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Se ow be betel ) om wie Ae % lu —=) S ae ye) hemes bia} Ct hw) foen ay 69m O — BY id iv) a (ay xm W =) RY iw) a AMA AMMMMAMVAMMMMMMAMcMaMVxwZ 11—8825 PRINTING CFFICE SOVERNISEN' RS - THE LONDON, EDINBURGH, axp DUBLIN PHILOSOPHICAL MAGAZINE AND ‘a JOURNAL OF SCIENCE. i CONDUCTED BY JOHN JOLY, M.A. D.Sc. F.R.S. F.GS. AND WILLIAM FRANCIS, F.L.S. “Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster vilior quia ex alienis libamus ut apes.” Just. Lips. Polit. lib. i. cap. 1. Not. VOL. XIX.—SIXTH SERIES. JANUARY—JUNE 1910. > co o> 5 3° e > > ) Ps ib. O85 o ’ Py > 20 © =e = 4 2 ) © © 5? r ’ 5 ¥,? 2 2 y a a0 ome a? »2 ne #753 5 sLOWDONY) > TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET, SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND CO., LD. SMITH AND SON, GLASGOW ;— HODGES, FIGGIS. AND CO., DUBLIN; YEUVE J. BOYVYEAU, PARIS ;—AND ASHER AND CO., BERLIN. “Meditationis est perscrutari occulta; contemplationis est admirari perspicua .... Admiratio generat questionem, questio investigationem, investigatio inventionem.”—Hugo de S. Victore. ——-“ Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, Cur caput obscura Phcebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina ccelo, Quo micet igne Iris, superos quis conciat orbes Tam vario motu.” J. B. Pinelli ad Mazonium. CONTENTS.OF VOL. XIX. SO Lo (SIXTH SERTBS). \. NUMBER CIX.—JANUARY_ 1910. Mr. William Sutherland on the -Fundamental Constant of Atomic Vibration and the Nature of Dielectric Capacity. . Mr. William Sutherland on Molecular Diameters.......... Mr. E. Gold on the Relation between Periodic Variations of Pressure, Temperature, and Wind in the Atmosphere.... Mr. Harold G. Savidge: Tables of the ber and bei and ker and bei Functions, with further Formule for their 8 eee 8 ea 4 O's wile. nim Ar) sal gp Mo WIRD oe Mr. J. 8. Dow on the Physiological Principles underlying DME P IWOLOMOCHET. 2. we he eee ew ween ac es Page Dr. J. W. Nicholson on the Effective Resistance and - ieee Of a Helical Coil .. 2.1... cee een ses ata Dr. C. V. Burton: Note on a Gravitational Problem ...... Lord Rayleigh on the Regularity of Structure of Actual NE Sid L ls k's G15 kine ok oR Ria tana nabein ee Woe wo aa an Dr. Walter Makower and Dr. Sidney Russ on the Recoil of freee trom: adigm Boos c... ok ek oe ee we eae bee oe Mr. W. A. Scoble on Ductile Materials under Combined TS DO ee ee Or ne ee ear Prof. Alexander McAulay on the Spontaneous Generation of Electrons in an Elastic Solid Atther ................ Dr. Geo. A. Campbell on Telephonic Intelligibility .... Dr. T. H. Havelock on the Instantaneous Propagation of Disturbance in a Dispersive Medium.................. Mr. Norman Campbell on the Principles of Dynamies...... Mr. Norman Campbell on the Atther.................... Prof. E. Rutherford on the Action of the a Rays on i ONS LS. in inci Bund ba dann AM Mt dome OCT «dace as Mr. P. V. Bevan on the Absorption Spectrum of Potassium SCE ACG ORM Fe 6 vine wiapa o's sao oom, win wine Hhumelabe gs a/e Mr. E. M. Wellisch on the Theory of the Small Ion in eI Sa Ses ces isl Si a RR Ee Cas ais ware a We wiles won 6 6 Dr. Alexander Russell on Alternating Current Spark et Eee LRN ae ale eave ta he d's deine os 96 100 116 129 . 152 160 168 181 192 195 201 903 1V CONTENTS OF VOL. XIX.—SIXTH SERIES. Dr. Robert A. Houstoun on the Damping of Long Waves in Page akevignemlan Trough |... \sceee daem elieees oh... 205 Proceedings of the Geological Society :— Messrs. R. H. Rastall and J. Romanes on the Boulders oLiherCanibridee Drttig ¥.\.e 4 ass... ). Os ee 206 Mr. A. M. Finlayson on the Nephrite and Magnesian Rocks of the South Island of New Zealand ........ NUMBER CX.—FEBRUARY., Prof. R. A. Millikan on a New Modification of the Cloud Method of Determining the Elementary Electrical Charge and the most Probable Value of that Charge............ 209 Dr. J. W. Nicholson on the Asymptotic Expansions of Bessel AE UNC HONUS We eee eres eye Gos ten) sys oe ce: teen 228 Mr. F. E. Smith on Cadmium Amaleams and the Weston Normal(Cell mabe: To. i. Sa oe eee ee 250 Dr. J. W. Nicholson on the Bending of Electric Waves COUNO RUN MM Ami sere end. eke: elie Chie eRe een ae 276 Mr. A. 8. Russell on some Variations observed in Electro- scopic Measurements and their Prevention ............ 278 Mr. T. Royds: Further Experiments on the Constitution of the Milechiae Spark. idate V2)... sacsueles oe eieuee ts ae 285 Mr. E. W. B. Gill on the Electrical Effect of the Ultra-Violet SPec GH UMA cia esa Su oD a Ae ae 290 Mr. F. J. Jervis-Smith on an Optical Method of Reading the Torsional Angele ot atvotahine Shatt .0 52.7. 2s. ee 300 Sir J. J. Thomson on a Theory of the Structure of the Electric Field and its Application to Rontgen Radiation and to Mie lat, OS cue wearer aM eh Relays so ss ea ee eee 301 Mr. Horace H. Poole on the Rate of Evolution of Heat by Pitchbletide : toi. 2 eens Lo... OTe kee ee 314 Prot. J. Joly jon /Pleochnore@etalosyer ts. 05... Sy 327 Notices respecting New Books :— H. Bounasse’s Cours’ de Physique... .............) aa 331 Results of Observations made at the Coast and Geodetic Survey Magnetic Observatory at Cheltenham, Mary- land; "1905. and 190 Ware eee tt, ain eee ee 331 C. W. Raffety’s Introduction to the Science of Radio- ACTIVI. “.'.0 4.6 fo be ae iim Cty fea 331 Proceedings of the Geological Society :— Mr. O. T. Jones on the Hartfell-Valentian Succession around Plynlimon and Pont Erwyd (North Cardigan- BNET) dee at tc Nk ee Pea TEA a hee aM 332 CONTENTS OF VOL. XIX.——SIXTH SERIES. Vv Page Mr. J. V. Elsden on the Geology of the Neighbourhood PMO SUISGOR)) iio obeys ois chun we nisl a x tela d eee oo 333 Messrs. C. T. Clough, H. B. Muff, and H. B. Bailey on the Cauldron Subsidence of Glen Coe and the Associated loneous Pheuomena .................. 334 Mr. C. Carus- Wilson on the Pitting of Flint-Surfaces.. 336 NUMBER CXI.—MARCH. Mr. C. A. Sadler on Homogeneous Corpuscular Radiation. MT ER. iy oN blirs SSL Meh, HRS haa 337 Mr. S. D. Chalmers on the Sine Condition in Relation to thewwama of Optical Systems ......0. 2.0... e eee eee ee 356 Mr. C. BE. Van Orstrand on Reversion of Power Series .... 366 Mr. J. L. Hogg on Friction in Gases at Low Pressures .... 376 Prof. W. M. Thornton on the Polarization of Dielectrics in a PPI, OF POTGE 4b PE es oe Vlas we ete 390 Mr. Robert E. Baynes: Saturation Specific Heats, &c., with van der Waals’s and Clausius’s Characteristics. (Plate VI.). 407 Dr. C. V. Burton on the Sun’s Motion with respect to the CS eee een) ke ee 2 ne ALT Sir J. J. Thomson on Rays of Positive Electricity .... 424 Dr. J. W. Nicholson on the Bending of Electric Waves UME OOO, as gana idlale stig i 9 eee la da Bv Od wele 435 Prof. Jean Perrin on the Charge of the Electron .......... 438 Prof. J. P. Kuenen on Pirani’s Method of Measuring Self- SR he vse ooo a Laos 36 cv olece sw Boksalwe a bles 439 Notices respecting Ne The Scientific Deda of Sir Wilham Hugaiad .)) 0.5: 441 J. Cavalier’s Lecons sur les Alliages meétalliques at aa 444 Q. Cramp and C. F. Smith’s Vectors and Vector Diagrams applied to the Alternating Current Circuit. 445 J. L. Coolidge’s Elements of Non-Huclidean Geometry.. 445 Dr. I. Kohlrausch’s Lehrbuch der Practischen Physik.. 446 Proceedings of the Geological Society :— Dr. J. E. Marr and Mr. W. G. Fearnsides on the Howeill ee and their Popography .%. i 03. Ua ee a ees 447 Dr. W. F. Hume on the Granite-Ridges of Kharga mes Intrusive or Tectonic? |... ot. ce eee 448 Intelligence and Miscellaneous Articles :— Correction to Mr. E. Gold’s Paper on “The Relation between Periodic Variations of Pressure, ‘temperature, aay Wind in the Atmosphere.” .......0..05 00064. 448 v1 CONTENTS OF VOL. XIX.—SIXTH SERIES; NUMBER CXII—APRIL. Page Prof. A. P. Chattock and Mr. A. M. Tyndall on the Pressure of the Electric Wind in Hydrogen containing traces of Oxygen. 449 Mr. J. M. Kuehne on the Electrostatic Effect of a Changing MacmetionWteld siuga iit curs ane elses 5 PME sis ¢\e-x seem 461 Dr. J. Robinson on Konig’s Theory of the Ripple Formation iy Kommelire MEWS. Unter ale sce neha takiceieie lt. als sami 476 Mr. R. Hargreaves on the Ignoration Problem. Explicit form UPR TE e Co al ails ah ase ke! 0: Ad al pe a an ae 486 Dr. P. Lowell on Photographs of Jupiter taken at the Lowell Observatory, April 1909, by Mr. E. C. Slipher and the Pirector. Or P. Prone 5 es ee a oe Oeste ee A488 Mr. A. Campbell on the Use of Mutual Inductometerg .... 497 Prof. C. H. Lees on the Laws regarding the Direction of Thermo-electric Currents enunciated: by M. Thomas .... 508 Prof. A. H. Gibson on the Variation of Disk Resistance with Temperature una Woaiber i)4/s0. sic 4) Re ate ee 513 Dr. J. W. Nicholson on the Bending of Electric Waves round Bi LOE PEO hecria tm heyecmialaiieiac hi se eha iS 4 taller 516 Prof. H. L. Callendar on Electrical Recording Thermometers EOE PO Mena TELL NV) OU ey ater Daehn Ges ieee eh ae ene 538 Prof. Harry C. Jones and Mr. W. W. Strong on the ne Spectra of certain Uranous and Uranyl ‘Companies ees Dr. W. C. McC. Lewis on the Energy of a ‘“ Double- layer” Condenser of Hlectronie’Origin +... 00) 40 see a73 Mr. H. Bateman on the Solution of the Integral Equation connecting the Velocity of Propagation of an Harthquake- Wave in the Interior of the Earth with the Times which the Disturbance takes to travel to the different Stations von ‘the arth is Surlace: 6... .ie ie Wh aes eee 576 Mr. H. Redmayne Nettleton on a New Method of Deter- mining Thermal Conductivity. (Plate VII.)............ 587 Dr. H. Wilde on a new Binary Progression of the Planetary Distances, and on the Mutability of the Solar System .... 597 Mr. W. A. Borodowsky on Absorption of § Rays from Radium by Solutions and Liquids .............0..0008 605 Mr. A. I. Steven on Anomalous Effects on First Loading a Wire, and some Hffects of Bending Overstrain in Soft Prom Wis iii. Oe Ren Bears J ee er 619 Dr. J. W. Nicholson on the Size of the Tail-particles of Comets, and their Scattering Effect on Sunlight ........ 626 Prof. J; Joly and Mr. A. L. Fletcher on Pleochroic Halos. (Plates: Valle & 1X9) 2a uly a ee ae ene 630 Mr. G. H. Berry on the Striking Point of Pianoforte Strings. laces Xe ke TW ei Sel chao has ee ata oa 648 Notices respecting New Books :— Physical Science in the time of Nero .............0-: 651 Proceedings of the Geological Society ..........6.... 653-656 CONTENTS OF VOL. XIX.—SIXTH SERIES. vil NUMBER CXIII.—MAY. Page Dr. A. 8S. Eve on the Effect of Dust and Smoke on the Lolear ii vit A EUS Sener a PA 657 Pro. erry on Telephone Circuits .. 2.000004. 00.5. 673 Mr. S. C. Laws on the Change of Resistance of Metals in a Magnetic Field at Different Temperatures.............. 685 Prof. P. Lowell on the Limits of the Oblateness of a Rotating Planet and the Physical Deductions from them ........ 700 Mr. 8. H. Burbury on Boltzmann’s Law of Probability e-*x. 712 Dr. G. C. Simpson on Earth-Air Electric Currents ........ 715 Mr. and Mrs. Soddy and Mr. A. 8. Russell on the Question of the Homogeneity of y-Rays. (Plate XII.) .......... 725 Dr. J. W. Nicholson on the Bending of Electric Waves round si) eis pital ware ye Sudo yea bal ola oh ANY ee aie 757 Prof. H. Rubens and Mr. H. Hollnagel on Measnrements in the Extreme Infra-Red Spectrum. (Plate XITI.) ...... 761 Dr. R. D. Kleeman on the Nature of the Forces of Attraction Between Atoms and Molecules .....655....6000-clges: 783 Prof. H. A. Wilson on the Relative Motion of the Earth Peet 8 da alee MOD. 809 Mr. W. Sutherland on the Theory of the Small Ion in Gases. 817 Notices respecting New Books :— Prof. J. Joly’s Radioactivity and Geology ........... 819 Proceedings of the Geological Society................ 820-823 Intelligence and Miscellaneous Articles :— Correction to Mr. Bateman’s Paper on the Reflexion of Light at an Ideal Plane Mirror moving with a Uniform Buea Of Peatisiation 0% lo. Se ee 824 NUMBER CXIV.—JUNE. Mr. G. N. Antonoff on Radium D and its Products of ENS OMN o655108 ac) p48 itn we Souivaaeapleks died wie eld «..- 3°332 13°43 17°38 5 (Gi PE a acd seautsae0. 8:00 13°5 13'2 36 11:2 0, ee ee 1°416 1°995 2°317 2°391 1°67 FR ee Evans «2 2:004 3981 5°367 5°718 2-789 RIE 1:750 2:297 5682 3°333 2°50 5007 -4944 “4974 5562 4594 In this table the data for the halogens are probably the best, and in this family the mean value of the ratio is 0°49395, from which the average departure is only 0:2 per cent. and the maximum departure 0°3 per cent. In this family there is the special relation that both B and wu have values as 1, 2, 3, 4. In the He family | have put for He w=? to indicate that there is no sure guidance as to what the value of u should be. A fractional value of w presents some difficulty of interpretation. If for He w=1/3 the ratio would be 0-460 in close agreement with the results for A, Kr, and X. As the table stands, the results for He and Ne are a good deal too small. In the table as a whole it will be noticed that those elements which have definitely ascertained values of B from the density in the solid state show a good approach to constancy in the ratio. For O the value of the ratio is large. For C and Si the available data allow only a rough determi- nation of the ratio. On the whole the table shows that the theory gives a good account of dielectric capacity in the non-metals. The interesting questions of the transition from metal to non-metal, and of the action of an element electri- cally positive in some compounds and negative in others have not been discussed, as they would lead into further complications hardly ready for discussion. The data for hydrogen should be considered here, namely, (N' —1) V=1°559 and B=3:18 from the kinetic theory, which make N=1:490, K=2 22, p=0°3145, and with u=4 the ratio (N'—1) Vp!/*/ K B?%u? is 0°3504, and with w=1/3 it is Phil. Mag. 8. 6. Vol. 19. No. 109. Jan. 1910. C 18 Mr. W. Sutherland on the 0°4592. These results for H are very like those for He. It may be mentioned that for oxygen if B=8-29 instead of 6°16, the ratio would become 0°4941. The evidence as to B for O and N in their compounds is on the whole towards making its value larger than that given by the kinetic theory. It is rather remarkable that the valency of the non-metallic elements does not influence their dielectric capacity. The order of the element in its family asserts itself by the ap- pearance of wu in (8), but the valency of the family does not appear. If in (8) we put e=M/B and (N’—1) V=(N—1)B, we get the simple result that (N—1)/N?=EM*u??. That is to say, that the index of refraction or the dielectric capacity of the stuff of an atom of non-metal is a function of only its atomic mass (weight) and u. In “The Dielectric Capacity of Atoms” (Phil. Mag. [6] vii. p. 402) it is shown that if K ig derived according to certain assumptions from the atomic conductivity of the atoms as ions in aqueous solution, then KB2/V is nearly constant for both metals and non- metals, V being valency. But in ‘Ionization in Solutions and Two New Types of Viscosity” (Phil. Mag. [6] xiv. 1907, p. 1) it appears that K thus derived in the case of the halogens is very different from N?, and must be interpreted as K for slow electric alternations, while N? is K for rapid alternations. The matter requires further investigation. At present we are dealing with K only as it appears in the electric alternations of light. In this section then it has been shown that dielectric capacity, both in atoms of combined metal and of non-metal, can be explained by means of internal electric fields in the atoms corresponding with the external electric fields which produce cohesional force, otherwise called molecular attrac- tion. The broad laws that the atom of metal is electrized so that its moment is proportional to its radius, while the atom of non-metal is electrized so as to have its electric moment proportional to its volume, have been found to be fundamental in determining the relations of dielectric capacity in the two classes of atoms. It is also profoundly significant that the radii of the atoms of combined metals should show simple numerical relationships like 2, 3, 4,5, 6 in the Li family, while the volumes of the atoms of non-metals show a tendency to similar simple relationships like 1, 2, 3, 4 in the halogens. It is evident then that electric conditions largely determine the architecture of atoms. The cohesion of the constitutive pairs of electrons with the aid of the special pair to form the atom demands investiga- tion. The pair of electrons is to the atom what the molecule J ' i ty i Jy is ( ji Nature of Dielectric Capacity. i 5 is to an ordinary piece of matter, it is a unit of structure. In “The Electric Origin of Molecular Attraction” (Phil. Mag. [6] xvii. 1909, p. 657) it is suggested, with the aid of diagrams, that in matter contiguous molecules adjust their axes so that these form axial lines, along each of which the axes are similarly directed, while the axial lines are alter- nately oppositely electrized. This arrangement is one of minimum electric potential energy and causes each molecule to attract its six nearest neighbours with repulsions and at- tractions beyond these six tending towards an average null effect. At the same time it gives no electric moment to any ordinary piece of matter, the sum of the electric moments of all the molecules being nothing. In the atom the chief difference from this state of affairs is that the cohering con- stitutive pairs have their electric axes similarly directed to the extent required to give the whole atom its characteristic electric moment es. This leads naturally to the conception of chains of pairs of electrons, the electric axes of all the pairs being similarly directed along the chain. Let us make a diagram of two such chains symmetrically situated with Fig. 1. regard to the electric axis of the whole atom, this axis being denoted by AB. The two chains being electrized in the same direction, on the average that of AB, repel one another. It seems to be the function of the special electron pair placed at AB to attract both of these repelling chains and to hold them in position. The special pair acts like a keystone to all the arches of the constitutive pairs of electrons. it appears then that the cohesion of the molecules of an ordinary piece of matter is the result of the action at more distant range of the same electric forces which cause the C2 20 Mr. W. Sutherland on a constitutive pairs of electrons to cohere in each chain. One of the effects of atomic disintegration must be the liberation of one of these chains, which, when free, will behave in many respects like the moving steel chain in gyroscopic experiments. If a chain is broken into its separate electron pairs each will form the pair which I have proposed to call a neutron, which seems to be a necessary constituent of the luminiferous ether to enable it to transmit electric and magnetic actions. Theoretically, then, there is reason to look for the free pair of electrons as a physical agent as it appears in Bragg’s hypothesis for the gammaand the Rontgen rays. If Ruther- ford’s view of the alpha particle is correct, namely, tHat it is. an atom of helium carrying two positive electrons, I would suggest for it the structure shown in the diagram, where a Fig. 2. bse + circular ring of electrons has collapsed into the form of two. parallel cohering straight chains enclosing a positive electron at each end. ‘These by their repulsion tend to keep the chains straight. If such a structure breaks away in the dis- integration of radium with a velocity nearly that of light, it will have many of the properties of an alpha particle. When the two positive electrons are removed the structure re- arranges itself in the form of the He atom. It ought to throw much light upon the internal structure of the atom if the reason is discovered for Bragg’s remarkable law that the stopping power of the atom of an element for the alpha particle varies as the square root of the atomic mass. It should be noted that each pair of electrons may be rotating round the average position of its electric axis, and may thus have a magnetic moment parallel to the electric. Each pair by virtue of its inertia has the properties of a gyrostat. Hach pair is a vibrating system whose period can be calculated on the principles here discussed. The idea of the electric gyrostat has already been used in a theory of the electric conductivity of metals in “The Electric Origin of Rigidity and Consequences.” yu + ue Kinematical Analysis ot Balmer’s Formula. 21 3. A Kinematical Analysis of Balmer’s Formula. Under this title in section 3 of “The Cause of the Structure of Spectra ” an attempt was made to trace Balmer’s formula to certain simple effects of relative motion. In that paper quite a number of numerical laws or indications of such laws are shown to imply the prominence of kinematical considerations. Yet since Rayleigh suggested in 1897 that Balmer’s formula points to kinematic, rather than dynamic, relations, many elaborate attempts have been made to find a dynamical explanation of the relations amongst the lines in a spectral series. To follow up the dynamical theory here advanced for the fundamental constant of atomic vibration in Balmer’s formula it seems desirable to make my former kinematical analysis of Balmer’s formula more definite in the following way. Consider a vibrating circle one point of which is constrained to be a node. The circumference of the circle is half a wave-length. The stationary wave is the re- sultant of two trains of waves which are travelling in opposite directions round the circle, and are always being reflected at the node. Let the angular velocity of each train be . Suppose now that the node A is caused to move with angular Fig. 3. ete o® © e %, velocity © because the constraint producing it so moves. Then relatively to A the one train is moving with angular velocity o—Q, and the other with —o—Q. Thus the con- straint will be met by the one train after time 27/(w—Q), and by the other after time 27/(#+), that is to say, the half wave-length of the one train which was originally 2c is changed to 2cw/(w+0), and that of the other to 2arcw/(w—©.). But upon reflexion each of these altered half waves is followed by the other. Thus each whole wave dX of one train is replaced by two parts Aw/2(@+) and Ao/2(@—Q.). The state of affairs is represented graphically in the diagram where AB=AC+CB and represents part of the circumference of the circle developed as an infinite Goos2 22 Mr. W. Sutherland on a straight line. AC=BD=22co/(o+Q); CB=DA=27co/ (o—Q). With «x as abscissa measured from A the curve in dots is composed as far as C of y= sin a/AC, and from © to B of y= sinza/CB, while the curve of dashes consists from A to D of y=sinaa#/DA, and from D to B of y=sin7z/BD. The full curve is got by summing the ordinates of these two. It represents a periodic, but not simple harmonic, function of wave -length AB=27cw{1/(@ +0) +1/(@—Q)} =4rew?/(o? — 2”) derived from the original wave-length 4c. With a= Th Rydberg’s extension of Balmer’s formula is n =)—6/(m+ p)?, reducing to Balmer’s when »=0. If we compare this with the formula just derived for wave-length we get, if V is velocity of light and v that of disturbance in atom, =V/4arev and (0/m)*/(m+p)=Q/o. « (9) Let us suppose that this disturbance travelling with angular velocities @ and —w is a deformation caused in the stuff of the atom, that is in the medium formed by the constitutive pairs of electrons, and that the circle of radius ¢ is the equator of the spherical atom whose poles are the ends of its electric axis, or a small circle parallel to the equator, giving a>. This deformation is caused by a displacement of the special pair of electrons which brings its negative electron nearer to the most deformed part of the spherical surface of radius R. This negative electron is the cause of the dis- turbance and also determines the node, so that its angular velocity is ©. Since v is the velocity of propagation of the disturbance through the stuff of the atom, then v=cw. Next consider the small circle parallel to the equator whose radius is c/m, where we shall first suppose m to be an integer. Then a disturbance travelling round this circle with velocity v will have angular velocities mw and —mw. In this way by con- sidering the special pair of electrons as cause of elastic dis- turbance in the atom, and as controlling the position of nodes in the circles of radii c/m, we can give a simple kinematical account of Balmer’sformula. To pass to Rydberg’s form we would need to find a kinematical reason for angular velocities which are proportional to m+vyp, where w is a fraction cha- racteristic of a spectral series. In my previous paper I have shown that m may take the form m+1/n where n is an integer, and even m+ p/g where p and q are integers. But for the present we shall confine our attention to the essentials as they appear in the formula of Balmer. We have a funda- Kinematical Analysis of Balmer’s Formula. 23 mental standing wave of length 4ze in the atom, which becomes 47cV/v or A in free zether. This gives an important relation between a linear dimension of the atom and the wave-length of its light. We can test this relation in the following way. We have v=(W/p)?= (2177/3)? (es/(2a)3p2) = (2277/3)? (M71/B2p):. Now in the non-metals we have B=10-* (M%/)2 nearly, and p of the same order of magnitude as 27/3, so that v is of the order 10°, and V/v is of the order 3x10. In the visible part of the spectrum 2 is of the order 6x 10~° and ¢ at its maximum will be R, but R is of the order 8 x 10-°, so that d/47rR is of the order 600. To make d/d4ae of the same order as V/v we should need to take ¢ to be only about 1/500 of R. In the combined metals of the Li family we have v= (20/3)?{1:27/0°65°(2, 3, 4, 5, 6)M}8, with a value whose order ranges from 10° for Li to 10° for Cs, so that c in Li would need to be about 1/50 of R, and for Cs it would need to be about 1/500 of R. If further investigation bears out this view of the kinematical origin of Balmer’s formula there will be special interest in seeking for the simplest kinematical conditions that will lead to Rydberg’s law of the relations between the principal and usual series in the spectra of the alkalies. 4, Summary. The central cause of the spectrum of an atom is a special pair of electrons situated within the atom on its electric axis. The atom is electrized, or electrically polarized, because it is formed of constitutive pairs of electrons which have their electric axes similarly directed so as to give the atom electric properties analogous to the magnetic ones of a uniformly magnetized sphere. The internal electric field of the atom corresponds with the external field which produces cohesion, and has been investigated in several papers on the laws of molecular attraction. The special electron pair consists of a positive electron which acts as though uniformly distributed through a sphere of radius R=0°795 x 10-8, which is the same for all atoms. This action may be due to the rapid random motion of a small positive electron through this sphere of constant size. The negative electron is situated at distance r along the electric axis of the atom from the centre of the sphere of radius R. As regards outside action the positive electron may be supposed altogether at the centre of 24. Mr. W. Sutherland on the sphere, so that the special pair of electrons has an electric moment er oppositely directed to that of the whole atom, which is denoted by es. This electric moment es produces the external electric field of cohesion, and it produces the internal field in which the special pair of electrons vibrates. The special negative electron is attracted by the positive with a force (e?/r?)(7°/R*)=e?r/R’, and comes to rest in the internal field at the distance defined by the equation 7/R°=s/a°, where a is radius of atom. If displaced in any direction at right angles to r the negative electron of inertia 2 oscillates with a frequency (e?/R°z)?/27. If displaced in the direction of » it oscillates with the same frequency. Thus the negative electron has three degrees of freedom in which its frequency is the same. Moreover, this frequency is the same for all atoms, if R is the same for all, and that is why it is stipulated above that R is to be an absolute con- stant. It is assumed that there is a fundamental period of vibration the same for all atoms, because in the Balmer- Rydberg formula 1/A=n=n,—b/(m+ yp)’ the parameter 6 has nearly the same value for many series of spectral lines in many elements. The frequency JV (V=velocity of light) is a fundamental constant of Nature, such that 1/V6 might yet furnish a natural unit of time, R being an asso- ciated natural unit of length. In the non-metals there is a tendency for es to be proportional to a’, so that if the frequency is constant and R also, then r is nearly constant in the non-metals. In the metals of the Li and Be families es is proportional to a, so that 7 is inversely proportional to a’, yet the fundamental frequency is constant. By means of the constitutive pairs of electrons and the special pair in the atom a theory of dielectric capacity is reached in the following way. The electric axes of the pairs in matter free from external electric action are distributed at random as regards their direction. Thus an average direction for a single representative pair when an electric intensity is applied to the matter is that at right angles to the intensity. The average pair of electrons is subjected to a shearing stress by the action of the external electric intensity, so also is the special pair, and also the atom as a whole. If in each case we introduce the corresponding rigidity we derive the associated strain which gives the whole electric displacement caused by the external electric intensity. In this way we obtain the dielectric capacity of any atom. For the atom of a metal in its compounds the result obtained is (6), namely, (N'—1)V=(N—1)B=CB% + DB (N?— 1)/p, Molecular Diameters. 25 where N’ is the index of refraction of the medium formed by a collection of the atoms asa gas such that a gramme-atom occupies volume V, and N is the index of the collection of atoms when reduced to the limiting volume B per gramme- atom, and K the dielectric capacity of the atom =N?. © and D are parameters characteristic of each family of metals, while p is the density of the atom. This formula is verified by the experimental data for the metals of the Li and Be families. For the non-metals the relation deduced is (8) (N’—1)V =(N—1)B= EKB*? v28/p12, where E is an absolute constant and uw is a number having the values 1, 2,3, 4 for the halogens from F to I and for the elements in other families corresponding with these halogens. These results are obtained by assuming that the internal electric fields of atoms are the counterpart of their external fields as ascertained in studies on cohesion. Thus the laws of molecular attraction are strongly confirmed by the present inquiry, while a very definite conception is obtained of the mechanism which is at the basis of radiation. This is as simple as a pendulum, or a galvanometer needle, or a piano string, but the complex of constitutive pairs of electrons associated with it must be very elaborate. From considera- tion of the cohesion of the parts of the atom certain con- ceptions as to the structure of the atom are indicated. The kinematical interpretation of Balmer’s formula is stated more definitely on the supposition that the vibrating special pair of electrons causes certain standing waves in the atom and controls the positions of their nodes. Melbourne, October 1909. Il. Molecular Diameters. By WiLu1AM SUTHERLAND *. A LREADY improvements in the fundamental data make it necessary to revise the list of molecular diameters given in the Phil. Mag. [6] xvii. 1909, Feb. p. 320. That list was derived from one calculated by Jeans (Phil. Mag. [6] viii. 1904, p. 692) who took for N, the number of mole- cules in a cm.® of gas under standard conditions, the value 4x 10°, which was the best obtainable at that date. As he had made noallowance for the effects of cohesional force, I applied that refinement to his calculations. But now from the study of the alpha particle Rutherford finds N=2:77 x 10%, in close agreement with 2°80 x 10!* found by Planck from * Communicated by the Author. 26 Mr. EH. Gold: Relation between Periodic Variations of the thermodynamics of radiation. Accordingly a recaleu- lation of molecular diameters becomes necessary. But it can be carried out very simply because the best source of values is data like that used for air by Jeans from its viscosity, namely, No?=3306 cm.’, where o is the diameter of a molecule of air. If we use for N the value of Rutherford in place of 4x10"%o must be multiplied by the factor (4/2°77)?. On applying this to the list previously given by me with allow- ance for the effect of cohesional force we get the following list of 10% times the molecular diameter in cm. Hy. He. Co. C,Hy. N,. Air, 217 1-92 2-74. 3-31 2:95 2:86 NO. O,. A. CO, N,0. Cl,. 2:59 271 266 2-90 3:33 3:76 III. The Relation between Periodic Variations of Pressure, Temperature, and Wind in the Atmosphere. By Ki. Goi, M.A., Fellow of St. John’s College, Cambridge*. ce purpose of the following paper is to find expressions for the variations in wind-velocity and in temperature which must be associated with a given periodic variation of pressure. The usual and perhaps natural method has been to attempt to find the pressure variation and the wind which must result from a given periodic temperature variation. The problem viewed in this way is inherently so difficult that it has been possible to solve it only on the assumption that the atmosphere was a very thin shell in which vertical variations could be neglected. The results were gratifying inasmuch as they furnished a reasonable explanation of the regularity of the semi-diurnal pressure-wave; but there was a great discrepancy between theoretical and observed phases. For frictionless motion, the phases of the diurnal pressure- and temperature-waves ought, according to the theory, to be the same; even for a large friction constant the difference of phase is only 33° at the equator, and diminishes to zero at the poles+. The observed phase-difference is with few ex- ceptions as much as 100°-150°, the mean value for 18 stations being 130°. For the semi-diurnal waves the phases ought by the theory either to be the same or to differ by 180°. Actually the mean observed difference for 18 stations is 83°. * Communicated by the Director of the Meteorological Office. + Margules, Sitzwngsberichte der Ak. der Wissenschaften in Wien, 1890, 1892, 1898. Pressure, Temperature, and Wind in the Atmosphere. 27 According to the results of the discussion in the following paper, the phase of the diurnal pressure-wave ought to exceed that of the temperature-wave by an angle between 90° and 180°; while the corresponding difference for the semi-diurnal waves is 90°, agreeing closely with the observed difference. This difference of phase arises from the change in the pressure-wave with altitude, a change neglected when the atmosphere is treated as a thin shell. The motion of the air in the pressure-wave is very different from what it would be if the pressure distribution were steady or changed slowly. In a recent paper by Professor Hann *, on the daily variation of wind force on mountain peaks in South India, he applies the formula v= eo shi metres/sec. to obtain an estimate of the semi-diurnal variation in wind velocity from the pressure distribution, by taking for AB the mean gradient from trough to crest in the pressure-wave. Now the formula used applies only if the motion is steady and perpendicular to the gradient, and it cannot be assumed that these con- ditions are fulfilled in the case of the semi-diurnal variation of wind. In fact, as the following investigation shows, the wind is at some parts of the pressure-wave along the gradient, while at others it is perpendicular to it as in ordinary steady motion. ie If the effect of the Harth’s rotation and the vertical motion are neglected, the equations for the motion associated with a pressure variation given by p=pol1+H, sin (nt +21) + BE, sin (2nt + 22.) are ag dp oy dp Pde da’? =O d#® ~~ ay where the axes of 2 and y are E. andS.and &, n are the displacements of air originally at (z, y). If ¢ is north polar distance and } east longitude Ber dahl hs i MER) OP dx Rsind@dr’ dy Rdd’ where R is the Harth’s radius. Therefore if E, E, are * Sitzungsberichte der Akad. der Wissenschaften in Wien, 1908; Met. Zeit, p. 220 (1908). 28 Mr. HE. Gold: Relation between Periodic Variations of independent of A, da? kT ae SE TR Sais “ [ E, cos (nt +24) + 2H, cos (2nt + 24) | d?n kT aaa EL Se *sin (nt +2.) + so sin (2nt+ 2n) | 5 where p=kpT. Hence dey, kT Ds oe es $ [ H, sin (nt +2) + E, sin (2nt + 2r2) | + constt. ot ee 1 Gos (nt +2) + : = 2 cos eat + 2X3) | + constt. On this Abb tee therefore, the bale of the westerly component differs by 180° from that of the presure, 2. e. east OE, oH, Od’ Op are in general positive, the phase of the N. component differs by 90° from that of pressure, 2. é. the N. wind is a maximum when pressure is increasing most rapidly. Butapproximately wind and high pressure occur together. Also since- 2a — ~ 8 e — — e —5 — 5 8 i a a it=—ibip4 107 cm.; w= or SRS EO 3.10-°, AT=8.10"{eme/ seem so that ee = 1:74 x 10* em./sec. If, further, it is assumed that E,« sin’? ¢, E,« sin*®@, agreeing approximately with observation, and if the pressure amplitudes at the equator are taken to be 0°6 mm. and 1°0 mm. for the diurnal and semi-diurnal waves respectively, it follows that u=—I14sin ¢$ sin (nt+A,) — 23 sin? sin (2nt + 2A) v= 28 sin d cos¢ cos (mt +A,) +34 sin? d cos g cos (2nt + 2A¢). At the equator v=0 and the amplitudes of the diurnal and semidiurnal changes in wu are 14 and 23 cm./sec., and are therefore large enough to be observed. At St. Helena * (lat. 16° 8.) the mean yearly semi-diurnal variation u 1s given according to observation, by u= —22 sin (2nt+158°), and in v by v=35 sin (Qnt + 237°). * J.S. Dines. Pressure, Temperature, and Windin the Atmosphere. 29 The phase of the semi-diurnal pressure variation is about 150°, so that the agreement with the simple theory is remarkable. The amplitude of v is, however, much greater than that calculated from the expression given above. The semi-diurnal variation near lat. 48° has been calcu- lated by Hann * from observations on Sintis, Obir, &e. The values given by him are u= —20 sin (2nt+ 177°) v=43 sin (2nt +2469), and here also, while there is a fair agreement in phase with the theoretical value, the amplitude of v is too large. EY, If the effect of the Earth’s rotation be taken into account, while the vertical motion and the friction are still neglected, the equations become du eng el kT oF +2nv cos d= ia Barge cos (nt + rj) + 2H, cos (2nt + 2X2) | ot —2nu cos d= — see _ = oF sin (nt +A,) + oiesin (2nt-+ 2r2) |. The solution of these equations is oF) kT o¢ nR sin d 1—4cos’ EK, + 2 sin d cos } u=Acosat+Bsinat— ; E, LT 2K, + sin ¢ cos ‘Se sin (2nt+ 2X2) Sida g an? 6 2K, cot dé + So v=—Beosat+Asin at+ Aig aed ae: cos (nt +A,) a: LT 2K, cot 6+ nR sin” d where a=2n cos 9. * Met. Zeit. 1903, p. 503. The amplitudes given by Hann are double because he has taken S.-N., W.-E. o¢ cos (2nt + 2A), sin (nt +4) 30 Mr. HE. Gold: Relation between Periodic Variations of The terms in cosat,sinat do not affect the diurnal and semi-diurnal oscillations except in the neighbourhood of lat. 30° and the poles, where a=n and 2n respectively. The effect of the Earth’s rotation on air moving in latitude } is to produce a periodic variation of period 2a/a. Near lat. 30° and at the poles the values of the period are 24 and 12 hours respectively, so that in these cases the effect of a force having a diurnal or semi-diurnal period would be unusually exaggerated. Neglecting these terms and sub- stituting for E,, E, the values taken above, we get 2 u=— 14sin or pars sin (nt+A,) —11°5 (243 cos? d) sin (2Qné + 2A,), v= ioe (nt +21) +57°5 cos @ cos (2nt + 2r,). The Earth’s rotation produces no effect, therefore, on the phases of the semi-diurnal components, but it increases the amplitudes. Thus in spite of the decrease in the amplitude of the semi-diurnal pressure variation as latitude increases, the resulting variation in wind velocity increases. North of lat. 42° the amplitude of v is slightly greater than that of «for the semi-diurnal component. For the diurnal component the amplitude of u is always greater than that of v. The infinite values for u and v at latitude 30° are of course impossible in the actual case. They indicate either that the form we have chosen for E, is inadmissible, or that the neglect of the vertical motion and of friction is no longer permissible. Margules determined the form of E, so that the motion remained finite near latitude 30°, but his value gives too large values to EH, in higher latitudes; his values for latitude 45° are more than double the equatorial values. A simple form for E, which makes wu, v finite for lat. 30° is d(sin@—#sin?d). It is clear that whatever form we determine for E, to satisfy this condition must be such that E, diminishes with the latitude in the neighbourhood of lat. 30°, and E must therefore have its maximum value N. of lat. 30° N. This does not agree with observation, and it follows that the vertical motion or friction, or both, cannot be neglected in the neighbourhood of lat. 30°. Pressure, Temperature, and Wind in the Atmosphere. 31 OR If the motion in the pressure-wave is resisted by a force proportional to the velocity, the equations of motion become du kT ae +2 nv cos ea Rain eo (nt +4) ) + 2K, cos (2nt + 2r2) } OL, ys * sin (nt +24) +9 —— sin (2nt + 2) | a af eee x dt If we write eTE, AD OE, _ Rand = 41, R Od =i, 2ncosd=a, @t+n?+P=X,, 2an=Yj, then for the diurnal terms in uw and v we obtain 2 2 = + { itn + Pa)? uh sin (nt +, + ;) 2 /? aa oe sca Br itt cos (nt +A, -+ 92), where ey — I(a,X,+B;Y;) tan 6,= a,(aY;—nX,) + By (aX, - nY,) tan 0,= + 1(B,X, +41) a,(nY¥;—aX;) +81(nX,—aY})’ those values of 6,, 0, being taken which give to sin @, cos 0 the same signs as the numerator and dencumuotin of these fractions: the radicals in w,, v¥; are then, and only then, to be taken with the + sign. For the seemediareell terms write 2kT ¢ - Oo, he Rene. 1 R O¢ 5 a*+4n?+?=X,, 4an= Y., and we get 2 24.2) % le (aB, ee Pay" \ sin (2nt + 2A. + W,) < Q a { SnPat oe By 8 ad (Qnt + 2n.-+bs) ie 32 Mr. E. Gold: Relation between Periodic Variations of where a —I(a,X_+ B,Y>) ee ile ato(a Yo— 2nX2) + B2(aXy—2nY.) et + I(agY_+ B2X>2) ears a,(2nY,—aXy) +B, (2nX,—aYo)’ and the values of Wy, yr. are chosen in the same manner as those of 6,, 0. The effect of friction is, therefore, to change both the phase and the amplitude of the motion. The amplitude is always diminished if a, 8), %, 8, are positive, 2. e. if the amplitudes of the variations of pressure always increase towards the equator. Trabert* treated //n as a small quantity and obtained approximate results for u, v according to which the amplitude is always increased by friction of this type. ex the equator v=0. The amplitude of the diurnal term 2\2 u, is diminished in the ratio 1: (1+ =) owing to the the aie l : friction, and the phase is increased by tan-" The ampli- 7A ah tude of we is diminished in the ratio 1: (1+ iz) , and the k oh,” At lat. 30° the diurnal terms no longer become infinite. The denominator X,’— Y,” is always a real positive quantity for all latitudes. For the form of E, chosen above phase increased by tan—* a are sin d, a=2 C sin ¢ cos ¢, so that we fla write Q 2 2 2 =F osi ng" ee a sin (nt +), + 41), 1 es — 0 sin @ cos SO) x af erty. 3 cos (nt +A, + A2), ome oe + 2YG COS i Ei nX1(4cos?@—1) ” 22 pape l(2X, cos @ + Yj) nY,(1—4 cos’ d) * Met. Zeit. 1903, pp. 554, 555, note. The error appears to arise. through the omission of the term 2/w sin p W' in the equation for S. Pressure, Temperature, and Wind in the Atmosphere. 33 Therefore from the equator to lat. 30°, @, increases from 180° + tan- at to 270°, and @, from tan— ee pe ee 2 52) t0 BOF From lat. a to the pole both 0, and @, increase at first, and after reaching a maximum value diminish very slightly indeed to their values at the pole, which are approximately 90° stants j in excess of their values at lat. 30°. At lat. 30° the amplitudes of w, and x, are equal, and each is 12 5 pom: /sec. The maximum value of the ratio »,/2, is 2 2/n2)3 Z ete 41+ (1+P/n*) y /2(14 aa If E,=F sin? ¢ as before, and the amplitude is 1 mm. at the equator, Up = n (2nt-+2d2+W,), 2kTn¥ ane [es p) +P La: fs 10kTnF sin?’ d cos 71 + 97?/100n? se hy R po ie ik —1{P +4n? Cit cos d)} n(3 cos? @—2) (4n? sin? 6 +1)’ 1{ 31? + 4n?(7 +3 cos? 6)} Bn Ye On (P + 20n? sin?) os (2nt+2A,+ Wy.) > tan = Thus yy; increases from 180+ tan 270°, when cos? $=2/3, and afterwards changes very little ; vr. increases from the equator to the pole, but is always less than 90°. The ratio v/v. increases from zero at the equator to a value slightly greater than unity when cos’ ¢=2/3, 7. e. in lat. 55°. Afterwards it diminishes very little and is always greater than unity The following table * gives the values of w, v, &e. in em./sec. for different latitudes, and for /=0, $n, n:— at the equator to * Cf. Margules, Sitzungsber iohte der k. Ak. der Wissenschaften Wien, Bd. cii. Abth. ii. Dee. 18038, pp. 1385, 1386. Phil. Mag. 8. 6. Vol. 19. No. 109. Jan. 1910. D 34 Mr. E. Gold: Relation between Periodic Variations of Tationde cs. o, 15% (30% 45° 60° [5c uaa a eae 20 14, eon u, for, l=in 12°5 Ly | 24 18 11 55 0 cee 12 its. 42. 1-0 180° 180° .... 360° 360° 360° 360° @, for I=3n 2072-280: 270° 31223292 33523379 I=n 225° «= 2440 9709-2979 318° 320° ~—- 320 08 oe Dk eID fe, he 8 12 59 podtor stati )/ 00) AIBC nek) 0 RO Fee as 8 2 i 7 385 H =0 23 25 32 40 49 55 57 au, for,l=in 22 24 28 31 23 <6 0 Jeet: 20°5 21 23 2) "136 3'8 0 s=0 180° 180° 180° 180° 180° 180°=saeas Wy, for, ¢=gn 194° 200° 225° 251° 272° - 273% eee (l= 207° 219° 241° '§259° | 272° = 27a? aes 1=0 y) 15 29 41 50 3-55 57 g,) for, f= An 0 14 25°5 3 25 76 0 l=n 0 12 20 21 13 39 0 /¢=0 0° 0° 0° 0° 0° 0° Fae wb, for =n 20° 21° 28° 40° 61° 81° 1SeR% ‘=n 37° 39° aye 59° 73° = 83° "Cae Hann * gives for the variation of the wind components at Sintis (lat. 47° N.) uy= 23 sin (98°+ nt), v1, =14 sin (124°+nt), U,=15 sin (829°42nt), ve=14 sin (240°+4 2nt). The value of A, is nearly 180°, while that of 2), is 123°. The values of u, &. corresponding to /=4n for lat. 45° are as follows:— uy=18 sin(132°+nt), v,=18cos(307°+nt) =18 sin (37°+nt), and Uy=31 sin (374°+4 2nt), vg=31 cos (163° + 2nt) = 31 sin (253°+ 2nt). * Loc. et. Pressure, Temperature, and Wind in the Atmosphere. 