  100°  C.  and  at  the 
temperature  of  the  room  was  to  show  that  the  temperature  effect  was  the 
same  as  for  hydrogen. 

*  Roy.  Soc.  Proc.  Vol.  LXVI.  (1900),  p.  68.     [Vol.  iv.  p.  452.] 

t  Roy.  Soc.  Proc.  Vol.  LIX.  (1896),  p.  207 ;  Vol.  LX.  (1896),  p.  56.     [Vol.  iv.  p.  225.] 
£  I  speak  doubtfully,  because  to  my  eye  the  interval  from  D1  to  D3  (helium)  appeared  about 
equal  to  that  between  D3  and  the  line  in  question,  whereas,  according  to  the  measurements  of 
Eamsay  and  Travers  (Roy.  Soc.  Proc.  Vol.  LXIII.  (1898),  p.  438),  the  wave-lengths  are— 

D! 5895-0 

Z)2 5889-0 

D3 5875-9 

D5 5849-6, 

so  that  the  above-mentioned  intervals  would  be  as  19 -1 : 26 -3.  [June  23. — Subsequent  observations 
with  the  aid  of  a  scale  showed  that  the  intervals  above  spoken  of  were  as  20  : 21.  According  to 
this  the  wave-length  of  the  line  seen,  and  supposed  to  correspond  to  D5,  would  be  about  5855 
on  Rowland's  scale,  where  D1  =  5896-2,  D2  =  5890-2,  Z>3  =  5876-0.]  I  may  record  that  the 
refractivity  of  the  gas  now  under  discussion  is  0-132  relatively  to  air. 

R.     IV.  31 


482        ON  THE   VISCOSITY   OF  GASES   AS   AFFECTED   BY  TEMPERATURE.        [259 

In  the  former  paper  the  results  were  reduced  so  as  to  show  to  what 
power  (n)  of  the  absolute  temperature  the  viscosity  was  proportional. 


n 

c 

Air 

0754 

111-3 

Oxygen 

0-782 

128-2 

Hydrogen  ) 

0-681 

72-2 

Helium     ( 
Argon              .    ... 

0-815 

150-2 

Since  practically  only  two  points  on  the  temperature  curve  were  ex- 
amined, the  numbers  obtained  were  of  course  of  no  avail  to  determine 
whether  or  no  any  power  of  the  temperature  was  adequate  to  represent  the 
complete  curve.  The  question  of  the  dependence  of  viscosity  upon  tempe- 
rature has  been  studied  by  Sutherland*,  on  the  basis  of  a  theoretical 
argument  which,  if  not  absolutely  rigorous,  is  still  entitled  to  considerable 
weight.  He  deduces  from  a  special  form  of  the  kinetic  theory  as  the  function 
of  temperature  to  which  the  viscosity  is  proportional 


....(I) 


c  being  some  constant  proper  to  the  particular  gas.  The  simple  law  #*, 
appropriate  to  "  hard  spheres,"  here  appears  as  the  limiting  form  when  6  is 
very  great.  In  this  case,  the  collisions  are  sensibly  uninfluenced  by  the 
molecular  forces  which  may  act  at  distances  exceeding  that  of  impact. 
When,  on  the  other  hand,  the  temperature  and  the  molecular  velocities  are 
lower,  the  mutual  attraction  of  molecules  which  pass  near  one  another  in- 
creases the  number  of  collisions,  much  as  if  the  diameter  of  the  spheres  was 
increased.  Sutherland  finds  a  very  good  agreement  between  his  formula  (1) 
and  the  observations  of  Holman  and  others  upon  various  gases. 

If  the  law  be  assumed,  my-  observations  suffice  to  determine  the  values 
of  c.  They  are  shown  in  the  table,  and  they  agree  well  with  the  numbers 
for  air  and  oxygen  calculated  by  Sutherland  from  observations  of  Obermayer. 


*  Phil.  Mag.  Vol.  xxxvi.  (1893),  p.  507. 


260. 


REMARKS   UPON  THE  LAW  OF  COMPLETE   RADIATION. 


[Philosophical  Magazine,  XLIX.  pp.  539,  540,  1900.] 

BY  complete  radiation  I  mean  the  radiation  from  an  ideally  black  body, 
which  according  to  Stewart*  and  Kirchhoff  is  a  definite  function  of  the 
absolute  temperature  0  and  the  wave-length  \.  Arguments  of  (in  my  opinion^) 
considerable  weight  have  been  brought  forward  by  Boltzmann  and  W.  Wien 
leading  to  the  conclusion  that  the  function  is  of  the  form 

0*<j>(0\)d\,   .................................  (1) 

expressive  of  the  energy  in  that  part  of  the  spectrum  which  lies  between 
X  and  A,  +  d\.  A  further  specialization  by  determining  the  form  of  the 
function  </>  was  attempted  later  J.  Wien  concludes  that  the  actual  law  is 


(2) 


in  which  Cj  and  c2  are  constants,  but  viewed  from  the  theoretical  side  the 
result  appears  to  me  to  be  little  more  than  a  conjecture.  It  is,  however, 
supported  upon  general  thermodynamic  grounds  by  Planck  §. 

Upon  the  experimental  side,  Wien's  law  (2)  has  met  with  important 
confirmation.  Paschen  finds  that  his  observations  are  well  represented, 
if  he  takes 

c2  =  14,455, 


*  Stewart's  work  appears  to  be  insufficiently  recognized  upon  the  Continent.     [See  Phil.  Mag. 
i.  p.  98,  1901 ;  p.  494  below.] 

t  Phil.  Mag.  Vol.  XLV.  p.  522  (1898). 
$  Wied.  Ann.  Vol.  LVIII.  p.  662  (1896). 
§  Wied.  Ann.  Vol.  i.  p.  74  (1900). 

31—2 


484  REMARKS   UPON   THE   LAW   OF   COMPLETE   RADIATION.  [260 

6  being  measured  in  centigrade  degrees  and  \  in  thousandths  of  a  millimetre 
(//,).  Nevertheless,  the  law  seems  rather  difficult  of  acceptance,  especially 
the  implication  that  as  the  temperature  is  raised,  the  radiation  of  given  wave- 
length approaches  a  limit.  It  is  true  that  for  visible  rays  the  limit  is  out 
of  range.  But  if  we  take  \  =  60  /JL,  as  (according  to  the  remarkable  researches 
of  Rubens)  for  the  rays  selected  by  reflexion  at  surfaces  of  Sylvin,  we  see  that 
for  temperatures  over  1000°  (absolute)  there  would  be  but  little  further 
increase  of  radiation. 

The  question  is  one  to  be  settled  by  experiment  ;  but  in  the  meantime 
I  venture  to  suggest  a  modification  of  (2),  which  appears  to  me  more  probable 
d  priori.  Speculation  upon  this  subject  is  hampered  by  the  difficulties 
which  attend  the  Boltzmann-Maxwell  doctrine  of  the  partition  of  energy. 
According  to  this  doctrine  every  mode  of  vibration  should  be  alike  favoured  ; 
and  although  for  some  reason  not  yet  explained  the  doctrine  fails  in  general, 
it  seems  possible  that  it  may  apply  to  the  graver  modes.  Let  us  consider 
in  illustration  the  case  of  a  stretched  string  vibrating  transversely.  According 
to  the  Boltzmann-Maxwell  law  the  energy  should  be  equally  divided  among 
all  the  modes,  whose  frequencies  are  as  1,  2,  3,  ....  Hence  if  k  be  the 
reciprocal  of  X,  representing  the  frequency,  the  energy  between  the  limits 
k  and  k  +  dk  is  (when  k  is  large  enough)  represented  by  dk  simply. 

When  we  pass  from  one  dimension  to  three  dimensions,  and  consider 
for  example  the  vibrations  of  a  cubical  mass  of  air,  we  have  (Theory  of  Sound, 
§  267)  as  the  equation  for  k2, 


where  p,  q,  r  are  integers  representing  the  number  of  subdivisions  in  the 
three  directions.  If  we  regard  p,  q,  r  as  the  coordinates  of  points  forming 
a  cubic  array,  k  is  the  distance  of  any  point  from  the  origin.  Accordingly 
the  number  of  points  for  which  k  lies  between  k  and  k  +  dk,  proportional 
to  the  volume  of  the  corresponding  spherical  shell,  may  be  represented 
by  k*dk,  and  this  expresses  the  distribution  of  energy  according  to  the 
Boltzmann-Maxwell  law,  so  far  as  regards  the  wave-length  or  frequency. 
If  we  apply  this  result  to  radiation,  we  shall  have,  since  the  energy  in  each 
mode  is  proportional  to  0, 

0k*dk,    ....................................  (3) 

or,  if  we  prefer  it, 

e\~*d\  ..................................  (4) 

It  may  be  regarded  as  some  confirmation  of  the  suitability  of  (4)  that  it  is  of 
the  prescribed  form  (1). 

The  suggestion  is  that  (4)  rather  than,  as  according  to  (2), 

\-*d\    ....................................  (5) 


1900]  REMARKS   UPON    THE   LAW  OF   COMPLETE   RADIATION.  485 

may  be  the  proper  form  when  X0  is  great*.     If  we  introduce  the  exponential 
factor,  the  complete  expression  will  be 

c10\-4e-c«/A»dX (6) 

If,  as  is  probably  to  be  preferred,  we  make  k  the  independent  variable, 
(6)  becomes 

Cl0k2e-c*edk (7) 

Whether  (6)  represents  the  facts  of  observation  as  well  as  (2)  I  am  not  in 
a  position  to  say.  It  is  to  be  hoped  that  the  question  may  soon  receive 
an  answer  at  the  hands  of  the  distinguished  experimenters  who  have  been 
occupied  with  this  subject. 

*  [1902.  This  is  what  I  intended  to  emphasize.  Very  shortly  afterwards  the  anticipation 
above  expressed  was  confirmed  by  the  important  researches  of  Rubens  and  Kurlbaum  (Drude 
Ann.  iv.  p.  649,  1901),  who  operated  with  exceptionally  long  waves.  The  formula  of  Planck, 
given  about  the  same  time,  seems  best  to  meet  the  observations.  According  to  this  modification 

of  Wien's  formula,  e~c^9  in  (2)  is  replaced  by  l-^(eC2/Xfl-l).     When  X0  is  great,  this  becomes 
X0/c2,  and  the  complete  expression  reduces  to  (4).] 


261. 


ON  APPROXIMATELY  SIMPLE   WAVES. 


[Philosophical  Magazine,  L.  pp.  135—139,  1900.] 


THE  _first_question  that  arises  is  as  to  the  character  of  absolutely  simple 
waves;  and  here  "it  may  be  well  to  emphasize  that  a  simple  vibration  implies 
infinite  continuance,  and  does  not  admit  of  variations  of  phase  or  amplitude, 
To  suppose,  as  is  sometimes  done  in  optical  speculations,  that  a  train  of  simple 
waves  may  begin  at  a  given  epoch,  continue  for  a  certain  time  involving  it 
may  be  a  large  number  of  periods,  and  ultimately  cease,  is  a  contradiction 
in  terms*."  A  like  contradiction  is  involved  if  we  speak  of  unpolarized  light 
as  homogeneous,  really  homogeneous  light  being  necessarily  polarized. 

This  much  being  understood,  approximately  simple  waves  might  be 
defined  as  waves  which  for  a  considerable  succession  deviate  but  little  from 
a  simple  train.  Under  this  definition  large  changes  of  amplitude  and  fre- 
quency would  not  be  excluded,  provided  only  that  they  entered  slowly  enough. 
More  frequently  further  limitation  would  be  imposed,  and  approximately 
simple  waves  would  be  understood  to  mean  waves  which  for  a  considerable 
succession  can  be  approximately  identified  with  a  simple  train  of  given 
frequency,  if  not  of  given  amplitude.  But  the  phase-f  of  the  simple  train 
approximately  representing  the  given  waves  would  vary  from  place  to  place, 
slowly  indeed  but  to  any  extent. 

Thus  if  we  take,  as  analytically  expressing  the  dependence  of  the  displace- 
ment upon  time, 

^,     ...........................  (1) 


*  Theory  of  Sound,  2nd  ed.  §  65  a,  1894. 

+  What  is  here  called  for  brevity  the  phase  is  more  properly  the  deviation  of  phase  from  that 
of  an  absolutely  simple  train  of  waves. 


1900]  ON  APPROXIMATELY  SIMPLE  WAVES.  487 

where  H  and  K  are  slowly  varying  functions  of  t,  the  frequency  may  be 
regarded  as  constant,  while  the  amplitude  \/(H2  +  K2)  and  the  phase 
tan"1  (K/H)  vary  slowly  but  without  limit.  It  scarcely  needs  to  be  pointed 
out  that  a  slow  uniform  progression  of  phase  is  equivalent  to  a  small  change 
of  frequency. 

(  /  In  one  important  class  of  cases  the  phase  remains  constant  and  then, 
since  a  constant  addition  to  t  need  not  be  regarded,  (1)  is  sufficiently  repre- 
sented by 

H  cospt (2) 

simply.     If  the  changes  of  amplitude  are  periodic,  we  may  write 

H  =  H0  +  Hl  cos  qt  +  H,'  sin  qt  +  H2  cos  2qt  +  H2'  sin  2qt  +  . . . ,    . .  .(3) 

in  which  q  is  supposed  to  be  small.  The  vibration  (2)  is  then  always 
equivalent  to  a  combination  of  simple  vibrations  of  frequencies  represented  by 

p,      p  +  q,      p  —  q,      p  +  2<£,      p  —  *2q,  &c. 


Under  this  head  may  be  mentioned  the  case  of  ordinary  beats,  so  familiar 
in  Acoustics.     Here 

H=H1cosqt,      ..............................  (4) 


and  Hcospt=%Hlcos(p  +  q)t  +  ^Hlcos(p-q)t  .............  (5) 

It  may  be  observed  that  although  the  phase  is  regarded  as  constant,  the 
change  of  sign  in  the  amplitude  has  the  same  effect  as  an  alteration  qf_  _ 
phase  of  180°. 

Another  important  example  is  that  of  intermittent  vibrations.     If  we  put 
H='2(l  +  cosqt\  ..............................  (6) 

the  amplitude  is  always  of  one  sign,  and 

H  cos  pt  =  2  cos  pt  +  cos  (p  +  q)  t  +  cos  (p  —  q)  t  ..........  (7) 

Three  simple  vibrations  are  here  required  to  represent  the  effect. 
Again  (Theory  of  Sound,  §  65  a),  if 

(8) 


we  have 

H  cospt  =  |  cospt  +  cos  (p  +  q)  t  +  cos  (p  —  q)  t 

+  \cos(p  +  2q)t  +  $cos(p-'2q)t  .......  (9) 

If  K  also  be  variable  and  periodic  in  the  same  period  as  H,  so  that  . 

K  =  Ko  +  Kl  cos  qt  +  KI  sin  qt  +  K*  cos  2qt  +  K2'  sin  2qt  +  .  .  .  ,  .  .  .(10) 
we  have  the  most  general  periodicity  expressed  when  we  substitute  these 


488  ON   APPROXIMATELY   SIMPLE   WAVES.  [261 

values  in  (1);  and  the  general  conclusion  as  to  the  periods  of  the  simple 
vibrations  required  to  represent  the  effect  remains  undisturbed. 

If  K  and  H  vary  together  in  such  a  manner  that  the  amplitude  \/(H*  +  K2) 
remains  constant,  the  sole  variation  is  one  of  phase.  My  object  at  present 
is  to  call  attention  to  this  class  of  cases,  so  far  as  I  know  hitherto  neglected, 
unless  an  example  (Phil.  Mag.  xxxiv.  p.  409,  1892  [Vol.  iv.  p.  16])  in 
which  an  otherwise  constant  amplitude  is  periodically  and  suddenly  reversed 
be  considered  an  exception. 

If  we  take 

H  =  cos  (a  sin  qt),         K  =  sin  (a  sin  qt),  ...............  (11) 

H  and  K  are  of  the  required  periodicity,  and  the  condition  of  a  constant 
amplitude  is  satisfied.     In  fact  (1)  becomes 

cos  (pt  —  a.  sin  qt)  ........................  ____  (12) 

Now,  since 

eiacose  =  JQ  (a)  +  2^  («)  cos  6  +  2»V,  (a)  cos  2^  +  ... 

+  2iV«(a)cosn0+..., 

we  have 

eiasmqt  =  J0  +  2tV1  sin  qt  +  2JZ  cos  2qt  +  2iJs  sin  3qt 

+  2/4  cos  4^+...,  ............  (13) 

and  thus 

cos  (a  sin  qt)  =  J0  (a)  +  2/2  (a)  cos  2qt  +  2/4  (a)  cos  4>qt  +  .  .  .  ,  .  .  .(14) 
sin  (a  sin  qt)  =  2Jt  (a)  sin  qt  +  2/3  (a)  sin  3gtf  +  ...  ,     ............  (15) 

where   J0,   J1}  &c.  denote  (as  usual)  the  Bessel's  functions  of  the  various 
orders.     In  the  notation  of  (3).  and 


H,  =  #3  =  ...  =,0,        #/  =  #/  =  #,'=...=<), 
H0  =  J0(a\        H2  =  2J2(a),        fT4=2J4(a),     &c., 
K0  =  K,  =  K2=  ...  =  0,        Kl  =  K^  =  ...  =  0, 
JST1'  =  2J1(a),         K3'=2J3(«),         K5'  =  2J5(a),     &c. 
Accordingly  (12),  expressed  as  a  combination  of  simple  waves,  is 
«/0  (a)  cos  pt  +  Jz  (a)  {cos  (p-  2g)  t  +  cos  (p  +  2q)  t] 


+  /j  (a)  {cos  (p-q)t-  cos  (p  +  q)  t} 

+  J3(a){cos(p-3q)t-cos(p+3q)t}  +  .............  (16) 


1900] 


ON   APPROXIMATELY   SIMPLE   WAVES. 


489 


n 

<M3) 

/.(«) 

Jn  (12) 

Jn  (18) 

Jn  (24) 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 

-  -26005 
+  -33906 
+  -48609 
+  -30906 
+  -13203 
+  -04303 
+  •01  139 
+  -00255 
+  -00049 
+  -00008 
+  -00001 

+  -15065 
-  -27668 
-  -24287 
+  •11477 
+  •35764 
+  -36209 
+  -24584 
+  -12959 
+  -05653 
+  -02117 
+  -00696 
+  -00205 
+  '00055 

+  -04769 
-  -22345 
-  -08493 
+  •19514 
+  -18250 
-  -07347 
-  -24372 
-•17025 
+  -04510 
+  -23038 
+  -30048 
+  -27041 
+  -19528 

-  -01336 
-  -18799 
-  -00753 
+  -18632 
+  -06964 
-  -15537 
-  -15596 
+  -05140 
+  -19593 
+  -12276 
-•07317 
-  -20406 
•17624 

-  -05623 
-  -15404 
+  -04339 
+  •16127 
-  -00308 
-  -16230 
-  -06455 
+  -13002 
+  -14039 
-  -03643 
-  '16771 
-  -10333 
+  -07299 

13 
14 



+  -00013 
+  -00003 

+  •12015 
+  -06504 

-  -03092 
+  •13157 

+  •17632 
+  •11803 

15 

+  -00001 

+  -03161 

+  '23559 

-  -03863 

16 

+  •01399 

+  -26108 

-  -16631 

17 

+  '00570 

+  •22855  ' 

-  -18312 

18 

+  -00215 

+  •17063 

•09311 

19 

+  -00076 

+  •11271 

+  -04345 

20 

+  -00025 

+  '06731 

+  •16191 

21 

+  -00008 

+  -03686 

+  '22640 

22 

+  •00002 

+  -01871 

+  -23429 

23 

24 

+  -00001 

+  -00886 
+  '00395 

+  •2031  3 
+  -15504 

25 

+  -00166 

+  -10695 

26 

+  -00066 

+  -06778 

27 

+  -00025 

+  -03990 

28 

+  •00009 

+  -02200 

29 

+  -00003 

+  •01143 

30 

+  -00001 

+  -00563 

31 

+  -00263 

32 

+  •001  18 

33 

+  •00050 

34 

+  -00021 

35 

+  -00008 

36 

+  -00003 

37 







+  -00001 

490  ON  APPROXIMATELY   SIMPLE  WAVES.  [261 

If  a,  representing  the  maximum  disturbance  of  phase,  be  small,  we  may 
write  approximately 


while  1/3  &c.  are  of  higher  powers  in  a  than  a2.     Thus  if  we  stop  at  the  first 
power  of  a,  we  are  concerned  only  with  the  multiples  of  t  represented  by 

p,        p-q,        p  +  q; 
while  if  we  include  a2  we  have 

p,        p-q,        p  +  q,         p-2q,        p  +  2q. 

But  when  a  is  not  small,  the  convergence  is  slow,  and  a  large  number 
of  terms  will  be  required  even  for  a  moderately  close  approximation.  The 
preceding  table,  due  to  Meissel,  is  condensed  from  Gray  and  Mathew's  Bessel's 
Functions.  So  far  as  TT  can  be  identified  with  3,  the  values  of  a  equal  to 
3,  6,  12,  18,  24  correspond  to  maximum  deviations  of  phase  (in  both  directions) 
equal  to  \,  1,  2,  3,  4  periods  respectively.  It  appears  that,  the  largest  value 
of  Jn  (a)  occurs  for  a  value  of  n  somewhat  less  than  a.  Indeed,  it  is  at  once 
evident  from  (12)  that  frequencies  in  the  neighbourhood  of  p  ±  qa.  will  be 
important  elements. 


262. 


ON  A  THEOREM  ANALOGOUS  TO  THE  VIRIAL  THEOREM. 


[Philosophical  Magazine,  L.  pp.  210—213,  1900.] 

As  an  example  of  the  generality  of  the  theorem  of  Clausius,  Maxwell* 
mentions  that  "  in  any  framed  structure  consisting  of  struts  and  ties,  the  sum 
of  the  products  of  the  pressure  in  each  strut  into  its  length,  exceeds  the 
sum  of  the  products  of  the  tension  of  each  tie  into  its  length,  by  the  product 
of  the  weight  of  the  whole  structure  into  the  height  of  its  centre  of  gravity 
above  the  foundations."  It  will  be  convenient  to  sketch  first  the  proof  of 
the  purely  statical  theorem  of  which  the  above  is  an  example,  and  afterwards 
of  the  corresponding  statical  applications  of  the  analogue.  The  proof  of  the 
general  dynamical  theorem  will  then  easily  follow. 

If  X,  Y,  Z  denote  the  components,  parallel  to  the  axes,  of  the  various 
forces  which  act  upon  a  particle  at  the  point  x,  y,  z,  then  since  the  system 
is  in  equilibrium, 


If  we  multiply  these  equations  by  x,  y,  z  respectively,  and  afterwards  effect 
a  summation  over  all  the  particles  of  the  system,  we  obtain  a  result  which 
may  be  written 

2£]  =  0  ......................  (1) 


The  utility  of  the  equation  depends  upon  an  alteration  in  the  manner 
of  summation,  and  in  particular  upon  a  separation  of  the  forces  R  (considered 
positive  when  repellent)  which  act  mutually  between  two  particles  along  their 

*  Nature,  Vol.  x.  p.  477,  1874  ;  [Maxwell's]  Scientific  Papers,  Vol.  n.  p.  410. 


492       ON  A  THEOREM  ANALOGOUS  TO  THE  VIRIAL  THEOREM.       [262 

line  of  junction  p.     If  x,  y,  z  and  x  ',  y',  z'  be  the  coordinates  of  the  particles, 
we  have  so  far  as  regards  the  above-mentioned  forces, 


or  with  summation  over  every  pair  of  particles  HRp.     The  complete  equation 
may  now  be  written 

0,    .....................  (2) 


where  in  the  first  summation  X,  Y,  Z  represent  the  components  of  the 
external  forces  operative  at  the  point  x,  y,  z.  In  Maxwell's  example  the  only 
external  forces  are  the  weights  of  the  various  parts  of  the  system  (supposed 
to  be  concentrated  at  the  junctions  of  the  struts  and  ties),  and  the  reactions 
at  the  foundations. 

The  analogous  theorem,  to  which  attention  is  now  called,  is  derived  in 
a  similar  manner  from  the  equally  evident  equation 


(3) 


We  have  to  extract  from  the  summation  on  the  left  the  force  R  mutually 
operative  between  the  particles  at  x,  y,  z  and  at  x,  y',  z'  ;  and  we  shall  limit 
ourselves  to  the  case  of  two  dimensions.  If  X,  Y  be  the  components  of  force 
acting  upon  the  latter  particle,  p  the  distance  between  the  particles,  and 
<p  the  inclination  of  p  to  the  axis  of  x,  we  have 


so  that   if  now  X,  Y  represent   the   total   external   force   acting   at   x,  y, 
(3)  becomes 

2£  =  0,  .....................  (4) 


where  the  first  summation  extends  to  every  particle  and  the  second  to  every 
pair  of  particles. 

If  the  external  force  at  x,  y  be  P  and  be  inclined  at  an  angle  a,  we  have 

X  =  Pcosa,         F=Psina; 
so  that,  if  x  =  r  cos  8,     y  =  r  sin  6     as  usual,  (4)  may  be  written 

0  ...................  (5) 


As  simple  examples  of  these  equations,  consider  the  square  framework 
with  one  diagonal  represented  in  Figs.  1  and  2,  and  take  the  coordinate  axes 
parallel  to  the  sides  of  the  square.  Since  sin  20  =  0  for  all  four  sides  of 
the  square,  the  only  R  that  occurs  is  that  which  acts  along  the  diagonal 
where  sin  20  =  -  1.  In  Fig.  1  opposed  forces  P  act  at  the  middle  points  of 
the  sides,  but  since  in  each  case  6  +  a  =  0,  the  terms  containing  P  disappear. 
Hence  R  =  0. 


1900] 


ON  A  THEOREM  ANALOGOUS  TO  THE  VIRIAL  THEOREM. 


493 


In  Fig.  2,  where  external  forces  P  act  diagonally  at  the  unconnected 
corners,  sin  (6  +  a)  =  —  1,  and  since  p  =  2r,  R  =  —  P,  signifying  that  the 
diagonal  piece  acts  as  a  tie  under  tension  P.  In  neither  case  would  the 
weight  of  the  members  disturb  the  conclusion. 

Fig.  1.  Fig.  2. 


The  forces  exercised  by  the  containing  vessel  upon  a  liquid  confined 
under  hydrostatic  pressure  p  contribute  nothing  to  the  left-hand  member 
of  (4).  The  normal  force  acting  inwards  upon  the  element  of  boundary 
ds  is  pds,  so  that 

X  =  —pdy,         Y  =  pdx, 
and  accordingly 


vanishing  when  the  integration  extends  over  the  whole  boundary. 

Abandoning  now  the   supposition  that  the  particle  at  x,  y  is  at  rest, 
we  have 

d2  (xy)  _    dx  dy        d2y        d2x 
dt*    ~     dt  dt     X~dt2     y  dt2  ' 


so  that  if  m  be  the  mass  of  the  particle,  X,  Y  the  components  of  force  acting 
upon  it, 

o™  dx  dy  - 


or  with  summation  over  all  the  particles  of  the  system, 


•(7) 


We  now  take  the  mean  values  with  respect  to  time  of  the  various  terms 
in  (7).     If  the  system  be  such  that 


does  not  continually  increase,  we  obtain,  as  in  the  case  of  the  virial  theorem, 


It  would  seem  that  this  equation  has  application  to  the  molecular  theory 
of  the  viscosity  of  gases,  analogous  to  that  of  the  virial  as  applied  to  hydro- 
static pressure. 


263. 

ON  BALFOUR  STEWART'S   THEORY  OF  THE  CONNEXION 
BETWEEN   RADIATION   AND   ABSORPTION. 

[Philosophical  Magazine,  I.  pp.  98—100,  1901.] 

ON  a  recent  occasion*  I  remarked  that  Stewart's  work  appeared  to  me 
to  be  insufficiently  recognized  upon  the  Continent.  One  reason  for  this  is 
probably  the  comparative  inaccessibility  of  the  Edinburgh  Transactions  in 
which  his  first  paper  appeared f.  Another  may  be  found  in  the  fact  that  the 
paper  itself  is  not  well  arranged,  and  that  the  principal  conclusion  is  put 
forward  in  the  first  instance  as  if  it  were  the  result  of  Stewart's  special 
experiments.  The  experiments  were  indeed  of  great  value ;  but  this  course 
gave  an  opening  to  Kirchhoff's  objection  that  "  this  proof  [of  the  law  that  the 
absorption  of  a  plate  equals  its  radiation  and  that  for  every  description  of 
heatl]  cannot  be  a  strict  one,  because  experiments  which  have  only  taught 
us  concerning  more  and  less,  cannot  strictly  teach  us  concerning  equality^." 
I  am  inclined  to  think  that  Stewart  would  have  received  more  recognition 
if  he  had  never  experimented  at  all ! 

While  yielding  to  no  one  in  admiration  for  Kirchhoff,  I  can  hardly  regard 
him  as  in  this  matter  an  impartial  critic.  In  a  paper||  which  should  be 
studied  by  the  historical  inquirer,  Stewart  himself  protests  against  some  of 
Kirchhoff's  remarks,  and  to  my  judgment  makes  out  his  case.  In  his  ex- 
cellent Handbuch  der  tipectroscopie,  recently  published,  Prof.  Kayser,  with 
evident  desire  to  be  impartial,  gives  Stewart  much,  but  not  all,  of  the  credit 
that  I  would  claim  for  him.  But,  so  far  as  I  have  seen,  neither  Stewart 
himself  nor  any  of  his  critics  favourable  or  unfavourable  have  cited  the 
paragraph  upon  which  he  mainly  relies.  It  may  be  of  service  to  readers  who 
are  unlikely  to  see  the  original,  if  I  reproduce  it  here,  exactly  as  it  stood : — 

*  Phil.  Mag.  S.  5,  Vol.  XLIX.  p.  539  (1900).     [Vol.  iv.  p.  483.] 

t  Edin.  Trans.  Vol.  xxn.  p.  1,  March  1858. 

J  The  italics  are  Stewart's. 

§  Kirchhoff,  "  On  the  History  of  Spectrum  Analysis,"  &c.,  Phil.  Mag.  Vol.  xxv.  p.  258  (1863). 

||  Phil.  Mag.  Vol.  xxv.  p.  354  (1863). 


1901]        ON  STEWART'S  THEORY  OF  RADIATION  AND  ABSORPTION.  495 

'20.     A  more  rigid  demonstration  may  be  given  thus: — Let  AB,  BC  be 
two  contiguous,  equal,  and  similar  plates  in  the  interior  of  a  substance  of 
indefinite  extent,  kept  at  a  uniform  temperature. 
The  accumulated  radiation  from  the  interior  im- 
pinges on  the  upper  surface  of  the  upper  plate ; 
let  us  take  that  portion  of  it  which  falls  upon 

the  particles  A,  in  the  direction  DA.     This  ray,      

in  passing  from  A  to  B  will  have  been  partly 
absorbed  by  the  substance  between  A  and  B\ 
but  the  radiation  of  the  upper  plate  being  equal 
to  its  absorption  (since  its  temperature  remains 

the  same),  the  ray  will  have  been  just  as  much  recruited  by  the  united 
radiation  of  the  particles  between  A  and  5,  as  it  was  diminished  in  intensity 
by  their  absorption.  It  will  therefore  reach  B  with  the  same  intensity  as  it 
had  at  A.  But  the  quality  of  the  ray  at  B  will  also  be  the  same  as  its 
quality  at  A.  For,  if  it  were  different,  then  either  a  greater  or  less  pro- 
portion would  be  absorbed  in  its  passage  from  B  to  C,  than  was  absorbed 
of  the  equally  intense  ray  at  A,  in  its  passage  between  A  and  B.  The 
amount  of  heat  absorbed  by  the  particles  between  B  and  C  would  therefore 
be  different  from  that  absorbed  by  the  particles  between  A  and  B.  But  this 
cannot  be ;  for,  on  the  hypothesis  of  an  equal  and  independent  radiation  of 
each  particle,  the  radiation  of  the  particles  between  B  and  C  is  equal  to  that 
of  the  particles  between  A  and  B,  and  their  absorption  equals  their  radiation. 
Hence  the  radiation  impinging  on  B,  in  the  direction  of  DB,  must  be  equal 
in  quality  as  well  as  quantity  to  that  impinging  upon  A  ;  and,  consequently, 
the  radiation  of  the  particles  between  A  and  B  must  be  equal  to  their  ab- 
sorption, as  regards  quality  as  well  as  quantity ;  that  is,  this  equality  between 
the  radiation  and  absorption  must  hold  for  every  individual  description  of 
heat*." 

Surely  this  goes  to  the  root  of  the  matter,  and  it  presents  the  argument 
in  its  most  natural  form.  Kirchhoff 's  independent  investigation  of  a  year 
and  a  half  later f  is  more  formal  and  elaborate,  but  scarcely  more  convincing. 

No  one  in  England  or  elsewhere  disputes  the  great  obligations  under 
which  Spectrum  Analysis  lies  to  Kirchhoff.  In  a  passage  quoted  by  Dr 
Kayser  (loc.  cit.  p.  92)  from  Lord  Kelvin — "  To  Kirchhoff  belongs,  I  believe, 
solely  the  great  credit  of  having  first  actually  sought  for  and  found  other 
metals  than  sodium  in  the  sun  " — the  force  of  "  solely"  seems  to  have  been 
misunderstood.  I  have  Lord  Kelvin's  authority  for  interpreting  this  to  mean 
that  the  entire  credit  of  the  discovery  mentioned  belongs  to  Kirchhoff,  not 
that  he  is  entitled  to  no  credit  in  other  directions. 

*  Edin.  Trans.  Vol.  xxn.  p.  13,  March  1858. 

+  Monatsbericht  der  Akad.  d.  Wiss.  zu  Berlin,  Dec.  1859. 


264. 


SPECTROSCOPIC  NOTES  CONCERNING  THE  GASES   OF  THE 
ATMOSPHERE. 


[Philosophical  Magazine,  I.  pp.  100—105,  1901.] 


On  the  Visibility  of  Hydrogen  in  Air. 

MY  first  experiments  upon  this  question  were  made  in  July  1897.  The 
sparks  were  taken  between  platinum  points  in  a  small  chamber  through 
which  dried  air  at  atmospheric  pressure  could  be  led,  and  the  spectrum  was 
examined  with  a  spectroscope  of  two  prisms.  The  (7-line  could  be  very 
nearly  obliterated  by  careful  drying.  But  if  jfa  part  by  volume  of  hydrogen 
were  added  and  the  mixture  passed  afterwards  through  the  phosphoric 
anhydride,  the  increased  visibility  of  C  was  very  marked.  At  that  time 
I  was  occupied  with  the  density  of  carbonic  oxide  and  interested  in  the 
question  as  to  whether  it  contained  appreciable  quantities  of  hydrogen*. 
When  carbonic  oxide,  prepared  from  prussiate  of  potash  and  dried  as  for 
weighing,  was  passed  through  the  apparatus,  the  (7-line  became  nearly  in- 
visible ;  but  the  test  with  carbonic  oxide  was  thought  to  be  less  delicate  than 
with  air  in  consequence  of  the  proximity  of  another  bright  line  in  the  former 


I  have  lately  resumed  these  experiments,  induced  thereto  principally  by 
the  remarkable  results  of  M.  Gautier.  This  observer,  working  by  chemical 
methods,  finds  that  air  normally  contains  about  10to0o  of  hydrogen  in  addition 
to  variable  amounts  of  hydrocarbons.  It  appeared  to  me  that  a  spectro- 
scopic  confirmation  would  be  interesting. 

*  Proc.  Boy.  Soc.  Vol.  LXII.  p.  205  (1897).    [Vol.  iv.  p.  348.] 


1901]   SPECTROSCOPIC   NOTES   CONCERNING  GASES   OF  THE   ATMOSPHERE.     497 

Using  the  old  apparatus,  in  which  the  tubes  conveying  the  gas  and  the 
electrodes  were  fitted  into  a  rubber  cork,  I  could  not  succeed  in  getting  quit 
of  C  from  the  spectrum  of  somewhat  powerful  sparks,  however  carefully  the 
air  were  dried.  The  coil  was  excited  with  five  Grove  cells  and  a  large  leyden- 
jar*was  connected  with  the  secondary  in  the  usual  way.  This  observation 
was  of  course  consistent  with  the  presence  of  hydrogen  in  the  atmosphere ; 
but  it  was  suspicious  that  the  best  approach  to  evanescence  was  obtained 
with  a  somewhat  brisk  rather  than  with  a  slow  current  of  air,  indicating 
that  the  source  of  the  hydrogen  was  in  the  apparatus  rather  than  in  the 
atmosphere.  As  it  seemed  desirable  to  apply  heat,  I  discarded  the  old 
apparatus,  substituting  for  it  a  simpler  one  consisting  merely  of  a  small 
bulbous  enlargement  of  the  gas-leading  tube.  Into  this  the  platinum  elec- 
trodes were  sealed.  The  gases  under  examination  were  stored  under  slight 
pressure  and  on  leaving  the  reservoirs  were  partially  dried  with  sulphuric 
acid.  A  three-way  tap  allowed  the  easy  substitution  of  one  gas  for  another. 
After  passing  this  tap  the  gas  was  further  dried  by  phosphoric  anhydride  on 
its  way  to  the  sparking-bulb. 

The  application  of  heat  to  the  bulb  and  to  the  short  length  of  tubing 
between  the  bulb  and  the  phosphoric  anhydride  led,  as  was  expected,  to 
a  recrudescence  of  0.  Subsequently  there  seemed  to  be  an  improvement. 
Not  only  was  C  less  conspicuous,  but  its  visibility  remained  about  the  same 
although  the  rate  of  flow  were  varied.  It  is  difficult  to  describe  in  words 
the  effect  upon  the  eye,  but  I  may  say  that  with  the  actual  spectroscopic 
arrangements  including  a  somewhat  wide  slit  the  line  could  be  certainly  and 
steadily  seen. 

The  above  was  the  appearance  with  a  stream  of  (country)  air.  When  air 
to  which  jr-^  part  of  hydrogen  (by  volume)  had  been  added  was  substituted, 
the  visibility  of  C  was  markedly  increased ;  and  the  difference  was  such  that 
one  could  easily  believe  that  the  proportion  of  hydrogen  actually  operative 
had  been  doubled.  This  conclusion  would  be  in  precise  agreement  with 
M.  Gautier,  could  we  assume  that  the  smaller  quantity  of  hydrogen  really 
accompanied  the  air.  But  the  facts  now  to  be  recorded  render  this  assump- 
tion extremely  doubtful. 

In  the  first  place  the  visibility  of  G  with  ordinary  air  was  not  perceptibly 
diminished  by  passage  of  the  air  over  red-hot  cupric  oxide  included  between 
the  sulphuric  acid  and  the  three-way  tap.  It  may  be  argued  that  cupric 
oxide  is  not  competent  in  moderate  length  to  remove  the  last  traces  of 
hydrogen  from  air,  even  though  the  air  be  passed  over  it  in  a  slow  stream. 
I  found,  however,  on  a  former  occasion*  that  hydrogen  purposely  introduced 

*  On  an  Anomaly  encountered  in  Determinations  of  the  Density  of  Nitrogen  Gas,  Proc.  Roy. 
Soc.  Vol.  LV.  p.  343  (1894).     [Vol.  iv.  p.  107.] 

E.    iv.  32 


498  SPECTKOSCOPIC  NOTES   CONCERNING  [264 

into  nitrogen  could  be  so  far  removed  in  this  way  that  the  weight  remained 
sensibly  unaffected,  although  10*000  of  residual  hydrogen  might  be  expected 
to  manifest  itself. 

Moreover,  when  air  purposely  contaminated  as  above  with  ^^5-  of  hy- 
drogen was  passed  over  the  copper  oxide,  the  additional  hydrogen  appeared 
to  be  removed,  the  visibility  of  C  reducing  itself  to  that  corresponding  to 
untreated  air. 

Being  desirous  of  testing  the  matter  as  far  as  possible,  I  have  ex- 
perimented also  with  nitrous  oxide  and  with  oxygen.  In  the  former  case 
the  general  appearance  of  the  spectrum  is  much  the  same  as  with  air.  The 
(7-line  was  thought  to  be,  if  anything,  more  visible  than  in  the  case  of  air, 
but  the  difference  could  not  be  depended  upon.  Two  samples  of  gas  were 
tried,  one  from  an  iron  bottle  as  supplied  commercially,  the  other  prepared 
in  the  laboratory  from  ammonium  nitrate.  Oxygen  from  permanganate  of 
potash  also  showed  the  (7-line  more  distinctly  than  air,  but  this  may  probably 
be  attributed  to  the  elimination  of  a  neighbouring  nitrogen  line.  It  is 
possible,  of  course,  that  these  gases  may  have  contained  traces  of  hydrogen, 
but  in  that  case  it  is  strange  that  the  proportion  should  be  so  nearly  the 
same  as  in  air. 

These  observations  certainly  seem  to  leave  a  minimum  of  room  for  the 
hydrogen  found  by  M.  Gautier,  but  I  should  be  unwilling  to  call  his  con- 
clusion in  question  on  the  strength  of  what  are  after  all  but  eye  estimates. 
I  have  not  been  able  to  find  a  detailed  account  of  M.  Gautier's  experiments 
or  of  what  precautions  he  took  to  assure  himself  that  the  water  collected 
could  not  have  had  its  origin  in  the  glass  or  copper  oxide  of  his  hot  tubes*. 
The  most  satisfactory  test  would  be  comparison  experiments  in  which  oxygen 
or  nitrous  oxide  is  substituted  for  air,  or,  perhaps  better  still,  in  which  air  is 
used  over  and  over  again. 

If,  as  I  should  suppose  were  I  to  judge  from  my  own  experiments  alone, 
the  residual  (7-line  was  not  wholly  or  even  principally  due  to  hydrogen  in  the 
air,  it  would  have  to  be  explained  by  hydrogen  evolved  from  the  glass  of  the 
sparking-chamber  or  from  the  platinum  electrodes.  In  view  of  what  is  known 
respecting  the  behaviour  of  vacuum-tubes,  such  an  explanation  does  not 
appear  improbable. 

Experiments  upon  the  visibility  of  C  in  vacuum-tubes  have  shown  a  much 
smaller  degree  of  sensibility.  The  tube  was  in  connexion  with  a  Topler 
pump  and  was  traversed  by  a  stream  of  air.  The  passage  from  high  (atmo- 
spheric) to  low  pressure  took  place  at  a  glass  capillary  which  allowed  about 
30  c.c.  per  hour  (reckoned  at  atmospheric  pressure)  to  leak  past.  When 

*  [1902.  See  Annales  de  Chimie,  xxii.  Jan.  1901.  Further  experiments  of  my  own  are  detailed 
in  Phil.  Mag.  in.  p.  416,  1902.] 