35 The agreement with the observed results is only approxi- mate, but it must be remembered that the variations are affected by vertical convection of air, and this cannot be altogether eliminated by taking the difference between the variations in oppositely directed components, because the winds from opposite directions do not occur with the same frequency and strength, nor under the same conditions of temperature and insolation. Vertical convection would tend to diminish the difference of phase between u and v, and the fact that this is the case in the above example may be taken as evidence that a considerable convection effect exists in the wind records from mountain peaks. EY, In the preceding sections the equation of continuity has been neglected, the assumption being tacitly made that the temperature variation and perhaps the small vertical motion would be such as to satisfy it. In this section it will be shown that if account be taken of the observed change in the pressure variation with change of altitude, the temperature variation which must be associated with the pressure varia- tion according to the hydrodynamical equations, is in fair agreement with the observed temperature variation. It will also be shown that the vertical velocity is small, of the order 10-?, compared with the horizontal velocity, and that between the equator and lat. 18° the maximum upward velocity in the semi-diurnal wave occurs before the passage of the trough of the pressure-wave, while between lat. 18° and the pole the maximum upward velocity occurs after the trough of the pressure-wave has passed. Margules* has dealt very fully with the oscillations of the atmosphere, and has deduced the pressure-waves resulting from given temperature-waves. ‘The inherent difficulties of the problem are, however, so great that it is impossible to solve the whole system of equations in this way, and conse- quently the vertical velocity and the equation for the vertical motion were neglected by him. But if the pressure variation is taken as given by observation, the whole system of equa- tions can be solved by approximation, and it is by this method that the results of the present section are reached. The equations for the small oscillations of the atmosphere, * Sitzungsberichte Akad. Wiss. Wien, 1890, 1892, 1893, Bd. 99, 101, 102. ~ D2 36 Mr. E. Gold: Relation between Periodic Variations of on the rotating Earth, expressed in polar coordinates, are, neglecting terms of the second order in u, v, w *, di 5 kT, O€ BEE ; =— pe te sin p+2nv cos b Ser i) as 2nu cos b = bee : = dw Oc dp sin oti where the Harth’s axis is the axis of reference, @ is polar distance, A is HE. longitude, u,v, w are Velocities in the diréction of increasing A, ¢, 7, p,¢is the pressure variation, Lit 4.4), temperature’ | 3 p, and T, are the pressure and temperature i in the undisturbed state, n is the angular velocity of the earth about its axis, k is the constant in the gas equation p=fpT. With these equations must be taken the equation of con- tinuity d(e—T) P| 1 oT Ow we "7 ve a kT, By, T 5) ann oe es ee ie) The observed value of ¢ for the semi-diurnal wave may be represented with fair accuracy by e= Hsin’ $ sin (2n¢+2\ + ce?) =H sin® ¢ sin y, say. Here z is height above M.S.L. and the term ce Fe..18 introduced to represent the diminution!fof phase with height. * It is fair to neglect pure viscosity terms of the form Les because (see enfra) the greatest of these would be & 2Py, vie tbeeee to the terms retained the ratio 1077" neatly. The*value of k/o found by experiment is 0°18 for air at 0° CO, Pressure, Temperature, and Wind in the Atmosphere. 37 This diminution is probably due in part to the greater re- sistance to motion near the earth’s surface ; it may be due in part also to a change in the phase of the semi-diurnal temperature variation in the free atmosphere. The value of E is taken to be independent of the height because the amplitude of the pressure variation p,e is found from observation to be nearly proportional to p,. With this value of ¢ the quantities u, v, w, 7 must be given by u=A,cosy+Agsin y, v= B,cos y+ B, sin x, w=C,cosy+C, sin x, T=T7, COS Y¥+7T, SiN y. Substitution in the equations of motion leads to the follow- ing equations for Ay, B,, &c. —2nA;+2n cos 6B, + 2x sin GC, =9, : 2kT if 2nAy+2n cos 6B + 2n sin ¢Cy= — r ‘E sin’ ¢, ad Seg —2n cos A,—2nB, = ——— Esin’ ¢ cos d, / fad —2n cos PA; + 2nB, ==) —2nsin dA, - —22nC,=979, —2nsin dA; +2nC,=97,+ EkT\cBe-* sin’ ¢, whence gt, = —EKTcBe- * sin? ¢, ET, . gt) = sin 6[2+3 cos’ ¢]. If the diminution of phase is taken to be 8° for 300 m. (Eiffel Tower) and 40° for 3000 m. (Sonnblick), c=0°84, 8=6.10-* cm.-! nearly. Also r=6°3.108 cm. nearly, so that 7, is large compared with 7, at the earth’s surface. The phase of the semi-diurnal temperature variation 1s therefore 90° less than that of the semi-diurnal pressure variation; the difference diminishes with increasing height and with ¢. 38 Mr. E. Gold: Relation between Periodic Variations of The following values for the phases of the semi-diurnal pressure and temperature variations have been calculated from published observations :— Kew. Valencia. Ben Nevis Ben Nevis Equator Pic du Midi (1343 m.). (Summer). (Ocean). (2860 m.). Pressure ......... 146° 140° 130° 116° 156° 124° Temperature .... 42° 49° 50° 86° 84° 80° Difference ...... 104° 98° 80° 30° 72° 44° The observations agree, therefore, as well as could be expected with the theoretical result. [ Note.—The value of 8° for the retardation of phase for the Eiffel Tower was taken from a paper by Jaerisch, Met. Zert. 1967, and appears. to have been calculated by Hann from the earliest observations. Later results give for the retardation only 2°, see Angot, Annales du Bureau Météorologique, 1894, Part I. This of course affects the values of ¢, 8. I have, therefore, taken the results given by Hann (Lehrbuch, p. 602) for high-level stations, and assumed the phase at sea-level corresponding to them to be 150°. I have also calculated for the same period (summer) the difference of phase between Ben Nevis and Fort William. The results are plotted in the accompanying diagram, in which the curve Connexion between the decrease of phase in the semi-diurnal pressure wave and the height. SONNBLICH 30) bén NEVIS 2 g, WEND MUNICH 1000 e000 3000 272 STE/IV FETARDATION OF PHASE IN SEMI OIURNAL WAVE represents the equation @6=c(1—e-§:) for c=15, B=2.10-§ The temperature amplitudes at sea-level would be reduced owing to this change in the values of c, 8 in the ratio of 3 to5. ‘The general con- clusions will not, however, be affected. The decrease in the semi- diurnal temperature amplitude with increase of height will be slower than that given in the table below. Pressure, Temperature, and Wind in the Atmosphere. 39 The following table, taken mainly from Hann, gives the amplitudes a, and phases A, of the semi-diurnal temperature- and pressure-waves at a number of places chiefly in low latitudes :— ! Barometer. Temperature. Place. | Ms | WA |) a igh Books Equatorial | 7 ae ae _ 9 a } a 158 ob. 82 97 | 0-23 Gi Batavia ......s+0- | 159 | 95 68 | 0-83 91 Trevandrum...... 158 FOS ||. 85 0-75 73 Manila ............ || 159 89 68 0°75 91 NIGGA) ce. dwsinss. | yen amet ga 76 0°98 82 POAT GSO ......... | 157 SO. 41) ~ 89 1°38 68 SPITE Satna 00s. 165 "84 61 0-74 104 Ascension ......... | 158 “41 87 0-86 71 *St. Helena......... 153 ‘74 43 0°66 110 Port au Prince... 163 ‘88 80 110 83 Am Gabun ...... 157 1:05 71 0:87 86 Singapore ......... | 156 “98 86 1:05 70 Bogmma,..:;........ 154 84 72 0:56 82 Dar-es-Salam ... | 156 ‘91 90 0-95 66 *Bismarckburg ... 160 "S4 BEY t- 14g 7 Allahabad ......... / 153 ‘89 of | 2S 96 Goalpara ......... 150 1:03 58 | 0°88 92 Greenwich ...... 143 "23 58 0°58 85 Elne Hill ......... es a i aa 95 0 | | 76 * Altitude above 500 m. The mean difference of phase is 83°, and is therefore in full agreement with the theoretical result. The temperature amplitude corresponding to T,;=280° A (=T° C.) is, according to the theory, 1°5 e~* sin @ degrees C. The mean value of a’, for the equatorial stations is 0°°9 C., which is rather less than the theoretical value*. Many of the stations, however, are island or coast stations, and it is probable that over the ocean the change in the phase of the pressure variation with height would be less than over the land. The value given by the formula for Kew and Valencia is 0°4 C., and this is also the observed value. For Ben Nevis the theoretical value is 0°11 C., and the observed value is 0°18 C. The difference of phase ought to dimish only slowly with the height ; for lat. 45° the decrease would only be 45° at a height of 10 km. The value of the temperature amplitude corresponding to * See, however, note above. 40 Mr. E. Gold: Relation between Periodic Variations of the assumed pressure variation is as follows for different latitudes and heights, if the vertical temperature gradient is 6° C, per km. Altitude | 0 1 Fite Bi) ve 5 km. | Ho | fo) fo) fo) fo) fo) | ees 172 | 092 048 | 025 | 013 | 00750 a0 152 | 62] 43} -22 | +12 | seep se 104) 55 29 | 15 | 08 | 042 ae O55 | 29! 16 | 08 | -O4 025 | ae |018 | -10'| 05 | 03 | 015] | -007 pes 0:025/ 014, -007| -003' -002| — -001 iene 0. Fe Bee CO ag rt ae Oi The amplitudes for the whole-day wave are given in the | | | | | | } e Or Oo 30° | 45° 60° | 15° | 90° ere eeeseeees ee reeetesere eee seseesese fe eewveoeeos | following table (vide infra p. 47) :— 0) 1 2 Otte ae 5 km, 206 1129 0x0 | O48 | O27 | O15C. 190 |120 | “73 | 44.| -95 | +14 146 |092 ) 56 | 33 | 19 | <1 | 093 |059 | 36 | 21] 13 |) -o7 | 043 |027 | -17 | 10] 06 | 03 O11 | 007 | 05 | -03.| 02 | -O1 0 0 ) | 0 0 0 Returning to the original equations, we find after substi- tuting for u,v, w, tT in the continuity equation, 2n|[ E sin? 6—7.| + (2/r—y—t,)C, + ot —BYC, and —2nt, + (2/r—y—t) Cy + 1 O(Bising) BL 101, my rsin fore) 1 r sin ~ 2 ~ o(B, sing) B,10T, 2A, o6 OC, ie T a¢ “+ PyrC, r Ti Og r sind esi ? rsin Pressure, Temperature, and Wind in the Atmosphere. Al where Pi ig Sent, See ah But C,=A, sin 6=B, tan ¢, C,=B, tan@d—3 Fsing, A,=—B, sec ¢+3 F sin’ ¢, where F = EKT,/nr. ‘One Also 1/r is small compared with Y or c8 and n’r is small compared with g, while £T,/r is about 1°3. Therefore, neglecting the small terms and eliminating Ay, ©;, As, Cz, we obtain OB, 5 WW . tan ee —B, tan d(y+t,) +3 Fy sin d—y B, tan 6=9, tan oes —B, tan O(y +t) -—3 FB ysin 6+ AWB, tan d=0. The solution of these equations is B,=3 Fceos $+ Tye”? [P cos W+Q sin |, B= Tye’ | —Q cosp+Psinw], where Z = : ydz. 0 But the vertical velocity must vanish at the earth’s surface, sO that C= Ca for 2= ). Therefore P=Q=0, and A, =0, Aj= —tF(2 + 2S cos" d), B,=5 F cos ¢, Bae 0. C.=0, ¢3=0 The motion is therefore exactly the same as that found in Section IT. In order to proceed to a further approximation and to obtain values for Cj, Cy, put B, — By/+3F Cos d, A, = A,’—4F(2+3 cos’ ¢), so that A, = —B,'/sec¢d, C, = B,' tan o. 42 Mr. EH. Gold: Relation between Periodic Variations of Then the equations for B,’, B, are tan oe — B,' tan 6(y +t) —BWB, tan d+ 2nH sin’ } — = F cosec (4 sin? 6 +5)=0, and B tan ox —B, tan d(y +t,) + BW By tan d+ me By sin? d6=0. Put Dy es B, = BIT, e. Then OL ot : 1 ’ a4 —MBw+ ore 72a sin’ d— ae cosec (4 sin? +5) ]=0, a + LOW + a Aw e~7 sin? d cos 6=0, whence b/ aM oh 12 Aneel aE M a2 2 p—Bz__ -Z— =e aE )+ c? Be PcBe 0, where Ek cos aS P=— sarang sin? 6+ 9) + ~7,—-sin® ¢ cos o. If kT, is taken to be independent a Z ae solution of the equations can be putinto a form that admits of calculation. In fact Z r ee Berp? ce y° eae (y+ 8) (y+ 28)(y+38) * ee 1 P Sa Ale eee Bp? L= amet ee ie! Gy 58 (y 428) ae 2Qnlik —Zain2 08 t e~*sin* d cos Here Q and R are constants to be determined from the con- dition L=M=0 when z=0. Since L=0 at z=0, it follows that Oi is always negative near the surface, and therefore M is always negative near the surface. Consequently B, and ©, are always negative near the surface. OL But 02 : 2nH sin’ ¢ = 9, cosec f(4 sin? +9), which gives 6=72° nearly. Pressure, Temperature, and Wind in the Atmosphere. 43 Thus N. of lat. 18°, L is positive near the surface, while between the equator and lat. 18° L is negative. Conse- quently B,, C, change sign and become positive N. of lat. 18°. Thus between the equator and lat. 18° the phase of the upward motion differs from that of the pressure variation by an angle between 180° and 270°, while N. of lat. 18° the difference of phase lies between 90° and 180°. If _— sin? ¢ cos $¢=D and P/yD=gq and the two series in brackets in the expressions for L, M are denoted by X, Y respectively, then, using the condition L=M=0 for z=0, we find L=DJ[cos(e—) —e-?| +qD[ — Xocos (e—W) — Yosin (ec— pp) + Xe“? ], M=—Dsin (ec—) + gD[ Xo sin (c—p) — Yo cos (c—p) + Ye“? } where Xo, Yo are the values of X, Y at z=0. Here D and gq depend on the latitude but not on the height, while their coefficients depend on the height but not on the lnmiudeme it G=6.10-5, c=0°84, kT, =8.108, the values of X; Y, c—, e~7 at various heights are as follows :— Bethe in Pier aoe | ekertn dk, ses dais | 7480 | 6297 | 1000 | 0 Mk 2 se | 9213 | 3721 | 887 | 21° 457 Bee cae | 9760 | 2081 | 7:87 | 33° 38' PePAUN | 9927 | 1149 | 698 | 40°10’ BRP Vevey 9-978 | 0632 | 619 | 43° 46! eed 9993 | 0347 | 549 | 45° 44 The values of g and DT, tan@ in different latitudes are given in the table : (Se i 15° 30° 45° 60° OE 116 1:07 0°68 —0°61 —6:97 DO TANG 65. c.ces 1°57 1:42 1:02 0°56 0°20 The following table gives for different latitudes and heights the amplitude of the upward motion in mm./sec., and the | excess of phase above that of the variation of pressure. — ss - . 44 Mr. E. Gold: Relation between Periodic Variations of Amplitude a in mm./sec. and Excess of Phase 6 for w *, Ho Reptud 0) of map eae 1 2 3 4 5km. Fpl O63, 103 139 «165 rae Equator | 97077" 194° 202° 2072 += 2092 ~—210° car ean G58. 083,12 148 alee Tat. 15° 4 5 192° 200° +2042 -—«208°—«207° | FAME. O41 068 O88 104 1:22 RSA ai 195° 190° «193° s198° 1928 025 044 O59 O74 0:89 (oe 164° 164° «160° ~=—s«157° ~~ —s«158° 022 045 ~ 069. O94:. 1:33 eG 123° «124° = «420°—s«d1:18°~—s«d1169 In the original equations terms of the form oe were r ap neglected. This:was quite fair in the first approximation, but these terms are of the order 74, of nQ,, nC, and may therefore produce some modification in the values found for Ci, Co. The pressure variation in the whole-day wave is not so regular and well defined as for the half-day wave. The principal characteristics are a general decrease in the ampli- tude from equator to pole and from the earth’s surface upwards. The decrease upwards continues until the amplitude vanishes. Afterwards it reappears with a change of phase of 180° and at first increases. The variation may with tolerable approximation be represented by e= Hsin? o( fe-” —e-™) sin (nt +X) =H’ sin? 6 sin (nt +X). If f=1°6, l=4.10-§, m=4.10-‘, the amplitude vanishes at about 1300 m., and afterwards increases until at 7000 m. it is 1:5 times the surface value. Above this height it diminishes again. * The corresponding values for c=1°5, B=2.10~° are as follows :— Je sede a Nae 1 2 3 4 5 km, rp en ak 046 O97 128 169 209 Borges alkenes 192°). 199°). 205°. “39119 eae Pepa a tae 042 O79 115°" 151 187 Tat. 10°] 5 190°. 197°): 208°,,, 208°, | eae pone yal, 030 O57 O84 109 1°84 EARS Ns 1g0° 186°: 191" 195° 1990 Uae 020 039 059 O78 0:98 Liat gO ria LN 153° «157° ~=—«159° «162° «1630 | 021 048 068 096 1:98 oe ec HIBS) APSO) 129%.) 15a Wi ie Pressure, Temperature, and Wind in the Atmosphere. 45 Writing as before u = A, cos x+ A, sin yx, v = B,cosy+B, sin y, w= C, cos ¥+ C2 sinx, F = 7, COSY + Tz Sin X, where y =nt+X, we obtain the following equations for A, &c. —nA,+2nsin d Co+2n cos d B,=0, —nA,+2nsin d C,+2ncos dé B} = —nF' sin ¢, —2ncos od A, —nB, = —2nlk" sin d cos ¢, —2ncos d Ay +nB, ss) —2nsind Av—nCy —kT, sin? as —2nsin d A, +nC, =9T1, Where iT nr From these we find oF 2 a e A,=— = 97,sIin d, an B,=— or sin @ cos 2 3n 1 hy rahe 1 C= — 377 (1—4 Cos >), | 2 : Ce 2kT, sin’ POH! | =— =~ 1¥F’ sin ‘i : A; 3 978 sin@ +E’ sin (144 cos?) + — a ae B= = 9 sin ¢ cos d—2 I’ sin d cos $ (4 cos? 6—2) ii Fs kT, sin® d cos oe , Q= 3" = gro(1— —4 cos? d) —2 1" sin? 6 (1+ 4 cos® d) OH ae fs STA = 2 ea. "4 f iT, sin? 6(1 — 4 cos? ¢) P| The continuity equation gives By a Je OC, 1 o(B sing) | Ja) on’ > ney +(— Te Mae Bie snd 0o¢ err OC, 1 1 0(Bzsing) _ Bit ke dc orsing ad rsin =—()s 2 760) +{ yh )Cs+ yr 46 Mr, E. Gold: Relation between Periodic Variations of But 1/r is negligible compared with y and n is negligible compared with yg/n, so that a8 — (y+4)O:, = 9, whence ©, =C,=0 to the first approximation since Cj =C,=0 at the earth’s surface. roe Thus 7, is large compared with 7, and is given approxi- mately by OL’ fos Therefore the difference of phase between the variations of temperature and pressure is 180° nearly. The following table gives the amplitudes and phases of the diurnal varia- tions of temperature and pressure at different places. 9T_ = kT, sin? $ | | Pressure. Temperature. Place | | AN. ay ak! ay’. | A,—A,'. | : emo cope . C. 3 Ocean os icseenecines | 308) ee OD 243 0-72 115 avaviapohtsieaecs 28) 564i "62 232 2°8 153 Trevandrum ...... | 2005.2) 39 240 2°8 140 OVParavlay yen tiecet ee es | "52 232 Pa) 149 IMaGYAS ...0h050250. S00)" “DO! We HM eer 3°0 123 SamdOS6 | .2.cs.00: | of 3 ain A599 4:0 145 SUT: pecans. | 23) 4 "34 256 aleve 127 Ascension ......... 19 28 251 2-9 128 St. Helena ...... i 322 | Mages 224 1-4 98 Port-au-Prince...) 24 | ‘Lh i ao 4:0 144 Am Gabun ...... i Se | Tio: yeh aaa 27 128 Singapore ......... | 26 "Oo. jl 240 32 146 Woands <.s.cc22. he. | 8 "85 =| = =229 1:4 139 Dar-es-Salam .../ 345 8 |} «~~ 240 | 30 105 Bismarckburg ...| 27 DD» oy Pees 3°2 145 Allahabad......... y 337 oh | 232. 5°4 105 Goalpara ......... | 347 “BO i) grado 3:2 128 Greenwich ...... | 336 glo MAY | Pg op) 2°8 109 Blue Hall yi... | se gay ER i 128 BGStOM fens. 2--- i 143 | The mean difference of phase is 130°, and the agreement is perhaps as good as could be expected, considering the variations in the diurnal pressure-wave which have been neglected. Pressure, Temperature, and Wind in the Atmosphere. 47 If there is a retardation of phase * with increasing height similar to that for the semi-diurnal pressure wave, then 7, is no longer necessarily negligible compared with 7,, and in fact gt = —H’ sin? kT, c'B’e°*, , EK’ In this case the difference of phase between the pressure and the temperature variation is 180°—0@, where ‘| OE rely If c’, 8’ have the same values as for the semi-diurnal wave, tan @=5/6 or @=40° at the earth’s surface, so that the phase-difference would be 140°. The agreement of this value with the mean observed value is sufficiently close to suggest that the retardation of phase with increasing height in the semi-diurnal wave is due to frictional influences, and that it is present also in the diurnal wave in the free atmosphere ; and further, that the semi- diurnal variation of temperature in the upper air is a con- sequence of the semi-diurnal pressure variation. If the mean amplitude of the pressure variation at the equator is taken as 0°6 mm., the value of E is unity, and with the values of e, 7, m, taken above, the values for the temperature variation given in the table on p. 40 have been found. The mean temperature amplitude for the places given above is 2°8C., which is rather greater than the theoretical value. According to Margules’ solution, on the assumption of a given temperature variation and negligible vertical changes, the phases of the variations of pressure and temperature ought to agree. If account is taken of friction, the phases differ slightly, the maximum difference for a large friction constant being 33°7. This is so far from agreement with the observed difference of phase that it appears certain that although the vertical motion may be small compared with the horizontal motion, the atmosphere cannot be treated as an infinitely thin shell in dealing with the diurnal variation tan d= —H'c’B’ e-F' * The results of observations at higher levels before the reversal of phase is reached give conflicting results. Munich and Peissenberg show an acceleration of phase, the Eiffel Tower a retardation. + Margules, Srtz, Ber, 1893, p. 1386. 48 Variations of Pressure, Temperature, and Wind. of the barometer. The changes in the variation with in- creasing height must be taken into account. In order to proceed to a further approximation put Ob’ gt,=kT, sin’ d ae oh Te i Then ' : 1+4 cos’? ¢@ LO ee gto =2F'’n sin oF Uae and this will be approximately correct except in the neigh- bourhood of lat. 30°, where the effect of viscosity will add to the denominator a term which will remain finite while 1—4 cos? decreases to zero. The value of C;, is still zero to this order of approximation and consequently 7; is also zero. But A,= —F’sin b(1 +4 cos” )/(1—4 cos’ ), B,=4F" sin ¢ coso/{(1—4 cos’). Thus near the earth’s surface A, is negative in the equatoriai regions and positive between lat. 30° and the pole: B, is positive near the equator, negative between lat. 30° and the pole. The nature of the horizontal motion in the two cases is shown in the accompanying diagram. Is Motion in Diurnal Pressure-Wave. 1, Lat. 0-80°. 2. Lat. 30°-90°. The arrows fly with the wind. Pee eS Te Tables of ber and bei and ker and:kei Functions. 49 The roughness of the approximation to the diurnal pressure variation makes it useless to proceed further in the deter- mination of the upward motion. Indeed the results for the horizontal motion will be affected so much by convection that it is doubtful if they will show any agreement with the results of observations taken near the earth’s surface. The phase of the motion changes with height in the same way as that of the pressure variation. IV. Tables of the ber and bei and ker and kei Functions, with further Formule for their Computation. By HaroLp G. SavipeE*. pena after the publication of Dr. A. Russell’s paper on the ber and bei and allied functions (Phil. Mag. April 1909), Professor Lees suggested that tables of the ker and kei functions, as defined by Dr. Russell, and tables of composite functions would be of great use to electricians and others; and the subjoined tables are the outcome of this suggestion. Other suggestions have been given in the course of the work by Professor Lees and Dr. Russell, for which the author desires to take this opportunity of expressing his thanks. Definitions. The functions tabulated may be defined as follows :— ber e+e beia = Iy(a Wt) t= Jy(uavs), bera—ctbeia = Jy)(vVvt) =I1,(urvs), kerv+cekeiz = Ko(a /t) = Yo(ur 4/2), kera—ckeia = Yj(¢ Vt) = K,(uvv 2), X(wv) = ber? a+ bei? a, V(@) = ber? z+ bei? 2, Z(#@) = ber x ber’ a + bei x bei’ a, W (2) = ber z bei'x —bei z ber’ a. * Communicated by the Physical Society : read November 12, 1909. + See Gray and Matthews, ‘ Bessel’s Functions.’ J and Y are the Bessel functions of the first and second kind respectively, and I and K the same functions for unreal values of the argument. Dr. Russell’s function Y(2) is here called V(v). ber’ x and bei'z stand for the differ- ential coefficients of ber z and beiz with respect to «. Phil. Mag. 8. 6. Vol. 19. No. 109. Jan. 1910. H 50 Mr. H. G. Savidge: Tables of the X,(x), V;(z), etc. are the corresponding ker and kei forms; MF X,(x) = ker? 2+ kei? z, etc. S(a) = ber’ x ker’ x+bei’ x kei’ z, T(#) = bei’ z ker’ #—ber' x kei’ x. The series a, 8, «', 8’ are also tabulated. They are useful for interpolation, and 8 appears in Dr. Russell’s formula, (107) (there called e). They may be defined by the equations : ber v2 = bei 2 = ee 21x kero =a / Zo cos 8’, kein =a/ me ‘sin 8’, where wz is not less than 6, and and fe List 2a Mee see Te ae ey ae B= i ee eae rr eee / 2 8 &fbe Wee 38 202% tae eg 2 Ene a J 2 8/ 2a BBA 2a" 12828) ] 1 25 bad ea ar ager 8 TS 75, 16a®” 3847 da! When 2 is small, we have a = tlog {27rxX(2)}, 8 = arctan (bei z/ber z), 5 log {2229 | , '= arctan (kei 2/ker #). a’ = tole Further Formule (« greater than 5). Since ber @+ibeia=I)(x/t), we can find the values of ber # and bei by putting w/e for a in 1? 3? 1”. OE: etre x. (eae B. = t a (Gray & Matthews: Bessel’s Functions, p. 68.) Making this substitution, noticing that tHh{/24 724+1V/2— V9}, ber and bei and ker and kei Functions. 51 and equating the real and imaginary parts we obtain erlV2 ( ber = 57 — | (V24+V2 e037 +/2— V2; V7 sin Sr +(Vv24V2 V2sinF5 —V2— f9 V 2eos75)s t, ee fon V5 EVE er= 2,/ ine ae == sin F cos 7) = (W24 V2 eos Fa +V2—V/2sin 7) s shy where ea i, Pere PR es ew er 7 822 = [3(8x)*/2 |4(8a)* 5(8x)/2 il Bee La 8 oe OP —BVo0" 2x? B(82yeV2 = [5(8ayPV2 These expressions reduce to ber a=f cos ( —$), | bei «=/ sin — ), q f V2 ? } where eee J/ dre SY2x 2562? 20480223 512?z**" 12—5,/2 2— 2—5¢ isn g=V5-1+ 1 bf 2 20/2 13 | 435./2—560 or Bey 2562" 819224 : From the ‘ig we obtain the following formule : e i 33 3594 i j= — + —— = ai ee =o =e) Qarx {1 An/2u O40" 956VW 9x3 12872*"*” $ aV2 3 9 75 4950 i, e Lat MMe WD, 523 V(2)= ae AV 2a Y 64a? T 256V 203 12872" } i es D2 Ae lS A c 630 () =o va 8@ 64/20? 512a°* 198%/ og } v3 ; e* 1 1 2 39 150 Wa) = Ft 82’ Ghv 2a? st st ga } ber’ «= ii COS @ iam J/ 21x e” bei’ z= ———sin a, 52 Mr. H. G. Savidge: Tables of the where eG, a 21 eae URAL er ee) ere abe re 3) 3 21 ea gees pe pee a 75+ 8 8/oa 16a * 198 9n% There are similar formule throughout for the ker and kei and composite functions: X=" j1— fe, 29 38 ee 1 RENE Anda 640° ' 956/23 128?! and similar formule for V,(2), Z,(z), and W,(«). Z,(x) and W,(#) may be found by means of the tables for Z(x} and W(«) and the formule :— 1 3 Z(a)Zy (a) =-5,(1 — pa) i dt W(@)Wi(e) =— (1455). In these’two series the coefficient of #-* is zero. They can be used when z is not less than § : for smaller values of the argument ithese functions are tabulated S(#) = esin e,/2—d cos a,/2 T(w) = — dsina,/2—c cos #,/2 where wal 3 i 75 15435 8 J2a? 512 Vat 2567 228 By aay 9 nae 2475 Dp AM ORae Lee ae and these may be written S(@)=—g cos T(a)=—g siny, ber and bei and ker and kei Functions. Ac where 1 Zt ome haar 207" por J/2 + zi via IS pal I at AN2e 64722? 2560 V22x° Vy ber a. | bei a. ker a. | kei x. otek ay aes / oT | ae x 1 jj+/ 10 x9844 +) 10 X2496 +) 10 x2867 ||—-| 10 %x4950 2 ||4| 107!x7-517 ||+) 10° 'x9-723 ||—| 107? x4166 ||—| 1077 x2-024 3 ||—| 10? 2-214 |/+) 1-938 |—| 107? x6-703 ||-| 10°? x5-112 4 ||— 2-568 ||-+/ 2-293 || 107” x3:618 | +| 107° 2-198 5 | 6-230 ||+ 107'x1-160 |—| 107? x1-151 | +] 107? x1-119 6 || 8-858 ||— 7335 |—| 107* x6531 ||+| 107° x7-216 7 ||- 3633 ||—| 10 2-124 ||+) 10-3 x1-922 ||+| 10°* x2-700 8 \l4}10 2-097 |—| 10 3502 |+) 107? x1-486 ||+| 10°* x3-696 9}4110 7-394 ||-| 10 x2-471 |+| 107* x6372 ||-| 10°* x3-192 10 ||4| 102 x1388 ||+) 10 x5637 |+! 107* x1-295 ||-| 10°* x3075 11 |]4| 102 x1-330 ||+) 102 x2:572 || 107° «4-779 |-| 107° «1-495 12 |_| 102 x1-285 +! 102 x5-470 |—| 107° xe308 |_| 107° x3'899 13 |_| 102 x88e7 ||+| 102 x6-466 ||| 107° x3-474 I+ 10” x5-387 14 ||| 10° x2:131 || 102 x1-609 ||—| 107° x1-088 ||+! 10°” x1-268 15 ||—| 10° x 2-967 Hh aor |sea-ges I 107® x1-514 {14} 107° 7-963 16 || 102 x6-595 ||—| 10° x8-191 |+/ 107° x2466 ||+) 107° x 2-895 17 4) 10° 9-484 |—| 10! 1-309 Hl 107° 1-797 || + 107 x 2-861 18 |+/ 10! x3-096 ||—| 10° 7-454 |+/ 10‘ x7-438 ||-| 10° x4:555 19 | +) 10! x5-625 |/+| 10' x2804 + 10’ x1-293 ||-| 107" «3-982 20 ||+| 10! x4-749 +) 10° x1-148 ||—| 107° x7-715 | 107* 1-859 21 ||| 10! x7-616 ||+| 10° 2-337 ||—| 107° x8-636 ||—| 107° x4-388 22 |i—| 10° x4:155 |i+| 10° x2-539 |- 107 x 4535 + 107" x 1-097 28 || 10° x9986 |—| 10° x1527 || 10” x1820 | +/ 107° x1-824 24 ||| 10° 1-242 ||—| 10° x1-460 |+] 10 8786 |+! 10°° «1-083 95 |4+| 10° 9798 ||| 10° 3809 |+/ 10> x3-723 |+| 107° x3-703 96 ||+| 10° x4-936 I 10° x5-744 ||+ 107” 2582 + 107 x1912 2% ||+| 10" x1-489 ||—/ 10° x2308 |+) 10° x9918 |—| 107 x7-207 98 ||+| 107 x2-553 ||-+) 107 x1378 ||+| 10° 'x1-367 |-| 10°” x5-791 29 ||4+| 107 » 1825 ||+| 107 x5-695 |—| 107" x1-820 |—| 107° x2:568 80 |—| 10" x4612 +) 10° x1-100 | — 10-1294 ||| 107" 5-290 ss a Wick at RE Ce eee | eve os : oo | | ee) 0 | 0 54 Mr. H. G. Savidge: Tables of the ory a. B. —a'. — 6’. 1 9343 "2483 ‘7844 1:0458 2 1:4717 9126 14560 17738 3 2°1362 16846 21499 2°4900 4 2°8473 2°4119 2°8501 3°2023 5 3°5532 3'1230 35529 39127 6 4°2572 38332 4°2574 46221 7 4:9622 4°5430 4:9623 53310 8 5°6678 52520 56678 6:0394 9 6'°3737 5°9606 637387 6°7475 10 7:0799 6°6689 70799 74555 id 77862 73769 77862 81633 12 8:4926 80848 84926 88710 13 9°1992 8°7925 9°1992 95786 14 9:9058 95001 9°9058 10°2862 15 10°6125 10:2077 10°6125 109937 16 11°3192 10-9152 11°3192 11°70 17 120260 11:6227 12-0260 12°4085 18 12°7328 12:3301 12°7328 131159 19 13°4397 13-0375 13°4397 13°8232 20 14°1465 13-7449 141465 145306 21 14:8534 144522 14°8534 15:2379 22 15°5604 15°1595 15°5604 15-9452 23 16:2673 15:8668 16°2673 166524 24 169742 16-5741 16°9742 17°3597 25 17°6812 17-2813 17-6812 18-0669 26 183882 17-9886 183882 18:7742 27 19:0951 186958 19-0951 19-4814 28 19-8021 19°4031 19-8021 2071886 29 20°5091 20°1103 20°5091 20°8958 30 21:2161 20°8175 21-2161 21-6030 00 Ce) ee) co 00 The following simple expressions for « and 8 give a four figure accuracy when 2 is not greater than 1: 5 and 1-4, respectively : 1/z ATS [ay s- ib Ga ee tr 2 eee +7(5 )- ian(3) + s5000( 5) ns) (obtained by logarithmic expansion), 4 log 27=:918939 ; ber and bei and ker and kei Functions. 55 a\2? 1 /a \s a &=(5) —5(5) + wala) a (obtained by expansion of arctan be =). ber x ae X(wv). ¥(@). Z(a). W(2). 0 1-000 0 0 0 1 1031 | 10°'x2513 | 10°°x6266 | 10°'x5:052 2 1510 1-084 | 107'x5:209 1-169 3 3:803 3-240 2-054 2-847 4 | 10 x1183 | 10 1-007 6-909 8-446 5 | 10 x3883 | 10 3375 | 10 2345 | 10 2757 6 | 10? x1323 | 102 x1177 | 10 8215 | 10 9-393 7 | 102 x4:643 | 102 x4203 | 102 2943 | 10: 3-294 8 | 10° x1666 | 10° x1:526 | 10° x1072 | 10° x1-181 9 | 10° x6077 | 10° x5:621 | 10° x3953 | 10° x4:305 | | 10 10' x2'245 | 10! x2:093 | lu' x1474 | 10 x1:590 3 11 | 10' x8383 | 10! x7-863 | 10! x5541 | 10! x5-935 12 10° x3'157 | 10° ~2977 | 10° x2:099 | 10° 2-234 13 10° x1197 | 10° x1134 | 10° x7:999 | 10° x8-472 14 10° x4568 | 10° x4:344 | 10° x3:065 | 10° ~3-233 15 107 x1752 | 107 x1-672 | 107 x1:180 | 107 x1-240 16 107 x6'752 | 107 x6461 | 107 x4:561 | 107 x4-777 : 17 10° x2612 | 10% x2:506 | 10° x1-770 | 10° x1-848 18 10° x1:014 | 10° x9:752 | 108 x6887 | 10° 7-175 | 19 10° «3950 | 10° x3-806 | 10° x2688 | 10° ~%2794 | 20 10! x 1543 10! «1-489 | 10'° x1:052 | 10° 1-091 | 21 106-041 | 10% x5:842 | 10! x4:127 | 100 ~%4-273 | 2 10% 2°371 | 10% x%2-296 | 10" x1:622 | 10" x1-677 23 10° 9326 | 104 ~9-:044 | 10" x6390 | 10" x6596 24 1023675 | 10 ~3568 | 10" x2521 | 1012 ~2:599 25 10% x1451 | 10% x1-410 | 102 x9-966 | 10% x 1-026 26 10" x 5736 103 5582 | 10 «3-945 | 103 ~4:057 27 1042-271 | 10% ~%2-213 | 10“ x1:564 | 10™ x1:6U6 28 10° 9-007 | 10“ 8-783 | 104 x6207 | 10" x6:370 29 10° %3:576 | 10% x3490 | 10% «2467 | 10% ~2:529 30 101° 1-422 | 10! x1:389 | 10% x9'799 | 10! x 1-005 ood Oo ie 0) fe 2) fo. @) ie 2) oO 56 Mr. H. G. Savidge: Zables of the We may write V(#), Z(v), W(«) in a form similar to 2 Dr. Russell’s formula X(#) = ae namely : & 3 21 27 - ———_ — 4 V2 8/22 f 128 f/22? 12d2* pon Vite) (= fs , where 7 = a. VO RZ). wx), ZIV, V/V, 0 0 0 0 0 00 1 ‘2437 06076 4899 ‘2494 2-0104 2 ‘T177 3449 ‘7738 ‘4806 10782 3 8518 5399 ‘TASH +6389 ‘8787 4 8512 5842 ‘7141 6863 8389 5 8691 6040 ‘7101 6950 ‘8171 6 8901 ‘6211 ‘7101 ‘6978 ‘7979 7 ‘9051 6339 7094 7004 7838 ‘9162 6433 ‘7088 7021 7736 9 9250 6505 7084 ‘7033 "7659 10 9321 "6562 ‘7081 7040 "7597 isl ‘9380 “6609 ‘7079 7046 ‘7547 12 ‘9430 6648 ‘7078 7050 | “7506 13 ‘9472 ‘6681 ‘1077 7054 | | ai 14 -9509 “6709 ‘7076 ‘7056 7449 15 9541 6734 ‘7075 "7058 ‘7418 16 | -9569 6755 ‘7075 7060 7394 17 9593 ‘6774 ‘7074 ‘7061 P3T4 18 ‘9615 ‘6791 *| 7074 7062 ‘W357 19 ‘9635 ‘6805 ‘7074 7063 "7342 20 9653 ‘6819 7073 7064 7328 21 9669 ‘6831 ‘7073 "7065 ‘7315 22 9684 6842 ‘7073 ‘7065 ‘7304 23 9698 6852 ‘7073 ‘7066 7293 24 ‘9710 ‘6861 ‘7078 7066 7284 25 ‘9721 ‘6870 ‘7073 7067 7275 26 ‘9732 ‘6877 "7072 7067 7267 27 9742 -6885 ‘7072 7067 7260 28 9751 6891 ‘7072 7068 7253 29 9759 6898 7072 7068 7247 | 30 9767 6903 7072 7068 ‘7241 ie 1 7071 ‘7071 7071 7071 ber and bei and ker and kei Functions. 57 ty ie) = ae Haier a pee py _ Ye MSP 8 Palit Z : 21x NZ). Bal 2a Los? 198 22? 1 1 23 Ws) = = Ee where » = sh SI 8 /2x te oh 384 2.203 =a 12874" wv. i (a); V (#). S(@). Dey 0 oa) oO 0 —0°5 1} 10° x3-272 | 10" x6-066 || + | 10°'x2:186 | — | 107'x3-285 2110 7 x4-270 | 10°” x5-968 || + | 10°’ 2541) + | 107° 1-063 3| 107? x7-106 | 10° «8-933 || + | 10°°x 4733) + | 10771-6834 4| 10-° x1:314 | 10°” «1-563 || — | 107'x1-104|| + | 1077x5-949 5| 10 * x2877 | 10. x2-962 || — | 10°°x6-254) — 10 °x7801 6 | 107° «5-250 | 10°” x5-901 | + | 10°°x5501 | — | 10°°x6-263 7 | 10°° x1-099 | 10°° x1-214 |) + | 10°?x6-086 | + | 10°-?x3-741 8 | 10°" x2:344 10. x2560 || — | 107?x2347) + | 10 "x 5793 9| 10°" x5-078 | 107" x5491 || — | 10° "xs-421/) — | 10°°x1-217 10 107! x 1113 107° X1195 |} + 107" x 2-910 — | 10° ” x 4-992 11} 10 x2464 | 10° x2628 |] + | 10°7°xas21)) — | 10° x 4-684 12 10 x 5500 10 X5'833 || + | 10- x 1:087 Ye 107" x 4-022 13 10 x1 236 1G Ay 1-305 || (| 20) "X3°506 | # 10" x1582 14 10279 ae «2-936 || — | 10— "x 1-967 a | 10 "x 2-981 15 10x o34 PO aii X 6°646 | boll) j "x 2-456 et. ty AO "2-254 16 | 10 X1446 | 10° x 1512 | + 10 7 x2451 L349), IDS "x 1-989 17 | 10 _ Xoald 10 x8-452 | — | 10° x 1-488) + | 107 X 2566 18 | 10° x7-608 | 10° x7-912 || — | 107 “X2607 PAL AD *x9'593 19 | 107? x 1-753 10" 1-820 ] + | 107x508 |) — | 107 “X2582 20 | 10°*x4051 | 10 x4-197 || + | 107 °x2-498|| + | 10° “x9-118 21 | 10-9383 he 9-704 | + | 107° 2-883) + | 1077x2363 22 10 x2178 10 x2250 || — | 10-2185) + | 10 °x6261 23 | 10° x5068 | 10° x5226 | — | 10.) x9195) — | 10 °x1-970 #4|10 "x1-181 | 10°" x1-216 | ei} 10, ., 751926 I |,10 ~* 1-166 95 | 10 *'x 2757 | 1077 x2:836 || + 10-1366 + | 10 "x 1461 26 | 10° *x6-447 | 1078x6625 | hy x 1182] es) Sg (xboly 27 107 *x1510 1078x1550 | — | 19°“ x1-621|, — | 10° x8-948 28 | 10° x3540 10° °x3631_ | + | 10° "x 6-073), — | 107 x L679 29 | 10° x8s12 10’ x8917 +} 10 -x1693|| + | 10° "x 3'253 30 | 10~ «1-954 | 10°~°x2:000 || — | 107° 3-196 || + | 10° ” 1666 fe hala vie A) Gah emer 00 0 0 | 0 0 | Vi) = 58 Mr. J. 8S. Dow on the Physiological Similarly for V,(«), Z,(x), and W,(~#), e. g. weer z 3 21 27 h = — ——_ ea mien Or Nay as We ae ce 128/22 1282 we: —Zi(x) | —W,(#). BD) = | =i 1] 10°'x3736 | 10 °x2-428 9.1. 10 A005 10°? 3-073 Se Or tees nea ate 31 10 °x6145 10° °x 5-072 | 4 | 10° °x1191 | 10°°x9:343 10° 2-070 10° *x 1-829 a | V. Some further Notes on the Physiological Principles under- lying the Flicker Photometer. By J.S. Dow, B.Se.* ie a paper read before the Physical Society of London in 1906 (see Phil. Mag. August 1906; Proc. Phys. Soe. vol. xx.), the author discussed some of ‘the phenomena of colour-vision which affect heterochromatic photometry, and referred to the theory, due to von Kries, ef the action of the minute light-percipient organs on the retina known as the “rods” and “cones,” by the aid of which some workers in photometry have endeavoured to explain the “ yellow spot”’ and Purkinje effects. It was pointed out that, when the comparison of two sources of light differing in colour was attempted, the reading of a photometer of the “equality of brightness”’ pattern would be found to depend upon :— (a) The obliquity at which rays from the illuminated surfaces strike the eye. (b) The distance away of the eye from the surfaces. (c) The size of the surfaces. It was, however, also found that, for some reason, the effect was much less readily perceptible in the case of a flicker photometer; and this suggested that the physiolcgical basis of such instruments might be found to difter from that * Communicated by the Physical Society : read November 12, 1909. Principles underlying the Flicker Photometer. 59 of photometers of the ordinary variety. In this communica- tion the author wishes to summarise the results of some further experiments on this point which seem to throw some light on the theory of the flicker photometer : this, however, us Mr. A. P. Trotter has justly remarked, is essentially physiological, and is therefore correspondingly difficult to investigate. Before proceeding further, however, it may be well to recapitulate briefly the main points in this theory of the action of rods and cones on the retina, as it will be frequently referred to in what follows. According to this theory, there exist on the retina two distinct varieties of minute light-perceiving organs, known, from their appearance, as the ‘‘ rods” and the “cones ” re- spectively. ‘The rods, it is thought, are sensitive to light but unable to perceive colour, as such; they are, however, most sensitive to bluish-green light, probably in the neigh- bourhood of 0°51 to 0°52 yu. Light of all wave-lengths seems to be usually perceived by these organs as white*. These organs are also sensitive to very weak light but, as the illumination is increased, they become as it were saturated and do not respond further to increased stimulus. The cones, on the other hand, perceive colour but are most sensitive to yellow-green light, probably usually near 0°58 pz, and, while they are insensitive to the very weak order of illumination to which the rods respond, they continue to become more active, under the influence of increased stimulus, once they have started, long after the rods have ceased to do so. As explained in the paper referred to above, this theory seems to account very satisfactorily for the Purkinje effect. Moreover, an experiment described therein also suggested that, as the Purkinje effect only became very noticeable to the author, for small retinal areas, at illuminations below about 0°2 lux, this might be the order of illumination at which the cones finally go out of action, and cone-vision is more or less completely replaced by rod-vision. In view of what follows this figure is rather important, and we may next briefly refer in passing to a confirmatory experiment which acts as a useful check upon it. According to the rod and cone theory, the yellow-spot effect is to be ascribed to the peculiar distribution of rods * According to arecent paper by Professor G. J. Burch (Proc. Roy. Soc. 1905) the retina, which, as a rule, seems to lack the power of colour- perception at very low iliuminations, regains it if kept in the dark for some hours. — —— -—- _——__- — ——————E—— eee 60 Mr. J. 8. Dow on the Physiological and cones over the retina. At the centre of the retina only cones appear to exist; portions of the retina more remote from the fovea are covered by a mixture of rods and cones, while the extreme peripheral portions of the retina contain rods only. It will be seen therefore, that when the bright- ness of two surfaces, illuminated by the aid of red and green light respectively, is compared, the result will depend very largely on the proportions of rods and cones on the portion of the retina on which the image of them is received. It therefore occurred to the author that it would be of interest to obtain curves connecting the distance away of the eye from the photometer-surfaces so illuminated, and the apparent relative brightness of the red and green, with a series of different illuminations of the photometer. This experiment was carried out in exactly the same manner as those of a similar description detailed in the paper before the Physical Society in 1906. Fig. 1. Showing Effect of varying distance of Eye from Joly Photometer at various illuminations of the photometer-blocks. (‘* Equality of Brightness ”’ Method.) Red Green Ratio of Candle-power, 0 20 40 60 80 100 Distance of Hye from blocks of Photometer, centimetres. The results are shown in fig. 1, and seem to bear out completely the suggestion that, whatever be its explanation, aS Sg en ee a i Principles underlying the Flicker Photometer. 61 a somewhat abrupt alteration in the nature of colour-vision occurs, in the case of the author’s eye, near 0-2 lux. The instrument was in this case a Joly photometer, the blocks of which were illuminated by red and green light furnished by two carbon-filament glow-lamps screened with ruby-red and signal-green glasses respectively. It will be seen that, as the eye is withdrawn, the reading invariably tends to favour the red light at ine expense - the green. This, according to the rod and cone theory, is to be explained on the supposition that, as the image falls more and more towards the centre of the retina, the number of rods, as com- pared with cones, in the area covered by it, continually decreases ; eventually, when the fovea eentnalis is reached, a more or less steady state of things should be arrived at since in this region of the retina only cones are believed to exist. For convenience in comparing the steepness of the curves at different illuminations, all the readings are reduced to a scale such that the value corresponding with the eye at a distance of 10 centimetres is unity. It will be observed that, as we should expect, we obtain in each case a curve which gradually rises but afterwards tends to become horizontal. But it will once more be seen how very much steeper the curve suddenly becomes when the illumination is reduced to the order of 0°2 lux (metre-candles). On the other hand, an increase in the intensity of illumination from 10 to 40 ‘lux has relatively little effect. All this is exactly what the theory would lead us to expect. It is apparently only at comparatively low illuminations that the struggle of the rods and cones becomes marked. And at an order of illumination about 0°2 lux (metre-candles) the cones might be supposed abruptly to cease exercising their powers, with the result that the relatively small number of rods in play become propor- tionally more effective. A small change in the retinal con- ditions would therefore be much more influential in disturbing the judgment of balance than at high illuminations, and the curve becomes correspondingly steeper. The above experiment, therefore, is interesting not only because of the confirmation it seems to afford to the provisional rod and cone theory, but also because it serves to bear out the suggestion that an abrupt change in the retinal conditions (seemingly adequately explained by the sujposition that the cones go out of action or at any rate suddeily decrease in sensibility) occurs at an illumination near 0:2 lux. This figure will subsequently be recalled in connexion with experiments on the flicker photometer. 62 Mr. J. 8. Dow on the Physiological Meantime it may be pointed out that other experiments of the author * suggest that there is a sudden diminution in visual acuity at just about the above order of illumination. This, therefore, is once more additional evidence that a profound physiological change in the retina occurs near this point. Let us now turn to the main question to be considered in this paper, namely, the bearing of the rod and cone theory and the physiological phenomena explained by its aid, on the flicker photometer. Whatever be the exact explanation of the functions of the rods and cones and the retina adopted (and physiolog sts seem as yet to differ considerably in their views on this matter), the physiological effects on which the theory is founded seem to be well authenticated. It is therefore of interest to inquire how far they can be considered applicable to the flicker photometer, which is now beginning to receive a considerable amount of attention as a means of comparing sources of light which differ in colour. As an illustration of the possibility of such instruments yielding results which differ from those obtained by the aid of photometers of the “ Hquality of Brightness ”’ or “ Contrast” types, an experi- ence recently related by Mr. L. Wild may be mentioned. This observer finds that, when an incandescent lamp, having a tungsten filament and run at a consumption of 1:5 watts per ©.P., is compared with a carbon-filament lamp running at 4 watts per candle, the result varies by 6°/) according as a flicker photometer or an instrument of the ordinary variety is used for the test. Apparently the tungsten lamp has a lower candle-power in the former case f. This matter was touched upon in the writer’s previous communication to the Physical Society on Colour-photo- metry {. On that occasion it was found that the readings of an Hverett-Edgcumbe flicker instrument, when used to compare red and green lights, did not differ, when the distance of the eye from the field of view of the instrument was altered, nearly so greatly as was found to be the case for an “ Equality of Brightness ” instrument. This suggested, therefore, that the behavour of the retina was in some way different, when judging disappearance of flicker, from that characteristic of it when judging brightness. It seemed, however, preferable to confirm this result by an experiment in which exactly the same portion of the retina. * ‘Tlluminating Engineer’ (London), vol. ii. April 1909, p. 287. + Electrician, July 16, 1909. { Phil. Mag. vol. xx. 1906. Principles underlying the Flicker Photometer. 63 was utilized in observing both brightness and flicker, and the illumination of the surfaces viewed was the same in each ease. Under these circumstances, the two sensations would be compared under exactly similar conditions, and if the physiological principle of both instruments is the same, should lead to similar results. Comparison of the Yellow-spot Effect for Equality of Brightness and Flicker Photometers. The Rood type of photometer is particularly convenient for the investigation of this point for, by keeping the lens still, one may use it as an ordinary instrument, and then, by oscillating the lens, compare the brightness of the two photo- metrical surfaces, still at the same distance from the eye, by a flicker method. For this purpose the writer made use of the arrangement shown in fig. 2. The surfaces of a Ritchie Wedge, W, con- sisting of fine unglazed cardboard, were illuminated by the Fig. 2. Arrangement of Rood Type of Photometer. > ial VE two heterochromatic sources of light to be studied, A and B. In front of the wedge a convex lens, L, is mounted on springs and attached to a cord which passes round the pin P mounted eccentrically on the pulley of the electric motor M. In front of the lens was placed a small black screen 8 pierced with an aperture 5 ems. high and 2 cms. wide. If, now, the motor is stationary, the observer balances the brightness of the red and green by the equality of brightness method. But when the motor is set in rotation a band of flicker is produced between the coloured surfaces, since the Red Green’ Ratio of Candle-power 64 Mr. J. 8. Dow on the Physiological dividing line between them appears to oscillate to and fro. By altering the position of the pin P the width of this band of flicker can be adjusted until it occupies exactly half the field of view, 7. e. has the same dimensions as either of the coloured surfaces. We thus have the exact equivalent of observing two coloured surfaces of equal size, either simul- taneously, when we judge their intensity by the equality of brightness method, or in rapid succession, when we judge their intensity by the absence of flicker. » = = 0 ee a ee Principles underlying the Flicker Photometev. 