1901] 


THE   GASES   OF   THE   ATMOSPHERE. 


moist  air  from  the  room  on  a  damp  day  (15°  C.)  was  admitted,  the  hydrogen 
(7-line  was  very  bright,  nearly  obliterating  one  of  the  dark  bands  of  nitrogen. 
On  drying  the  air  with  phosphoric  anhydride,  the  (7-line  disappeared.  Air 
mixed  with  1  per  cent,  of  hydrogen  showed  it  doubtfully,  1£  per  cent,  plainly, 
2  per  cent,  perhaps  equally  with  the  moist  air.  The  much  smaller  sensibility 
(about  50  times)  in  these  experiments  may  be  partly  due  to  the  less  favour- 
able character  of  the  ground  upon  which  the  hydrogen  line  has  to  show  itself. 


Demonstration  at  Atmospheric  Pressure  of  Argon  from  very  small 
quantities  of  Air. 


Success  in  reducing  the  necessary  amount  of  air  depends  a  good  deal 
upon  the  form  of  tube  employed.     That  sketched  (Fig.  1)  allows  a  minimum 
residue  to  be  sparked  and  examined.     In  some  ex- 
periments the  tube,  standing  over  weak  alkali,  was  Fi8-  *• 
charged  with  5  c.c.  only  of  air.     The  first  part  of 
the  sparking  is  with  electrodes  ending  in  platinum 
points  and  brought  up  in  U-shaped  tubes  of  which 
the  bends  are  filled  with  mercury.     A  Ruhmkorff 
actuated  by  two  or  three  Grove  cells  is  employed 
at  this  stage,  and  the  sparks  pass  just  under  the 
shoulder  of  the  containing  tube,  oxygen  being  sup- 
plied as  required  from  a  small  electrolytic  gene-          /      \  Scale  f  • 
rator.      When  the  volume  is  sufficiently  reduced 
and   most   of  the   nitrogen    has   disappeared,   the 
electrodes  above  spoken  of  are  removed. 

In  the  next  stage  the  sparks  are  taken  between 
a  sealed-in  electrode  at  the  top  of  the  containing 
tube,  and  another  sealed  into  the  top  of  a  single 
U-tube  brought  round  through  the  alkali,  and 
rising  (as  shown)  through  the  narrow  part  of  the 
containing  tube.  In  order  to  avoid  splashing  and 
consequent  risk  of  fracture  from  sudden  cooling  of 
the  heated  glass,  it  was  thought  an  advantage  that 

the  tubes  through  part  of  their  length  should  make  a  tolerably  close  fit. 
But  the  most  important  precaution  appears  to  be  the  use  of  very  short 
sparks  and  a  reduction  of  the  battery  to  two  cells.  When  it  is  desired  to 
observe  the  spectrum,  a  small  jar  must  be  connected  in  the  usual  way. 

The  spectroscope  employed  had  two  prisms,  and  the  sparks  were  focused 
upon  a  somewhat  wide  slit  by  a  2-inch  lens.     A  low-power  eyepiece  was 

favourable. 

32—2 


500  SPECTROSCOPIC   NOTES   CONCERNING  [264 

The  group  of  lines  in  the  argon  spectrum  first  observed  by  Schuster* 
[for  figure,  see  Vol.  iv.  p.  199]  is  easily  seen.  Owing  to  the  warmth  F  is 
very  diffuse,  sometimes  nearly  filling  up  the  interval  between  4879  and  4847. 
On  one  occasion  when  the  original  air  taken  was  only  5  c.c.,  the  group  was 
almost  as  distinct  as  with  pure  argon.  The  residual  gas,  measured  cold,  was 
probably  no  more  than  1  c.c.  This  was  rather  an  extreme  case,  and  it  would 
not  have  been  possible  to  renew  the  sparking  without  an  addition  of  oxygen, 
to  be  afterwards  removed  by  careful  additions  of  hydrogen.  But  the  argon 
spectrum  shows  fairly  well  even  when  the  gas  is  diluted  with  two  or  three 
times  its  volume  of  oxygen. 

I  have  described  this  experiment  at  some  length  because  I  think  that  it 
would  make  a  good  exercise  for  students,  requiring  no  special  apparatus  but 
what  they  should  be  able  to  construct  for  themselves.  Although,  as  I  have 
said,  5  c.c.  of  air  is  ample,  a  novice  would  naturally  begin  with  10  or  15  c.c. 


Concentration  of  Helium  from  the  Atmosphere. 

In  a  footnote  [p.  266]  to  a  paper  on  the  Separation  of  Gases  by  Diffusionf. 
I  suggested  that  the  lighter  constituent  of  a  mixture  might  be  concentrated 
by  causing  it  to  diffuse  against  a  stream  of  an  easily  absorbable  gas,  such  as 
carbonic  acid.  In  Jan.  1899  a  good  many  trials  upon  these  lines  were  made 
with  the  object  of  putting  in  evidence  the  helium  of  the  atmosphere,  and 
a  certain  degree  of  success  was  attained.  A  stream  of  carbonic  acid  (pre- 
pared from  marble  and  hydrochloric  acid  and  reckoned  at  3  litres  per  hour) 
is  maintained  for  say  14  hours  through  a  diffusion-tube  open  above  to  the 
atmosphere.  This  tube,  placed  vertically,  is  about  40  cm.  long  and  of  about 
5  cm.  diameter.  The  gases  of  the  atmosphere  diffuse  downwards  into  the 
tube,  but  the  heavier  constituents  are  held  almost  entirely  at  bay  by  the 
stream  of  carbonic  acid.  If  we  draw  off  continuously  a  supply  from  a  point 
say  halfway  down  the  diffusion-tube,  we  shall  obtain  carbonic  acid  with  a 
small  admixture  of  atmospheric  gases  in  which  the  lighter  ingredients,  e.g. 
water,  hydrogen,  and  helium,  are  relatively  much  concentrated.  In  my  ex- 
periments the  lateral  stream  was  about  250  c.c.  per  hour,  and  was  manipulated 
with  the  aid  of  a  Sprengel  pump.  Between  the  pump  and  the  diffusion-tube 
was  interposed  a  short  length  of  tobacco-pipe  through  the  walls  of  which  the 
gas  had  to  pass  and  which  presented  the  right  degree  of  obstruction.  After 
passage  to  the  low-pressure  side,  the  bulk  of  the  C02  was  absorbed  with 
alkali,  and  the  residual  gases  collected  over  alkali  at  the  foot  of  the  Sprengel 
in  the  usual  way. 

*  Rayleigh  and  Ramsay,  Phil.  Trans.  186,  p.  224  (1895).     [Vol.  iv.  p.  169.]    See  also  Nature, 
Vol.  m.  p.  163  (1895).     [Vol.  iv.  p.  199.] 

t  Phil.  Mag.  Vol.  XLII.  p.  493  (1896).     [Vol.  iv.  p.  266.] 


1901]  THE   GASES   OF   THE   ATMOSPHERE.  501 

The  subsequent  treatment  for  removal  of  nitrogen  by  the  electric  dis- 
charge was  conducted  as  usual,  towards  the  close  in  the  tube  described  and 
figured  above.  The  final  residue  on  the  occasion  when  D3  was  best  seen 
(under  the  jar  discharge)  was  about  '25  c.c.  Argon  was  also  plainly  visible 
and  probably  constituted  the  greater  part  of  the  bulk.  When  the  volume 
was  doubled  by  addition  of  oxygen,  Dg  was  seen  less  well. 

Success  depended  a  good  deal  upon  precautions  to  avoid  the  presence  of 
gases,  and  especially  of  argon,  which  had  not  undergone  diffusion.  It  was 
necessary  to  eliminate  the  dissolved  gases  of  the  dilute  hydrochloric  acid 
with  which  the  C0.2  was  prepared,  and  to  keep  an  atmosphere  of  C02  in  the 
supply- vessel.  Until  these  precautions  were  taken,  D3,  though  frequently 
suspected,  was  not  clearly  and  steadily  seen.  Even  at  the  best,  good  measure- 
ments could  hardly  have  been  taken ;  but  the  line  appeared  to  be  in  the  right 
place  for  helium,  as  distinguished  for  example  from  neon. 


265. 


ON  THE  STRESSES  IN  SOLID  BODIES  DUE  TO  UNEQUAL 
HEATING,  AND  ON  THE  DOUBLE  REFRACTION  RESULT- 
ING THEREFROM* 


[Philosophical  Magazine,  I.  pp.  169—178,  1901.] 

THE  phenomena  of  light  and  colour  exhibited  in  the  polariscope  when 
strained  glass  is  interposed  between  crossed  nicols  are  well  known  to  every 
student  of  optics.  The  strain  may  be  of  a  permanent  character,  as  in  glass 
imperfectly  annealed  or  specially  unannealed,  or  it  may  be  temporary,  due 
to  variations  of  temperature  or  to  mechanical  force  applied  from  without. 
One  of  the  best  examples  under  the  last  head  is  that  of  a  rectangular  bar 
subjected  to  flexure,  the  plane  of  the  flexure  being  perpendicular  to  the 
course  of  the  light.  The  full  effect  is  obtained  when  the  length  of  the 
bar  is  at  45°  to  the  direction  of  polarization.  The  revival  of  light  is  a  maximum 
at  the  edges,  where  the  material  traversed  is  most  stretched  or  compressed, 
while  down  the  middle  a  dark  bar  is  seen  representing  the  "  neutral  axis." 
It  is  especially  to  be  noted  that  the  effect  is  due  to  the  glass  being  unequally 
stretched  in  the  two  directions  perpendicular  to  the  line  of  vision.  Thus 
in  the  case  under  discussion  no  force  is  operative  perpendicular  to  the  length 
of  the  bar.  Under  a  purely  hydrostatic  pressure  the  singly  refracting 
character  of  the  material  would  not  be  disturbed. 

When  a  piece  of  glass,  previously  in  a  state  of  ease,  is  unequally  heated, 
double  refraction  usually  ensues.  This  is  due,  not  directly  to  the  heat, 
but  to  the  stresses,  different  in  different  directions  and  at  different  places, 
caused  by  the  unequal  expansions  of  the  various  parts.  The  investigation 
of  these  stresses  is  a  problem  in  Elasticity  first  attacked,  I  believe,  by 

*  From  the  Lorentz  Collection  of  Memoirs. 


1901]  ON  STRESSES   DUE  TO  UNEQUAL   HEATING.  503 

J.  Hopkinson*.  It  will  be  convenient  to  repeat  in  a  somewhat  different 
notation  his  formulation  of  the  general  theory,  and  afterwards  to  apply  it 
to  some  special  problems  to  which  the  optical  method  of  examination  is 
applicable. 

In  the  usual  notation  f  if  P,  Q,  R,  S,  T,  U  be  the  components  of  stress ; 
u,  v,  w  the  displacements  at  the  point  x,  y,  z ;  \,  /*  the  elastic  constants ; 
we  have  such  equations  as 

'du     dv      dw\      .    du 

¥dy+d,)  +  ^di'    (1) 

•dw    dv^ 


These  hold  when  the  material  is  at  the  standard  temperature.     If  we  suppose 
that  the  temperature  is  raised  by  6  and  that  no  stresses  are  applied, 

du     dv      dw        n 

-T~  =  j-  =  rr-  =  *0> 
dx     dy      dz 

while  dw/dy  &c.  vanish.     The  stresses  that  would  be  needed  to  produce  the 
same  displacements  without  change  of  temperature  are 


Hence,  so  far  as  the  principle  of  superposition  holds  good,  we  may  write 
in  general 

..........  <3> 


with  similar  equations  for  Q,  R,  T,  U. 

If  there  be  no  bodily  forces,  the  equation  of  equilibrium  is 

^  +  f  +  f=0,  ..............................  (5) 

dx      dy      dz 

with  two  similar  equations  ;  or  with  use  of  (3)  and  (4) 


=  <>  .............  (6) 

if 

(7) 


One  of  the  simplest  cases  that  can  be  considered  is  that  of  a  plate,  bounded 
by  infinite  planes  parallel  to  xy,  and  so  heated  that  6  is  a  function  of  z  only. 

*  Mess,  of  Math.  Vol.  vm.  p.  168  (1879).    [1902.    From  a  notice  by  W.  Konig  (Beiblatter,  1901) 
I  gather  that  some  of  the  problems  here  dealt  with  had  already  been  treated  by  Neumann  in  1841.] 
t  See,  for  example,  Love's  Theory  of  Elasticity,  Cambridge  University  Press,  1892. 


504      ON   THE   STRESSES   IN   SOLID   BODIES   DUE   TO   UNEQUAL   HEATING,     [265 

If,  further,  6  be  symmetrical  with  respect  to  the  middle  surface,  the  plate 
will  remain  unbent;  and  if  the  mean  value  of  0  be  zero,  the  various  plane 
sections  will  remain  unextended.  Assuming,  therefore,  that  u,  v  vanish  while 
w  is  variable,  we  get  from  (3)  and  (4) 


(8) 


P=Q  =  X-70,  .................................  (9) 

S  =  T=  U=0  .....................................  (10) 

In  (8)  R  is  assumed  to  vanish,  since  no  force  is  supposed  to  act  upon  the 
faces.     From  (8),  (9) 


If  the  plate  be  examined  in  the  polariscope  by  light  traversing  it  in 
the  direction  of  y.  the  double  refraction,  depending  upon  the  difference 
between  R  and  P,  of  which  the  former  is  zero,  is  represented  simply  by  (11). 
Dark  bars  will  be  seen  at  places  where  6  =  0.  If  the  direction  of  the  light 
be  across  the  plate,  i.e.  parallel  to  z,  there  is  no  tendency  to  double  refraction, 
since  everywhere  P  =  Q. 

In  the  above  example  where  every  layer  parallel  to  xy  remains  unextended, 
the  local  alteration  of  temperature  produces  its  full  effect.  But  in  general 
the  circumstances  are  such  that  the  plate  is  able  to  relieve  itself  to  a 
considerable  extent.  A  uniform  elevation  of  temperature,  for  instance,  would 
entail  no  stress.  And  again,  a  uniform  temperature  gradient,  such  as  would 
finally  establish  itself  if  the  two  surfaces  of  the  plate  were  kept  at  fixed 
temperatures,  is  compensated  by  bending  and  entails  no  stress.  In  such 
cases  before  calculating  the  stress  by  (11)  we  must  throw  out  the  mean  value 
of  6  so  as  to  make  fPdz  =  0,  and  also  such  a  term  proportional  to  the  distance 
from  the  middle  surface  as  shall  ensure  that  fPzdz=0.  Otherwise  the 
edges  of  the  plate  could  not  be  regarded  as  free  from  imposed  stress  in 
the  form  of  a  force  or  couple. 

The  assumption  in  (1),  (2)  that  u  =  v  =  Q  is  now  replaced  by 

u  =  (a  +  /3z)a;,         v  =  (a  +  $z)y,    ..................  (12) 

and  w  =  w'  -  \$  (of  +  t/2),     ........................  (12') 

where  w'  is  a  function  of  z  only.     We  find 


(13) 


U=0  ........................................  (15) 


1901]   AND  ON  THE  DOUBLE  REFRACTION  RESULTING  THEREFROM.     505 

Since  R  is  supposed  to  vanish,  we  get 


In  (16)  a  and  /3  are  to  be  determined  by  the  conditions 


or,  which  comes  to  the  same,  we  are  to  reject  from  6  such  linear  terms  as 
will  leave 

Q  .........................  (17) 


Since  w'  and  6  are  independent  of  x  and  y,  the  equations  of  equilibrium  (5) 
are  satisfied. 


It  is  of  interest  to  trace  the  influence  of  time  upon  the  double  refraction 
of  the  heated  plate  when  light  passes  through  it  edgeways,  e.g.  parallel  to  y. 
Initially  6  may  be  supposed  to  be  an  arbitrary  function  of  z,  while  the  faces 
of  the  plate,  say  at  0  and  c,  are  maintained  at  given  temperatures.  Ultimately 
the  distribution  of  temperature  is  expressed  by  a  linear  function  of  z,  say 
H'  +  Kz  ;  and,  as  is  known  from  Fourier's  theory,  the  distribution  at  time 
t  may  be  expressed  by 

0  =  H'+Kz  +  ^Ane-P''f  sin  (nirz/c),     ...............  (18) 

where  n  is  an  integer  and  pn,  depending  also  upon  the  conductivity,  is 
proportional  to  n2.  After  a  moderate  interval,  the  terms  corresponding 
to  the  higher  values  of  n  become  unimportant. 

In  the  subsequent  calculation  it  is  convenient  to  take  the  origin  of  z 
in  the  middle  surface,  instead  of  as  in  (18)  at  one  of  the  faces.  Thus 

0  =  H  +  Kz  +  A.e-P^  cos  —  -  Aze~^  cos  —  +  ... 

-A^sin—  +  A<tr**aa—  -  .............  (19) 

c  c 

If  0'  represent  the  value  of  0  when  reduced  by  the  subtraction  of  the 
proper  linear  terms  as  already  explained,  we  find 

* 


V  =  Alf"  (cos  ™  -  *)  -  A,r»  (cos  ?=  +  £ 

\         c        TT)  \          C         BTT 

......  (20) 


After  a  moderate  time  the  term  in  A  l  usually  acquires  the  preponderance, 
and  then  0'  =  0  when  cos  (trz/c)  =  2/w.  When  the  plate  is  looked  at  edge- 
ways in  the  polariscope,  dark  bars  are  seen  where  z  =  ±  '280  c,  c  being  the 
whole  thickness  of  the  plate. 


506      ON   THE   STRESSES   IN   SOLID   BODIES   DUE  TO   UNEQUAL  HEATING,    [265 

As  a  particular  case  of  (19),  (20)  let  us  suppose  that  the  distribution 
of  temperature  is  symmetrical,  or  that  K  vanishes  as  well  as  the  coefficients 
of  even  suffix  A^,  At,  &c.  H  then  represents  the  temperature  at  which  the 
two  faces  are  maintained,  and  (19)  reduces  to 

- 


(21) 
c  c 

If  we  suppose  further  that  the  initial  temperature  is  uniform  and  equal 
to  ®,  we  find  by  Fourier's  methods 

A  =  ^(®-#),       AZ  =  ^(®-H},       4,=  l(e-J50,     &c-     -(22) 
and 

2 


..,    .........  (23) 

where  also 

p3  =  9Pl,        p5  =  25Pl,        &c  ...................  (24) 

At   the   middle   surface,  where   z  =  0,   the   right-hand  member  of  (23) 
becomes 

e-,,(1_?)_je-w(1  +  J?.)  +  ................  (25) 

Initially 


as  was  required.     If  we  put  e~p^  =  T,     (25)  may  be  written 

...;  .........  (26) 


and  (26)  may  be  tabulated  as  a  function  of  T,  and  thence  of  t.  It  vanishes 
when  T=l  and  when  T=0.  The  maximum  value  occurs  when  jT='747. 
When  T  is  less  than  this,  which  corresponds  to  an  increased  value  of  t,  only 
the  first  two  or  three  terms  in  (26)  need  be  regarded.  The  above  value 
of  T  gives 

pj  =  -292  ; 

and  if,  as  for  glass,  the  diffusivity  for  heat  in  C.G.s.  measure  be  '004,  we  get 

(27) 


Thus  if  a  plate  of  glass  be  one  centimetre  thick,  so  that  c  =  1,  the  light 
seen  in  the  polariscope  at  the  centre  of  the  thickness  is  a  maximum  about 
7£  seconds  after  heat  is  applied  to  the  faces. 


1901]    AND  ON  THE  DOUBLE  REFRACTION  RESULTING  THEREFROM. 


507 


The   following   small   table   will   give  an  idea  of  the  relation  between 
(26)  and  T. 


T 

(26) 

T 

(26) 

o-o 

o-oooo 

0-6 

0-2139 

0-1 

0-0363 

0-7 

0-2381 

0-2 

0-0727 

0-8 

0-2371 

0-3 

0-1090 

0-9 

0-1823 

0-4 

0-1453 

1-0 

o-oooo 

0-5 

0-1809 

In  his  paper  above  referred  to,  Hopkinson  considered  the  strains  produced 
by  unequal  heating  in  a  spherical  mass,  under  the  supposition  that  the 
temperature  was  everywhere  the  same  at  the  same  distance  from  the  centre. 
A  similar  analysis  applies  in  the  two-dimensional  problem,  which  is  of  greater 
interest  from  the  present  point  of  view.  We  suppose  that  everything  is 
symmetrical  with  respect  to  an  axis,  taken  as  axis  of  z,  and  that  6  is  a 
function  of  r,  equal  to  *j(a?  +  y2),  only.  The  displacements  in  the  directions 
of  z  and  r  will  be  denoted  by  w  and  u  ;  in  the  third  direction,  perpendicular 
to  z  and  r,  there  is  supposed  to  be  no  displacement. 

We  may  commence  with  the  strictly  two-dimensional  case  where  w  =  0 
throughout.  This  implies  a  stress  R  whose  magnitude  is  given  by 


in  which 


represents  the  dilatation. 

The  other  principal  stresses  operative  radially  and  tangentially  are 


(30) 
(31) 


The  equation  of  equilibrium,  analogous  to  (5),  is  obtained  by  considering 
the  stresses  operative  upon  the  polar  element  of  area.     It  is 


508      ON  THE  STRESSES   IN  SOLID   BODIES   DUE  TO  UNEQUAL  HEATING,     [265 

Substituting  from  (30),  (31),  we  get 

d?u     1  du      u  _      7       dO 

dr*  +  ^r  dr~r*  =  \  +  fy  dr  ' 
so  that 

du  u         yd 


where  a.  is  an  arbitrary  constant.     Integrating  a  second  time  we  find 

..................  (34) 


fr 

J  o 


in  which,  however,  £  must  vanish,  if  the  cylinder  is  complete  through  r  =  0. 
From  (34) 

\Terdr,  .  ...(35) 


a.-r^J    Mr-^T.>  (36> 

and 


*-**&&-*£ 


It  is  on  (P  —  Q)  that  the  double  refraction  depends  when  light  traverses 
the  cylinder  in  a  direction  parallel  to  its  axis. 

In  (35),  (36),  (37) 

-  I 0rdr 


represents  the  mean  temperature  (above  the  standard)  of  the  solid  cylinder 
of  radius  r.  It  is  to  be  remarked  that  the  double  refraction  of  the  ray  at 
r  is  independent  of  the  values  of  0  beyond  r,  and  also  of  any  boundary- 
pressure.  If  0  increases  (or  decreases)  continuously  from  the  centre  out- 
wards, the  double  refraction  never  vanishes,  and  no  dark  circle  is  seen  in  the 
polariscope. 

In  the  above  solution  if  the  cylinder  is  terminated  by  flat  faces,  we 
must  imagine  suitable  forces  R,  given  by  (28),  to  be  operative  over  the  faces. 
The  integral  of  these  forces  may  be  reduced  to  zero  by  allowing  a  suitable 
expansion  parallel  to  the  axis.  Regarding  dw/dz  as  a  constant  (not  necessarily 
zero),  independent  of  r  and  z,  we  have  in  place  of  (28) 


The  additions  to  P  and  Q  are  \dw/dz,  while  (P  -  Q)  remains  unchanged. 

If  the  cylinder  is  long  relatively  to  its  diameter,  the  last  state  of  things 
may   be   supposed   to  remain  approximately  unchanged,  even   though   the 


1901]        AND   ON   THE    DOUBLE    REFRACTION    RESULTING   THEREFROM.  509 

terminal  faces  be  free  from  applied  force.  In  the  neighbourhood  of  the  ends 
there  will  be  local  disturbances,  requiring  a  more  elaborate  analysis  for  their 
calculation,  but  the  simple  solution  will  apply  to  the  greater  part  of  the 
length. 

The  case  of  a  thin  plate  whose  faces  are  everywhere  free  from  applied 
force  is  more  difficult  to  treat  in  a  rigorous  manner,  but  the  following  is 
probably  a  sufficient  account  of  the  matter.  By  supposing  R  =  0  in  (38) 
we  get 

(X4^g^-x(**?);    (39) 

and  using  this  value  of  diujdz, 


Comparing  these  with  (30),  (31),  we  see  that  the  only  difference  is  that 
X  and  7  of  those  equations  are  now  replaced  by 


, 
' 
Hence,  instead  of  (37),  we  should  have 

(42) 


and  the  same  general  conclusions  follow. 

In  the  preceding  calculations  we  have  supposed  that  the  solid  is  free 
from  stress  at  a  uniform  standard  temperature  when  u,  v,  w  vanish.  In 
the  case  of  unannealed  glass,  it  would  require  a  variable  temperature  to 
relieve  the  material  from  stress.  To  meet  this,  6  in  the  above  equations 
would  have  to  be  reckoned  from  the  variable  temperature  corresponding 
to  the  state  of  ease,  rather  than  from  a  uniform  standard  temperature. 

Some  of  the  questions  above  considered  are  easily  illustrated  experi- 
mentally. A  slab  of  glass  about  8  cm.  square  and  1  cm.  thick,  polished 
upon  opposite  edges,  when  placed  in  the  polariscope  shows  but  little  revival 
of  light  so  long  as  the  temperature  is  uniform.  The  contact  of  the  hands 
with  the  two  faces  suffices  to  cause  an  almost  instantaneous  illumination, 
rising  to  a  maximum  at  the  middle  of  the  thickness  after  a  few  seconds. 
Dark  bands  situated  about  halfway  between  the  middle  and  the  faces  are 
a  conspicuous  feature.  After  about  30  or  40  seconds  the  light  fades  greatly, 
a  result  more  rapidly  attained  if  the  hands  be  removed  after  10  or  20  seconds' 
contact.  In  the  earlier  stages  of  the  heating  thfe  outside  layers  are  the 


510  ON   STRESSES   DUE  TO   UNEQUAL  HEATING.  [265 

warmer,  and  being  prevented  from  expanding  fully  are  in  a  condition  of 
compression.  The  inner  layers  at  the  same  time  are  in  tension,  a  conclusion 
that  may  be  verified  by  interposition  of  another  piece  of  glass,  of  which 
the  mechanical  condition  is  known,  and  of  which  the  effect  may  be  either 
an  augmentation  or  a  diminution  of  the  light. 

An  examination  in  the  polariscope  of  the  so-called  toughened  glass,  intro- 
duced a  few  years  ago,  is  interesting.  It  was  understood  to  be  prepared 
by  a  sudden  cooling  in  oil  while  still  plastic  with  heat.  When  it  is  examined 
through  the  thickness  of  the  sheet,  a  great  want  of  uniformity  is  manifested. 
In  spite  of  the  shortness  of  the  distance  traversed,  there  is  in  places  con- 
siderable revival  of  light  with  intermediate  irregularly  disposed  dark  bands. 
The  course  of  these  bands  is  altered  when  by  fracture  any  part  is  relieved 
from  the  constraining  influence  of  neighbouring  parts.  To  make  an  examina- 
tion by  light  transmitted  edgewise,  it  was  necessary  to  immerse  the  glass  in  a 
liquid  of  nearly  equal  refractivity  (benzole  with  a  little  bisulphide  of  carbon) 
contained  in  a  small  tank.  The  width,  traversed  by  the  light,  was  about 
]  cm.  In  this  way,  and  with  the  aid  of  a  magnifier,  the  condition  of  the 
various  layers  could  be  well  made  out.  The  dark  bands  of  no  double  refraction 
seemed  to  be  nearer  to  the  faces  than  according  to  the  calculation  made 
above,  but  the  whole  thickness  is  so  small  that  this  observation  is  scarcely 
to  be  relied  upon.  The  interior  was  in  a  state  of  tension,  and  the  double 
refraction  was  nearly  sufficient  at  the  middle  to  give  the  yellow  or  brown 
of  the  first  order.  By  the  action  of  hydrofluoric  acid  on  the  lower  end  of  one 
of  the  strips  the  outermost  layers  were  dissolved  away.  This  caused  a  drawing 
together  of  the  dark  bands  towards  the  middle,  and  though  a  good  deal 
remained  the  light  was  much  reduced. 

The  cause  of  the  toughening  has  been  sought  in  a  special  crystalline 
condition  due  to  the  sudden  cooling.  There  may  be  something  of  this  nature  ; 
but  it  would  seem  that  most  of  the  peculiarities  manifested  may  be  explained 
by  reference  to  the  known  condition  of  stress.  The  fracture  of  glass  is  usually 
due  to  bending,  and  the  failure  occurs  at  the  surface  which  is  under  tension. 
If,  initially,  the  superficial  layers  are  under  strong  compression,  a  degree 
of  bending  may  be  harmless  which  otherwise  would  cause  fatal  results.  It 
seems  possible  also  that  the  superficial  compression  may  be  the  explanation 
of  the  special  hardness  observed. 

A  short  length  of  glass  rod  in  its  natural  imperfectly  annealed  condition 
may  be  used  to  illustrate  symmetrical  stress.  The  ends  may  be  ground,  and 
[then]  either  polished  or  provided  with  cover-glasses  cemented  with  Canada 
balsam.  In  the  specimen  examined  by  me  the  colours  varied  from  the 
black  of  the  first  order  on  the  axis  to  the  red  of  the  second  order  near  the 
surface.  The  length  of  the  cylinder  was  T6  cm.  and  the  diameter  T8  cm. 


266. 

ON  A  NEW  MANOMETER,  AND  ON  THE  LAW  OF  THE 
PRESSURE  OF  GASES  BETWEEN  1-5  AND  O'Ol  MILLI- 
METRES OF  MERCURY. 

[Phil.  Trans.  CXCVIA.  pp.  205—223,  1901.] 
Received  January  15, — Read  February  21,   1901. 

Introduction. 

THE  behaviour  of  air  and  other  gases  at  low  densities  is  a  subject  which 
presents  peculiar  difficulties  to  the  experimenter,  and  highly  discrepant 
results  have  been  arrived  at  as  to  the  relations  between  density  and  pressure. 
While  Mendeleef  and  Siljerstrom  have  announced  considerable  deviations 
from  Boyle's  law,  Amagat*  finds  that  law  verified  in  the  case  of  air  to  the 
full  degree  of  accuracy  that  the  observations  admit  of.  In  principle  Amagat's 
method  is  very  simple.  The  reservoir  consists  mainly  of  two  nearly  equal 
bulbs,  situated  one  above  the  other  and  connected  by  a  comparatively  narrow 
passage.  By  the  rise  of  mercury  from  a  mark  below  the  lower  bulb  to 
another  on  the  connecting  passage,  the  volume  is  altered  in  a  known  ratio 
which  is  nearly  that  of  2  :  1.  The  corresponding  pressures  are  read  with  a 
specially  constructed  differential  manometer.  Of  this  the  lower  part  which 
penetrates  the  mercury  of  the  cistern  is  single.  Near  the  top  it  divides  into 
a  U,  widening  at  the  level  of  the  surface  of  the  mercury  into  tubes  of 
2  centims.  diameter.  Higher  up  again  these  tubes  re-unite  and  by  means  of 
a  three-way  tap  can  be  connected  either  with  an  air-pump  or  with  the  upper 
bulb.  Suitable  taps  are  provided  by  which  the  two  branches  can  be  isolated 
from  one  another.  During  the  observations  one  branch  is  vacuous  and  the 
other  communicates  with  the  enclosed  gas,  so  that  the  difference  of  levels 
represents  the  pressure.  This  difference  is  measured  by  a  cathetometer. 

It  is  evident  that  when  the  pressure  is  very  low  the  principal  difficulty 
relates  to  the  measurement  of  this  quantity,  and  that  the  errors  to  be  feared 
in  respect  to  volume  and  temperature  are  of  little  importance.  Amagat, 
fully  alive  to  this  aspect  of  the  matter,  took  extraordinary  pains  with  the 
manometer  and  with  the  cathetometer  by  which  it  was  read.  An  insidious 
*  Ann.  de  Chimie,  Vol.  xxvni.  p.  480  (1883). 


512         ON    A   NEW   MANOMETER,   AND   ON   THE   LAW  OF   THE   PRESSURE      [266 


error  may  enter  from  the  refraction  of  the  walls  of  the  tubes  through  which 
the  mercury  surfaces  are  seen.  But  after  all  his  precautions  Amagat  found 
that  he  could  not  count  upon  anything  less  than  -^  millim.,  even  in  the 
means  of  several  readings.  It  may  be  well  to  give  his  exact  words  (p.  494): — 
"  Dans  les  experiences  dont  je  donnerai  plus  loin  les  resultats  numeriques, 
les  determinations  sont  faites  en  ge"ne"ral  en  alternant  cinq  fois  les  lectures 
sur  chaque  menisque ;  les  lectures  etaient  faites  au  demi-centieme,  et  les 
divergences  dans  les  series  regulieres  oscillent  ordinairement  entre  un 
centieme  et  un  centieme  et  demi ;  en  prenant  la  moyenne,  il  ne  faut  pas 
compter  sur  plus  d'un  centieme ;  et  cela,  bien  entendu,  sans  tenir  compte  des 

causes  d'erreur  independantes  de  la  lecture  cathetome'trique 

Les  resultats  numeriques  consignes  aux  Tableaux  que  je  vais  donner  main- 
tenant  sont  eux-memes  la  moyenne  de  plusieurs  experiences ;  car,  outre  que 
les  lectures  ont  et^  faites  en  general  cinq  fois  en  alternant,  on  est  toujours, 
apres  avoir  reduit  le  volume  a  moitie,  revenu  au  volume  primitif,  puis  au 
volume  moitie :  chaque  experience  a  done  ete  faite  aux  moins  deux  fois,  et 
souvent  trois  et  quatre." 

The  following  are  the  final  results  for  air : — 


Pression 

initiale 
en  millims. 

pv 
p'v' 

Pression 
initiale 
en  millims. 

pv 
p'v' 

Pression 
initiale 
en  millims. 

pv 
p'v' 

millims. 

millims. 

millims. 

12-297 

0-9986 

3-770 

1-0019 

1-377 

1-0042 

12-260 

1-0020 

3-663 

0-9999 

1-316 

1-0137 

10-727 

0-9992 

3-165 

1-0015 

1-182 

1-0030 

7'462 

1-0013 

2-531 

1-0013 

1-140 

1-0075 

7-013 

1-0015 

2-180 

1-0015 

1-100 

0-9999 

6-210 

1-0021 

1-898 

1-0050 

0-978 

1-0160 

6-160 

1-0025 

1-852 

0-9986 

0-958 

1-0100 

4-946 

1-0010 

1-751 

[1]-0030 

0-860 

1-0045 

4-275 

1-0048 

1-457 

1-0150 

0-295 

0-9680 

3-841 

1-0027 

1-414 

1-0143 

Since,  as  it  would  appear,  the  "  initial "  pressure  is  the  smaller  of  a  pair, 
the  lowest  pressure  concerned  is  about  "3  millim.  of  mercury,  and  the  error  at 
this  stage  is  about  3  per  cent.  It  is  not  quite  clear  which  is  which  of  pv 
and  p'v.  For  while  it  is  expressly  stated  that  p  is  smaller  than  p,  the 
value  of  v'jv  is  given  at  2'076.  I  think  that  this  is  really  the  value  of  vfv'. 
But  any  lingering  doubt  that  may  be  felt  upon  this  point  is  of  no  con- 
sequence here,  inasmuch  as  Amagat's  comment  upon  the  tabular  numbers  is 
"  On  ne  saurait  done  se  prononcer,  ni  sur  les  sens  ni  meme  sur  1'existence  de 
ces  ecarts." 


1901]      OF   GASES    BETWEEN    1'5    AND   O'Ol    MILLIMETRES   OF    MERCUKY.        513 

After  such  elaborate  treatment  by  the  greatest  authority  in  these  matters, 
the  question  would  probably  have  long  remained  where  Amagat  left  it,  had 
not  C.  Bohr  found  reason  to  suspect  the  behaviour  of  oxygen  at  low  pressures. 
This  led  to  a  prolonged  and  apparently  very  careful  investigation,  of  which 
the  conclusion  was  that  at  a  pressure  of  7  millim.  of  mercury  the  law  con- 
necting pressure  and  volume  is  subject  to  a  discontinuity. 

"  1.  Bei  einer  Temperatur  zwischen  11°  und  14°  C.  weicht  der  Sauerstoff 
innerhalb  der  beobachteten  Druckgrenzen  von  dem  Boyle-Mariotte'schen 
Gesetze  ab.  Die  Abhangigkeit  zwischen  Volumen  und  Druck  fiir  einen 
Werth  des  letztgenannten,  grosser  als  070  mm.,  kann  man  annahernd  durch 
die  Formel 

(p  +  0109)  v  =  k 

ausdriicken,  wahrend  die  Formel  fur  Werthe  der  Drucke,  welche  kleiner  als 
070  mm.  sind : 

(p  +  0-070)  v  =  k 

ist. 

2.  Sinkt  der  Druck  unterhalb  070  mm.,  so  erleidet  der  Sauerstoff  eine 
Zustandsveranderung ;  er  kann  wieder  durch  ein  Erhb'hen  des  Druckes  bis 
liber  070  mm.  die  urspriingliche  Zustandsform  ubergefuhrt  werden*." 

Fig;  l. 


I  O  2O  3O  4O  SO  80  70  8O  9O  IOO 


Fig.  1  is  a  reproduction  of  one  of  Bohr's  curves,  in  which  the  ordinate 
represents  pv  and  the  abscissa  represents  p  on  such  a  scale  that  1  millim.  of 
mercury  corresponds  to  the  number  20.  It  will  be  seen  that  at  the  place  of 
discontinuity  a  change  of  pv  to  no  less  than  -^  of  its  amount  occurs  with  no 
perceptible  concomitant  change  in  the  value  of  p.  In  the  neighbourhood  of 
the  discontinuity  the  pressure  is  uncertain.  Thus  (p.  475)  "  Wenn  man  bei 
einer  gewissen  Sauerstoffmenge  im  Rohre  a  das  Quecksilber  erst  in  der  Art 
einstellt,  dass  der  Druck  einen  etwas  geringeren  Werth  als  070  millim.  hat, 
und  dann  durch  Verringern  des  Volumens  den  Druck  tiber  0*70  millim. 
steigert  (z.B.  bis  0'8  millim.),  so  zeigt  sich,  dass  dieser  Druck  nicht  constant 
bleibt,  sondern  im  Verlaufe  von  3 — 5  Stunden  bis  zu  einem  Werthe  sinkt,  der 
ungefahr  10  Proc.  kleiner  ist,  als  der  ursprtingliche." 

*  Wied.  Ann.  Vol.  xxvn.  p.  479  (1886). 
R.   iv.  33 


514        ON   A   NEW   MANOMETER,   AND   ON   THE   LAW   OF  THE   PRESSURE        [266 

So  far  as  I  am  aware,  no  attempt  to  repeat  Bohr's  difficult  and  remarkable 
experiments  has  been  recorded,  but  some  confirmation  of  anomalous  behaviour 
of  oxygen  in  this  region  of  pressure  is  afforded  by  the  observations  of  Ramsay 
and  Baly*.  Sutherland!  interprets  the  results  as  a  "  Spontaneous  Change  of 
Oxygen  into  Ozone  and  a  Remarkable  Type  of  Dissociation,"  and  connects 
therewith  some  observations  of  Crookes  relating  to  radiometer  effects  in 
oxygen  gas.  On  the  other  hand,  chemical  tests  applied  by  Professor  Threlfall 
and  Miss  Martin  J  failed  to  indicate  the  presence  of  ozone  in  suitably  expanded 
oxygen. 

Improved  Apparatus  for  Measuring  very  small  Pressures. 

In  spite  of  the  interest  attaching  to  the  anomaly  encountered  by  Bohr, 
I  should  hardly  have  ventured  to  attack  the  question  experimentally  myself, 
had  I  not  seen  my  way  to  what  promised  to  be  an  improved  method  of  dealing 
with  very  small  pressures.  In  operations  connected  with  the  weighing  of 
gases,  extending  over  a  series  of  years,  I  have  had  much  experience  of  a 
specially  constructed  manometric  gauge  in  which  an  iron  rod,  provided  above 
and  below  with  suitable  points,  is  actually  applied  to  the  two  mercury 
surfaces  arranged  so  as  to  be  situated  in  the  same  vertical  line§.  Although 
two  variable  quantities  had  to  be  adjusted — the  pressure  of  the  gas  and  the 
supply  of  mercury — no  serious  difficulty  was  encountered ;  and  the  delicacy 
obtained  in  the  observation  of  the  approximation  of  a  point  and  its  image  in 
the  mercury  surface  with  the  assistance  of  an  eye-lens  of  25  millims.  focus 
was  very  satisfactory.  In  order  to  get  actual  measures  of  the  delicacy,  a 
hollow  glass  apparatus  in  the  form  of  a  fork  was  mounted  upon  a  levelling 
table.  The  stalk  below  was  terminated  with  a  short  length  of  rubber  tubing 
compressible  by  a  screw.  This  allowed  the  supply  of  mercury  to  be  adjusted. 
The  mercury  surfaces  in  the  U  were  about  20  millims.  in  diameter,  and  were 
exposed  to  the  air.  They  were  to  be  adjusted  to  coincidence  with  needle 
points,  rigidly  connected  to  the  glass-work,  by  suitable  use  of  the  compressor 
and  of  the  screw  of  the  levelling  table.  Readings  of  the  latter  in  successive 
and  independent  settings  showed  that  a  degree  of  accuracy  was  attainable 
much  superior  to  the  limit  fixed  by  Amagat  for  the  best  work  with  the 
cathetometer.  It  is  unnecessary  to  record  the  numbers  obtained  at  this 
stage  of  the  work,  inasmuch  as  the  final  results  to  be  given  below  prove  that 
the  errors  of  setting  are  considerably  less  than  y^1^  millim. 

It  will  now  be  possible  to  form  a  preliminary  idea  of  the  proposed  mano- 
meter. The  readings  of  the  levelling  screw,  obtained  as  above,  may  be 

*  Phil.  Mag.  Vol.  xxxvm.  p.  301  (1894). 
+  Phil.  Mag.  Vol.  xmi.  p.  201  (1897). 
J  Proc.  Roy.  Soc.  of  New  South  Wale*,  1897. 

§  "  On  the  Densities  of  the  Principal  Gases,"  Proc.  Roy.  Soc.  Vol.  LIH.  p.  134, 1893.  [Vol.  iv. 
p.  39.1 


1901]     OF   GASES    BETWEEN    1'5    AND    O'Ol    MILLIMETRES   OF   MERCURY.         515 


regarded  as  corresponding  to  the  zero  of  pressure,  or  rather  of  pressure 
difference.  If  the  pressures  operative  upon  the  mercury  surfaces  be  slightly 
different,  the  setting  is  disturbed ;  and  the  change  of  reading  at  the  screw 
required  to  re-establish  the  adjustment  represents  the  difference  of  pressures. 
In  order  to  interpret  the  result  absolutely  it  is  only  necessary  to  know  further 
the  pitch  of  the  levelling  screw,  the  leverage  with  which  it  acts,  and  the 
distance  between  the  points  to  which  the  mercury  surfaces  are  set.  If  the 
space  over  one  mercury  surface  be 
vacuous,  the  change  of  reading  at 
the  levelling  screw  represents  the 
absolute  pressure  in  the  space  over 
the  other  mercury  surface. 