15 so used is mainly dependent on the impulses received through the cones. The portion of the retina employed in a flicker photometer is usually restricted to the central region, where there are comparatively few rods, and therefore these organs have probably not much to do with its action. Partly on this account, but chiefly because the correct speed for a flicker photometer used to compare widely different heterochromatic lights is so limited, it does not seem likely that informa- tion as to the theory of flicker photometers can readily be obtained by studying the effect of speed on the actual readings of the instrument. It is true that Lauriol * has been quoted as having recorded very marked variations in the reading of a flicker photometer at different speeds; but on closer examination it appears that this observer's results establish rather the difference between readings of a flicker photometer and a photometer of the equality of brightness pattern. Jor the difference recorded is that between the readings of a flicker instrument when used as such and when used with the flicker-wheel motionless. The writer has sought to determine whether the per- missible range of speed of a flicker instrument used to com- pare moderately heterochromatic sources could be responsible for any marked error, but without detecting any distinct variations due to this effect. Colour-blind Observers and the Flicker Photometer. Another possible direction in which an inquiry into the theory of the flicker photometer might be pushed is the examination of people having defective colour-vision. Some data are available on this subject, but they lead to somewhat uncertain conclusions. It appears, however, that in some eases colour-blind individuals have been observed to obtain correspondingly abnormal results with both flicker and equality of brightness photometers +, and it would certainly Jead one to doubt the soundness of the principle underlying such instruments were this not the case. In reality there seem to be so many different types of colour-blindness, and the existing knowledge of the subject is still so incomplete, that it is difficult for anyone but a physiologist to draw any valid conclusions on the theory of * Bull. Soc. Ins. des Electriciens, 1904. t+ See Brochun, Gas World, Feb. 15. 1908; Dow, Electrician, Feb. 1 & 8, 1907; Tufts, Phys, Review, Dec. 1907, &c. 76 Physiological Principles underlying Flicker Photometer. the flicker photometer by an examination of colour-blind individuals. The study of the principles underlying this instrument must, it would seem, be conducted in the light of the modern theories of the action of the retina. Physiologists, however, seem to have found some difficulty in reconciling many forms of colour-blindness with the usual theory of the action of the rods and cones. As an illustration of the com- plexities of the subject it may be mentioned that according to Edridge Green* some forms of colour-blindness arise, not through imperfect retinal apparatus, but through con- fusion at the centres of perception of colour in the brain ; the same observer states that colour-blindness frequently accompanies some forms of insanity. Concluding Remarks. In conclusion the writer would like to say that he is conscious that the data presented are in many respects incomplete, and that other and more adequate experiments are needed before any decision as to the correctness of the suggestions brought forward can be taken. It may be noted, for instance, that these results described were, like those described in previous papers, obtained for the author’s eye only. For this reason they are strictly comparable with the latter and in some respects seem very consistent. But naturally in order to be conclusive experiments in photometry should be based upon the observations of a large number of individuals, and upon a number of different types of instruments. These experiments were mainly carried out at the Central Technical College, South Kensington, during 1906 and 1907, and the author had hoped to make the series of researches described in this paper somewhat more complete. Circum- stances having prevented this, however, it occurred to him that their present publication might lead others to investigate the phenomena described more closely from the standpoint of photometry; the need for gaining some insight into the physiological facts underlying the flicker photometer may, perhaps, give these experiments a certain suggestive value. The writer is also well aware that the problem investigated is so essentially physiological in character as to be strictly only capable of being adequately studied by the combined efforts of the physiologist and the physicist interested in the problem from the photometrical side. It may be stated, too, that many points in connexion with the theory of the action * “Colour-Blindness and Colour-Perception.’ Liffective Resistance and Inductance of a Helical Coil. 77 of the rods and cones on the retina (only the main essentials of which are referred to in this paper), seem to be still the subject of much discussion among physiologists to whom we must look for the exact interpretation of their bearing on problems in photometry. Dr. Edridge Green, for instance, in a recent lecture before the Optical Society *, expressed his dissent from the theory of vision based on the supposed he- haviour of the rods and cones, and expressed the view that visual impulses are only received through the latter organs. But the general phenomena on which the theory mainly rests, and which have been utilized in this paper, seem to be well authenticated. VI. The Effective Resistance and Inductance of a Helical i tae problem of the propagation of alternating currents in wires has been solved in but few cases. For a single wire isolated in space, a solution has been given by Sommerfeld ft. If the return current be conducted along a concentric sheath, instead of by the dielectric, the problem is that treated earlier by Sir J. J. Thomson§. The case in which the return current flows along a parallel wire has been discussed by Mie ||, and to a first approximation, in a simpler manner, by Morton *{, who has given, in other papers **,a detailed examination of the distribution of forces, and applied his method also to the case of a greater number of wires. The influence of electrostatic capacity in wires renders the mathematical investigations very complicated, but from a practical point of view, the main object to be attained is a knowledge of the effective inductance and resistance of the wires when the capacity and leakage are not so great as to produce a sensible attenuation of the wave in a short distance. The concentration of the current in the outer regions of the wires, when the frequency is high, completely changes _the character of the inductance and resistance, and a know- ledge of their values for steady currents is frequently of * Optical Society (London) Oct. 21, 1909. t Communicated by the Physical Society: read November 26, 1909. t Wied. Ann. lxvii. p. 233 (1899). § Recent Researches, p. 262. || Ann. der Phys. ii. p. 202 (1900), 4} Phil. Mag. Dec. 1900. ** Phil. May. May 1901, Sept. 1902, June 1903. 78 Dr. J. W. Nicholson on the Effective little service. But even with the restriction to wires with small capacity and leakage, mathematical solutions have not hitherto progressed far. The problem of the concentric main was discussed by Maxwell* and Heaviside f, and afterwards for very high frequencies by Lord Rayleigh f. A practical formula for “the resistance of the inner conductor was given by Lord Kelvin §, whose solution is not essen- tially” different from that of Heaviside. Formule for the effective inductance in this case have also been obtained by Sir J.J. Thomson ||, and an exhaustive treatment of the whole problem by more simple methods, in which the ranges of application of all the previous formulze are critically examined, has been given by Russell . The investigation of the effective inductance and resist- ance of two parallel wires, one being the return of the other, was made by the author **, and developed, ina later paper +t, in a form capable of more immediate practical use. The upper limit necessary to the capacity was at the same time examined. The main problem of importance in which the wires are bent is that of the single helical coil of considerable length. Solutions have hitherto dealt with the case in which the wire is closely wound round a circular cylinder, forming a helix whose pitch is very small. Wien tt first discussed the problem, but assumed that the current distribution is the same at all points equidistant from the axis. Battelli showed that this assumption was not lawful, and the results do not agree with experiment. Shortly afterwards, Sommerfeld §§ investigated the problem in so far as the resistance is con- cerned, and his method was applied later by Coffin |||| to a determination of inductance. Picciati Il] gave an independent solution, but like those of Sommerfeld and Wien, its agree- ment with experiment was not very good. The best solution in this respect is that of Cohen***, which appears to be quite * Elec. and Mag. vol. i. § eae Tt Electrical Papers, i. p. 353; i. p. G4 et sey. t Phil. Mag. xxii. p. 381 (1886) ; Scientific ‘Papers, ll. p. 486. § Journ. Inst. Elec. Eng. xxili. p. 4 (1889) ; Math. & Phys. Papers, 1, p. 491. || Recent Researches, p. 295. q Phil. Mag. April 1909 ; Proc. Phys. Soc. xxi. ** Phil. Mag. Feb. 1909; Proc. Phys. Soc. xxi. choo tl. Maz. Sept. 1909; Proc. Phys. Soc. xxii. tt Ann. der Phys. xiv. p. fl (1904). §§ Ann. der Phys. xv. p.673 (1904). ||| Bulletin of Bureau of ae li. p. 275. 44 Il Nuovo Cimento (5) i. p. 35. *** Bulletin of Bureau of Siael vids, iv. no. 2. Resistance and Inductance of a Helical Coil. 79 satisfactory for all cases in which the pitch of winding is very small. The present paper deals with the corresponding formule for a helical wire whose pitch is not very small, wound on a cylinder whose radius is large compared with that of the section of the wire, which is of course circular, and cannot be treated as square in the manner applied by Cohen to the other extreme case. A rigorous solution would be very difficult, but it is shown that some very simple approximate results are sufficient for practical purposes. ‘The coil is to be regarded as sufficiently long for the effects of the ends to be neglected. A connexion between these formule and those of Cohen cannot be made without a consideration of the difficult intermediate case in which the pitch is moderately small. Choice of Coordinates. Let the axis of the cylinder on which the wire is wound be chosen as that of z in a Cartesian system. Defining the central curve as the locus of the centres of all normal sections of the wire, this curve will be a helix, and if u is the radius of the cylinder, and @ the twist in a plane perpendicular to z of the radius in that plane measured from a line parallel to w, the coordinates of a point on the helix become ese’, y=asn’, s=altana. . (1) where « is the angle of the helix. Let (pp) be polar coordinates in the plane of the normal section at the point defined by @. The initial line from which ¢ is to be measured is perpendicular to z and to the tangent of the helix. Thus if < were vertical, the initial line in any section would be the horizontal line. Now the direction cosines, at the point 0, of the tangent to the helix are (—sin@cosa, cos@cosa, sing). A line perpendicular to this and to the axis of z (001) is readily shown to have a direction (cos@, sin 6, 0). This is the line ¢=0 in the section defined by @. Let (A, #, v) be the direction cosines of the radius vector p in the section. It is at an angle ¢ to the above line, and is perpendicular to the central curve, so that d cos 0+ sin 6 = cos 4, *—Asin Ocos «+cos Ocosa+tyvsina = 0, with M+w +r? = 1. 80 . Dr. J. W. Nicholson on the Effective Thus 7 l—v? = 0? +p? = (Acos 0+ sin 0)?+ (pu cos O—D sin 8)? = cos’ d+’ tan? a, or v= + cosesin ¢. We choose the positive sign, and deduce > = cos Ocosd+sin asin @ sind, (2) = sin Ocosd—sin a cos @ sin g, 7 vy =cosasin J as the direction of the radius vector. The coordinates of any point of space are therefore x = acos0+pr ) =(a+pcos d) cosO+p sina sin @ sin ¢, y= asin 0+ py =(a+p cos d) sin @—psin «cos @sin d, 4 (3) | z= a0tana+py = af tana+p cos asin d ) We shall call (9 @) the helical coordinates of any point of space. If 6s (which only involves 6@) be an element of length perpendicular to a normal section at the point (p, ¢, @), then the space elements 6s, dp, pd are mutually perpen- dicular, and the surfaces p=const. are those of helical wires with a common central curve. This system of coordinates is therefore appropriate to problems in which surface con- ditions must be satisfied at the boundaries of such wires. Property of Orthogonal Systems. In the problem here contemplated, in which an alternating electromotive force He” is applied to the terminals of a long helical coil, the electric and magnetic vectors in and near the coil will not depend on 9, if the other portion of the circuit is kept at a distance. Thus 9/d00=0. Consider now a general orthogonal system in which the components of all vectors are independent of one coordinate. Denoting the three coordinates by (9 @@), let the corre- sponding space elements along their instantaneous directions of increase at any point be é (ds,, dsp, dss) = (dp/p,, dP/p2, dO/p;)... . (A) Let (a, 8, y), (#, y, 2) be the components of magnetic and Resistance and Inductance of a Helical Coil. 81. electric force along these directions. Since the operation 0/09 annuls any component, the solenoidal conditions become. Ol re ) es ee = op P2P3 bik o¢ P\Ps { (5) a half pac —— = (), — OP P2P3 | OP Pips a -Itis assumed also that (p; p2p;) are independent of 0, as is the case in all important applications. [unctions y and w may be introduced, where X= pops0x/0b, Y = pips 0x/Op, | 2=popsd¥/d¢, B= pips ov/dp By Faraday’s law, if « be the permeability, poy ox 0 Xx Pip; ot Opp, = Ob pr _ 0 Pip ox 4 0 PaPs OX, (7) Op ps Od pi Od’ and by Ampére’s law, in a pein eat SOx q | ws a= = wees s iil va he, eas Ged whence, on reduction, x, andjsimilarly wy, satisfy the equation eo: Pips oY 4 0 Paps Oy _ Ps or (9) Op po Op. Of ~r OP Vi~rps BF?" * where V is the velocity of propagation of electromagnetic disturbances. Thus any disturbance in which the vectors are independent of one coordinate of an orthogonal system must be made up of two types, in which the electric and magnetic forces along the direction of increase of that coordinate are respectively zero everywhere. If the wave-length of the oscillation, supposed simply periodic, be 2z7/k, the equation for yr becomes (6) 0 PipsOv , 0 Peps OW Wh ne ea Op po op Of pm Od | pyr | and for a conducting medium, it is only necessary to change k?, In the present problem the oscillation defined by y exists, Phil. Mag. 8. 6. Vol. 19. No. 109. Jan. 1909. G 82 Dr. J. W. Nicholson on the Effective but not that defined by y. The vectors are given by A= Y=0, a=y=0, =H, p=—ae Sh. . a) The electric force is along 6 increasing, or parallel at any point to the corresponding tangent of the central curve. The _ magnetic force is along ¢ increasing. At the surface of any wire of the helical system, both forces are tangential, as they should be. Solution as a Fournier Series. If ds, is an element of are along p increasing, Bej=(3)aRY+Ryaeveroren so that p,=1. Moreover, Ose y=(3 2 =) Oz ) (33) (35) +33) #85 =(—p sin ¢ cos 8+/ sin # sin @ cos ¢)” +(psin d sin@+~ sin a cos 6 cos $)? +p” cosa cos? 6 = p* on reduction, and pp=p~'. Similarly p3= 4a? sec?a + 2ap cos d + p?(cos? d + sin?a sin? g)+—*. . (12) If the wire be thin, p3=a-! cose, and we are led to the usual solution for the straight wire of great length. The differential equation may be reduced to DO ev, Lowe .. ov 1 Ops , Loy 1 Op — px-+-— +Mpbt+pat — 24+ —-2 =0.. (13) Op” dp p OF" pw POP ps Op | p Od P; Ob oo Now 1 Ops _ ap sin d+)? sin ¢ cos ¢ cos" Ps OP =a” sec’a + Zap vos p + p*(cos*h + sina sin’) : . Desi ; = E sin d cos*a — 9q7 Sn 2 costa +...... A on expansion in powers of p/a, replacing the trigonometric functions by those of multiple angles. Again, 2 : s Ps =— F cos’a cos 6 — a cos’a(2—3 cos*a —co:*a cos 2h). . (15) We shall not retain p*/a® or higher powers. 5 p ; = (2,2 \dp " Op _ g- costa{2 cos PA,’ +(1+ cos 2p) Ay’ +(cos 6+ cos 3p) Ao’ +. Resistance and Inductance of a Helical Coil. The equation (13) has a solution of the form w=A,+A,cos 6+A, cos 26+...... the accent denoting 0/dp, fo) ; ee 2 _ il cos’a(2—3 cos? z)(Ao’ +cos PA,’ + cos 26Ay/ +...... ) 2a? 2 + = costa{2 cos 26Ay + (cos d+ cos 3) Ay’ + (1+ cos 4g) Ad! + 2 — 95 {(1—c0s 2h) Ay + 2(cos $—cos 3h)A2 + 3(cos 26—cos 46)As-+ 2 4+ —P- costa{ (cos d —cos 3p) A; + 2(1—cos 4) Ay + 3(cos 6—cos 5) A; + 4a? we deduce the following system of equations, where Se a = dp? dp + kp, (17) DAo— ‘3 cos’aA,'— Po cos’a (2—3 cos*a«)A,! + £ costaA,’ Ay 2 pAs Tirgy Poe ¢ + Fe cos‘a=0, Ay Pe deh? REE aa Si ees a. — a a(2Ay + A,’) =_ 2a? Cos a(2 3 cos a) A, ef cos*a(A,' + A;’)— = cos*a + ee ;costa(A,+3A*)=0, 4a? and soon. But it is evident that Ao, Aj, Ag...... 83 (16) where the functions A are independent of ¢. By substitution, form a sequence, each member of which is of a higher order in a7! than those before, Kor the purpose of this investigation, A is required to order p?/a?, and therefore, as an intermediary, A, to order p/a only, and A, not at all. Therefore the equations may be written more simply in the forms dA, _cos’a d = p> a aaah <2 mr re Nea cubated eat DA, 3q2 8 a(2—3 cos’) rs iia ticks (pA,), and A, must be eliminated. G 2 : + (19) 0 seeeee . (18) 84 Dr. J. W. Nicholson on the Effective It is convenient to take a new variable kp=., so that 2 2 ol ee oY 2 2—3 cos? l d ia cos’a(2—3 cos*«) Ay aS = (2A,). ae es, Ayo Bee nak Ge Now (20) may be written nie di1d pias: va dA, ka dx’ dx xax 1 l #14 oa piss cos’a d to (~ 4, dx” iF 2 haw da” dis and the elimination with (21) may be effected at once, a or += 41-— dz? * x dx: iaelginee 2) amet cosa lid id A, Se eA TENE pt 2 2h2a? re vaan 0 (22) and it is at once evident that Ao is only altered to the second order by the bending of the wire. It is henceforth supposed that not only a, but ‘ha is large, as this case includes nearly all of practical interest. When ka is small, a special investi- gation is necessary. Ignoring the effect of bending, the differential equation becomes (vee ee, cos’a(2—3 cosa) (S43 2 - +1) = 7, bo @ ,1¢@ ee av dx so that, for a solution finite on the central curve, d? an ie <4 ee x Ja) ; and finally Ayp=AJ,(2) + Bed, (x), where A and B are arbitrary. In determining the self- induction and resistance, we only require the modification, inside the metal, of the form Jo(x), and in the outer medium, of the form K,( 12). The solution «J,(7) may be discar ded, as it corresponds to nothing in the case of the straight wire, to which the formule must Pina when a is infinite, or «=0. Resistance and Inductance of a Helical Coil. 85 Writing therefore A,=Jd (2) + cos” ey meee Oe where wu is a new variable, only required in its first term, not involving (ka), then on substitution, and rejection of (ka)-? in the result, | y cos*« d ial a fe Jy + ss +1 i Mig” ee ae 3eosta)(S a ++ — +1) 0 By properties of the Bessel function of zero order, this becomes Bay yolid dds : oats. 41) u=+(1—5sin%a)J,(2); . (24) whence 2 i MMe oe “3 L)u= + $(1—5 sin*«)2J,(2), w dx ignoring the complementary function. Finally, =—4(1~—5sin’x)a7J,(z),. . » «= (25) again ignoring a complementary function. The total form thus ignored is AwJ,(7) + BJ,(x), of which the first term pertains to the redundant solution of the original equation. The second has no relation to the original J,(a v) with unity as ‘coetiicient. We write therefore cos?a(1—5 sin?a)z Ag= 4 1- Kate “16/2a2 et Sole) ae Oe (26) as the value of A, to the second order. Substitution in the differential equation directly verifies this solution, and it therefore corresponds to the function Jo(z) of the unbent wire. Solution for the Dielectric. The above values hold for the interior of the wire of resis- tivity o, provided that k°=—4apep/o, where p/27 is the frequency of oscillation. For the dielectric, the appropriate function outside a straight wire would be K (chp), where h=p/V takes the place of k. The function is defined by K,,(vx) = | dd cosh npe— cosh $ Rea tia (2 7) 0 86 Dr. J. W. Nicholson on the Effective and possesses the properties ne uw’ K,(u) = —u"K,_-1(w), 2 Kasa(0) —Ka-a(u) = < Ka(w), eee Ky4i(u) + K,1() = — 2 | J l a K,(u) = — Ky (wu). The analysis required in determining the corrected function is very similar to that above, and leads finally to the function Or gets ra 1— © a(1—5 sin?«) 12 Bo(uhp)= { tea Bed p* | Ko(shp). (29) Determination of Inductance and Resistance. The terms of the Fourier series involving ¢@ do not con- tribute to the total current across a section of the wire. Let Ke? be the mean value across a section, of the line integral of impressed electric force per unit length perpendicular to the section, and parallel to the centralcurve. If we'?* be the total current, and (L, R) the effective self-induction and resistance per unit length of wire, (lup+Rjo=B. . . . . ee Now at this point, the problem becomes identical in process with that of the straight wire, provided that the appropriate functions Ap(kp) and B,(ckp) are used for the wire and di- electric respectively, in place of Jo(kp) and K)(chp) for the straight wire. Quoting therefore the result in the latter case, as developed in a former paper ™, namely, if 7 be the radius of the wire, EB Quip gdo(kr) — k Ky (thr) oy si kr nx Jo(kr) phe Ko’ (thr) b Meer sy. «> (31) we find for the helix ae ame f Ay (kr) & Bo (thr) 39) ‘ye Were Uo Ger)’ ahs Bol (thr) Jee ( Dw * Phil. Mag, Feb. 1909, p. 259. The result requires a factor ~1/a. Resistance and Inductance of a Helical Coil. a7 Let foo— cos atsina—l), .)'. ..' . (38) y pe Mhen ole) = (14+) Jo (en 67” 267 A,’ (kr) = (1 + = ) Jo (kr) + qe (kr), and to order 1/h?a” A, (kr) _ Jy (kr) fx 25r Jy (her) Ay(hr) ~ Sol(kr) ka? Tyler) S Finally, if Baer) 1 ay had ol (kr) ae thr Ky!(chr) Pr+eQ, + (34) a) 3 fh 2677 = Zep (P+6Q) are na & +10) } — Jip (P’ +Q’) {1= ce (P’ + iQ) b. (35) In determining the effect of the concentration of current upon the inductance and resistance, which in the former case is all that is sought, only the terms involving P and Q (and thus the resistivity) are needed. The effective resistance is altered from its steady current value of o/mr? to the real part of E/a or R=—2upQ+ eT cos? a (1—5sin?a) PQ, . (36) and the self-induction is L=2uP— S cos’ a (1—5 sin? a)(P?—Q?). . (37) The values of P and Qare well known. With Lord Kelvin’s definition of the functions ber w, bei 2, where e=t-ikvr=r (Ampp/c)?, Jy(Ar)=bera+ibeiz, . . . . (38) and on reduction, — #Q = (ber x bei’ x —bei « ber’ x)/{ (ber’ x)? + (bei! #)?} bt 39 vP = (ber « ber’ «+ bei a bei’ x)/{ (ber' 2)? + (bei! x)?} J * sa 88 Dr. J. W. Nicholson on the Effective The values of these functions have been tabulated, yet as Russell has shown*, the tables need revision. But it is preferable, in place io a laborious tabulation, to use the asymptotic formule for these functions, which Russell develops in the same paper. We may distinguish two im- portant cases, and it is convenient to recapitulate, at this stage, the meanings of the various symbols. The oscillation is assumed to be sine-shaped, and of frequency p/27, and are the permeability and resistivity of the wire. the wire has a radius 7 of section, and is wound on a cylinder of radius a, such that 7/a is small. The angle of the helix is @. Let n be the number of turns of the helix in a length z parallel to its axis. Since, in a helix, -=a@ tan a, we have n=0/22=2/27a tan a, and therefore a=stan'(2/2¢7na). . 2... Case of Small Frequency. When #=7(4%yup/c)? is less than 2,we may write 1 =jef{i— as aye 135 (5 ve 647 eae aay 24 he ic 2%, 360.56 + 439 : x y 1 (2 oe a fit ley 180 o(3) = me J and therefore a= get t+ al) —atanl2) } P—a'= ig{1-ia(5) + sean(3) “3 1+ 6(3) —s40(9) } no higher order being necessary in the last two, as they will be multiplied by 7?/a’. The resistance becomes 1 (rupr’ 1 (pp? 11 Tupr?\§ Reaew ig (e") Jet ( 7) ) up : y 2414 rere [BO e129. e 30\ one att 25 ein? I (mppry _ Tae TOC. Che, , (41) Resistance and Inductance of a Helical Coil. 89 and the self-induction is changed by the concentration of current to an amount LL Cy 4 2B (ent (aoe =; Sie x(—e Gao — 127360.5 oe 1 (mppr 60 (mppr rae 64a = COs? ae( 1 — sin’ a ){1- 12\ Cc y i 3640 3 i o Bes if meee _i (appr aaa >a(1—5 sin’ ay {145 ) 5 (™ i ae (43) In these results it is not possible to proceed to the limit of zero frequency, for it has been assumed that k’a° is large. Although, therefore, the expression for R does not become o/7r? when p is made zero, no error is involved on this account. Near p=0 another form of expression must be used. The limitations of these results are determined by the con- siderations that 4r shall be less than 2, and that 7°/a’ and ka~* shall be negligible. As an average practical case, we take a copper wire of radius 2 millimetres, and determine the requisite conditions for a three figure accuracy. In the first place, 7*/a* cannot be ignored, unless a is greater than 127 approximately. This condition is usually fulfilled. Even if it is not, the order of accuracy may extend to two figures for a much greater value of r/a, so that this limitation is not of great moment. Secondly, kv is less than 2, so that i in the present case, if f be the frequency, and c= 1696 c.G.s. units, approximately ae 1 ptt e& « pus : Rs Fae or, | ay se TQ, where 7 is in centimetres. The upper limit in frequency is therefore about 2500 per second. Thirdly, (4a)~* may be neglected if ka>10, so that, to determine the lower limit, there is an inequality af® > 50, or if a=12r in the most unfavourable case, 7 is about 400 per second. The formule, therefore, range between fre- quencies of 400 and 2500 for a wire of 2 millimetres radius, wound on an appropriate cylinder. In general the conditions for a three figure accuracy are Gee Lee az? > 50, of2—_10,. . . (44) the radii being in centimetres. 90 = Hffective Resistance and Inductance of a Helical Coil. Case of High Frequency. We pass to the case of a higher frequency, whose lower limit is determined later. When A= 222 = 2p (Qarpp/o)3 is not less than 8, a four figure accuracy may be obtained by the use of the formule * 3 3 AP=1— —; ee | ae Denne | OE WA Yee als ah Q=1+ se aa a small but convenient change in the notation having been introduced. Therefore : Lor 49 2/2 i dMAD ee ee wP-Q)=-(5 5 54 a3), and finally _ ( upa\? Picasa ae R=(5") {1 a (aa) rf A. aaete cutie MEE! Wake Chi ai MONE aes Poe ies —3) 397-2 08 a(1—5 sin? a) tae aaa A\ Saupe Goo “(otta} {1- Sgt esta i \2arpre + sya — 2\ 8arppr?) J o cos’ a (1—5 sin? « ) {a4 Sean ED 32arpra* (s=) e) +] = A three-figure accuracy in these results will be secured if the conditions aS 12a nye = (07%. 6 ee are satisfied, in the case of copper wires. No further con- dition as regards a is here needed, as it is satisfied by virtue of the others. In the limiting case of very high frequency R= Pee) = 575 cos? a (1—5 sin? 2). re es _{ om \? o cos? a (1—5 sin? at o he 2. ow \ L=(-o) 6 d2rpra’ Qor up 39 ee Qarpr? ») (90) * Russell, 7. ¢. p. 582. On a Gravitational Problem, 91 and the change in resistance due to twisting tends to become independent of both the frequency and the radius of the .wire,.whereas the change in self-induction tends to vanish. For the intermediate case,in which 7f? lies between the limits of about 10 and 70, the formule (36, 37) must be used, provided that a satisfies the proper condition. These formule require the functions P and Q when 2 lies between 2 and 7. In a paper recently presented to the Physical Society *, Mr. H. G. Savidge has given tables of these functions suitable for the case in question, together with graphs of the functions —7Q, 2#P of (39). The formule (36), (37) are therefore rendered sufficient for practical use. VII. Note on a Gravitational Problem. By C. V. Burton, D.Se.t i, JITHOUT actual loss of generality, the question ' here considered may be stated as tollows :—/Jf the action of gravity were intermittent in character, the mutual attraction of any two bodies fluctuating between zero and twice tts mean value, how long could the period of the fluctuation be without giving rise to observable periodic effects? At a first glance we might be tempted to reply that the period could only be a very small fraction of a second; but though this answer would be justified if the earth were perfectly rigid, the yielding of the earth to pressure and tu shearing-stress completely changes the aspect of the problem, and the results obtained are so different from what might have been hastily assumed, that a brief indication of them may possess some interest. 2. Let us suppose that the gravitation-constant G repre- sents merely the mean value of G(1+ Ae"), where A is a numerical constant ; so that, at any time ¢, the attraction between masses m, m’ at a distance r apart is G(1+ Ae‘) mm’/7?. For the sake of simplicity the earth will now be regarded as spherical and sensibly homogeneous ; and accordingly the intensity of the gravitative field at any point of the surface at time ¢ will be g(1+Ae); . BY) abi ey Ate * Supra, p. 49. t+ Communicated by the Physical Society : read November 26, 1909. 92 . Dr. C. V. Burton on a and at any interior point distant 7 from the centre Ze Ade). a 3. At any interior point, a mass-element free to move under the periodic term of the field-intensity (2) would execute oscillations given by Sr = AC wg ee Consider then the radial vibrations defined by (3), and the forces which would be needed to maintain them. At any given instant, every radius vector 7 has acquired an incre- ment 6r proportional to 7, the whole sphere being thus symmetrically and homogeneously expanded or contracted, and the proportional increment of linear dimensions being —g/ap* . Ae’, corresponding to the proportional increment of volume —sg/ap?. Aer... kw i ee In the sequel the discussion is restricted to the case where Ap? is small enough to make the expression (4) a small fraction. 4, Thus if every mass-element suffers the periodic changes of position defined by (3), there will also be for each element periodic (proportional) changes of volume indicated by (4). The changes of position we may consider to be provided for by the periodic term in (2), while throughout the mass of the planet, the fluctuations of pressure needed to produce the changes of volume of the various elements are supplied by the reaction of those elements on one another. At the free surface, however, we should further require a periodic pres- sure P given by P= Skolap? Ae”... to complete the system of forces under which the vibrations (3) would be executed ; & being the bulk-modulus of elasticity of the substance of the planet. 5. For a moment suppose the periodic pressure P to be applied to the earth’s surface, in addition to the partly periodic character of gravitation expressed by the factor 1+Ac?’. Then every mass-element of the earth would be executing a vibration expressed by (3), precisely as if it were free to move under the periodic part of the earth’s attraction, and a relatively small body resting on the earth’s surface would execute, under gravitative influence alone, vibrations exactly Gravitational Problem, 93 agreeing with those of the surface. There would accordingly be no periodic term in the reaction between the body in question and the ground upon which it rested ; the weight of the body would appear perfectly constant, and, generally, no observations concerned wholly with gravitating bodies at the earth’s surface * would give any indication of periodic variations iy the gravitation * constant.” 6. Now further add to the system of forces acting on the earth a superficial periodic pressure —P; thus the sole periodic influence at work is that due to the term GAe?* in the gravitation “constant,” while the only effects observable at the earth’s surface are such as arise from the pressure —P. The result of such a periodic pressure is a radial movement of the surface of period 27/p, and of an amplitude which, under certain simplifying assumptions, can be readily ex- pressed in terms of the elastic constants of the earth’s substance. 7. Homogeneousness has already been assumed; let us now further postulate some small degree of viscosity, sufi- cient to ensure that compressional waves of short period (one second or less) propagated inward from the surface, would be sensibly extinguished before reaching the centre, though not sufficient to produce any perceptible damping effect in the course of a few wave-lengths. We have thus only to consider the waves travelling inward from the surface under the in- fluence of the periodic pressure —P ; and since, in the cases to which this inquiry is restricted, the wave-length of the disturbance is very small compared with the earth’s radius, the curvature of the surface may be disregarded, and the problem of the motion near the surface reduces to one of plane waves. 8. Let a—r, the depth of any point below the surface, be denoted by h: the waves due to the pressure —P and propa- gated in the direction of /-increasing may be represented by Re EET eis ag ge we (6) where the constant C has to be found and is not necessarily real; while the velocity V is given by VPs (eS wy pis Na ET €T) Here fk, as already defined, is the bulk-modulus of elasticity of the earth’s substance, while w is its rigidity, and p its * The attractions of such bodies for one another are here excluded from consideration. 94 Dr. C. V. Burton on a density. The radial stress at any point, reckoned as a pres- sure, 1s —(k+ 4p) dbh/di’ ; and, equating the value of this for h=0 to —P given by (5), we readily find ag dekgA 8 = Sma /[(k+aopy °° where n=p/27, the frequency of the periodic variation of gravity. 9. The rigidity ~ of the earth as a whole is not very dif- ferent from that of steel*, and if we assume the com- pressibility also to be something like that of steel, we may take as roughly approximate numerical values k = 15x10" dynes/sq. em., mw = °82 x 10” dynes/sq. cm. ; moreover, p the density is about 6°6 grams per cc., and thus (8) gives for the semi-amplitude of the apparent motion of the earth’s surface in centimetres 0068A/n°. 10. As an example, take A equal to unity, and n equal to one second. The gravitative attraction between any two bodies would then vary between zero and twice its mean value, the period of the variation being one second. The actual periodic displacement of the earth’s surface would amount to about 25 cm. above and below its mean position ; but the observable effects would, with our assumptions, be the same as if (with gravity invariable) the earth’s surface were pulsating with a period of one second and with a semi- amplitude of ‘0068 cm. The maximum acceleration involved in this apparent pulsatory movement would be about -35 em./sec’, or roughly 1/2500 part of the mean acceleration of gravity ; and refined observations would be required to discern any periodic effects. If the freqnency n of the fluctuations were changed in a given ratio m (their amplitude remaining the same), the amplitude of the apparent surface pulsation would be changed in the ratio m-*, and the maximum acceleration involved in that pulsation would be changed in the ratio m7? f. 11. It appears, then, that the interaction of the earth with * A somewhat vague statement, though definite enough for the purpose bi + "The amplitude and extreme acceleration of the apparent motion are of course proportional to A, which, according to any pulsatory theory of gravitation, would be a large number. The value unity has here been given to A merely by way of illustration. Gravitational Problem. 95 bodies at its surface is far from providing a delicate test for the existence of rapidly periodic fluctuations of gravity. The simplifying assumptions which have been made above are not such as to alter the essential aspect of the problem ; no doubt the substitution of more correct (mean) values for the elastic constants involved would lead to somewhat different results, but the general character of the conclusions would remain as before. 12. The apparatus used in the Cavendish experiment, as carried out, for instance, by Mr. Boys, would indicate rapid fluctuations of gravity with far greater sensitiveness ; on the other hand, Prof. Poynting’s form of the experiment, wherein a beam-balance was used, would be practically unaffected by such a state of things. Kennington, near Oxford, June 26, 1909. Postscript: Nov. 17, 1909. I have been reminded by Dr. Chree, President of the Physical Society, that my simplifying assumption of an elas- tically homogeneous earth must necessarily be far from the truth, the pressure at the centre being something like 2800 tons per square inch. This I am most ready to admit ; but although the assumption requires modification, we do not know to what extent. Tidal phenomena afford some infor- mation as to the rigidity of the earth as a whole, but it seems. very probable, as Dr. Chree points out, that the bulk-modulus of elasticity of the more central portions is increased by the pressure thereon to a far greater extent than is the rigidity- modulus. This would reduce the amplitude of the actual compressional waves discussed in the paper, the observable effects at the earth’s surface being thus correspondingly increased. On the other hand, it appears that my assumed values for k and pw give for Young’s modulus a value consi- derably higher than is sometimes assumed, and this would affect the conclusions in the contrary sense. The problem was one which presented itself in the course of some speculations regarding the mechanism of gravitation ; its solution was not afterwards found to be important in that connexion, but it seemed that, as a mere dynamical curiosity, the matter might be worth a short note. So in the above discussion the assumptions made were the simplest possible, and instead of the most recent authorities being consulted for geophysical data, Lord Kelvin’s original rough comparison between the earth and a sphere of steel was made to serve . For the presentit hardly seems that labour would be profitably expended on a more elaborate treatment. i ak VIII. Note on the Regularity of Structure of Actual Crystals. By Lord Rayurien, 0.M., PRS | dls question must often have presented itself as to how far the mathematical regularities dealt with by the crystallographer are realized in actual crystals. That the natural faces of crystals tend to be plane is fundamental ; on the other hand, it is well known that in practice it is difficult to get any but very small faces to stand the roughest optical test. Explanations of the discrepancy may readily be suggested. The ideal conditions under which alone the tendency to flatness could fully assert itself may be scarcely attainable in practice. The case of surfaces obtained by cleavage would seem to offer a better chance. To test this one naturally refers to mica. Mr. Boys, I think, has somewhere remarked upon the fact that a piece of mica held in front of the object-glass of a telescope does not disturb the definition in the way that a piece of glass does, unless the latter be carefully worked. Mica thin enough to be convenient for such tests is of course too flexible for an examination of flatness. And it is easy to recognize that flexibility is not the only cause of deviation. There are also local irregularities, due possibly to particles of foreign matter or to strains which have exceeded the elastic limit. But these irregularities do not seem seriously to disturb the thickness of a thin plate; and the inquiry sug- gests itself as to how far the thickness of well split mica is really uniform. A very delicate test of such uniformity is afforded by reflexion in the mica of a soda-flame. For a preliminary examination, at any rate, it is best to dispense with ali optical accessories, simply holding the mica in the hand and observing the reflexion of the flame at various parts of the surface and at various moderate angles of incidence. The interposition of a piece of card to shield the eye from the direct light of the flame is convenient. In this way I have examined a number of sheets of superior mica, obtained man years ago from a photographic warehouse. Most of these exhibit serious irregularities in the splitting. The surface is divided into patches where the reflexion varies, the patches themselves appearing uniform and the boundaries sharp to an eye focussed upon the plate. In some cases the boundary- lines cross one another—a feature difficult to understand until it is remembered that the irregularities may be upon * Communicated by the Author. Regularity of Structure o, Actual Crystals. 97 both sides of the plate. The absence of irregularities must not be concluded too hastily. It will happen occasionally that at a particular angle of incidence there is little differ- ence to be recognized upon the two sides of a line of division which is conspicuous enough at a somewhat different incidence of the light. The reason will be apparent to every one familiar with interference phenomena. In some sheets a considerable area appears uniform; and there were two or three on which no abrupt changes of brightness could be detected however the incidence was varied. One of these was submitted to further observation by the more elaborate method described in ‘ Scientific Papers,’ iv. p. 56, in which provision is made for maintaining constant a small angle of incidence. No differences of brightness could be perceived; but I am not sure that in dealing with a flexible and not perfectly flat sheet the method is really more searching. The conclusion that I felt justified in drawing was that there is no abrupt change of thickness capable of producing a shift of } or 1, of an interference-band, and no gradual change giving a shift of say + of a band. The interpretation is discussed at the close of this note. In order to submit the equality of thickness to a further test, I divided the sheet of mica into approximately equal parts, cutting it with a pen-knife guided by a straight-edge. A comparison of the relative areas and relative weights of the two parts would then give material for a comparison of relative thicknesses, or at least of relative densities reckoned superficially. The weights (w, w’) of the two parts distinguished as “plain” and “ marked” respectively were easily found, the difference being determined with special care by weighing them against one another. The weight of plain was *2466 gm., and the difference was ‘00828 gm., so that w! —w ee "00828 el w "2466 The comparison of areas was a more difficult matter, if only because the edges were not everywhere well defined. Both pieces were very nearly rectangular in shape, the shorter side (a) of plain being 49:1 mm., and the longer (4) being 086 mm. The comparison was effected with the aid of a reading microscope, which was used to measure, not the whole width, but always the difference. For this purpose the two pleces were approximately superposed in such a manner that the edges were nearly parallel but sufficiently separated to Phil. Mag. 8. 6. Vol. 19. No. 109. Jan. 1910. H 98 Lord Rayleigh on the Regularity of avoid confusion. Thus alon o the shorter side measurements of the overlap were made at the ends L, Nand at the middle M, and on the parallel side at P,Q, R. The differences of these overlaps, suitably averaged by the formule of the calculus of finite differences, give b'—b. It is hardly necessary to explain the process further. The results were 2 ‘— b’—b = 2) 039, —< = +'0379 ; and thence all’ —ab a’ =a A ome (a’—a)(b’—)) ab a b ab = +:0379 —:0032 —:0001=:0346. This is the fraction by which the area of marked exceeds that of plain. The difference between this fraction and that found above for (w’—w)/w indicates that marked is thinner than plain by the fraction :0010. The total thickness is most easily found from the weight and area with the aid of an assumed specific gravity (2°8). For plain we get thickness ="0306 mm.=:00120 inch. The difference of thickness in the two pieces resulting from the observations on weight and area is thus little over a millionth part of an inch, and might perhaps be considered as devoid of significance. But it is difficult to admit that either the weights or the areas could be in error to the extent of a thousandth part. J am disposed to think that the dis- crepancy is real in the sense that it would not be eliminated by repetition. Possibly it may be attributed to differences in the condition of the surfaces which had been handled more than was good for them. ‘There are also scratches and other minor irregularities to be considered. Structure of Actual Crystals. 99 As regards grease it may be worth while to mention the result of a few observations on greasing one side of a “lantern ” glass plate of 68 sq. cm. area (3} in. x 34in.). Two such were opposed in the balance, one face of each being greased alter- nately over its whole area with heavy marks from a finger which had touched the hair. The difference between the weights of a plate with one face thus greased and clean came out about+!,mg.* The greased area here would exceed that of both faces of one of the mica plates. Not feeling quite satisfied with the edges formed by the knife, which under the microscope compared unfavourably with the original edges, I determined on another set of measurements in which the new edges were made by scissors. The objection in this case is that it is less easy to cut straight, but I thought that the prejudicial effect might be obviated by cutting the two pieces together superposed. The mea- surements were conducted as before except that now four measurements of the overlap were taken along each edge. But the discrepancy resulting from a comparison of the relative weights and areas was not removed, being indeed a little greater than had been found before from what were practically the same pieces. The total thickness above reckoned is about 50 wave- lengths (A) of soda light, so that a difference of a thousandth part corresponds to ;),’. But as regards the formation of interference-bands in the optical examination, this difference must be multiplied by 3, being doubled in virtue of the double passage of the light reflected by the hinder surface, and increased again in the ratio 3: 2, or thereabouts, in virtue of the refractive index of the material. The discrepancy sug- gested by the measurements of weight and area thus amounts to about + of a band, and is accordingly not much below the limit marked out on the ground of the merely optical examination. The conclusion is that some plates of mica are uniform in an extraordinarily high degree and that there is perhaps no reason for doubting that the thickness over finite areas may be as uniform as is consistent with a molecular structure. The stringency of the optical test might probably be increased a few times by silvering the surfaces after the manner of Fabry and Perot (compare Phil. Mag. xii. p. 489, 1906). * The mean thickness of the layer of grease would thus be about 3A. H 2 lool IX. The Recoil of Radium C from Radium B. By WALTER Maxower, W.A., D.Sc., Assistant Lecturer and Demon- strator, and SIDNEY Russ, D.Sc., Demonstrator in Physics in the University, Manchester *. INTRODUCTION. ‘TT has been shown in a previous paper f, that during a radioactive transformation involving the expulsion of an a particle, the residue of the atom from which the @ particle has been expelled recoils in an opposite direction to that in which the a particle is emitted and can travel a considerable distance through a gas if the pressure is sufficiently low. A similar effect was also demonstrable in the case of the trans- formation of radium B into radium C, although this change is supposed to be accompanied by the expulsion of only 8 rays, no a rays having ever been detected. (It has recently been shown by Hahn and Meitner that radium C is complex ; but it is only the product whose period is 19 minutes with which we are concerned in what follows f.) Now it was thought that this transformation was worthy of further study, since it is possible to investigate it by itself without any complications arising from other radioactive processes taking place simultaneously. Ifa plate is exposed as the negative electrode in an electric field to radium emanation for a long time, radium A, radium B, and radium C will be found on the plate after removal from the emanation; but if the plate is then left for half or three-quarters of an hour, the radium A on the plate will have diminished to a small fraction of its initial value, and we are left with only radium B and radium OC, which are being transformed respectively to radium C and radium D. A disk which is suspended above sucha plate therefore receives radium C and radium D projected on it as a result of the recoil during the transformation of radium B and radium (©. Now, since the time period of radium D is exceedingly long, the activity of the disk due to the presence of this substance after exposure to the active plate is inappreciable, and we should therefore be in a position to study the disintegration of radium B by itself. The phenomena were, however, soon found to be more complicated than had been anticipated, and it was found possible only under certain conditions to study the recoil of * Communicated by the Physical Society: read November 26, 1909. + Russ & Makower, Proc. Roy. Soc. A, vol. lxxxii. 