The  difficulty,  which  will  at  once 
present  itself  to  the  mind  of  the 
reader,  in  the  use  of  a  manometer  on 
this  plan,  is  the  necessity  for  a  flex- 
ible connexion  between  the  instru- 
ment and  the  rest  of  the  apparatus, 
such  as  the  air-pump  and  the  vessel 
in  which  the  pressure  is  required  to 
be  known.  With  the  aid  of  short 
lengths  of  rubber  tubing  this  re- 
quirement could  be  easily  met,  but 
the  class  of  work  for  which  such  a 
manometer  is  wanted  would  usually 
preclude  the  use  of  rubber.  In  my 
apparatus  the  requisite  flexibility  is 
obtained  by  the  insertion  of  con- 
siderable lengths  (3  metres)  of  glass 
tubing  between  the  manometer  and 
the  parts  which  cannot  turn  with 
it.  Although  the  adjustment  was 
made  by  the  screw  of  a  levelling 
table  as  described,  the  actual  readings 
were  taken  by  the  mirror  method, 
the  supports  of  the  mirror  being 
connected  as  directly  as  possible 
with  the  pointswhose  angular  motion 
is  to  be  registered.  In  this  way  we 
become  independent  of  the  rigidity 
of  the  glass-work,  and  are  permitted  to  use  wood  freely  in  the  levelling 
table  and  in  its  supports.  It  frequently  happened  that  an  adjustment  left 
correct  was  found  to  be  out  after  an  interval.  The  screw  had  not  been 

33—2 


516         ON    A   NEW   MANOMETER,   AND   ON   THE   LAW   OF   THE   PRESSURE       [266 

moved,  but  the  mirror-reading  was  altered.  On  resetting  by  use  of  the 
screw,  the  original  mirror-reading  was  recovered  within  the  limits  of 
error. 

Fig.  3. 


The  essential  parts  of  the  manometer,  as  finally  employed,  are  shown 
(Fig.  2)  in  elevation  and  plan,  and  the  general  scheme  of  the  mounting  is 
indicated  in  Fig.  3.  At  A  is  the  stalk  of  the  glass  fork,  of  such  length  that 
the  mercury  in  the  hose  below  is  always  at  a  pressure  above  atmosphere ; 
B,  B  are  bulbs  of  about  25  millims.  diameter,  at  the  centres  of  which  are 
situated  the  points.  These  are  of  glass  *,  which  need  not  be  opaque ;  and 
they  must  be  carefully  finished  upon  a  stone.  A  considerable  degree  of 
sharpness  is  desirable,  but  similarity  is  more  important  than  the  extreme  of 
sharpness.  In  the  actual  apparatus  complete  similarity  was  not  attained,  and 
in  the  first  trials  the  difference  was  rather  embarrassing.  However,  after  a 
little  practice  the  eye  becomes  educated  to  set  the  mercury  to  each  point  in 

*  At  first  iron  needle  points  were  tried. 


1901]     OF   GASES    BETWEEN    T5    AND   O'Ol    MILLIMETRES   OF   MERCURY.         517 

a  constant  manner,  and  this  is  all  that  is  really  required.  The  same  con- 
sideration shows  that  minute  outstanding  capillary  differences  should  not 
lead  to  error.  It  may  be  remarked  that  the  mercury  is  always  on  the  rise  at 
the  time  of  adjustment,  and  in  fact  it  was  found  best  to  make  it  a  rule  not 
to  allow  the  points  to  be  drowned  at  any  time  when  it  could  be  avoided. 
After  such  a  drowning  it  was  usually  (perhaps  always)  found  that  the 
mercury  surface  was  disturbed  by  the  proximity  of  the  points  without  actual 
contact,  an  effect  attributed  to  electrification. 

The  presentation  of  the  point  to  the  mercury,  or  rather  of  the  point  to 
its  image  as  seen  by  reflection  in  the  mercury,  was  examined  with  the  aid  of 
two  similar  eye-lenses  (not  shown)  of  22  millims.  focus.  The  illumination, 
from  a  small  gas  flame  suitably  reflected  by  mirrors,  was  from  behind,  and  it 
and  the  lenses  were  so  arranged  that  both  points  could  be  seen  without  a 
motion  of  the  head.  Precautions  were  required  to  prevent  the  radiation 
from  the  gas  flame  and  from  the  observer  from  producing  disturbance, 
especially  by  unequal  heating  of  the  two  limbs  of  the  U.  The  U  itself  was 
well  bandaged  up,  and  between  it  and  the  observer  were  interposed  sheets 
of  copper  and  of  insulating  material  so  as  to  ensure  that  at  all  events 
there  should  be  no  want  of  symmetry  in  any  heating  that  might  take  place. 

The  adjustment  itself  is  a  double  one,  requiring  both  the  use  of  the 
levelling  screw  J  and  an  accurate  feed  of  mercury.  The  hose  terminates 
as  usual  in  a  small  mercury  reservoir  D.  This  facilitates  the  preliminary 
arrangements,  but  in  use  the  reservoir  is  cut  off  by  a  screw  clamp  E  just 
below  it.  The  rough  adjustment  of  the  supply  of  mercury  is  effected  by 
a  large  wooden  compressor  F.  The  fine  adjustment  required  for  the  actual 
setting  is  a  more  delicate  matter.  The  first  attempts  were  by  fine  screw 
compressors  acting  upon  the  pendent  part  of  the  hose,  but  the  tremors  thence 
arising  were  found  very  disturbing.  A  remedy  was  eventually  applied  by 
operating  upon  the  part  of  the  hose  which  lies  flat  upon  the  floor  or  rather 
on  the  bottom  of  a  mercury  tray.  The  compressor  is  shown  at  G,  Fig.  3 ; 
the  screw  being  provided  with  a  long  handle  H  to  bring  it  within  con- 
venient reach.  The  advantage  accruing  from  this  small  device  would  scarcely 
be  credited. 

The  glass-work  is  attached  by  cement  to  a  board,  which  hangs  down- 
wards in  face  of  the  observer  and  is  itself  fixed  rigidly  to  the  levelling  stand 
K.  This  is  supported  at  two  points  /,  which  define  the  axis  of  rotation,  and 
by  a  finely  adjustable  screw  J,  within  reach  of  the  observer.  The  whole 
stands  in  a  very  steady  position  upon  the  floor  of  an  underground  cellar  in 
my  country  house. 

The  arrangements  for  the  connexion  of  the  mirror  must  now  be  de- 
scribed. The  glass  stems,  whose  lower  extremities  form  the  "points,"  are 
prolonged  upwards  by  substantial  tubing,  and  terminate  above  in  three 


518         ON   A   NEW   MANOMETER,   AND   ON   THE   LAW   OF   THE   PRESSURE 

slightly  rounded  ends,  L,  L,  suitable  for  the  support  of  the  mirror  plat- 
form N.  The  two  supports  necessary  on  the  left  are  obtained  by  a 
symmetrical  branching  of  the  tube  on  that  side.  The  platform  is  of  worked 
glass,  so  that  a  slight  displacement  of  the  contacts  has  no  effect  on  the 
slope  of  the  mirror.  The  latter  is  of  worked  glass  silvered  in  front.  Suitable 
stops  are  provided  to  guide  the  mirror  platform  into  the  right  position  and  to 
prevent  accidents,  but  these  exercise  no  constraint. 

The  axis  //  about  which  the  apparatus  rotates  is  horizontal  and  parallel 
to  the  face  of  the  mirror,  so  that  the  sine  of  the  angle  6  of  rotation  from  the 
zero  position  represents  the  difference  of  levels  of  the  mercury  surfaces. 
The  axis  //  lies  approximately  in  the  mirror  surface  and  at  about  the  middle 
of  the  height  of  the  operative  part.  The  rotation  of  the  mirror  is  observed 
in  the  usual  way  by  means  of  a  telescope  and  vertical  millimetre  scale.  The 
aperture  of  the  object-glass  is  30  millims.,  and  the  distance  from  the  mirror 
3150  millims.  The  readings  can  be  taken  to  about  '1  millim. 

In  many  kinds  of  observation  the  zero  can  only  be  verified  at  intervals, 
as  it  requires  the  pressures  over  the  mercury  to  be  equalised.  On  the  whole 
the  zero  was  tolerably  constant  to  within  two  or  three-tenths  of  a  millimetre 
of  the  scale.  A  delicate  level  was  attached  to  the  telescope  to  give  warning 
of  any  displacement  of  the  stand  (all  of  metal)  or  of  the  ground. 

The  differences  of  pressure  to  be  evaluated  are  not  quite  in  simple  propor- 
tion to  the  scale  reading  from  zero.  The  latter  varies  as  tan  20,  while  the 
former  varies  as  sin  6.  The  correcting  factor  is  therefore 

f  lm~20  =  X  -  1*9*  approximately. 

If  the  zero  reading  (in  millimetres)  be  a,  and  the  current  reading  as,  D  the 
distance  between  telescope  and  mirror, 

a      x  —  a  .      .  . 

8  =  —nj)     approximately; 

so  that  the  correcting  factor  is 


The  actual  correction  to  be  applied  to  (x  —  a)  is  thus 


In  practice  (x  —  a)  rarely  exceeded  350,  for  which  the  correction  would 
be  —  1'6.  When  (x  —  a)  falls  below  120,  the  correction  is  insensible. 

The  next  question  is  the  reduction  to  absolute  measure.  What  (cor- 
rected) scale-reading  corresponds  to  1  millim.  actual  difference  of  mercury 
levels  ?  The  distance  between  the  points  is  27'3  millims.,  so  that  1  millim. 


1901]      OF   GASES   BETWEEN    1'5    AND   O'Ol    MILLIMETRES   OF    MERCURY.        519 

mercury  corresponds  to  231   millims.  of  the  telescope  scale.     The  highest 
pressure  that  could  be  dealt  with  is  about  1^  millims.  of  mercury. 

The  above  reckoning  proceeds  upon  the  supposition  that  the  distance 
between  the  points  can  be  regarded  as  invariable.  Certain  small  discrepancies 
manifested  at  the  higher  slopes  of  the  apparatus  induced  me  to  examine  the 
question  more  particularly,  for  it  seemed  not  impossible  that  owing  to  the 
bending  of  the  glass-work  some  displacement  might  occur.  But  a  rather 
troublesome  measurement  of  the  actual  distance  in  various  positions  by  means 
of  microscopes  negatived  this  idea.  I  would  however  recommend  that  this 
point  be  kept  specially  in  view  in  the  design  of  any  subsequent  apparatus  of 
this  kind. 

Experiments  to  determine   the   Relation   of  Pressure   and  Volume   at  given 

Temperature. 

In  order  to  test  Boyle's  law  one  of  the  lateral  branches  C  is  connected  to 
the  air-pump  and  the  other  to  the  chamber  in  which  the  gas  is  contained. 
The  pump  is  of  the  Topler  form,  and  is  provided  with  a  bulb  containing 
phosphoric  anhydride.  No  tap  or  contracted  passage  intervenes  between  the 
pump-head  and  B.  A  lateral  channel  communicates  with  a  three-way  tap,  by 
which  this  side  of  the  apparatus  can  be  connected  with  the  gas-generating 
vessel.  The  third  way  leads  to  a  blow-off  under  mercury  more  than  a 
barometer-height  below. 

The  two  sides  of  the  apparatus  are  connected  by  a  cross-tube  which  can 
be  closed  or  opened  by  means  of  a  tap.  The  plug  of  this  tap  is  provided 
with  a  wide  bore.  When  it  is  intended  to  read  the  zero,  the  tap  is  open. 
If  desired,  the  mercury  may  be  raised  in  the  Topler  so  as  to  prevent  the 
penetration  of  gas  into  the  pump-head.  When  pressures  are  to  be  observed, 
the  tap  of  the  cross-tube  is  closed,  and  a  good  vacuum  is  made  on  the  pump 
side.  No  particular  difficulty  was  experienced  with  the  vacuum.  In  the  use 
of  the  Topler  the  mercury  was  allowed  to  flow  out  below,  and  was  trans- 
ferred at  intervals  to  the  movable  reservoir.  The  latter  was  protected  from 
atmospheric  moisture  by  a  chloride  of  calcium  tube.  When,  after  standing 
five  or  ten  minutes,  the  mercury  was  put  over,  and,  on  impact,  gave  a  hard 
metallic  sound  with  inclusion  of  no  more  than  a  small  speck  of  gas,  the 
vacuum  was  nearly  sufficient,  and  no  further  change  could  be  detected  at  the 
manometer.  The  capacity  of  the  pump-head  was  two  or  three  times  that  of 
the  remaining  space  to  be  exhausted. 

In  the  earlier  experiments  the  gas-containing  tube,  placed  vertically,  was 
graduated  to  50  cub.  centims.  at  intervals  of  10  cub.  centims.  Prolonged 
below  by  more  than  a  barometer-height  of  smaller  tubing,  it  terminated  in  a 
hose  and  mercury  reservoir,  the  latter  protected  by  chloride  of  calcium.  In 


520         ON    A    NEW   MANOMETER,   AND   ON   THE   LAW   OF   THE   PRESSURE       [266 

order  to  get  rid  of  most  of  the  adherent  moisture  and  carbonic  anhydride, 
the  tubes  on  both  sides  of  the  apparatus  were  heated  pretty  strongly  in  a 
vacuous  condition.  The  first  trial  was  with  oxygen,  in  the  hope  of  at  once 
obtaining  a  confirmation  of  Bohr's  anomaly;  but  not  succeeding  in  this, 
I  fell  back  upon  nitrogen  and  hydrogen.  With  a  vacuum  on  the  pump  side, 
readings  of  pressure  were  taken  with  the  mercury  in  the  chamber  at  0  and 
at  50  cub.  centims.,  and  the  ratio  of  pressures  (about  2:1)  was  deduced. 
When  this  had  been  repeated,  some  of  the  gas  was  allowed  to  escape  by 
opening  the  cross-tap,  the  zero  was  again  observed,  and  the  vacuum  re- 
established on  the  pump  side.  Another  ratio  of  pressures  could  now  be 
obtained,  corresponding  to  the  same  (unknown)  volumes  as  before,  but  to  a 
different  total  pressure. 

In  utilising  the  ratios  of  pressure  thus  obtained,  it  was  of  course  necessary 
to  consider  how  far  the  temperature  could  be  assumed  to  be  unchanged 
within  each  pair  of  pressures  brought  into  comparison.  The  general  tem- 
perature of  the  cellar  was  extremely  uniform,  and  no  difference  could  be 
read  upon  a  thermometer  worth  taking  into  account.  Passing  over  this 
question  for  the  present,  we  may  consider  how  far  the  results  conformed  to 
Boyle's  law.  The  agreement  of  the  ratios,  except,  perhaps,  at  the  highest 
pressures  of  about  1^  millims.  of  mercury,  was  sufficiently  good,  and  of  itself 
goes  a  long  way  to  confirm  Boyle's  law.  In  strictness,  all  that  the  constancy 
of  the  ratio  can  prove  is  that  the  relation  between  pressure  (p)  and  density 
(p)  is  of  the  form 

p  =  *pn, (1) 

where  n  is  some  numerical  quantity.  To  limit  n  to  the  value  unity,  the 
constancy  of  the  ratios  might  be  followed  up  into  the  region  of  pressure 
for  which  Boyle's  law  is  known  to  hold,  but  this  can  scarcely  be  said  to 
have  been  done  here.  Otherwise,  we  need  to  know  what  the  ratio  of  densities 
in  the  two  positions  of  the  mercury  really  is,  and  not  merely  that  it  remains 
constant. 

In  the  case  of  the  original  volume  chamber  the  first  was  the  method 
employed.  The  smaller  volume,  defined  by  the  upper  mark  in  the  volume 
tube  and  by  the  "point"  in  the  manometer,  was  filled  with  dry  air  at  a 
known  atmospheric  pressure.  The  included  air  was  then  isolated  and 
expanded  until  it  occupied  the  larger  (approximately  double)  volume,  and 
the  new  pressure  determined  by  observation  of  the  difference  of  levels  in 
the  tube  and  in  a  mercury  reservoir  similarly  fashioned.  The  operation  was 
rather  a  difficult  one,  and  the  result  was  only  barely  accurate  enough.  The 
ratio  of  volumes  thus  determined  by  use  of  Boyle's  law,  as  applied  to  air 
at  atmospheric  and  half  atmospheric  pressures,  agreed  sufficiently  well  with  the 
ratio  of  pressures  found  by  the  manometer  for  rare  hydrogen  and  nitrogen ; 


1901]     OF   GASES    BETWEEN    1'5    AND   O'Ol    MILLIMETRES    OF    MERCURY.         521 

and  thus  Boyle's  law  may  be  considered  to  be  extended  to  these  rare  gases. 
The  rarefaction  was  carried  down  to  a  total  pressure  of  only  '02  millim.  At 
this  stage  discrepancies  of  the  order  of  5  per  cent,  are  to  be  expected. 

Having  obtained  fairly  satisfactory  results  with  hydrogen  and  nitrogen, 
I  returned  to  oxygen,  fully  expecting  to  verify  the  anomalous  behaviour 
described  by  Bohr.  In  this  I  have  totally  failed.  The  gas  was  prepared  by 
heating  permanganate  of  potash,  dried  by  phosphoric  anhydride,  and  may  be 
regarded  as  fairly  pure.  The  region  of  pressure  round  '7  millim.  was  carefully 
examined,  use  being  made  of  the  intermediate  divisions  of  the  50  cub. 
centims.  range  of  volume.  No  unsteadiness  of  the  kind  indicated  by  Bohr, 
or  appreciable  departure  from  Boyle's  law,  was  detected.  And  when  the 
pressures  were  diminished  down  to  a  few  hundredths  of  a  millimetre,  there 
was  no  falling  off  in  the  product  of  pressure  and  volume.  The  observations 
were  repeated  a  second  time  with  a  fresh  supply  of  oxygen. 

The  experience  gained  up  to  this  date  (August,  1900)  showed  that  the 
manometer  worked  well,  and  that  there  was  no  difficulty  about  the  vacuum, 
but  I  was  not  altogether  satisfied  with  the  way  in  which  the  volumes  had 
been  determined.  There  was  some  want  of  elegance,  to  say  the  least,  in 
using  Boyle's  law  for  this  purpose,  and  barely  adequate  accuracy  in  the  appli- 
cation itself.  The  latter  objection  might  have  been  overcome  by  the  use  of 
a  suitable  cathetometer,  but  such  was  not  to  hand.  The  most  direct  method 
by  actually  gauging  with  mercury  the  spaces  concerned  being  scarcely 
feasible,  I  devised  another  method  which  has  the  advantage  of  easy  execution 
and*  is  practically  independent  of  the  assumption  of  Boyle's  law.  The 
opportunity  was  taken  to  increase  the  range  over  which  the  volume  could  be 
varied. 

The  new  chamber,  composed  mainly  of  tubing  of  18  millims.  diameter,  is 
graduated  at  intervals  of  10  cub.  centims.  over  a  total  range  of  200  cub. 
centims.  It  is  prolonged  above  and  below  by  narrow  tubing  in  order  to 
connect  it  with  the  sloping  manometer  bulb  and  with  the  hose  and  mercury 
reservoir  as  before.  The  zero  mark  is  situated  on  the  upper  tube  a  few 
centimetres  above  its  junction  with  the  wider  one.  It  is  scarcely  necessary 
to  say  that  no  rubber  was  employed  except  for  the  hoses,  and  that  these 
were  always  occupied  by  mercury  under  a  pressure  above  atmosphere.  The 
mercury  reservoirs  themselves  were  protected  against  damp  by  chloride  of 
calcium. 

If  we  call  the  ungauged  volume  (from  the  zero  mark  to  the  bulb  of  the 
sloping  manometer  with  "point"  set)  V,  and  the  gauged  volume  v,  the  total 
volume  occupied  by  the  gas  is  V  +  v ;  and  the  problem  is  how  to  determine  V. 
If  we  may  assume  the  correctness  of  Boyle's  law  for  rare  gases  and  may  rely 
upon  the  sloping  manometer,  the  process  is  simple  enough.  We  have  only  to 
find  the  pressures  exerted  by  the  included  gas  at  volumes  V  and  V  +  v,  whence 


522    ON  A  NEW  MANOMETER,  AND  ON  THE  LAW  OF  THE  PRESSURE   [266 

by  Boyle's  law  the  ratio  of  these  volumes  is  known  and  thus  V  determined 
in  terms  of  v.  In  order  to  avoid  the  use  of  Boyle's  law,  further  observations 
are  necessary. 

The  requisite  data  can  be  obtained  by  changing  the  quantity  of  gas. 
Suppose  that  with  the  original  quantity  of  gas  certain  pressures,  P,  P',  corre- 
spond to  total  volumes,  V  +  vlt  V  +  v2,  and  that  with  a  reduced  amount  of  gas 
the  same  pressures  are  recorded  with  volumes  V+v3,  V  +  v4.  Since  the 
pressure  is  a  function  of  the  density,  whether  Boyle's  law  be  applicable  or  not, 
it  must  follow  that 

V  +  v,      V+v3 

r+^-FTv  ..............................  9 

whence  V  is  determined  in  terms  of  the  known  volumes  vlt  v2,  v3,  v4.  It  may 
be  remarked  that  this  argument  does  not  assume  even  the  correctness  of  the 
scale  of  pressures. 

In  carrying  out  the  method  practically  it  was  necessary  to  work  to  the 
fixed  marks  of  the  volume  chamber,  and  thus  the  same  pressures  could  not  be 
recovered  exactly.  But  the  use  of  Boyle's  law  in  order  to  make  what  is 
equivalent  to  small  corrections  is  unobjectionable. 

With  this  explanation  it  may  suffice  to  give  the  details  of  an  actual 
determination  executed  with  nitrogen.  With  the  original  quantity  of  gas, 
volumes  F+70,  F+170  gave  pressures  proportional  to  345'4,  184'9. 
Sufficient  gas  was  now  removed  to  allow  the  remainder  to  give  nearly  the 
same  higher  pressure  as  before  with  v  =  0.  Thus,  corresponding  to  volumes 
F+  0,  V  +  40  the  pressures  were  344'9,  183'3.  We  have  now  only  to  calcu- 
late V  from  the  equation 

V  +  40     344-9    184-9  F+170 


345^4    F+70  ' 

or  F»  +  110F+  2800  =  1-0072  (F2  +  170F)  ; 

whence  F  =  45'5  cub.  centims. 

The  adopted  value,  derived  from  observations  upon  nitrogen  and  hydrogen,  is 
F  =  45  '6  cub.  centims. 

In  charging  the  apparatus,  the  first  step  is  to  make  a  good  vacuum 
throughout,  the  cross-tap  being  open.  The  gas  supply  being  started,  the 
first  portions  are  allowed  to  blow  off  from  under  mercury,  and  then,  by  use 
of  the  three-way  tap,  a  sufficiency  is  introduced  into  the  apparatus  to  an 
absolute  pressure  of,  perhaps,  10  centims.  of  mercury.  The  gas-leading  tube 
would  then  be  sealed  off.  Ultimately  the  remainder  of  the  supply  tube  and 
the  blow-off  tube  were  exhausted  to  diminish  the  risk  of  leakage. 


1901]     OF   GASES    BETWEEN    1'5    AND   O'Ol    MILLIMETRES   OF   MERCURY.        523 

The  "  nitrogen"  was  prepared  from  air  by  passage  over  red-hot  copper  and 
desiccation  with  phosphoric  anhydride.  Accordingly  it  contained  argon  to 
the  amount  of  about  1  per  cent. 

In  taking  a  set  of  observations  the  procedure  would  be  as  follows. 
Assurance  having  been  obtained  that  the  vacuum  was  good,  the  next  step 
would  be  to  set  the  mercury  in  the  volume  chamber  so  that  v  =  190  cub. 
centims.,  then  after  a  few  minutes  to  adjust  the  sloping  manometer  and  to 
read  the  telescope  scale.  It  was  of  course  necessary  to  ensure  that  sufficient 
time  was  allowed  for  uniformity  of  pressure  to  establish  itself,  and  observa- 
tions were  frequently  renewed  after  a  quarter  of  an  hour  or  longer.  In  the 
case  of  oxygen,  to  be  considered  later,  several  hours  were  sometimes  allowed. 
If  operations  were  leisurely  conducted,  with  first  a  rough  setting  of  the 
volume  and  then  a  rough  setting  of  the  manometer  followed  by  accurate 
settings  in  the  same  order,  little  or  no  change  could  afterwards  be  detected. 
Indeed  I  was  rather  surprised  to  find  how  rapidly  equilibrium  seemed  to  be 
established.  The  next  smaller  volume,  e.g.,  v  =  150,  would  then  be  observed, 
and  so  on  until  v  =  40.  In  observations  to  be  used  for  the  examination  of 
Boyle's  law  v  was  not  further  reduced,  as  too  much  stress  might  thereby  be 
thrown  upon  the  accuracy  of  V.  The  same  observations  were  then  repeated 
in  reverse  order  and  the  mean  taken.  The  numbers  recorded  are  thus  the 
mean  of  two  settings  only  of  the  manometer. 

The  next  step  was  to  allow  about  half  the  gas  to  escape.  The  mercury 
at  the  pump  was  allowed  to  rise  so  as  to  cut  off  the  pump-head  and  V  +  v  was 
so  adjusted  as  to  be  equal  to  the  volume  remaining  upon  the  other  side,  about 
130  cub.  centims.  The  cross-tap  was  then  opened,  and  after  a  sufficient  interval 
of  time  the  zero,  corresponding  to  no  pressure,  was  read.  In  the  course  of  the 
observations  upon  nitrogen,  extending  over  ten  days,  the  zero  varied  from 
43'5  to  43'8.  Whenever  possible  the  zero  used  for  a  set  was  the  mean  of 
values  found  before  and  after. 

The  annexed  tables  give  the  results  for  nitrogen  in  detail.  In  Table  I., 
dealing  with  the  highest  quantity  of  gas,  the  first  column  gives  the  volume 
( V  =  45'6  cub.  centims.) ;  the  second  represents  the  pressure,  being  the  mean 
of  the  two  actually  read  numbers  (expressing  millimetres  of  telescope  scale) 
less  the  zero  reading  437  and  corrected  to  infinitely  small  arcs  as  already 
explained.  The  third  column  is  the  logarithm  of  the  product  of  the  first 
two,  and  should  be  constant  if  Boyle's  law  holds.  The  fourth  column  gives 
the  approximate  value  of  the  pressure  in  millimetres  of  mercury;  the 
fifth  the  deviation  of  pv  from  the  mean  taken  as  unity.  In  the  sixth 
column  is  shown  the  amount  by  which  the  observed  value  of  p  exceeds 
that  requisite  in  order  to  make  pv  constant,  expressed  in  millimetres  of 
mercury. 


524         ON   A   NEW   MANOMETER,   AND   ON   THE   LAW   OF   THE   PRESSURE       [266 


TABLE  I. — Nitrogen. 
November  9-11,     Zero  =  43'7. 


Volume  in 
cub.  ceiitims. 

Pressure  in 
scale  divisions 

Log.  product 

Pressure  in 
millims.  Hg 

Deviation 
of  pv 

Error  of  p 
in  millims. 

V+   70 

345-4 

•6013 

1-49 

+  •0002 

+  •0003 

V+  80 

318-3 

•6018 

1-38 

+  •0014 

+  •0019 

V+  90 

294-1 

•6007 

1-27 

-•0012 

-•0015 

F+110 

256-8 

•6016 

1-11 

+  •0009 

+  •0010 

F+130 

227-4 

•6013 

•98 

+  •0002 

+  •0002 

F+150 

203-7 

•6004 

•88 

-•0018 

-•0016 

F-f  170 

184-9 

•6005 

•80 

-•0014 

-•0011 

F+190 

169-8 

•6021 

•73 

+  -0021 

+  •0015 

•6012 

TABLE  II. — Nitrogen. 
November  11-12,     Zero  =  437. 


Volume  in 
cub.  centims. 

Pressure  in 
scale  divisions 

Log.  product 

Pressure  in         Deviation 
millims.  Hg    \         of  pv 

Error  of  p 
in  millims. 

F+     0 

344-9 

•1966 

1-49 

+  •0007 

+  •0010 

F+   10 

282-3 

•1958 

1-22 

-•0012 

-•0015 

F+   20 

239-5 

•1962 

1-04 

-•0002 

-•0002 

F+  40 

183-3      . 

•1956 

•79 

-•0016 

-•0013 

F+  60 

148-8 

•1963 

•64 

•0000 

•oooo 

F+  80 

125-2 

•1966 

•54 

+  •0007 

+  •0004 

F+110 

101-1 

•1968 

•44 

+  O012 

+  •0005 

F+150 

80-2 

•1955 

•35 

-•0018 

-•0006 

F+190 

66-9 

•1976 

•29 

+  •0030 

+  •0009 

•1963 

1901]     OF   GASES    BETWEEN    1'5    AND   O'Ol    MILLIMETRES   OF   MERCURY.        525 

TABLE  III. — Nitrogen. 
November  13,     Zero  =  43'6. 


Volume  in 
cub.  centims. 

Pressure  in 
scale  divisions 

Log.  product 

Pressure  in 

inillinis.  Hg 

Deviation 
of  pv 

Error  of  p 
in  millims. 

V+  40 

91-1 

•892 

•394 

•ooo 

•oooo 

V+   60 

73-9 

•892 

•320 

•ooo 

•oooo 

V+   80 

62-3 

•893 

•269 

+  •002 

+  •0005 

F+110 

50-2 

•893 

•217 

+  •002 

+  •0004 

F+  150 

39-6 

•889 

•171 

-•007 

-•0012 

F+190 

33-1 

•892 
•892 

•143 

•ooo 

•oooo 

TABLE  IV.— Nitrogen. 
November  14,     Zero  =  43*5. 


Volume  in 
cub.  centims. 

Pressure  in 
scale  divisions 

Log.  product 

Pressure  in 
millims.  Hg 

Deviation 
of  pv 

Error  of  p 
in  millims. 

F+   40 

46-0 

•595 

•199 

+  •005 

+  •0010 

F+  60 

37-1 

•593 

•160 

•ooo 

•oooo 

F+  80 

31-1 

•592 

•135 

-•002 

-•0003 

F+110 

25-1 

•592 

•109 

-'002 

-•0002 

F+150 

20-1 

•595 

•087 

+  •005 

+  •0004 

F+190 

16-5 

•590 

•071 

-•007 

-•0005 

•593 

TABLE  V. — Nitrogen. 
November  16,     Zero  =  43'5. 


Volume  in 
cub.  centims. 

Pressure  in 
scale  divisions 

Log.  product 

Pressure  in 
millims.  Hg 

Deviation 
of  pv 

Error  of  p 
in  millims. 

F+   40 

22-8 

•290 

•099 

•ooo 

•oooo 

F+   60 

18-6 

•293 

•081 

+  •007 

+  •0006 

F+   80 

15-6 

•292 

•067 

+  •005 

+  •0003 

F+110 

127 

•296 

•055 

+  •014 

+  •0008 

F+150 

9-9 

•287 

•043 

-•007 

-•0003 

F+190 

8-15 

•283 
•290 

•035 

-•016 

-'0006 

526          ON   A   NEW   MANOMETER,   AND   ON   THE   LAW   OF   THE   PRESSURE      [266 


TABLE  VI.— Nitrogen. 
November  17-18,     Zero  =  43'7. 


Volume  in 
cub.  centims. 

Pressure  in 
scale  divisions 

Log.  product 

Pressure  in 
millims.  Hg 

Deviation 
oipv 

Error  of  p 
in  millims. 

V+   40 

11-40 

•989 

•049 

+  •005 

+  '0002 

V+   60 

9-10 

•983 

•039 

-•009 

-  -0004 

V+  80 

7-65 

•983 

•033 

-•009 

-•0003 

7+110 

6-25 

•988 

•027 

+  •002 

+  •0001 

7+150 

5-10 

•999 

•022 

+  028 

+  •0006 

7+190 

4-05 

•980 

•017 

-•016 

-  '0003 

•987 

TABLE  VII.— Nitrogen. 
November  18-19,     Zero  =  43'8. 


Volume  in 
cub.  centims. 

Pressure  in 
scale  divisions 

Log.  product 

Pressure  in 
millims.  Hg 

Deviation 
of  pv 

Error  of  p 
in  millims. 

F+  40 

5-90 

•703 

•026 

+  •014 

+  •0004 

7+  60 

4-60 

•686 

•020 

-•026 

-•0005 

7+  80 

4-15 

•717 

•018 

+  '047 

+  •0008 

7+110 

3-10 

•683 

•013 

-•033 

-•0004 

7+150 

2'55 

•698 

•on 

+  •002 

•oooo 

•697 

In  the  second  set  the  quantity  of  gas  had  been  adjusted  to  give  a  suitable 
pressure  with  v  =  0.  It  is  from  it  and  from  Table  I.  that  the  data  were 
obtained  for  the  calculation  of  V  already  given. 

These  tables  give  a  fairly  complete  account  of  the  behaviour  of  nitrogen 
from  a  pressure  of  about  1'5  millims.  down  to  '01  millim.  of  mercury.  In 
each  set  the  range  of  pressure  is  nearly  in  the  ratio  of  3  :  1,  and  overlaps  the 
range  of  the  preceding  and  following  sets.  An  examination  of  the  fifth 


1901]     OF   GASES   BETWEEN    1'5    AND   O'Ol    MILLIMETRES   OB^    MERCURY.        527 

column  shows  no  indication  of  departure  from  Boyle's  law.  The  sixth 
column  allows  a  judgment  to  be  formed  of  the  degree  of  accuracy  to  which 
the  law  is  verified.  It  gives  the  amount  by  which  p  exceeds  the  value 
necessary  in  order  that  pv  should  be  absolutely  constant,  expressed  in  milli- 
metres of  mercury.  The  errors  thus  exhibited  include  not  only  those  arising 
in  the  setting  of  the  manometer  and  the  reading  of  the  telescope,  but  also 
those  entailed  in  the  measurements  of  volume,  and  in  consequence  of  fluctua- 
tions of  temperature.  The  latter  source  of  error  is  of  course  more  important 
at  the  higher  pressures.  It  will  be  seen  that  the  accuracy  attained  is  very 
remarkable.  Even  at  the  higher  pressures  the  mean  error  is  only  about 
•001  millim.,  while  at  the  lower  pressures  of  Tables  III. — VII.  the  mean  error 
is  less  than  '0004  millim.  And  it  must  be  remembered  that  the  numbers  to 
which  these  errors  relate  are  the  means  of  two  observations  only. 

As  a  means  of  dealing  with  very  small  pressures,  the  sloping  manometer 
has  proved  itself  in  a  high  degree  satisfactory,  the  performance  being  some 
twenty-five  times  better  than  Amagat's  standard.  It  could  hardly  have  been 
expected  that  the  mean  error  would  prove  to  be  less  than  one  wave-length  of 
yellow  light*.  Considered  as  a  pressure,  the  mean  error  corresponds  to  the 
change  of  barometric  pressure  accompanying  an  elevation  of  4  millims. 

On  hydrogen  more  than  one  series  of  observations  have  been  carried 
out.  The  specimen  that  will  be  given  is  not  in  some  respects  the  most 
satisfactory,  but  it  is  chosen  as  having  been  pursued  to  the  greatest  rare- 
factions. The  gas  was  dried  carefully  with  phosphoric  anhydride  and  was 
introduced  into  the  apparatus  as  already  described.  It  is  thought  sufficient 
to  record  only  numbers  corresponding  to  the  three  last  columns  of  Tables 
I. — VII.,  the  first  column  giving  the  pressure  in  millims.  of  mercury,  the 
second  the  deviation  of  pv  from  the  mean  value  of  the  set  taken  as  unity, 
the  third  the  error  in  p  from  what  would  be  required  to  make  pv  absolutely 
constant. 

In  several  of  the  sets  of  observations  recorded  in  Table  VIII.,  there  would 
seem  to  be  a  tendency  for  the  positive  errors  to  concentrate  towards  the 
beginning,  i.e.,  for  pv  to  diminish  slightly  with  p.  It  was  at  this  stage  that 
a  suspicion  arose  that  the  distance  between  the  glass  points  of  the  mano- 
meter might  not  be  quite  constant,  but,  as  has  been  related,  the  suspicion 
was  not  verified.  It  is  just  possible  that  at  the  higher  pressures  and  smaller 

*  I  had  at  one  time  contemplated  an  apparatus  from  which  a  further  ten-fold  increase  in 
accuracy  might  be  expected.  Two  beams  of  light,  reflected  nearly  perpendicularly  from  the 
mercury  surfaces,  would  be  brought  to  interference  by  an  arrangement  similar  to  that  used 
in  investigating  the  refractivity  of  gases  (Proc.  Roy.  Soc.  Vol.  LIX.  p.  200,  1896  [Vol.  iv.  p.  218] ; 
Vol.  LXIV.  p.  97,  1898  [Vol.  iv.  p.  364]).  Preliminary  trials  proved  that  the  method  is  feasible ; 
but  the  delicacy  is  excessive  in  view  of  the  fact  that  according  to  Hertz  the  pressure  of  mercury 
vapour  at  common  temperatures  itself  amounts  to  '001  millim. 


528          ON    A   NEW   MANOMETER,   AND   ON   THE   LAW   OF   THE   PRESSURE      [266 


TABLE  VIII.— Hydrogen. 

October— November,  1900. 


Pressure  in 
millims.  Hg 

Deviation 
of  pv 

Error  of  p 
in  millims. 

Pressure  in 
millims.  Hg 

Deviation 
of  pv 

Error  of  p 
in  millims. 

1-43 

+  •0025 

+  0036 

1-44 

+  O018 

+  •0026 

1-31 

+  •0030 

+  •0039 

1-18 

-•0005 

-•0006 

1-20 

+  •0002 

+  •0002 

i-oo 

+  •0009 

+  •0009 

1-11 

-0012 

-•0013 

•87 

+  •0007 

+  •0006 

•97 

-•0005 

-•0005 

•77 

+  •0005 

+  •0004 

•86 

-•0002 

-•0002 

•62 

•0000 

•oooo 

•77 

-•0016 

-•0012 

•57 

-•0028 

-•0016 

•70 

-•0025 

-•0017 

•52 

-•0009 

-•0005 

•64 

+  0007 

+  •0004 

•48 

-•0018 

-•0009 

— 

— 

- 

•42 

+  •0018 

+  •0008 

•769 

+  •0021 

+  •0016 

•386 

•0000 

•oooo 

•624 

+  •0028 

+  •0017 

•315 

+  •0044 

+  '0014 

•524 

•0000 

•0000 

•264 

+  •0023 

+  •0006 

•423 

+  •0002 

+  •0001 

•213 

+  •0014 

+  •0003 

•335 

-•0037 

-•0012 

•168 

-•0072 

-•0012 

•279 

-•0018 

-•0005 

•140 

-•0014 

-•0002 

•196 

+  •0079 

+  •0015 

•098 

-•009 

-•0009 

•158 

+  •0046 

+  •0007 

•080 

•000 

•oooo 

•133 

+  •0053 

+  •0007 

•068 

+  •005 

+  •0003 

•106 

-•0053 

-•0006 

•055 

+  •007 

+  •0004 

•085 

-•0037 

-•0003 

•044 

+  •007 

+  O003 

•070 

-•0083 

-•0006 

•036 

-•005 

-•0002 

•051 

+  •007 

+  •0004 

•027 

-•047 

-•0013 

•041 

+  •002 

+  •0001 

•023 

+  •016 

+  •0004 

•034 

-•009 

-•0003 

•018 

-•054 

-•0010 

•027 

-•023 

-•0006 

•016 

+  •021 

+  •0003 

•023 

+  •036 

+  •0009 

•013 

+  •040 

+  •0005 

•018 

-•014 

-•0003 

•010 

+  •019 

+  •0002 

volumes  the  temperature  changes  were  not  insensible.  It  is  probable  that 
they  would  operate  in  the  direction  mentioned,  inasmuch  as  at  the  smaller 
volumes  a  larger  proportion  of  the  gas  would  be  in  the  connecting  tubes 


1901]     OF   GASES    BETWEEN    1'5    AND   O'Ol    MILLIMETRES   OF    MERCURY.        529 

at  a  higher  level  in  the  room,  and  therefore  warmer.  Considerable  pre- 
caution was  taken,  and  I  was  not  able  to  satisfy  myself  that  disturbance  due 
to  temperature  really  existed.  In  another  series  of  observations  on  hydrogen 
the  tendency  was  scarcely  apparent,  and  it  remains  doubtful  whether  there 
is  any  real  indication  of  departure  from  Boyle's  law.  It  may  be  noted  that 
interest  was  concentrated  rather  upon  the  lower  pressures,  and  that  perhaps 
less  pains  were  taken  over  the  readings  of  the  higher  pressures,  where  in 
any  case  the  error  would  be  a  smaller  proportion  of  the  whole.  Also  some 
of  the  observations  were  not  repeated.  Another  point  that  may  be  noted 
is  that  the  means  are  chosen  with  respect  to  the  values  of  pv,  and  that  a 
different  choice  would  in  many  cases  materially  reduce  the  mean  error  in 
the  last  column. 

Having  thoroughly  tested  the  apparatus  and  the  method  of  experimenting 
with  hydrogen  and  nitrogen,  I  returned  with  curiosity  to  the  case  of  oxygen. 
Special  pains  were  taken  to  ensure  that  the  gas  should  be  pure  and  above 
all  dry.  To  this  end  glass  tubes  were  prepared  containing  permanganate  of 
potash  and  phosphoric  anhydride,  and  these  were  connected  by  sealing  to 
one  of  the  branches  of  the  three-way  tap.  A  high  vacuum  having  been  made 
throughout,  heat  was  gradually  applied,  and  some  of  the  oxygen  allowed  to 
blow  off.  The  phosphoric  tube  (of  considerable  capacity)  was  then  allowed 
to  stand  full  of  gas  for  some  little  time,  after  which  the  necessary  gas  to  a 
pressure  of  about  10  centims.  was  allowed  to  enter  the  apparatus  by  means 
of  the  three-way  tap.  With  regard  to  the  maintenance  of  the  purity  of  the 
gas  under  rarefaction,  it  may  be  remarked  that  the  method  of  experimenting 
was  favourable,  inasmuch  as  the  last  stages  were  not  reached  until  the 
apparatus  had  been  exposed  to  the  gas  under  trial  for  a  week  or  two.  Any 
contamination  that  might  be  communicated  from  the  glass  during  the  first 
few  days  would  for  the  most  part  be  removed  before  the  final  stages  were 
reached. 

Before  the  regular  series  was  commenced,  special  observations  extending 
over  several  days  were  made  in  the  region  of  pressure  (from  1  millim.  to 
•5  millim.)  where  Bohr  found  anomalies.  No  unsteadiness  could  be  detected. 
Whatever  reading  was  obtained  within  a  few  minutes  of  a  change  of  pressure 
was  confirmed  after  an  interval  of  an  hour  or  more.  For  example,  on 
November  29,  at  12h  25m  the  pressure  which  had  stood  for  some  time  at  '80 
millim.  was  lowered  to  '65  millim.  At  8h  Om  the  pressure  was  unaltered.  In 
no  case  was  the  behaviour  in  any  way  different  to  that  which  had  been 
observed  with  the  other  gases.  It  is  true  that  when  the  observations  were 
reduced  one  preliminary  set  showed  an  excess  of  pressure  at  the  smaller 
volumes  similar  to  that  recorded  in  the  case  of  hydrogen,  but  the  tendency  is 
scarcely  visible  in  the  regular  series  now  to  be  given,  which  extended  from 
November  27  to  December  9. 