1909. + Hahn & Meitner, Phys. Zeitschr. x. no. 20, pp. 697-703 (1909). The Recoil of Radium C from Radium B. 101 radium C from radium B without the interference of secondary disturbances. Before discussing these conditions it will be necessary to briefly describe the apparatus used and the method of con- ducting an experiment, which were essentially similar to those employed in the experiments quoted above. The glass vessel V (fig. 1), of 1°9 cms. diameter, was Fig. 1, GAEDE PUMP. provided with stoppers at both ends. A brass support A was fixed as shown in the diagram, and an active plate P which was to serve as the radiating surface could be placed upon it. From the other stopper a disk was suspended by a wire and exposed to the radiation from the plate mountedon A. After an exposure of the required time with the air in V at the desired pressure, the disk B could be removed and tested for its activity in the usual manner, either by means of a quadrant electrometer or a sensitive a ray electroscope. 102 Dr. W. Makower and Dr. 8. Russ on the The method of experimenting was as follows :—A platinum plate, mounted on an iron disk to give it rigidity, was exposed in an electric field to radium emanation over mercury for at least three hours to allow the maximum amount of active deposit to be collected on the platinum surface. After removal from the emanation the iron-platinum plate was allowed to stand for a time varying in different experiments from half an hour to three-quarters of an hour before mounting on the support A (fig. 1). This interval was used to get rid of the emanation dragged out, from the vessel containing the emanation, by the disk and adhering to it. It was no easy matter to do this satisfactorily, but after a number of trials the best plan was found to be to put the plate in an evacuated tube kept at about 360° C. by an electrically heated furnace. Air was occasionally admitted and pumped out again to remove the emanation as it was given off by the plate. In this way, the adhering emanation was almost completely removed from the plate without appreciable loss of radium B by volatilization. The quantity of active deposit on the plate after this treatment was then tested by a y ray electroscope in the usual manner by com- parison with a standard quantity of radium. Conditions for obtaining Pure Radium C by Recoil. Attention has already been drawn to the fact that it is only under special conditions that the recoil of radium C from radium B can be studied by itself without the interference of disturbing causes. It was soon found that if the disk B (fig. 1) was exposed to the active plate P in a vacuum, there was in general radiated to the disk, not only radium C, as was to be expected, but also a certain amount of radium B. Now, since in all the experiments to be described in this paper, a long interval of time was allowed to elapse between withdrawing the active plate from the emanation and mounting it for an experiment, the proportion of radium A left on the plate must have been very small. It was therefore some- what surprising that any radium B should reach the disk, and the facts seem capable of only two explanations. Hither — the proportion of radium C particles shot off from an active plate by recoil is exceedingly small compared with the total number of radium B particles breaking up on the plate, or the radium B is in some way detached from the plate and carried away during the recoil of radium D from radium C. If the first explanation is correct, then, although the ratio of the number of radium A particles on the plate to that of radium B is small, the actual number of radium B particles a Recoil of Radium C from Radium B. 103 leaving the plate by recoil from radium A might be com- parable with the number of radium C particles. It will be seen later that at least one-tenth of the radium B particles which recoil from radium A actually leave the plate. The second explanation of the phenomenon seems probable if there is deposited on the plate sufficient active deposit to form a double layer of atoms, in which case a radium B particle, if deposited on the top of a radium C particle, might easily be mechanically carried off the plate during the formation of radium D. It will be seen later that each of the two causes suggested is in all probability partly responsible for the ejection of radium B from the active plate. On account of these considerations it was thought that in order to obtain pure radium C on a disk exposed to an active plate, as long a time as possible should be allowed after removing the plate from the emanation to allow the radium A on it to decay to an inappreciable quantity. Moreover, it seemed desirable to use the smallest quantities of emanation that would permit of the subsequent measurements being made with sufficient accuracy. The following series of experiments shows how a condition can thus be reached in which the active plate radiates practically pure radium C to a disk exposed to it in a high vacuum. The plate to be rendered active was exposed for some hours to a considerable quantity of emanation, after which it was withdrawn and heated in a vacaum for 27 minutes to remove any adhering emanation. The quantity of radium C on the plate was then measured on av ray electroscope, and found to be equal to the amount in equilibrium with 8:05 mgms. of radium bromide. Six minutes later it was mounted on the support A (fig. 1), and a disk suspended a few millimetres above it to receive the radiant active matter emitted by it. After an exposure of 20 minutes the disk was removed, and the rate of decay of the activity collected on it measured by a quadrant electrometer. The results obtained are shown in fig. 2, curve I. (p. 104). It will be seen that the activity of the disk fell to half value in about 33 minutes, indicating that there were both radium B and radium C on it. After 80 minutes from the time at which the active plate was removed from the emanation, a second disk was exposed to the radiating plate, and a similar series of observations made with it after an exposure of 20 minutes. The results are given in curve II. and show that even after this time the disk had received some radium B as well as radium (, for the activity took 244 minutes to fall to half its initial value, although the 104 Dr. W. Makower and Dr. 8. Russ on the radium A on the plate must have fallen to a hundred millionth of its initial value by the time this exposure was begun. A third series of observations was therefore made with another disk beginning 184 minutes after the removal Log.of Acmvirr rs Fig. 2 25 T= 244 ming I . T' = 20 MLNS. h VG) aO 3O 40 MINUTES. of the plate from the emanation. By this time the quantity of active deposit on the plate had become so small, that on testing the disk its activity was found to fall to half value in 20 minutes, indicating that the matter radiated to the disk now consisted of practically pure radium C (see curve III.). The experiments just described indicate that even when there was practically no radium A on the plate, a certain amount of radium B reached the disk suspended above it in vacuo; and, as we have seen, this phenomenon can be explained if we suppose that radium B is mechanically removed from the plate during the recoil of radium D from radium C. Since, in addition to the interest of the pheno- menon itself, this action is lable to produce disturbances when studying the recoil of radium C from radium B, the matter was investigated in greater detail. Now, since the recoil of radium D from radium © takes place as the result of the emission of an @ particle of high velocity, whereas in the transformation of radium B into radium C no such particles have been detected, it is evident that the energy with which the radium D residue leaves the disintegrating atom must far surpass that of radium C when expelled from radium B. If, therefore, using the same apparatus as previously, the pressure of the air between the radiating plate and the disk were increased to such a value that all the radium C leaving the plate by recoil from radium B was prevented from reaching the disk by the intervening Recoil of Radium C from Radium B. 105 air, any radium B or radium © found on the disk after exposure could only have reached it by secondary mechanical projection during the more energetic recoil of radium D trom radium ©. Moreover, the number of particles of radium B and radium UC expelled from the plate should have been in the ratio of the number of particles of radium B to that of radium C on the plate at the time of an experiment. The point was tested in the following way :—A disk was suspended 7 millimetres above a strongly active plate with no radium A left on it and exposed to the radiation from it for 10 minutes, the pressure of the air between the plate and disk being maintained at 3°3 mm. of mercury. After exposure, the disk was removed and the time period of the activity received by it tested. As may be seen from fig. 3, e ~/ : 7 in © tag : S | 13 | 10 &0 50 ere ay LYINUTES. the activity was found to fall to half value in about 40 minutes, and this may easily be shown to mean that the numbers of radium B and radium © particles on the disk, when tested, were nearly equal. This was to be expected since, under the experimental conditions, the amounts of radium B and radium C present on the plate must also have been almost equal. It will be noticed that experiments were made with the plate charged with positive and the disk with negative : electricity and vice versa, and also with the plate and disk at the same potential. This was done because it was thought that possibly the radium C produced from radium B_ by recoil might become charged after being stopped by the air. 106 Dr. W. Makower and Dr. S. Russ on the If no field existed between the plate and disk, this radium C would diffuse to the disk and plate and to the walls of the containing vessel, whereas in an electric field it might be attracted to one or other electrode. Any such action would have complicated the results, and the effect of an electric field of 340 volts per centimetre was therefore tried. It will be seen, however, that the field had very little effect, the time period of the activity received by the disk being almost independent of the direction of the field, as may be seen from curves II. and ITI. (fig. 3). The result just arrived at, that radium B and radium C are mechanically projected from the plate as a secondary effect due to the recoil of radium D from radium C, is not a little surprising when we consider the possibility of the formation of a double layer of atoms of active deposit. Tor the area of the plate was about one square centimetre, and if we take the diameter of a molecule to be 2x 10-° cm., we see that it would require about 3x10” active deposit particles to completely cover the plate with a single layer of atoms. But since the quantity of radium C in equilibrium with 1 gram of radium emits 3°4 x 10!° a particles per second”, it follows that in this quantity of radium C there are 5°77 x 10 particles. Hence, since in the above experiments there were about the same number of radium B and radium C atoms ‘on the plate, it would have required the amount of these two products in equilibrium with 26 grams of radium to form a single complete layer of atoms on the plate. This is certainly 3000 times as much as was present on the plate in the experiments, and to explain the observed effect it seems necessary to suppose that the active deposit is not evenly distributed over the plate but deposited in heaps at certain places on the plate, the greater portion of which is entirely free from deposit. W hether this localization of the activity is due to the non-uniformity of the electric field during the exposure of the plate to the emanation caused by the uneven- ness of its surface, or to some other unknown cause, remains a matter for speculation, but the existence of this aggregation of the deposit is certainly striking. It may be ‘mentioned that an active glass surface behaved quite similarly to the platinum surfaces generally used, so that the phenomena do not appear to depend on the smoothness of the surface. To emphasize the extreme difficulty of getting absolutely pure radium © by the recoil method, an experiment may be cited in which a plate was exposed to a small quantity of * Rutherford & Geiger, Proc. Roy. Soc. A. vol. Ixxxi. p. 141. Recoil of Radium C from Radium B. 107 radium emanation for some hours, when it was removed and transferred to an evacuated vessel placed in a furnace at 360° C. for nearly half-an-hour to remove emanation. The plate was then tested by its y radiation and found to be equivalent in activity to ‘066 milligram of radium bromide. A disk was then suspended above the plate in vacuo, and after an exposure of 20 minutes, removed and tested by a sensitive a-ray electroscope. The decay of the activity on the disk is shown in fig. 4. It will be seen that the activity Fig. 4. ae p S bg <2 6 > NS & = 8 zs - 70 70 30 40 50 CO 7 BO IV INUTES at first began to decay with the period of Ra, viz. 19 minutes, which gradually became longer as time proceeded, indicating that even in these circumstances there was present on the disk a small quantity of radium B. The Proportion of Radium C projected from a Plate / ’ projected f by Fecoil. In a previous paper * it was shown that when a disk was exposed above some emanation condensed by liquid air at the bottom of a tube, the amount of radium A and radium B projected on to the disk was about one-eleventh ot the quantity which would have reached it if every a particle had been effective in causing a recoil as it left a disintegrating atom. Having regard to the easy absorption of this radiant active matter, this was perhaps as large a fraction as was to be expected, for any slight film of condensed water-vapour or other gas over the surface of the condensed emanation would tend to prevent the escape cf the active matter from the surface. A similar result has since been obtained for * Loe. cit. 108 Dr. W. Makower and Dr. S. Russ on the the recoil of radium B from radium A deposited on a plate. In the case, however, of the recoil of radium C from radium B, which is much less energetic, it was at once seen that the fraction of the total number of particles of radium C escaping from an active plate to the total number formed was very small. This was clear without accurate measure- ment for, whereas the activity of the plate could be measured in a y-ray electroscope, the activity on a disk exposed to it for 20 minutes could be measured only by careful observations with a sensitive «-ray electroscope. It was therefore necessary to carry out a series of quantitative measurements to determine the fraction of the number of recoils of radium (C which resulted in its removal from the surface of the plate. For this purpose, the activity of the radiating plate was compared with that of a standard quantity of radium by means of a y-ray electroscope in the usual manner. The activity on a disk exposed in vacuo at a known distnnce from the plate for 20 minutes, was measured on a sensitive a-ray electroscope and compared with that of a polonium standard * giving outa known number of « particles per second. In this way the number of « particles emitted from the active disk would be found and the number of atoms of radium C on it deduced. In a long series of experiments in which the activity of the radiating plate was equal to that of 2 milligram of radium bromide, the quantity of radium C collected on a disk suspended 2°5 cm. above it for 20 minutes in vacuo varied between the equivalent of 5x 10-7 and 1:4x 10-° milligram of radium bromide. Now, if half the radium C particles produced had been projected upwards from the plate, the disk, which was of diameter 1°7 cm., should have had on it the number of radium C particles equivalent to about 4x 10-3 milligram of radium bromide at the end of the exposure. Thus the quantity of radium C reaching the disk varied from s000 tO 3) of the total quantity which could have reached it. The smallness of the amount of radium C escaping from an active surface by recoil has recently been noted by Hahn and Meitner +, who obtained little more than one millionth of the total obtainable quantity of radium C on a plate exposed to a surface covered with radium B. This is a far smaller quantity even than we have found ; but the experi- ments of Hahn and Meitner were carried out at atmospheric * This standard was kindly lent to us by Dr. Geiger, who determined the number of a particles emitted by it per second. + Hahn & Meitner, Phys. Zeitschr. x. 1909, pp. 697-703. Recoil of Radium C from Radium B. 109 pressure, which may explain the smallness of the effect obtained by them. The lack of constancy of the amount of radium C emitted from the surface upon which radium B had been deposited is remarkable, seeing that the plates used were always treated in an exactly similar manner. But it was perhaps even inore surprising to find that, working with a single surface, the power of emitting radium C varied considerably if the surface was allowed to stand untouched. After making due allowance for the decay of radium B, it was found that an active surface after standing for half an hour or an hour, altered its power by emitting radium C, sometimes becoming a more powerful and on other occasions a less powerful radiator. The cause of these changes in the surface is very obscure, and it is difficult in the present state of our knowledge to advance any satisfactory explanation of it. Returning to a consideration of the small amount of radium C which was in all cases emitted from the active surfaces, it seems not unreasonable to imagine that, on account of the smallness of the energy of recoil of the radium C particles, it is only those particles which come off normally or in directions making small angles with the normal to the plate, that succeed in getting away from the plate beyond the range of molecular attraction. If this were so, all particles emitted from the plate in directions making large angles with the normal to the surface of the plate would be drawn back, and therefore not be detected as radiant matter. To test the correctness of this view, experiments were undertaken to study the variation with distance of the amount of radium © received by a disk exposed in vacuo at different distances from the radiating plate. The apparatus used was that shown in fig. 1. The results obtained are shown in fig. 5 (p. 110). On the same diagram are also plotted the curve showing the variation of activity with distance assuming that particles emitted in all directions from the plate are equally likely to escape from the surface, and also acurve assuming a cosine law for the radiation such as could be applicable to the ease of light *. It will be seen that the experimental curve agrees with neither of these hypotheses, indicating that the radiation from the plate falls oti as the angle which it makes with the normal to the plate increases. The deviation of the experimental curve from that obtained on the supposition that * These curves have been plotted from values as yet unpublished obtained by Mr. H. Bateman. 110 Dr. W. Makower and Dr. S. Russ on the the particles are radiated in all directions at random seems, however, insufficient to entirely account for the smallness of the fraction of radium C particles which succeed in getting free from the radiating surface. Fig. 5. 100; / Zz P| 4- a Note to Curve :— Line thus === represents the Experimental Curve. Inverse Square Law Curve. Cosine Law Curve, y wie ” ” ) ) ” The Absorption of Radium C by Air. When radium B is transformed into radium C, the process is supposed to be accompanied only by the emission of 8 particles of low velocity *. It therefore follows that the energy possessed by a radium C atom after recoil should be exceedingly small compared with that of an atom of radium B when produced from radium A or with that of any atom which recoils as the result of the emission of an & particle. On this account, the power of penetrating matter possessed by an atom of radium C should be very much smaller than that, say, of an atom of radium A or radium B when projected from a dis- integrating atom. For if we take the mass of an atom of radium A to be 218 and that of an « particle to be 4, and its * H. W. Schmidt, Phys. Zeitschi. vi, p. 897, and Annalen der Physik, xxi. p. 609. Recoil of Radium C from Radium B. UML velocity to be 1:77 x 10° centimetres per second *, then, from the equation of momentum, the velocity with which the particle of radium B must be travelling after recoil will be 3°25 x 10’ centimetres per second. It is evident thatan atom of radium A when formed from the emanation will recoil with approximately the same velocity. On the other hand, since the mass of an atom of radium B is 214 and that of a 8 particle 755, then, if we assume the velocity of the 8 particle from radium B to be 101° centimetres per second f, the velocity of expulsion of the radium C atom formed by the process will be 2°75 x 10* centimetres per second. It therefore follows that an atom of radium B will recoil with 1:39 x 10° times the energy of an atom of radium UC. The penetration of matter by radium C should therefore be less than one- millionth of that of radium B, if the power of an atom of penetrating matter is proportional to its energy. Now the latter radiation is exceedingly easily absorbed, being cut down to one-tenth of its initial value by passing through 6°5 centimetres of air at a pressure of one millimetre of mercury. It might therefore be expected that even a minute quantity of air would be sufficient to entirely stop radium C from escaping any distance from a deposit of radium B. ‘This is, however, not the case ; for the radium C expelled by recoil from radium B can penetrate appreciable quantities of air before being stopped, and it was even found possible to measure its absorption by air, though the experi- ments were difficult and the accuracy of the results not great. The apparatus used was that shown in fig. 1. In view of the very large absorption to be expected, and on account of the necessity of working with small quantities of radium B in order to get pure radium C projected on to the disk B, it was at first thought desirable to bring the disk close to the radiating plate. The first experiments were therefore made with the disk suspended 2 millimetres above the active plate, and the radiation received by the disk after an exposure of 20 minutes to the plate measured, the pressure of the air.in the vessel V being varied. It was, however, soon found that working in this wav very irregular results were obtained, and it seemed that the disturbances were produced by secondary causes due to the diffusion of the radium C particles when stopped by the air ; for if a certain proportion of the atoms of radium C projected from the active plate are stopped by impact with * Rutherford, Phil. Mag. Oct. 1906. + There appear to be no available data regarding the velocity of the 6 particles emitted by radium B; but the value assumed is such as to give an upper limit to the velocity of expulsion of radium C. 12 Dr. W. Makower and Dr. S. Russ on the air molecules before reaching the disk suspended above, then it becomes a matter of chance whether such, particles diffuse on to the disk or back again on to the plate. In fact a small current of air passing between the plate and disk might make all the difference in deciding whether the disk received much or little radium C from the plate. The distance between the disk and plate was therefore increased as far as was possible ‘without rendering the amount of radium C reaching the disk unmeasurably small. With the disk 2°5 centimetres from the plate, and using the active deposit equivalent to *2 milligram of radium bromide on the surface of the plate, it was found just possible to make the required measurements. Hven in these circumstances the experiments were difficult to perform with any precision, for, as has been mentioned above, the amount of radium © expelled from a plate coated with a definite quantity of radium B is a variable quantity, and it was therefore necessary not only to test the quantity of active deposit on the plate before starting an experiment but also to determine its power of radiating radium C. Unfortunately, the measurements were still further complicated by the changes in the radiating power of the surface when left to stand ; the surface had therefore to be tested after as well as before an experiment, to make sure that it had not changed enough to completely vitiate the results. Surfaces have never yet been obtained which maintained their power of radiating radium C perfectly satisfactorily, but the changes were erratic, being sometimes very large and sometimes comparatively small. Only those experiments were made use of in which the changes of surface were reasonably small ; other experiments were rejected. The method of conducting an experiment consisted in exposing a platinum plate, mounted on an iron disk to give it rigidity, to some radium emanation for several hours. The plate was then removed from the emanation and allowed to stand for three-quarters of an hour to allow radium A to decay to an inappreciable amount, the emanation adhering to the plate being driven off in the manner already described. The activity of the plate having been measured by a y-ray electroscope, it was mounted on the support A and the disk B was suspended 2°5 cm. above it to receive the radium C expelled from the plate. The air in the vessel V was then pumped out as quickly as possible by a Gaede pump, and after 20 minutes the disk B was removed and tested by a sensitive a-ray electroscope. It was noticed that both sides of the disk were radioactive when tested, though the activity of the front side of the disk which had faced the radiating plate was always greater than that of the back. Now no active matter could Recoil of Radium C from Radium B. L13 have reached the back by direct radiation, and its activity was at first attributed to traces of emanation left on the plate and gradually escaping from it. But it was subsequently shown that this was not the case ; for when the disk was exposed to an active plate for a long time, so that all the radium B and radium C present had decayed, neither side of the disk showed the slightest activity. No doubt the real explanation of the effect is that a portion of the radium C emitted by the plate is stopped before reaching the disk by the residual air left even at very low pressures and then diffuses round to the back of the disk. In order to find the amount of radium C directly shot on to the disk, the activity of the back of the disk was subtracted from that on the front. After exposing a disk in this way another similar one was then suspended above the plate, and the pressure of the air in V adjusted to any desired value which was read on a McLeod gauge. After 20 minutes this disk was removed and tested,and a third one was inserted and exposed to the radiation for 20 minutes iz vacuo. Allowing for the decay of radium B on the plate, it was easy to calculate the amount of radium OC which should have reached this third disk in terms of the quantity which reached the first disk exposed. If the quantity of radium C found on the disk proved to be very different from the calculated amount, the experiment was rejected ; in other cases, the mean of the two readings obtained in vacuo was taken. In this way it was possible to get some idea of the TABLE I, | Whe wid | Activity in Arbitrary Units. Percentage of | } te A PRE ee of Mereury. Front. Back. ea Readings. | ack. q | ao eae SSE tee eshis 2 teed Oll4 85 36 | 49 88 | 019 1040 182 | 858 61 020 259 Be he ggasi es 64 028 226 61 165 | 72 036 347 65 282 53 O41 186 66 | 35 043 88 | 44 44 | 4G O45 833. | 145 188 49 066 89 31 ) o4 | | 073 pice | 38 BS) 4) 14 i Phil. Mag. 8. 6. Vol. 19. No. 109. Jam. 1910. I 114 Dr. W. Makower and Dr. S. Russ on the law of absorption of radium C by air. The results are given in Table TI. (p. 113) and fig. 6. The absorption at different pressures can be seen from'the last column, which gives the quantity of radium C reaching the disk, the mean of the two vacuum readings being taken as 100. Fig. 6.—Absorption of RaC by Air. 100 Although the results are not very consistent, the order of magnitude of the absorption can be seen from the numbers iven. By a comparison with the absorption of the radiant maiter from the emanation condensed at the bottom of a tube in liquid air, studied in a previous paper, it will be seen that radium A and radium B, when expelled as the result of a recoil from an « particle, can penetrate only about 40 times as much air without being stopped as radium C when it recoils as the result of the emission of a @ particle from radium B. Now it has already been calculated that the energy possessed by radium C should be less than one- millionth of that possessed by radium A or radium B after recoil. The order of magnitude of the absorption of radium C is therefore far smaller than was to be anticipated. It is difficult to explain this discrepancy ; if an @ particle of slow velocity were expelled during the transformation of radium B Recoil of Radium C from Radium B, 115 into radium C, the difficulty would be removed. But it must be admitted that the possibility of the existence of an un- detected « particle in this transformation seems remote in view of other evidence. There still remains the question whether all the excess of activity on the front of the disk over that on the back at the higher pressures is due to radium C projected directly on to it, or whether some of it finds its way there by diffusion after having been stopped by the residual gas left even at the highest obtainable vacuum. To test this possibility the following experiment was made :—According to the procedure already detailed, a brass disk was fixed 1:4 centi- metre above a radiating surface, but shielded from direct radiation by a copper screen placed between the surface and the disk so that any activity obtained on the disk must have reached it by diffusion. The exposure lasted half- an-hour in vacuo, and on removal the disk was found to have a very small activity both on the front and back, the activity on the two sides being almost exactly equal. This showed that the excess of activity obtained on the front of the disk over that on the back in previous experiments is not due to diffusion, but to radium C projected directly on to the disk, The Electrical Charge of Radium C. During the course of this work several attempts have been made to ascertain whether radium C becomes charged with electricity at the moment of its formation from radium B. Assuming that radium B emits only @ particles, a simple calculation shows that if radium C is charged, a field of 500 volts per centimetre between the plate and disk (fig. 1) should suffice to cause the recoiling atom to return to the plate from which it started after travelling z}> millimetre. All attemps to stop radium C when it recoils by an electric field have failed, indicating that radium C when formed from radium B either remains electrically neutral or, as is sug- gested by the absorption experiments just described, the energy of a recoiling atom of radium C is, for some unknown reason, greater than has been supposed. We have again to thank Professor Rutherford for his kindness in supplying us with the necessary radium emanation and also for his interest in this work. I 2 poeag i | X. Ductile Materials under Combined Stress. By Water A. Scosie, A.R.C.Se., B.Sc., Whitworth Scholar*. [Plate I.] Introduction. 4 eae theory of combined stress, and the results recorded by earlier writers, were discussed in a previous com- munication +, and the literature of this subject has since been very fully reviewed, from an engineering standpoint, by Mr. L. B. Turner+ and Mr. C. A. Smith§. The most im- portant experimental! results obtained by other observers are given later. The object of this paper is to further consider the results of the writer’s earlier experiments, and to record the data which were obtained from some later tests made on tubes. The results are also compared with those obtained by other observers, who employed different methods and combinations of loading. Further Consideration of the Earlier Tests. Specimens and Apparatus.—The first test bars were of steel, ? inch diameter, and 36 inches long. The ends were of square section, 3 inch side, to allow a torque to be applied to the bar. One squared end was held ina special clamp which prevented this end of the bar from rotating under the torque, but allowed it to take its natural slope as a supported beam under a bending load. The bar was also supported, at 30 inches from the centre of the clamp, on V rollers, which formed the other support under the bending load, and offered no resistance to the torsion of the bar. A bar was bent as a beam 30 inches long, supported at its ends, by a dead load directly applied midway between the supports ; it was twisted by means of a wooden pulley which fitted on the other squared end of the bar. Two flexible wire ropes were attached to the pulley, and exerted a couple upon it because of the weights which they were made to carry. Therefore it will be noticed that when a bar was twisted, it was subjected to a uniform torque all along its length, but the maximum bending moment acted on one section only of the bar. Within the elastic limit of the metal, the shear stress produced by the torque varied from zero at the axis of the bar, to a maximum all over the surface. Similarly the * Communicated by the Physical Society : read November 26, 1909. + Proc. Phys. Soc. London, vol. xx., and Phil. Mag. Dec. 1905, t Engineering, Feb. 5, 1909. § Ingineering, Aug. 20, 1909. Ductile Materials under Combined Stress. A of maximum stresses due to bending were a compression at the top of the bar, and a tension only at its lowest point ; and these were confined to the mid-section, which was under the maximum bending moment. This system of loading repre- sented the most common example of combined stress in engineering practice, which is a shaft subjected to combined bending and torsion, and it had the further advantage that the critical stresses were produced by comparatively small loads. The bending deflexion was measured by. means of a scale which rested on the bar, and was guided in a slide provided with a vernier. ‘This simple arrangement was quite satis- factory on such a long beam. The torsion was measured by a pointer clamped to the bar, which moved over a fixed, finely-divided circle. The twist was also measured at three points by clamping mirrors to the bar; fixed telescopes and vertical scales were used in connexion with these mirrors. The Criterion of Strength. The theory of elasticity is based on Hooke’s law, that strain is proportional to stress. Beyond the elastic limit of a material this law is no longer strictly true, and therefore the usual formule for calculating the stresses cannot be applied. Consequently it is clear that the elastic limit is the correct criterion of strength. But it isnow the usual practice to adopt the yield-point. Guest * stated that Hooke’s law holds to the yield-point, and he considered that the first deviation from proportionality of stress to strain was caused by local variations in the material, which altered the stress distribution, and caused local yielding. Thus he assumed that stress was proportional to strain for the main part of the material until the yield-point was reached, and therefore he selected the yield-point as the criterion of strength. Un- doubtedly the yield-point of a material is less affected by special treatment than the elastic limit, and it is more easily determined, consequently other observers, following Guest, have adopted the yield-point. Unfortunately opinions differ concerning the exact location of the yield-point, and for comparative purposes, in order to have a well-defined point, the writer also neglected the intermediate state between perfect elasticity and complete yield, and obtained the critical loads from the intersections * Proc. Phys. Soc. London, vol. xvii. Sept. 1900. 118 - . Mr. W. A. Scoble on Ductile of the continuations of the lines which represent these two limiting states on the stress-strain curves. It is probable that Guest’s assumption is partly true, because, with the loading adopted by the writer, with which the stresses are unequally distributed, the ratio of the strain to the stress is constant until it increases very rapidly just before the complete breakdown of the specimen. The slight deviation from Hooke’s law, which is noticed between the elastic limit and the yield-point in a simple tension test, is not observed. It therefore appears that for this kind of test the elastic limit and the yield-point practically coincide, and therefore the stresses are now taken, in all these tests, from the point where the strain ceases to be proportional to the stress. The Quantities Tabulated. The strains have not been tabulated because the maximum strain theory is not supported by engineers in this country. The formule relating to combined stress which are given in the text-books are based on the maximum stress theory. Recently, however, the shearing stress theory, or the stress difference theory as elasticians prefer to name it, has rapidly gained favour with engineers. It is, therefore, clear that the maximum stress and the maximum shearing stress are most important froma practical standpoint, but the maximum strain hypothesis is indirectly considered later, when the deviations from the shear-stress law are discussed. Calculation of the Stresses. The maximum tensile stress, p, in the material due to bending is calculated from the formula pio yor in which M is the maximum bending moment ; I is the moment of inertia of the area of the section about its neutral axis, in this case a diameter ; yis the greatest distance of a point in the section from the neutral line, and equals half the diameter of the section.” The maximum shear stress, 8, caused by the torque, is Materials under Combined Stress. 119 calculated from the formula in which T is the torque ; D is the diameter of the bar. For the tubes employed in the later tests the formula becomes otter Dt Sth a Dd 32 in which D is the external diameter of a tube ; d is its internal diameter. Having found p and 8, the maximum and mininum prin- cipal stresses are represented by the expressions baal e+ one of which is positive, and the other is negative. The third principal stress is zero, because in the case of solid bars the stress due to the shearing force ou the section is zero where the bending stress is a maximum. Since the tubes are bent by couples, there is no direct shearing force on a cross-section. The stress difference is the difference between the maxi- mum and minimum principal stresses, and therefore equals 2 / Ps, G2 as 3.5% zo . 4 It is twice the maximum shearing stress. The maximum strain is given by P,;—n(P,+P2) iD >] in which P, is the maximum principal stress ; P, and Py are the other principal stresses ; 27) 18 Poisson’s Ratio ; E is Young’s Modulus of Elasticity. 120 Mr. W. A. Scoble on Ductile In these tests there are stresses in one plane only, and the expression becomes Py —nP,, If the maximum strain be constant, then P,—7P; must be constant, so that this hypothesis comes between the maximum stress law, that P, is constant, and the stress difference or shear-stress theory, that P,—P3 is constant, but it is nearer the former. The Results from Steel Bars. Taste A.—Original Tests of Solid Steel Bars. | | i “Tensile | Shear | yp Mininim | toate Number | Bending Twisting Stress | Stress pee Pri at pe Ea | Moment. Moment. due to | due to | ae g ns Mastae Bar. | lbs. ins. Ibs. ins. Bending.’ Torque. Ib reeca ib acs Sh | Ibs./s. in.| Ibs./s. in,| g./S. 1n. S/S. I. | ae | | Stress. a oe 2190 | 0 26600 26600 | —26600 : 53200 | IV...., 667-5} 2130 | 16220 25880 35260 | —19040 . 54300 Teh LAO 2035 | 28200 | 24700 42520 | —14320 | 56840 | V....| 133b | 1985 | 32850 | 24100| 45210 | —12860 ; 58070 | XID... 2000 | 1360 | 48600 | 16520; 53700 | — 5100 ; 58800 VI..... 2000 | 1630 | 48600) 19800| 55650 | — 7050 ; 62700 IX....| 2020 | 1335 | 49000! 16220; 53900 | — 4900 58800 WIIL, :..|\, 2320 645 58400 | 56300 7840 57350 | — 1050 vids y..| 2420 | 980 58750 | 11900 61075 H if | — 9895 | 63400 1 Only one test was made on each bar. The bending and twisting moments at yield are plotted in fig. 1 (Pl. 1.). If the maximum principal stress were constant at the yield-point, the twisting moment under pure torque would be double the bending moment required to cause the bar to yield. If the maximum shearing stress were constant, the plotted points would lie on a circle described with the origin as centre. Fig. 1 confirms the writer’s original conclusion that the maximum shear stress is approximately constant at failure, but that an ellipse lies between the points better than a circle, and that the bending moment is greater than the torque. The maximum stress varies between 26,600 and 61,075 Ibs./s. in., it certainly is not constant. The stress difference is not exactly constant; it varies from 53,200 to 63,400 Ibs./s. in., but it increases steadily with the bending moment, consequently the deviation from the stress difference, or shear stress law, is still further away from a constant maximum stress, so that the maximum strain also varies considerably. ge ee a a Materials under Combined Stress. Lt The Apparatus for the Tests on Tubes. The tests on tubes were made in a machine which is part of the equipment of the engineering laboratory at the City and Guilds Technical College, Finsbury. The machine was designed by Prof. E. G. Coker, and it has been recently described *. A tube to be tested was sw eated, and when it | was necessary pinned, on two steel holders, or mandrils, each of which was provided with two keyways. One holder fitted into a wormwheel so that this end of the tube could be twisted, but the wormwheel casing was pivotted on roller bearings so that there was no resistance to bending. A lengthening piece was fitted on the holder to carry an over- hung load, which produced a bending moment on the spe- cimen. The other holder was held in a special fitting which was supported on a spindle of small diameter, so that the resistance to torsion was negligible. The fitting was also arranged to carry an ov erhanging load at this end, and an arm projected at right angles to the specimen to support the load which measured the torque. This latter arm was always kept horizontal by turning the wormwheel as the twisting . load was increased. The ‘loading in this machine was also by dead weights. The torque was produced by the twisting load and the extra supporting force which it required at its support. The torque was uniform along the tube. The bending moment was caused by a couple “at each end, com - posed of the supporting force and the overhanging load ; was constant along the specimen. ‘There was no own hy force on a section due to the bending loads. The twist of a tube was measured ‘by Prof. Coker’s torsion- meter, and the bending deflexion by an adaptation of the Ayrton-Perry twisted sirip. The latter apparatus was designed for use during some bending tests in which the deflexion was extremely small, and it was therefore very sensitive for the purpose of the present tests. It was necessary to measure the strains separately, because the writer has : shown that when a ductile material is under combined load- | ing, it does not always yield first in the way which is indicated by the iner easing load. The first yield is probably determined by the loading which produces the greatest shearing stress. The Tests on Steel Tubes. Solid drawn steel tubes were tested, and the yield during each test was kept very small so that several ‘experiments could be made with each tube. All these specimens were * Proc. Phys. Soc. London, vol. xxi., and Phil. Mag. April 1909. 122 Mr. W. A. Scoble on Ductile The results are collected cut from the same length of tube. below. Internal diameter 0°818 in. TABLE B.—Tests of Solid Drawn Steel Tubes. External diameter 0°885 in. | : : eu Stress EN te ata | He nsile Shear || Maximum | Minimum | igeeemeen Bending | Twisting) Stress | Stress Pe cioal | Pinot al Test. | Moment, Moment., due to | due to Str ect ee - | : : : ress. Stress. | Maximum lbs. ins. | lbs. ins. Bending., Torque. Teles 5 ling tel Shale TA ea wee s./s. in. | lbs./s. in. ear wis: | Stress, | | | | ‘Se A eek {shoe SAS | glee | | | | Baer. 0 | 940 | 0 25600 | 25600 | —25600 51200 | BS. ae 0 L030) «i 0 28050 28050 | —28050 56100 | ees 300 | 1090 | 16380 | 29700 38970 | —226380 61600 | LOR Nees 600 | 970 | 32700 | 26400} 47400 | —14700 62100 | 1D eae 800 | 600 43550 | 16330 48980 | — 5420 54400 | Oh eae 810 700 | 44100 | 19100 51250 | — 7150 58400 (2) aan 1100 400 59800 | 10900 61770 | — 1870 63640 12ie Nasa 1120 800 61000 | 21800 68000 — 7000 75000 Me 1250 | 0 68100 | 0 68100 0 68100 | ae 1330 O | 72400 0 72400 0 72400 | Bids once 1400 O | 76200 0 76200 ) 76200 | WF Ola. .s: 1400 | 3800 | 76200 8165 77000 — 800 77800 | The maximum stress has 25,600 and 77,000 Ibs./s. inch for its extreme values, and the stress difference varies from 51,200 to 77,800 Ibs./s.in. The bending and twisting moments are plotted in fig. 2, in which the numbers of the tests are shown against the points. ‘The letter refers to the tube, and the number indicates the order in which the test was made on that specimen. The figure is interesting because it shows very clearly the effect of repeated loading. An ellipse is drawn to le evenly between the points. Specimens cut from the same length of material are not exactly alike, and an error is possible in locating the yield-point ; but after allowance is made for these facts, it is evident that the vield-point was raised’ by a previous test. Nevertheless, the results clearly indicate that the maximum shear stress is more nearly constant than the principal stress, and that the bending moment is again greater than the torque, so that the shearing stress and the stress difference increase with the bending moment. The Tests on Copper Tubes. Tests were also made on solid drawn copper tubes, of 0°79 inch internal and 0°881 inch external diameter. The specimens were given a set to correct for their defective ee ae ee ee ee ee eer SS ee Materials under Combined Stress. 