B.    iv.  34 


530         ON   A   NEW  MANOMETER,   AND   ON  THE   LAW  OF  THE   PRESSURE 


An  examination  of  the  numbers  in  the  Table  IX.  shows  that  Boyle's  law 
was  observed,  practically  up  to  the  limits  of  the  accuracy  of  the  measure- 
ments, and  in  particular  that  there  was  no  such  falling  off  in  the  value  of  pv 
at  low  pressures  as  was  encountered  by  Bohr.  What  can  be  the  cause  of  the 
difference  of  our  experiences  I  am  at  a  loss  to  conjecture.  I  can  only  suppose 
that  it  must  be  connected  somehow  with  the  quality  of  the  gas,  complicated 
perhaps  by  interaction  with  the  glass  or  with  the  mercury. 


TABLE  IX.— Oxygen. 


Pressure  in 
millims.  Hg 

Deviation 
of  pv 

Error  of  p 
in  millims. 

Pressure  in 
millims.  Hg 

Deviation 
ofpv 

Error  of  p 
in  millims. 

1-53 

+  •0016 

+  •0024 

•580 

-•0035 

-•0020 

1-17 

-•0012 

-•0014 

•472 

+  •0005 

+  •0002 

•95 

+  •0005 

+  •0005 

•396 

-•0007 

-•0003 

•80 

+  •0007 

+  •0006 

•321 

+  •0016 

+  •0005 

•65 

+  •0012 

+  •0008 

•255 

+  •0012 

+  •0003 

•57 

-•0009 

-•0005 

•212 

+  •0016 

+  •0003 

•51 

-•0014 

-•0007 

— 

— 

— 

•47 

-•0014 

-•0007 

— 

— 

— 

•43 

+  •0009 

+  •0004 

— 

- 

— 

•288 

+  •002 

+  •0007 

•142 

+  •005 

+  •0007 

•233 

•ooo 

•0000 

•115 

+  •009 

+  •0011 

•196 

•ooo 

•oooo 

•094 

-•019 

-•0018 

•159 

+  •005 

+  •0008 

•077 

•ooo 

•oooo 

•125 

-•002 

-•0003 

•062 

+  •012 

+  •0007 

•103 

-•009 

-•0010 

•051 

-•012 

-•0006 

•068 

-•002 

-•0002 

•034 

•ooo 

•oooo 

•056 

+  005 

+  •0003 

•029 

+  •059 

+  •0017 

•048 

+  •019 

+  •0009 

•022 

-•042 

-0009 

•038 

+  •009 

+  •0004 

•019 

+  •023 

+  •0004 

•029 

-•019 

-•0005 

•014 

-•035 

-•0005 

•025 

-'009 

-•0002 

— 

— 

— 

The  final  result  of  the  observations  on  the  three  gases  may  be  said  to 
be  the  full  confirmation  of  Boyle's  law  between  pressures  of  1*5  millims.  and 
•01  millim.  of  mercury.  If  there  is  any  doubt,  it  relates  to  the  case  of 
hydrogen,  which  appears  to  press  somewhat  in  excess  at  the  highest 


1901]     OF   GASES   BETWEEN    1'5   AND   O'Ol    MILLIMETRES   OF   MERCURY.        531 

pressures.  But  when  we  consider  the  smallness  of  the  amount  and  the 
various  complications  to  which  it  may  be  due,  as  well  as  a  priori  probabilities, 
we  may  well  hesitate  to  accept  the  departure  from  Boyle's  law  as  having  a 
real  existence. 

So  far  as  the  present  results  can  settle  the  question,  they  justify  to  the 
full  the  ordinary  use  of  McLeod's  gauge  within  the  limits  of  pressure 
mentioned  and  for  nitrogen  and  hydrogen  gases.  The  same  might  be  said  for 
oxygen;  but  until  the  discrepancy  with  the  conclusions  of  Bohr  can  be 
explained,  the  necessity  for  some  reserves  must  be  admitted. 

In  any  case  the  new  manometer  has  done  its  work  successfully,  and  is 
proved  to  be  capable  of  measuring  small  pressures  to  about  -%^  of  a  milli- 
metre of  mercury.  It  was  constructed  under  my  direction  by  Mr  Gordon. 


34—2 


267. 


ON  A  PROBLEM  RELATING  TO  THE  PROPAGATION  OF 
SOUND  BETWEEN  PARALLEL  WALLS. 


[Philosophical  Magazine,  I.  pp.  301—311,  1901.] 

THE  influence  of  viscosity  and  heat  conduction  in  modifying  the  propaga- 
tion of  sound  in  circular  tubes  of  moderate  dimensions  has  been  treated  by 
Kirchhoff  *  in  his  usual  masterly  style,  but  he  passes  over  the  case  when  the 
diameter  is  very  large.  In  my  book  on  the  Theory  of  Sound,  2nd  edition, 
§  348,  I  have  given  a  full  statement  of  Kirchhoff's  theory,  and  have  indicated 
the  alterations  required  when  the  boundary  is  supposed  to  take  the  form  of 
two  parallel  planes  instead  of  a  cylindrical  surface.  In  any  case  the  action 
of  the  wall  is  supposed  to  be  such  as  to  annihilate  variation  of  temperature, 
and  tangential  as  well  as  normal  motion.  In  connexion  with  the  problem 
of  the  propagation  of  sound  over  water  I  recently  had  occasion  to  extend 
the  analysis  to  the  case  of  a  layer  of  very  great  thickness;  and  though,  as 
the  result  showed,  the  solution  fails  to  answer  the  question  which  I  had 
then  in  view,  it  is  of  some  interest  in  itself.  In  this  case  the  practical 
question  differs  somewhat  from  that  proposed  by  KirchhofF,  who  assumes 
not  only  complete  periodicity  with  respect  to  time,  but  also  a  quasi-periodicity 
with  respect  to  x,  the  direction  of  propagation,  all  the  functions  being  sup- 
posed proportional  to  emx,  where  m  is  a  complex  constant,  and  not  otherwise 
to  depend  upon  x.  This  assumption  is  retained  in  the  present  paper.  It 
seems  advisable  to  give  a  brief  recapitulation  of  Kirchhoff's  theory,  referring 
for  more  detailed  explanation  to  the  original  paper  or  to  the  account  of  it 
in  Theory  of  Sound. 

*  Pogg.  Ann.  Vol.  cxxxiv.  1868;  Collected  Memoirs,  p.  540. 


1901]      ON   THE   PROPAGATION   OF   SOUND   BETWEEN    PARALLEL   WALLS.         533 

The  condition  of  the  gas  at  any  point  x,  y,  z  being  defined  by  the 
component  velocities  u,  v,  w,  and  6',  where  6'  is  proportional  to  the  excess 
of  temperature,  the  equation  for  0'  is  found  to  be 

KB'  -{a?  +  h  O'  +  p"  +  v)}  V20'  +  j-  {b2  +  h  (//  +  /')]  V40'  =  0.    . .  .(1) 

In  this  equation  V2  stands  for  d2fdx2  +  d*/dy*  +  d*ldz* ;  h  is  such  that  all  the 
variables  (u,  v,  w,  9')  are  proportional  to  e**;  a  is  the  velocity  of  sound  as 
reckoned  on  Laplacean  principles,  6  the  corresponding  Newtonian  value ; 
p,  p",  v  are  coefficients  of  viscosity  and  of  heat  conduction. 

A  solution  of  (1)  may  be  obtained  in  the  form 
where  Q1}  Q2  are  functions  of  x,  y,  z  satisfying  respectively 

Xj,  X2  being  the  roots  of 

Aa  -  (a2  +  h  (p!  +  p"  +  v)}  X  +  \  {62  +  h  (p!  +  //')}  X2  =  0 ;   (4) 

fl 

while  Alt  Az  denote  arbitrary  constants. 

In  correspondence  with  this  value  of  0',  particular  solutions  are  obtained 
by  equating  u,  v,  w  to  the  differential  coefficients  of 

AQi  +  B*Q*, 

taken  with  respect  to  x,  y,  z.   The  relation  of  the  constants  Blt  B2  to  Alf  A.2  is 

h 


More  general  solutions  may  be  obtained  by  addition  to  u,  v,  w  respectively 
of  u',  v',  w',  where  u',  v',  w'  satisfy 

W  =  4w',         W  =  A«',        VW  =  ^w'  ..........  (6) 

fJL  H  p. 

Thus 

u  =  u  +  B.dQ.jdx  +  B,dQ,ldx, 


v=  v'  +  B1dQl/dy  +  B2dQ,/dy,    ,  ..................  (7) 

w  =  w  +  B.dQ./dz  +  B2dQ,/dz, 
where  Blt  B2  have  the  values  given  above. 
It  appears  that 


534  ON   A    PROBLEM   RELATING   TO   THE   PROPAGATION  [267 

These  results  were  applied  by  Kirchhoff  to  the  case  of  plane  waves, 
supposed  to  be  propagated  in  infinite  space  in  the  direction  of  +x,  and 
it  may  conduce  to  clearness  to  deal  first  with  this  case.  Here  v'  and  w 
vanish,  while  u',  Q1}  Q2  are  independent  of  y  and  z.  It  follows  from  (8)  that 
u'  also  vanishes.  The  equations  for  Ql  and  Q.2  are 


\Q1>        ffiQt/da?  =  \&;  ..................  (9) 

so  that  we  may  take 


where  the  signs  of  the  square  roots  are  to  be  so  chosen  that  the  real  parts 
are  positive.     Accordingly 


(11) 


(12) 


in  which  the  constants  A^,  A2  may  be  regarded  as  determined  by  the  values 
of  u  and  6'  when  x  =  0. 

The  solution,  as  expressed  by  (11),  (12),  is  too  general  for  our  purpose, 
providing  as  it  does  for  arbitrary  communication  of  heat  at  x  =  0.  From 
the  quadratic  (4)  in  X  we  see  that  if  //,  p",  v  be  regarded  as  small  quantities, 
one  of  the  values  of  A,,  say  \,  is  approximately  equal  to  h2/a2,  while  the  other 
(\a)  is  very  great.  The  solution  which  we  now  require  is  that  corresponding 
to  \i  simply.  The  second  approximation  to  \  is  by  (4) 


that 


If  we  now  write  in  for  h,  we  see  that  the  typical  solution  is 

...........................  (14) 


where 


In  (14)  an  arbitrary  multiplier  and  an  arbitrary  addition  to  t  may,  as 
usual,  be  introduced  ;  and,  if  desired,  the  solution  may  be  realized  by  omitting 
the  imaginary  part. 


In  passing  on  to  consider  the  influence  of  walls,  by  which  gas  is  confined, 
upon  the  propagation  of  sound,  it  is  here  proposed  to  take  the  case  of  two 
dimensions,  rather  than  the  tube  of  circular  section  treated  by  Kirchhoff. 


1901]  OF   SOUND   BETWEEN   PARALLEL   WALLS.  535 

The  analysis,  however,  is  nearly  the  same.  We  suppose  that  sound  is  propa- 
gated in  the  layer  of  gas  bounded  by  fixed  walls  at  y  =  ±  y1  ,  so  that  w  =  0, 
while  u,  v,  0'  are  functions  of  x  and  y  only.  The  like  may  be  assumed 
respecting  u',  v',  Ql}  Qz.  We  suppose  further  that  as  functions  of  x  these 
quantities  are  proportional  to  einx,  where  m  is  a  complex  constant  to  be 
determined.  The  equations  (3)  for  Q1}  Q2  become 


(\l-m*)Ql,        d«Qa/djf  =  (*,-!»«)&.    ...(16,17) 

For  u'.  v'  equations  (6),  (8)  give 


*wj/  4-  —0  (~\R    1Q    9fh 

— >  —  7/&    I  ct  .  ~^; — ~    —  \        /   —  '/f     i  f  j         //e-u-    ~t*      7^"   —  v/.     •  *  » •»*i  -LO,  X  t/.  ^W  I 

fi          I  dyz      \p          J    '  dy 

These  three  equations  are  satisfied  if  u'  be  determined  by  means  of  the 
first,  and  v'  is  chosen  so  that 

v^-j-^--  ~,    (21) 

h/fju  -  m?  dy 

a  relation  obtained  by  subtracting  from  (19)  the  result  of  differentiating  (20) 
with  respect  to  y.  The  solution  of  (18)  may  be  written  u'  =  AQ,  in  which 
A  is  a,  constant,  and  Q  a  function  of  y  satisfying 


Thus,  by  (5),  (7), 

AH                 (k 

dy 

Qi  —  A%m 

f±-pk 

•••v-/ 
...(23) 

m 

dQ 

A  (k 

U.     )  * 

\  dQl       ,    (h        \  e 

^2       ("^ 

h/fj,  —  m? 

dy 
6' 

Ms 

J  dy            \\2       / 
A,Q2. 

dy  '  -(24) 
...(25) 

On  the  walls  at  y  =  ±  yl}  u,  v,  9'  must  satisfy  certain  conditions.  It  will 
here  be  supposed  that  there  is  neither  motion  of  the  gas  nor  change  of 
temperature  ;  so  that  when  y  =  ±yi,u,v,  0'  vanish.  The  condition  of  which 
we  are  in  search  is  thus  expressed  by  the  evanescence  of  the  determinant 
of  (23),  (24),  (25),  viz.  : 


which  is  to  be  satisfied  when  y—  ±  yv. 

Since  u  is  an  even  function  of  y,  we  have  from  (3),  (22), 
Q  =c 


536  ON   A   PROBLEM   RELATING  TO  THE   PROPAGATION  [267 

From  (23),  (25),  and  from  the  fact  that  w  =  0  when  y  =  yi,  we  get  as 
the  general  value  of  u,  without  regard  to  the  constant  multiplier, 

M=s'5(£)  +  A/x1-A/x1  of^O'Ajx^A/x,  sferr    -(28) 

In  equation  (26)  the  values  of  \1}  X^  are  independent  of  ylt  being  deter- 
mined by  (4).  In  the  application  to  air  under  normal  conditions  /u/,  p",  v 
may  be  regarded  as  small,  and  we  have  approximately 

X1==A2/a2,         X-2  =  ha2/vb? (29) 

A  second  approximation  to  the  value  of  X^  is  given  in  (13).  It  is  here 
assumed  that  the  velocity  of  propagation  of  viscous  effects  of  the  pitch  in 
question,  viz.  V(2/A'w),  is  small  in  comparison  with  that  of  sound,  so  that 
inp'/a?,  or  hp'/a?,  is  a  small  quantity — a  condition  abundantly  satisfied  in 
practice. 

In  interpreting  the  solution  we  limit  ourselves  here  to  the  case  which 
arises  when  //,  ft",  v  are  treated  as  very  small — so  small  that  the  layer  of  gas 
immediately  affected  by  the  walls  is  but  an  insignificant  fraction  of  the  whole. 
When  //  &c.  vanish,  we  have 

Xj  ==  h?ja?,        m2  =  A2/a2, 

so  that  unless  y  be  great  y^/(iftii  —  \l)  is  small.  On  the  other  hand, 
yiV/(m2  —  A //A'),  2h  v'  (m2  —  X,)  are  large.  For  the  moment  we  leave  the  value 
of  y*/(m?  -  \)  open,  and  merely  introduce  the  simplifications  arising  out 
of  the  largeness  of  the  arguments  in  Q  and  Q2. 

If  z  be  a  complex  quantity  of  the  form  £  +  iy,  we  have  in  general 

cos  z  =  cos  £  cosh  r\  —  i  sin  £  sinh  ?;,  (30) 

_  sin  2f  +  i  sinh  2 17 
~  cos  2f+  cosh  2*,  ' 
Thus,  when  i)  is  large, 

d  log  cos  z 

— ^ =  -  tan  z  =  —  i ; 

dz 

so  that  when  y  =  yi,  since  A  is  a  pure  imaginary, 
d\ogQ_     //A\          dlogQ,_ 


The  introduction  into  (26)  of  these  values  and  those  of  Xj  and  X,  from 
(29),  gives 

dlogQ1=     y'tf 
dy  a?  ' 

where 

y^vy  +  ^/fc-WV*;    .....................  (33) 


1901]  OF   SOUND   BETWEEN   PARALLEL   WALLS.  537 

or,  if  z  =  yl  ^(m?  —  Xj), 

ztanz^y'tf-y./a-  ............................  (34) 

This  is  the  equation  by  which  z,  and  thence  m2,  is  to  be  determined. 

In  the  case  corresponding  to  that  treated  by  Kirchhoff,  yl  is  not  so  large 
but  that  the  right-hand  member  of  (34)  is  a  small  quantity.  The  solution 
of  (34)  is  then 

*•  =  •/%!/«•;  ..............................  (35) 

whence 

(36) 


We  now  write  k  =  ni,  so  that  the  frequency  is  71/2-Tr.     Thus 

V/*=V(i~).(l+»)  ...........................  (37) 

and 

m  —  ±  (m'  +  im"),    ...........................  (38) 

where  by  (36) 


The  solution  differs  from  that  found  by  Kirchh6ff  for  a  circular  tube 
of  radius  r  merely  by  the  substitution  of  ^yl  for  r*. 

So  far,  then,  as  it  depends  on  t  and  x,  the  typical  solution  is 

aint  g—m'x—im"x 

or  when  realized, 

e~m'x  cos  (nt  -  m"x),      ........................  (40) 

where  m,  m"  have  the  values  given  in  (39).     This  is  for  waves  travelling 
in  the  positive  direction. 

As  a  function  of  y,  u  is  given  by  (28);  but  this  may  now  be  simplified 
in  virtue  of  the  supposition  that  the  layer  directly  influenced  by  the  viscosity 
is  but  a  small  fraction  of  ylt  By  (27) 

cos  (y^n/f  .  V^t)  =  cos  {yx  Vn/V  .  (1  -  »)} 


irin(*V&)\ (41) 

use  being  made  of  (30),  in  which  17  is  large.  In  consequence  of  (41), 
Q(y)  +  Q(yi)  vanishes  unless  y  be  nearly  equal  to  ylt  viz.  unless  the  point 
considered  be  within  the  frictional  layer.  In  like  manner,  and  under  the 
same  restriction,  Q2  (y)  -r-  Q.z  (y^  may  be  neglected.  Except  in  the  immediate 
neighbourhood  of  the  walls,  (28)  now  reduces  to 


Theory  of  Sound,  2nd  edit.  §  350. 


538  ON   A   PROBLEM   RELATING  TO  THE   PROPAGATION  [267 

In  the  case  considered  by  Kirchhoff,  where  the  argument  of  Ql  is  small, 
we  have  from  (27)  approximately 

&(</)  =  &  (2A)  =1, 

and  accordingly  u  =  —  1.  To  this  approximation  the  velocity  is  uniform 
across  the  whole  section  until  it  begins  to  fall  off  as  the  walls  are  closely 
approached. 

As  a  first  step  towards  the  consideration  of  what  occurs  when  yl  is  great, 
we  may  proceed  to  a  second  approximation.     Thus,  from  (27), 


-^)=i-^2;   ...............  (43) 

*/]- 

so  that 


yI 


This  equation  expresses  the  dependence  of  u  upon  y.     The  dependence 
on  t  and  a;  is  given  by  the  factors  already  considered,  viz. 


That  (44)  is  complex  indicates  that  the  phase  varies  with  y.     The  realized 
expression  will  be 


from  which  we  infer  (i)  that  the  intensity  is  least  in  the  middle  where 
y  =  0,  and  increases  towards  the  walls  until  the  frictional  layer  is  approached; 
(ii)  that,  as  y2  increases  from  the  centre,  constancy  of  phase  demands  a 
diminishing  x,  or,  in  other  words,  that  the  wave-surface  is  convex  towards 
+  x  and  the  wave  divergent. 

We  have  now  to  trace  the  solution  of  (34)  when  the  right-hand  member 
becomes  large.     Writing  it  in  the  form 


(46) 


we  have  to  find  such  a  complex  value  of  z,  say  £  +  irj,  that  the  function 
on  the  left  is  real.     Initially,  when  yl  is  small, 

%  +  i<n=p  (cos  0  +  i  sin  0)  =  p  (cos  67£°  +  i  sin  67£°), 
and 


1901]  OF   SOUND    BETWEEN    PARALLEL   WALLS.  539 

If  \ve  retain  the  angle  67£°  and  increase  p,  we  find,  calculating  by  means 
of  (31),  that  i~**.  ten*  becomes  complex  with  imaginary  part  positive. 
Thus  if  p  =  1,  we  get 

i~*z.  tan  z  =  "80  (cos  9°  54'  +  i  sin  9°  54'). 
This  is  a  sign  that  6  must  be  reduced.     If  we  take  p  =  1,  0  =  60°,  we  find 

i-*z  .  tan  z  =  '83  (cos  2°  7'  -  i  sin  2°  7'). 
If  p  =  1-5,  while  0  =  60°,  we  get  in  detail 

z  =  £  +  irj  =  -75  +  i  x  1-299  ; 
sin  2£  =  sin  85°  57'  =  '998,         sinh  fy  =  6'695, 

cos  2£  =  cos  85°  57'  =  "071,         cosh  ty  =  6'769, 
whence 


so  that 

i-*s  .  tan  $  =  1-5  (cos  6°  31'  +  t  sin  6°  31'). 

The  course  of  the   calculation   makes   it   clear   that   as   z   increases,  tan  z 
approaches  the  limit  i,  so  that  ultimately  B  =  ^TT,  or  the  angle  reduces  from 
67£°  to  45°.     Hence 

ft-tVnfy*/*1'  ..............................  (47) 

and 

rn?  =  h*la?-  indict,      ........................  (48) 

independent  of  y^ 

In  order  to  obtain  u  as  a  function  of  y,  we  have  now  to  interpret  (42) 
for  the  case  where  yl  is  great.     It  may  be  written 


where  e,  given  by  (47),  is 


„  —  ,  ...(49) 

COS  3 


By  (30),  since  17  is  large,  we  have  approximately 

cos  z  =  ^(cos  f  +  i  sin  £),     (51) 

where 


Accordingly,  cos  2  is  large.     If,  as  near  the  middle  of  the  layer  of  gas,  y  be 
not  large,  cos  (2y/yl)  =  1,  and 

u--  I/cos  z,    (53) 

a  small  quantity.     When  y  is  so  large  that  2y/ya  is  large,  as  well  as  z,  we 
may  write 

M  =  -e*'-1>.ei<*'-*>,    (54) 

where 


540  ON    A   PROBLEM   RELATING   TO   THE   PROPAGATION  [267 

As  the  walls  are  approached  u  rapidly  increases,  and  at  last  e^'-v  becomes 
nearly  equal  to  unity.  We  must  bear  in  mind,  however,  that  (42)  and  there- 
fore (54)  must  not  be  applied  within  the  frictional  layer  lying  quite  close 
to  the  walls,  so  that  we  are  not  at  liberty  to  suppose  y  actually  equal  to  yt. 

Under  normal  conditions  the  thickness  of  the  frictional  layer  is  very  small. 
If  in  C.G.s.  measure  we  take  /*'  =  '16,  w  =  2?r  x  256,  we  find  V(™/  V)  =  67. 
Thus  if  we  suppose  the  thickness  of  the  frictional  layer  in  (41)  to  be 
defined  by 


we  get 

y\  —  y  =  '15  millim. 

If  the  point  under  consideration  be  a  few  multiples  of  this  (say  1  millim.) 
from  the  walls,  the  ratio  Q  (y)  -r-  Q  (y^)  may  be  neglected. 

The   thickness   of  the   layer   through   which    Q2  (y}  +  Q2  (y^   is  sensible 
is  of  the  same  order  of  magnitude. 

Let  us  next  consider  what  value  of  (yl  —  y}  makes  (?/  —  77)  in  (54)  equal 
to  unity.     By  (52),  (55) 


If  we  take  p  ='16,  v  =  '256,  we  find  from  (33)  7'  ='6;  and  a  =33200;  so 
that  for  a  frequency  of  256  we  get 

V  2.  33200  2 


For  air  and  for  a  sound  of  this  pitch  the   falling  off  becomes  important 
at  a  distance  of  about  400  metres  from  the  walls. 


As  has  already  been  suggested,  this  solution  fails  to  answer  the  practical 
question  for  the  sake  of  which  it  was  originally  attempted.  It  was  desired 
to  know  whether  in  the  propagation  of  sound  for  long  distances  over  smooth 
water,  there  was  any  important  shadow  formed  near  the  surface  under  the 
influence  of  viscosity  and  heat  conduction.  It  would  apparently  be  a  matter 
of  some  difficulty  to  formulate  and  solve  a  definite  problem  in  which  this 
question  is  involved.  But,  as  Lord  Kelvin  has  pointed  out  to  me,  a  sufficient 
answer  to  the  practical  question  may  be  arrived  at  by  very  simple  reasoning 
on  the  basis  of  a  solution  originally  given  by  Stokes  (see  Theory  of  Sound, 
§  347).  If  U  be  the  tangential  velocity  of  a  plane  vibrating  rigidly  in  an 
atmosphere  of  viscous  fluid  with  a  frequency  H/^TT,  the  work  required  to 
maintain  the  motion  is,  for  unit  of  area, 


1901]  OF   SOUND   BETWEEN   PARALLEL  WALLS.  541 

where  Um  denotes  the  maximum  value  of  U  during  the  period.  The  same 
expression  may  be  applied  to  find  the  work  lost  by  the  presence  of  a 
fixed  plane  in  air  vibrating  with  velocity  U.  The  energy  of  this  motion 
is,  per  unit  of  volume,  $pUm*,  or  for  a  stratum  of  height  y  resting  upon  unit 
of  area, 


If  we  equate  the  two  expressions  we  get  a  superior  limit  to  the  thickness 
of  the  stratum  whose  energy  could  be  absorbed  in  time  t.     We  find  thus 


yrsV  \*P 

or,  if  we  take  n  =  ZTT  x  256,  and  as  for  air  p/p  =  p  =  '16,  T/  =  lit. 

Thus  in  9  seconds  the  thickness  of  a  stratum  of  shadow  could  not  reach 
1  metre,  and  must,  in  fact,  be  very  much  less.  It  would  appear  therefore 
that  this  effect  may  be  neglected  in  practice,  unless  it  be  in  the  case  of 
an  observer  extremely  close  to  the  water. 


268. 

POLISH. 

[Proceedings  of  the  Royal  Institution,  xvi.  pp.  563—570,  1901 ; 
Nature,  LXIV.  pp.  385—388,  1901.] 

THE  lecture  commenced  with  a  description  of  a  home-made  spectroscope 
of  considerable  power.  The  lens,  a  plano-convex  of  6  inches  aperture  and 
22  feet  focus,  received  the  rays  from  the  slit,  and  finally  returned  them  to 
a  pure  spectrum  formed  in  the  neighbourhood.  The  skeleton  of  the  prism 
was  of  lead;  the  faces,  inclined  at  70°,  were  of  thick  plate-glass  cemented 
with  glue  and  treacle.  It  was  charged  with  bisulphide  of  carbon,  of  which 
the  free  surface  (of  small  area)  was  raised  above  the  operative  part  of  the 
fluid.  The  prism  was  traversed  twice,  and  the  effective  thickness  was 
5£  inches,  so  that  the  resolving  power  corresponded  to  11  inches,  or  28  cm., 
of  CS2.  The  liquid  was  stirred  by  a  perforated  triangular  plate,  nearly 
fitting  the  prism,  which  could  be  actuated  by  means  of  a  thread  within  reach 
of  the  observer.  The  reflector  was  a  flat,  chemically  silvered  in  front. 

So  far  as  eye  observations  were  concerned,  the  performance  was  satis- 
factory, falling  but  little  short  of  theoretical  perfection.  The  stirrer  needed 
to  be  in  almost  constant  operation,  the  definition  usually  beginning  to  fail 
within  about  20  seconds  after  stopping  the  stirrer.  But  although  the  stirrer 
was  quite  successful  in  maintaining  uniformity  of  temperature  as  regards 
space,  i.e.  throughout  the  dispersing  fluid,  the  temperature  was  usually  some- 
what rapidly  variable  with  time,  so  that  photographs,  requiring  more  than 
a  few  seconds  of  exposure,  showed  inferiority.  In  this  respect  a  grating  is 
more  manageable. 

The  lens  and  the  faces  of  the  prism  were  ground  and  polished  (in  1893) 
upon  a  machine  kindly  presented  by  Dr  Common.  The  flat  surfaces  were 
tested  with  a  spherometer,  in  which  a  movement  of  the  central  screw  through 
!  60*000  inch  could  usually  be  detected  by  the  touch.  The  external  surfaces 


1901]  POLISH.  543 

of  the  prism  faces  were  the  only  ones  requiring  accurate  flatness.  In  polish- 
ing, the  operation  was  not  carried  as  far  as  would  be  expected  of  a  professional 
optician.  A  few  residual  pittings,  although  they  spoil  the  appearance  of 
a  surface,  do  not  interfere  with  its  performance,  at  least  for  many  purposes. 

In  the  process  of  grinding  together  two  glass  surfaces,  the  particles  of 
emery,  even  the  finest,  appear  to  act  by  pitting  the  glasses,  i.  e.  by  breaking 
out  small  fragments.  In  order  to  save  time  and  loss  of  accuracy  in  the 
polishing,  it  is  desirable  to  carry  the  grinding  process  as  far  as  possible,  using 
towards  the  close  only  the  finest  emery.  The  limit  in  this  direction  appears 
to  depend  upon  the  tendency  of  the  glasses  (6  inches  diameter)  to  seize,  when 
they  approach  too  closely,  but  with  a  little  care  it  is  easy  to  attain  such 
a  fineness  that  a  candle  is  seen  reflected  at  an  angle  of  incidence  not  exceed- 
ing 60°,  measured  as  usual  from  the  perpendicular. 

The  fineness  necessary,  in  order  that  a  surface  may  reflect  and  refract 
regularly  without  diffusion,  viz.  in  order  that  it  may  appear  polished,  depends 
upon  the  wave-length  of  the  light  and  upon  the  angle  of  incidence.  At 
a  grazing  incidence  all  surfaces  behave  as  if  polished,  and  a  surface  which 
reflects  red  light  pretty  well  may  fail  signally  when  tested  with  blue  light  at 
the  same  angle.  If  we  consider  incidences  not  too  far  removed  from  the  per- 
pendicular, the  theory  of  gratings  teaches  that  a  regularly  corrugated  surface 
behaves  as  if  absolutely  plane,  provided  that  the  wave-length  of  the  corruga- 
tions is  less  than  the  wave-length  of  the  light,  and  this  without  regard  to  the 
depth  of  the  corrugations.  Experimental  illustrations,  drawn  from  the  sister 
science  of  Acoustics,  were  given.  The  source  was  a  bird-call  from  which 
issued  vibrations  having  a  wave-length  of  about  1'5  cm.,  and  the  percipient 
was  a  high-pressure  sensitive  flame.  When  the  bird-call  was  turned  away, 
the  flame  was  silent,  but  it  roared  vigorously  when  the  vibrations  were  re- 
flected back  upon  it  from  a  plate  of  glass.  A  second  plate,  upon  which  small 
pebbles  had  been  glued  so  as  to  constitute  an  ideally  rough  surface,  acted 
nearly  as  well,  and  so  did  a  piece  of  tin  plate  suitably  corrugated.  In  all 
these  cases  the  reflection  was  regular,  the  flame  becoming  quiet  when  the 
plates  were  turned  out  of  adjustment  through  a  very  small  angle.  In  another 
method  of  experimenting  the  incidence  was  absolutely  perpendicular,  the 
flame  being  exposed  to  both  the  incident  and  the  reflected  waves.  It  is 
known  that  under  these  circumstances  the  flame  remains  quiescent  at  the 
nodes  and  flares  most  vigorously  at  the  loops.  As  the  reflector  is  drawn 
slowly  back,  the  flame  passes  alternately  through  the  nodes  and  loops,  thus 
executing  a  cycle  of  changes  as  the  reflector  moves  through  half  a  wave- 
length. The  effects  observed  were  just  the  same  whether  the  reflector  were 
smooth  or  covered  with  pebbles,  or  whether  the  corrugated  tin  plate  were 
substituted.  All  surfaces  were  smooth  enough  in  relation  to  the  wave-length 
of  the  vibration  to  give  substantially  a  specular  reflection. 


544  POLISH.  [268 

Finely-ground  surfaces  are  still  too  coarse  for  perpendicular  specular  re- 
flection of  the  longest  visible  waves  of  light.  Here  the  material  may  be 
metal,  or  glass  silvered  chemically  on  the  face  subsequently  to  the  grinding. 
But  experiment  is  not  limited  by  the  capabilities  of  the  eye ;  and  it  seems 
certain  that  a  finely  ground  surface  would  be  smooth  enough  to  reflect  with- 
out sensible  diffusion  the  longest  waves,  such  as  those  found  by  Rubens  to  be 
nearly  100  times  longer  than  the  waves  of  red  light.  An  experiment  may  be 
tried  with  radiation  from  a  Leslie  cube  containing  hot  water,  or  from  a 
Welsbach  mantle  (without  a  chimney).  In  the  lecture  the  latter  was  em- 
ployed, and  it  fell  first  at  an  angle  of  about  45°  upon  a  finely  ground  flat 
glass  silvered  in  front.  By  this  preliminary  reflection,  the  radiation  was 
purified  from  waves  other  than  those  of  considerable  wave-length.  The 
second  reflection  (also  at  45°)  was  alternately  from  polished  and  finely  ground 
silvered  surfaces  of  the  same  size,  so  mounted  as  to  permit  the  accurate  sub- 
stitution of  the  one  for  the  other.  The  heating-power  of  the  radiation  thus 
twice  reflected  was  tested  with  a  thermopile  in  the  usual  manner.  Repeated 
comparisons  proved  that  the  reflection  from  the  ground  surface  was  about 
•76  of  that  from  the  polished  surface,  showing  that  the  ground  surface  re- 
flected the  waves  falling  upon  it  with  comparatively  little  diffusion.  A  slight 
rotation  of  any  of  the  surfaces  from  their  proper  positions  at  once  cut  off  the 
effect.  It  is  probable  that  the  device  of  submitting  radiation  to  preliminary 
reflections  from  one  or  more  merely  ground  surfaces  might  be  found  useful  in 
experiments  upon  the  longest  waves. 

In  view  of  these  phenomena  we  recognise  that  it  is  something  of  an 
accident  that  polishing  processes,  as  distinct  from  grinding,  are  needed  at  all ; 
and  we  may  be  tempted  to  infer  that  there  is  no  essential  difference  between 
the  operations.  This  appears  to  have  been  the  opinion  of  Herschel*,  whom 
we  may  regard  as  one  of  the  first  authorities  on  such  a  subject.  But, 
although,  perhaps,  no  sure  conclusion  can  be  demonstrated,  the  balance  of 
evidence  appears  to  point  in  the  opposite  direction.  It  is  true  that  the  same 
powders  may  be  employed  in  both  cases.  In  one  experiment  a  glass  surface 
was  polished  with  the  same  emery  as  had  been  used  effectively  a  little  earlier 
in  the  grinding.  The  difference  is  in  the  character  of  the  backing.  In 

*  Enc.  Met.,  Art.  Light,  p.  447,  1830:  "The  intensity  and  regularity  of  reflection  at  the 
external  surface  of  a  medium  is  found  to  depend  not  merely  on  the  nature  of  the  medium,  but 
very  essentially  on  the  degree  of  smoothness  and  polish  of  its  surface.  But  it  may  reasonably  be 
asked,  how  any  regular  reflection  can  take  place  on  a  surface  polished  by  art,  when  we  recollect 
that  the  process  of  polishing  is,  in  fact,  nothing  more  than  grinding  down  large  asperities  into 
smaller  ones  by  the  use  of  hard  gritty  powders,  which,  whatever  degree  of  mechanical  comminu- 
tion we  may  give  them,  are  yet  vast  masses,  in  comparison  with  the  ultimate  molecules  of  matter, 
and  their  action  can  only  be  considered  as  an  irregular  tearing  up  by  the  roots  of  every  projection 
that  may  occur  in  the  surface.  So  that,  in  fact,  a  surface  artificially  polished  must  bear  some- 
what of  the  same  kind  of  relation  to  the  surface  of  a  liquid,  or  a  crystal,  that  a  ploughed  field  does 
to  the  most  delicately  polished  mirror,  the  work  of  human  hands." 


Fig.  1. 


Fig.  2. 


1901]  POLISH.  545 

grinding,  the  emery  is  backed  by  a  hard  surface,  e.g.  of  glass,  while  during 
the  polishing  the  powder  (mostly  rouge  in  these  experiments)  is  imbedded  in 
a  comparatively  yielding  substance,  such  as  pitch.  Under  these  conditions, 
which  preclude  more  than  a  moderate  pressure,  it  seems  probable  that  no 
pits  are  formed  by  the  breaking  out  of  fragments,  but  that  the  material  is 
worn  away  (at  first,  of  course,  on  the  eminences)  almost  molecularly. 

The  progress  of  the  operation  is  easily  watched  with  a  microscope,  pro- 
vided, say,  with  a  £-inch  object-glass.  The  first  few  minutes  suffice  to  effect 
a  very  visible  change.  Under  the  microscope  it  is  seen  that  little  facets, 
parallel  to  the  general  plane  of  the  surface,  have  been  formed  on  all  the  more 
prominent  eminences*.  The  facets,  although  at  this  stage  but  a  very  small 
fraction  of  the  whole  area,  are  adequate  to  give  a  sensible  specular  reflection, 
even  at  perpendicular  incidence.  On  one  occasion  five  minutes'  polishing  of 
a  rather  finely  ground  glass  surface  was  enough  to  qualify  it  for  the  formation 
of  interference  bands,  when  brought  into  juxtaposition  with  another  polished 
surface,  the  light  being  either  white  or  from  a  soda  flame ;  so  that  in  this 
way  an  optical  test  can  be  applied  almost  before  the  polishing  has  begun  f. 

As  the  polishing  proceeds,  the  facets  are  seen  under  the  microscope  to 
increase  both  in  number  and  in  size,  until  they  occupy  much  the  larger  part 
of  the  area.  Somewhat  later  the  parts  as  yet  untouched  by  the  polisher 
appear  as  pits,  or  spots,  upon  a  surface  otherwise  invisible.  Fig.  1  represents 
a  photograph  of  a  surface  at  this  stage  taken  with  the  microscope.  The 
completion  of  the  process  consists  in  rubbing  away  the  whole  surface  down 
to  the  level  of  the  deepest  pits.  The  last  part  of  the  operation,  while  it 
occupies  a  great  deal  of  time,  and  entails  further  risk  of  losing  the  "  truth " 
of  the  surface,  adds  very  little  to  the  effective  area,  or  to  the  intensity  of  the 
light  regularly  reflected  or  refracted. 

Perhaps  the  most  important  fact  taught  by  the  microscope  is  that  the 
polish  of  individual  parts  of  the  surface  does  not  improve  during  the  process. 
As  soon  as  they  can  be  observed  at  all,  the  facets  appear  absolutely  structure- 
less. In  its  subsequent  action  the  polishing  tool,  bearing  only  upon  the  parts 
already  polished,  extends  the  boundary  of  these  parts,  but  does  not  enhance 
their  quality.  Of  course,  the  mere  fact  that  no  structure  can  be  perceived 
does  not  of  itself  prove  that  pittings  may  not  be  taking  place  of  a  character 
too  fine  to  be  shown  by  a  particular  microscope  or  by  any  possible  microscope. 
But  so  much  discontinuity,  as  compared  with  the  grinding  action,  has  to  be 
admitted  in  any  case,  that  one  is  inevitably  led  to  the  conclusion  that  in  all 
probability  the  operation  is  a  molecular  one,  and  that  no  coherent  fragments 

*  The  interpretation  is  facilitated  by  a  thin  coating  of  aniline  dye  which  attaches  itself 
mainly  to  the  hollows. 

t  With  oblique  incidence,  as  in  Talbot's  experiments  (see  Phil.  Mag.  xxvm.  p.  191,  1889 
[Vol.  in.  p.  308]),  achromatic  bands  may  be  observed  from  a  surface  absolutely  unpolished, 
but  this  disposition  would  not  be  favourable  for  testing  purposes. 

R.    iv.  35 


546  POLISH.  [268 

containing  a  large  number  of  molecules  are  broken  out.  If  this  be  so,  there 
would  be  much  less  difference  than  Herschel  thought  between  the  surfaces 
of  a  polished  solid  and  of  a  liquid. 

Several  trials  have  been  made  to  determine  how  much  material  is  actually 
removed  during  the  polishing  of  glass.  In  one  experiment  a  piece  6  inches 
in  diameter,  very  finely  ground,  was  carefully  weighed  at  intervals  during  the 
process.  Losses  of  '070,  '032,  '045,  '026,  '032  gms.  were  successively  registered, 
amounting  in  all  to  '205  gms.  Taking  the  specific  gravity  of  the  glass  as  3, 
this  corresponds  to  a  thickness  of  3'6  x  10~4  cm.,  or  to  about  6  wave-lengths 
of  mean  light,  and  it  expresses  the  distance  between  the  original  mean 
surface  and  the  final  plane.  But  the  polish  of  this  glass,  though  sufficient 
for  most  practical  purposes,  was  by  no  means  perfect.  Probably  the  6  wave- 
lengths would  have  needed  to  be  raised  to  10  in  order  to  satisfy  a  critical  eye. 
It  may  be  interesting  to  note  for  comparison  that,  in  the  grinding,  one  charge 
of  emery,  such  as  had  remained  suspended  in  water  for  seven  or  eight 
minutes,  removed  a  thickness  of  glass  corresponding  to  2  wave-lengths. 

In  other  experiments  the  thickness  removed  in  polishing  was  determined 
optically.  A  very  finely  ground  disc  was  mounted  in  the  lathe  and  polished 
locally  in  rings.  Much  care  was  needed  to  obtain  the  desired  effect  of  a  ring 
showing  a  continuously  increasing  polish  from  the  edges  inwards.  To  this 
end  it  was  necessary  to  keep  the  polisher  (a  piece  of  wood  covered  with 
resin  and  rouge)  in  constant  motion,  otherwise  a  number  of  narrow  grooves 
developed  themselves. 

The  best  ring  was  about  half  an  inch  wide.  When  brought  into  contact 
with  a  polished  flat  and  examined  at  perpendicular  incidence  with  light  from 
a  soda  flame,  the  depression  at  its  deepest  part  gave  a  displacement  of  three 
bands,  corresponding  to  a  depth  of  1£X.  On  a  casual  inspection  this  central 
part  appeared  well  polished,  but  examination  under  the  microscope  revealed 
a  fair  number  of  small  pits.  Further  working  increased  the  maximum  depth 
to  2£\,  when  but  very  few  pits  remained.  In  this  case,  then,  polish  was 
effected  during  a  lowering  of  the  mean  surface  through  2  or  3  wave-lengths, 
but  the  grinding  had  been  exceptionally  fine. 

It  may  be  well  to  emphasize  that  the  observations  here  recorded  relate  to 
a  hard  substance.  In  the  polishing  of  a  soft  substance,  such  as  copper,  it  is 
possible  that  material  may  be  loosened  from  its  original  position  without 
becoming  detached.  In  such  a  case  pits  may  be  actually  filled  in,  by  which 
the  operation  would  be  much  quickened.  Nothing  suggestive  of  this  effect 
has  been  observed  in  experiments  upon  glass. 