123 elasticity, and to allow several tests to be made on each tube. As in the case of the steel tubes, the yield during each test was kept small. The data obtained from these tests are tabulated below. TaBLE C.—Tests of Solid Drawn Copper Tubes. External diameter 0°8812 inch. Internal diameter 0°790 inch. | | sa | Stress : _.. | Tension, 2°" | Maximum | Minimum | Difference Bending /Twisting | Stress ay er SEE ae 2 | due to | Principal | Principal == JEG Test. |Moment.} Moment. Hades | due to Sires es oe Maxi pee aredbs ans, (228 ™S-Ipwistine. ress. Stress. | Maximum | sages lbs./s. in. ihe ee lbs./s.in. | lbs./s. in. Shear / S./8. ae! | Stress. lean: 0 1150 0 24200 24200 | —24200 48400 Bes odes 300 1180 12620 | 24800 31910 | —19290 |; 51200 Oa 500 | 1100 | 21040 23150 | 35920 | —14880 50800 Oe 800 910 33660 19140) 42350 | — 8670 51000 te ee 1000 730 42080 15360 47080 | — 4000 52080 Oi cane 1100 600 46300 = 12620 49510 | — 5210 52720 EY Ziadie 1300 300 54700 s- 6810 | = 55450 =| — =750 56200 B'S aint 1330 0 56000 0 | 66000 | 0 56000 Here again the stress difference is approximately con- stant, and the deviation from this law is opposed to a constant maximum stress, because the stress difference increases steadily with the bending moment. The bending moment and the torque are plotted in fig. 3. Deviations Jrom the Shear-Stress Law. The stress difference is given in the tables, because it is the difference between the maximum and minimum principal stresses, which it follows. It is now more convenient to deal with the maximum shear stress, which is half the stress difference. The stress difference and maximum shear-stress laws are therefore practically alike, but the former does not indicate the existence of the shearing stress which appears to cause the actual fracture of a ductile material. . and call e the curl. For finite strains the curl thus defined is not correctly described as the double rotation. We shall speak of x and e as of the first order because they are of the first degree in 7; similarly T is of the zeroth and first orders, v of the first and second orders. The ratio ds'/ds of the strained element ds’ of volume to the standard value ds will be denoted by m; it might be called the total dilatation or total expansion, m—1 being called the dilatation or expansion. The three parts of m of first, second, and third orders will be denoted by m’, m”, m'”’. Thus (7) ma=ds'/ds=1+m!' +m!" +m!" i =45ViV2V5Sp1'pe'p3' m =—SV9, m”= —ESVV VeVi, m” =4ASViV2Vs8m MMs (8) m’ will be called the divergence and —m’ the convergence. [Thus in this paper the curl, the convergence and the divergence stand for the curl of 9, the convergence of 7 and the divergence of 7. | The density in the standard and strained states will be denoted by n, n’, so that nds=nds, n =m nn, 2: nn A vector element of surface in the standard and strained states will he denoted by d=, d’, so that dp! =Tdp, dy =mT'-1d, ds’ =mds. . I) The potential energy of the element, ds, will be denoted by wds; also if convenient by w'ds’. The characteristic of a fluid is that w is a function of m only ; of a solid that w is a function of v, that is of T’T, which is not a mere function of m. If z is a scalar function of v, the differential operator is defined by dzg=—Sdvtd2el. . .. ae Thus @ is a symbolic self-conjugate linity capable of explicit =_—_so. of Electrons in an Elastic Solid Ether. 131 expression. If < is a scalar function of y, the differential operator Cis defined by ees a A a Oo) Thus C is a symbolic linity capable of explicit expression. w is a function ol v, and is therefore a function of y. For such a function there is of course a definite connexion between C and d. It is easily proved to be lee ie A ti, ee hae Sela a EP) Since g is self-conjugate it follows that not @ but AT" is self-conjugate and T~'C is self-conjugate. It may be shown that the necessary and sufficient condition for z to be a mere function of v is that T—'C< is self-conjugate. In (11) and (12) %,¢ are used with their usual bilinear meanings given by Qe, V=QAE t) +Q(7,7) + QA, &), Q(o1, Ci Oo, o) =Q(i, i, l, v) + Q(?, ty JJ) ana: < The meanings now defined of the symbols qd, d, K, F, G, e, f, g,m,m’,m”, m'", n, 2', w, w’, V7, 1, dx, a2, / ! €, £7, p, p’, ds, ds’, v, X; @, will be rigorously adhered to in this paper, and will not be explained as required. The same remark applies to the use of the dash for the conjugate of a linity, so that, for instance, x’ is the conjugate of x; and to the bar for the self-conjugate part of a linity, so that for instance v is the self-conjugate part of v. Note that there are three several uses of the dash above ; (1) linity self-conjugacy ; (2) orders as with m!, m’’, m'''; (3) reference to the actual as opposed to the standard state as with n’, w’, dd’, p’, ds’. The use of numerical suffixes is mainly confined, but we need not be rigorous in this custom, to the purpose of denoting the symbol to which a éifferential operator such as Y or C applies. &, &. &, & may be regarded as of this nature, since they may be replaced by Vo P1> V2 P2 K 2 132 Prof. A. MceAulay on Spontaneous Generation Il. Enunciation of Mathematical Theorems. Th. 1. If Q(a, @) is linear in & and @ then Q(f 2) =Q($'S, ), where ¢ is any vector linity. N B.—A quaternion linity is a linear quaternion function of a quaternion. A vector linity is a linear vector function of a vector. When linity is used without qualification in this paper it is understood in the more restricted sense of vector linity. The logical custom would no doubt be} the sonverse. Th. 2. If @ is an arbitrary vector linity and Wu, Ws are vector linities, then if Sort = Speed vra will necessarily be equal to yy. If the same relation holds with ¢@ an arbitrary self-conjugate linity it follows that Vra=W. Th. 3. In operating on a function of uv equation (13) above holds, GQ=Ta.e ~ "a0 a) Gee ee This is proved by first noting that v=3(T’T—1), and then showing that for < a function of v dz= —SdT6dzz = —SdTzT az. Th. 4. If the linity ¢ is given by dw = — PSaa—P'Swa’— ... = —TPS8aa, then QE $2) ==Q(a, 8). In particular Q(g, xo) = Q(Vip m1)» Q(o, X51, So x2) = Q(Vis M, V2 No), and so on. Thus we may always replace a pair Va, ma by Za, Xo, and conversely. Th. 5. If @ is any quaternion linity the following are the line-surface and surface-volume theorems of integration. (edp=((¢iVazy,°. . . 6 ae \Whd>—=(di Vids... eo. In (14) d= is an element of an arbitrary surface, and dp of Electrons in an Elastic Solid ther. tea an element of the boundary in the positive direction round a near d>. In (15) ds isan element of an arbitrary volume, and d= an element of the boundary in the direction which points away from the volume bounded. Th 6. ‘he bodily equation of motion of any elastic body; solid or fluid; with strains which are large or small; with density in the standard state varying from point to point or constant; with w varying in form from point to point or constant in form; is Hy CO We ee re CEG Bodily forces other than those due to the stress are not here included, merely because below they are not required. They may, however, be included in the following without any difficulty. (16) and the corresponding surface traction equation are obtained from the dynamical principle 8\ Ldt =0, where ¢ is the time and L the Lagrangian function. In our case MEV (RNG + in)ds, wh wee ee (LY) 6L=— i\y (nSndn—SdyFd wt) ds = — (d/at) \) nSiSnds+ MN) (nSp8n + Sin, AwY,)ds = —(djdt) \{\ nSninds+ {\ S8CwdS + {\) Sonlnm— (Cw): Va]ds. Th. 7. If v is any term in w, whether or not a mere function of v, either of the following two equivalent rules suffices to determine (v the corresponding part of Cw. First rule.—Express dv in the form Sdyf&%, where ¢ is a vector linity. Then dv is —@. Second rule.— Express dv in the form SdmV;, where is a vector linity. Then Cv is —@. For instance, if the term v=—V*Vn=—-—é occurs we put dvu= —2Sede= —2Sdn, VeVi, and deduce Cra=2Vew, (dv), Vi=—2VVe. Th. 8. Apply (14) and (15) to the statements that (dp’ =() 134 Prof. A. MeAulay on Spontaneous Generation for a closed curve, and (Vaz =() for a closed surface. We obtain TVG ASO (nT 1) Oe hed sy (18), which may also be written Vi VoVi=0, e ° e ° . e (20) is deducible directly from the definition of T or of x. (19) is deducible directly from the following known fact:— Ph). mY’ —10= —1Vp,;'p.SoViV2_ - - - (dl) >! = —3V_;'p2'SdV1V2- 5 e ° (22) Phy 20. GmamnPtp al a oo ae Th. 11. If any one of the four scalars m, m’, m'’, m!”, occurs as a term in w it produces no effect whatever on the bodily equation of motion. Some ten or twenty other “known” theorems, similar in their fundamental nature and importance to the above eleven, could be enunciated, but they.are not required below. Would it require one hand or two hands to furnish fingers sufficient to count the physicists of the world who know them? III. Description of first Aether Hypothesis. Some steps are taken below in the direction of rendering in- telligible the fundamental assumptions of Sir Joseph Larmor’s system of the universe as explained in his ‘ Adther and Matter.’ He finds, largely from the researches of MacCullagh and Maxwell, that those natural records which we have most successfully read well-nigh compel us to assert that the sether behaves as if it were a “rotationaily elastic” medium. In an admirable manner, the consequences of postulating this much and as little as possible more, are developed in the treatise. The development, which seems to have been Sir Joseph Larmor’s chief object, and which certainly occupies the greater part of the treatise, does not concern us here so much as the foundation. ' Occasionally Sir J. Larmor seems, in a half apologetic manner, to invite us to regard the “rotationally elastic” property as something of an ultimate nature, behind which there should be no urgent demand to go. Whether or not this invitation is really intended, most of us cannot tolerate the mental attitude here alluded to with comfort. We have the same instinctive objection to an elastic resistance to absolute rotation of a medium, as we have to action between bodies at a distance ; both alike we seek to “explain” by —_—ii = - of Electrons in an Elastic Solid Atther. 135 something beneath, something much nearer to the direct sug- gestions of our muscular sense. If we cannot hope and need not seek, as Sir J. Larmor seems sometimes to imply, to probe to a lower depth than the action principle (6 | Ldt=0), by all means let us readmit the discredited action at a distance, as fine an example of the principle as we could desire. Sir Oliver Lodge seems to strike the correct note when he, somewhere, calls upon us to explain everything in the physical world by pushing and pulling, or indeed by pushing and not pulling. Here we seem to tind cause for the fasci- nation exercised by the vortex theory of matter. Carry the pushing view to its logical conclusion. The now familiar perfect liquid is alone left from which to extract our bricks and straw wherewith to build the world as it appears. If this liquid is our ultimate goal then most assuredly the simplest thing we know, free ether, is simple only in appearance for its properties are not those of a liquid at rest. HKyven granting (without conceding any imperative necessity to do so) that this simplest thing ultimately may come to be explained by the simplest constitution, that of a perfect liquid; it appears to me that the most hopeful intermediate step to take is to assign to the zether a constitution next in simplicity to that of a perfect liquid, namely, that of an isotropic solid. Probably most people think that the elastic solid constitu- tion of the zether has been applied to modern views and found wanting. Now, however dubious some of our contentions may be, one result comes out with certainty from the dis- cussion below, and that is that the elastic solid theory has not as yet been so much as tried. There is a good reason for this, namely, that for nearly forty years, if not longer, there has appeared to be an insuperable difficulty in recon- ciling any elastic solid theory with the facts of optical re- flexion and refraction at the boundary of a transparent body. That difficulty has been surmounted (so we claim) in a paper on an elastic solid zther appearing in the Phil. Mag. for April 1909, p. 553. It has been surmounted on the old statical lines, where the difficulty alone really existed, by a reconsideration of the transition layer at the boundary. Moreover, those statical lines are not the lines along which the train of modern speculation mainly runs. We here accept the general picture presented to us by recently acquired knowledge of atoms and electrons, and the attendant hypothesis that the atom is what we know it to be largely because formed of orbitally moving electrons, and we accept the view that free ether is stagnant. If free eether is a stagnant isotropic elastic solid what can 136 Prof. A. M*Aulay on Spontaneous Generation be the general nature of those freely mobile electrons? The answer is almost axiomatic. The free ether is a stable form of the solid. By finite straining (we suppose below by finite small straining) the stability is left behind. There is a second state, providing for electrons, which may or may not be associated with a second absolute minimum value of the volume potential energy. ‘There is a critical condition such that if the strains in an assigned manner exceed it, the medium instead of tending to return to the free ether state tends to deviate still further; and though this critical con- dition is not one of absolute instability, we may refer to it as the unstable state. The accompanying diagram, though an imperfect repre- sentation of w asa function of e, f, g, much aids the com- prehension of what we have now to say. There are two of Electrons in an Elastic Solid ther. 137 hypotheses treated, the second very briefly, below. By differing interpretations the one diagram is made to represent both, but we will speak of it here principaily in connexion with the first. The curves of the diagram are the contour lines at equal vertical intervals (the dotted curves excepted) of the surface 2=2b(e«—y)? +c(a+2y)?—6ary?,. . . (24) where } and ¢ are positive constants, and in the actual drawing c is taken equal to 2b. The axis of zis perpendicular to the plane of the paper and upwards; « will be spoken of as pointing east and y as pointing north. In our first hypothesis we suppose ¢ really to be very large compared with }, say c is thousands or millions of times ), but so long as c is greater than b the characteristics of the surface do not for our purposes change. The shaded region of the figure wil] be called the lake, the regions V and W the north and south valleys, A the amphitheatre, and H the ridge. The points 8; and 8, will be called the north and south saddles. In order that the height of such a surface or mapped country may represent w, w must be a function of two variables only; this is the imperfection of the representation. It is assumed that f=g, and the height of the surface repre- sents w asa function of eand f. wis proportional to z; in the second hypothesis e and f are proportional to 2 and ¥ respectively ; but in the first hypothesis e=+2y, f=g=F(aty). »« « + (25) The ambiguity of sign implies an ambiguity accepted at first in the first hypothesis. [ Beyond the limits of the paper we may suppose the true surface to turn upwards, by adding to the right of (24) such a term as h(2? + 27")*, h being positive and as small as we please. | 7 We suppose that throughout vastly the greater portions of space, the elastic solid is in the state represented by a point very near to the lake bottom at the origin. The oscii- lations about that state serve to convey light and, perhaps, gravitation. In certain localities the state is represented by a point in the south valley, and it is never represented by a point in the north valley, because the north saddle is so high that in the most violent disturbances that ever take place the north saddle is not crossed. The original cause of the south valley conditions may be supposed to be an initial state of violent irregular motion throughout the solid generally. 138 Prof. A. MeAulay on Spontaneous Generation We have now to describe what conceivably follows this chaotic or nebulous stage. It is conceivable of course that a state might evolve in which large finite regions of the solid forming continuous volumes were characterized by south valley conditions. Instead we suppose, as stated just now, that throughout the vastly greater portion of space the lake conditions come to prevail, while such exceptional singularities given by the south valley as occur concentrate about points, “lines, and surfaces; these three singularities may be called knots, threads, and webs. If one of these conditions occurs it will give rise to two opposing tendencies, and we have to make the pure assumption that these result in a stable state for the ether asa whole. The local tendency at a point of the solid where the south valley conditions hold will be for the corresponding representative point on the diagram to run down the valley. But it can only do so by distorting neighbouring regions in which lake conditions hold, and that the first mentioned diagram point may run down the valley, the points representing the lake conditions will have to rise towards the south saddle. Thus local con- ditions tend in one direction and the reaction of the esther as a whole tends in the opposite, the result being by our assumption stability for the ether as a whole. A remarkable and very suggestive fact is indicated by the dotted straight lines which divide the diagram into four quadrants. The north-east quadrant contains the whole lake, the south-east the whole south valley. It is proved below that for all points in the north-east quadrant there is an elastic resistance to the establishment of curl, and for all points in the other quadrants the tendency is the converse of this, or curl is attracted to regions of space represented hy points in the south valley. As will appear in the argument below this is precisely the condition required to make Sir J. Larmor’s intrinsic radial cur! intelligible. Threads and webs inherently tend to shrink. It seems certain, therefore, that they cannot exist permanently except when projecting from knots. Finally, we come to believe electrons to be knots connected in systems by threads (or webs which seem to behave like collections of threads) pulling them together. The undoubted shrinking tendencies of webs and threads, and the undoubted attraction of curl to the middle of webs and threads follows from no highly artificial constitution for the solid, but from what can only be called as simple a con- stitution as it 1s possible to give. We will explain this statement in some detail. of Electrons in an Elastic Solid A@ther. 139 w is a symmetrical function of e, 7, ¢, the same in form at all points of space. e and Xe (meaning e+/+g) are not simple functions of » and its space derivatives, though %(e+4e?) is sucha simple symmetrical function. Another is det dfotefg=m'+m/+m'’=m say. Any rational function of m,, e+4e?, f+4/?, 9+49? which is symmetrical in e, f, g is expressible in terms of 7 and its space derivatives, and the degree in 7 is the same as the degree ine, 7, g. We therefore seek the three simplest symmetrical functions of the kind. These are s’ of the first and second orders, s’’ of the second and third orders, and s" of the third and fourth orders, where Saeee rie. Ae. (26) s/=3' —m=X(4e—fo)—efg, . .« ~ (27) sM= 5 —S [4 (e+ he?)?— (f+ 47") (9 +39°)] =32[-6 + fof +9) —2efy] + 32(—e + 2f%9") ¢- 8) =3(—e+f+y9)(e—f+g) (e+f—g) (1+ te) We are guided to s” by getting rid of the first order terms and to s’” by getting rid of the second order terms by sub- tracting an appropriate function from s”. Let w be as simple a function of s’, s'’, s’’’ as it can be. If w is a mere multiple of s’ the consequences are remarkable and unexpected, and seem to have a distinct bearing on the properties of actual ether. I have been acquainted with the properties of the medium thus constituted for a long time, but till lately had supposed them to be of mere mathe- matical interest. For the present let it suffice to say, that though very suggestive, they do not altogether meet present requirements. We will be influenced in our choice of w chiefly by the ordinary assumption that wis a true minimum for e=f=g=0. This excludes a mere multiple of s’ and suggests the sum of multiples of s” and s’. This carries us necessarily as far as the fourth order, so that it would be an artificial restriction to exclude a term in s'”, The form which it is convenient to use is wi A= +38" + 02Fi— gH) +4c(Ser)", « + (29) where A, 6, ¢ are positive constants, and q=etse, A=f(t3P, n=gtiy. .« ~ (30) 140 Prof. A. M¢Aulay on Spontaneous Generation It is easy to see that (29) gives the most general form of w, consisting of terms in s®, s’, and 8’, which is a true minimum for e=f=g=0. We can make the state corresponding to the south valley truly stable instead of unstable by adding to the right of (29), £(2e,")*, or a higher power of Ze’, k being a positive constant as small as we please (so as not to alter the values of w for small values of ¢, 7, 9). Our hypothesis requires that there should be at least one set of small values for e, f, g besides e=f=q=0, giving a stationary value of w. For such small values we may neglect orders above the third, and in the third we may replace é,f,9 by 45 fi, 91» Do this and put —ath += + 2p, a—fitn= + 29, ¢ +A-n= + -- (31) +ej=qt+?; +f=rtp, +n=ptg. Thus we get w/A=—6pqr+bd(q—r)?+c(Sp)?, . . (32) For the stationary values Uy = Wy = W,=0. Using these in the form Wp=0, Wy—wz=0, w,—w,=0, it can be easily shown that a stationary value necessarily requires two of p.g,7 to be equal. Taking these as gq, 7, we find as the only cases besides p=q=7r=0 the following for stationary values of w; either P=GO=T= 6, wie 61) 5) Cy or else Q=f= —), P= tye ay a. i where | a=b(6+2¢)/(2b4-6),.... «., «see so that a is positive. c might be finite or even much larger than unity, and still if b were sinall there would be a stationary value for a small strain besides the minimum value at p=q=r=0. In this paper, however, we will suppose that ¢1is small and 0 very much smaller. The surface (24) is obtained by putting =p, y=q=7, z=w/A: The quadrants mentioned above are determined by y=—bandx«=—b. These two straight lines pass through the extreme south and west points of the lake. [If b v, Cf Je In this paper the latter course is almost exclusively followed, but we do not consider ourselves confined tO 1b. i Tf We first find the simplest forms of s’, s'’, s‘’’ in terms of 7 and its space derivatives. We have my=m +m! +m" =—-SVn—-4t8VV1V2V mn +38ViV2V 35min, - - (88) s= —Swug= —Soue= —SV7+38ViV 8mm. . (39) From these we form by the same process as in (26), (27), (28), s’” and s’” in succession thus s' ‘== s! =, =38ViVomm—J8SViVoV 381723 , (40) x sl — ESE vl ls, for s'’’ was obtained by subtracting from s’’ that definite ae function of e+e’, &., which cancelled the second order terms. The only quadratic function of v which can effect this cancelling is that one which is of the same form in v as the quadratic terms of s’’, namely ISViV omn2= $8 C150 51N S05 are in x. We now show that So Sudiul. =Soioudues, by putting v= vu+4Ve( ), and showing first that the linear terms in e,and secondly that the quadratic terms in e vanish. As to the linear terms we have (Th. 1)/ SOf20G,v,= Nai Cou'f,u! 2=% (SE Soudiul, ata So Su'Su' fs) . of Electrons in an Elastic Solid ther. 143 Putting in the last v=v+}Ve(), v =uv—-3Ve() the linear terms in € clearly disappear, ‘and the quadratic terms in e =186,6,( Vet.) (Vets) = 186,(&8 Vet, Vet. — VetiSo,e, + VebaSonel;) =(), Thus we have MAGUSIS es SO.Gv6\ul,= SC,Gu'Giu'o. . (41) In the last expression put vo= —Vi8@(m—3V 3573) a V25e (n2— 3 V Som) and apply Th. 4. We get So, S.uovo=SV1V 0m — 3 VsSmins) (m2 — 3 V Sons) =SViVem— SViV2VsnSnins iz SViVeVsVaSnins82M14- Hence s"=AS8ViV2V8Smn3— 8 ViV2V38m72N3) —SViVoV3VaSmISnM- (42) We now transform s’, s", s! so as to be expressed in terms of (1) divergence m/’ and curl e, (2) m'’ and m'"’ because of Th. 11, (3) Seve. Our choice of these is largely determined by counting the simplest scalar functions of orders first, second, third, that can be formed from them, thus : Number of Order. Scalar functions. scalar functions. First. m, i. Second. m2, m'’, e*. 3. Third. m®?, mm", m'e®, n!’’, Sexe. oe Now it is easy to transform each of these, and also s’ s‘" (except for the fourth order part of s’”’ the following :— eile: ) into terms of Number of Order. Scalar functions. scalar functions. First. SV». 1, Second. (SV), SViV Sn SVimSV om- 3. Third, (SV! SVBViV 8mm The CEs SViV SV 3780103; SV 1728 Von Van 144 Prof. A. MeAulay on Spontaneous Generation Hence the quantities m/!, e, m’’, m'’’, Sexe are found to be adequate to express s’, s’, s!", always excepting the fourth order part of s’”’, which is left unmodified below. | (43) and (44) below are given for reference. (45), (46) (47) contain the modifications we seek. (SV 9) =m? SViV2mn=m?—2m"'—e\, . . . (48) SV inSV on =m? — 2m” (SV7)=—m? SVS ViV 8ning= —m'(m'? — 2m"! — &2) SV SV nS V y= — m'(m!? — 2m’) SViV8VsnS8nins= —m’? + m!(3m" + &) — 3m!" — Sexe | SV nS VonsS Vam = — m2? + 3m! m” — 3m!" ] s' =m! +}(m?—-2m"—), . 1 (45) gl 3(m?—4m"—e?)— ml", . . (46) sl" = (—}m? + 2m'm” + dle —4m'" —Séye) —SV1V2V3VSannsSnom ee) The third order terms of s'" are equal to —4 times the discriminant of ~— 3m‘ that is of —$VV,( )m. [The dis- eriminant of any linity @ means 180,66;86666.66]. For b the discriminant of y+. is known to be \-—-~ » (44) m+ aml! +22m! + 2, so that the discriminant of y—Jm’ is m!"’ —Im'm" + km?, Also the discriminant of ¢@ is known to exceed the discri- minant of @ by gSdAgA, where A is the curl Vépg of ¢. Putting 6=x— 3m’, rX becomes e. There is probably here some easy method involved of proving (47), but I do not see what it is. [What is required is an easy proof that ISViV2V 3728173 —GS ViV2V 38NNeN3 Is —4 times the discriminant of ~—m’. [Pure Mathematical Note-—The whole theory of linities of the nth order, and especially such work as the present, may be based simply and very instructively, on the study of the scalar n-linear symmetrical function which in the case of n=3 is given by m(Apv) = $SHiF CSAC wove, of Electrons in an Elastic Solid Ether. 145 where X, ws, v are any three linities. We may take Ao= —SoV. 24, ho=—SoV. 8, vo= —SoV. y, so that m(ApV)=3ASV1V2V 3541 B273- Thus we have the fundamental theorem md+v, p+, d+) = m(Poh) + 3m(Phy) + 3m(py) +m(Wryyr). Here as a special case we may put =a scalar wv, p=y, when we get + am + em! +m"= x +32°m(11y) +3um(1yx) +m(XXX)- Or we may put d=v, ~=4Ve(), 6+Y=v and so on. The cubie of yx is xe— x7’ + yin! —m'"=0, and it may be written x3 — dy. m(11xy)+3x .m(lyx)—m(xxx) =0. | w of equation (29) may be expressed in terms of s', s", 3'’’ thus. From (26), (27), (28) we have: ¥(e?—2 F191) — 2 (s’—s'"), Ye" + 2fhigy) =s5"", Hence LeP= hs? +8!’ — 5” 22 fig=ys?—s" +8", and w/A= 43s" + b(48? 4 38" — 3s!) + des”, or w/A=3bs'’ + (Lb +4c)s?2+3(41-—2b)s". . (48) Th. 11 shows that for our purposes we may omit from w any constant multiples of m’, m'’,m’’. Let us do this and also omit fourth order terms, so that now for the first time must we assume the strain small. Put v for the modified value of w. Thus v/A = 3b(m™ —e*) + (fb+ 40) m’ [m’ + (m?—2m" — &)] + ?(41—2b)(—m? +4m'm' + m'e — 2Sexe), Phil. Mag. 8. 6. Vol. 19. No. 109. Jan. 1910. L 146 Prof. A. MeAulay on Spontaneous Generation or restoring the meanings of m’, m", and € ofA=380(8Vn-V?V9)+Gbtte)[S'Vn | —SVn(S’V7+8VViV2Vim— VV 9)] (49) 1 #( ae bes 2b) [SV a(SPV n ee VN 1V2V nn | pa —VWV)—28(VVnxVVn)] J For free ether the strains are small compared with 0, and we have: (free eether) v/A = (26 “= L)S°V 7 = BDVPV/n. so iUatae (50) This equation is known to give wave propagation of curl with velocity ,/[(A/n)3b] and wave propagation of divergence with velocity ./[(A/n)(46+4c)]. The first is necessarily less than the second, and familiar considerations incline us to put it as very much less by supposing, as is necessary for the purpose, that } is very much less than ec. We now make this assumption ; partly, of course, because otherwise (49) seems intractable. With c/b thus large there is always the possibility some time of explaining gravitation as due to ¢. The discussion above, by aid of the diagram, shows that in the middle of a web or thread or knot we must suppose south valley conditions to hold. As we pass in space from the middle, the representing point on the diagram passes by some path near the saddle towards the lake bottom, and the mean condition of matter in bulk will be represented by a point probably between the lake bottom and the saddle. Thus in (49) in the neighbourhood of an electron we must suppose V7 and y to be of the order b, while SV/7y is of a negligibly smaller order such as 0°/c. Rejecting all but the most important terms thus given we get v/A= —20WVV 0+ 38(VV0)x(VV 07), 8Vn=0. (1) The condition SV7=0 must be taken account of by inde- terminate multipliers when applying the principle SI dt. It is easy to see that this gives a term —\/p in ny, p being a scalar, that is we must include the effect of a hydrostatic pressure in our equation of motion. We digress here to comment on the general nature of (51). Tt was above asserted that the elastic solid theory had not been so much as tried in the light of modern views concerning the ether. (51) has been arrived at in a most inartificial manner. We have simply examined the general nature of the modern views, and have then postulated as simple a con- stitution as can be well thought of for our solid, with no peek of Electrons in an Elastic Solid Atther, 147 more detailed assignment of the minimum number, three, of constants which inevitably present themselves than is implied by the words large and small. The result is that the solid behaves in an unmistakable manner, as if it were rotationally elastic and practically incompressible. When we remember how these properties have been forced on our unwilling acceptance in studying the ether both optically and elec- trically, this first indubitable result of our elastic solid theory is of the highest significance, and goes far to persuade us | that we are seeking in the right direction. (51) may be written a pA shia la At the south saddle when c/b is large two of the roots of ) +¥ are each equal to —b and the third equal to 24. As a point passes on the diagram from the origin straight to the saddle and beyond it, each of the first two roots of b+y beginning with the positive value ) diminishes to zero at the saddle and becomes negative beyond. Hence at the middle of a thread or web the corresponding term in v/A is negative, near the middle where the saddle condition holds, this nega- tive value passes through zero, and further from the middle takes the usual or free ether sign, namely, positive. In so far as the term depends on the curl e¢ it varies as the square of «. Hence e¢ tends to increase in two of the three principal directions at the middle of the singularity ; but at a minute distance from the middle the usual free ether elastic re- sistance to the establishment of curl sets in. If then a thread projects radially from a knot curl will be attracted to the thread. Other curl though elastically resisted must arise to close the circuit of the first. There will thus come about radial intrinsic elastically resisted curl. Here we have a complete rational account of Sir J. Larmor’s fundamental eether and electron requirements. For free wether we have as 1s required v/A=—3bV’V/7; for electrons we have as is required radial intrinsic curl; and the singularities forming these electrons are perfectly mobile through the ether as is required. This forms very strong evidence that an elastic solid con- F stitution of the ether can meet all demands. Strangely, as it appears to me, every one of these satisfactory results Hows also from the elastic solid constitution of our second hypo- thesis. This is an additional encouraging feature, because it implies that if our fundamental assumption of the possibility of stable existence of the singularities should prove invalid L2 148 Prof, A. M°Aulay on Spontaneous Generation there are other constitutions to try, where the primary con- ditions are also fulfilled. This fundamental assumption and others that for simplicity have been tacitly made are to be examined by aid of the equation of motion which flows from j(al). The equation of motion is at once written down by aid of Th. 6 and Th. 7. Noting that ye=ye and that SadyB =— Sdn (2@SBV 1) we obtain dv/3A= —Sdml - VViO4X) VV 9 FaVV9 SIV V0) Vid. d Be (52) With the upper sign, from the last term, we have a tension along the lines of curl equal to —}V?Vn; with the lower a pressure. [A tension quite really implies a tendency of a line of curl to shrink because it implies that shrinkage causes decrease in potential energy.] We therefore assume that the Maxwellian tension along the lines of electrostatic induction determines that we must select the upper sign; we henceforth drop the lower sign. In passing note that the existence of this stress is yet another confirmation that we are on the right lines. Our equations of motion now are mn[BA=—VV(b4+x)VVn-3SVnV.VV9n- Vee (53) SV7=0, yo= —SoV.7, , + the pressure term — Vp being necessitated as we have already seen by the condition SV7=0 in connexion with 4 { Ldt=0. Putting 7=0 we have the equation which must eventually decide for or against the validity of our assumptions if the provisional constitution is accepted. In this first plea for an elastic solid eether we leave this apparently intractable differentia] equation alone. Perhaps some mathematician more familiar with differential equations than the present writer may be persuaded to attempt its treatment. V. Second Atther Hypothesis. The origin of the present investigation was by aid of another tentative constitution; this origin will now be briefly described. I was aware that the term m=EFG in w produced no effect on the bodily equation of motion. I was also acquainted with the very remarkable medium, capable of strains of any magnitude, which is given by wA=L>r WS ITT l=s' 43. . a ; b P. of Electrons in an Elastic Solid Atther. 149 Its bodily equation of motion is Mp pel WS ag Be CSS) An isolated finite portion of the medium tends to shrink to evanescence, but if the medium extend through all space, and x and A be constant everywhere [this means merely that the standard, not the actual density is the same for all points], (55) shows that it will be stable, because it shows that 7 will be propagated according to the familiar laws of ordinary wave motion. This linear differential equation, and these deductions therefrom, are in the present case true for finite displacements of any magnitude. In surmounting the older statical optical difficulty con- nected with reflexion, the term ef (e, 7, g, being now small) in w had appeared as of fundamental significance. I had for several years entertained the idea that an elastic solid with two possible states was the medium by which to explain the properties of ether and matter. It occurred to me to unite these contributing items into the one hypothesis that w consisted of three terms, multiples of SE2, EFG, efy. The results were essentially the same as have already been arrived at by another constitution, and seemed so encouraging that that other constitution was tried, because it involved the simplest Gn degree) expression of w as a function of 7 and its space derivatives. The fact that both constitutions led, for our purposes, to essentially the same result, came as a surprise*, for they by no means agree in general mathe- * At the moment of despatching the paper I see why the results of the two hypotheses are for our purposes identical. In the first we assume, to make progress, Syn=0 or e+f+g=0. Now when this con- dition holds the cubic term (—e+f+y)()( ) of the first hypothesis becomes a multiple of efy, as in the second hypothesis. This indicates the essential reason, though for a complete account we ought to mertion the second order terms also. The case of w equal to the general sym- metrical rational function of degrees two and three in e+e, f+3/", g+zg seems worthy of study. For stationary values other than é, =/,=9, =0 we must have two of e,, /,, 7, equal unless e,+f,+ 9, 1s a factor of the cubic term; if it is a factor there are an infinite number of values for which wz is stationary; in the last case w is a function of the two expressions Se,, Sf,g,, and the infinite number of values belong to +e, = one constant, Sf,g,= a second constant. Putting this case aside, and taking the quadratic terms as d3(f,--g,)*+e(3e,)*, assuming ¢/b large, and therefore Se, small, or say zero, we see that this constitution also gives us the same results as the two hypotheses of the text. Ina word, those results flow from taking the cubic terms into account and assuming the zther almost incompressible, and that one of the stationary values occurs with a small strain; no more detailed assumption is required. But note that the second hypothesis stands out peculiar in that we do not with it require to postulate approximate incompressibility. 150 Prof. A. M¢eAulay on Spontaneous Generation matical interpretations. This approximate identity of results, for our purposes, is probably due to the third order terms of w in both cases being a multiple of the discriminant of xX +km' where k is a constant. efg is not a rational function of s', s’, and s’”, and we naturally take efg really to be merely the third order term of some function of third and higher orders. This generalized form of efy need not go beyond the fifth order, since s +s’ = —Aefg + tole +6 29° —Ae' (f+ 9) — lea) — Tega but it is simplest to let it go to the sixth order by generalizing efg to efg(L +g) (L+3f) (L+ 29) =GShibobsSul vleuls =1S6,60,Sutubtvt, + Seve hee =m!" (1+3m!'+4m" + dm") +48eve. The last transformation follows by noting that since v=(1+43y’)x, the discriminant of v is the product of the discriminants of 1+3y' and y. If we neglect terms above the third order (56) reduces to efg=ml! + iSeve.. . «oe Consistently with the above, and initially making no assumption that 7 is small, put /B= 3S EH? — (1—k) EFG— (1/81) (E?—1) (F2—1) (G2—1) (58) where B, 4, and / are given constants. To get stationary values of w equate wz, &e., to zero. E—(1—4)FG— (1/4) E(F?—1)(G?—-1)=0. Multiply by E and write the next similar equation. E*— (1—k)EFG — (1/41) E°(F? —1)(G?—1) =0, F’—(1—k) EFG— (1/41) F2(G? — 1)(E?—1)=0. Subtract and reject the factor E+ F. (H— F)| 1+ (G?—1)/41] =0. Similarly | (F—G) Pha el) 40 = (), of Electrons in an Elastic Solid ther, 151 Hence for a stationary value two of E, F, G are equal, say FandG. Thus either E=F=G, 1—(1-—/)E—(E?—1)?/41=0, or else F=G=v(1—41), E=1—k. Expressing in terms of e, 7, g we have, either 7) — 7 (et+te)+ (l—k)l.e—kl=0, .. (59) or else i os, a ee ee (1) At this stage take & and / small, and, to get the same j notation as before* take the roots of (59) to be —h, —h+e, and the solution (60) to be f=g=—h—b, e=—h+a. Finally, express everything in terms of b,c. Thus a=b(b+2c)/(2b +c) 7 Sh—=)-+4 vh=b-+ 2c Und 10M) gay 3k=(b+2c)(—b+¢)/(2b+¢) | . 31=2b+¢, zt . oe Pere | pee w/B=3SE"—(1 soba ERG 3 | ee ES mee. 2 pe 62 and the stationary values give eth=f+h=g+h=0 orc, . . . (68) or else fth=gth=—), et+h=a=b(b+ 2c)/(2b+c). (64) Use (57) ; also reject m’, m”, the modified value of w. Thus o/B=3(SV 9 — V?V 0) —3/4(2b +c). S(VV 0) x(V V7). (65) Here not »=0, but »=—hp, gives the condition for free ether. Put »+hp=; again reject constants and multiples of SV Xe. from v and eall the modified form 1; 3D 3 10-432) V™) — Tap Eo SWVV mdxolV Vn): (66) On account of the great similarity between the results of m'"' from w; also put v for %/B=F(8°V 1 — * By comparing our present cubic term efy with the former (—e+f+ 9)( )( ) we must expect our present e to correspond to our former —e+f+ 9. 152 Dr. G. A. Campbell on the second hypothesis and those of the first, we refrain from further detail. The second hypothesis has one advantage over the first. There is no longer the same necessity to” suppose ¢ very large compared with b. The equations resulting from c/b finite are reasonably tractable. Hence our diagram includes both saddles. University of Tasmania, July 5, 1909. AIL. Telephonic Intelligibility. By Guo. A. CAMPBELL, Ph.D.* N the August 1908 number (vol. xvi. p. 242) of the ‘Philosophical Magazine’ a paper by Lord Rayleigh contains some notes upon the acousticon, in the course of which he says :— “The reproduction of speech, given at about one foot away from the microphone, was better than anything I had ever heard before. The first impression was that all the consonantal sounds were completely rendered, but this turned out to be an illusion. In listening to the numerals, given in order, the observer would feel confident that he heard the / in jive and the sin siz. But if the initial sound was prolonged —fffive, sssix, the observer could not tell until he heard the sequel which it was going to be. Further, if the sounds were given as ss ive, f fia, they were heard normally as five and six. It was plain that there was no difference in the rendering of f and s. I am informed that this is a well- known difficulty in ordinary telephoning, and that in spelling ite containing 7 or s it is usual to say ‘/ for Friday’ or ‘s for Saturday.’ But the articulation of the acousticon is so superior that it was surprising to find the failure complete. The characterization of sh was not much better, though after a little practice I could distinguish it from s or f, bat pro- bably only by a greater loudness. “These failures might have been ascribed to my rather defective hearing, but other observers with normal hearing did no better.” This statement of telephonic intelligibility is typical of the impression made by the general use of the telephone. Most of us feel that we understand almost everything transmitted by the telephone or next to nothing, according as we are listening to a familiar voice talking upon some well-known subject, or as we are obliged to listen to disconnected syllables spoken possibly in a strange tone of voice. This is obviously a case for the statistical analysis of results. Some time ago the American Telephone and Telegraph * Communicated by the Author. ; 4 Telephonic Intelligilility. 153 Company began an investigation of the distortion introduced by the telephone instruments and transmission network. The work has not advanced sufficiently to warrant general statements as to the telephonic intelligibility of the 5000 phonetic syllables employed in the English language, but what has been done seems to show the order of intelligibility of the consonantal sounds with some definiteness. For the tests which will be described the following twenty syllables, each ending in long 7 and preceded by one of the simple consonant sounds, were used :— bi as in bee ni as in knee chi Chi(le) pi pea di de(pot), dee ri re(bus), rei fi fee si see feat Gi(zeh) shi she hi he ti tea ji gece thi the(ory) ki key vi ve(nus), vee hi lee ! ye mi me Zl ze(bra), zee. The list contains seventeen words and three syllables which occur in compound words. According to the scientific alphabet used in the Standard Dictionary, there are but twenty-four elementary English consonant sounds. All of these consonant sounds are included except dh, ag, w, and eh; dh is somewhat like th, and was similarly recorded in an earlier series of tests; w was recorded correctly in 99 per cent. of the cases in a set of twelve thousand records ; ny and zh are not of frequent occurrence, and do not seem to be used as syllables with merely long 7 following. Ten lists of one hundred syllables each were prepared, every one of which contained the twenty consonants five times in a perfectly haphazard manner. All of the ten lists were spoken over the telephone connexion and a record made of the consonant heard. The tests were then continued, the experimentors changing places. Ali records were made in a quiet room and vyer quiet lines. Before this series of tests was begun the observers had made at least fifteen thousand records, and had become perfectly familiar with the method of test. The syllables were never repeated for any one given record, and the observer always made a record however uncertain he might feel as to its correctness. While the sequence of the lists and of the syllables was perfectly hap- hazard, the observer knew that each syllable occurred the same number of times in each list of one hundred syllables. This may have led in some cases of uncertainty to recording a syllable merely for the reason that it did not seem to have 154 Dr. G. A. Campbell on received its proper share of records, It does not seem pro- bable that this influenced the results to any considerable extent. As far as such an effect occurred it would corre- spond in a general way to the condition obtaining in the practical use of the telephone, where the subject matter and the preceding syllables give some clue as to the syllable which may be expected. The telephonic network included regular common battery subscribers’ sets at each end, common “battery transformers and cord circuits at the central offices, and a line of about one hundred miles of loaded No. 16 or No. 14 B. and 8. gauge cable, with the addition of from three to sixteen miles of non-loaded No. 19 B. and 8. gauge cable. The series of tests over this connexion included twelve thousand records. For comparison, tests were also made with direct trans- mission through the air, the same observers facing each other with no intervening obstruction but not looking at each other. The observers were fifteen feet apart in a quiet room. This series contained two thousand records. Table I. shows the distribution of all records. Where there are two figures in a square the upper refers to the test made through the air. The syllables have been arranged in such an order as to bring those which are most frequently mistaken for each other as near together as possible. This brings the surds or voiceless consonants, 7, s, h, sh, ch, k, t, and p together. The remaining consonants are all voice consonants with the exception of th. The occurrence of th among the voice consonants may be accounted for by the fact that th is liable to confusion with the voice consonant dh as in thee, and it is possible that in these tests the distinction between the two was not properly made. In an earlier series of tests which included dh in place of th the same mistakes were made as in the present series with th, that is, dh was confused with v on one side and < on the other. In the table the voiceless explosives k, ¢, and p come together, also the voice explosives g, d, and 6, and the nasals m and n. A blank square indicates that no record was made by either observer corresponding to that case. The table shows 156 such squares, and therefore of the possible incorrect eases only 59 per cent. occurred. A zero indicates that not over one-half of one per cent. of the total records for that syllable fell within that square. Had the distribution of records been entirely haphazard the average number in each square would have been 5 percent. The table actually shows only 34 squares with 5 per cent. or more incorrect records, therefore only 9 per cent. of the incorrect cases occurred with a frequency equal to or greater than the probability corresponding to a perfectly haphazard distribution of records. Telephonie Intelligibility. 155 The consonants may be divided into three groups, as indicated by the heavier lines in the table, which are sufficiently distinct so that no consonant in one group is mistaken for any consonant in the other groups as frequently as 5 per cent. of the time. The totals in the last column of Table I. show that the records were not evenly divided among the twenty syllables. Thus knee was recorded only 27 per cent. as many times as it was called, while tea was recorded nearly twice as often as called. The scattering of the records increases somewhat with the total number of records, but not proportionately. In general each consonant was much more frequently mistaken for some one consonant than for any other. Table II. gives the correct records and the leading mistakes for transmission through the air, and in Table ILI. the same records are given for telephonic transmission, The last columns of these tables show that 96 per cent. of the records were correct for transmission through the air, and 59 per cent. correct for telephonic transmission. The leading mistake made for air transmission occurred in 3 per cent. of the cases (total correct and incorrect records), while for telephonic transmission the leading mistake amounted to 26 per cent. of the cases. The mistakes made through the air were also made over the telephone, but additional mistakes also occurred, and often these are the more frequent ones. The most common mistake made over the telephone was in recording me when knee was called. This occurred in 67 per cent. of the cases when knee was called. nee is thus naturally the syllable least often recorded correctly, occurring in only 24 per cent. of the cases. Lee was recorded correctly in 98 per cent. of the cases, a record excelling the average for direct transmission through the air. The records tabulated in the above tables were all made by Mr. W. L. Richards and Mr. E. C. Molina. They dif- fered considerably in the distribution of records, but pre- sumably the averages are about normal as far as the personal element goes. With the best possible articulation the intel- ligibility would undoubtedly be considerably higher. Our tests indicate that the greater part of the distortion occurs in the subscriber’s set, and that a short length of line may actually improve intelligibility as compared with zero line. As the length of line reaches the commercial limit the intelligibility gradually falls off. The tests indicate, as would be expected, that changes in the telephone apparatus and the line affect the different syllables quite differently. 1¢ will be seen that the syllables fee and see are confused with each other in telephonic transmission. They were interchanged in 27 per cent. of the cases, and recorded cor- rectly in 58 per cent. of the cases, which makes f and s an average pair of consonants, as the twenty consonants were 156 Dr. G. A. Campbell on recorded correctly in 59 per cent. of the cases, and the leading mistake was made in 26 per cent. of the cases. Hven as the records stand, it will be seen that 7 and s are correctly re- corded twice as frequently as they are interchanged with each other. While these results are not directly applicable to the acousticon, which is the instrument referred to by Lord Rayleigh, the difference should be in favour of the acousticon, for our tests were made with commercial instru- ments over long commercial lines. As the conditions of our tests differed from those of Lord Rayleigh in almost every particular, further tests, summarized in Table IV., have been made which bring the conditions more nearly into agreement. The number of syllables in each list was here reduced to two, and the six distinct series of tests included Lord Rayleigh’s test syllables five and sive ; fix and six, and also fee and see; ohm and own; am and an; me and knee. Hach list contained one hundred syllables, and was varied each time it was called. The lists were called over the telephone by six persons, here designated as H, I, G, H, 1, J; of these [and J were young women. Four telephone receivers were arranged in series, parallel at the receiving station, and four persons A, B, C, D, listened simultaneously and made independent records. This method materially reduced the time and labour of making this set of 14,400 records, and it also enables us to compare directly the records of different observers with any one person calling or of any one observer with different persons calling. Observer Hi listened directly to H, I, and J, and these records are added to the table. The ten persons who assisted in these tests were familiar with the use of the telephone, but they had had little or no practice in syllabic tests. The telephone con- nexion from the talking to the listening station was made through the private branch exchange of the office, and therefore the length of line included was almost negligible. It will be seen that the records made by E for direct receiving thro ugh the air were in general nearly perfect, but that for “ohm” and “own” they - were In some cases poorer than the telephonic records. Comparison of the original records for each calling of the lists shows that the distribution of mis- takes was, on the whole, purely accidental, and could not be traced to definite irregular ities in the pronunciation ofthe lists. For Lord Rayleigh’s syllables fix and six, five and sive, about 85 per cent. of the records were correct; haphazard records would have averaged but 50 per cent. correct. In one case, i’ calling and D recording, a perfect record was made for the “‘ five-sive ”’ list. While it is obvious that the telephone seriously distorts speech-waves, nevertheless even those consonants which most nearly resemble each other are not distorted sufficiently to be indistineuishable. Telephonic Intelligibility. 157 TaBueE I. Distribution of Telephonic Records in Percentages (Air Records above). CALLED | 100, | 10} 2 34 69| 28) 17| 3 ee eS Oe ee 1) 23) 158 Dr. G. A. Campbeil on TasLe IT, — Air-Records. ie CALLED iF OS aie let pig i a@\|bty Plane EB wm | Ul | » GAv. Correct: .........] 97100, 99 88 97/100 98100 87, 87 97) 99) 98) 90, 881100 100| 94) 99 100] 96 1st incorrect ... * ThA aie Rats e) a ree See ee fm fe | a 2 1) 10) 2 1) | 18) 12) "2\ a1 a) 6) 8 5) 1 Al others .....) 1/0] of 2/ a/ of i| ol o a] a) of a] al al of of 1) Of Oo TasLe I1l.—Telephonic Records. CALLED s h|sh|ch| k Cp ND ag ae age yi vith) zi m|n|t Correct | Records ... 69 45) 33| 41) 45} 70) 45) 61) 47) 91! 46) 76) 41 41) 72] 90) 24) 98 USE... Fs \ el Me) Eb) 3p) 05) 9 Aa NO er ane, | en | z|th| | m|m 20 28) 35) 45) 49) 25) 45) 17) 23) 3) 40 9} 32) 25) 15] 5] 67) 1 2) 2nd ,.. ch | ch | s|t|pl|k|dl\eh| tlp|p}zle|v}o]Jullir Be 9,17; 5) 5 4 4 65) 11) 38] 4) 4) 19) 24) J 3 Ga tT 58 = Syscar aaa etn aml ae Nes pee Ge eeepc er} Srd ... shiy|h|s th) slob hy es) aed ed a aia 3, ah si fi she Oy a 2) AN TO} 21 S193 2 1S a Si a at All 215) 7| 6 1| o| 4/13] 9] 1 y s! 6 7| 41 a] & oO others | TABLE IV. Me Knee. El 66 | 86 | 96 | 96 | 86 38 | 68 | 72 | 84 | 65 Fi go | 92 | 70 | 68 | 77 80 | 881921 68 | 82 Gl 73 | 90 | 92 | 56 | 79 88 98 | 96 | 82] 91 Hl 94 | 96 | 82 | 90 | 90/100 1861 98/ 96 | 921 93 | 100 Tl 52° | 64 | 54 | 78 | 591 94 T50| 84 | 76 | 74) 7a x| go | 88 | 72 | 84 \76| 90 | 46! 72 | 74 | 641 641 98 78 78 Alm. An , ; ie Xx | B | il ap ea ledposde ane ta aoa ap iG a flediuliwea | a6 | laa | 71 70! 78/62190| 75] Fls@ho) -¥6 |st | 98 178 64| 52/50 | 64 | 57 Gilet 74 | ee | 54. | 67 92| 83 | 94 | 66 | 85 Hi 90 | 94 | g8 | s2 |ss| 98 |9u|] 98/98] 92/1 941] 94 1] 63 | c2 | zw | 74 1691 96 1961 94/961 96! 951 84 sl so | so | 64 | 76 175| 92 190! 84184186186] 68 74 “82 Telephonic Intelligibility. 159 TABLE LV. (continued). Obm. Own. Oo) Dy fas) ETA | B fol Day! & a oan eee eae 66 | 9 | 82 68 | 66 | 88 | 54 | 69 72 | 86 | 82 90} 90 | 92/94) 91 86 | 90 | 90 74 82 98 | 94 87 82 | 9% | 8| 76 | 96} 90/94/86] 91] 80 64 | 76 | 74| 72 196] 88/98] 88|92] 82 68 | 90 |79| 90 | 94] 90/92) 96) 938] 54. | 8 | Ron Pee 88 | 98 | 78 | 87 80 | 80 80 86 70 SL | 100° 96 | 90 | 90! 100 | 80| 60! 74/84/74: 96 | | 98 | 44 | 76 74| 96 | 74 | 98 | 85 | 80 | 40 | 66 741 62/96 | 92] 81 | 94 | 76 | 87 76 | 961 96 | 90 | 89 | 76 | 60 | 66 100 | 82| 84) 90| 66] 80/ 98 84 | 70 |72/)100 |66| 44 | 621661 59 | 100 ss | 86 | 751100 |68| 64] Go| 74 | 66 | 100 74 77 4 AS ut eee de Lee Fix. Six. C | D |ay.| E [A | Bio | D |Av.| = 92 | 58 | 86. 92| 841921781 86| 96 | 72 | 88 76 | 84) 94190! 86 94 | 78 | 90 36 ici ae Pi eR asd aad. GD! fay. OBA Be po rD ay. E 100 | 100 | 94 | 96 86} 92} 92] 92 | 90 98 | 100 | 100 | 99 | 88 | 100 | 96 1100 96 96 on.) 4} Ot 82 | 98 | 92 |100 | 93 100 | 68 | 78 | 85/100 | 72] 80/90] 96| 84| 98 74 | 90 | 70 86 | 100 | | ( WigeaboOe | XIII. On the Instantaneous Propagation of Disturbance in a Dispersive Medium. By T. H. HAVELOCK, M.A., Dose Armstrong College, Newcastle-on- Tyne™ iM fee a recent paper in this Magazine (July 1909) Lord Rayleigh discussed some cases of instantaneous pro- pagation of a limited initial disturbance, and pointed out that, although the physical assumption which permits an infinite velocity may be obvious, there is an apparent paradox when the disturbance can be analysed by the Fourier method into simple waves of which the maximum wave-velocity is finite. In the following note two further cases which admit of exact solution are examined; the mathematical expressions are modified by a further Fourier analysis into periods so as to express more suitably a dis- turbance which has a definite beginning in time, and the result appears to emphasize the connexion of the phenomenon with the dispersive character of the medium. Let y denote the effect at a position w and time é of an initial disturbance in a dispersive medium; for an initial displacement cos «, with no initial velocity, the disturbance is given by y = eos(KVt) coske, . .. En where V is supposed a known function of «. Generalizing by Fourier’s method we have for an initial displacement iia) . 10 ast yah If the initial disturbance is limited practically to a line through the origin it is usual to write dk COs eve| {(w) coskK(w—x)do. . (2) 0 1 ; 2 |, cos (x Vt) cos(xz)dx, . 3am where for convergency a factor e~*’ may be introduced under the integral sign, and the limit taken for 7 y Zero. The chief regular features of the disturbance may be obtained by interpreting (3) in terms of groups of waves travelling out from the origin. The same expression (3) was used by Lord Rayleigh in discussing the initial motion due to a limited disturbance. He showed that in general the effect begins at all points without delay, even when— * Communicated by the Author. Presented to the British Association at Winnipeg. ee Se j } a Propagation of Disturbance in a Dispersive Medium. 161 as in water of finite depth—the wave velocity V has a finite limiting value; a non-dispersive medium (V_ constant) appears as a particular case in which the effect may begin abruptly at each point after suitable delay. Lord Rayleigh remarks that in the ordinary theory of waves on water the velocity of propagation of waves of expansion is regarded as infinite, and doubtless some such dynamical assumption can be found in most cases; but the mathematical expressions may be modified and another point of view is suggested. The previous expression (3) gives equal values for negative as for positive values of ¢ ; in fact the whole disturbance is built up of standing waves and may be regarded as a vibration in which at the instant ¢=0 the displacement is confined to the region near «=O and the velocity is every- where zero. The appearance of an effect at every point immediately after this instant appears connected with the occurrence of a disturbance at each point before the instant in question, apart from considering velocity of propagation. If we attempt to form an expression which shall be zero for negative values of ¢ and shall give suitable values for ¢ positive, we find again that the effect begins without delay at all points; but this now appears connected with the fact that for certain periods no wave motion is possible. In other words, a Fourier analysis with respect to period divides the disturbance into two types; to each period there corre- sponds in the one case a wave motion and in the other a disturbance established in equal phase throughout. Hach art of the expression implies an effect beginning without delay at all points, but it is suggested that the existence of both types is essential for this result—the character of the dispersion of the medium being dependent upon the range of period and the manner in which wave motion is impossible. In order to avoid some complication of formulee in localizing the initial disturbance, illustrations are taken in propagation along connected systems of similar bodies. 2. Consider the case of a stretched string itself without mass, but carrying loads (w) at equal intervals (a) ; let T; be the tension of the string and let y, be the transverse dis-- placement of a particle. The potential and kinetic energy MK functions are ™ = qT, 2 qT 2 V= Soeene + da (Yr—Yr—1) + 2a (Yr +1 —yr) Sree ee Fr, P (4) T= 2huy, | * Lord Rayleigh, Scientific Papers, iv. p. 342. Phil. Mag. 8S. 6. Vol. 19. No. 109. Jan. 1910. M 162 Dr. T. H. Havelock on the Instantaneous If y» is proportional to e“#*+*"2), the velocity V of waves of length 27/« is given by T,a\ sinixa fs vag /(=*) 2%, a provided p?<4T,/ua. If this condition is not satisfied, simple wave motion is not possible. Instead of building up solutions from this simple type, we may solve directly a particular problem and then analyse the solution into simple harmonic components. ~ Suppose that initially a particle (y) is held displaced through unit distance and then released, so that when ¢=0 we have y, and ¥, zero for all values of 7 except that yp=1. The subsequent disturbance is symmetrical with respect to the particle 7g, and the equations of motion are Tip = Yr — 2Y yr $Y PSI e m oem aot oie Ne (6) T Yo = 241-205 3 where 7a? c= (Tia/p), From these equations, with the initial conditions, we may solve for yr as a series in t; we find the general result r= 3(-)aa () > et n! (n+ 2r)!\7 oto) a=0 This can be expressed by a Bessel function, as in fact follows directly from the equations (6), and we have for all values of 7, jp = do(Qi/t). «|.» 4 Thus the vibrations of the central particle are given by Jo(2t/rT), and those of the particles on either side by successive even Bessel functions of the same argument. Using tables of Bessel functions we may represent graphically the motion of the particles. The curves in fig. 1 show how the dis- placement of certain particles varies with the time, while in fig. 2 we have the positions of the particles on one side of the origin at instants given by t/7=0, 2, 4, 6,11. We may notice how the front part of the disturbance remains a crest as it travels out, becoming of greater effective wave- length in the process. 4 a 4 : £ a ‘ Propagation of Disturbance in a Dispersive Medium. 163 Fig. 1. meer - wm eee eo ew ow eo t t t t U : i t ' 6 L} ‘ Using an integral expression for Jo,, and writing w for ra we have from (8), Y= J orja(2ct/a) si 2 a (2 ree | = — cos 5 —(wd—ct sin d aw \, ae ) ¢ Pi oy) x T =") cos 4«(a—Vt) idk, ae epee ee (9) T Vo . __ sin (3«a) = CO apemacries 2K M2 where 164 Dr. T. H. Havelock on the Instantaneous We have also the group velocity U J=d(«V)/de=c cos (xa). In the range for « from 0 to z/a, both U and V are positive and vary between 0 and ¢; fig. 3 shows the curves for U and V as functions of the wave-length A, and their limiting positions for a zero. When a is diminished indefinitely we approach as a limit the case of a uniform string with V and U equal to «. For an initial displacement / at the origin, we have y given by (9) multiplied by 1; as a approaches zero / must become infinite in order that we may obtain the divergent integral (3) which is used to represent the effect of a concentrated initial disturbance. Fiz. 3 Jn the integral (9) the solution (8) is expressed in terms of simple harmonic components of wave-lengths ranging from 2atow. The effect begins without delay at each point, and the solutions give equal values for equal positiv eand ee e values of ¢. Suppose we regard the central particle (yp) as a source of displacement. Then if 2 yo is zero for all negative values of ¢, and equal to /(t) for positive values, we can write 1 co 3 i) . Vox ar a a (w)cosn(m—t)do, . . (i) with y=4/(0+) for t=0. If we put f(t) equal to Jo(2t/r) we must cut out of the integration with respect to the region in the vicinity of n=3t/t ; in this way, or writing down the result dir ectly, Propagation of Disturbance in a Dispersive Medium. 165 we have Ly 72 eusigin My SELL E a mre a poe ees CLL oh alo Vag eae a pe eae a er) T a z wo PONE beste a = nyt Ir. 9 Se i: cos («Vt)de+ ={, sim, (eV bya, = (12) where V=c sin (4«a)./Lea ; | V'=c cosh (4e/a)./3ea. In (12) each integral is equal numerically to $Jo(2t/7) 5 3 hence yp is zero for t “hegative, J(2t/7) for t positive, and $ for é zero." ” Now we may write down a similar expression y, which is zero for t negative and equal to J2,(2t/7) for ¢ positive. We have * ie hs 1 ‘er neal cos (2 sin b—2n6)dp+ Lf” sin (u+, a a re r+ 7 v a —S—— ; f a2 WHr\t Las \a —K'2 3} \J! ! ‘ Fre) conten Vt) jde-+(=—1) ao v sin («'V't)dk (13) with the same notation as in (12). Comparing (12) and (13) we see how the central par ticle may be regarded as a source of displacement which is sud- denly cre eated at a certain instant. Physically, the idea is simpler if y is a velocity suddenly created by an impulse, but the analytical expressions are less direct. ‘The first integrals in (12) and (13) correspond to periods (>t) for which simple wave-motion is possible, the greatest wave-velocity being finite ; the second integrals arise from com ponent periods less than wr. Of the value of y, at any time one half is associated with each type of integral. In the ordinary analy sis, (12) and (13) would be replaced by twice the first integral in each case, with 0 and © as the limits, thus assuming the possibility of representing the effect of a suddenly created source of disturbance by simple wave- trains of all possible wave-lengths ; the above illustration suggests that the existence of two types of integral may be necessary in a dispersive medium and that the apparent instantaneous propagation is connected with this fact. In this illustration the infinite velocity enters dynamically by the neglect of the inertia of the string. If we supposed this to be of uniform density we should ‘have something analogous oO to two interacting media, the particles being connected by * Sonine, Math. Ann. xvi. p. 15, 166 Dr. T. H. Havelock on the Instantaneous stretches of non-dispersive medium along which a disturbance could travel with an abrupt front. 3. Another illustration which may be treated in the same way is taken from Lord Rayleigh’s paper, already quoted, on propagation of waves along connected systems of similar bodies. Let each mass be connected to its immediate neigh- bours on the two sides by an elastic rod capable of bending but without inertia. The potential energy function is of the form Vi oe FBO 2Yn—1 —Yr—2 — Yr)” + BO (2Yr— Yr -1— Yr)” + 4b (2yr41—-Yr— Ye)? + stave wialeunime e ° ° (14) With the same initial conditions, the coordinate 7 unity, we have for the subsequent motion—writing t?=a?/c?=p/4l, Ap? y = —Yy—2 + 4Yr—1— 6 Yr FAY 41 — Yr ?> "e Ar’y, = 4y)— Ty, + 4y2—Ys, (15) Ar*yy= — 6 yo + 841 — 2y2-- J Forming a series for y, in terms of ¢ we find this can be expressed for all values of rv in the form t= I )os(rF ~*). a Writing this in an integral form, with « for va, we have ya Fal Jooo(Si— Z) = =f" COs {: (db —ct sin?d) hag diy T ai [eos tele-Vo hae, 0 with V=< eee oka Further, we have the group velocity U=csin (ka). In (17) the disturbance is expressed in simple trains of wave- lengths ranging from 2a to » ; in this range both V and U are positive and their variation with the wave-length is shown in fig. 4. The hyperbolic curves in the same diagram show the limiting forms of U and V if we wish to pass to the case of a continuous beam by making a small; the ordinary theory gives V = (const.) «, hence ¢ must become infinitely large so that in the limit the product ac is finite. Propagation of Disturbance in a Dispersive Medium. Fig. 5 is obtained by plotting the positions of the particles at the instants t/r=0, 2, 6, 11; we can trace the progress of the disturbance outwards from the centrai particle. Fig. 5 168 Mr. N. Campbell on the may be compared with fig. 2 of the previous case ; here, owing to smaller wave-lengths being associated with larger velocities, we may observe ‘the front “part of the disturbance alternating between advancing as a crest and as a trough. If we wish to regard the origin as the seat of a suddenly created disturbance we must analyse yp and y into periods and we find as before two types of integrals. Using certain known integrals involving Bessel functions * we may reduce the expressions to a form suitable for our purpose: we find in this way = = s cos («Vt)de+ al sin (x/Vt)d«', . 7) 2 “ 100 WA OER Nise T= aah cos {x(a—Vt)}de+ mil e~*™ sin (x Vi)de ae where V=c sin? (4«a)./Sea 3 V'=c sinh?($x'a)/Sk'a. Comparing the expressions in (17), (18), and (19) we see that they allow of the same inter pretation as the similar forms in the previous illustration. XIV. The Principles of Dynamics. By NorMAN CAMPBELL, Fellow of Trinity College, Cambridge ft. & Introduction. . Definition of the problems discussed. z The relation of mathematics and physics. 4, An example of this relation. 5. The relation “ A.” 6, 7,8. The fundamental conceptions of dynamics. 9. The application of dynamics to experiment. 10. The assumptions that must be made to make such application possible. li. “ Absolute and relative motion.” 12. “ The velocity of the sun in space.” 18. “ Absolute translation and absolute rotation.” $1. [T is doubtless rash nowadays to attempt to offer any remarks on a subject which has been so much discussed as the basis of dynamics with the hope that they shall be at the same time novel and useful. Ever since the publication of the brilliant treatise of Prof. Mach on the history of Mechanics the questions which he raised have been canvassed eagerly by inquirers of every nation, and it might * Nielsen, Cylinderfunktionen, p. 195. + Communicated by the Author. Principles of Dynamics. 169 seem that every possible and relevant observation on the subject had been offered long ago. But so long as complete agreement is not attained the discussion cannot be closed. It is not long since Mr. Bertrand Russell, in his ‘ Principles of Mathematics,’ denounced Prof. Mach’s conclusions, and asserted that after all Newton’s statement of the case was superior to all the attempts that had been made to displace it; and, though probably all students of physics would agree as to the main lines on which the principles of dynamics should be stated, they display in approaching the questions of Absolute and Relative Motion a hesitancy and uncertainty which suggests that their ideas are after all not quite so clear as they pretend. 2. There are two distinct problems, which are often confused, which require solution. The first is “In what manner are the fundamental propositions from which the conclusions of theoretical dynamics are deduced to be stated ; what are the conceptions employed in those propositions, and what are the relations stated between them”? The second is ‘* In what manner are these propositions applied to experi- ment ; how do we measure the magnitudes associated with the fundamental conceptions ?”” I believe that the difficulty which has been found in answering these questions arises from some confusion as to the relation between mathematics and physics, and accordingly 1 propose to open the discussion with some remarks on this relation. § 3. The usefulness of mathematics to the student of ex- perimental science arises from the following fact :—It is sometimes possible to form mathematical equations containing a certain number of quantities which are all capable of being regarded as variables such that, when for some of the variables are substituted numbers arising from measurement and the equations solved, the resulting values of the other variables represent the numbers resulting from certain other measurements. J do not propose here to diseuss how the equations are to be formed, or in what cases they can be formed : it will be sufficient for our purpose to note that they are sometimes possible. § 4. Let us take as an example, simpler in some respects than that offered by mechanics, Van der Waals’s equation for the state of a gas (r+° 2) eb) Jeo RT. There are six different quantities involved in this equation, p, v, I, a, b, Ky; in order that the equation may be solved so 170 Mr. N. Campbell on the as to give a numerical result for one of them, numerical values must be substituted for the other five. But it appears immediately that from the physical, though not from the mathematical standpoint, there is a great difference between the nature of these quantities. Three kinds of physical measurements are connected with the equation—measure- ments of pressure, volume, and temperature, and the measurement of these three properties for any mass of gas can be effected independently of each other, and of any knowledge of Van der Waals’s equations. Correlated with these measurements are the three quantities p, v, T, so that when the results of the measurements are known the values of these quantities which are to be substituted in the equation are also known. But the other quantities a, ), R are not so correlated with any physical measurement; no amount of physical measurement alone can decide what values are to be attached to these quantities; their values can only be found by making three independent series of measurements of pressure, volume, and temperature, and forming and solving three resultant equations in a, b, and R. § 5. It will be noted that I have been careful to speak of the quantity “p” (e. g.) being correlated with the pressure measured experimentally. I want to insist very strongly that the relation is not one of identity, that ‘‘p” is not the same thing as the measure of the pressure. This statement should be sufficiently obvious, and yet it contains, I believe, the solution of the difficulties of dynamics. Two concepts are identical only if they are defined in the same way, or if their definitions can be shown to be logically equivalent. Now I do not propose to define either the quantity “‘p’’ or the measure of the pressure, but nobody can reasonably doubt that any satisfactory definitions would not be the same or logically equivalent; they would contain totally different conceptions. The definition of ‘“‘p,” at least if defined by Mr. Russell, would probably contain something about “ well-ordered continua,” and continuity is, as M. Poincaré has often pointed out, foreign to experiment; the definition of ‘ pressure ” would contain much about sense-impressions, and sense-impressions, since the exposure of Mill at least, have been held universally to be foreign to mathematics. But if anyone doubts the diversity of the two classes of concepts, let him consider the quantity “v.’? Van der Waals’s equation, when numerical values are inserted for the other quantities, is a cubic in v; for each set of values of p, T, a, 6, R we deduce three values for v. But experiment shows that, for any values of the pressure and temperature of a given mass of gas, there are Principles of Dynamics. Let never three possible values for the volume——usually there is only one, sometimes there are two. Accordingly, “v’’ can have values which the volume cannot have, and it cannot be held that ‘vy ” and the volume are the same concept. The pressure and the quantity “p” are, then, perfectly dis- tinct concepts ; but there is a relation between them, into the exact nature of which it is unnecessary for our present purpose to inquire further. For the sake of brevity in what follows I shall call this relation “ relation A,” and I shall denote the mathematical quantity which bears relation A to a physical measurement by the name of the measurement written with a capital letter. Thus “p” will be called the Pressure, “T” the Temperature, and “ v” the Volume. Now let us consider for a moment the quantities a, 6, R. It is clear that these are quantities of the same nature as the Pressure, Volume, and Temperature, and are not similar to the pressure, volume, and temperature ; but they differ from those quantities in the fact that they do not bear the relation A to any physical measurements. But, it may be said, é is (or is proportional to) the volume of the molecules of the gas. This is just the fallacy I wish to expose: 6 may be the Volume of the molecules, but it is not a volume at all. I do not deny that it is very useful to call b the Volume of the molecules; the expression suggests a relation between 6 and quantities occurring in other equations, but we must not let the name conceal the fact that the term “volume of the molecules’? is meaningless, and that there is no physical measurement which is correlated with 8 as the volume of the gas is correlated with v. “” cannot be defined without re- ference to Van der Waals’s equation, but “v” can be defined without such reference by means of the relation A which it bears to the volume. , § 6, Now, after this preliminary discussion, let us turn to dynamics. Dynamics is the study of motion, or the change of distance with time. The only physical measurements which are con- cerned in the study are those of distance and time*. Qur object in stating the fundamental principles of theoretical dynamics is to formulate equations such that, when for certain quantities are substituted values depending on dis- tances and times and the equations solved, some of the roots should have values which depend in the same way on other distances and times. * It is not relevant to our purpose to inquire.how distance and time are measured ; it is sufficient to note that there are certain conventional methods which are familiar to all. 72 Mr. N. Campbell on the We must first note that the ‘“ distance between two bodies of finite size”? is, in general, meaningless; distances can in strictness enly refer to the smallest portions of a body which are distinguishable. Accordingly, in order that any propo- sition about the motion of a system of bodies may be stated, it must be supposed that they are subdivided into the smallest portions which are distinguishable. These portions are called particles. § 7. It is obvious that among the quantities occurring in the equations which we are going to establish must be (i) a quantity ¢, the Time, which bears relation A to the time, and (2) Distances 7,2... 712... mn -.. So correlated with the distances between the various particles (supposed to be distinguished by numbers 1, 2, ... m, ... n). The briefest inspection shows, however, that these quantities are not sufficient ; it may be possible to deduce for any particular system the distances at one time from the distances given at another,. but the form of the relation between ¢, 79... ?nn will certainly be different for different systems. In order to obtain relations of the same form for all systems, we must introduce quantities which depend upon the nature of the particles of the particular system. It is the first postulate of dynamics that, in order to establish equations applicable to all systems, the number of such quantities which must be introduced is equal to the number of particles, so that one quantity may be assigned to each particle. These quantities will be termed the Masses, and will be denoted by my,... 1). A further postulate is made—for excellent reasons into which we need not inquire here——as to the form which the equations are to take. It is stated that the equations are to be linear in the m’s, and that the quantity ¢ is to occur only in G27 (Pin, tan) dt" § 8. When we endeavour to frame with these quantities and with these postulates equations which shall fulfil the fundamental purposes which are demanded of them, we find that the task is impossible. In order to attain success we must either reject one of the postulates, and so change the form cf our equations, or we must introduce new quantities. Again we need not inquire into the reason of our choice, which falls s upon the latter alternative. We introduce a number of new quantities functions of the form S12 Jigs hyo saiereieil J dng g lig hin, Bick caleoine Imig lives Ly, Yi cy ep leb ciel ier ce Ung Yirs en 5 Principles of Dynamics. 173 and state that the following relations hold between the old and the new quantities :— d? iat ie fink Ft (Nien sl Vou 2S dé #m) eal, ne Jimi ae Im2+ peeeee Ymn == dt? (7 Hi ( , : - . ( f A abe Na Rind ae Dinot nae e (a = ie (2g and : i (fm —2n)? + (Yn —Yn)? + En—2n)2=Pmn > (2) We now state the fundamental propositions of theoretical dynamics in a series of equations, of which the following are typical :— Mm Jnn + Mn fnm=0, and so for g’s and f’s, . (3) i mn __ Um Un Imn fi m —“Yn ToT 9 = ° ° . (4) Jmn Ym Yn Ain on Zn These equations are the answer to the first of the questions of §2; they represent the fundamental propositions of theoretical dynamics. Before proceeding to discuss the second, and more important question, a few remarks as to: the nature of the quantities involved will be in place. Only the quantities *,,, and ¢ bear relation A to the measures of physical quantities ; correlated with these Relative ; Distances and Times there are relative distances and times. | The other quantities are not so correlated. The quantities Wm; Ym, 2m are termed the Absolute Coordinates of the par- | ) » | | “a Bets ae ticles, the quantities ap (2m Ym, 2m) their Absolute Accele- rations, the quantities 7,,, their Partial Accelerations. These : quantities can only be defined by means of the equations of dynamics ; so far as we have scen at present there are no | absolute coordinates or absolute accelerations to which they ; bear the relation A™*. : | § 9. In the problems which are usually discussed by theoretical dynamics a knowledge of the values of some of the Masses and Partial Accelerations is assumed; the others are eliminated by the introduction of certain belaeians between * It is to be observed that there has been no necessity to introduce Absolute Times, related to the Relative Time ¢ as the Absolute Coordinates are related to the Relative Distances, because ¢ enters into the equations only in the differential coefficients ; we are only concerned with differences of ¢. 174 Mr. N. Campbell on the the Absolute Coordinates. These values must, of course, be chosen so as to satisfy equations (3) and (4); these equations play no further part in the discussion, which consists in finding the relations between the Absolute Coordinates and the Time by means of equations (1), the number of which is equal to the number of Absolute Coordinates. The Relative Distances can then be expressed as functions of the Time by means of the equations (2). But in the use of theoretical mechanics for the purposes of experimental science the most important qnestion is that of which the answer is assumed in dealing with such problems; the object of the investigation is to find out what values are to be assigned in any special case to the Masses and the Partial Accelerations, and these values, if they are to be determined at all, must be deter- mined from the fundamental equations. For, as has been pointed out, they cannot be determined directly from measure- ment by means of our relation A, nor can they possibly be determined by the solution of any other equation, since the equations of dynamics are fundamental and logically prior to any others involving the same concepts. We must decide then whether it be possible, and, if so, by what means it is possible, to determine the Masses and Partial Accelerations from a knowledge of the relation between the Relative Dis- tances and the Time, such as is deduced from the observed relation between the relative distances and the time. It is clear immediately that a unique determination of these quantities 1s not possible in general. For suppose that we have made N observations on a system of 2 particles, giving N sets of corresponding values of 7,,, and t. Then we have 3Nn equations of the type (1), N(3n—5) of the type (2) (see § 12 below), eS n(n—1) y the type (3), and Nn(n—1) of the type (4), in “all ( n+ — Go H5)N, On the otlier hand, the number of unknown PON ee which must be treated as independent variables, is 3n?N-+n—1, that is 3Nn(n—1) Partial Accelerations, 3Nn Absolute Coordinates, and »—1 Masses, since we are content with a knowledge of the ratios of the Masses * It is to be observed that the number of equations will be greater than that of the independent variables for some values of N and n, and will be less for others. * Really the number of equations ought to be somewhat greater than is stated, for it is assumed implicitly that all the masses are positive quantities. The estimation of tie additional number of equations to be added on this account, if it is possible at all, ig very difficult and would not affect the result of the ar sument. Principles of Dynamics. 175 § 10. Now we do not know either N or n. For the equations (3) and (4) claim perfect generality ; they must be true for all the particles in the universe and all the observations that have been or will be made. It is true that for certain values of the Partial Accelerations certain of the bodies can be left entirely out of consideration, but we cannot tell which these bodies are till we know what the Partial Accelerations are, and this is precisely what we are trying to discover. Accordingly, in the first instance, we cannot put for N or m any number less than that of all experiments and all particles. Moreover, our measurements must include the relative distances of all particles at each of the times considered. In this form the problem is absolutely intract- able; in practice the matter is simplified by the introduction of a very large number of assumptions which eliminate com- pletely most of the particles and reduce greatly the number of variables. The assumptions made are of three kinds :— (1) It is assumed that the particles may be divided into two groups, one of which includes only a small number of par- ticles, and that all Partial Accelerations, the suffixes of which are taken from different groups, are nil. Assumptions of this kind are made in all estimations of masses. (2) It is assumed that some of the Partial Accelerations are known functions of the Absolute Coordinates: such assumptions are made in astronomical estimates of Masses. (3) It is assumed that the values of some of the Absolute Coordinates are known: such assumptions are made in the measurement or estimation of Masses by ballistic methods in the laboratory. The number of independent equations has doubtless not been estimated quite accurately above, but it is certain at least that, for very large numbers of particles such as would be concerned if our observed system included the whole physical universe, tlhe number of independent variables would be greater than the number of equations, and the problem of determining the Masses and Partial Accelerations would be indeterminate. But the assumptions introduced in practice are always so numerous as to make the number of equations greater than the number of variables. Some of the assumptions are precarious and might be proved to be inconsistent with the others or with the propositions of dynamics: nor can we avoid this precariousness by making exactly the right number of assumptions, for we do not know how many assumptions to make. Theoretical mechanics is useful in experimental science only because certain fortunate people possess that remarkable quality called ‘ scientific intuition,’ which enables them, in a very small number of 176 Mr. N. Campbell on the guesses, to devise assumptions of which none are actually ever proved to be inconsistent with each other. Even the greatest genius fails sometimes at the first guess, as in the case of the Foucault pendulum, but then, as in that case, the second guess is successful. Much of this is trite and has been said in other words often before. But the conclusion is important that the pro- position of theoretical dynamics (even if relation A between the Distances and Times on the one hand and the distances and times on the other is maintained strictly) cannot be either proved or disproved by experiment. In order to apply those propositions to experiment it is necessary to introduce a certain number of assumptions, which are wholly inde- pendent of those of theoretical dynamics: the connexion between theoretical and practical dynamics will be broken only when it is impossible to find assumptions, suitable for the purpose, which are not inconsistent with those adopted on other grounds. § 11. Let us now consider the vexed questions of dynamics —those concerning ‘absolute and relative motion.” It is sometimes asked, ‘“‘ Is motion absolute or relative ”’? From what has been said it is clear that the question does not admit of a definite answer without further explanation. The word ‘motion’? might mean any one of three things : (1) a concept defined by the Absolute Coordinates and the Time; (2) a concept defined by the Relative Distances and the Time; (3) a concept defined by the relative distances and the time. There is no doubt that the word is often used in the senses (1) and (2), which we may term Absolute Motion and Relative Motion respectively; but I do not think that I have ever seen it used in the sense (3). The “motion” round which almost all discussion takes place is defined in terms of mathematical quantities, to which the operations of mathematics can be applied significantly. But “ motion” in the sense (3) is not a mathematical quantity: it is defined in terms of conceptions which are wholly foreign to that study. lt is no more significant to talk of adding (in the mathema- tical sense) two distances than of adding two colours. All that we can do in this direction is to define Distances bearing the relation A to the two distances, add these Distances, and define a new distance to which the sum bears the relation A. Now Absolute Motion and Relative Motion are both valid concepts: they are related, but the relation is neither identity nor contradiction; they happen to be commonly called by the same name because they have developed from the same Principles of Dynamics. 177 ideas. Absolute Motion is of fundamental importance in theoretical mechanics, Relative Motion is of fundamental importance in experimental science. As a natural conse- quence, a student of applied mathematics, when asked of the nature of “motion,” thinks of Absolute Motion, and replies that ‘‘ motion is absolute ;” and the physicist when he meets the same question thinks of relative motion, and replies that “‘motion is relative.” But there is no difference of opinion between them, only a mutual misunderstanding. An exact analogy can be imagined. The word “rule” is, for historical reasons, used in two senses, first as a precept, ‘and secondly as an instrument (slide-rule) ; and there can be a relation between the two meanings, as in the phrase “the rule for the use of the slide-rule.” Now suppose two people were asked whether the rule was accurate: one might think that the question concerned the soundness of the instructions given for the use of the instrument, the other might think that it concerned its construction. They might arrive at formally contradictory conclusions, and no amount of argu- ment would settle the controversy, until they began to dispute in Esperanto. But another, and much more interesting, question is raised by such a statement as the following (put i in very rough and loose language) :—‘‘ Absolute motion is a mere mathematical abstraction, whereas relative motion is a real thing.” Of course, if (as is usually the case) Absolute Motion and Re- lative Motion are meant, the statement is absurd: for both coneepts are mathematical quantities, and if one is “real” so is the other. But I think that anyone who made such a statement would have different, and much more valuable, ideas in his mind: he would mean to point out that, whereas Absolute Motion can be given a meaning only by use of the fundamental propositions of dynamics, Relative Motion can be given a meaning without employing those propositions and by using only experimental results and relation A, and that the ideas thus involved appear to him much more fundamental and necessary than those of dynamics. . It is, therefore, very interesting to inquire whether, by any means whatsoever, the Absolute Coordinates can be defined inde- pendently of dynamics (i. e¢. equations (3) and (4)), and by the use of only those concepts which are necessary to define Relative Motion, or of those which are equally necessary*. * It would lead us much too far to discuss why we regard one con- ception as more fundamental or necessary than “another, or whether there is any justification for the view. But I think almost every physicist would agree in regarding relation A as on a different plane to the relations asserted in equations (3) and (4). Phil. Mag. 8. 6. Vol. 19. No. 109. Jan. 1910. N 178 Mr. N. Campbell on the The object might be thought to be achieved if we could express the Absolute Coordinates as algebraical functions of the Relative Distances and the Times by means of equations (1) and (2). But, since there are more variables than there are equations, we cannot do so unless we introduce more equations relating the variables. Now, whatever form these additional equations are given, they will introduce, implicitly or explicitly, Partial Accelerations. But Partial Accelera- tions are concepts which have no meaning apart from dynamics ; hence the additional propesitions which we in- troduce in order to render equations (1) and (2) determinate are dynamical propositions of some form or other: the ideas contained in them differ just as much in their nature from ideas which are necessary for the definition of Relative Motion as those contained in the ordinary equations (3) and (4). We must conclude, therefore, that it is impossible to define Absolute Motion without the use of concepts which appear less fundamental than those which are necessary for the definition of Relative Motion. To sum up. If it be asked, Is “ motion” absolute or relative ?, the answer is, Both. If it be asked, “Is Absolute Motion a concept as fundamental and necessary for the student of physics as Relative Motion?” or “Can it be related in as simple a way to the results of observation?” then the answer is, No. § 12. In discussions* concerning “absolute motion” a concept is often introduced which is called “the velocity of the solar system in space,” or ‘“‘relative to the stars.” It is desirable that a brief consideration should be given to the nature of this concept. The introduction of this concept is usually prefaced by the statement that we can (ideally at least) determine the velocity of the sun relative to any star, both in and at right angles to the line of sight. There seems a tendency to ov erlook the obvious fact that, if by “ the relative velocity of two particles ” is meant change of their distance with time, there can be no sense In speaking of a determination of the relative velocity of two particles ‘in or at right angles to”’ the line joining them. Tf only these two particles are known the idea of direction is impossible; the particles change their distance, but it is meaninyless as yet to inquire Ww hether their motion is in or at right angles to the line joining them. The measurements which are “actually made on the sun and stars concern the rate of change of the relative distances of two stars or of * See e.g. Schuster, “ A Plea for Absolute Motion,” Nat. Ixxiii. p. 462. Prof. Schuster’s remarls show very clearly the confusion between Motion aud motion to which attention has been drawn. Principles of Dynamics. 