Another  method  of  operating  upon  glass  is  by  means  of  hydrofluoric  acid. 
Contrary  to  what  is  generally  supposed,  this  action  is  extremely  regular,  if 
proper  precautions  are  taken.  The  acid  should  be  weak,  say  one  part  of 
commercial  acid  to  two  hundred  of  water,  and  it  should  be  kept  in  constant 


1901]  POLISH.  547 

motion  by  a  suitable  rocking  arrangement.  The  parts  of  the  glass  not  in- 
tended to  be  eaten  into  are,  as  usual,  protected  with  wax.  The  effect  upon 
a  polished  flat  surface  is  observed  by  the  formation  of  Newton's  rings  with 
soda-light.  After  perhaps  three-quarters  of  an  hour,  the  depression  corre- 
sponds to  half  a  band,  i.e.  amounts  to  ^X,  and  it  appears  to  be  uniform  over 
the  whole  surface  exposed.  Two  pieces  of  plate  glass,  3  inches  square,  and 
flat  enough  to  come  into  fair  contact  all  over,  were  painted  with  wax  in 
parallel  stripes,  and  submitted  to  the  acid  for  such  a  time,  previously  ascer- 
tained, as  would  ensure  an  action  upon  the  exposed  parts  of  J  X.  After 
removal  of  the  wax,  the  two  plates,  crossed  and  pressed  into  contact  so  as  to 
develop  the  colours,  say  of  the  second  order,  exhibited  a  chess-board  pattern. 
Where  two  uncorroded,  or  where  two  corroded  parts,  are  in  contact,  the 
colours  are  nearly  the  same,  but  where  a  corroded  and  an  uncorroded  surface 
overlap,  a  strongly  contrasted  colour  is  developed.  The  combination  lends 
itself  to  lantern  projection,  and  the  pattern  upon  the  screen  [shown]  is  very 
beautiful,  if  proper  precautions  are  taken  to  eliminate  the  white  light  reflected 
from  the  first  and  fourth  surfaces  of  the  plates. 

In  illustration  of  the  action  of  hydrofluoric  acid,  photographs*  were 
shown  of  interference  bands  as  formed  by  soda-light  between  glass  surfaces, 
one  optically  flat  and  the  other  ordinary  plate,  upon  which  a  drop  of  dilute 
acid  had  been  allowed  to  stand  (Fig.  2).  Truly  plane  surfaces  •  would  give 
bands  straight,  parallel,  and  equidistant. 

Hydrofluoric  acid  has  been  employed  with  some  success  to  correct  ascer- 
tained errors  in  optical  surfaces.  But  while  improvements  in  actual  optical 
performance  have  been  effected,  the  general  appearance  of  a  surface  so  treated 
is  unprepossessing.  The  development  of  latent  scratches  has  been  described 
on  a  former  occasion  f. 

A  second  obvious  application  of  hydrofluoric  acid  has  hitherto  been  less 
successful.  If  a  suitable  stopping  could  be  found  by  which  the  deeper  pits 
could  be  protected  from  the  action,  corrosion  by  acid  could  be  used  in  sub- 
stitution for  a  large  part  of  the  usual  process  of  polishing. 

In  connexion  with  experiments  of  this  sort,  trial  was  made  of  the  action 
of  the  acid,  upon  finely  ground  glass,  such  for  example  as  is  used  as  a  backing 
for  stereoscopic  transparencies,  and  very  curious  results  were  observed.  For 
this  purpose  the  acid  may  conveniently  be  used  much  stronger,  say  one  part 
of  commercial  acid  to  10  parts  of  water,  and  the  action  may  be  prolonged 
for  hours  or  days.  The  general  appearance  of  the  glass  after  treatment  is 
smoother  and  more  translucent,  but  it  is  only  under  the  microscope  that  the 
remarkable  changes  which  the  surface  has  undergone  become  intelligible. 
Fig.  3  is  from  a  photograph  taken  in  the  microscope,  the  focus  being  upon 
the  originally  ground  surface  itself.  The  whole  area  is  seen  to  be  divided 
*  The  plates  were  sensitised  in  the  laboratory  with  cyanine. 
t  Proc.  Roy.  Inst.  March  1893.  [Vol.  iv.  p.  59.] 

35—2 


548 


POLISH. 


[268 


into  cells.  These  cells  increase  as  the  action  progresses,  the  smaller  ones 
being,  as  it  were,  eaten  up  by  the  bigger.  The  division  lines  between  the 
cells  are  ridges,  raised  above  the  general  level,  and  when  seen  in  good  focus 
appear  absolutely  sharp.  The  general  surface  within  the  cells  shows  no 
structure,  being  as  invisible  as  if  highly  polished. 

That  each  cell  is  in  fact  a  concave  lens,  forming  a  separate  image  of  the 
source  of  light,  is  shown  by  slightly  screwing  out  the  object-glass.  Fig.  4 
was  taken  in  this  way  from  the  same  surface,  the  source  of  light  being  the 
flame  of  a  paraffin  lamp,  in  front  of  which  was  placed  a  cross  cut  from  sheet- 
metal. 

The  movement  required  to  pass  from  the  ridge  to  the  image  of  the  source, 
equal  to  the  focal  length  (/)  of  the  lens,  may  be  utilised  to  determine  the 
depth  (t)  of  a  cell.  In  one  experiment  the  necessary  movement  was  '005  inch. 
The  semi-aperture  (y)  of  the  "  lens  "  was  '0015  inch,  whence  by  the  formula 
y*  =ft,  we  find  t  =  '00045  inch.  This  represents  the  depth  of  the  cell,  and  it 
amounts  to  about  8  wave-lengths  of  yellow  light. 


Fig.  5. 

The  action  of  the  acid  seems  to  be  readily  explained  if  we  make  the  very 
natural  supposition  that  it  eats  in  everywhere,  at  a  fixed  rate,  normally  to 
the  actual  surface.  If  the  amount  of  the  normal  corrosion  after  a  proposed 
time  be  known,  the  new  surface  can  be  constructed  as  the  "  envelope "  of 
spheres  having  the  radius  in  question  and  centres  distributed  over  the  old 
surface.  Ultimately,  the  new  surface  becomes  identified  with  a  series  of 
spherical  segments  having  their  centres  at  the  deeper  pits  of  the  original 
surface.  The  construction  is  easily  illustrated  in  the  case  of  two  dimensions. 
In  the  figure  A  is  supposed  to  be  the  original  surface ;  B,  C,  D,  E  surfaces 
formed  by  corrosion,  being  constructed  by  circles  having  their  centres  on  A. 
In  B  the  ridges  are  still  somewhat  rounded,  but  they  become  sharp  in  D 
and  E.  The  general  tendency  is  to  sharpen  elevations  and  to  smooth  off 
depressions. 


Fig.  3. 


Fig. 


269. 

DOES   CHEMICAL  TRANSFORMATION  INFLUENCE   WEIGHT? 
[Nature,  LXIV.  p.  181,  June,  1901.] 

CAREFUL  experiments  by  Heydweiller,  published  in  the  last  number  of 
Drude's  Annalen  (Vol.  v.  p.  394),  lead  their  author  to  the  conclusion  that  in 
certain  cases  chemical  action  is  accompanied  by  a  minute,  but  real,  alteration 
of  weight.  The  chemical  actions  here  involved  must  be  regarded  as  very 
mild  ones,  e.g.  the  mere  dissolution  of  cupric  sulphate  in  water,  or  the  sub- 
stitution of  iron  for  copper  in  that  salt. 

The  evidence  for  the  reality  of  these  changes,  which  amount  to  0*2  or 
0'3  mg.,  and  are  accordingly  well  within  the  powers  of  a  good  balance  to 
demonstrate,  will  need  careful  scrutiny;  but  it  may  not  be  premature  to 
consider  what  is  involved  in  the  acceptance  of  it.  The  first  question  which 
arises  is — does  the  mass  change  as  well  as  the  weight  ?  The  affirmative 
answer,  although  perhaps  not  absolutely  inconsistent  with  any  well  ascer- 
tained fact,  will  certainly  be  admitted  with  reluctance.  The  alternative — 
that  mass  and  weight  are  not  always  in  proportion — involves  the  conclusion, 
in  contradiction  to  Newton,  that  the  length  of  the  seconds'  pendulum  at 
a  given  place  depends  upon  the  material  of  which  the  bob  is  composed. 
Newton's  experiment  was  repeated  by  Bessel,  who  tried  a  number  of  metals, 
including  gold,  silver,  lead,  iron,  zinc,  as  well  as  marble  and  quartz,  and  whose 
conclusion  was  that  the  length  of  the  seconds'  pendulum  formed  of  these 
materials  did  not  vary  by  one  part  in  60,000.  At  the  present  day  it  might 
be  possible  to  improve  even  upon  Bessel,  or  at  any  rate  to  include  more 
diverse  substances  in  the  comparisons ;  but  in  any  case  the  accuracy  obtain- 
able would  fall  much  short  of  that  realized  in  weighings. 

As  regards  Heydweiller's  experiments  themselves,  there  is  one  suggestion 
which  I  may  make  as  to  a  possible  source  of  error.  Is  the  chemical  action 
sufficiently  in  abeyance  at  the  time  of  the  first  weighing  ?  If  there  is  copper 
sulphate  in  one  branch  of  an  inverted  U  and  water  in  the  other,  the  equi- 
librium can  hardly  be  complete.  The  water  all  the  time  tends  to  distil  over 
into  the  salt,  and  any  such  distillation  must  be  attended  by  thermal  effects 
which  would  interfere  with  the  accuracy  of  the  weighing. 

[See  further  Nature,  May  15,  1902.] 


270. 

ACOUSTICAL  NOTES.— VI. 

[Philosophical  Magazine,  n.  pp.  280—285,  1901.] 

Forced   Vibrations. 

IF  free  vibrations  be  represented  by  cos  nt,  and  if  the  forced  vibration 
due  to  a  force  acting  in  a  very  long  period  be  cospt,  then  the  actual  forced 
vibration  will  be 

n2  cos  pt 
ri1  -p2  ' 
It  is  here  implied : — 

(1)  That  in  all  cases  the  forced  vibration  takes  its  period  from  the  force, 
whatever  may  be  the  natural  period. 

(2)  That  if  the  forced  vibration  be  the  slower,  viz.  if  p  <  n,  the  phase  is 
the  same  as  if  the  vibration  were  infinitely  slow,  in  which  case  the  vibrator 
would  be  situated  at  any  instant  of  time  in  the  position  where  the  momentary 
force  would  permanently  maintain  it. 

(3)  That  if  the  forced  vibration  be  the  quicker  (p  >  n),  the  phase  of  the 
actual  vibration  is  the  opposite  of  that  defined  in  (2). 

(4)  That  if  the  force  have  nearly  the  period  of  the  free  vibrations,  the 
effect  is  much  enhanced.     Indeed,  according  to  the  formula  it  would  become 
infinite,  which  means  that  forces  of  a  viscous  character,  never  really  absent, 
must  now  be  brought  into  the  reckoning. 

So  far  as  I  am  aware,  illustrations  of  this  important  theory*  have  usually 
been  wanting  in  lecture  demonstrations,  except  as  regards  (4).  I  have  found 
that  if  we  employ  as  vibrator  a  magnet  with  attached  mirror,  as  used  for 
example  in  Thomson  galvanometers,  the  whole  may  readily  be  brought  before 
a  large  audience. 

*  Young's  Lectures  on  Natural  Philosophy,  p.  578  (1807). 


1901]  ACOUSTICAL  NOTES.  551 

With  the  aid  of  an  external  magnet,  whose  distance  could  be  varied,  the 
frequency  of  (complete)  vibration  was  adjusted  to  10  per  minute,  the  vibra- 
tions being  manifested  by  the  motion  of  a  spot  of  light  reflected  from  the 
mirror  on  to  a  scale  in  the  usual  manner.  The  force  brought  to  bear  upon 
the  vibrator  had  its  origin  in  the  revolution  of  a  rather  long  permanent 
magnet,  situated  at  some  little  distance,  and  so  mounted  as  to  be  capable  of 
rotation.  No  particular  situation  is  necessary,  but  the  action  of  the  magnet 
is  simplest  in  certain  special  cases,  as  when  its  centre  is  at  the  level  of  the 
suspended  magnet  and  in  the  direction  of  the  screen.  The  plane  of  revolu- 
tion being  horizontal,  the  deflecting  action  is  then  greatest  when  the  revolving 
magnet  points  towards  the  suspended  magnet.  In  one  of  these  positions, 
say  when  the  spot  is  deflected  to  the  right,  a  bell  rings  automatically. 
Uniform  rotation  at  any  desired  speed  is  maintained  by  hand  with  the  aid 
of  gearing,  diminishing  the  speed  in  the  ratio  of  5:1,  and  of  a  metronome 
set  as  required. 

To  illustrate  propositions  (1)  and  (2)  the  long  magnet  is  caused  to  rotate 
with  a  frequency  of  8  per  minute,  i.e.  with  a  frequency  somewhat  less  than 
that  natural  to  the  suspended  system.  At  first  the  phenomenon  is  com- 
plicated by  the  interaction  of  natural  and  forced  vibrations ;  but  the  former 
soon  die  away.  It  is  then  recognised  that  the  vibrations  observed  upon  the 
screen  are  isochronous  with  the  revolution  of  the  magnet,  and  that  the  bell 
rings  at  the  moment  when  the  spot  of  light  attains  its  greatest  elongation 
towards  the  right. 

In  the  next  experiment  the  speed  of  revolution  is  altered  to  12  per 
minute,  so  as  to  bring  about  the  condition  of  things  contemplated  in  (3). 
After  a  little  interval  of  settling  down  the  bell  rings  always  at  the  moment 
when  the  spot  is  most  deflected  to  the  left,  showing  that  the  phase  has  been 
altered  by  half  a  period. 

To  illustrate  (4)  the  speed  of  revolution  may  now  be  adjusted  to  8  per 
minute.  The  arc  of  vibration  is  seen  gradually  to  increase  until  it  reaches 
a  large  value,  the  bell  now  ringing,  not  at  either  extreme  elongation,  but  as 
the  spot  passes  from  left  to  right  through  its  position  of  equilibrium. 


Vibrations  of  Strings. 

At  the  Royal  Institution  it  is  usual  to  illustrate  this  subject  by  ex- 
periments after  the  method  of  Melde  and  Tyndall.  The  string  is  connected 
with  a  large  tuning-fork,  whose  prongs  stand  vertically,  and  the  vibrations 
are  maintained  electrically  in  the  well-known  manner.  The  electric  contact 
is  between  solids  (of  platinum),  one  attached  to  the  prong,  the  other  forming 
the  point  of  an  adjustable  screw  carried  by  the  framework. 


552  ACOUSTICAL  NOTES.  [270 

The  string,  10  feet  long,  is  stretched  horizontally  and  the  tension  is 
adjusted  until  a  vigorous  vibration  ensues,  which  happens  when  one  of  the 
modes  of  vibration  has  a  period  in  simple  relation  to  that  of  the  fork.  There 
is  here  an  important  distinction  according  as  the  length  of  the  string  is 
parallel  or  perpendicular  to  the  motion  of  the  point  of  attachment.  In  the 
latter  case  the  vibrations  are  of  the  character  commonly  classified  as  forced, 
and  the  period  is  the  same  as  that  of  the  fork.  But  if  the  fork  be  so  situated 
that  the  motion  of  the  point  of  attachment  is  along  the  length  of  the  string, 
the  vibrations  are  of  an  entirely  different  character,  and  are  executed  in 
a  period  the  double  of  that  of  the  fork.  The  theory  of  vibrations  of  this 
class  was  discussed  in  a  paper  on  Maintained  Vibrations*  published  many 
years  ago,  reference  to  which  must  here  suffice. 

A  convenient  device  for  demonstrating  the  relationship  of  periods  is  to 
illuminate  the  string  by  sparks  synchronous  with  the  vibrations  of  the  fork 
itself.  For  this  purpose  an  induction-coil  is  included  in  the  circuit  by  which 
the  fork  is  driven,  so  that  every  break  at  the  fork  causes  a  spark  between  the 
secondary  terminals,  to  which  a  small  jar  is  connected  in  the  usual  manner. 
If  then  the  vibrations  of  the  string  be  isochronous  with  the  fork,  and  there- 
fore with  the  sparks,  the  intermittent  illumination  exhibits  what  is  ordinarily 
seen  as  a  gauzy  spindle  resolved  into  the  appearance  corresponding  to  a  single 
phase  of  the  vibration ;  that  is,  the  string  is  seen  apparently  fixed  (in  a  dis- 
placed position)  and  single.  But  if,  as  when  the  point  of  attachment  moves 
parallel  to  the  length  of  the  string,  the  vibrations  are  only  half  as  fast  as 
those  of  the  fork,  the  string  is  found  in  two  (opposite)  phases  at  the  moments 
of  illumination,  and  is  consequently  seen  double.  The  effect  is  improved  by 
a  piece  of  ground  glass,  which  may  be  held  either  between  the  sparks  and 
the  string,  or  between  the  string  and  the  eye.  In  the  latter  case  it  is  a 
shadow  that  is  seen.  It  is  desirable  to  retain  enough  continuous  light  to 
allow  the  form  of  the  gauzy  spindle  to  remain  visible.  In  this  way  the 
difference  between  the  two  kinds  of  vibration  may  be  exhibited  to  many 
persons  at  once.  [1902.  The  stroboscopic  method  of  observation  had  already 
been  very  similarly  applied  to  this  experiment  by  Costing,  Onder  houden 
trillungen  van  gespannen  draden,  Helder,  1889.] 

A  detail  of  some  importance  relates  to  the  use  of  the  condenser,  associated 
as  usual  with  the  primary  circuit  of  the  coil.  If  its  poles  be  connected 
simply  with  the  outer  terminals  of  the  fork-apparatus  regarded  as  an  in- 
terrupter, the  secondary  sparks  will  be  inferior  or  may  fail  altogether.  The 
explanation  is  to  be  sought  in  the  self-induction  of  the  magnet  associated 
with  the  fork,  which  apparently  interferes  with  the  suddenness  of  the  break. 
The  poles  of  the  condenser  should  be  connected  as  directly  as  possible  with 
the  two  pieces  of  metal  between  which  the  break  takes  place.  In  the 

*  Phil.  Mag.  Vol.  xv.  p.  229  (1883) ;  Scientific  Papers,  Vol.  n.  p.  188. 


1901]  ACOUSTICAL   NOTES.  553 

apparatus  at  the  Royal  Institution  it  makes  all  the  difference  on  which  side 
of  the  small  electromagnet  the  pole  of  the  condenser  is  attached. 

Beats  of  Sounds  led  to  the  Two  Ears  separately. 

When  two  approximately  pure  tones,  of  equal  intensity  and  of  approxi- 
mately equal  frequency,  are  conveyed  to  one  ear,  beats  are  perceived  according 
to  a  well-known  elementary  theory,  the  frequency  of  the  beats  being  the 
difference  of  the  frequencies  of  the  tones.  When  the  beats  are  somewhat 
slow,  the  phase  of  silence  is  distinctly  recognisable,  and  indeed  the  moment 
of  the  occurrence  of  this  phase  is  capable  of  being  fixed  with  great  accuracy. 

The  question  whether  the  beats  are  still  audible  when  one  sound  is  led 
to  one  ear  alone,  and  the  second  sound  to  the  second  ear  alone,  is  of  great 
importance.  A  careful  experiment  of  this  sort  is  described  by  Prof.  S.  P. 
Thompson  *,  in  which  the  sounds  were  conveyed  to  the  ears  by  rubber  tubes ; 
and  the  conclusion  was  that  in  spite  of  all  precautions  the  beats  were  most 
distinctly  heard,  although  there  was  no  phase  of  "  silence,"  such  as  is  per- 
ceived when  both  sounds  are  conveyed  to  the  same  ear. 

I  have  lately  tried  a  somewhat  similar  experiment,  using  telephones  and 
electrical  conveyance,  by  which  perhaps  the  risk  of  the  sounds  reaching  the 
wrong  ears  is  reduced  to  a  minimum.  Two  entirely  independent,  electrically 
driven,  forks  of  about  128  vibrations  per  second  were  the  sources  of  sound. 
Near  the  electromagnet  of  each  fork  was  placed  a  small  coil  of  wire  in 
connexion  with  a  telephone.  The  higher  harmonics  were  greatly  moderated 
by  the  interposition  of  thick  sheets  of  copper ;  but  the  sounds  were  doubtless 
no  more  than  rough  approximations  to  pure  tones.  Both  forks  were  placed 
at  a  great  distance  from  the  observer ;  and  in  one  case  the  double  connecting 
wire  was  passed  through  a  hole  in  a  thick  wall  specially  arranged  many  years 
ago  for  this  sort  of  experimenting.  When  the  telephones  were  pressed  closely 
to  the  ears,  the  utmost  possible  was  done  to  secure  that  each  sound  should 
have  access  only  to  its  proper  ear. 

The  results  depended  somewhat  upon  the  frequency  of  the  beats.  When 
this  exceeded  one  per  second,  the  beats  were  very  easily  audible.  When,  on 
the  other  hand,  the  frequency  was  reduced  to  \  or  \  beat  per  second,  the 
beats  were  not  easily  perceived  at  first.  After  a  little  while  the  attention 
seemed  to  concentrate  itself  upon  the  variable  element  in  the  aggregate 
effect,  and  the  cycle  became  clear.  But  even  after  some  practice  neither 
Mr  Gordon  nor  I  could  hear  slow  beats  during  the  first  10  or  15  seconds 
of  observation. 

The  general  results  of  the  experiments  do  not  appear  to  me  to  exclude 
the  view  that  the  comparatively  feeble  beats  heard  under  these  conditions 

*  Phil.  Mag.  Vol.  iv.  p.  274  (1877). 


554  ACOUSTICAL  NOTES.  [270 

may  be  due  to  the  passage  of  sound  from  one  ear  to  the  other  through  the 
bones  of  the  head  or  perhaps  through  the  Eustachian  tube. 

Loudness  of  Double  Sounds. 

Observations  upon  the  double  syrens  (with  separate  horns)  used  by  the 
Trinity  House  have  given  the  impression  that  as  heard  from  a  distance  the 
two  syrens  are  no  better  than  one,  even  though  the  horns  are  parallel,  and 
the  observer  situated  in  the  direction  of  the  axis.  Dr  Tyndall's  experience 
was  similar.  In  his  Report  of  1874  he  remarks  (June  2),  "  There  was  no 
sensible  difference  of  intensity  between  the  single  horn  and  the  two  horns  " ; 
and  again  (June  10),  "  Subsequent  comparative  experiments  even  proved 
the  sound  of  the  two  horns  to  be  more  effective  than  that  of  the  three." 

These  conclusions  are  rather  startling,  suggesting  the  query  as  to  what 
then  can  be  the  use  of  multiplying  pipes  in  an  organ  or  voices  in  a  chorus. 
In  order  to  clear  the  ground  a  little,  I  have  recently  tried  some  small-scale 
experiments  with  organ-pipes. 

Two  stopped  pipes  of  pitch  about  256  were  mounted  near  the  window 
of  a  room  on  the  ground-floor.  When  the  window  was  open  the  sounds  could 
be  heard  (over  grass)  to  about  200  metres ;  but  when  the  window  was  closed 
the  range  was  much  less.  Some  difficulty  was  experienced  in  getting  equal 
effects  from  the  two  pipes.  According  to  the  instructions  of  the  observer, 
one  or  other  supply-pipe  was  more  or  less  throttled  with  wax. 

With  approximate  equality  of  intensities  and  with  such  tuning  that  the 
beats  were  at  the  rate  of  about  two  per  second,  the  results  were  very  distinct. 
The  beats  were  much  more  easily  audible  than  either  of  the  component 
sounds.  Doubtless  part  of  the  advantage  was  due  to  the  contrast  provided 
by  the  silences ;  but  it  was  thought  that,  apart  from  this,  the  swell  of  the 
beat  was  distinctly  louder  than  either  sound  alone. 

The  result  of  the  experiment  is,  of  course,  just  what  was  to  be  expected 
from  a  mechanical  point  of  view.  According  to  theory  the  intensity  (reckoned 
according  to  energy  propagated)  at  the  loudest  part  of  the  beat  should  be 
four  times  that  of  the  (equal)  component  sounds  heard  separately. 

In  another  set  of  experiments  the  pipes  were  mistuned  until  the  interval 
was  about  a  minor  third,  no  distinct  beats  being  audible.  In  this  case  the 
intensity  of  the  compound  sound  might  be  expected  to  be  double  of  that  of 
the  (equal)  component  sounds.  The  impression  upon  the  observer  hardly 
corresponded  to  this  anticipation.  It  was  difficult  to  say  that  the  compound 
sound  was  decidedly  the  louder ;  although  the  accession  of  the  second  sound 
as  an  addition  to  the  first  could  always  be  distinguished,  and  this  whether 
the  higher  or  the  lower  sound  were  the  one  added.  It  may  be  remarked 
that  the  question  involved  in  this  experiment  is  partly  physiological,  and  not 
merely  mechanical  as  in  the  case  of  sounds  nearly  in  unison. 


271. 


ON  THE  MAGNETIC  ROTATION  OF  LIGHT  AND  THE  SECOND 
LAW   OF  THERMODYNAMICS. 


[Nature,  LXIV.  pp.  577,  578,  1901.] 

IN  a  paper  published  sixteen  years  ago  I  drew  attention  to  a  peculiarity 
of  the  magnetic  rotation  of  the  plane  of  polarisation  arising  from  the  cir- 
cumstance that  the  rotation  is  in  the  same  absolute  direction  whichever  way 
the  light  may  be  travelling.  "A  consequence  remarkable  from  the  theoretical 
point  of  view  is  the  possibility  of  an  arrangement  by  which  the  otherwise 
general  optical  law  of  reciprocity  shall  be  violated.  Consider,  for  example, 
a  column  of  diamagnetic  medium  exposed  to  such  a  force  that  the  rotation  is 
45°,  and  situated  between  two  Nicols  whose  principal  planes  are  inclined  to 
one  another  at  45°.  Under  these  circumstances  light  passing  one  way  is 
completely  stopped  by  the  second  Nicol,  but  light  passing  the  other  way  is 
completely  transmitted.  A  source  of  light  at  one  point  A  would  thus  be 
visible  at  a  second  point  B,  when  a  source  at  B  would  be  invisible  at  A ;  a 
state  of  things  at  first  sight*  inconsistent  with  the  second  law  of  thermo- 
dynamics." (Phil.  Trans.  CLXXVI.  p.  343,  1885;  Scientific  Papers,  Vol.  II. 
p.  360.)  It  is  here  implied  that  the  inconsistency  is  apparent  only,  but  I  did 
not  discuss  it  further. 

In  his  excellent  report  ("  Les  Lois  theoriques  du  Rayonnement,  Rapports 
presented  au  Congres  International  de  Physique,"  Paris,  1900,  Vol.  II.  p.  29), 
W.  Wien,  considering  the  same  experimental  combination  of  Nicols  and 
magnetised  dielectric,  arrives  at  a  contrary  conclusion.  It  may  be  well  to 
quote  his  statement  of  the  case.  "  La  rotation  magne'tique  du  plan  de 
polarisation  constitue  un  cas  exceptionnel  digne  de  remarque,  et  Ton  pourrait 
ici  imaginer  un  dispositif  qui  mettrait  en  echec  le  principe  de  Carnot  s'il 
n'existait  pas  une  compensation  inconnue. 

*  The  italics  are  in  the  original     That  magnetic  rotation  may  interfere  with  the  law  of 
reciprocity  had  already  been  suggested  by  Helmholtz. 


556  MAGNETIC    ROTATION   OF    LIGHT.  [271 

"  Faisons,  en  effet,  les  suppositions  suivantes :  Deux  corps  de  temperature 
egale  sont  entoures  d'une  enveloppe  adiabatique.  Les  rayons  qu'ils  s'en- 
voient  r^ciproquement  traversent  deux  prismes  de  nicol.  Entre  ces  prismes 
se  trouve  une  substance  non  absorbante  sur  laquelle  agissent  des  forces 
magnetiques  qui  font  tourner  le  plan  de  polarisation  d'un  angle  determine. 
La  radiation  emanant  du  corps  1  penetre  dans  le  nicol  1.  Nous  supposerons 
que  le  rayon  subissant  la  reflexion  totale  n'est  pas  absorbe,  mais  renvoye" 
dans  sa  propre  direction  par  des  miroirs  convenablement  disposes.  Admettons 
que  le  plan  de  polarisation  soit  tourne"  de  45°  par  les  forces  magnetiques. 
La  section  principale  du  deuxieme  nicol  etant  orientee  dans  la  direction 
parallele  au  plan  de  polarisation  du  rayon  emergent,  toute  la  lumiere  trans- 
raise  par  la  substance  absorbante  (sic)  traversera  le  nicol.  Par  consequent, 
la  moiti£  des  rayons  e"mis  par  le  corps  1  frappera  le  corps  2. 

"  Les  rayons  emis  par  le  corps  2  se  divisent  en  deux  parties  egales,  dans 
le  nicol  2.  Une  moitie  est,  comme  precedemment,  renvoyee  par  reflexion. 
L'autre  moitie,  apres  que  son  plan  de  polarisation  a  subi  une  rotation  de  45° 
dans  le  meme  sens  que  les  rayons  emis  par  le  corps  1,  vient  frapper  le  premier 
nicol.  La  section  principale  de  ce  nicol  etant  perpendiculaire  au  plan  de 
polarisation,  aucune  radiation  ne  le  traverse,  et  nous  pouvons  renvoyer  toute 
la  lumiere  au  corps  2. 


"  Le  corps  2  re9oit  ainsi  trois  fois  plus  d'energie  que  le  corps  1.  [That  is, 
2  receives  the  whole  of  its  own  radiation  and  the  half  of  that  of  1,  while  1 
receives  only  the  half  of  its  own  radiation.]  L'un  de  ces  corps  s'echauffera 
par  consequent  de  plus  en  plus  aux  depens  de  1'autre." 

Wien  then  suggests  certain  ways  of  escape  from  this  conclusion,  but  it 
appears  to  me  that  the  difficulty  itself  depends  upon  an  oversight.  It  is  not 
possible  to  send  back  to  2  the  whole  of  its  radiation  in  the  manner  proposed. 
The  second  half,  which  after  passage  of  Nicol  2  is  totally  reflected  at  Nicol  1 
and  then  returned  upon  its  course,  on  its  arrival  at  Nicol  2  is  not  transmitted 
(as  Wien  seems  to  suppose)  but  is  totally  reflected.  When  again  returned 
upon  its  course  by  a  perpendicular  reflector,  and  again  reflected  through  45° 
by  the  magnetised  medium,  it  is  in  a  condition  to  be  completely  transmitted 
by  Nicol  1,  and  thus  finds  its  way  to  body  1,  and  not  to  body  2  as  the 
argument  requires.  The  two  bodies  receive  altogether  the  same  amount  of 
radiation,  and  there  is  therefore  no  tendency  to  a  change  of  temperature. 

Although  I  have  not  been  able  to  find  any  note  of  it,  I  feel  assured  that 
the  above  reasoning  was  present  to  my  mind  when  I  wrote  the  passage 
already  cited. 


272. 


ON   THE  INDUCTION-COIL*. 


[Philosophical  Magazine,  n.  pp.  581—594,  1901.] 

ALTHOUGH  several  valuable  papers  relating  to  this  subject  have  recently 
been  published  by  Oberbeck-f-,  Walter  J,  Mizuno§,  Beattie||,  and  KlingelfussIT, 
it  can  hardly  be  said  that  the  action  of  the  instrument  is  well  understood. 
Perhaps  the  best  proof  of  this  assertion  is  to  be  found  in  the  fact  that,  so  far 
as  I  am  aware,  there  is  no  a  priori  calculation,  determining  from  the  data  of 
construction  and  the  value  of  the  primary  current,  even  the  order  of  mag- 
nitude of  the  length  of  the  secondary  spark.  I  need  hardly  explain  that 
I  am  speaking  here  (and  throughout  this  paper)  of  an  induction-coil  working 
by  a  break  of  the  primary  circuit,  not  of  a  transformer  in  which  the  primary 
circuit,  remaining  unbroken,  is  supplied  with  a  continuously  varying  alter- 
nating current. 

The  complications  presented  by  an  actual  coil  depend,  or  may  depend, 
upon  several  causes.  Among  these  we  may  enumerate  the  departure  of  the 
iron  from  theoretical  behaviour,  whether  due  to  circumferential  eddy-currents 
or  to  a  failure  of  proportionality  between  magnetism  and  magnetizing  force. 
A  second,  and  a  very  important,  complication  has  its  origin  in  the  manner  of 
break,  which  usually  occupies  too  long  a  time,  or  at  least  departs  too  much 
from  the  ideal  of  an  instantaneous  abolition  of  the  primary  current.  A  third 
complication  arises  from  the  capacity  of  the  secondary  coil,  in  virtue  of  which 
the  currents  need  not  be  equal  at  all  parts  of  the  length,  even  at  the  same 

*  From  the  Jubilee  volume  presented  to  Prof.  Bosscha. 
t  Wied.  Ann.  LXH.  p.  109  (1897);  LXIV.  p.  193  (1898). 
J  Wied.  Ann.  LXH.  p.  300  (1897) ;  LXVI.  p.  623  (1898). 
§  Phil  Mag.  XLV.  p.  447  (1898). 
||  Phil.  Mag.  L.  p.  139  (1900). 
f  Wied.  Ann.  v.  p.  837  (1901). 


558  ON  THE   INDUCTION-COIL.  [272 

moment  of  time.  If  we  ignore  these  complications,  treating  the  break  as 
instantaneous,  the  iron  as  ideal,  and  the  secondary  as  closed  and  without 
capacity,  the  theory,  as  formulated  by  Maxwell*,  is  very  simple.  In  his 
notation,  if  x,  y  denote  the  primary  and  secondary  currents,  L,  M,  N  the 
coefficients  of  self  and  mutual  induction,  the  energy  of  the  field  is 

(1) 


If  c  be  the  primary  current  before  the  break,  the  secondary  current  at  time  t 
after  the  break  has  the  expression 


(2) 


8  being  the  resistance  of  the  secondary  circuit.     The  current  begins  with 
a  value  c  .  M/N,  and  gradually  disappears. 

The  formation  of  the  above  initial  current  is  best  understood  in  the  light 
of  Kelvin's  theorem,  as  explained  by  me  in  an  early  paperf.  For  this 
purpose  it  is  more  convenient  to  consider  the  reversed  phenomenon,  viz.,  the 
instantaneous  establishment  of  a  primary  current  c.  The  theorem  teaches 
that  subject  to  the  condition  x  —  c  the  kinetic  energy  (1)  is  to  be  made 
a  minimum;  so  that 

Me  +  Ny  =  0 

gives  the  initial  secondary  current.     In  the  case  of  the  break  we  have  merely 
to  reverse  the  sign  of  y. 

Immediately  after  the  break,  when  x  —  0  and  y  has  the  above  value,  the 
kinetic  energy  is 


Immediately  before  the  break  the  kinetic  energy  is  \  Z/c2,  so  that  the  loss  of 
energy  at  break  —  the  energy  of  the  primary  spark  —  is 


vanishing  when  the  primary  and  secondary  circuits  are  closely  intertwined  — 
the  case  of  no  "  magnetic  leakage." 

If  we  maintain  the  suppositions  as  to  the  behaviour  of  the  iron  and  the 
suddenness  of  the  break,  the  above  calculated  secondary  current  may  be 
supposed  to  be  instantaneously  formed,  even  although  the  secondary  circuit 
be  not  closed.  This  is  most  easily  seen  when  a  condenser,  such  as  a  leyden- 

*  "  Electromagnetic  Field,"  Phil.  Trans.  1864  ;  Maxwell's  Scientific  Papers,  i.  p.  546. 
t  "  On   some   Electromagnetic   Phenomena   considered   in   connexion   with   the   Dynamical 
Theory,"  Phil.  Mag.  XXXMII.  p.  1  (1869)  ;  Scientific  Papers,  i.  p.  6. 


1901]  ON   THE    INDUCTION-COIL.  559 

jar,  is  associated  with  the  ends  of  the  secondary.  Even  when  no  jar  is 
applied,  the  capacity  of  the  secondary  itself  acts  in  the  same  direction  and 
allows  the  formation  of  the  current.  Whether  partly  due  to  a  jar  or  not,  it 
will  be  convenient  for  the  present  to  regard  the  capacity  as  associated  with 
the  terminals  only  of  the  secondary  wire.  Under  these  circumstances  the 
secondary  current  follows  the  laws  laid  down  by  Kelvin  in  1853,  the  same  in 
fact  as  govern  all  vibrations  in  which  there  is  but  one  degree  of  freedom.  If 
the  resistance  is  not  too  high,  the  current  is  oscillatory.  After  the  lapse  of 
one  quarter  of  a  complete  period  of  these  oscillations,  the  current  vanishes, 
and  the  whole  remaining  energy  is  the  potential  energy  of  electric  charge. 
If  the  resistance  of  the  secondary  wire  can  be  neglected  (so  far  as  its  influence 
during  this  short  time  is  concerned),  the  potential  energy  of  charge  is  the 
equivalent  of  the  original  energy  of  the  secondary  current  at  the  moment 
after  the  break.  In  the  case  of  no  magnetic  leakage,  this  is  again  the  same 
as  the  energy  of  the  primary  current  before  break. 

On  these  principles  it  is  easy  to  calculate  a  limit  for  the  maximum 
potential-difference  at  the  terminals  of  the  secondary,  or  for  the  spark-length, 
so  far  as  this  is  determined  by  the  potential-difference.  For  if  q  be  the 
capacity  at  the  secondary  terminals,  V  the  maximum  potential-difference,  the 
energy  of  the  charge  is  ^  q  V2,  and  this  can  never  exceed  the  energy  of  the 
primary  current  before  break,  viz.,  \Lc-.  The  limit  to  the  value  of  V  is 
accordingly 

V=c^(L/q\    (4) 

and  it  is  proportional  to  the  primary  current. 

So  long  as  the  iron  can  be  treated  as  ideal,  the  above  formula  holds  good, 
and  upon  the  supposition  of  a  sufficiently  sudden  break  there  seems  to  be  no 
reason  why  it  should  not  afford  a  tolerable  approximation  to  the  actual 
maximum  value  of  V.  The  proportionality  between  spark-length  and  primary 
current  was  found  to.  hold  good  in  Walter's  experiments  over  a  considerable 
range. 

When  the  core  is  very  long  in  proportion  to  its  diameter,  or  when  it 
approximates  to  a  closed  circuit,  the  behaviour  of  the  iron  may  deviate 
widely  from  that  described  as  ideal,  and  the  quantity  denoted  by  L  has  no 
existence.  But  the  principle  remains  that  the  energy  of  charge  at  the 
moment  preceding  the  secondary  spark  cannot  exceed,  though  it  may  some- 
what closely  approach,  the  energy  of  the  primary  current  before  break. 

We  have  next  to  consider  how  the  energy  of  the  primary  current  is  to 
be  reckoned,  and  here  we  encounter  questions  as  to  which  opinion  is  not 
yet  undivided.  The  general  opinion  would,  I  suppose,  be  that  the  bodily 
magnetization  of  the  iron  represents  a  large  store  of  available  energy.  If 
this  be  correct,  the  inference  would  be  irresistible  in  favour  of  a  very  long, 


560  ON  THE   INDUCTION-COIL.  [272 

or  a  completely  closed,  iron  core.  Some  years  ago*,  reasoning  on  the  basis  of 
the  theory  of  Warburg  and  Hopkinson,  I  endeavoured  to  show  that  highly 
magnetized  iron  could  not  be  regarded  as  a  store  of  energy  —  that  the  energy 
expended  in  producing  the  magnetization  was  recoverable  but  to  a  small 
extent,  or  not  at  all.  Although  this  conclusion  does  not  appear  to  have  been 
accepted,  perhaps  in  consequence  of  an  erroneous  application  to  alternating 
current  transformers,  I  still  see  no  means  of  escape  from  it.  The  available 
energy  of  a  highly  magnetized  closed  circuit  of  iron  is  insignificant.  If  the 
length  be  limited,  there  is  available  energy,  in  virtue  of  the  free  polarity  at 
the  ends. 

The  theory  is  best  illustrated  by  the  case  of  an  ellipsoid  of  revolution 
exposed  to  uniform  external  magnetizing  force  «£)'  acting  parallel  to  the  axis. 
"  If  3  be  the  magnetization  parallel  to  the  axis  of  symmetry  (2c),  the  de- 
magnetizing effect  of  3  is  N%,  where  N  is  a  numerical  constant,  a  function 
of  the  eccentricity  (e).  When  the  ellipsoid  is  of  the  ovary  or  elongated  form, 

a  =  b  =  c  V(l  -  e*), 


becoming  in  the  limiting  case  of  the  sphere  (e  =  0), 

N=%7T- 

and  at  the  other  extreme  of  elongation  assuming  the  form 

2c 


/e,x 
(5) 


"  The  force  actually  operative  upon  the  iron  is  found  by  subtracting 
from  that  externally  imposed,  so  that 


and  if  from  experiments  on  very  elongated  ellipsoids  (N  =  0)  we  know  the 
relation  between  .£)  and  3,  then  the  above  equation  gives  us  the  relation 
between  .£)'  and  3  for  any  proposed  ellipsoid  of  moderate  elongation.  If  we 
suppose  that  $  is  plotted  as  a  function  of  3,  we  have  only  to  add  in  the 
ordinates  N%,  proper  to  a  straight  line,  in  order  to  obtain  the  appropriate 
curve  for  «£)'." 

The  work  expended  in  magnetizing  the  iron  is  per  unit  of  volume 


*  "On  the  Energy  of  Magnetized  Iron,"  Phil.  Mag.  xxn.  p.  175  (1886);  Scientific  Papers, 
n.  p.  543. 


1901]  ON   THE   INDUCTION-COIL.  561 

if  we  reckon  from  the  condition  of  zero  magnetization.  The  first  part  is 
practically  wasted ;  the  second,  which  in  most  cases  of  open  magnetic  circuits 
is  much  the  larger,  is  completely  recovered  when  the  iron  is  demagnetized. 

If  it  appear  paradoxical  that  the  large  integral  electromotive  force  which 
would  accompany  the  disappearance  of  high  magnetization  in  a  closed  iron 
circuit  should  be  so  inefficient,  we  must  remember  that  the  mechanical  value 
of  electromotive  force  depends  upon  the  magnitude  of  the  current  which  it 
drives,  and  that  in  the  present  case  the  existence  of  more  than  a  very  small 
current  is  inconsistent  with  that  drop  of  magnetization  upon  which  the 
electromotive  force  depends. 

The  considerations  above  explained  are  of  interest  in  the  present  question 
as  affording  a  limit  depending  only  upon  the  iron  core  and  the  secondary 
capacity.  For  3  cannot  exceed  a  value  estimated  at  about  1700  C.G.S.,  what- 
ever may  be  the  magnetizing  force  of  the  primary  current.  Thus  if  v  be  the 
volume  of  the  core,  the  maximum  energy*  is 

i^xvx  17002; 
and  the  limit  to  V  is  found  by  equating  this  to  \qV*,  so  that 

) (6) 


I  have  made  a  rough  application  of  this  formula  to  a  coil  in  my  possession, 
with  results  that  may  be  here  recorded.  The  core  had  a  diameter  of  3  cm. 
and  a  length  of  27  cm.,  so  that  v  =  180  c.c.  From  (5),  properly  applicable 
only  to  an  ellipsoid,  we  get  by  setting  2a  =  3,  2c  =  27,  N  =  '30. 