179 the sun and one star*. There are thus obtained “\” ~~? Distances, each of which bears the relation A to one of the measurements. Now a long past experience, to which there has been no exception, has revealed that whenever we _ nrn—l : obtain laa Distances, ?n»..., by measurement on 7 a particles at the same time, these distances are always roots of Asam 1) —(3n—5) equations of the form /4(”,,. ... )=0, where ‘8 forms of the functions are the same for all sets of measurements; so that, if 3n—5 of the Distances are known, the others can be found. This is the proposition which we mean to assert when we say that physical space has three dimensions: if the number of such equations were Aa a —(Nn—2N +1), we should say physical space had N Gites... Accordingly we proceed to find 8n—5 independent varia- bles, suitable functions of which we propose to introduce 4nto our equations in place of the Distances. In practice, for sake of symmetry, we find 3n independent variables (2m; Ymy 2m --- ) from 3n equations of the type (Um— in)? + (Ym—Yn)” + ( + i) (13) This definition also makes J_, (z)=(—)' J, (z) when n is integral. By comparison with Hankel’s expansions the formule (8,9) are obviously the most suitable for the ex- pansion of (7, Ys) in the forms R3(sinp. cos p) in general, * Hankel, 2. c. (14) (15) 232 Dr. J. W. Nicholson on the Asymptotic when x ceases to be small. As defined above, J» (z) admits the integral formula* aD TS y a= { cos (z sin @—76)d@—sin oe | ag. e-n0—c eink, aaa 0 0 z being positive, whilst 7J_, (z) is represented by the same expression with the sign of x changed. If n be an integer we select, as standard solutions, n=) os i... a y=—(Z) Ye), | fa where Y,, (z) is Hankel’s second solution of Bessel’s equation, defined by Jn a ‘pew & Yn @=(° a 1_(-p? = Sees, : Cu. ‘By proceeding to the limit when 2 is integral in the former case, it is at once obvious that this substitution is the natural continuation of the first. Thus it is again true that yon’ —yyel=1, - - . i and, therefore, when x is integral, the expansions deduced for (J_n (2) — cosnm Jz (z)) cosecnm will remain valid for a ~ Yn (z). This is an obvious property of Hankel’s expan- sions when » is small. When z is small, Y,(z) may be written * Ya(2)=2J,(2)4 9+ log 5 ~6) (or+4-6) + — z aS. ~ nt28, +Sr41 ¢ —(5) = +(5) neil il, “ieee (22) iL y=—'OT7 cory Sa=1l+d+44+ coe + n where and negative factorials are to be taken as zero. The asymptotic expansion when < 1s large isf z\2 7 OT : RT sn (2) ¥a(2)=— Uns) 08 (-- OF +g) + Vas) sin (z- = +7 ),(23) . where U,, V, are as before. * e.g. vide Whittaker, Modern Analysis, p. 281. + Hankel, Math. Ann. 1. p. 494. EHapansions of Besselj Functions. 233 Solutions of the Associated Equation. Values of R, p. If y denote y, or yin the substitutions for Bessel functions, (yz)! + (1- ) yz =0. 21 oe oo at EE} and the associated equation becomes Sul + (42 —4n2z+2)u'+(4n?—l)u=0. . (25) Writing v= De Bre"; 7 pene fi i aa 2 the relation between successive coefficients becomes rPt+1li.r+2—2n.74+242n. ap4o= —Ara,, with an indicial equation s—1l.s—14+2n.5s—1—2n=0. The following series solutions therefore exist, An?—12 1.3 dn? —1?, dn? —3? ie ate 5) Pay (427)? + eee . . . © “ ioe “ng ait ball ee eee 9 4 fit og nt CD ES a eee ee: ) = ( n+l 2. 2n417 mtl.nt+2 2.4.2n+1.2n42°°) | ge a eS) where if » be an integer, uw. must be multiplied by the evanescent factor of the denominators, thereby ceasing to be distinct from w;. For positive real values of n, us and us are convergent for all finite values of z, but w is ultimately divergent except when 2n is an odd integer, in which case it terminates. Proceeding to an examination of w, it is seen that when z is infinite in comparison with n, u;=1. But comparing (8, 9) and (12, 23), in this case, y,? + y.?=1, or R=1 from the first asymptotic substitution. Thus u,=R when z is very great, and being always a linear combination of R, 8, and T, which are of similar magnitudes, it must always be R either identically or in an asymptotic sense, not necessarily that of Poincaré. 234 Dr. J. W. Nicholson on the Asymptotic When x and < are both great, a first approximation to R is obviously 9° gh OURS ae 22 2 (A. ue or z R=———- Z>n, which leads to one of Lorenz’s expansions, when 2 is half an odd integer. But the error involved would be very doubtful. Now writing as in usual notation for Bessel functions of imaginary argument, _yX Kyo(A)=] e-Acoshodd, . . . . (29) 0 then it may be shown * that when the series terminates be ( : Ny == = =) Ky (22 sinh ¢) cosh 2né dt, (30) 0 or, as a reversible double integral, — a | e—esinhécosheosh 2ntdidy. (31) OF 40 But this expression remains finite and determinate when n is not half an odd integer, and it may, moreover, be proved by direct substitution that it is still a solution of the same differential equation. Accordingly, it is still the value of R, as it also takes the correct value when <=. Thus for all real values of » and gz, ies zi ( K, (22 sinht) cosh 2ntdt, . . (82) “0 and when 2n is not an odd integer, the series w,, though divergent ultimately, may be used for the computation of the integral. Hixpressing the value of R in terms of the Bessel functions, we deduce, when » is not an integer, ao ‘ | J2(2) +d _,7(2) —23,(2)J_,,(z) cosn7= —esin” nar K,(22 sinh ¢) cosh 2né di, 0 : e (33) i and when z is an integer, Vi(2) + 7d 32) =8 K,(2z sinh ¢) cosh 2né dé. (34) «0 * Phil. Mag. Dec. 1907. OO —— << ee : Expansions of Bessel Functions. 235 Some of the special cases in which 2n is an odd integer are very interesting, as the integral can then be expressed in terms of trigonometric functions. Since by (4, 16) and when z is very great in comparison with x, the usual expansions yield CONE |) iain | eeacnel vent ss is 4? it follows that in general nT ey a: C aoe | (G-1)a - . (89) _ Second Solutions. Values of T, t when n is not integral. The second series solution, when » is not integral, is of the absolutely convergent form 3 Bes 2 aie oe Se i ag mn ee a eee + ie it ‘ ee? Felt n?—3? tr ra | sin 2nwv sin?” 2 dx, 0 where 7 is not, and m is an integer, then it is readily shown that —2m.2m—1 ao. 2. 5. @ Um~1 4A.n?—m ""” (—)” 2m! wr ep eT re area ae sin 2ne dz, 92m? — m?, n?—m— 17. ... n?=—1? 0 e whence 2nz (22 sin v)* 22 Bi 2). ik a ( pie a ( at Ra ... )sin 2ne dz 9) apes v PAY | ie ‘ : > = > ———5— ] sin 2na.J,(2z:sinz) dz. (36) 1— cos 2n7 J, It remains to identify w.. Now making a substitution (a modification of the third asymptotic substitution), n=(F) HO, = (5) cosecnm.J_n(2); nys=Tr (87) 236 Dr. J. W. Nicholson on the Asymptotic Then, by direct multiplication, an ascending series is obtained whose leading terms are ~> ~ Ay eee a SS8 On 1 2 8n. oe 2 and which, moreover, must satisfy the associated equation. Thus by comparison of series Uy =2nT), and v7 T = ee SI vu 22 I & wv. e ie anes =. sin 2naJ9(2z sin x) dv (38) _ This result was known to Lorenz for the case in which 2n is an odd integer. The substitution (36) is more convenient than (9) when 2 is non-integral and greater than z. Thus for all real values of z and n, the latter not being an integer, JZ 7 2)= a) sin 2nz .Jo(2¢sinaz)da. (39) W?sin nr When 7 is an integer, evaluating the form then presented, by an obviously legitimate process, 1 ee é (—)Ji(e)= a 2 sin 2nz .Jo(2¢ sin 2) da= a sin 27 0 On (—)* « fe s £ oa srl 2x cos 2ZneJ,(22 sin x) dx (=) af" : re { 2 cos 2nvJo(2z sin x) da, 0 or 2 Lee é S Jn(Z)= =| sin 2auw.Jd(2zsinx)dz, nu=integer. (40) 0 This follows otherwise from a result given by Neumann*. The determination of T (T, being infinite) when z is integral is somewhat difficult, for ug and w3 cease to be distinct. For this determination a more direct method is useful f. It was shown that if ¢ be defined by the third asymptotic substitution, a Beret * Cf. Gray and Matthews, p. 28. + Cf. the formula for Ja(z) Yn(z), fra. Pe ne ee ee ee ee ee Expansions of Bessel Functions. 237 Now near 2=0, e =4 becomes zero like ga c=V, Y2 2° (n)T(n +1)’ or near -=0, 1, Pet Re: a °8 nT*(n) , and therefore in general 1 7 ad Se aS — _ $$$ =e D2 Ye — —_ — ~ i= 318 STC) n log 2 +n log +f (5 ds, (41) this being the only possible function satisfying all the con- ditions. Ina similar way, if t, corresponds to T, as ¢ to T,, t=n log 42+ 1 7 —7 te ee i= 2 Stn) —nlog2+nlog +f (sy — 2) 4. (42) These two relations cease to be distinct when n is very great in comparison with z, for it will appear that T, and ‘T only differ by an amount which is exponentially evanescent when n tends to infinity. The third series satisfying the associated equation is oni] 2n+1 2 2n+1.2n+3 ot | . mi 2.2nt+1" ntl.nt2 2.4. 9n41.2.042 ) T 2 tn 2n+2 pra Se is 2 ant 70 a4 gin22+49 — con+l 2-82.) 7S Si aR _ - sn Co bw aa a es i! (sin g L!2n+1 + 919n41. Ind? ..)a0- sin?” 6 d@ + 0 _T@m+il) ee ay Ie aCe 9 ail att) ayy Jon(22 sin 8) dé. But w2n+1 we = ae +3 Wwe a & y2= — J(2)= aa an | tL - s t ... 1 aay n(2) 22n+1T3(n+1) 2nd : and therefore by comparison, and by the differential equation, 9 TUsg Tv Y= FHIP(n +1) = ap Jon (22 sin 8) dé, by the use of a well-known property of gamma functions. Here 2n must be greater than —1, in order that the integral may be finite. Thus for all real values of z, and values of n greater than —3, Tv | rIie)= | Jon (@esind)de . . (44) 0 which is a known result for integral values of n. 238 Dr. J. W. Nicholson on the Asymptotic The value of §, in the second asymptotic substitution, may now be expressed, but this substitution is of small importance, as it is identical with the third in all useful cases. Again, by (44), if n<4, ae { J» (2esin 8) dé.) «+. 4) 0 Thus, if the argument of the functions be constantly ") = ("{2 sph Oe a— 2 on 2esin 8) +d8, or, when n is made cao its only aus value being zero, Tv wJo(2)¥olz)={ Yo(2ssin)d@, . . (46) 0 A more general result is proved in the next section. The Formula for Jn(z) Y,(2). An expression for the product of two Bessel functions of different ty pes, when n is integral, has been given by Neumann*. But it is somewhat unsuitable for our purpose, and an alternative is now developed which can, however, be formally identified with that of Neumann. By (43), the argument being < unless otherwise specified, Odn sae) ; 27J,,— a =| aA Jon (2z sin 8) dé. But wT Td on (w= | cos (wsin 6—2nd) dd 0 — sin 2rer| dd e~ 2" —w sinh 0 * Bessels'che Functionen, p. 65. . Fe Expansions of Bessel Functions. 239 Thus when 2 is an integer, Tv 7D Jan(w) = 2( ¢ sin (wsin 6—2n) dd 0 Hh anf dd e— 2g —w sinh ¢, 0 so that on reduction wT 9 T 12d an a | cos 2nd Ho(2z sin d) dp — =) P sin 2nd Jo(22 sin hd) dp . 0 0 2 nig — 2nd —2z sin @ sinh —= ( didpe dla) Me T 0 4 a) where Hy (w) is Struve’s function * defined by Hy (w) = = sin(wsin@)d@. . . . (48) 0 Again, 7 sin 2n v 0 sin nm so that when x is integral, OF an MY sin Qn a Jams = +(— yng, 2 *) = arc sin @.—-———__ dg 07 2 sin n7?r (2x cos Snes sin nw —7 COS nT sin 2nx) | = Sol 22 sin 2) st sin? x7 The Meena takes the form ; when x is integral. Therefore evaluating in the usual way, which is obviously legitimate, we obtain , Oo cil OJ Ae Sa bak , (-)'Jn t+dn arr ek, (7? —42*) sin 2nw Jo(2z sin x) da. By Be isien with (47), and with the help of the results, true for integral values of 2, vis | Moeaed;(22sinz)da-0. . . . .« «ss. = (49) = ris \ a? sin Qnw Jg(2e sin 2) do=n( sin 2naJ(2esine) de, . . (50) 0 0 it appears that Tr Sale e)Y,(2) =F] cos 2nex Ho( (Qesin a)de— "asf dé e —2ng —22 sin @ sinh¢ 0 (21) * Cf. Struve, Wied. Ann, Bd. xvi. 1882, p. 1008; Lord Rayleigh, ‘ Theory of Sound,’ § 202. 240 Dr. J. W. Nicholson on the Asymptotic The value of T when x is integral may now be deduced, and thence t by (41). Expansions of the First Type. Returning to the value of R given in (381), it is first to be noticed that the evaluation (asymptotic) of the integral, given in a previous paper *, was in no way dependent upon the restriction of 2n to an odd integer. Accordingly, this evaluation may be used in the general case. The same applies to the subsequent treatment of p as given by (35) of the present paper. Thus quoting the values of R and p previously obtained, we obtain the following asymptotic expansions when 7 is less than z, and z—n is not very small :— When 2 is not an integer Jale)= ~*) sin a(<)=(—— ) sin p 2R): ’ J_,(2)— cos nm Jn(z) =(—) cos p Sin 27, Tz and when 2 is an integer, Bay \s,. In(2\= =~) sin ne) Neen p 2arR\2 Va) a l= ea COS Pp; where if n==. Also F(0)=0, and F’ must not vanish in the ranges except perhaps at ¢=0. In the present case w= , and n is less than 2, so 7 z cosh that EF’ =v’ cannot vanish. The other conditions are obviously satisfied. Mt a gee Ga ; Now when ¢=0, v=0, and — }) — is zero when 7 is : v' dt] v odd, and when * is even, it is given by 5-A**? w,41, where u,41 represents the term in R next after the final one retained ; i : (in the second expression for R) with —.—~, substituted : ML +) for oe and the summation and integration prefixed. Thus the remainder after + terms only differs by an order A! from 7 £f area ie 0) eh alt Or ay ( NUps1 ahs 0 r ‘when v is odd. When « is even its order is less. If the 2th term of R be therefore denoted by 7U,(2?—n?)-2, where U; is a certain operation, the error involved in stopping at the rth term is of magnitude 1 if 1 ate Pen OME ARO. ea aU al ¥(- cosh w—n + cosh rane or il re | — Ups Vins * atten and is therefore of the same magnitude as the term next after the last retained. The expansion of R is thus asymptotic in the proper sense. That of p will also be asymptotic, but less convergent. ee SC ae ee Ee a Expansions of Bessel Functions. 243 Eapansions when n is greater than a. When 2 is non-integral, in the notation of (36), ~ Z 7 , e i a ( sin 2a. J o( 22 sin z)dx 1 —cos 2n7 ; eg where be eee ( sin 2n(w typ sin a)de, (63) e O a z sin b 1 i y fe) “7 ’ 7 + . ad 7 . Ge 1 Now ae (w+ sin x) is never zero, ora very small quantity for any possible values of # and » in the double range of integration. Therefore the integration of I, and I, may be effected by the method of the last section, and in so far as the leading terms are concerned, len 1 I, =| eae (1+ pcos x) sin 2n(e + sin x )de oe (* digs 1—cos 2ar ear es ~ Qn+22sing Thus E ~ dor ay I6( 2 ane n—z sin Sys wiley yf 2 (nc?) on reduction. The leading term of ol is therefore (n?—2°)-#. Let. its expansion as a series be of the form ae, * & =+atst.. Be eename Cd) where w= (n?—<*)é. It is a solution of the same differential equatiou as which had the value when n <, the real form iE ee tg... awe) ib is a solution of the differential equation for x By com- 27, parison, it represents ——~ if v2,=(—)*A2.3; and therefore if w= (n?—z*)3, and the coefficients X have the same values as in expansion A, oT = s de = = ta —..; . ie and if the coefficients w are also given by expansion A, Alene Ho 4. ba ) ne “(1 bag Be) By We may now prove the existence of the identical relation (59)... Mor a = BE EY (sir). =2n(1 P + z= lh But by the integral (37), and therefore fa ay Beg | This is readily verified to any order by direct substitution of the values of pw calculated from (53). Now, with the use of the relation (a) =2n, it follows from (41) that Vo T fe z 1 L (ies t,=4 log ST"(n) nlog 2+ (ar) Jee Z +{ d (sy iss Ga ) or ti: +nlog2—n log z ae See ss (n) = {od 1G z) =n({— tanh @(tanh B—1)dB— 5 f dB Pa “cosh? B+. Nie (68) 8 where -=7sech #, v=” tanh £. ———=— a - - lUr oS . \ a—nlogn— Slog =1l loo 2+ Cn Expansions of Bessel Functions. 24 But ( tanh 8 (tanh 8—1)d8=— lo B and thus of Edel B + tanh 8+log2— ty —nlogs—$l in+tn log. - 28 P@) 3 —ntanh B 1 ib = sf dB\ 3 Fy cosech? 8 eee “? cosech? Both’ 8+. S) id by the use of the tapatical Taheadl and finally =(&-£ a ieee 5 fn (5 -coth B- 5 — a coth® Bi. 3 But by Stirling’s series, n being large, B, su: B, ky Bs Z 8S nT™Gn) a Loe. 3.473 5.6.8 where the B’s are Bernoullian numbers *. And finally tj=n(tanh B—8)—3 log 2— (4s 3am ) n 3 n? Be B, 1! 5B, 2! + = (x, coth 8— “%coth g.. jt ah tas ate (70) It appears at once that J,(z) is ultimately rapidly evan- escent when x greatly exceeds z. Accordingly, T, and T become identical, as also ¢, and ¢. Thus when x is not integral, and the coefficients X, w in the expansion are identical with those of expansion A, then if rae n> s==2 003 PB, s(t ——| gat J_,(2) — cos nowd,(2)=sin NT. (=) ec we have where 2T sinh B= 1—*coth? 8 + * ~teotht@—...3; . (71) * BP. g. vide Whittaker, ‘ Modern Analysis,’ p. 194. (69) 246 Dr. J. W. Nicholson on the Asymptotic and the second, fifth, ... terms are each two orders in x smaller than those preceding, and if i t=n(tanh B—8)—} log, 2— = (m.— 3 Ps += = =) n- oo 1 a Booe) aee “Wha val .coth B— Pcoth’ 4 .. jae a at aan where 1, on-2, wan-*, decrease in order by n~? at each third member of the series. This will be called expansion B. This value of Jx(z) continues to hold when x is an integer, and it may be expected that ~£Y.() will then replace {J -n(z) —cos nid »(<)} cosec nz ; but the formal proof is necessary owing to the use of T, in place of T. This proof is given in the next section. Expansions when nis integral and greater than 2. In order to verify the result last suggested, concerning the expression for Y,(2), it is only necessary to prove that if WG = (=) 2T 3 j=- ( So ame Then the value of T is, to a first order, given by OT ee (oz) For if this be true, the subsequent proof follows the lines above. Now by (50) IAY,()=4| eas 2ne . H(22 2 sin v)de i 2 a dd { de e —2nG—2z2 sind sinh @ 0 TY) 0 2 0 2 =i],- mee (say). Ste . ° e ° > . e ° e (74) The first approximation to I,, found in the usual manner, is, when ”> 2, n is large, and not too close to z in value, Leet 1 if: : =f a—csnd n+zsndg = th el Sine) ha E a es Lapansions of Bessel Functions. 247 To the same order, | C= -| "10 : | (en + 2z sin 6 cosh d)~!e~ 248-2" | 0 a) dé 2 “ n+czsin @ =e —sin-= | ,on reduction,. . . (76) Thus Ly Ama 1 a4 Jn(z)Y ate) =t1,- is L= Sass Jn — 22 ai ie (7 7) giving the proper value for T. Finally, in expansion B, when z is integral, we write 5.(e)= (=e Y,(2=- aye; 2 list eee and the functions (T, ¢) are those previously given. The limits of accuracy of expansion B will be the same as those for A, with n and z interchanged. The transition between different forms of expansion. It will now be shown that there is no range of large values of x or z for which expansions are yet to be determined, when n and zare real, and that each expansion passes naturally into the other, without the necessity of an intermediate expansion. We shall first define expansion C as that of a previous paper, where »—z is not large in comparison with 23, viz. 5) =2(2) Ae) ; ey == (°){ fi(p) cos nar +f2(p) +f,(p) sin nz}, 2 not integral. BNP : ¥n(2) =- (2) {AW +A), Sp a a RS 1) 248 Dr. J. W. Nicholson on the Asymptotic where =e) nie={- cos (w + pw)dw, 0 JAp)= ( sin (w* + pw)dw, 0 /s(p)= | dwewtee, . 2. , 3 es 20 Now Stokes * has shown that when o is not small, an asymptotic value of i(—2)-a3-0)=| die: 0 is given by so that Al—2)= 340% (2(5) — 27) PY aos pea seria (2 (5) -37). et where o is negative. Moreover, when p is positive, and not too small, (3)\ cos 5 (w° + pw)dw= 5 Bip? Ve . (82) The values of Jn(z) and Y,(z) of expansion C, when these formule are valid. become, on reduction, identical with 2R\: . J,(2)= (=) sin p, ¥,()=— (72) cos p, yy es * Camb. Phil. Trans. ix., Math. and Phys. Papers, ii. p. 529 ef seq. Expansions of Besse! Functions. 249 provided that 1 Zz 2 ree) 2 .2—n/) ’ pode t3(e—mi(=), - Bs a when 7 is less than <. The corresponding values from expansion A are, retaining the leading terms R= (2a? p=(22—n*)2 +0 sin-1“ —inr+tin, . . (85) But when 2 and z are nearly equal, if sin-} i = vs —e. Then cos €= = pelt a : (c—n)z(c+n)23 and p=2(sin e—e cose) +}a7 =r +3(2—0)1(Z),. Termine) on reduction, which is the value furnished by expansion C. Accordingly there exists a region in which either A or C may be used. Similarly, such a region exists for C and B. Finally, the scheme of expansions A, B, C is complete for a real argument. For the case of purely imaginary argument, only one set of expansions is necessary. These may be expressed in terms of the same coefficients (A, ~) as A and B*. Trinity College, Cambridge. * Cf. British Association Report, Dublin, 1908. PO 04 XXIV. On Cadmium Amalgams and the Weston Normal Cell. By ¥F. EB. Suita, A.R.C.Se.* (From the National Physical Laboratory. ) [Plate ITI. | JITHIN the past few years a number of researches have been made on the electromotive properties of Bt mereurous sulphate, and as a result the standard cell is now reproducible with a very high degree of accuracy. How- ever, there are many ineanees of abnormal behaviour of such | cells both with regard to constancy of E.M.F. and with respect to change of E.M.F. with varying temperature. The : want of constancy has been attributed by Hulett to slow : hydrolysis of the mercurous sulphate, and by Steinwehr® to a change in the size of the mercurous sulphate crystals. | Irregular behaviour with changing temperature has been observed by Tinsley?’, by Janet’ and Jouast!’°, and more recently by Cohen ‘and Kruyt”, and all of these attribute the observed differences to the cadmium amalgam. | Dearlove! first showed in 1893 that attention must be | paid to the percentage of cadmium in the amalgam, and from 1898 to 1900, Cohen®, Kerp and Boetitger*, Bijl’, Puschin’, | and Jaeger? made independent investigations on such amalgams. The research of Bijl’s was especially complete and its value appears to have been overlooked in much of the recent work on the standard cell. Possibly the reason is that the electro- | motive pr nf ag ties of the amalgams were examined at 20°, 25°, 50° and 75° C. only. However, from other data given by | Bijl—as peel out by C ohen and Kruyt—the conclusion : may be drawn that a 124 per cent. amalgam cannot be usefully employed below ne 14° C. Jaeger made observations at room temperature (20° C. approx.) on amalgams varying in content from 1 to 20 per : cent. of cadmium, and showed that at that temperature all : amalgams containing from 5 to 14 per cent. of cadmium could be used in the anode limb of the Weston cell with the same resulting E.M.F. In a note on Bijl’s work Jaeger’ emphasizes the fact that the cells made by Lindeck® and himself with amalgams containing 12 and 13 per cent. of cadmium showed no irregularities ‘when cooled. No objection appears fo have been raised against the 125 per cent. amalgam until 1905, when the author’ stated * Communicated by the Physical Society : read October 22, 1909. X : ys J J, ‘ | Cadmium Amalgams and the Weston Normal Cell. 251 that it was not wise to use such an amalgam at low tempe- ratures (0° C.) but that an 8 per cent. amalgam could be so used. A few experiments were subsequently made with 10 and 121 per cent. amaigams, but the results were incon- sistent. However, an amalgam of less cadmium content than 124 was preferred for low temperatures and in con- sequence most of the Weston normal cells at the National Physical Laboratory’ contain 10 per cent. amalgams. Pending further investigation it was thought wise not to specify the limits of temperature between which a cell con- taining 124 per cent. cadmium amalgam might be used, and in preliminary specifications’ for the standard cell, issued in 1908, no limits of temperature are fixed. Tinsley ?*, in 1908, obtained some results with Weston cells which led him to believe that cells containing 124 per cent. amalgam should be used with caution below 10° C. However, the cell with 124 per cent. amalgam had the lower E.M.F. at the lower temperatures, which is not in accordance with the author’s observations. Janet and Jouast’’, '° obtained results which were opposed to those of Tinsley. They found a 124 per cent. amalgam cell to be normal at low temperatures (0°5 C.) and a 10 per cent. amalgam cell to behave abnormally. This conclusion also is directly opposed to our own experience. Wolff" at the National Bureau of Standards, Washington, used 200 Weston cells to determine the relation between E.M.F. and temperature, the range being 0° C. to 40° C. All of these cells were set up with 124 per. cent. amalgams and only a few behaved abnormally. In November 1908, Cohen and Kruyt!’ found differences of 0°04, 0°23, 0°13. 0°16, 0°20, and 0:20 millivolt between amalgams of 10 and 124 per cent. of cadmium when the electrolyte was a solution of cadmium sulphate and the tem- perature was maintained at 0° C.; the E.M.F.’s of the cells containing the 124 per cent. amalgams were the greater. Cohen suggests the use of an 8 per cent. amalgam for general use. The exact constitution of cadmium amalgams is doubtful and is not discussed in the present paper, but there is no doubt that the electromotive properties of the amalgams depend on whether one or two phases are present. When, by increasing the temperature a solid amalgam is converted into a mixture of liquid and solid phases, there is an im- mediate change in the electromotive temperature coefficient of the amalgam, and a similar change results when the coexisting phases are changed entirely into the liquid state. ~~ ——-— —--— — ——— ed 252 Mr. F. E. Smith on Cadmium Amalgams If the temperature of an amalgam, containing the mixed phases, is raised, the liquid phase is increased and the solid phase diminished : ; when stable, however, the E.M.F. of an amalgam towards a solution of cadmium sulphate does not depend on the relative amounts of the two phases. Hence, stable amalgams containing different percentages of cadmium, but possessing the two phases, have the same H.M.F. towards a cadmium sulphate solution. With amalgams all solid or all liquid, the E.M.F. varies with the cadmium content. Bil determined the percentage composition of amalgams possessing the 2-phase system from experiments on their electromotive properties. His observations were made at 20°, 25°, 50°, and 75° C. Other experiments on the dilatation of the amalgams, rate of cooling, &c., were made at tempe- ratures ranging from —39° to “B21°. The present investigation was commenced in October 1908. The objects in view were :—(1) To trace the cause of the electromotive difference, which a existed and which was sometimes absent, between 10 and 124 4 per cent. amalgams. (2) To determine the limits of temperature between which amaleams of various concentrations could be usefully employed. (Requests for information on this point had on several occasions been received from India.) (3) To explain the abnormal E.M.F'’s of cells made by various observers, when the amalgams contained from 124 to 14 per cent. of cadmium. (4) As the recent International Conference on Electrical Units and Standards (London, 1908) specify a 125 per cent. amalgam, it was important to decide whether the limits of temperature (0°—40° C.) specified, required amending or not. The Method. The method consisted in directly measuring the difference of E.M.F. between two amalgams in contact with a saturated solution of cadmium sulphate. The materials in contact with the amalgams were identical with those used in the Weston normal cell, and any amalgam could at any time be used as the negative element of such a cell. The “glass vessels used were of the form shown in fig. 1. Each vessel has four vertical limbs connected by means of a tubular cross-piece. A platinum wire is sealed through the bottom of each of the limbs and passes into a narrow ’ side- tube containing mercury. A 10 per cent. amalgam was placed in the bottom of one of the limbs, and amalgams of different percentages of cadmium were placed in two of the and the Weston Normal Cell. 253 other limbs; in the remaining limb, mercury and a depolarizer consisting of a paste of mercurous sulphate and cadmium Fie. 1, es oe eee | ==) Pa ee sulphate erystals and solution were introduced. Cadmium sulphate crystals were added so as to form a deep layer over the amalgams and paste, and afterwards the vessel was nearly filled with a neutral saturated solution of cadmium sulphate. The vessel was then hermetically sealed. Such a cell will be seen to consist of one positive element and three negative ones, one of which consists of a 10 per cent. amalgam. The advantages of such a cell are :— (1) All of the constituents of the Weston normal cell are 7 present. (2) The electromotive force of any combination having the mereury as the positive element and one of the amalgams as the negative element, may be directly compared with the electromotive force of any standard cell. 254 Mr. F. E. Smith on Cadmium Amalgams (3) A large number of the vessels may be grouped together and the positive poles connected. The difference in E.M.F. between any two negative poles, even though they be in different cells, may thus be directly measured. (4) If there is any doubt about small differences in E.M.F. and a suspicion that they are due to slight differences in the depolarizers, the 10 per cent. amalgam poles may be connected together, and the differences in E.M.F. measured between these and other amalgams. (Preliminary observations had proved the reliability of the 10 per cent. amalgam.) (5) The cells may be immersed in ice, water, oil, or other liquid, and they cannot leak through the junction of platinum with glass. Preparation of the Materials. The cadmium sulphate, the mercury, and the mercurous sulphate were prepared by methods which are fully described elsewhere. The cadmium was from various sources. Some was purchased from Messrs. Baird and Tatlock ; another lot was from Messrs. Harrington’s and further samples were procured from Kahlbaum’s and Merck’s. In addition, a considerable quantity was prepared by electrolysing pure cadmium sulphate, an anode of commercially pure cadmium and a cathode of platinum being used. The Cells. The cells experimented with ave divided into groups, the division being due partly to the heat treatment of the amalgams, and in part to the source of the cadmium. Group I1.—Cadmium from Baird and Tatlock. Range of amalgams :—1 to 20 per cent. of cadmium. Manufacture :—Three amalgams were first made; these contained 20, 15, and 10 per cent., respectively, of cadmium. All oxide and dross was removed by washing with very dilute sulphuric acid while the amalgams were hot and completely liquid. Each amalgam was then chilled with ice-cold water and broken into small pieces. From the 20 per cent. amalgam other amalgams estimated to contain from 19 to 16 per cent. of cadmium were made; the 15 per cent. amalgam was and the Weston Normal Cell. 255 used to prepare others having from 14 to 103 per cent. of cadmium, and from the i0 per cent. amalgam the 9 to 1 per cent. amalgams were made. The amalgams were introduced into the limbs of the vessels and melted either in a water bath or over a bunsen flame; the cells and amalgams were allowed to cool ina normal manner. Afterwards, the other constituents of the cells were introduced and the vessels were hermetically sealed ; they were subsequently immersed in paraftin oil and kept at a constant temperature of 20° C, for 48 hours, after which they were placed in crushed ice for 8 days. Ten Weston normal cells of the usual H form were con- tinually maintained at a constant temperature of 20° C. and the E.M.F.’s of the various combinations of the elements in the 4 limb vessels were compared with these 10 cells from time to time. When the experimental cells were at 0° C. the E.M.F.’s were compared every day for 8 days; afterwards further observations at 5°, 10°, and 15° C. were made, but the first two of these temperatures were not maintained constant within less than 0°-1 C., and this for 3 hours only. The temperature of 15° C. was maintained constant for 24 hours before the first comparison of E.M.F.’s was made, and the whole lot of cells did not deviate from 15° C. by more than 0°5 during the following 3 months. From time to time the temperature was adjusted to 15° C. and the cells com- pared. After the 3 months had elapsed other comparisons were made at intervals of 5° from 20° to 45°, these higher temperatures being maintained constant for at least 4 hours before making the observations. ‘The oil baths were heated electrically, and a toluene thermostat sensitive to 0°01 was used. l*or temperatures above 30° the cells were immersed in crude olive oil. Results. The results of the E.M.F. comparisons are given in Tables I. and IT. Table I. gives the E.M.F.’s of amalgams containing from 1 to 13°5 per cent. of cadmium. When maintained at a constant temperature the E.M.F.’s of these amalgams (with the exception of that containing 6 per cent. of cadmium) were found to be constant also within 1 or 2 parts in 100,000. It is not necessary, therefore, to give more than one value at any one temperature. Y Mr. F. EH. Smith on Cadmium Amalgams 256 eee Ee ole | pas | PhL— | PPL— | GhI— | GhI— | OFT | SFI— | GhL— | 9PL— | SPI— | 962— | GE9— | FO8— |I8OI—|GTFI—|Le8I—lorFs—| “| ofFl6oer | '@L1— | SII— | S11— | St— | 11 i. PII— | S1g— | sso— |ecs— lpett—|rect—|zoes—| “| OF (60ST | | | et Oe OS NE - 9] a hee Sek el ee (08— |F6— |G9S— \829— |P96— |G9eT—jeg6I—| “| G8 60'S | ‘Gh |8h— [eh |F— FF |8F— [Sb— [6h— |6P— |6F— |OS— |opI—|LoF— lezk— [zert—lk1Zt-| “| ocelsosrs | jd 8 Dimes ll ac | tee See SLs Gl (eGa 1664 86> 1SG=el fom | 76-41 be 98I— |408— \O1I6— |ssrI—-| ~~ o£6 |606 61 | | = | 3 LG | GG G6 GG Gs G1, |. ORS seep can sI 8T LT LT GL— —|PoP=_ reOlS| =": oGT |60 01 66 |9PG—~\86L—-~) o0I |80°T1'6 cia (agai |) a a ie of |s0TT9 He cet (cog— 161-0 00 30'01'86 peers 4 >| ss le TS PS oe S03) cette 6elte Sl ss F89— joesI—| “| 00% |60°s'ST | | BL | “81 | Gl} eL | ‘GIT | | eee Re oe Oe : ieee a eee ey ee de |_| ee ‘aLV(T ‘WUIVo[CUL 9} UL WHIUIpTd Jo osvjudII0g "4JOA B JO sijpuesnoy3-perpunt UI SULIq SeOUALOIp oY} ‘o[qvy, O44 UI USAIS sooUaIOYIp oy} 4- ‘s}TOA “GUL OFSLO-L = 8A IN'W poatosqg ‘uMMUpvo Jo ‘Judd aod G.ET 0} [T WOAZ YIM survGeure SULULL}UOD S]]90 [BUALOU UOyso AA JO SoaNqvioduie, snoIIeA qe S/T Peatesqo oy} Surarg—*] a1avy, 257 and the Weston Normal Cell. + ce | os g | 0gI— | ae GS - i Si 2S oe 60'2'F Si cb tot | aS ie SS ae 4 ee ee | S91 | 009% | eo'e 098 Te la sI—- | ot 6oI— | OLI- Pte fo Sos | 196 FFG | 08 = SOR |. Ue BO he OE ay Ee a 60'S T ae ee a 52's | 6L— ead ape oy Pa Se a a a 2 a rn Saar ee ee b= “Sh= TL— SL— FL— ss | ‘D088 | 60'S'SG Si = i eee =} BL | eI Ob— ce 28. ~ ie Si a eee GIL | £69 | 9! col oe. | SS Se iS ee ae ee LL8 O9L | egg | Le eae | 0 9 Saas = Re / 60G6I [Ss 918 == “G02 Leg | Lee 0 0 io a 8t=) 2-9 = * ae 60SLT | padt= 7). G86 = 912, 88h OL eg i 3E ee sate Se BE: ‘See ae ee 8) 88h ed ag | GL ee ‘0.008 | — GO'B'IT ssl | c4or | 18 SFO FFG rate eit St = "| = S00 bart | = Fell =| “028 689 = eee | eae — | oe eae Ooff | SO1lst | eesl | O81 | S20l 0&8 et 4s aes cl eS Se : SOITOL | _- eta OLE | tent =| ose we fae So we! Peco '0o01 | sore |S Qt | Obl solr | 696 | 18h eey 688 St aE = fh 2 oh See 9461 | 6IFI colt =|: 96 [f= Sr -— -} 9 Se 6 |= OE St a eee (:) a 2 “as SS es eee ey, «| ea OG 2 | =" arr i eee ek | ROOT = SP St Oe. OS SS eet? | et See S57 Saas en eee ES] ROSE LEVERS nce Gio. ~ SiS 4) ee BER Sg at ‘O | iS hl = ee | 7 \"S aE a elie eee ee te Be i ae Se ee ee eS | ‘aNay, | ‘ava |= "MVS[VULG OY} UL WNIWUpVd Jo oSBjUII.Ig | | a | | |R D> ‘JOA B JO SY4pUBsNoOy-poipuny S UI SULOG SOOUdTEYTP OYyy ‘o[QUy, OY} UI WOAIS seoUOTOPIp oy + A OFSTO-T = SLID poatosqg — ‘wintuIpLd jo yuo Jed QZ 0} FT Wosz YIM suuvsyvure = SOULULLJUOD ST] [VUILOU UO}SOA\ JO soanzuioduto} SNONVA JV STG Peadosqo oy} SuIALQ—']T] ATAVY, x ¥ 258 Mr. F. E. Smith on Cadmium Amalgams With regard to the 6 per cent. amalgam, at 30° C. the following values were observed :— Date. E.M.F. =1:01840 v. + D2 Oe — 000059 v. A ay —0:°00127 DA Dia nis —0°00140 Da ee —0:00140 This change is shown hereafter (p. 271) to be due to the closeness of the 6 per cent. amalgam at 30° C. to the transi- tion temperature (about 28° C.), at which the two phase system changes into a one phase (liquid) system. The tran- sition appears to take place slowly, and the H.M.F. of the cell does not therefore immediately assume a constant value. It is purely accidental that such a change was only observed with the 6 per cent. amalgam. Many of the cells containing amalgams with from 14 to 20 per cent. of cadmium did not remain constant when the temperature was unchanged. ‘The variations will be seen from Table II., which gives the initial and final values at the different temperatures. Of the cells dealt with in Table I. the E.M.F.’s of ali the cells, with the exception of that containing the 15 per cent. amalgam, increased in value when maintained at 0°. During the 3 months that the cells were at 15° C. the 15, 17, and 19 per cent. amalgam cells fell in E.M.F., and the 14, 144, and 16 per cent. amalgam cells rose in value. At the higher temperatures there was very little change, but it is worthy of notice that the 18 and 19 per cent. amalgam cells exhibited declining values. After the cells had remained at 45° C. for 2 days they were again laced in crushed ice and kept there for 6 weeks. The E.M.F.’s were measured from time to time, and were far from constant for the amalgams of higher concentration. The following table (II1.) gives the results, and for the sake of comparison the original values at 0° C. are also given. The cells containng the amalgams with from 2 to 11°5 per cent. of cadmium may be regarded as quite constant. Amalgam cells with from 12 t) 13°5 per cent. of cadmium have tallen in E.M.F. by a few bundred-thousandths of a volt, while those with amalgams richer still in cadmium (with the exception of the 20 per cent. one) show considerable changes which in two instances (14:5 and 16) amount to 0-V04 volt. With increasing time the H.M.F.’s of these cells are observed also to increase, but even after 6 weeks the final and the Weston Normal Cell. 259 Tasie III. Giving the observed E.M.F.’s at 0° C. of cells containing from 2 to 20 per cent. of cadmium in the amalgams. Observed E.M.F.’s=1:01840 int. volts+ diffs. in hundred- thousandths of a volt. | | | Percentage of 0° ©. 0° C. 0° C. OF @: | Max. Cadmium in é low 9 9 . fet Sralenrn: 5.11.1908. | 6.2.1909. | 27.2.09. | 20.3.09. | difference. Beh lank, —365 — 368 | ee eee SOR: +35 35 | 35 34 1 ee. c, a = a teed 35 1 Banat Ae 9. in en Sao) 9136 35 0 Rstanaeicyiss)s oe, |e 35 35 36 1 | OS ee pa | ae. 36 36 36 0 ak Ae a ae 36 37 36 l ROS... 36 50 36 Sap Le ai 1 . Oa Be ih Es 36 36 37 1 ae 37 | os 36 | 36 37 1 oy ap ee 37 | 2s 36 | 36 37 1 or) Oa 39 | SB ad a 37 2 ce 40 S& 26) Wo 3G 39 | 4 Bt, .4.... 41 | 7a] 39 38 3 3 Sees ¢a05-- 47 | 32 40 40 40 7 Te... 50 | Sh 40 | 38 41 12 as 49 |.& Baia heyhite 86 211 oo ae em 48 | 58 216 409 ae 673 | Ss 443 567 626 | 230 a 560 Ee 163 179 498 | 397 ee 1101. |= | 1006 | 1042 | 1049 95 |. eee 105 1260 1266 | 1266 | 35 eA... 1564 | 3 1493 | 1498 | 1507 71 a 1613 1611 | 1616 | 16i7 | 4 values are considerably lower than the original values. The only cell dealt with in Table II. which had practically the same initial and final values at 0° C. (see Table III.) is the 20 per cent. amalgam cell, and this also is the only cell which was practically constant in H.M.F. when any temperature from ‘0° to 45° C. was maintained constant. The graphs which result on plotting the values of the percentages of cadmium and the corresponding E.M.F.’s at various temperatures, are shown in fig. 2 (PI. II].)*. For the purpose of the figure the final values of the E.M.F.’s given in Table IT. were chosen. The lower ascending portion of a curve is taken to indicate that the amalgam is completely liquid ; the nearly horizontal portion indicates that a two- phase system—part solid and part liquid—is present, and the * In some cases the plotted values lie at considerable distances from the corresponding curves. 8 2 260 Mr. F. BE. Smith on Cadmium Amalgams upper ascending curves show that the amalgam is completely solid. It is evident, however, from the results given in Tables I. to III. that the electromotive property of an amalgam may depend to some extent on its previous thermal treatment. Whereas, originally, the H.M.F.’s of cells at 0° C. with amalgams containing from 3 to 13°5 per cent. of cadmium, were “nearly identical, we find atter heating to 45° C. and subsequent cooling that the limits are 3 and 145 per cent. Similarly at 15° C. an approximate constancy of E.M.F. was originally obtained with 5 to 14 per cent. amalgams, but after the cells were raised to 45° C. and then cooled to 15° the range was extended to 16 per cent. amalgams. Flence it appears possible that if an amalgam rich in cadgien’ is raised to a temperature at which the solid and liquid phases are present, it may, on cooling, have different electromotive properties owing to the persistency of the two phases. This persistency may be due to lack of homogeneity of the amalgam. ‘The curves in fig. 2 show that the amalgams containing from 9 to 17 per cent. of cadmium were partly liquid and partly solid at 45° C., while the 18 and 19 per cent. amalgams were fairly close to their transition ea ia Referring to Table III. and to the curves in fig. 2, we see that the amalgams which show the greatest changes in their electro- motive properties (these are the amalgams containing from 14 to 17 per cent. of cadmium) are par ‘tly liquid and partly solid at 45° C., but are normally quite solid at 0° C. Other amaloams containing 14, 15,16,and 17 per cent. of cadmium were prepared, and their general appearances before and after heating to 45° C. were carefully noted; there is no doubt that these amaleams appeared to be more liquid after the hea treatment. T he jinal appearance of the amalgams as explained if we suppose that a layer of amalgam of compa- ratively low cadmium concentration sur rounds a core of higher cadmium concentration. The curves in fig. 2 are worthy of notice. The slopes of the higher ascending curves are almost identical for the temperatures 15° C. to 45° C., but only the upper part of the higher curve at 0° C. is approximately parallel to the others. (To avoid confusion the curves for 5° and 10° are not given.) Now if the upper part of the curve at 0° is continued so as to be approximately parallel to the curves at 20°, &e., it meets that part of the curve corresponding to the two- phase system at a point indicated by 11 per cent. of cadmium. It is possible that the curve thus drawn was not Lar ’ | and the Weston Normal Cell. 261 experimentally realized because ef some difference in the thermal treatment of the amalgams, for it has already been shown that the electromotive properties of an amalgam vary with such treatment. If so we conclude that cells with amalgams containing more than 11 per cent. of cadmium may have E.M.F.’s at 0° C., differing very considerably from the E.M.F.’s of cells with from 3 to 11 percent. of cadmium. The results also lead one to suggest that the transition temperatures of the amalgams containing from 11°5 to 13°5 per cent. of cadmium are below 20° C. but above 0° C. Experiments to test this point were immediately proceeded with, but before describing them the results obtained with other groups of cells are of interest. The new groups of cells contain cadmium from Har- rington, Kahlbaum, Merck, and from an_ electrolytic source. Each of the new amalgams was made by adding metallic cadmium to mercury, and all of them were melted inside the glass vessels and allowed to cool in a normal manner, The E.M.F.’s of these cells were compared at many temperatures, but it will be sufficient for the present purpose to give the H.M.EF.’s at 0° C. (p. 262). In order that comparisons may be readily made the original E.M.I*.’s at (°C. are also given for the first group of cells. The results obtained with the four new groups of cells confirm the conclusions arrived at with the cells of Group L., and furthermore suggest that the maximum values for the H.M.F.’s of the cells with amalgams containing high per- centages of cadmium have not yet been obtained. If, as suggested on p. 260, the result of slowly cooling an amalgam to a temperature at which it should (if homogeneous) be in a solid state is the production ofan outer shell of amalgam of comparatively low cadmium concentration, covering a solid core of higher concentration, then different rates of cooling should produce different results. Also, it seems probable that by raising the temperature until the amalgam is com- pletely liquid, and then chilling, the differences of concen- tration will be largely avoided; but if the amalgam is not in the completely liquid condition at the high temperature, or nearly so, the effect of chiliing may not be marked. To test the first effect, a 14 per cent. amalgam was made and divided into two parts; these were placed in two limbs of the same vessel. One of the parts was completely melted and allowed to cool in a water bath down to a temperature of about 16°; the other part was completely melted and chilled by quickly immersing the limb containing it in ! | 6 bo iw Mr. F. EB. Smith on Cadmium Amalgams TABLE LV. Giving the observed E.M.F.’s at 0° C. of various cells con- taining amalgams with from 1 to 25 per cent. of cadmium. Observed E.M.F.’s =1-01840+the differences given in the table, the differences being in hundred-thousandths of a volt. | Percentage of Cadmium. eecewroce) e@ecccccce eerccscces eoeeesece Grovr I. Cadmium from Baird and Tatlock. —1197 — 365 +35 34 Grovr II. Electrolytic Cadmium. (2) | (2) — 1203) —1205 —310\— 372 +35\+ 34 doji+ 34 35+ 35 35) 36 36) 36 36) 510) 37 ol OT Oi 38 7 39) 3 40) 38 72) 40) 38 454) 65 451) 45 78 48 455) 49 1047's 866 1612 1580 Grovpr LIT. Cadmium from Kahlbaum., Grovr IV. Cadmium from Harrington. Group V. Cadmium from Merck. N.B.—An interval of 86 hours elapsed between observations (a) and (0). In this interval the temperature of the cells was raised to 65° C. for 1 hour, They were then placed in ice and the observations (/) taken 6 hours afterwards. and the cells allowed to cool very slowly in the bath. aleohol cooled with solid CO,, the temperature being about —50° C. The other constituents of the cell were added, the vessel hermetically sealed and immediately afterwards it was placed in crushed ice. as follows :— The observed E.M.F.’s at 0° C. were and the Weston Normal Cell. 263 TABLE V. Giving the observed E.M.F.’s at 0° C. of cells containing 14 per cent. amalgams, one of which was chilled and the other slowly cooled. The E.M.F.’s=1-:01840 v.+ values given in Table. 