The  capacity  of  the  secondary  is  more  difficult  to  deal  with.  In  modern 
coils  the  greater  part  would  appear  to  arise  from  the  positive  and  negative 
potentials  at  the  ends  of  the  coil  as  opposed  to  the  zero  potential  of  the 
primary  wire.  The  capacity  between  the  primary  and  secondary  wires,  con- 
sidered as  poles  of  a  condenser,  can  be  calculated  and  in  many  cases  de- 
termined experimentally.  The  axial  dimension  of  the  secondary  of  the  coil 
above  referred  to  is  about  18  cm.,  and  the  external  diameter  of  the  primary 
wire  is  about  5  cm.,  making  the  area  of  each  of  the  opposed  surfaces 
270  sq.  cm.  The  interval  between  the  primary  and  secondary  wires  is 
•25cm.;  so  that,  taking  the  specific  inductive  capacity  of  the  intervening 
layer  at  3,  we  get  for  the  capacity  in  electrostatic  measure  of  the  condenser 
so  constituted 

J-x3x|5 

*  The  energy  of  the  primary  current  without  a  core  is  here  neglected. 

t  Another  coil  by  Apps,  in  which  the  insulation  was  sufficiently  good  to  allow  the  application 
of  electrostatic  methods,  was  tested  experimentally.  The  capacity  between  primary  and  secondary 
wires  was  thus  found  to  be  120  cm.,  less  than  the  half  of  that  calculated  for  the  first  coil.  But 
in  this  case  an  ebonite  tube  separated  the  two  wires. 

36 


562  ON  THE   INDUCTION-COIL.  [272 

Only  a  fraction  of  this,  however,  is  operative  in  the  present  case.  On  the 
supposition  of  a  coil  constructed  in  numerous  sections,  the  potential  in  the 
middle  will  be  zero,  the  same  as  that  of  the  primary  wire,  and  will  increase 
numerically  towards  either  end.  The  factor  of  reduction  on  this  account 

r+i 
will  be   I      #2  dx,  or  -^  ,  so  that  we  may  take  as  the  value  of  q  in  (6)  about 

23  cm.  —  probably  rather  an  underestimate.  Calculating  from  these  data,  we 
get  in  (6) 

F=2600. 

This  is  in  electrostatic  measure.  The  corresponding  volts  are  7'9  x  105.  If 
we  reckon  33,000  volts  to  the  cm.,  the  spark-length  will  stand  at  24  cm. 
The  coil  in  question  is  supposed  to  be  capable  of  an  8  or  10  cm.'  spark,  and 
doubtless  was  capable  when  new.  It  is  remarkable  that  the  limit,  fixed  by 
the  iron  and  secondary  capacity  alone,  should  exceed  so  moderately  the 
actual  capability  of  the  coil. 

The  limiting  formula  (6),  in  which  neither  the  value  of  the  primary 
current  nor  the  number  of  secondary  windings  appears,  is  arrived  at  by 
supposing  the  iron  to  be  magnetically  saturated.  It  illustrates,  no  doubt 
with  much  exaggeration,  the  disadvantage  of  too  great  a  length.  If  a  be 
given,  while  c  varies,  v  and  q  are  both  proportional  to  c,  so  that  V  oc  *JN. 
And  JN  oc  c"1  nearly.  In  somewhat  the  same  way  the  increase  of  effective 
capacity  explains  the  comparative  failure  of  attempts  to  increase  spark-length 
by  combining  similar  coils  in  series,  in  spite  of  the  augmented  energy  at  the 
moment  of  break*. 

If  the  object  be  a  rough  estimate  rather  than  a  limit,  a  more  practical 
formula  will  be  obtained  by  substituting  for  3  in  (6)  its  approximate  value 
N\  so  that 


«£)'  denoting  the  external  magnetizing  force,  due  to  the  primary  current. 
The  actual  magnetizing  force,  required  to  magnetize  the  soft  iron,  is  here 
regarded  as  relatively  negligible.  According  to  (7)  the  spark-length  is  pro- 
portional, cceteris  paribus,  to  the  primary  current  ;  and  it  increases  with  the 
length  of  the  coil,  since  N  now  occurs  in  the  denominator.  The  application 
must  not  be  pushed  into  the  region  where  the  iron  becomes  approximately 
saturated. 

In  the  above  discussion  the  capacity  q  of  the  secondary  will  probably  be 
thought  to  play  an  unexpectedly  important  part,  and  the  question  may  be 
raised  whether  it  is  really  this  capacity  which  limits  the  spark-length  in 

*  I  am  indebted  to  Mr  Swinton  for  the  details  of  some  experiments  in  this  direction  made 
for  Lord  Armstrong. 


1901]  ON   THE   INDUCTION-COIL.  563 

actual  coils.  It  is  not  difficult  to  prove  by  experiment  that  capacities  of  the 
order  above  estimated,  applied  to  the  secondary  terminals,  do  in  fact  reduce 
the  spark-length,  though  not,  so  far  as  I  have  seen,  to  the  extent  demanded 
by  the  law  of  q~*.  But  we  must  remember  that  this  law  has  been  obtained 
on  the  assumptions,  not  to  be  fulfilled  in  practice,  of  absolute  suddenness  of 
break  and  of  entire  absence  of  eddy-currents  in  the  iron.  If  under  these 
conditions  secondary  capacity  were  also  absent,  it  would  seem  that  there 
could  be  no  limit  to  the  maximum  potential  developed.  The  experiments 
of  Prof.  J.  J.  Thomson*  may  be  considered  to  show  that  even  in  extreme 
cases,  such  as  the  present,  the  iron,  as  a  magnetic  body,  would  not  fail  to 


As  regards  the  eddy-currents,  it  may  be  well  to  consider  a  little  further 
upon  what  their  importance  depends.  If  there  were  no  secondary  circuit, 
the  magnetism  of  each  wire  of  the  iron  core  would  be  continued  at  the 
moment  after  break,  supposed  infinitely  sudden,  by  a  superficial  eddy-current. 
A  secondary  circuit,  closely  intertwined,  with  the  primary,  would  transfer 
these  eddy-  currents  to  itself,  and  so  continue  for  the  first  moment  the  mag- 
netism of  the  core.  But  a  little  later,  as  the  magnetism  diminished,  eddy- 
currents  would  tend  to  be  formed,  and  their  importance  for  our  purpose 
depends  upon  their  duration.  If  this  be  short,  compared  with  the  time- 
constants  of  the  secondary  circuit,  their  influence  may  be  neglected.  Other- 
wise the  electromotive  force  of  the  falling  magnetism  lags,  and  acts  to  less 
advantage.  The  time-constant,  viz.  the  time  in  which  the  current  falls  in 
the  ratio  e  :l,  for  the  principal  eddy-current  in  a  cylinder  of  radius  R  is 
given  by 

torfiCK  /ox 

=  (2=464?'   •' 

where  C  represents  the  conductivity  and  fi  the  permeability  f.  If  d  be  the 
thickness  of  a  thin  sheet  having  the  same  time-constant  as  the  wire  o'f  radius 
R,  it  is  easily  shown  in  the  same  way  that 

d-.R^-jr:  2-404. 
If  we  take  for  iron  in  C.G.S.  measure 

(7=1/9611,.  /A  =  500, 

we  get  approximately 

....................................  (9) 


so  that  for  a  wire  of  1  mm.  diameter  T  =  -^fa  second.  It  may  be  doubted 
whether  this  would  be  small  enough  to  prevent  the  eddy-currents  reacting 
injuriously  upon  the  secondary  circuit. 

*  Recent  Researches,  p.  323. 

t   Brit.  Assoc.  Rep.  p.  446  (1882)  ;  Scientific  Papers,  n.  p.  128. 

36—2 


564  ON   THE    INDUCTION-COIL.  [272 

We  will  now  consider  the  third  of  the  causes  which  impose  a  limit  upon 
the  secondary  spark,  viz.  want  of  suddenness  in  the  break,  supposed  for  the 
present  to  be  unprovided  with  a  condenser.  After  the  cessation  of  metallic 
contact  the  primary  current  is  prolonged  by  the  formation  of  a  sort  of  arc, 
the  duration  of  which  depends  among  other  things,  such  as  the  character  of 
the  metals,  upon  the  magnitude  of  the  current  itself.  If  we  again  suppose 
the  behaviour  of  the  iron  to  be  ideal,  we  may  treat  the  secondary  circuit  as 
a  simple  vibrator,  upon  which  acts  a  force  (  U)  proportional  to  the  rate  of  fall 
of  the  primary  current.  The  equation  of  such  a  vibrator  is,  as  usual, 

£  +  -*+-- *'    <10> 

and  the  solution  corresponding  to  u  =  0  (no  charge),  dujdt  =  Q  (no  current), 
when  £  =  0,  is* 

u  =  -1,  re-fc  <«-«')  sin  n'  (t  -  t') .  Udt',  . . .(11) 

n  J0 
where 

ri  =  J(n*-iK*) (12) 

The  various  elements  of  (11)  represent  in  fact  the  effects  at  time  t  of  the 
velocities  Udt'  communicated  (t  —  t')  earlier.  In  the  present  case  we  are  to 
suppose  that  U  is  positive  throughout,  and  that  fUdt'  is  given. 

The  integral  simplifies  in  the  case  of  K  =  0,  that  is  of  evanescent  secondary 
resistance.  We  have  then  n'  =  n,  and 


-  I' 
n  Jo 


.(13) 


It  is  easy  to  see  that  the  integral,  representing  the  potential  at  the  secondary 
terminals,  is  a  maximum  when  U  is  concentrated  at  some  one  time  t',  and 
t  is  such  that 

sinn(t-t')=l, 

that  is,  when  the  break  is  absolutely  sudden  and  the  time  considered  is  one 
quarter  period  later.  If  the  break  be  not  sudden,  sin  n  (t  —  t')  will  depart 
from  its  maximum  value  during  part  of  the  range  of  integration,  and  the 
highest  possible  value  of  u  will  not  be  attained. 

The  theory  is  substantially  the  same  if  K  be  finite.     There  is  some  value 
of  (t  —  t')  for  which 


is  a  maximum  ;  and  the  greatest  value  of  u  will  be  arrived  at  by  concen- 
trating U  at  some  time  t',  and  by  so  choosing  t  that  (t  —  t')  has  the  value 
above  defined.  The  conclusion  is  that  if  the  primary  current  fall  to  zero 

*  Theory  of  Sound,  Vol.  i.  §  66. 


1901]  ON   THE   INDUCTION-COIL.  565 

from  its  maximum  value  without  oscillation,  the  potential  at  the  secondary 
terminals  will  be  greatest  when  this  fall  is  absolutely  sudden,  and  that  this 
greatest  value  begins  to  be  sensibly  departed  from  when  the  break  occupies 
a  time  comparable  with  one  of  the  time-constants  of  the  secondary  circuit. 
In  the  case  of  no  resistance  we  have  to  deal  merely  with  the  time  of 
secondary  oscillation ;  but  if  the  resistance  is  high,  the  other  time-constant, 
N /S,  may  be  the  smaller  (see  equation  (2)}. 

It  is  here  that  the  character  of  the  secondary  coil,  especially  as  regards 
the  number  of  its  windings,  enters  into  the  question.  On  the  supposition  of 
an  absolutely  sudden  break,  we  arrived  at  the  rather  paradoxical  conclusion 
that  the  limit  of  spark-length  depended  only  upon  the  capacity  of  the 
secondary  without  regard  to  the  number  of  windings — a  number  which  could 
be  changed  in  a  high  ratio  without  sensibly  influencing  the  capacity.  We 
see  now,  at  any  rate,  that  a  reduction  in  the  number  of  windings,  and  the 
accompanying  diminution  in  the  time  of  oscillation,  would  necessitate  a 
greater  and  greater  suddenness  of  break,  if  the  full  effect  is  to  be  retained. 

We  will  now  consider  the  action  of  the  primary  condenser — a  question,  the 
reader  may  be  inclined  to  think,  already  too  long  postponed.  For  it  is  well 
known  that  in  most  actual  coils  the  condenser  is  an  auxiliary  of  the  utmost 
importance,  increasing  the  spark-length  5  or  10  times,  even  when  the  break 
is  made  at  pieces  of  platinum.  And,  although  it  has  been  customary  to  say, 
no  doubt  correctly,  that  the  condenser  acts  by  absorbing  into  itself  the 
primary  spark,  and  so  increasing  the  suddenness  of  break,  it  is  usual  to 
attribute  to  it  a  further  virtue,  and  not  unnaturally  when  it  is  remembered 
that  the  effect  may  be  not  merely  to  stop,  but  actually  to  reverse,  the  primary 
current.  If,  however,  the  theory  of  the  foregoing  pages  is  correct,  we  shall 
be  constrained  to  take  a  different  view. 

The  action  of  the  condenser,  and  especially  the  most  advantageous 
capacity,  has  been  studied  experimentally  by  Walter  and  by  Mizuno.  That 
there  must  be  a  most  advantageous  capacity  is  evident  beforehand,  inasmuch 
as  a  very  small  capacity  is  continuous  with  no  condenser  at  all,  and  a  very 
large  capacity  is  continuous  with  an  uninterrupted  flow  of  the  primary 
current.  It  is  more  instructive  that  the  former  observer  found  the  most 
advantageous  capacity  to  vary  with  the  manner  of  break  (whether  in  air  or 
under  oil),  and  that  the  latter  found  a  dependence  upon  the  strength  of  the 
primary  current,  a  larger  current  demanding  a  larger  condenser. 

When  a  condenser  is  employed,  it  is  important  that  it  be  connected  as 
directly  as  possible  with  the  points  between  which  the  break  is  made  to 
occur.  A  comparatively  small  electromagnet,  included  between  one  of  the 
break-points  and  the  associated  condenser-terminal,  suffices  to  diminish,  or 
even  to  annul,  the  advantage  which  the  use  of  the  condenser  otherwise 


566  ON  THE   INDUCTION-COIL.  [272 

presents*.  The  explanation  is,  of  course,  that  the  current  in  an  electro- 
magnet so  situated  tends  to  flow  on  across  the  break -gap,  and  so  to  establish 
an  arc,  with  a  force  which  the  condenser  is  powerless  to  relieve. 

Returning  to  the  theoretical  aspect  of  the  question,  and  inquiring  whether 
there  is  any  reason  for  expecting  a  condenser  to  give  an  advantage  as 
compared  with  an  absolutely  sudden  cessation  of  the  primary  current,  it 
is  difficult  to  see  ground  for  other  than  a  negative  answer.  In  the  case  of 
no  magnetic  leakage,  somewhat  closely  approached,  one  would  suppose,  in 
practice,  an  instantaneous  abolition  of  the  primary  current  throws  the  whole 
available  energy  into  the  secondary  circuit,  and  thus,  doing  all  that  is  possible, 
allows  no  room  for  an  improvement.  Under  such  conditions  a  condenser  can 
only  do  harm. 

In  the  opposite  extreme  case  of  but  a  relatively  small  mutual  induction 
between  primary  and  secondary,  it  is  indeed  conceivable  that  the  action  of 
a  condenser  may  be  advantageous.  The  two  currents  would  then  be  com- 
paratively independent  and,  if  the  resistances  were  low,  they  might  execute 
numerous  oscillations.  If  the  primary  current  were  simply  stopped,  the 
effect  in  the  secondary  would  be  small ;  whereas,  especially  if  there  were 
synchronism,  the  vibrations  of  the  primary  current  rendered  possible  by  the 
condenser  might  cause  an  accumulation  of  effect  in  the  secondary.  The  case 
would  be  that  of  "intermittent  vibrations f,"  such  as  may  occur  when  a  large 
tuning-fork  is  clamped  in  a  vice.  A  vibration,  started  by  a  blow,  in  one  prong 
gradually  transfers  itself  to  the  other.  But  it  is  difficult  to  believe  that  any- 
thing of  this  sort  occurs  in  an  induction-coil  as  actually  used. 

I  do  not  know  how  far  the  theoretical  arguments  here  advanced  will 
convince  the  reader  that  the  use  of  a  condenser  in  the  primary  circuit  should 
offer  no  advantage  as  compared  with  a  sufficiently  sudden  simple  break ;  but 
I  may  confess  that  I  should  have  hesitated  to  put  them  forward  had  I  not 
obtained  experimental  confirmation  of  them.  My  earlier  attempts  in  this 
direction  were  unsuccessful.  A  quick  break  was  constructed  in  which  a 
spring,  bearing  upwards  against  a  stop,  could  be  knocked  away  by  a  blow 
with  a  staff,  or  by  a  falling  weight.  Although  the  contacts  were  of  platinum, 
but  little  advantage  was  gained  in  comparison  with  the  ordinary  platinum 
break  of  the  coil.  Thus  in  one  set  of  experiments,  where  the  coil  was  excited 
by  a  single  Grove  cell,  a  break  made  quickly  by  hand  gave  a  spark  about 
8  mm.  long.  The  use  of  a  weight,  hung  by  a  cotton  thread,  and  falling 
through  about  12  feet  when  the  thread  was  burned,  increased  the  length  only 
to  8£  mm.  This  was  without  a  condenser.  When  the  condenser  was  applied, 
the  spark-length  was  14  mm.,  and  it  made  no  perceptible  difference  whether 
or  not  the  falling  weight  was  employed.  Considering  that  the  velocity  of  the 

*  PkiL  Mag.  Vol.  n.  p.  282  (1901).     [Vol.  iv.  p.  552.] 
t  Theory  of  Sound,  Vol.  i.   §  114. 


1901]  ON   THE   INDUCTION-COIL.  567 

weight  at  impact  must  have  been  about  30  feet  per  second  and  that  its  mass 
was  large  compared  with  that  of  the  spring,  these  results  were  far  from  pro- 
mising. With  a  stronger  primary  current  the  advantage  gained  from  the 
condenser  was  much  greater,  and  the  utility  of  the  quicker  break,  with  or 
without  condenser,  seemed  to  be  nil. 

But,  in  spite  of  the  failure  of  the  quick  break,  one  or  two  observations 
presented  themselves  which  seemed  worthy  of  being  followed  up.  It  was 
noticed  that,  with  one  Grove  cell  in  the  primary,  the  spark,  although  very 
inferior  when  no  condenser  at  all  was  employed,  was  improved  when  the 
usual  condenser  (of  large  capacity)  was  replaced  by  a  single  sheet  of  coated 
glass  (Franklin's  pane).  And,  what  was  perhaps  more  instructive  still,  when 
the  already  weak  primary  current  was  further  reduced  by  the  insertion  of  one 
or  two  ohms  extra  resistance,  the  spark-length  (now  very  small)  was  less  with 
than  without  the  usual  coil  condenser.  This  observation  was  repeated,  with 
like  result,  upon  another  coil  (by  Apps)  and  its  associated  condenser.  At 
any  rate  in  the  case  of  very  weak  primary  currents,  the  usual  condenser  did 
harm  rather  than  good. 

The  view,  suggested  by  the  foregoing  results,  that  while  the  ordinary 
break  was  quick  enough  in  the  case  of  weak  currents  to  allow  a  condenser 
to  be  dispensed  with,  the  superior  arcing  power  of  strong  currents  demanded 
a  much  more  rapid  break,  encouraged  further  efforts.  An  attempt  to  secure 
suddenness  by  forcibly  breaking  with  a  jerk  a  length  of  rather  thin  copper 
wire,  forming  part  of  the  primary  circuit,  failed  entirely,  as  did  also,  perhaps 
for  want  of  sufficiently  powerful  appliances,  an  attempt  to  blow  up  a  portion 
of  the  primary  circuit  by  electric  discharge.  Another  method,  however,  at 
once  allowed  an  advance  to  be  secured.  This  consisted  in  cutting  the  primary 
circuit  by  a  pistol-bullet ;  and  it  was  found  that  this  form  of  break  without 
condenser  was  about  as  efficient  as  the  usual  platinum  break  with  condenser, 
although  the  primary  current  was  increased  to  that  supplied  by  three  or  four 
Grove  cells  and  the  spark-length  to  40  mm.,  that  is,  under  about  the  ordinary 
conditions  of  working. 

A  further  improvement  was  effected  by  cutting  away  about  half  of  the 
bullet  with  the  intention  of  raising  its  velocity.  The  following  results  were 
recorded  with  an  Apps'  coil  excited  by  three  Grove  cells.  The  spark-gap 
being  50  mm.,  the  usual  platinum  break  and  condenser  were  not  able  to  send 
a  spark  across.  Even  with  the  somewhat  more  efficient  break  provided  by 
a  pot  of  mercury  well  drowned  in  oil  and  condenser,  only  about  one  break  in 
fifteen  succeeded.  On  the  other  hand,  of  three  bullets  fired  so  as  to  cut  the 
primary  wire  (no  condenser)  two  succeeded ;  while  for  the  failure  of  the  third 
there  was  some  explanation.  The  bullet  without  condenser  was  now  dis- 
tinctly superior  to  the  best  ordinary  break  with  condenser. 

The  next  step  was  the  substitution  of  a  rifle-bullet,  fired  from  a  service 
rifle.  Here  again  the  bullets  were  reduced  to  about  one-half,  and  after 


568  ON  THE  INDUCTION-COIL.  [272 

cutting  the  wire  were  received  in  a  long  box  packed  with  wet  sawdust.  At 
60  mm.,  while  the  mercury-under-oil  break  with  condenser  gave  only  feeble 
brush-discharges,  good  sparks  were  nearly  uniformly  obtained  from  the  bullet 
working  without  a  condenser.  At  70  mm.  the  bullet  without  condenser  was 
about  upon  a  level  with  the  mercury-under-oil  break  with  condenser  at 
60  mm.  As  regards  the  strength  of  the  primary  current,  if  there  was  any 
difference,  the  advantage  was  upon  the  side  of  the  ordinary  break  with 
condenser,  inasmuch  as  in  the  case  of  the  bullet  the  leads  were  longer  and 
included  about  8  cm.  of  finer  copper  wire  where  the  bullet  passed. 

In  the  next  set  of  experiments  upon  the  same  Apps'  coil  excited  by  three 
Groves,  the  bullet  was  used  each  time,  and  the  comparison  was  between  the 
effect  with  and  without  the  usual  coil  condenser.  At  55  mm.  the  bullet 
without  condenser  gave  each  time  a  fair  or  a  good  spark,  while  with  the 
condenser  there  was  nothing  more  than  a  feeble  brush  scarcely  visible  in 
a  good  light. 

The  single  pane  of  coated  glass  was  next  substituted  for  the  usual  con- 
denser of  the  coil,  with  the  idea  that  possibly  this  might  be  useful  although 
the  larger  capacity  was  deleterious.  But  no  distinct  difference  was  detected 
when  the  bullet  was  fired  with  this  or  without  any  condenser. 

In  the  last  set  of  experiments  now  recorded  the  primary  current  was 
raised,  six  Grove  cells  being  employed  partly  in  parallel,  and  the  wire  was 
cut  each  time  by  a  rifle-bullet.  At  90  mm.  no  spark  could  be  got  when  the 
coil  condenser  was  in  connexion ;  when  it  was  disconnected,  a  spark,  good  or 
fair,  was  observed  nearly  every  shot. 

Altogether  these  experiments  strongly  support  the  view  that  the  only  use 
of  a  condenser,  in  conjunction  with  an  ordinary  break,  is  to  quicken  it  by 
impeding  the  development  of  an  arc,  so  that  when  a  sufficient  rapidity  of 
break  can  be  obtained  by  other  means,  the  condenser  is  deleterious,  operating 
in  fact  in  the  reverse  direction,  and  prolonging  the  period  of  decay  of  the 
primary  current.  It  is  hoped  that  the  establishment  of  this  fact  will  inspire 
confidence  in  the  theory,  and  perhaps  suggest  improvements  in  the  design  of 
coils.  The  first  requirement  is  evidently  the  existence  of  sufficient  energy  at 
break,  and  this  implies  a  considerable  mass  of  iron,  well  magnetized,  and  not 
forming  a  circuit  too  nearly  closed.  The  full  utilization  of  this  energy  is 
impeded  by  want  of  suddenness  in  the  break,  by  eddy-currents  in  the  iron, 
and  (in  respect  of  spark-length)  by  capacity  in  the  secondary.  It  is  to  be 
presumed  that  in  a  well-designed  coil  these  impediments  should  operate 
somewhat  equally.  It  would  be  useless  to  subdivide  the  iron,  or  to  reduce 
the  secondary  capacity,  below  certain  limits,  unless  at  the  same  time  the 
break  could  be  made  more  sudden.  It  would  not  be  surprising  if  it  were 
found  that  the  tentative  efforts  of  skilful  instrument-makers  have  already 
led  to  a  suitable  compromise,  at  least  in  the  case  of  coils  of  moderate  size. 
The  design  of  larger  instruments  may  leave  more  to  be  accomplished. 


CONTENTS    OF   VOLUMES   I.-IV. 

CLASSIFIED  ACCORDING  TO   SUBJECT. 
MATHEMATICS. 

PAGE 

ri 
On   the   Values   of  the   Integral   I    QnQn'dfj,,    Qn,    Qn'   being 

Jo 

Laplace's  Coefficients  of  the  orders  n,  n',  with  an  appli- 
cation to  the  Theory  of  Radiation.     Art.  3.     1870      .         .       Vol.  I.    21 
On  a  Correction  sometimes  required  in  Curves  professing  to 

represent  the  connexion  between  two  Physical  Magnitudes. 

Art.  12.     1871  .         .        ...         .         .        .        .  „       135 

Notes  on  Bessel's  Functions.     Art.  15.     1872          .         .         .  ,,140 

Note  on  the  Numerical  Calculation  of  the  Roots  of  Fluctuating 

Functions.     Art.  28.     1874 ,,190 

A  History  of  the  Mathematical  Theories  of  Attraction  and  the 

Figure  of  the  Earth  from  the  time  of  Newton  to  that  of 

Laplace.     By  I.  Todhunter,  M.A.,  F.R.S.     Two  Volumes. 

(London,  Macmillan  &  Co.,  1873.)     Art.  29.     1874    .         .  „       196 

On  the  Approximate  Solution  of  Certain  Problems  relating  to 

the  Potential.     Art.  39.     1876 ,,272 

Questions   from   Mathematical   Tripos  Examination  for  1876. 

Art.  41.     1876 ,,280 

On  the  Relation  between  the  Functions  of  Laplace  and  Bessel. 

Art.  51.     1878 ,,338 

A    simple    Proof  of  a    Theorem    relating   to   the    Potential. 

Art.  54.     1878 ,,347 

On  the  Resultant  of  a   large  number  of  Vibrations   of  the 

same  Pitch  and  of  arbitrary  Phase.     Art.  68.     1880  .        '.  „       491 

On  Point-,  Line-,  and  Plane-Sources  of  Sound.    Art.  147.     1888    Vol.  III.    44 
On   James   Bernoulli's   Theorem   in  Probabilities.     Art.    243. 

1899  ..      .        •    Vol.  IV.  370 


570  CONTENTS   OF    VOLUMES   I. — IV. 

GENERAL  MECHANICS. 

PAGE 

Some  General  Theorems  relating  to  Vibrations.    Art.  21.     1873  Vol.  I.  170 
Section  I.    The  natural  periods  of  a  conservative  system,  vibrating 
freely  about   a  configuration  of  stable    equilibrium,    fulfil  the 

stationary  condition            .    ,                 .          .          .                     .  „         170 

Section  II.     The  Dissipation  Function    .          .          .           '.'".''*.  „          176 

Section  III.     [The  Reciprocity  Theorem]          .          .          ,          .  ,,176 

On  the  Vibrations  of  Approximately  Simple  Systems.     Art.  24. 

1873,  1874 ,,185 

On  the  Fundamental  Modes  of  a  Vibrating  System.     Art.  25. 

1873 ,,186 

A  Statical  Theorem.     Art.  32.     1874,  1875    .          .         .         .  „       223 

General  Theorems   relating   to   Equilibrium   and   Initial   and 

Steady  Motions.     Art.  34.     1875      .....  „       232 

Uniformity  of  Rotation.     [Phonic  Wheel]     Art.  56.     1878      .  „       355 

On  Maintained  Vibrations.     Art.  97.     1883     .         .        ..         .     Vol.  II.  188 
The  Soaring  of  Birds.     Art.  98.     1883     .         .         .         .       -f.  „       194 

Investigation  of  the  Character  of  the  Equilibrium  of  an  In- 
compressible Heavy  Fluid  of  Variable  Density.  Art.  100. 
1883 ,,200 

Suggestions   for   facilitating   the   Use  of  a  Delicate  Balance. 

Art.  104.     1883          .         .         .        .-       ...        .        .  ,,226 

A    Theorem    relating    to    the    Time-Moduli    of    Dissipative 

Systems.     Art.  125.     1885         .         t;        ..:/...         .  „       428 

Testing  Dynamos.     Art.  133.     1886         ....         .  ,,474 

The  Reaction  upon  the  Driving-Point  of  a  System  executing 
Forced  Harmonic  Oscillations  of  Various  Periods,  with 
Applications  to  Electricity.  Art.  134.  1886  .  .  .  „  475 

On  the  Maintenance  of  Vibrations  by  Forces  of  Double 
Frequency,  and  on  the  Propagation  of  Waves  through  a 
Medium  endowed  with  a  Periodic  Structure.  Art.  142. 
1887  .  .  .  .'  '  .  .  "  .  .  .  .  Vol.  III.  1 

The  Sailing  Flight  of  the  Albatross.     Art.  159.     1889.         .          „         267 
On  the  Vibrations  of  an  Atmosphere.     Art.  166.     1890          .         „         335 


CONTENTS   OF   VOLUMES   I. — IV.  571 

PAGE 

On    Huygens's  Gearing  in   Illustration   of  the    Induction    of 

Electric  Currents.     Art.  171.     1890          ....  Vol.  III.  376 

The  Bourdon  Gauge.     Art.  172.     1890 „         379 

Experiments     in     Aerodynamics.        [Review      of     Langley's] 

Art.   184.     1891 ,,491 

Superheated  Steam.     Art.  188.     1892 ,,538 

Heat  Engines  and  Saline  Solutions          .           .          .           .           .  „           539 

Heat  Engines  and  Saline  Solutions          .....  „           540 

Remarks  on  Maxwell's   Investigation   respecting  Boltzmann's 

Theorem.     Art.  190.     1892 ,,554 

Grinding  and  Polishing  of  Glass  Surfaces.     Art.  205.     1893  .  Vol.  IV.    74 

On   the  Propagation  of  Waves  along  Connected  Systems  of 

Similar  Bodies.     Art.  235.     1897 „         340 

On  Iso-periodic  Systems.     Art.  242.     1898    .         .         .         .  „        367 

On  the  Calculation  of  the  Frequency  of  Vibration  of  a  System 
in  its  GraVest  Mode,  with  an  Example  from  Hydro- 
dynamics. Art.  249.  1899 „  407 

The  Law  of  Partition  of  Kinetic  Energy.     Art.  253.     1900       .  „         433 

The  Mechanical  Principles  of  Flight.     Art.  257.     1900     .         .  „         462 

On  a  Theorem  analogous  to  the  Virial  Theorem.     Art.  262. 

1900           .' ,,491 

Polish.     Art.  268.     1901  .  542 


ELASTIC  SOLIDS. 

On  the  Nodal  Lines  of  a  Square  Plate.  Art.  22.  1873  .  Vol.  I.  182 

Vibrations  of  Membranes.  Art.  26.  1873  .  .  .  .  „  187 
On  the  Infinitesimal  Bending  of  Surfaces  of  Revolution. 

Art.  78.     1881  .........  551 

On  Waves  propagated  along  the  Plane  Surface  of  an  Elastic 

Solid.    [With  reference  to  Earthquakes]    Art.  130.    1885.    Vol.11.  441 

On  the  Bending  and  Vibration  of  Thin  Elastic  Shells,  especially 

of  Cylindrical  Form.     Art.  152.     1888       .         .         .         .  Vol.  III.  217 

Note  on  the  Free  Vibrations  of  an  Infinitely  Long  Cylindrical 

Shell.     Art.  155.     1889     ,  244 


572  CONTENTS   OF   VOLUMES   I. — IV. 


PAGE 


On  the  Free  Vibrations  of  an  Infinite  Plate  of  Homogeneous 

Isotropic  Elastic  Matter.     Art.  156.     1889      .         .         .  Vol.111.   249 

On  the  Uniform  Deformation  in  Two  Dimensions  of  a 
Cylindrical  Shell  of  Finite  Thickness,  with  Application 
to  the  General  Theory  of  Deformation  of  Thin  Shells. 
Art.  162.  1889 ,,280 

On  Bells.     Art.  164.     1890     .         .         .         .         .         .         .         „         318 

Appendix  :    On  the  Bending  of  a  Hyperboloid  of  Kevolution       .  „  330 

The  Bourdon  Gauge.     Art.  172.     1890 ,,379 

On  the  Stresses  in  Solid  Bodies  due  to  Unequal  Heating, 
and  on  the  Double  Refraction  resulting  therefrom. 
Art.  265.  1901  .  Vol.  IV.  502 


CAPILLARITY. 

The  Instability  of  Jets.     Art.  58.     1879 Vol.1.  361 

The    Influence    of    Electricity    on    Colliding    Water    Drops. 

Art.  59.     1879 „  372 

On  the  Capillary  Phenomena  of  Jets.     Art.  60.     1879           .  „  377 

Appendix  I.     [Vibrations  about  a  Cylindrical  Figure]         .           .  „  396 

Appendix  II.     [Vibrations  about  a  Spherical  Figure]          .           .  „  400 

Further   Observations    upon  Liquid   Jets,  in  Continuation  of 
those  recorded  in  the  Royal  Society's  '  Proceedings '  for 

March  and  May,  1879.     Art.  85.     1882  ....    Vol.  II.  103 
On  some  of  the  Circumstances  which  influence  the  Scattering  of 

a  nearly  Vertical  Jet  of  Liquid             .....  „  103 

Influence  of  Regular  Vibrations  of  Low  Pitch            ...  „  106 

The  Length  of  the  Continuous  Part        .           .           .           .  „  110 

Collision  of  Two  Resolved  Streams          .  .  .  .          .  ,,112 

Collision  of  Streams  before  Resolution    .          .          .          .          .  „  115 

On   the   Equilibrium  of  Liquid  Conducting  Masses  charged 

with  Electricity.  Art.  90.  1882 „  130 

On  the  Crispations  of  Fluid  resting  upon  a  Vibrating  Support. 

Art.  102.     1883         .         .         .         ...         .         .  ,,212 

On  Laplace's  Theory  of  Capillarity.  Art.  106.  1883  .  .  „  231 

The  Form  of  Standing  Waves  on  the  Surface  of  Running 

Water.  Art.  109.  1883 ,,258 

On  the  Tension  of  Recently  Formed  Liquid  Surfaces. 

Art.  167.     1890  .    Vol.  III.  341 


CONTENTS   OF    VOLUMES   I. — IV.  573 

PAGE 

Measurements  of  the  Amount  of  Oil  necessary  in  order  to 
check  the  Motions  of  Camphor  upon  Water.  Art.  168. 
1890 Vol.111.  347 

Foam.     Art.  169.     1890 .         .         .  „         351 

On  the  Superficial   Viscosity  of  Water.     Art.  170.     1890     .  „         363 

Instantaneous  Photographs  of  Water  Jets.     Art.  174.     1890  „         382 

On  the  Tension  of  Water  Surfaces,  Clean  and  Contaminated, 

Investigated  by  the  Method  of  Ripples.     Art.  175.     1890  „         383 

Postscript.     [Optical  Effect  of  greasy  Contamination]          .          .  „            394 

On  the  Theory  of  Surface  Forces.     Art.  176.     1890       .         .  „         397 

Some  Applications  of  Photography.     Art.  179.     1891     .         .  „         441 

On  Reflexion  from  Liquid  Surfaces  in  the  Neighbourhood  of 

the  Polarizing  Angle.     Art.  185.     1892  496 

On  the  Theory  of  Surface  Forces.  II.     Compressible  Fluids. 

Art.  186.     1892         .                 ,,513 

Experiments  upon  Surface- Films.     Art.  192.     1892         .         .  „         562 

The  Behaviour  of  Clean  Mercury             ......  562 

Drops  of  Bisulphide  of  Carbon  upon  Water  „            563 

Movements  of  Dust     .........  564 

Camphor  Movements  a  Test  of  Surface-Tension  „            565 

Influence  of  Heat         ........  „           567 

Saponine  and  Soap       ........  „           568 

Separation  of  Motes     .........  569 

The  Lowering  of  Tension  by  the  Condensation  of  Ether  Vapour  „            570 

Breath  Figures  and  their  Projection        ......  570 

On   the   Theory   of    Surface    Forces.   III.      Effect   of    Slight 

Contaminations.     Art.  193.     1892 „         572 

On   the   Instability  of  a  Cylinder   of  Viscous  Liquid   under 

Capillary  Force.     Art.  195.     1892 ,,585 

On  the  Instability  of  Cylindrical  Fluid  Surfaces.     Art.   196. 

1892 ,,594 

Investigations  in  Capillarity.     Art.   251.     1899       .         .         .  Vol.  IV.  415 

The  Size  of  Drops ,,415 

The  Liberation  of  Gas  from  Supersaturated  Solutions        .          .  „          420 

Colliding  Jets ,,421 

The  Tension  of  Contaminated  Water-Surfaces  „           425 

A  Curious  Observation                                           ....  „           430 


574  CONTENTS   OF   VOLUMES   I. — IV. 


HYDRODYNAMICS. 

PAGE 

Notes  on  Bessel's  Functions.     Art.  15.     1872         .         .         .      Vol.  I.  140 

Vibrations  of  a  Liquid  in  a  Cylindrical  Vessel.     Art.  37.     1875  „       250 

On  Waves.     Art.  38.     1876     .         .....         .  „       251 

The  Solitary  Wave       .          .          .        • .          .       -  ....       "  .  „         256 

Periodic  Waves  in  Deep  Water ,,261 

Oscillations  in  Cylindrical  Vessels  .          .          ;          .          .          .  „         265 

On  the  Approximate  Solution  of  Certain  Problems  relating  to 

the  Potential.     Art.  39.     1876         .....  „       272 

On  the  Resistance  of  Fluids.     Art.  42.     1876         .         .         . "          „       287 

Notes  on  Hydrodynamics.     Art.  43.     1876      .       '.'.'.  „       297 

The  Contracted  Vein   .          .          .          .          .          ...  „         297 

Meeting  Streams  .          .          ...          .          .          .          .   ".         „         302 

On  Progressive  Waves.     Art.  47.     1877.         .         .         .         .  ,,322 

Note  on  Acoustic  Repulsion.     Art.  52.     1878         .        .        .  „       342 

On  the  Irregular  Flight  of  a  Tennis-Ball.     Art.  53.     1877    .  „       344 

On  the  Instability  of  Jets.     Art.  58.     1879   .         .        ,    .     ,  „       361 
The    Influence    of    Electricity    on    Colliding    Water    Drops. 

Art.  59.     1879.         ........  ,,372 

On  the  Capillary  Phenomena  of  Jets.     Art.  60.     1879  .         .  „       377 

Appendix  I.     [Vibrations  about  a  Cylindrical  Figure]         .           .  „         396 

Appendix  II.     [Vibrations  about  a  Spherical  Figure]          .          .  „         400 

On   the   Stability,  or  Instability,  of  Certain  Fluid   Motions. 

Art.  66.     1880 ,,474 

Further  Observations  upon  Liquid  Jets,  in  Continuation   of 
those  recorded  in  the  Royal  Society's  '  Proceedings '  for 
March  and  May,  1879.     Art.  85.     1882  .         .'       .         .     Vol.11.  103 
On  some  of  the  Circumstances  which  influence  the  Scattering  of 

a  nearly  Vertical  Jet  of  Liquid             .          ,          .          .          .  „         103 

Influence  of  Regular  Vibrations  of  Low  Pitch            ...  „         106 

The  Length  of  the  Continuous  Part        .          ...          .          .  ,,110 

Collision  of  Two  Resolved  Streams          .     .     .          .     '-..,..  „         112 

Collision  of  Streams  before  Resolution     .          .          .          . ...  {  •  ,  „         115 

On  the  Equilibrium  of  Liquid  Conducting  Masses  charged 

with  Electricity.  Art.  90.  1882 „  130 

On  the  Dark  Plane  which  is  formed  over  a  Heated  Wire  in 

Dusty  Air.  Art.  93.  1882  .  .  .  .  .  .  .  „  151 


CONTENTS   OF    VOLUMES   I. — IV.  575 

PAGE 

The  Soaring  of  Birds.     Art.  98.     1883   .....  Vol.  IT.    194 

Investigation  of  the  Character  of  the  Equilibrium  of  an  In- 
compressible Heavy  Fluid  of  Variable  Density.  Art.  100. 
1883 .  .  .  .• „  200 

On  the  Vibrations  of  a  Cylindrical  Vessel  containing  Liquid. 

Art.  101.     1883         .  ,,208 

On  the  Circulation  of  Air  observed  in  Kundt's  Tubes,  and  on 

some  Allied  Acoustical  Problems.     Art.  108.     1883         .         „         239 

The    Form   of  Standing  Waves  on  the  Surface  of  Running 

Water.     Art.  109.     1883  .         .         .         ."  .         .         „         258 

On  the  Stability  or  Instability  of  Certain  Fluid  Motions.    II. 

Art.  144.     1887         .         .         .        .        .        .         .         .  Vol.  III.     17 

The  Sailing  Flight  of  the  Albatross.     Art.  159.     1889  .         .         „         267 
On  the  Vibrations  of  an  Atmosphere.     Art.  166.     1890         .         „         335 

On  the  Tension  of  Water  Surfaces,  Clean  and  Contaminated, 

Investigated  by  the  Method  of  Ripples.     Art.  175.     1890         „         383 

Some  Applications  of  Photography.     Art.  179.     1891     .         .         „         441 

Experiments     in     Aerodynamics.        [Review     of     Langley's] 

Art.  184.     1891 ,,491 

On   the   Question   of  the   Stability   of  the    Flow   of  Fluids. 

Art.  194.     1892 ,,575 

On   the   Instability  of  a   Cylinder  of  Viscous  Liquid  under 

Capillary  Force.     Art.  195.     1892 ,,585 

On  the  Instability  of  Cylindrical  Fluid  Surfaces.     Art.  196. 

1892 ,,594 

On  the  Influence  of  Obstacles  arranged  in  Rectangular  Order 

upon  the  Properties  of  a  Medium.     Art.  200.     1892      .    Vol.  IV.     19 

On  the  Flow  of  Viscous  Liquids,  especially  in  Two  Dimensions. 

Art.  208.     1893 „          78 

On  the  Stability  or  Instability  of  Certain  Fluid  Motions.   III. 

Art.  216.     1895       .........        ,        .        .        •        •        •         »        203 

On  the  Propagation  of  Waves  upon  the  Plane  Surface  separ- 
ating Two  Portions  of  Fluid  of  Different  Vorticities. 
Art.  217.  1895  ,,210 


576  CONTENTS   OF   VOLUMES   I. — IV. 

PAGE 

On  some  Physical  Properties  of  Argon  and  Helium.     Art.  218. 

1896 '     .    Vol.  IV.  215 

Density  of  Argon          .          .          .          .          .          .           .           .  „         215 

The  Refractivity  of  Argon  and  Helium   .....  „         218 

Viscosity  of  Argon  and  Helium      ......  „         222 

Gas  from  the  Bath  Springs   .......  „         223 

Buxton  Gas            .           .                   .          .          .           .          .          .  ,,223 

Is  Helium  contained  in  the  Atmosphere  ?          .          .        '  '. '   '     .  ' - v        „         224 

On  the  Viscosity  of  Hydrogen  as  affected  by  Moisture. 

Art.  234.  1897  .  .  .  .  .  .  .  .  ,,336 

On  the  Calculation  of  the  Frequency  of  Vibration  of  a  System 
in  its  Gravest  Mode,  with  an  Example  from  Hydro- 
dynamics. Art.  249.  1899  ......  ,,407 

On  the  Viscosity  of  Argon  as  affected  by  Temperature. 

Art.  254.  1900 ,,452 

The  Mechanical  Principles  of  Flight.     Art,  257.     1900 .'      .  „       462 

On  the  Viscosity  of  Gases  as  affected  by  Temperature. 

Art.  259.  1900 ,,481 

SOUND. 

Remarks  on  a  Paper  by  Dr  Sondhauss.     Art.  4.     1870         .      Vol.  I.     26 

On  the  Theory  of  Resonance.     Art.  5.     1870          .         .         .  „         33 

Introduction         .........  ,,33 

Part  I. ,,37 

Several  Openings       ........  ,,39 

Double  Resonance     .           .           .           .           .           .           .           .  „           41 

Open  Organ-pipes      ........  ,,45 

Long  Tube  in  connexion  with  a  Reservoir     ....  ,,48 

Lateral  Openings       ........  ,,50 

Part  II ,,51 

Long  Tubes ,,51 

Simple  Apertures       ........  ,,52 

Cylindrical  Necks       ........  ,,53 

Potential  on  itself  of  a  Uniform  Circular  Disk        ...  ,,55 

Nearly  Cylindrical  Tubes  of  Revolution         ....  ,,62 

Upper  Limit    .........  ,,62 

Application  to  Straight  Tube  of  Revolution  whose  end  lies  on 

two  Infinite  Planes          .......  ,,64 

Tubes  nearly  Straight  and  Cylindrical  but  not  necessarily  of 

Revolution    .........  ,,64 

Tubes  not  nearly  Straight            .          .                     .          .          .  „           66 

Part  III '     .  -       .         ...  ,           67 

Experimental    .........  ,,67 


CONTENTS   OF    VOLUMES    I. — IV.  577 

PAGE 

On  the  Vibrations  of  a  Gas  contained  within  a  Rigid  Spherical 

Envelope.     Art.  13.     1872 Vol.  I.    138 

Investigation   of    the    Disturbance   produced   by    a   Spherical 

Obstacle  on  the  Waves  of  Sound.     Art.  14.     1872          .  „       139 
Some    General    Theorems    relating    to    Vibrations.     Art.   21. 

1873 „       170 

Section  I.    The  natural  periods  of  a  conservative  system,  vibrating 
freely  about  a   configuration    of   stable  equilibrium,    fulfil  the 

stationary  condition            .......  „         170 

Section  II.     The  Dissipation  Function     .....  „         176 

Section  III.     [The  Reciprocity  Theorem]          ....  „         179 

On  the  Nodal  Lines  of  a  Square  Plate.     Art.  22.     1873       .  „       182 
On  the  Vibrations  of  Approximately  Simple  Systems.     Art.  24. 

1873,  1874 ,,185 

On  the  Fundamental  Modes  of  a  Vibrating  System.     Art.  25. 

1873 ,,186 

Vibrations  of  Membranes.     Art.  26.     1873      ....  ,,187 

Harmonic  Echoes.     Art.  27.     1873 ,,188 

Mr  Hamilton's  String  Organ.     Art.  33.     1875        ...  „       230 

Vibrations  of  a  Liquid  in  a  Cylindrical  Vessel.     Art.  37.     1875  „       250 

On  Waves.     Art.  38.     1876 ,,251 

The  Solitary  Wave ,,256 

Periodic  Waves  in  Deep  Water       .          .           .           .          .           .  ,,261 

Oscillations  in  Cylindrical  Vessels ,,265 

Our  Perception  of  the  Direction  of  a  Source  of  Sound.   Art.  40. 

1876 ,,277 

Questions   from  Mathematical  Tripos  Examination   for  1876. 

Art.  41.     1876 ,,280 

On  the  Application  of  the  Principle  of  Reciprocity  to  Acoustics. 

Art.  44.     1876 ,,305 

Acoustical  Observations.  I.     Art.  46.     1877    ....  „       314 

Perception  of  the  Direction  of  a  Source  of  Sound     ...  » 

The  Head  as  an  Obstacle  to  Sound » 

Reflection  of  Sound      ......••  »         316 

Audibility  of  Consonants       .....••  » 

Interference  of  Sounds  from  two  unisouant  Tuning-forks  .          .  „ 

Symmetrical  Bell           ......••  »> 

Octave  from  Tuning-forks     ....•••  » 

Influence  of  a  Flange  on  the  Correction  for  the  Open  End  of  a  Pipe  „ 

The  Pitch  of  Organ-pipes ,,320 

37 


578  CONTENTS   OF    VOLUMES   I. — IV. 

PAGE 

On  Progressive  Waves.     Art.  47.     1877.       ....        .         .      .  «     Vol.  I.  322 

On  the  Amplitude  of  Sound- Waves.     Art.  48.     1877     .         .  „  328 

Absolute  Pitch.     Art.  49.     1877      .        .        .         .         .         .  „  331 

Note  on  Acoustic  Repulsion.     Art.  52.     1878         .         .         .  „  342 

The  Explanation  of  certain  Acoustical  Phenomena.     [Singing 

Flames,  &c.]     Art.  55.'    1878    .         .         -.         .         .         .  ,,348 

On   the   Determination   of  Absolute    Pitch   by  the   Common 

Harmonium.     Art.  57.     1879    .         .         .         .        .        .  „  357 

On  the  Instability  of  Jets.     Art.  58.     1879    .        .        .        .  „  361 

On  the  Capillary  Phenomena  of  Jets.     Art.  60.     1879  .         .  „  377 

Appendix  I.     [Vibrations  about  a  Cylindrical  Figure]           ;.          .  „  396 

Appendix  II.     [Vibrations  about  a  Spherical  Figure]          .           .  „  400 

AcousticalObservations.il.     Art.  61.     1879.         .    '     .  ••••-.••:        „  402 

Pure  Tones  from  Sounding  Flames           .....  „  402 
Points   of   Silence   near   a   Wall   from    which   a   Pure    Tone    is 

reflected ..........  „  403 

Sensitive  Flames .  •  „  406 

Aerial  Vibrations  of  very  Low  Pitch  maintained  by  Flames         .  „  407 

Rijke's  Notes  on  a  large  scale         ......  „  408 

Mutual  Influence  of  Organ-pipes  nearly  in  Unison     .      ...  „  409 

Kettledrums ,,411 

The  .Eolian  Harp ,,413 

On  Reflection  of  Vibrations  at  the  Confines  of  Two  Media 

between  which  the  Transition  is  Gradual.     Art.  63.     1880  „  460 

Acoustical  Observations.  III.     Art.  65.     1880          .         .         .  „  468 

Intermittent  Sounds     ........  „  468 

A  New  Form  of  Siren  .  .  .          .          .          .  ,,471 

The  Acoustical  Shadow  of  a  Circular  Disk        ....  „  472 

On   the   Stability,   or   Instability,   of  certain    Fluid    Motions. 

Art.  66.     1880 ,,474 

On  the  Resultant  of  a  large  number  of  Vibrations  of  the  same 

Pitch  and  of  arbitrary  Phase.     Art.  68.     1880  .         ;        .  „  491 

On    a    New    Arrangement    for    Sensitive    Flames.     Art.    70. 

1880.         .                                    .         .         .         .         .     .    .  „  500 

The  Photophone.     Art.  71.     1881    .        .....       '.        .        .  „  501 

On   the    Infinitesimal    Bending    of    Surfaces    of    Revolution. 

Art.  78.     1881  .  551 


CONTENTS   OF   VOLUMES   I. — IV.  579 

PAGE 

Acoustical  Observations.  IV.     Art.  84.     1882  Vol.  II.     95 

On  the  Pitch  of  Organ- Pipes            .          .       /.          .          ;          .  „           95 
Slow    versus    Quick    Beats    for    comparison    of    Frequencies    of 

Vibration          .*.......  ,,97 

Estimation  of  the  Direction  of  Sounds  with  one  Ear           .  ,,98 

A  Telephone-Experiment        .                *     .          .          .         ".          .  )?           99 

Very  High  Notes.     Rapid  Fatigue  of  the  Ear  .          .          .          .  „          99 

Sensitive  Flames            ........  „         100 

Further  Observations  upon  Liquid  Jets,  in  Continuation   of 
those  recorded  in  the  Royal  Society's  '  Proceedings '  for 

March  and  May,  1879.     Art.  85.     1882  .         .         .         .  „       103 

On  some  of  the  Circumstances  which  influence  the  Scattering  of 

a  nearly  Vertical  Jet  of  Liquid    .          .           .           .           .          .  ,,103 

Influence  of  Regular  Vibrations  of  Low  Pitch  ....  ,,106 

The  Length  of  the  Continuous  Part ,,110 

Collision  of  Two  Resolved  Streams           .          .          .          .          .  ,,112 

Collision  of  Streams  before  Resolution     .           .           .           .          .  ,,115 

On  an  Instrument  capable  of  Measuring  the  Intensity  of  Aerial 

Vibrations.     Art.  91.     1882                .                           .         .  ,,132 

On  Maintained  Vibrations.     Art.  97.     1883    ....  „       188 

On  the  Vibrations  of  a  Cylindrical  Vessel  containing  Liquid. 

Art.  101.     1883 ,,208 

On  the  Crispations  of  Fluid  resting  upon  a  Vibrating  Support. 

Art.  102.     1883 ,,212 

On  Porous  Bodies  in  Relation  to  Sound.     Art.  103.     1883  .  „       220 

On  the  Circulation  of  Air  observed  in  Kundt's  Tubes,  and  on 

some  Allied  Acoustical  Problems.     Art.  108.     1883         .  „       239 

Acoustical  Observations.  V.     Art.  110.     1884          ...  „       268 

Smoke-jets  by  Intermittent  Vision           .....  „         268 

Srnoke-jets  and  Resonators              ......  ,,         269 

Jets  of  Coloured  Liquid ,,270 

Fish-tail  Burners ,,272 

Influence  of  Viscosity             .......  „         273 

On  Telephoning  through  a  Cable.     Art.  115.     1884                .  „       356 

On  Waves  propagated  along  the  Plane  Surface  of  an  Elastic 

Solid.     [With  reference  to  Earthquakes]    Art.  130.     1885  „       441 

The  Reaction  upon  the  Driving-Point  of  a  System  executing 
Forced   Harmonic   Oscillations  of  Various  Periods,  with 

Applications  to  Electricity.     Art.  134.     1886  .         .    '     .  „       475 

37—2 


580  CONTENTS   OF   VOLUMES    I. — IV. 

PAGE 

On  the  Maintenance  of  Vibrations  by  Forces  of  Double 
Frequency,  and  on  the  Propagation  of  Waves  through  a 
Medium  endowed  with  a  Periodic  Structure.  Art.  142. 
1887 Vol.  III.  1 

Diffraction  of  Sound.     Art.  145.     1888  .         ...         .         .         „  24 

On   Point-,   Line-,   and    Plane-Sources   of  Sound.     Art.    147. 

1888 .         .         .         .         .         „  44 

On  the  Bending  and  Vibration  of  Thin  Elastic  Shells,  es- 
pecially of  Cylindrical  Form.  Art.  152.  1888  .  .  „  217 

Note  on  the  Free  Vibrations  of  an  Infinitely  Long  Cylindrical 

Shell.     Art.  155.     1889     .        .  ' „         244 

On  the  Free  Vibrations  of  an  Infinite  Plate  of  Homogeneous 

Isotropic  Elastic  Matter.     Art.  156.     1889       .         .         .         „         249 

On  the  Uniform  Deformation  in  Two  Dimensions  of  a  Cylind- 
rical Shell  of  Finite  Thickness,  with  Application  to  the 
General  Theory  of  Deformation  of  Thin  Shells.  Art.  162. 
1889 .  .  .  .  ....  „  280 

On  Bells.     Art.  164.     1890 ,,318 

Appendix  :    On  the  Bending  of  a  Hjperboloid  of  Revolution        .  „  330 

On  the  Sensitiveness  of  the  Bridge  Method  in  its  Application 

to  Periodic  Electric  Currents.     Art.  180.     1891       .         .         „         452 

On  the  Instability  of  a   Cylinder   of  Viscous   Liquid   under 

Capillary  Force.     Art.  195.     1892 ,         585 

On  the  Instability  of  Cylindrical  Fluid  Surfaces.     Art.  196. 

1892 594 

On  the  Influence  of  Obstacles  arranged  in  Rectangular  Order 

upon  the  Properties  of  a  Medium.     Art.  200.     1892       .    Vol.  IV.     19 

On   the   Reflection   of    Sound    or   Light   from   a   Corrugated 

Surface.     Art.  206.     1893         .         .  .         .         ,         „  75 

On  the  Minimum  Current  audible  in  the  Telephone.    Art.  211. 

1894 ,,109 

An   Attempt   at   a   Quantitative   Theory   of    the    Telephone. 

Art.  212.     1894        . „        119 

On    the    Amplitude    of   Aerial    Waves    which    are    but   just 

Audible.     Art.  213.     1894  125 


CONTENTS   OF   VOLUMES   1. IV.  581 

On  the  Passage  of  Waves  through  Apertures  in  Plane  Screens, 

and  Allied  Problems.     Art.  227.     1897      .  '   Vol  IV    283 

Perforated  Screen.— Boundary  Condition  d(j)/dn=0    .          .  284 

Boundary  Condition  0=0      .           .          .  2gfi 

Reflecting  Plate.—  d^/dn  =  0  ....  28g 

Reflecting  Plate. — 0  =  0          ....  289 

Two-dimensional  Vibrations             ....  290 

Narrow  Slit. —Boundary  Condition  d^/dn  =  0    ...  291 

Narrow  Slit. — Boundary  Condition  0  =  0            .          .          .  293 

Reflecting  Blade. — Boundary  Condition  d<j>/dn  =  0        .          .          .  294 

Reflecting  Blade. — Boundary  Condition  0  =  0    .           .           .          .  295 

Various  Applications    ........  295 

The  Limits  of  Audition.     Art.  228.     1897      .         .         .        .  „      297 

On  the  Incidence  of  Aerial  and  Electric  Waves  upon  Small 
Obstacles  in  the  Form  of  Ellipsoids  or  Elliptic  Cylinders, 
and  on  the  Passage  of  Electric  Waves  through  a  Circular 

Aperture  in  a  Conducting  Screen.     Art.  230.     1897          .  „      305 

Obstacle  in  a  Uniform  Field  .......  „         306 

In  Two  Dimensions       ........  „         309 

Aerial  Waves       .........  310 

Waves  in  Two  Dimensions    .          .           .          .          .           .          .  ,,314 

Electrical  Applications            .          .           .           .          .          .          .  ,,317 

Electric  Waves  in  Three  Dimensions        .....  „         318 

Obstacle  in  the  Form  of  an  Ellipsoid ,,323 

Circular  Aperture  in  Conducting  Screen  .....  ,,         324 

On   the    Propagation  of  Waves  along  Connected  Systems  of 

Similar  Bodies.     Art.  235.     1897 „      340 

Some  Experiments  with  the  Telephone.     Art.  239.     1898      .  „      357 

On  Iso-periodic  Systems.     Art.  242.     1898    ....  ,,367 

On  the  Cooling  of  Air  by  Radiation  and  Conduction,  and  on 

the  Propagation  of  Sound.     Art.  244.     1899     ...  „      376 

On  Approximately  Simple  Waves.     Art.  261.     1900      .         .  „      486 

On  a  Problem  relating  to  the  Propagation  of  Sound  between 

Parallel  Walls.     Art.  267.     1901 ,,532 

Acoustical  Notes.  VI.     Art.  270.     1901           ....  „      550 

Forced  Vibrations ,,550 

Vibrations  of  Strings   ........  „         551 

Beats  of  Sounds  led  to  the  Two  Ears  separately        ...  „         553 

Loudness  of  Double  Sounds  .......  „         554 


582  CONTENTS   OF    VOLUMES   L— IV. 


THERMODYNAMICS. 


PAGE 


On  the  Dissipation  of  Energy.     Art.  35.     1875       .       \      .   .     Vol.  I.    238 

On  the  Work  that  may  be  gained  during  the  Mixing  of  Gases. 

Art.  36.     1875  .......        '..          „       242 

On  a  Question  in  the  Theory  of  Lighting.     Art.  76.     1881  .  „       541 

On  the  Tension  [Pressure]  of  Mercury  Vapour  at  Common 

Temperatures.     Art.  87.     1882          .         .         ...      Vol.  II.  125 

On  the  Theory  of  Illumination  in  a  Fog.     Art.  121.     1885       .  „       417 

On     the     Thermodynamic     Efficiency    of    the     Thermopile. 

Art.  129.     1885         .        .        ...        ...        .  „       438 

Notes,  chiefly  Historical,  on  some  Fundamental  Propositions 

in  Optics.     Art.  137.     1886     ...        ...  „       513 

The  History  of  the  Doctrine  of  Radiant  Energy.     Art.  154. 

1889  .         .         .         .        ...        .        .        .   Vol.  III.  238 

On   the   Character   of  the   Complete   Radiation  at   a   Given 

Temperature.     Art.   160.     1889        .         .-       .         .         .  „       268 

Superheated  Steam.     Art.  188.     1892     .        .        .        .        .  „       538 

Heat  Engines  and  Saline  Solutions          .          .          .          .          .  „         539 

Heat  Engines  and  Saline  Solutions          .  .          .          .  „         540 

Remarks   on   Maxwell's  Investigation  respecting  Boltzmann's 

Theorem.     Art.  190.     1892 554 

The  Theory  of  Solutions.     Art.  224.     1897    ...        .    Vol.  IV.  267 
Liquid  Air  at  One  Operation.     Art.  240.     1898     .         .         .  „       360 

The  Law  of  Partition  of  Kinetic  Energy.     Art.  253.     1900    ,,,.,."        „       433 

Remarks  upon   the  Law    of  Complete   Radiation.     Art.  260. 

1900  .        .        .        .        .        .        ....        .        ..          „       483 

On    Balfour   Stewart's   Theory    of    the    Connexion    between 

Radiation  and  Absorption.     Art.  263.     1901    .      ...»       ..  „       494 

Does  Chemical  Transformation  influence  Weight?     Art.  269. 

1901  .         .         .         .         .   "      .      *.         .         .         .  „       549 

On  the  Magnetic  Rotation  of  Light  and  the  Second  Law  of 

Thermodynamics.     Art.  271.     1901  .         .         .         .         .  ,,557 


CONTENTS   OF   VOLUMES   I.— IV.  583 

DYNAMICAL   THEORY   OF   GASES. 

Note  on  a  Natural  Limit  to  the  Sharpness  of  Spectral  Lines 

Art.  23.  1873  .  .  .  .  Vol.  I.  183 

On  the  Work  that  may  be  gained  during  the  Mixing  of  Gases. 

Art.  36.  1875 '  ,,242 

On  the  Dark  Plane  which  is  formed  over  a  Heated  Wire  in 

Dusty  Air.  Art.  93.  1882 Vol.  II.  151 

On  the  Limit  to  Interference  when  Light  is  Radiated  from 

Moving  Molecules.  Art.  157.  1889  .  .  .  .  Vol.  III.  258 
On  Van  der  Waals'  Treatment  of  Laplace's  Pressure  in  the 

Virial    Equation:    Letters   to  Professor   Tait.     Art.  181. 

1891  . m       465 

On    the    Virial    of    a    System    of    Hard    Colliding    Bodies. 

Art.  182.     1891 ,,469 

Dynamical  Problems  in  Illustration  of  the  Theory  of  Gases. 

Art.  183.     1891 ,,473 

Introduction          .........  „         473 

Collision  Formulae          ........  „         473 

Permanent  State  of  Free  Masses  under  Bombardment       .          .  „         474 

Another  Method  of  Investigation     ......  „         479 

Progress  towards  the  Stationary  State     .....  „         480 

Pendulums  in  place  of  Free  Masses           .....  „         485 

Remarks  on  Maxwell's  Investigation  respecting  Boltzmann's 

Theorem.  Art.  190.  1892 ,,554 

On  the  Physics  of  Media  that  are  composed  of  Free  and 
Perfectly  Elastic  Molecules  in  a  State  of  Motion. 
[Introduction  to  Waterston's  Memoir.]  Art.  191.  1892.  „  558 

The  Law  of  Partition  of  Kinetic  Energy.     Art.  253.     1900     .    Vol.  IV.  433 
On    the    Viscosity    of    Gases    as    affected    by    Temperature. 

Art.  259.     1900 ,,481 

PROPEKTIES   OF  GASES. 

On  the  Relative  Densities  of  Hydrogen  and  Oxygen.  (Pre- 
liminary Notice.)  Art.  146.  1888  ....  Vol.  III.  37 

On  the  Composition  of  Water.     Art.  153.     1889  „       233 

On  the  Relative  Densities  of  Hydrogen  and  Oxygen.  II. 
Art.  187.  1892 

Density  of  Nitrogen.     Art.   197.     1892  ....    Vol.  IV.      1 


584  CONTENTS   OF   VOLUMES   I. — IV. 

PAGE 

On  the  Densities  of  the  Principal  Gases.     Art.  201.     1893    .     Vol.  IV.    39 

The  Manometer .          .           .  „           40 

Connexions  with  Pump  and  Manometer       .....  „           43 

The  Weights        .          .          .          ...          .          .          .  „           44 

The  Water  Contents  of  the  Globe ...           .          .          .  „           45 

Air    .           .          ......          .          .          .          .  „           46 

Oxygen       .          .           .           .          .           .          .                     .          .  „           47 

Nitrogen     .           .          .          .          .        -.     ...;..       ,          ...•     .....•;        „  48 

Reduction  to  Standard  Pressure      .           .                    -,.        .          ,  „           50 
Note  A.      On    the    Establishment    of   Equilibrium    of    Pressure 

in  Two  Vessels  connected  by  a  Constricted  Channel       .           .  „           53 

On    an    Anomaly    encountered    in    Determinations     of     the 

Density  of  Nitrogen  Gas.     Art.  210.     1894    ...  „       104 
Argon,    a   New   Constituent    of    the   Atmosphere.     By   Lord 

Rayleigh  and  Prof.  William  Ramsay.     Art.   214.     1895 .  ,,130 

Density  of  Nitrogen  from  Various  Sources        .                •     .          .  „         130 

Reasons  for  Suspecting  a  hitherto  Undiscovered  Constituent  in  Air  „         135 

Methods  of  Causing  Free  Nitrogen  to  Combine           .          ...  „         138 
Early  Experiments  on  sparking  Nitrogen  with  Oxygen  in  presence 

of  Alkali ,,141 

Early    Experiments   ori    Withdrawal    of    Nitrogen   from    Air   by 

means  of  Red-hot  Magnesium     .          .          .          .          .          .  ,,144 

Proof  of  the  Presence  of  Argon  in  Air,  by  means  of  Atruolysis  „         150 
Negative  Experiments  to  prove  that  Argon  is  not  derived  from 

Nitrogen  or  from  Chemical  Sources     .          .          .           ...  ,,153 

Separation  of  Argon  on  a  large  scale      .           .           .          .          .  ,,155 

Density  of  Argon  prepared  by  means  of  Oxygen      ...  „         165 

Density  of  Argon  prepared  by  means  of  Magnesium          ..           .  „         167 

Spectrum  of  Argon         ........  „         168 

Solubility  of  Argon  in  Water „         170 

Behaviour  at  Low  Temperatures    .          .          .          .          .          .  ,,173 

The  Ratio  of  the  Specific  Heats  of  Argon      .          .                    .  ,,174 

Attempts  to  induce  Chemical  Combination      .          .          .          .  ,,176 

General  Conclusions „         180 

Addendum,  March  20  (by  Prof.  W.  Ramsay)  .          .          .          ,  ,,184 

Addendum,  April  9      ........  ,,187 

Argon.     Art.  215.     1895 ,,188 

On  some  Physical  Properties  of  Argon  and  Helium.     Art.  218. 

1896           .         .       V        .         .       'i         .         .         .         .  „       215 

Density  of  Argon          .          .          .          .          .                     .          .  „         215 

The  Refractivity  of  Argon  and  Helium    .          .        ..          .          .  „         218 

Viscosity   of  Argon  and  Helium    .          .'    '    .          .        '.          .  „         222 

Gas  from  the  Bath  Springs            .          ....          .          .  „         223 

Buxton  Gas         .           .          ....          .           .           .          .  ,,223 

Is  Helium  contained  in  the  Atmosphere  ?                   .           .           .  „         224 

On  the  Amount  of  Argon  and  Helium  contained  in  the  Gas 

from  the  Bath  Springs.     Art.  219.     189(i         .         .     ;\  „       225 


CONTENTS   OF   VOLUMES   I. — IV.  585 

PAGE 

Theoretical  Considerations  respecting  "the  Separation  of  Gases 

by  Diffusion  and  Similar  Processes.     Art.  223.     1896     .   Vol.  IV.  201 

The  Theory  of  Solutions.     Art.  224.     1897    .         .        ".  267 

Observations  on  the  Oxidation  of  Nitrogen  Gas.    Art.  225.    1897  270 
On  the  Viscosity  of  Hydrogen  as  affected  by  Moisture.    Art.  234 

1897 „  888 

On  the  Densities  of  Carbonic  Oxide,  Carbonic  Anhydride,  and 

Nitrous  Oxide.     Art.  236.     1897 „  347 

Carbonic  Oxide    ..........  347 

Carbonic  Anhydride     .........  349 

Nitrous  Oxide      ...                     ......  350 

Liquid  Air  at  One  Operation.     Art.  240.     1898     .  360 

On    the   Character  of  the  Impurity  found  in  Nitrogen  Gas 
derived  from  Urea  [with  an  Appendix  containing  Details 
of  Kefractometer].     Art.  241.     1898          .         .         .         .         „         361 

Details  of  Refractometer         .  .  .  .  .          .  .  „    -       364 

On  the  Cooling  of  Air  by  Radiation  and  Conduction,  and  on 

the  Propagation  of  Sound.     Art.  244.     1899  .         .         .         „         376 
On  the  Conduction  of  Heat  in  a  Spherical  Mass  of  Air  confined 

by  Walls  at  a  Constant  Temperature.     Art.  245.     1899     .         „         382 
On  the  Viscosity  of  Argon  as  affected  by  Temperature.   Art.  254. 

1900 ,,452 

On  the  Passage  of  Argon  through  Thin  Films  of  Indiarubber. 

Art,  255.     1900 ,,459 

On  the  Weight  of  Hydrogen  desiccated  by  Liquid  Air.    Art.  256. 

1900 ,,461 

On  the  Viscosity  of  Gases  as  affected  by  Temperature.    Art.  259. 

1900 ,,481 

Spectroscopic  Notes  concerning  the  Gases  of  the  Atmosphere. 

Art.  264.     1901         ; ,,496 

On  the  Visibility  of  Hydrogen  in  Air    .....  »,  496 

Demonstration  at  Atmospheric  Pressure  of  Argon  from  very  small 

quantities  of  Air       ...  .....  »  499 

Concentration  of  Helium  from  the  Atmosphere        .  .  .  „  500 

On  a  New  Manometer,  and  on  the  Law  of  the  Pressure  of 
Gases   between    T5    and    O'Ol    Millimetres    of    Mercury. 

Art.  266.     1901 ,,511 

Introduction         ...          .          .          •  •          •          •  »  &H 

Improved  Apparatus  for  Measuring  very  small  Pressures  .  „  514 

Experiments  to  determine  the  Relation  of  Pressure  and  Volume 

at  given  Temperature  .  •          •          '          •          •  >  519 


586  CONTENTS   OF    VOLUMES    I. — IV. 

ELECTEICITY   AND   MAGNETISM. 

I'AOK 

On  some  Electromagnetic  Phenomena  considered  in  connexion 

with  the  Dynamical  Theory.     Art.  1.     1869      .         .         .       Vol.  I.       1 

On  an  Electromagnetic  Experiment.     Art.  2.     1870        .         .  ,,14 

On  the  Theory  of  Resonance.     Art.  5.     1870          .         .         .  „         33 

Introduction         .          :                     .          .          .        . '.                     .  „           33 

Part  I ~if    ..--.:.•  „           37 

Several  Openings       ........  ,,39 

Double  Resonance     ........  ,,41 

Open  Organ-pipes     ........  ,,45 

Long  Tube  in  connexion  with  a  Reservoir     ....  ,,48 

Lateral  Openings        ........  ,,50 

Part  II ,,51 

Long  Tubes .  „           51 

Simple  Apertures       .           .           .           .                     *.          .          .  „           52 

Cylindrical  Necks      ........  ,,53 

Potential  on  itself  of  a  uniform  Circular  Disk      ...  „           55 

Nearly  Cylindrical  Tubes  of  Revolution        .          .                    .  „           62 

Upper  Limit     .........  ,,62 

Application  to  straight  Tube  of  Revolution  whose  end  lies  on 

two  infinite  Planes           .......  ,,64 

Tubes   nearly   Straight   and   Cylindrical  but  not  necessarily  of 

Revolution .*.-..  „           64 

Tubes  not  nearly  Straight           ......  ,,66 

Part  III ,,67 

Experimental    .........  „           67 

An  Experiment  to  illustrate  the  Induction   on   itself   of  an 

Electric  Current.     Art.  20.     1872    .         .                  .         .  167 

Some  General  Theorems  relating  to  Vibrations.    Art.  21.    1873  „       170 
Section  I.     The  natural  periods  of  a  conservative  system,  vibrating 
freely  about    a   configuration   of  stable  equilibrium,    fulfil    the 

stationary  condition              .           .           .           .           .           .           .  ,,170 

Section  II.     The  Dissipation  Function    .           .           .           .           .  ,,176 

Section  III.     [The  Reciprocity  Theorem]          .        -.          .      .    .  „         179 

On  the  Approximate  Solution  of  Certain  Problems  relating  to 

the  Potential.     Art.  39.     1876.         ..     .  .        .  .'.'..         .  „       272 

Questions  from  Mathematical   Tripos  Examination  for  1876. 

Art.  41.     1876.         .        .       ...        .,    ,    ,        .        .  ..'.-..  „       280 

On  a  Permanent  Deflection  of  the  Galvanometer-Needle  under 
the  influence  of  a  rapid  series  of  equal  and  opposite  In- 
duced Currents.  Art.  45.  1877  .  .  .  .  '.  .  „  310 

Uniformity  of  Rotation.     [Phonic  Wheel]     Art.  56.     1878     .  „       355 


CONTENTS   OF   VOLUMES    I. — IV.  587 

PAGE 

The  Influence  of  Electricity  on  Colliding  Water  Drops.    Art.  59. 

1879 .        .        .        .        .     Vol.  I.    372 

Note  on  the  Theory  of  the  Induction  Balance.     Art.  69.     1880  „       497 

On  the  Electromagnetic  Theory  of  Light.     Art.  74.     1881    .  „       518 
On  the  Determination  of  the  Ohm  [B.A.  Unit]  in  Absolute 
Measure.    By  Lord  Rayleigh  and  Arthur  Schuster.    Art.  79. 

1881 Vol.  II.      1 

Part  I.      By  Lord  Rayleigh „             1 

Part  II.    By  Arthur  Schuster ,,20 

Adjustment  of  the  Instruments  and  Determination  of  Constants  „           20 

The  Observations      ........  ,,24 

Air  Currents ,,28 

Reduction  of  Observations           ......  ,,30 

Results ,,34 

Experiments  to  Determine  the  Value  of  the  British  Association 

Unit  of  Resistance  in  Absolute  Measure.     Art.  80.     1882  „         38 

Measurements  of  Coil  .           .          .           .           .          .          .          .  „           51 

Calculation  of  GK „           53 

Calculation  of  L ,,53 

Theory  of  the  Ring  Currents          ......  ,,54 

L  by  Direct  Experiment        .......  ,,55 

Correction  for  Level ,,63 

Correction  for  Torsion            .......  ,,64 

Value  of  GK  corrected  for  Level  and  Torsion .          .          .          .  „           64 

Calculation  of   U          ........  ,,64 

Measurement  of  tan  p            .......  ,,64 

Measurement  of  D       ........  ,,65 

Reduction  of  Results     ........  ,,66 

Comparison  with  the  Standard  B.  A.  Units       ....  ,,75 

On  the  Specific  Resistance  of  Mercury.     By  Lord  Rayleigh 

and  Mrs  H.  Sidgwick.     Art.  81.     1882    .         .         .         .  „         78 

On  a  New  Form  of  Gas  Battery.     Art.  83.     1882         .         .  '  „         94 

On  the  Absolute  Measurement  of  Electric  Currents.     Art.  88. 

1882 ,,126 

On  the  Duration  of  Free  Electric  Currents  in  an  Infinite 

Conducting  Cylinder.  Art.  89.  1882  .  .  .  .  ,,128 
On  the  Equilibrium  of  Liquid  conducting  Masses  charged  with 

Electricity.  Art.  90.  1882 ,,130 

Comparison  of  Methods  for  the  Determination  of  Resistances 

in  Absolute  Measure.  Art.  92.  1882  .  .  .  .  ,,134 

Kirchhoff's  Method,  Maxwell's  Electricity  and  Magnetism,  §  759  „  135 

Weber's  Method  by  Transient  Currents,  Maxwell,  §  760  .  .  -„  137 


588  CONTENTS   OF    VOLUMES    1. — IV. 

PAGE 

Comparison  of  Methods  for  the  Determination  of  Resistances 

in  Absolute  Measure  (continued)       .....     Vol.  II.  134 

Method  of  Revolving  Coil      .          .          .'                   .           .          .  ,,139 

Method  of  Foster  and  Lippmann    .                     .          .          .          .  „         143 

Weber's  Method  by  Damping ,,145 

Lorenz's  Method            ........  „          145 

Experiments,  by  the  Method  of  Lorenz,  for  the  Further  Deter- 
mination of  the  Absolute  Value  of  the  British  Association 
Unit  of  Resistance,  with  an  Appendix  on  the  Determi- 
nation of  the  Pitch  of  a  Standard  Tuning-Fork.  By  Lord 

Rayleigh  and  Mrs  H.  Sidgwick.     Art.  94.     1883        .         .  „       155 
Details  of  Measurements  : 

Diameter  of  Disc        .          .           .           .           .           .           .          .  ,,167 

The  Inductioii-Coils ,,168 

The  Distance- Pieces  ........  „         169 

The  Induction-Coefficients            .           .          .           .          .          .  ,,170 

The  Resistance-Coils ,,171 

Appendix :    Frequency  of  Vibration  of  Standard  Fork         .          .  „         177 
Second  Appendix  :    On  the  Effect  of  the  Imperfect  Insulation  of 

Coils ,,182 

On  the  Mean  Radius  of  Coils  of  Insulated  Wire.     Art.  95. 

1883 ,,184 

On  the  Imperfection  of  the  Galvanometer  as  a  Test  of  the 

Evanescence  of  a  Transient  Current.     Art.  105.     1883  .  „       228 

On  the  Measurement  of  Electric  Currents.     Art.  107.     1883  „       237 
On  the  Measurement  of  the  Electrical  Resistance  between  Two 

Neighbouring  Points  on  a  Conductor.     Art.  111.     1884    .  „       276 
On  the  Electro-Chemical  Equivalent   of  Silver,  and   on  the 
Absolute  Electromotive  Force  of  Clark  Cells.     By  Lord 

Rayleigh  and  Mrs  H.  Sidgwick.     Art.  112      1884 .         .  „       278 

The  Fixed  Coils  .........  „         289 

The  Suspended  Coil ,,290 

Determination  of  Mean  Radius  of  Suspended  Coil     .          .          .  ,,291 

Calculation  of  Attraction        .......  „         295 

The  Silver  Voltameters ,,297 

Appendix.     [Mathematical  Table]   .           .          .          .          .          .  ,,327 

Explanation  of  Figures           .......  „         328 

Notes: 

Note  to  §  25.     [Effect  of  Temperature  on  Silver  Deposits]          .  „         329 

Note  to  §  26.     [Mascart's  revised  Calculation]           ...  „         329 

Note  to  §  27.     [Copper  and  Silver]          .....  „         330 

Note  to  §  30.     [Clark  Cells] ,,331 

Note  to  §  32.     [Post  Office  Daniells] ,,331 

Note  1  to  §  37.     [Clark  Cells] ,,331 

Note  2  to  §37.     [Clark  Cells]        .          .          .          ...  „         332 


CONTENTS   OF    VOLUMES   I. — IV.  589 


PAGE 


A  Lecture  Experiment  on  Induction.     Art.  114.     1884          .    Vol.  II.   355 

On  Telephoning  through  a  Cable.     Art.  115.     1884       .         .  „        356 

On  a  Galvanometer  with  Twenty  Wires.     Art.  116.     1884    .  „        357 

On  Clark's  Standard  Cells.     Art.  117.     1884  .         .         .  „        359 

On  the  Constant  of  Magnetic  Rotation  of  Light  in  Bisulphide 

of  Carbon.     Art.  118.     1885 ,,360 

The  Helix ,,367 

Correction  for  Finite  Length  ......  „  368 

Appendix:    Notes  on  Polarimetry  in  general     ....  „  378 

Postscript.     [Work  of  H.  Becquerel] ,,383 

Uber  die  Methode  der  Dampfung  bei  der  Bestimmung  des 

Ohms.  Art.  120.  1885 ,,415 

Self-induction  in  Relation  to  Certain  Experiments  of  Mr  Wil- 

loughby  Smith  and  to  the  Determination  of  the  Ohm. 

Art.  123.  1885 ,,422 

A  Theorem  relating  to  the  Time-Moduli  of  Dissipative 

Systems.     Art.  125.     1885 ,,428 

On  the  Thermodynamic  Efficiency  of  the  Thermopile.    Art.  129. 

1885 ,,438 

On  Professor  Himstedt's  Determination  of  the  Ohm.  Art.  131. 

1886 ,,448 

On  the  Clark  Cell  as  a  Standard  of  Electromotive  Force. 

Art.  132.     1886 ,,451 

Testing  Dynamos.     Art.  133.     1886 ,,474 

The  Reaction  upon  the  Driving-Point  of  a  System  executing 
Forced  Harmonic  Oscillations  of  Various  Periods,  with 
Applications  to  Electricity.  Art.  134  1886  .  .  .  „  475 

On  the  Self-Induction  and  Resistance  of  Straight  Conductors. 

Art.  135.     1886 ,,486 

Notes  on  Electricity  and  Magnetism.    I.     On  the  Energy  of 

Magnetized  Iron.     Art.  139.     1886 „        543 

Notes  on  Electricity  and  Magnetism.   II.     The  Self-Induction 

and  Resistance  of  Compound  Conductors.    Art.  140.    1886          „        551 
The  Interrupters  ........  „  553 

The  Induction-Compensators  ......  „  555 

Appendix. — The  Induction-Compensators  [p.  557]  „  577 

Notes  on  Electricity  and  Magnetism.  III.  On  the  Behaviour 
of  Iron  and  Steel  under  the  Operation  of  Feeble  Magnetic 
Forces.  Art.  141.  1887  .  579 


590  CONTENTS   OF    VOLUMES    I. — IV. 

PAGE 

Is  the  Velocity  of  Light  in  an  Electrolytic  Liquid  influenced 
by  an  Electric  Current  in  the  Direction  of  Propagation  ? 
Art.  151.  1888  . Vol.  III.  213 

The  Clark  Standard  Cell.     Art.  165.     1890    .         .         .         .         „  333 

On   Huygens's   Gearing   in   Illustration  of  the   Induction   of 

Electric  Currents.     Art.  171.     1890          .        v"--     .         .         „  376 

On  the  Sensitiveness  of  the  Bridge  Method  in  its  Application 

to  Periodic  Electric  Currents.     Art.  180.     1891         .    .     .         „  452 

On  the  Influence  of  Obstacles  arranged  in  Rectangular  Order 

upon  the  Properties  of  a  Medium.     Art.  200.     1892       .    Vol.  IV.     19 

On  the  Minimum  Current  audible  in  the  Telephone.    Art.  211. 

1894.         .         .         .,       .         . ,  109 

An   Attempt   at   a   Quantitative   Theory   of    the   Telephone. 

Art.  212.     1894        .       ".        .        .        .        ..       .         .         „  119 

The  Electrical  Resistance  of  Alloys.     Art.  221.     1896    .         .         „  232 

Observations    on    the    Oxidation    of  Nitrogen    Gas     [by   the 

Electric  Flame].     Art.  225.     1897   ....     jtgfe     „  270 

On   the   Passage   of  Electric    Waves   through    Tubes,  or  the 

Vibrations  of  Dielectric  Cylinders.     Art.  226.     1897       .         „  276 

General  Analytical  Investigation     .......  276 

Rectangular  Section       .........  279 

Circular  Section  ..........  280 

On  the  Passage  of  Waves  through  Apertures  in  Plane  Screens, 

and  Allied  Problems.     Art.  227.     1897      .  283 

Perforated  Screen.— Boundary  Condition  d<f>/dn  =  Q                                    „  284 

Boundary  Condition  <£  =  0      .           .           .          .          .          .           .           „  286 

Reflecting  Plate.— d(f)/dn  =  0  ........  288 

Reflecting  Plate.  — <£=0          ........  289 

Two-dimensional  Vibrations  .          .          .          ;     :     .          .          .           „  290 

Narrow  Slit. — Boundary  Condition  d(f)/dn  =  0                                            „  291 

Narrow  Slit. — Boundary  Condition  $  =  0            .          .          .          .           „  293 

Reflecting  Blade.— Boundary  Condition  d(f>/dn=.0       .                     .           „  294 

Reflecting  Blade. — Boundary  Condition  $  =  0    .          .          .          .           „  295 

Various  Applications     .           .           .           ....'.          .           „  295 

On  the  Measurement  of  Alternate  Currents  by  means  of  an 
obliquely  situated  Galvanometer  Needle,  with  a  Method 

of  Determining  the  Angle  of  Lag.     Art.  229.     1897      .         „  299 


CONTENTS   OF    VOLUMES    1. —  IV.  591 

PAGE 

On  the  Incidence  of  Aerial  and  Electric  Waves  upon  Small 
Obstacles  in  the  Form  of  Ellipsoids  or  Elliptic  Cylinders, 
and  on  the  Passage  of  Electric  Waves  through  a  Circular 
Aperture  in  a  Conducting  Screen.     Art.  230.     1897  .    Vol.  IV.  305 

Obstacle  in  a  Uniform  Field  ......  „  306 

In  Two  Dimensions       ........  „  309 

Aerial  Waves       .          .          .          .          ...          .          .          .  „  310 

Waves  in  Two  Dimensions     .......  „  314 

Electrical  Applications  ........  „  317 

Electric  Waves  in  Three  Dimensions        ......  318 

Obstacle  in  the  Form  of  an  Ellipsoid      .  .  .          .  „  323 

Circular  Aperture  in  Conducting  Screen.  „  324 

On  the  Propagation  of  Electric  Waves  along  Cylindrical  Con- 
ductors of  any  Section.     Art.  231.     1897         .  327 

The  Electro-Chemical  Equivalent  of  Silver.     Art.  232.     1897  .  „         332 

Note  on  the  Pressure  of  Radiation,  showing  an  Apparent  Failure 

of  the  Usual  Electromagnetic  Equations.    Art.  238.     1898  „         354 

Some  Experiments  with  the  Telephone.     Art.  239.     1898     .  „         357 

The  Mutual  Induction  of  Coaxial  Helices.     Art.  252.     1899  „         431 

On  the  Magnetic  Rotation  of  Light  and  the  Second  Law  of 

Thermodynamics.     Art.  271.     1901 „         555 

On  the  Induction-Coil.     Art.  272.     1901  557 


OPTICS. 

Note  on  the  Explanation  of  Coronas,  as  given  in  Verdet's 
Legons  d'Optique  Physique,  and  other  works.  Art.  6. 
1871 Vol.  I.  76 

Some  Experiments  on  Colour.     Art.  7.     1871        .         .         .  „         79 

Yellow        ... „          85 

On   the   Light   from   the    Sky,   its   Polarization   and   Colour. 

Art.  8.     1871 ,,87 

Appendix    ..........  ,,96 

On  the  Scattering  of  Light  by  Small  Particles.     Art.  9.     1871  „       104 

On   Double  Refraction.     Art.  10.     1871           .         .         .         .  ,,111 
On  the  Reflection  of  Light  from  Transparent  Matter.     Art.  11. 

1871  120 


592  CONTENTS   OF   VOLUMES    I. — IV. 

PAGE 

On  a  Correction  sometimes  required  in  Curves  professing  to 
represent  the  connexion  between  two  Physical  Magnitudes. 
Art.  12.  1871.  .  ..  .  -;  .  :-...  .  .  .  Vol.1.  135 

On  the  Reflection  and  Refraction  of  Light  by  Intensely  Opaque 

Matter.     Art.  16.     1872 ,,141 

Preliminary  Note  on  the  Reproduction  of  Diffraction-Gratings 

by  means  of  Photography.     Art.  17.     1872        .  .  „       157 

On   the   Application    of    Photography    to    copy    Diffraction  - 

Gratings.     Art.  18.     1872          .         .         .         .         .         .  ,,160 

On  the  Diffraction  of  Object-Glasses.     Art.  19.     1872    .         .  „       163 

Note  on  a  Natural  Limit  to  the  Sharpness  of  Spectral  Lines. 

Art.  23.     1873 ,        .  .,       183 

On   the    Manufacture    and    Theory    of    Diffraction-Gratings. 

Art.  30.     1874 .         .  ,,199 

Insects  and  the  Colours  of  Flowers.     Art.  31.     1874    .       ... .          „       222 

Investigations  in  Optics,  with  special  reference  to  the  Spectro- 
scope. Art.  62.  1879,  1880 „  415 

Resolving,  or  Separating,  Power  of  Optical  Instruments     .          .  „  415 

Rectangular  Sections ,,418 

Optical  Power  of  Spectroscopes       .          .           .           .           .          .  „  423 

Influence  of  Aberration           .......  „  428 

On  the  Accuracy  required  in  Optical  Surfaces            ...  „  436 

The  Aberration  of  Oblique  Pencils            .....  „  440 

Aberration  of  Lenses  and  Prisms   ......  „  444 

The  Design  of  Spectroscopes           ......  „  453 

On  Reflection  of  Vibrations  at  the  Confines  of  two  Media 
between  which  the  Transition  is  Gradual.  Art.  63. 
1880 '.  .  .  .  ,,460 

On  the  Minimum  Aberration  of  a  Single  Lens  for   Parallel 

Rays.     Art.  64.     1880 ,,466 

On  the  Resolving-Power  of  Telescopes.     Art.  67.     1880        .  „       488 

On  the  Resultant  of  a  large  number  of  Vibrations  of  the  same 

Pitch  and  of  arbitrary  Phase.     Art.  68.     1880  .         .         .  „       491 

On   Copying   Diffraction-Gratings,  and   on   some  Phenomena 

connected  therewith.     Art.  72.     1881  .         .         .  „       504 

On  Images  formed  without  Reflection  or  Refraction.     Art.  73. 

1881  513 


CONTENTS   OF   VOLUMES   I. — IV. 


593 


PAGE 

On  the  Electromagnetic  Theory  of  Light.     Art.  74.     1881    .     Vol.  I.     518 

On  the  Velocity  of  Light.     Art.  75.     1881     .        .        .        .  „       537 

On  a  Question  in  the  Theory  of  Lighting.     Art.  76.     1881 .  „       541 

Experiments  on  Colour.  Art.  77.  1881  .  .  542 
The  Use  of  Telescopes  on  Dark  Nights.  Art.  82.  1882  .  Vol.  II.  92 
On  the  Invisibility  of  Small  Objects  in  a  Bad  Light.  Art.  96. 

1883 ,,187 

Distribution  of  Energy  in  the  Spectrum.  Art.  99.  1883  .  ,,198 
On  the  Constant  of  Magnetic  Rotation  of  Light  in  Bisulphide 

of  Carbon.     Art.  118.     1885 ,,360 

The  Helix ,,367 

Correction  for  Finite  Length           ......  „         368 

Appendix :   Notes  on  Polarimetry  in  general     ....  „         378 

Postscript.     [Work  of  H.  Becquerel]          .          .          .          .          .  „         383 

Optics.     Art.  119.     1884 ,,385 

On  the  Theory  of  Illumination  in  a  Fog.  Art.  121.  1885  .  ,,417 
A  Monochromatic  Telescope,  with  Application  to  Photometry. 

Art.  122.     1885 ,,420 

On   the   Accuracy   of  Focus   necessary   for    Sensibly   Perfect 

Definition.     Art.  126.     1885 ,,430 

On   an  Improved   Apparatus   for   Christiansen's    Experiment. 

Art.  127.     1885 ,,433 

Optical  Comparison  of  Methods  for  Observing  Small  Rota- 
tions. Art.  128.  1885 ,,436 

On  the  Colours  of  Thin  Plates.     Art.  136.     1886           .         .  „       498 
Notes,  chiefly  Historical,  on  some  Fundamental  Propositions 

in  Optics.     Art.  137.     1886     ......  ,,513 

On   the  Intensity  of  Light    Reflected  from  Certain  Surfaces 

at  Nearly  Perpendicular  Incidence.     Art.  138.     1886      .  „       522 

Description  of  Apparatus       .......  „         525 

Prism  of  Crown  Glass  (I) ,534 

Prism  of  Crown  Glass  (II) ,537 

Plate  Glass  Silvered  Behind ,538 

Silver-on-Glass  Speculum       .......  ,         539 

Mirror  of  Black  Glass ,539 

On  the  Maintenance  of  Vibrations  by  Forces  of  Double 
Frequency,  and  on  the  Propagation  of  Waves  through  a 
Medium  endowed  with  a  Periodic  Structure.  Art.  142. 
1887 


R.     IV. 


Vol.  III.     1 
38 


594  CONTENTS   OF   VOLUMES   I. — IV. 

PAGE 

On  the  Existence  of  Reflection  when  the  Relative  Refractive 

Index  is  Unity.     Art.  143.     1887    +  .-. Vol.  III.    15 

Wave  Theory  of  Light.     Art.  148.     1888       ....  ,,47 

Plane  Waves  of  Simple  Type          .          ...          .          .          .  „           49 

Intensity     ........*..  „           51 

Resultant  of  a  Large  Number  of  Vibrations  of  Arbitrary  Phase  „           52 

Propagation  of  Waves  in  General   .          .          .    .      »       '  .          .  „           54 

Waves  Approximately  Plane  or  Spherical          .          .          .          .  „           56 

Interference  Fringes     .          ...          .          ,          .          .  „           59 

Colours  of  Thin  Plates           .          .;        .'.".-.          .          .  „           63 

Newton's  Diffusion  Rings       .          *          »          .          •          .          .  ,,           72 

Huygens's  Principle.     Theory  of  Shadows          .          .          .          •  „           74 

Fraunhofer's  Diffraction  Phenomena         .                     .          .          .  „           79 

Theory  of  Circular  Aperture „           87 

Influence  of  Aberration.     Optical  Power  of  Instruments     .          .  „         100 

Theory  of  Gratings       .           ...',.          .          .-         .  „         106 
Theory  of  Corrugated  Waves           .          ...          .          .          .        _     „         117 

Talbot's  Bands ,         "     „         123 

Diffraction  when  the  Source  of  Light  is  not  Seen  in  Focus         .  „         127 

Diffraction  Symmetrical  about  an  Axis    .          .          .          .          .  „         134 

Polarization          .           .          .          .          .          .                     .        .»  „         137 

Interference  of  Polarized  Light .  „         140 

Double  Refraction ,,148 

Colours  of  Crystalline  Plates           .          .          .          .                    .  „         156 

Rotatory  Polarization   .           .          .                  •  .          ,          .          .  „         159 

Dynamical  Theory  of  Diffraction    ......  „         163 

The  Diffraction  of  Light  by  Small  Particles     .          .          .          .  ,,170 

Reflexion  and  Refraction        .......  ,,176 

Reflexion  on  the  Elastic  Solid  Theory      .           .          .           .           .  ,,181 

The  Velocity  of  Light ,,187 

On   the  Reflection  of  Light  at  a  Twin  Plane  of  a  Crystal. 

Art.  149.     1888         .         .         .....         .  ,,190 

Equations  of  a  Dialectric  Medium,  of  which  the  Magnetic  Per- 
meability is  Unity  throughout     ......  „         190 

Isotropic  Reflexion        ........  ,,192 

Propagation  in  a  Crystal        ....        . .          .          .          .  „         194 

Reflexion  at  a  Twin  Plane    ........  „         194 

Incidence  in  the  Plane  of  Symmetry       .          .          .          .          .  „         195 

Plane  of  Incidence  perpendicular  to  that  of  Symmetry      .          .  „         197 

Doubly  Refracting  Power  Small      .           .           .          .          .          .  „         200 

Plate  bounded  by  Surfaces  parallel  to  Twin  Plane    .          .   ,      .  „         200 

On  the  Remarkable  Phenomenon  of  Crystalline  Reflexion 

described  by  Prof.  Stokes.  Art.  150.  1888  .  .  .  „  204 

Is  the  Velocity  of  Light  in  an  Electrolytic  Liquid  influenced 
by  an  Electric  Current  in  the  Direction  of  Propagation  ? 

Art.  151.  1888  213 


CONTENTS   OF   VOLUMES   I. — IV.  595 

PAOB 

The  History  of  the  Doctrine  of  Radiant  Energy.     Art.  154. 

1889  ..." Vol.111.  238 

On  the  Limit  to  Interference  when  Light  is  Radiated  from 

Moving  Molecules.  Art.  157.  1889 ,  258 

Iridescent  Crystals.  Art.  158.  1889  .  .  .  •  .  „  264 
On  the  Character  of  the  Complete  Radiation  at  a  Given 

Temperature.  Art.  160.  1889 „  268 

On  the  Visibility  of  Faint  Interference-Bands.  Art.  161. 

1889 ,,277 

On  Achromatic  Interference-Bands.     Art.  163.     1889     .         .  „  288 

Introduction         .........  „  288 

Fresnel's  Bands „  289 

Lloyd's  Bands „  292 

Limit  to  Illumination  ........  „  294 

Achromatic  Interference-Bands        ......  „  296 

Prism  instead  of  Grating       .......  „  299 

Airy's  Theory  of  the  White  Centre „  301 

Thin  Plates ,,303 

Herschel's  Bands           .          .          .          .          .          .          .          .  „  309 

Effect  of  a  Prism  upon  Newton's  Rings  .          .           .           .           .  „  311 

Analytical  Statement „  314 

Curved  Interference-Bands „  316 

On  Defective  Colour  Vision.     Art.  173.     1890        .         .         .  „  380 

Instantaneous  Photographs  of  Water  Jets.     Art.  174.     1890  .  „  382 

On  Pin-Hole  Photography.     Art.  178.     1891  429 

Some  Applications  of  Photography.     Art.  179.     1891     .         .  „  441 

On  Reflexion  from  Liquid  Surfaces  in  the  Neighbourhood  of 

the  Polarizing  Angle.     Art.  185.     1892  496 

Postscript  (October  11) „  511 

Aberration.     Art.  189.     1892 ,,542 

On  the  Intensity  of  Light  reflected  from  Water  and  Mercury 

at  nearly  Perpendicular  Incidence.     Art.  198.     1892       .    Vol.  IV.       3 

Appendix.     [Curvature  due  to  Capillarity]  „  13 

On   the  Interference  Bands  of  Approximately  Homogeneous 
Light;    in   a   Letter   to    Prof.    A.  Michelson.     Art.  199. 

1892 „  15 

On  the  Influence  of  Obstacles  arranged  in  Rectangular  Order 

upon  the  Properties  of  a  Medium.     Art.  200.     1892       .  „  19 

Interference  Bands  and  their  Applications.     Art.  202.     1893  „  54 


596  CONTENTS   OF   VOLUMES   I. — IV. 

- 

PAGE 

On  the  Theory  of  Stellar  Scintillation.     Art.  203.     1893      .    Vol.  IV.    60 
Astronomical  Photography.     Art.  204.     1893 ....          „  73 

Grinding  and  Polishing  of  Glass  Surfaces.     Art.  205.     1893  .          „  74 

On   the   Reflection   of  Sound   or   Light   from   a    Corrugated 

Surface.     Art.  206.     1893         .         .         .    •    .      .  .         .          „  75 

On  a  Simple  Interference  Arrangement.     Art.  207.     1893    .          „  76 

On  some  Physical  Properties  of  Argon  and  Helium.     Art.  218. 

1896 „         215 

The  Kefractivity  of  Argon  and  Helium   .          .         •,      •-,.. •'•.',     .  „  218 

The  Reproduction  of  Diffraction  Gratings.     Art.  220.     1896  .          „         226 

On  the  Theory  of  Optical  Images,  with  special  reference  to  the 

Microscope.     Art.  222.     1896    .         .         .         .     '    .         .          „         235 

On  an  Optical  Device  for  the  Intensification  of  Photographic 

Pictures.     Art.  233.     1897        .        ...        .         .         „         333 

Rontgen  Rays  and  Ordinary  Light.     Art.  237.     1898    .      . ;  .          „         353 

On   the  Character  of  the  Impurity  found  in  Nitrogen  Gas 

derived  from  Urea.     Art.  241.     1898         .         .         .         .         „         361 
Details  of  Refractometer „  364 

Transparency  and  Opacity.     Art.  246.     1899.         .         .         .          „         392 

On  the  Transmission  of  Light  through  an  Atmosphere  con- 
taining Small  Particles  in  Suspension,  and  on  the  Origin 
of  the  Blue  of  the  Sky.  Art.  247.  1899  .  397 

The  Interferometer.     Art.  248.     1899     .         .''.'.        .  „  406 

The  Theory  of  Anomalous  Dispersion.     Art.  250.     1899        .  „  413 
On  the  Law  of  Reciprocity  in  Diffuse  Reflexion.     Art.  258. 

1900 '••'"•       V       •        •        •  ,,  480 

Remarks  upon  the  Law  of  Complete  Radiation.   Art.  260.    1900  „  483 

On  Approximately  Simple  Waves.     Art.  261.     1900      .        ,  „  486 

On  the  Stresses  in  Solid  Bodies  due  to  Unequal  Heating,  and 
on  the  Double  Refraction  resulting  therefrom.  Art.  265. 
1901 ,,502 

Polish.     Art.  268.     1901.      ..'       .        v        i        »        >        -i         „         542 

On  the  Magnetic  Rotation  of  Light  and  the  Second  Law  of 

Thermodynamics.     Art.  271.     1901.    •     .    •     .-    .    •-•  ,-         „         555 


CONTENTS   OF   VOLUMES   I. — IV.  597 


MISCELLANEOUS. 

PAGE 

On  a  Correction  sometimes  required  in  Curves  professing  to 
represent  the  connexion  between  two  Physical  Magnitudes. 
Art.  12.  1871 Vol.  I.  135 

A  History  of  the  Mathematical  Theories  of  Attraction  and 
the  Figure  of  the  Earth  from  the  time  of  Newton  to  that 
of  Laplace.  By  I.  Todhunter,  M.A.,  F.R.S.  Two  Volumes. 
(London,  Macmillan  &  Co.,  1873.)  Art.  29.  1874  .  .  „  196 

Insects  and  the  Colours  of  Flowers.     Art.  31.     1874      .         .  „       222 

Questions  from  Mathematical  Tripos  Examination   for   1876. 

Art.  41.     1876 ,,280 

On  Mr  Venn's  Explanation  of  a  Gambling  Paradox.     Art.  50. 

1877 ,,336 

Uniformity  of  Rotation.     [Phonic  Wheel]     Art.  56.     1878     .  „       355 

Address  to  the  Mathematical  and  Physical  Science  Section  of 

the  British  Association.     Art.  86.     1882.         .         .         .   Vol.11.    118 

On  the  Dark  Plane  which  is  formed  over  a  Heated  Wire  in 

Dusty  Air.     Art.  93.     1882 ,,151 

Suggestions  for  facilitating  the  Use  of  a   Delicate   Balance. 

Art.  104.     1883 ,,226 

Presidential  Address.     [Montreal]     Art.  113.     1884        .         .  „       333 

Professor  Tait's  "  Properties  of  Matter."     Art.  124.     1885      .  „       424 

The   History  of  the  Doctrine  of  Radiant  Energy.     Art.  154. 

1889 Vol.  III.  238 

Clerk-Maxwell's  Papers.     Art.  177.     1890  .         .         .  „       426 

Experiments  in  Aerodynamics.    [Review  of  Langley's]    Art.  184. 

1891 ,,491 

The  Scientific  Work  of  Tyndall.     Art.  209.     1894          .         .  Vol.  IV.     94 

The  Theory  of  Solutions.     Art.  224.     1897     ....  ,,267 

Liquid  Air  at  one  Operation.     Art.  240.     1898      ...  „       360 

Polish.     Art.  268.     1901  ........  ,,542 

Does  Chemical  Transformation  influence  Weight  ?     Art.  269. 

1901. ,,549 

38—3 


INDEX    OF    NAMES. 


Abbe,     II  412,  519,    IV  236,  239 

Abney,     II  346,  421,     III  173,  439,     IV  73 

Adams,  .T.  C.     Ill  2 

Agamennone,     III  236 

Airy,  G.      I  166,  253,  255,  256,  261,  416,  417, 

428,     II  122,  501,     III  61,  87,  90,  123,  180, 

292,  301,  544 
Airy,  H.     Ill  267 

Aitken,     II  154,    III  358,  365,  368 
Amagat,     IV  oil 
Ampere,     III  151 
Andre,  C.     I  418,    III  95 
Angstrom,    I  160 
Appun,     I  331 
Arago,     II  498,     III  34,  102,    139,  140,  156, 

159,     IV  60,  69 
Armstrong,  Lord,     IV  562 
Auerbach,     II  210 
Austen,  Boberts-,     IV  549 
Ayrton,     II  469,     IV  117,  144 


Baines,    III  267 

Balfour,  F.  M.     I  547 

Balfour,  G.  W.     I  548 

Balmer,     IV  345 

Baly,     IV  169,  514 

Barrett,     II  101 

Barry,     I  500 

Bartoli,     IV  354 

Barus,    III  569 

Basset,     III  389,  578,  593 

Beattie,     IV  557 

Becquerel,    H.     II    338,   346,    361,   365,    377, 

382,  383 

Beer,    I  123,  131 
Beetz,     I  372 
Bell,  C.    Ill  382 
Bell,  G.    I  501,    II  288,  349 
Bell,  L.    Ill  116 
Bellati,    I  313 
Berthelot,     III  399,    IV  197 
Bertrand,    I  232 
Bessel,    I  140,  338,     IV  549 


Bichat,     II  377 

Bidone,     I  377 

Bidwell,     II  341 

Billet,     I  77 

Biot,     III  102 

Bohr,     IV  513,  529,  530 

Boltzmann,    I  329,    II  346,    IV  125,  128,  354, 

433,  444,  483 
Borda,     I  299 
Bosanquet,     I  230 
Bosscha,     I  354,     II  291 
Bottomley,     IV  378 
Bourdon,     III  379 
Boussinesq,     I  271 
Boys,    III  382 
Brewer,    I  188 
Brewster,     I  210,  455,    II  122,  240,  348,  396, 

III  123,  138,  148,  159,  212,  266 
Bridge,     III  81 
Briegleb,    IV  140 
Brillouin,     II  448 
Brough,     IV  109 
Briicke,    I  99,    HI  170,    IV  102 
Bryan,     III  554,    IV  433,  438 
Buff,    I  379,  393 
Bunsen,    IV  170 
Burbury,     III  555,     IV  434 
Burnside,     III  555 


Carhart,     II  332,  473,     III  333 

Caron,     IV  139,  140 

Cauchy,     I    111,    115,    122,   131,     141,     145, 

150,  460,  522 
Cavaille-Coll,    I  320 
Cavendish,    IV  96,  136 
Cayley,     I  194,     IV  27,  28 
Cazin,    II  280 
Chaulnes,    HI  72 

Chladni,     I  174,  351,     II  212,     III  319 
Christiansen,    I  142,    II  433,   III  15,   IV  392 
Christie,    I  454,  455 
Chrystal,    I  310,     II  157,  166,  168,  449 


600 


INDEX   OF   NAMES. 


Clark,  L.    II  287,  340,  359,  451,    III  333 

Clarke,    II  33 

Clausiua,     I  99,     II  346,  521,     III  170,   561, 

IV  181,  491 

Clerk  Maxwell.     See  Maxwell. 
Coddington,     I  466 
Common,     IV  56,  542 
Conroy,  J.     II  523,  539,    IV  3 
Cooke,     III  37,  43,  236,  524,     IV  52 
Cornu,    1 537,    II  347,  348,    in  62,  112,  132, 

303,  552 

Cotes,    II  513,     III  56 
Cotterill,    III  538 
Cottrell,    I  308 
Crafts,    IV  51 
Crookes,    II  125,  340,  345,    III  266,    IV  159, 

168,  170,  193,  198,  336 
Cross,    IV  118 
Culverwell,    IV  450 
Czapski,     IV  243 


Dallinger,     IV  236 

Darwin,  C.     I  222,     III  243 

Darwin,  G.     II  344,  441,  593 

Darwin,  H.     II  4 

Davy,  H.     IV  96,  270 

Dawes,     I  416,     in  92 

De  Coppet,     II  461 

De  la  Provostaye,     I  143,  149 

De  La  Rive,     I  1 

De  La  Rue,     II  320,  340,     IV  109 

De  Pontigny,    I  411 

De  Vries,     I  263 

Debray,     I  241 

Delisle,     HI  78 

Desains,     I  143,  149,    HI  179 

Deville,     IV  139,  140 

Dewar,     II  301,  347,     HI  354,  448,     IV  232, 

359,  461,  481 
Dittmar,     III  525 
Donkin,     I  177 
Dora,     II  415 
Draper,  H.     I  207 
Draper,  J.  W.     HI  238 
Drude,     IH  497 
Duff,  W.     IV  377 
Dunkin,     I  166 
Dupre,     in  346,  359,  364,  402,  412,  421,  422, 

448,     IV  416 
Duprez,     HI  570 
Dvorak,     I  342 


Earnshaw,    I  257 

Ebert,     III  258 

Eiseulohr,     I  123,  141,  146,  150,  522 


Ellis,     I  331,  333 

Encke,     I  194 

Ettingshausen,     II  332 

Evans,  M.     II  459,     III  334 

Everett,     I  82,  229,     IV  63,  242 

Ewing,     II   543,    547,    579,    585,     587,    589, 

IV  121 
Exner,  K.     IV  72 


Fabry,     III  66 

Faraday,     I     351,     II    193,    212,    239,    360, 

III  161,  384 
Ferranti,     II  339 
Ferraris,     IV  109,  117 
Ferrers,     I  121,  338 

Fitzgerald,  G.     I  518,    III  132,     IV  342 
Fitzgerald,  M.     IV  477 
Fizeau,     I  537,     III  60,  543,     IV  59 
Fleming,     II  24,  37,  88,  458,  463,  467,  472, 

IV  232 

Forbes,  G.     I  537,     II  348,  458,     III  188 

Forel,     II  344 

Foster,  C.     II  24,  88,  143 

Foucault,     I    164,    417,    488,    538,     II    406, 

HI  60,  384 

Frankland,  E.     H  152,     IV  271 
Franklin,     III  357 
Fraunhofer,     I   160,   416,  488,     U   411,   519, 

HI  79,  99 
Fresnel,    1 112,  117,  120,  125, 218,  460,    II  498, 

HI  33,  50,  59,  127,  139,  140,  149,  156,  177, 

183,  288,  544,     IV  3 
Froude,  W.     I  261,  290,  299,  322,  324,    II  343, 

HI  492,  494,    IV  472 
Fuchs,     HI  407 


Galton,  F.     I  308,  473,     H  98 

Gauss,     III  400 

Gautier,  A.     IV  496 

Gernez,     II  460,     IV  420 

Gerresheim,     IV  141 

Gerstner,     I  261 

Geuther,     IV  140 

Gibbs,  W.    I  540,    H  342,    III  189,  190,  359, 

364,     IV  37 
Gilbert,    HI  129 
Gill,    I  538 
Giltay,     I  313 

Glaisher,  J.  W.     H  263,     III  2 
Glazebrook,     I  489,     II  49,  57,  120,  137,  157, 

164,    168,    229,    288,   362,    365,    414,    542, 

HI  114,  154,  190 

Gordon,  J.  E.  H.     II  238,  339,  361,  383 
Gore,    H  279 
Gouy,     III  263,  270,     IV  353 


INDEX   OF   NAMES. 


601 


Graham,    IV  262 

Gray,  A.     IV  255,  258 

Gray,  M.    Ill  539 

Green,     I  90,  97,  111,  113,  121,  124,  126,  145, 

218,  255,  460,    III  140,  183,  186,  251 
Greenhill,     I  346 
Griffiths,     IV  332 
Gripon,     I  411 
Grubb,     I  454V    III  56 
Guthrie,     I  267,  269 


Hagen,     II  125 

Haidinger,     IV  59 

Hall,     II  340 

Hamilton,  B.     I  231 

Hamilton,  W.     I  443,     II  517,     III  89,  155 

Hampson,     III  539,     IV  360 

Hanlon,    I  297 

Hansen,     I  166 

Harcourt,  A.  V.     IV  104,  188 

Hastings,     III  154 

Haughton,     I  110,  123,  133 

Haweis,     III  327 

Heaviside,     II   484,   551,   572,     III   452,  457, 

459,  464,     IV  327 
Heine,    I  339 
Helmholtz,     I  28,  33,  68,   97,  181,  287,  291, 

298,  305,  319,  364,  417,  518,     II  100,  122, 

323,  342,  351,  379,  396,  412,  459,  463,  513, 

516,    517,    521,      III    116,    190,    277,   327, 

IV  78,  202,  235,  243 
Henderson,     III  525 
Henry,     I  13,  305,  306,     IV  101,  298 
Herapath,     III  560 
Herschel,  J.     I  85,     II  121,  405,  499,  508,  520, 

HI  73,  90,  111,  161,  170,  240,  271,     IV  544 
Herscliel,    W.     I     416,     417,      II     92,    411, 

III  242,  309 

Hertz,     III  537,     IV  321 
Heydweiller,     IV  549 
Hicks,  W.  M.     II  240,  343 
Higgs,     IV  73 
Hilger,    I  457 
Hill,     III  2,  4,  6,  7,  12 
Himstedt,     II  448 

Hockin,     II  79,  85,  237,  276,  469,     IV  243 
Hodgkinson,     III  208 
Hbek,     III  547 
Holden,     II  92 
Holman,     IV  454,  482 
Holmgren,     III  380 
Hopkinson,  J.     I  427,     II  393,  459,  474,  543, 

548,     III  306,    IV  503 
Huffaker,     IV  463 
Huggins,     II  347,     III  547 
Hughes,     I  499,     II  339,  349,  486,  551 
Hunt,  A.  B.     II  344 


Hunt,  B.     Ill  239 

Huygens,     III  74,  77,  148,  376 


Ibbetson,     III  281,  284 


Jacobi,    IV  27 

Jamin,      I    120,     129,    141,    144,    152,    522, 

HI  180,  496,  503,  511,    IV  3 
Japp,     III  537 
Jellet,     III  162,  224 
Jevons,     II  200,     IV  69 
Jolly,  v.     IV  44,  45.  51 
Jones,  V.     IV  431 
Joule,     II  280,     III  561,     IV  96 


Kaiser,     IV  422 

Kayser,     IV  345,  494 

Keiser,     III  233,  525 

Kelland,     I  255,     III  82 

Kelvin,  I  2,  6,  16,  170,  228,  232,  294,  323, 
325,  338,  346,  365,  474,  II  10,  120,  218, 
258,  266,  343,  351,  361,  475,  517,  III  16, 
17,  155,  162,  185,  241,  256,  343,  383,  401, 
402,  413,  522,  554,  556,  577,  580,  582, 
IV  204,  209,  342,  433,  450,  495,  540,  559 

Kirchhoff,  I  288,  291,  295,  299,  300,  II  135, 
222,  251,  513,  516,  III  268,  -t92,  IV  377, 
483,  494,  532,  537 

Klingelfuss,     IV  557 

Knockeuhauer,     III  128 

Knowles,     I  290 

Kohlrausch,  F.  II  1,  47,  120,  126,  145,  237, 
279,  280,  310,  340,  IV  332 

Kohlrausch,  W.     IV  298 

Konig,  B.     I  331 

Koosen,     II  582 

Korteweg,     I  263,     IV  78 

Kundt,  I  142,  156,  II  239,  251,  338,  345, 
IV  176,  337 

Kurlbaurn,     IV  485 

Kurtz,     I  122,  123 


La  Cour,     I  356,     II  8,  179 

Lagrange,     II  513,  515,     IV  243 

Lamb,     I   475,     II   442,   446,    571,    III   250, 

280,     IV  20(5,   287,  294,  408 
Langley,     I  293,     II  194,  199,  346,     III  188, 

238,  269,  276,  491,     IV  468 
Laplace,     I  338,     HI  397,  402,  417,  466,  515 
Larmor,     I  146 
Laurent,     III  162 
Lecher,     IV  330 
Leconte,     IV  99 


602 


INDEX    OF   XAMES. 


Leduc,     IV  51,  352 

Lenard,    I  393,     III  392 

Leslie,    II  121 

Linde,     IH  539,     IV  360 

Lippich,    I  134,     II  378,     HI  110,  163,  258 

Lippmann,     II  143,     III  13 

Lipschitz,     in  44 

Lissajous,     H  218 

Liveing,     I  455,     II  347 

Lloyd,  H.     I  123,     III  61,  155,  241,  292,  295, 

297 

Lodge,  A.    Ill  591 

Lodge,  O.     I  497,     II  154,  424,  443,     IV  298 
Lommel,     I   166,     III  34,  88,  90,   132,    134, 

432,     IV  207 
Lorentz,     I    518,    522,     III    190,    470,    551, 

IV  19 
Lorenz,     I   120,   130,   131,     II   50,   120,   145, 

155,  276,     IV  19 
Love,     I   536,     III   218,   227,   244,  280,   285, 

IV  503 

Lummer,     IV  59 
Lupton,     IV  104 


Macaulay,     I  540 

MacCullagh,     I  111,  125,     III  150 

Macdonald,     IV  330 

Macdougall,     IV  198,  223 

Mach,     IV  335 

Madan,     UI  211,  265 

Magnus,     I  143,  344,  379,  391,  393 

Mallet,     IV  140 

Mallock,     I  304,     II  44,  348 

Malus,     III  139 

Maquenne,     IV  140 

Marangoni,  III  341,  358,  361,  364,  412,  448, 
562 

Mascart,  II  126,  237,  280,  298,  310,  329,  414, 
450,  III  112,  289,  309,  548,  IV  59,  60, 
65,  307,  332,  364 

Masson,     I  144 

Mather,     IV  117 

Mathews,     IV  255,  258 

Matthiessen,  A.     I  65,     II  78,  85,  237,  276 

Matthiessen,  L.     II  193,  212,     III  384 

Maxim,     IV  475 

Maxwell,  I  1,  13,  43,  60,  79,  156,  168,  226, 
235,  237,  276,  297,  471,  498,  518,  U  11, 
80,  99,  128,  170,  185,  228,  280,  281,  288, 
290,  345,  346,  350,  396,  420,  480,  486,  492, 
498,  561,  572,  IU  49,  68,  190,  376,  380, 
398,  401,  426,  470,  476,  517,  540,  554, 
IV  32,  112,  304,  307,  397,  402,  413,  433, 
491,  558 

Mayer,  A.  M.     I  331,  342,  468 

Mayer,  J.  R.     IV  96 

McKichan,     II  280 


McLeod,     I  331,  360,     II  33 
Mehler,    III  45 

Melde,    II  190,     III  1,     IV  551 
Melloni,     in  242 
Mendeleef,     IV  202,  511 
Meyer,  0.  E.     I  156,     IV  222,  454 
Michell,     I  299 

Michelson,   A.     I  538,     U  348,     III   60,   6< 
188,  189,  213,  543,  549,     IV  15,  59,  406 
Michelson,  W.     Ill  268,  275 
Mizuno,     IV  557,  565 
Moissan,    IV  139 
Montigny,     IV  61,  66 
Morley,  E.  W.     Ill  189,  525,     IV  352 
Moulton,     II  340 
Mnir,     m  11 
Muirhead,     II  473 
Munro,     I  85 


Necker,    III  134 

Neumann,     I  111,     IV  503 

Newall,     IV  422 

Newcomb,     III  188 

Newton,     I  94,     II  414,  498,  509,     III  24,  65, 

68,  70,  170,  289,  303,  311,  491,     IV  96 
Nicol,  W.     II  461 
Niven,  W.  D.     U  13,  54,     III  426 
Nobert,     I  157,  160 
Noyes,     IH  525 


Oberbeck,     U  571,    III  365,  367,     IV  557 
Obermayer,    IV  482 
Olszewski,     IV  174 
Costing,     IV  552 
Ouvrard,     IV  139 


Page,  IV  118 

Parkinson,  I  466,  II  414 

Paschen,  IV  483 

Pasteur,  III  161 

Peal,  II  194,  m  267 

Peirce,     III  111 

Penaud,    HI  494,     IV  472 

Perot,     III  66 

Petzval,    I  517,     III  432,  451 

Pickering,  E.  C.     I  453,     II  532 

Place,     LEI  70,  311 

Planck,     IV  483 

Plateau,     I  373,  388,  391,  395,     II  110,  269, 

348,     UI  341,  360,  363,  370,  374,  585 
Pochhammer,     IV  276 

Pockels,  A.  (Miss),     HI  375,  572,  573,     IV  425 
Poincare,     IV  434 
Poisson,     I  460,  472,     II  498,     UI  33,  78 


INDEX   OF   NAMES. 


603 


Poynting,     II  363,     III  162 

Preece,     II  331,     IV  109,  124 

Prehlinger,     IV  141 

Preston,     II  414,     III  305,     IV  406 

Preyer,     I  331,     IV  298 

Priestley,     IV  137 

Provostaye,     III  179 


Quincke,  I  152,  155,  215,  387,  504,  II  231, 
236,  III  111,  350,  367,  371,  383,  392,  412, 
497,  562 


Ramsay,     III  472,     IV  1,  130,  184,  192,  195, 

215,  217,  222,  223,  224,  260,  265,  272,  351, 

361,  481,  514 
Randall,     IV  222 
Rankine,     I   113,  261,  324 
Reade,     III  239 
Regnanlt,     III  37,  43 
Reinold,     II  349,  511,     III  349,  425 
Respighi,     IV  61 
Reusch,     III  212,  266 
Reynolds,  0.     I  316,  323,   324,     II   273,  344, 

III  365,  575,     IV  101,  298 
Richards,     III  37,  236,   524,     IV  348 
Richardson,     IV  225 
Ridout,     I  500 
Riess,     I  354 
Rijke,     I  353,  408 
Riley,     II  231,  233 
Robinson,     I  454 
Robison,     III  491 
Roiti,     III  213 
Rontgen,     II  338 
Rood,     II  522,  533,     III  179,  442 
Roscoe,     I  110 
Rowland,     II  1,  16,  47,  76,  120,  135,  339,  341, 

347,  587,     III  110,  113,  116,    IV  73,  87 
Rubens,     I  144,     III  189,     IV  485 
Riicker,     II  349,   511,     III  349,  425,     IV  202 
Rudberg,     III  155 
Rumford,     IV  96 
Runge,     IV  345 
Russel  (Major),     I  161 
Russell,   Scott,     I  256,  261,     II  258 
Russell.     IV  363 
Rutherfurd,     I  207,  458,  505 
Rydberg,     IV  346 


Schoeder  van  der  Kolk,     II  91 

Schulze,     IV  222 

Schuster,  I  310,  II  1,  20,  40,  43,  63,  75, 
98,  340,  398,  III  276,  IV  170,  199,  346, 
353,  369,  500 

Schutzenberger,     IV  139 

Schwendler,     III  452 

Schwerd,     III  82 

Scott,  A.     II  302,     III  37,  43,  234,     IV  348 

Seebeck,     I  403 

Sellmeyer,     I  143,  156,     IV  413 

Shaw,     II  83,  446 

Shields,     IV  184 

Sidgwick,  Mrs,  II  43,  47,  63,  78,  103,  115, 
155,  271,  273,  278,  471,  540,  IV  332 

Siemens,  C.  W.     II  334 

Siemens,  Werner,     II  78,  90,  91 

Siljerstrom,     IV  511 

Simpson,     I  205 

Smith,  A.     Ill  75,  152,  169 

Smith,  F.  J.     IV  113 

Smith,  M.     Ill  384 

Smith,  R.     II  409,  513,     III  56 

Smith,  W.     II  422 

Sommerfeld,  III  163 

Sondhauss,     I  26,   36,  69,  351,     III  342 

Soret,     III  78 

Spottiswoode,     I  458,     II  340,     IV  160 

Stas,     IV  133 

Stefan,     IV  354 

Stephanelli,     III  448 

Stewart,  B.  Ill  241,  268,  552,  IV  378,  483, 
494 

Stokes,  I  50,  89,  91,  96,  99,  101,  113,  117, 
141,  191,  220,  255,  257,  263, 322, 531,  II 121, 
241,  273,  344,  403,  419,  479,  498,  III  49, 
62,  66,  68,  72,  86,  89,  92,  123,  146,  154,  163, 
179,  181,  204,  240,  272,  340,  569,  575,  588, 
595,  IV  78,  101,  209,  298,  321,  353,  376, 
409,  540 

Stoletow,     II  339,  587 

Stoney,     II  198,     IV  237 

Strutt,  R.  J.     IV  223 

Struve,  H.     Ill  63,  92,  127 

Stuart,     II  9 

Sumpner,     IV  117,   144 

Sutherland,     IV  482,  514 

Swan,     III  148 

Swinton,  A.  C.     IV  562 

Sylvester,     III  428 


Salmon,     III  150 

Sande  Bakhuyzen,     II  365 

Savart,  F.     I  373,  388,  390,     II  239,  269 

Savart,    N.     I  403,  404,    405 

Scheibler,     I  331 


Tait,  I  170,  228,  232,  338,  II  361,  424,  475, 
517,  III  256,  383,  465,  556,  593,  IV  109, 
124 

Talbot,  F.  I  507,  II  362,  III  51,  69,  123, 
134,  289,  IV  545 

Tate,     IV  415 


604 


INDEX   OF   NAMES. 


Taylor,  S.     II  240 

Taylor,     II  469 

Thiesen,     I  291 

Thollon,     I  456,  544,     II  347,  552 

Thompson,  S.  P.     IV  353,  553 

Thomson,  J.  J.  I  518,  547,  II  44,  449, 
488,  IV  276,  318,  323,  327,  353,  354.  503 

Thomson,  James,     II  154,     III  516,     IV  63 

Thomson,  W.     See  Kelvin. 

Thorpe,     IV  130 

Threlfall,     I  547,     II  458,     IV  514 

Todhunter,     I  22,  196,  338,  492,     IV  370 

Tomliuson,  C.     Ill  347,  566,     IV  420 

Topler,     I  328,     II  406,     IV  125,  128 

Tower,     II  344 

Travers,     IV  481 

Tyndall,  I  87,  101,  109,  305,  307,  316,  394, 
531,  541,  II  100,  151,  190,  220,  269, 
III  1,  25,  170,  IV  94,  379,  551,  554 


Van  der  Mensbrugghe,     III  347,  353,  412,  565 

Van  t'  Hoff,     IV  267 

Venn,     I  336 

Verdet,  *I    76,   164,   200,   417,   491,     II   414; 

in  99,  143,  146,  161,  291 
Vince,     I  293 


Waals,  van  der,     III  398,  421,  423,  465,  470, 

471,  516 

Walker,  J.     Ill  291 
Walker,     I  530 
Walter,     IV  557,  559,  565 
Warburg,     II  345,  544,     IV  176,  337 
Waterston,     III  477,  558,     IV  433 


Watson,     IV  434 

Weber,  H.  F.     II  1,   49,  120,     III  63,  268,  275 

Weber,  W.     II  120,  553 

Wenham,    III  492,     IV  467 

Wertheim,     I  29,  36,  320 

West,     IV  363 

Wheatstone,     I  182 

Whewell,     III  243 

Whitaker,     IV  225 

Whitehead,     IV  87 

Wiedemann,  G.     II  134 

Wien,  M.     IV  110,  125 

Wien,  W.     IV  354,  483,  555 

Wild,     II  279,  415 

Williams,  P.     IV  146 

Wohler,     IV  139 

Worthington,     I  387,     III  392,  398.  412 

Wright,  A.     II  314,  323,  342,  454,  458,  459, 

462,  472 

Wright,  H.  R.     IV  480 
Wright,  L.     IV  237 
Wiillner,     I  152,     IV  12 


Young,  J.     I  537,     II  348,     III  188 

Young,   S.     Ill  472 

Young,  T.  I  460,  II  235,  425,  498,  516, 
III  72,  100,  111,  139,  177,  238,  271,  397, 
400,  404,  414,  417,  419,  423,  544,  IV  %, 
550 


Zamminer,     III  329 
Zech,     I  127 
Zecher,     III  213 
Zollner,     II  138,  141 


CAMBRIDGE  :     PRINTED   BY   J.    AND   C.    F.    CLAY,    AT   THE    UNIVERSITY   PRESS. 


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