14 per cent. slowly 14 per cent. | cooled. | chilled. | | | i | +0:00058 v. | +0°00606 v. ae | 40 | 656 a | 40 | 686 : | 40 691 RU See. c sus | 40 693 ae | 41 696 ae | 41 695 EET aie | 41 694 The results show in a decisive manner that differences of very considerable amounts may arise owing to differences in the rates of cooling of the amalgams. The 14 per cent. slowly cooled amalgam cell had an E.M.F. only 5 parts in 100,000 greater than that of a cell containing a 5 per cent. amaleam, wiereas the 14 per cent. chilled amalgam had an E.M.F. which is considerably greater than that of any cell with 14 per cent.amalgam which is reported on in Table IV. It was thought possible that the chilled and slowly cooled amalgams might differ owing to local differences in con- centration other than that mentioned on p. 260. When a 14 per cent. amalgam—or one of higher concentration— is completely melted and slowly cooled in a water bath, the solid crystals which first separate out are lighter than the mother liquor, and rise to the upper portion of the amalgam. Hence, after the amalgam has cooled, it is probable that the concentration of the cadmium in the upper portion is greater than in the lower portion. When a 16 per cent. amalgam was melted in a long glass tube and cooled slowly, this difference was detected by inspection, the lower half of the amalgam having an appearance approaching that of liquid mercury, while the upper portion had a frosted appearance. To test whether there was any marked difference in the electromotive properties of the two ends a platinum wire was sealed through the middle portion of a closed glass tube, TT LE Ey ee —— 264 Mr. F. E. Smith on Cadmium Amalgams 3 mm. in diameter and 30 cm. long ; the tube was then filled with a 16 per cent. amalgam. After the amalgam had been completely melted and slowly cooled in a water bath, the lower end of the tube was broken away so as to expose part of the amalgam’s surface. The amalgam rod was now used as the negative element of two cells, the platinum wire serving as the negative lead. In one of the cells the upper part of the surface of the amalgam made contact with cadmium sulphate crystals and solution, and in the other cell the lower part of the amalgam’s surface was in contact with a similar mixture. After three hours at 15° GC. the: - cell containing the lower part of the amalgam was the higher | in E.M.F. by about 0°0001 volt, but the difference was constant, and both cells were increasing in value. Ve Ponulde, therefore, that the large differences observed between the chilled and slowly cooled amalgams are not due to es local differences of concentration as inaee been indicated. | or. a) = Cells with Chilled Amalgams. 1: Another series of cells were now prepared with ala some of which were melted and afterwards slowly coole in a water bath, and others were melted and chilled by immersion in alcohol cooled by means of solid CO, to about — 50°C. After being chilled the amalgams were not allowed to rise in temperature above 0° C., the cell being prepared with the glass vessel immersed in a mixture of ice and salt. After sealing and swilling with ice-cold water the cells were placed in ice. With amalgams containing from 1 to 11 per cent. of cadmium no difference greater than two hundred-thousandths of a volt was observed between the electromotive effects of chilled and slowly cooled amalgams of the same percentage composition. With amalgams of higher concentration the chilled amalgam cell had the higher E.M.F., and its value increased for from 7 to 14 days, after which a slight fall was generally observed. In Table VI. the E.M.F.’s of the various cells are given after the cells had been kept at a constant tem- perature of 0° C. for 1, 5, 10 and 30 days. The E.M.Fs of the cells with the slowly cooled and chilled amalgams—1 to 11 per cent. of cadmium—were practically identical with those given in Table [. (at 0° C.), and so they are not repeated here. How ‘ever, in order that comparisons may be made with a two-phase system amalgam the E.M.F.’s of the 10 per cent. amalgam cells are given in the Table. and the Weston Normal Cell. 265 TaBLe VI. Giving the observed E.M.F.’s at 0° C. of cells containing slowly cooled and chilled amalgams. Chilled amalgam _ cells are indicated by “c.” Observed E.M.F.’s = 1:01840 int. volts + the differences given, in hundred-thousandths of a volt. Percentage of | Cadmium in Ist day. , 5th day. 10th day. 30th day. Amalgam. | | Le nn 56) Vi) 36 36. | 36 Lie ie 36 36 "a6 36 Pg... 40 | 39 so | 39 sane 50 183 272 OBB oe... a 39 39 oa) ae 41 340 366 360 | oa 42 | 44 3 Pe aes ae 450 650 464 | 460i a 37 42 Bt as) Mss ssbn ne 606 691 695 cot ae 44 | 110 337 544 CO ae 583 897 920 M00 | BO ele a, 123 592 692 ily oe (i a i We 1112 rie.) (ae 50 680 800 STH | | ae 1238) |i) 1297 1295 1275 | BN caves 73 936 988 LON 3 Brive: 1406 1509 1508 1505 | ae 945 1238 a ea BOM 2 als. 160) fh. 1672 1676 1668 | BMT. Ph ROO Taney 1580 a a A ce 1840 1826 | MR. hoy: 9378 | 2370 2410) |) BBY OR lois ie-> 2496 | 2529 2550 | 2530 | When amalgams of the same cadmium content were melted and chilled, the cells containing them did not always have the same E.M.F. even after several weeks. However, the differences were comparatively small for the amalgams containing more than 14 per cent. of cadmium ; thus after 30 days at 0° C. two cells containing 16 per cent. chilled amaloams had E.M.F.’s of 102947 v. and 1:02931 v., and two other cells with 18 per cent. amalgams had E.M.F.’s of 1:03335 v. and 1:03310 v. In two instances there were very rapid rises in E.M.F. and subsequent rapid falls ; one instance is the 13 per cent. amalgam cell of Table VI., and in the second case a 16 per cent. chilled amalgam cell had an initial E.M.F. of 1:04097 v., 266 Mr. F. E. Smith on Cadmium Amalgams which fell in 5 days to 1:03032 After 30 days this latter cell had the value 1:02964 v., w Mon is almost identical with the value given for 16¢ in Table VI. Seven cells were prepared with 124 per cent. amalgam. The first four contained amalgams bien had been chilled with aleohol cooled with solid GO. ; ; the fifth was chilled by immersion in a mixture of ice and salt at —10°C. ; ; the sixth was chilled in water at 0° C., and the seventh was allowed to cool in a normal manner. The E.M.F.’s of the seven cells are given in Table VII. TasBLe VII. Giving the observed H.M.I’s at 0° C. of cells containing 123 per cent. amalgams. | Observed EMF. = 1:01840 +. Amalgam. Ast day. | Sthday. | 30th day. Chilled to about —50° C. ...... | 006036 000238 000222 patter o's hi) is Romana Muerkic rete | ca ee 273 310 %» eh en ee nee | 4] 340 360 > Pape wok taee | 116 83 80 SERB ois (OS lelags eee 5 52 63 176 Pears bch) ti cnapsuteane Genie 44 45 44 Waoled normally’ ...2s225.. cece. 40 | 40 41 The E.M.F.’s of the chilled amalgam cells dealt with in Tables Vi. and VII. cannot be regarded as constant even after 30 days. The small decline in E.M.F. from the 10th to the 30th day (Table VI.) was possibly the commencement of a fall which would have lasted a considerable time, and may be due to the chilling producing a too highly con- centrated amalgam in the outer shell. However, the uniformity of the results (see fig. 4) leads us to believe that the maximum values of the H.M.F’s for homogeneous amalgams are not very different from those observed on the 30th day. With regard to the slowly cooled amalgams (p. 265), it will be seen that the low initial E.M.F. is explained if the and the Weston Normal Cell. 267 outer shell of the amalgam is of lower cadmium concentration than the mean concentration. With continued diffusion the outer skin of the amalgam will become richer in cadmium, and thus the E.M.F. of the cell will increase. Such increases are recorded in Table VI., and have been previously observed by Jaeger? and by Bijl®. If, after a cell has been hermetically sealed, a previously slowly cooled amalgam (forming the negative element of the cell) is raised above its first transition temperature, but not above the second, subsequent chilling of the amalgam does not, in general, result in an increased E.M.F. .. but a lowering may result. A cell containing a 14 per cent. amalgam was raised to 50° © and then chilled with ice-water ; its E.M.F. was thus lowered from 1:02260 v. to 102220 v. This lowering was no doubt due to the outer shell of the amalgam becoming even less concentrated in cadmium than before. However, in such experiments it was repeatedly observed that the rate of increase of H.M.I*. of the cell with time was accelerated by such treatment. After remaining in ice for 30 days, the cells with the chilled and ely. cooled amalgams (Table VI.), and others containing from 2 to 10 per cent. cadmium amaloams, were gradually increased in temperature, and the B.M.F’s observed at intervals of 5° up to 65°. The temperatures 5° and 10° were maintained very nearly constant for about 1 hour, and all other temperatures were kept constant for at least 4 hours. The results are given in the following Table (p. 268). After the cells had been raised in temperature to 65 they were again immersed in ice, and after 14 hours a comparison of their H.M.F’.’s led to the results given in the last column of Table VIII. (compare with Table TEL). With the exception of these final values at 0°, the E.M.F.’s &c,, given in Table VIII. have been plotted, and the resulting curves appear in fig. 3(PI.III.). The dotted portions of the curves were not experimentally realized, as the cells were unstable at temperatures very near to the first transition points of the amalgams. A comparison of the effects of temperature changes on chilled and slowly cooled amalgams is now of interest. The whole of the slowly cooled ‘amalgam cells tabulated in Table VI. were raised in temperature from 0° to 65° C., but it is not necessary to give all the results. The following (Table IX.) is a typical case. Here, the three amalgams in one vessel were 10 per cent., 17 per cent. chilled, and > Ge Mr. F. E. 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[S0E= oTr tT — LE8T— OFFG— ee ‘WJOA B JO SYJPULSNOY}-porpunt GG tiie = 91— Li $1— GL GL aL 61 - 08 ~ +6 EG — 829 — #26 ~ cogl— 6°61 — C691 606 OOL GO9 LLE BIG Ta oF -- 6cL— 6c Aa 1911 9L01 £6 £91, CEC ERe OT ol To 1e-- co co (2 OG $E> a SY Cee te toe 98L— LO O16— CFI 0961 o¢aT 6301 CZG 004 CEC O&S& GG cos € ‘és ; ITTG 9OFT FEL csor €¢8 00L LEP 196 cP OG GT GT ef SI ST UL SUToq sooudtoyip oy} ‘oTquy, Ul WATS sooMoIOyIp + sjpOa “JUL OFSTO-T ‘stuesyeue parpigo Ajsnotvord Sururezwoo syjoo Jo s\T'L' Pesdosqo oy} Sutaty—'TIT A FAV, LOGS Eccl O6ET CEE! 266 GFS8 8&9 GCP GS] OW & , 16 | Cc i C ag Of OG OF 8686 c69T GET €Zél CPT CRG SLL O9G LOE O&G OET LE Le mes Lg os LE OF 9€ Cg of 08¢— Ogg Car Qgoy-b coer t+)" Cleats Ott 0164 | FQ (oe ooeg-+ i" G. eGo Se e Gp oft : 0) tl 9¢+ ceo a re+ ea ee Se be ae ST IV GL PoAtosq¢d, “TUNIUIP RY <)> Ok © = “i ed Ean) | jo 988]U90.10 1 | i d) and the Weston Normal Cell. 269 17 per cent. slowly cooled. The temperature was altered comparatively rapidly, the whole range 0°-40° being gone through in a little less than 8 hours. TABLE TX. Observed E.M.F.’s = 1:01840 + the differences in hundred-thousandths of a volt, given in the Table. | b | | | Amalgam. Date. | Time. | Temp. i ae | Pe 17. ° | | Slowly cooled. | Chilled. | 10%, | til one be EE SE ‘ab Vics h th aaS tale a Aah Pre aR 20.5.09 ee | 9.30a.s.| 0° | 872 | 1230 26 | : 14°52 | 467 oF 2 ii =O 50-0 | 294 tan 2 12.30 p.u.| 300 | 23 330 —49 | 1.0 | 300 | 6 | 333 —49 20 | 930°0 —S8 300 —49 3.0 | 35:0 | ae 165 | —79 4.0 36-0 te —83 5.0 | oP |} +195 | 1205 | 35 —_ See ey ea eee we _——_—_ 30.5.09 | 9.0 am | 0° 799 | 1222 36 10 p.m.| 20°0 215 650 0 3 295°0 51 500 — 23 4.30 0° 791 | 1218 | 3. 31.5.09| 9.0 ax.| 0° | ee ee 36 12. ew.) 40°) —]04 70 —111 2.0 45° —142 —140 — 146 4.0 mo? |: 178 +162 ~181 5.0 0° | od 43 33 1.6.09 1 go | 540 Shae Chee ead 10.6.09 | 0° | 850 1088 | 36 10.7.09 | 0° | 914 1113 | ob —— It wit be Getcred that the. E. M. Fr "s of the cells with the amalgams 17 and 17c changed with the rise of temperature from 0° to 20° by almost identical amounts. At 30° the E.M.F. of the cell with the slowly cooled amalgam was near to that of the 10 per cent. amalgam cell, and with increasing time, the temperature remaining constant, the two E.M.F.’s * got nearer and nearer together. On cooling to 0° a marked change was evident, the cell with the ‘slowly cooled amaigam ‘hay ing fallen by 67 millivolts. In 10 days the E.M.F. had risen again within 0°8 millivolt of the original value, and no further ‘change of importance was produced on warming the cell to 25° and subsequent cooling to 0°. During the whole period the change in E.M.F. of the 270 Mr. F. E. Smith on Cadmium Amalgams chilled amalgam cell did not exceed 0°2 millivolt. Both amalgams changed considerably on raising the temperature to 50° ©. and subsequent cooling, brt although at 50° both cells had nearly the same H.M.F., they differed very con- siderably at 0°. In all cases cells containing amalgams in a presumably solid state, whether chiiled or not, could be raised in temperature and subsequently cooled without appreciable change in H.M.F., unless with increasing tem- perature there was an abrupt change in the temperature coefficient. In such a case the temperature coefficient appears to diminish at first, but afterwards the E.M.F. of the cell rapidly falls to a value which is nearly identical with that of a cell containing a stable two-phase amalgam. In our experiments the change usually occurred when the H.M.F, was from 0°4 to 2°5 millivolts greater than that of a normal cell. In all cases subsequent cooling showed that a change in the amalgam had taken place as the new E.M.F. was less than the previous value at the same temperature. At temperatures near to but below the first transition temperature the diftusive processes in an unstable amaloam are no doubt accelerated, and the outer shell becomes richer in cadmium with a corresponding increase in the H.M.F. of the cell; even so the diffusion is very slow, and in general before the first transition temperature is reached the outer shell becomes partly liquid, and a two-phase system appears at a lower temperature than would be the case if the amalyam were homogeneous. On subsequent cooling a new outer shell, with even less cadmium than before, is formed, but the inward concentration gradient is greater than before, and hence the cell is more unstable. The results given in Table IX. are easiy explained on such an hypothesis. If a stable amalgam is taken in which the solid and liquid phases are certainly present, increase of temperature wiil diminish the solid and increase the liquid phase. When the liquid phase alone is present the temp, coef. of a cell con- taining the amalgam is very different from that of a cell with a two phase system. This is evident from fig. 3. It was surprising to find that even when the temperature of a cell was maintained several degrees above thie transition tempera- ture (corresponding to the change to the all liquid phase), several hours elapsed before the transformation was compiete. Tor example, the 8 per cent. amalgam appears to be completely liquid «at all temperatures above 41° C., yet when a cell containing such an amalgam was kept at 45° for four hours its E.M.I*. was in exact agreement with that of a cell containing a 10 per cent. amalgam, of which the transi- tion point is about 51°. At the end of five hours the E.M.F. and the Weston Normal Cell. Ze began to fall, and at the end of two more hours it became constant but 14 millivolt lower than previously. This liquid amalgam cell was now perfectly stable and could ve raised in temperature to 65° or lowered to 41° without any permanent change such as was observed with the solid amalgams (see also data, following Table I., with regard to a 6 per cent. amalgam). Probably the reason of this behaviour lies in a difference of concentration of cadmium in the amalgam in the upper and lower parts, such as was indicated on p. 263. In a two-phase system, since the solid crystals are lighter than the mother liquor they naturally rise to the upper part of the amalgam, and it is this part which is in contact with the cadmium sulphate solution. When the transition temperature corresponding to the ainalgam is reached, the lower half will be unsaturated with cadmium and therefore completely liquid, while the upper part will contain an excess of cadmiuin and be part solid and part liquid. If the temperature is raised the E.M.F. of the cell will correspond to one with a two-phase system until the upper part of the amalgam is completely liquid. We will now fix certain limits of temperature, between which various amalgams may be most usefully employed in the standard cell. Fig. 3 (PI. III.) gives the desired infor- mation and the results are tabulated below. TABLE X. Limits of Temperature for use Maseastace of Cadmium in Weston Normal Cell. in the Amalgam. Lower Limit. Upper Limit. a below 0° C, 4 ” 5 ” 6 9) | ” 8 ” 9 * 10 ” IL about. 0° C, 12 SP-7 a5 12°°] 13 IG*1 14 24°) 15 32°°5 16 39°'7 Ly 45°:3 18 52°'6 19 B70 272 Mr. F. E. Smith on Cadmium Amalgams These values have been plotted and give the curves shown in fig. 4 (Pl. [11.). Effect of Temperature Changes on the Weston Normal Cell. Tt must not be concluded from the curves shown in fig. 3 that the change in the E.M.F. of a two-phase amalgam towards a cadmium sulphate solution has a very small temperature coefficient. The curves indicate the change in E.M.F. with change of temperature of a Weston normal cell; that is. the difference between the change of H.M.F. of mercury towards a cadmium sulphate solution (plus mercurous sulphate), and the change of E.M.F. of a cadmium amalgam towards the solution. Because of its small temperature coefficient the Weston cell is sometimes used in an unprotected state with respect to temperature changes, and instances have been brought to notice in which one limb of a cell has been 3° higher in temperature than the other limb. This produced a change in E.M.F. of nearly 1 part in 1000. Rapid changes in the temperature of a cell are of little importance if both limbs are at the same temperature. To determine the temperature coefficient of each limb we have used H form vessels in which the two limbs are 8 inches apart. The tube connecting the limbs is bent in the form of U near to one end; this construction prevents diffusion of the hot liquid from one limb to the cold liquid of the other. Four such cells were made and their electrodes consisted of (1) Cadmium 10 °/o. . Cadmium 10 %p. (2) “ fe . Hg.SO, and Mercury. (3) 39 9 x 79 39 (4) Mercury and Hg.SO, . yu 5 At the beginning the cells were placed in ice and their E.M.F.’s determined. The one limb was then placed in a bath which could be raised in temperature, while the other limb was kept at 0°. The results are contained in the following table and show the importance of keeping a cell screened, in order that the difference of temperature of the two limbs shall not be appreciable. The difference between the two H.M.F.’s at the same temperature is almost identic:l with the difference calculated by the use of the formula given by Wolft™. and the Weston Normal Cell. 273 TaBLE XI. Amalgam cell no. (1). Hot limb +. || Mercury cell no. (4). Hot limb +. H.M.F.’s at various temperatures. E.M.F.’s at various temperatures, Cold limb. | Hot limb. Cold limb. | Hot limb. | 0° 0° 000000 v. 0° 0° 000000 v. 0° oF 144 0° 5° 145 oe 10° 297 0° 16° 291 0° 15° 459 ee Neh dae at aaa oF 20° 630 a? 20° 593 OF 25° 809 0° 25° 750 0° 30° 996 0° 30° 907 Comparison of Results. A comparison of some of our results with those of other observers is of interest. Jaeger made measurements on various amalgams at about 20° C.? and at this temperature, cells containing amalgams with from about 5 to 14 per cent. of cadmium were found to give nearly identical results. A 20 per cent. amalgam cell gradually increased in H.M.F. until the latter was 11°5 milli- volts greater than the .M.F. of normal cells. This differeace is not far removed from the difference 12°5 millivolts for the 20 per cent. chilled amalgam (Table VIII.), and the difference 10°4 millivolts for the slowly cooled amalgam (Table II.). Jaeger made no observations at 0° C. Bijl’s® experiments on the electromotive properties of cadmium amalgams were made at 20°, 25°, 50° and 75°. From the published results it appears that at 25° amalgams containing from about 5°8 per cent. (=9°9 atomic per cent.) to 15:7 per cent. (=25 atomic per cent.) of cadmium possess the two-phase system. From fig. 4 it will be seen that our own experiments fix the limits at 25° as 5°6 and 14 per cent. The lower limits may be taken as identical. At 50° we conclude from curves given by Bijl that the limits are about 9°9 per cent. (16°4 atomic per cent.) and 20 per cent. (380°8 atomic per cent.) of cadmium, whereas fig. 4 of this paper gives the limits at 50° as 9°7 per cent. and 17°7 per cent. Again the lower limits may be taken as Phil. Mag. 8. 6. Vol. 19. No. 110. Feb. 1910. ph 274 Mr. F. E. Smith on Cadmium Amalgams identical. Probably, the upper limits are different because the amalgams used by Bijl were not chilled. Bijl remarks that the quite solid amalgams reached their final values within a few days except in the case of those amalgams which lay in the neighbourhood of the end of the nearly horizontal portion of the curve (see fig. 2 of this communication), 7. e. the amalgams which were near the transition point. He states also that on passing to lower temperatures these mixtures will certainly become completely solid with a corresponding change in the E.M.F. of the cell. He repeatedly noticed that in such a case the E.M.F. attains its final value very slowly, and ascribes this to the last quantity of liquid amalgam (the surface being solid) requiring a long time to thoroughly mix with the other portion. No observations on the electromotive properties were made at 0°, but one or two observations on the melting-points of the amalgams were made in the neighbour- hood of 0°. From these latter observations we may infer (as recently pointed out by Cohen’) that the amalgams with from 2°4 per cent. to about 9 per cent. of cadmium may be usefully employed at 0°. Our own results fix the limits as 26 and 11 per cent. As one of the curves given by Bijl was obtained by extrapolation below 8° the agreement may be considered satisfactory. Cohen and Kruyt’s! observations at 0° C. were made with 10 and 124 per cent. cadmium amalgams. At 0° the maximum observed difference between cells containing the amalgams was 0°25 millivolt. With chilled amalgams our own observations show that the difference may be more than 10 times as great, the difference observed being 3°30 milli- volt. Cohen also measured differences between two 10 per cent, amalgams at 0° C., but the differences were sometimes positive and at other times negative. Conclusions. 1. Cadmium amalgams of such a composition that if homogeneous * they would be completely solid below certain temperatures, may not in a Weston normal cell have that, H.M.F. towards a cadmium sulphate solution corresponding to such a solid. The general result is a lowering of the H.M.F. of the cell and is due to lack of homogeneity of the amalgam, the outer shell of which is of low cadmium con- centration and the centre of the mass of high cadmium concentration. Diffusion tends to restore uniformity, and in * The term is used in a relative sense. and the Weston Normal Cell. 275 consequence the cell is unstable for a very considerable time—the E.M.F’. rising. - When the amalgam is at a temperature near to, but below, the first transition point, the difference of concentra- tion between the inner and outer parts of the amalgam need be only small to enable the outer shell to be a two-phase system. Owing to the small difference of cadmium concen- tration throughout the mass the diffusion process will be slow and the E.M.F. of the cell may remain constant for a very long time. There is, however, in general, a small difference in E.M.F. between such a cell and one containing a normal two-phase system. 3. Cadmium amalgams of such a composition that, if homogeneous, they “would be all liquid above certain temperatures, may not in a standard cell have that E.M.F. towards the solution corresponding to a homogeneous amaleam. In sucha case the H.M.F. of the cell is higher than usual. The effect is due to the upper part of “the amalgam being a two-phase system and the lower part an all liquid system of less cadmium concentration than the upper part. Diffusion quickly equalises the concentration and the E.M.F. falls to a normal value. 4, The 125 per cent. cadmium amalgam at present used in the Weston normal cell may be completely solid at tempera- tures below 12°, and cells containing it may have a higher E.M.F. than normal if used below that temperature. Fortunately, even at temperatures such as 0° C. the amalgam is comparatively near to its first transition point, and “the conditions are similar to those notedin (2). In he eg except in rare cases, cells containing the 124 per cent. amalgam will behave at 0° as though they contained an amalgam which was always in a two-phase : system at that temperature. d. As a 10 per cent. amalgam at 0° is above its first transition temperature this may be used in standard cells at low temperatures. Such cells may also be used at tempera- tures as high as 51°. We suggest that the 124 per cent. amalgam in ‘the Weston normal cell be replaced by a 10 per cent. “amalgam, REFERENCES. DearLove. The Electrician, xxxi. p. 645, 1895. . W. JAEGER. Wied. Amn,, Ixv. p. 106, 1898. . E. Conen. Zeitschr. f. physik. Chemie, xxxlv. p. 69, 1900. . Kerp and BorrrGer. Zeitschr, f. anorg. Chem., xxv. p. 1, 1900. 2 Pw vp ho ~l lor) Dr. J. W. Nicholson on the Bending of . W. Jarcer and St. LinpEcx. Zeiischr. f. Instrumentenk., xxi. p. 33 and p. 65, 1901. H. C. Bru. Zeitschr. f. physik. Chemie, xli. p. 641, 1902. . W. Jarcer. Zeitschr. f. physik. Chemie, xlii. p, 682, 1903. . Puscuin. Zeitschr. f. anorg. Chem., xxxvi. p. 201, 1908. H. von Srrinwenr. Zeitschr. f. Instrumentenk., xxv. p. 205, 1905. 10. F,E. Smiru. Report Elec. Stands. Committee: B. A. Report, 1905. 11. F. E. Smitu. Report Elec. Stands. Committee: B. A. Report, 1908. 12, F. E.Smirn. Phil. Trans. Roy. Soc., cevii. p. 393, 1908. 13. H. Trnstry. The Electrician, 1x1. p. 321, June 12, 1908. 14. F. A. Wotrr. Bull. Bureau of Standards, vol. v. no. 2, p. 309, 1908. 15. P. Janet and R. Jovasr. Bull de la Société internationale des Electriciens. Aug.—Sept. 1908. 16. R, Jouast. Comptes Rendus, exlvii. p. 42, July 6, 1908. 17. EK, Cowen and H. R. Krvyr. Zeitschr. f. physik. Chemie, Ixy. p- 359, 1909. Oo ONIN KN XXV. On the Bending of Llectric Waves round the Earth. By J. W. Nicuouson, A7.A4., D.Sc.* CONSIDERABLE amount of discussion has taken place in recent years concerning the extent to which the results of experiments in wireless telegraphy can be ex- plained by the diffraction round the surface of the earth, of the waves sent out by an oscillator. No satisfactory con- clusion has as yet been reached. About eighteen months ago, in the course of some investigations on diffraction of electric waves, the writer completed an estimate of the intensity, to be expected from theory, caused by an oscillator placed on the surface. This investigation has not yet been published, but as a treatment of the same problem by M. Poincaré, which contains a flaw in the mathematical work rendering the results erroneous even in their order of magnitude, has recently been given +,it seems desirable to state at once the results which follow when this flaw is avoided. The method adopted by M. Poincaré is in the first place to express the force round the surface of the earth, regarded as a sphere of perfect conductivity, in the form of an integral of Fredholm’s type. This is subsequently reduced to a series, proceeding in zonal harmonics, the coefficient of a typical harmonic of order n being a complicated function of the two kinds of Bessel function of order m=n+4, and of their derivates. If} be the wave-length of the vibration, a the * Communicated by the Author. . +t Comptes Rendus, Séance de March 2th, 1909, and other notes. j : | Electric Waves round the Earth. DG radius of the sphere, and kX =2z, the argument of the Bessel functions is z=ka. This series can be shown to be formally identical with a particular case of the general series derived by the writer in the investigation mentioned above, without the use of Fredholm’s integral as an intermediary. The particular case is that in which the electric effect is only required on the surface of the sphere. In the summation of this series lies the error of M. Poincaré’s treatment. Noting that when the order and argument (m and z) of a Bessel function are large, and z>m, the asym- ptotic: expansion contains a factor (z*—m?)—*, and that if z er re er ee em ee 282 Mr. A.S. Russell on some Variations observed in The normal leak is obtained also by weak ionization over a correspondingly longer time, and indeed the natural ionization of the electroscope’s air is itself sufficient if the whole instru- ment be left for 4 to 6 hours. It is not necessary that the leaf be charged. The normal leak because of its constancy for any one disposition, and for the ease and rapidity with which it may be obtained, was used throughout in the work on the absorption of the y rays of uranium X and radium. The effects described have been regularly and clearly shown though in varying degree by every system whether weighted or not, tried in electroscopes of the ordinary pattern. It is immaterial (1) what is the nature of the electroscope walls or base ; (2) what is the nature of the ionizing source, whether it be 8 or y rays, or from radium or uranium; (3) whether the source be intense, or made weak by absorbing screens ; (4) whether the potential be just great enough to ensure saturation or much greater. If A, denotes the percentage of maximum leak over normal leak, and A, that of normal over minimum, a series of experiments has proved that : (1) A, is much greater for a weighted leaf than for an ordinary gold leaf, varying from 4 per cent. with the first to 38 per cent. in the second for typical cases ; (2) for an unweighted gold leaf, A, increases as strength of source decreases, for a given potential ; also A, is practi- cally constant between a divergence of tip of leaf from its discharged position from 7 mm. to 13 mm. For a weighted leaf : (1) A, varies in same direction as potential of leaf if strength of source be constant: thus, Source = 50 div. per min. ? By Ve Potential in volts} 300 350 400 460 530 600 2 ee eee 21:3 20°5 8 | 28:0 | 30-2 35°8 | (2) A, decreases as strength of source, S, increases, if potential be constant : Potential 460 volts. ye at A s40 | 1298 | 1736 | 469 | 483 | 800 | j } 7 j 1 K Electroscopic Measurements and their Prevention. 283 Similar results were obtained for potentials of 400 and 390 volts. (3) A, always varies directly as A,, decreasing as strength increases for constant potential, and increasing with potential for a constant strength of source. Subsidiary effects as well as the chief effect are shown particularly with unweighted leaves where they are not masked so much by the chief effect. The magnitude of any leak for one fixed disposition varies slightly (1 to 3 per cent.) according as (1) the leaf is charged so that it comes on to the scale in the eyepiece within a short time, say 15 to 30 seconds ; the leaf is charged up too much, and (2) it is brought almost on to the scale by an extra source of radium which is then removed ; or (3) it is allowed to come on to the scale under the influence of the ionization which is being measured, taking about 2 to 4 minutes to do so. Method (1) was used generally and in making all measurements shown in fig. 1. The effect of (2) is, as would be expected, to give a leak slightly less than that obtained by (1). Thus a normal leak obtained by charging up as in (2) would be slightly slower than that made in the ordinary way. The effect of (3) is the reverse of (2). In general the leaks obtained by successively charging up to the same potential according to (1) lay on a smooth curve extending from the normal to the maximum 70 re e K 3 > 66 aS elles ~ ¥ | e | 2 | \ | Q | | ; = 62 | : | Fal er | on | | | pan pe 58 are | 2 | iene 0 20 4U 0) OU 11mME& IN MINUTES leak, such as shown in fig. 3. The influence of the methods (2) and (3) was to give a point below the smooth curve and above it respectively. After making such a measurement 284 Variations observed in Electroscopic Measurements. the next made in accordance with (1) lay on the opposite side of the curve, while further measurements made by (1) caused points to resume their place on the smooth curve. In fig. 3, the points obtained by (2) and (3) are marked. The points not marked were obtained by (1). The reason for the existence of these effects was sought next. It was found that the placing of an earthed ring round the sulphur, as employed by C. T. R. Wilson in his original work on the subject of electroscopes (Proc. Roy. Soc. 1901, Ixvili. p. 152), minimized the effect. Leaf systems, however, were made up with weighted leaves and no earthed ring, which gave the effect considerably less than other leaf systems. It was found eventually that only the leaf systems containing a sulphur rod from 2°5 to 4:4 cm. in length gave the effects, leaf systems with a small bead of sulphur only, with or with- out the earthed ring, hardly showing the effect at all. and n is unknown, but quartz is said to cut off all waves shorter than 7X=185 py; this, however, is not in agreement with the results obtained by the electrical method. Rubens’ curve is drawn for the ordinary ray, but after refraction through quartz the extraordinary ray is not very far sepa- rated from the ordinary, and is, moreover, very feeble in its effects. The absorption by quartz undoubtedly increases as the waves get shorter, as is also the case with the absorption by air. The apparatus used to measure the electrical effects of the different parts of the spectrum was as follows. The source of light was an oscillatory electrical discharge passed between two aluminium terminals; the length of the spark-gap was 4 mm. and the terminals about 1 mm. in diameter. The spark-gap was connected to the secondary of an 8-inch Ruhmkorff coil in parallel with a leyden-jar. The current through the primary was about 3 amperes, and a hammer interrupter was used. This source of light, besides being very rich in ultra-violet rays, was of very steady intensity. the furthest points The spectrum was formed as in fig. 2. The spark-gap S was placed in a metal box A, this, as usual, being necessary to shield the rest of the apparatus from induction effects. The light entering the collimator tube through a vertical slit at B was rendered parallel by the first quartz lens D and fell upon the prism E; this prism was of quartz having angles of 60 degrees. With such a prism working at minimum deviation no trouble can be experienced from To ee 294 Mr. E. W. B. Gill on the Electrical light internally reflected. The light, after passing through the prism, was brought to a focus by the second quartz lens D, exactly similar to the first, at the vertical slit G, which was the entrance to the detector F to be described later. It was found better in practice to have the spark-gap vertical and not horizontal as shown in the figure. The tubes carrying the lenses D, D were telescopic, allowing the lenses to be focussed for light of any refractive index within certain limits. The apparatus was mounted on a strong graduated spectroscope table which allowed the prism and the receiver to be rotated through any desired angle. The rays in travelling from the spark-gap 8 to the detector F traversed a distance of about 90 cm. of air (at atmospheric pressure), and the central ray passed through about 3 cm. of quartz, The tubes were blackened on the insides to prevent light being reflected from the sides, but as it seemed uncertain whether lampblack absorbed all the rays the sides of the tubes were cut away in places and two diaphragms C, C’ inserted to stop oblique light. The Electrical Measurements. The light which entered the detector fell upon a zine plate. and caused a number of negative ions to be set free at its surface. The charge which thus escapes from the plate is very small since the slits at G and B (fig. 2) are narrow, so that in order to detect the effect of the light it is necessary to magnify the charge. Professor Townsend has shown that if the zinc plate be arranged opposite and parallel to a second plate at distance d, and if the electric force between the plates is X, then if ions be set free at the zine plate the total number of ions x reaching the positive plate is given by the equation | (a) oe-Px a—P e(%—B)4 > where « and 8 are functions of X and the pressure p only. (a— 8) do-P a — BoB large as we please by suitably adjusting X and p. During an experiment p was kept at 4 mm. of mercury, and X was about 700 volts perem. This gave a big factor and was not too near the sparking potential to produce irregularities. The charge transferred was measured by a quadrant electro- meter, and the ratio of the readings of the electrometer gave in consequence the ratio of the numbers of ions set free for i—), The multiplying factor can be made as Liffect oy the Ultra-violet Spectrum. 295 two different portions of the spectrum. The actual arrange- ment of the detector is shown in fig. 3. It was circular in | Fig. 3. > TB © TEIZAA\A\Y WITT \ section except for the slit G, through which the light entered, which was vertical. The light having entered through a quartz plate CC passed through a silver wire gauze AA and fell upon the zinc plate ZZ, distant 6 mm. from the gauze. The plate ZZ was connected by a brass rod with the electro- meter, and the silver gauze was kept at a potential of 400 volts by being connected through the outside brass tube to a battery of small storage-cells. To prevent electrical leakage a tube BB kept constantly at zero potential was inserted. The shaded portions repre- sent ebonite which was fastened by black wax to the brass tubes. All the joints were made air-tight and the apparatus was exhausted to a pressure of 4 mm. through a tube (not shown in the figure). The method of actually carrying out an experiment was as follows :—Suppose the effect of light of a given refractive index n was to be measured. As the focal lengths of the lenses D, D (fig. 2) vary with the refractive index it was necessary to find the focal length corresponding to x from 1 : the formula f« ony (the extreme values of f were 36 cm. for n=1°54 and 27°7 cm. for n=1°71, this being the range over which electrical effects were detected). The lenses were then focussed for parallel light for the particular value of f. All experiments were conducted at minimum deviation and the angles of incidence and deviation were calculated for this value of n for minimum deviation ; which gave the proper setting for the prism and receiver. With this arrangement the ray of refractive index n is finally brought to a focus at the centre of the slit of the detector. In order to obtain a measurable effect it was found convenient to have this slit about 5 or 6 mm. in width (5°5 mm. actually) and so other rays slightly out of focus were also gathered in on either side of the central ray. If the amount of spectrum 296 Mr. E. W. B. Gill on the Electrical entering the slit be the portion between n+3dn to n—$dn it is easy for the particular dimensions of the apparatus used to show that a ee vA ia dn = 36° J — ri approximately, the lenses being properly focussed and the prism having an angle of 60°. The extreme observations were given by n=1:56 and n=1°71, and the corresponding values of dn were ‘009 and ‘0105; it is thus seen that dn never differs much from ‘01 and as readings were taken for n = 1°56, 1°57 .....1'71 the whole spectrum was embraced in these 16 readings. A few intermediate readings were also taken near points of interest on the resulting curve. It must be borne in mind that although the range of spectrum dn gathered in is approximately constant yet the number of different wave-lengths dX gathered in by the slit is by no means constant. The apparatus being properly adjusted for minimum deviation for refractive index x the light was turned on for 10 seconds and the corresponding deflexion of the electro- meter noted. The prism was then turned through some angle « to the position corresponding to some other index of refraction n' and the detector was turned through 2a, thus being put in the minimum deviation position for n', the lenses were refocussed for refractive index 2’ and the deflexion again noted in 10 seconds. The ratio of the two readings gave the relative number of ions given off by the two ranges of spectra n—°05 to 2+°05, and n’—:05 to n’+°05. Care must be taken lest in the second focussing we alter the cone of light received by the apparatus from the spark. To keep this constant 8, B, and C (fig. 2) were kept fixed in space ; and in order that the length of air traversed by the rays should be constant G was kept at a fixed distance from the prism. To prevent cumulative errors three standard positions were taken corresponding to values of n of 1°57, 1°61, and 1°65 respectively, and the ratios of the numbers of ions set free at these very carefully determined. All other ratios were referred to one or other of these positions; for example, the effect at m=1°59 was compared with that at n=1°61. - General Results. A large number of observations were made and the mean taken, the result being shown in the curve given in fig. 4. The method of construction of the curve will be understood Effect of the Ultra-violet Spectrum. 297 from the following example; the ordinate at the point n= 1°64 represents in some arbitrary units the number of ions given off in 10 seconds by the zine plate when light having “55 1-56 1-58 1-60 1-62 1-64 1-66 1-68 1-70 VisiBLE SPECTRUM refractive indices between 1°635 and 1°645 fell upon it. The units are of necessity arbitrary because the actual number of ions given off is unknown. The limitations of the curve may be briefly recapitulated here. In the first place it simply refers to the spectrum produced by a quartz prism, no correction having been made either for the absorption of the quartz or of the air through which the rays passed, the absorption in both cases increasing with x. Secondly, in order to produce a measurable effect a comparatively large portion of the spectrum is included in each reading, and the curve gives no information about the effect of lines or narrow bands in the spectrum. Thirdly it should be noticed that the curve gives the connexion between the number of ions set free and the refractive index. A curve giving the connexion between the number of ions liberated and the wave-length would be very different. Rubens’ curve shows that = for the visible a) spectrum is very big, decreasing very rapidly as x increases, and if therefore a curve connecting n and ) were drawn it would show that the effects of the small wave-lengths were very much more marked than even the effect of large values of m in fig. 4. This curve has not, however, been drawn 298 Mr. FE. W. B. Gill on the Electrical because it would have to be obtained by combining Rubens’ curve (fig. 1) with fig. 4, and his measurements do not go as far as the electrical ones ; it would be in addition slightly misleading because for any particular value of n we have a mixture of two wave-lengths corresponding respectively to the ordinary and extraordinary rays, though for quartz these wave- lengths are nearly equal and the extraordinary ray is of very feeble intensity. The curve given represents the mean of a number of observations, but observations made at different times showed variations greater than experimental errors which have not yet been explained. There are three possible causes of variation, the state of the zinc plate may vary, the light emitted by the spark may vary, or the absorption by the air may vary. ‘Taking these in order it is true in the first place that a freshly polished zinc plate is very active in giving off ions, and as it gets dirty, 7.e. oxidized, the number of ions given off (other things being equal) decreases, but a steady state is soon reached. The only thing which appears to disturb this steady state is the passage of a spark-discharge from the wire gauze to the zinc plate, but even then the original steady state is regained after 2 or 4 hours. In the second place the light from the spark-gap was kept as constant as possible by always having the same current through the primary, the same length of spark, and approximately the same adjustment of the hammer interrupter. On the whole it is improbable that either of the above causes is responsible for the variations, a supposition that is partly confirmed by the fact that on any one day the readings were consistent, the variation being from day to day; and it seems likely that the absorption by the air is the determining factor. The published results for air-absorption by different investigators are very inconsistent, and it may be that the absorption is more due to the moisture in the air than to the air itself, which would explain the variation from day to day. A study of the curve obtained (fig. 4) shows that the electrical effect is just beginning about the middle of the visible spectrum and increases up to a first maximum for refractive index n=1°66, sinks to a minimum at 1-67, and reaches a second maximum for n=1°68, rapidly decreasing after that. The position of the minimum was always exactly the same for every experiment and the second maximum at n=1°'67 was fairly steady, but the first maximum on different occasions varied, sometimes appearing at n=1°65. In this case besides the above considerations there is an additional explanation of this. Rubens’ curve shows that in the. Effect of the Ultra-violet Spectrum. 299 neighbourhood of n=1°65 a comparatively large change of refractive index involves only a small change in wave-length, and therefore the wave-lengths used at n=1-65 differ only slightly from those at n=1°66. With regard to the minimum a rather closer study of the region of this point was made by taking readings for the ranges of spectrum with centres at n=1°665 and n=1'675; both of these readings were larger than for n=1°67 showing that the actual minimum cannot be very far from n=1'67. The wave-length corre- sponding to n=1°67 is not known, Rubens’ readings only going as far as n=1°65. An absorption band is, however, known to exist in this region, and quartz is supposed to cut off all rays of wave- length less than 185 pu. Wood (Physical Optics, p. 324) calculates the ultra-violet absorption band at 103 pegs) seems likely that the electrical minimum at n=1°'67 is coincident with the absorption band in quartz, in which case the statement that quartz is opaque to rays less than 185 wy in wave-length is erroneous, for the second maximum which was always well marked is beyond this limit. In any case it is highly improbable that for »=1:70, at which the electrical effect was measurable, the wave-length should not be less than 185 pp, for it is unlikely that from n=1°65 to n=1°70 the wave-length should not change by more than 15 pp. It is interesting to note that in Pfliiger’s research on the energy of the ultra-violet spectrum, conducted under very much the same conditions as the present electrical one, except that he used a fluorite prism, the energy commencing in the visible spectrum was very small until he reached X=190 py when it rose to a very “strongly marked maximum, This would coincide very nearly with the first maximum in the electrical curve. Too much stress should not, however, be laid upon this, as for reasons indicated previously there is ‘not to be expected any simple relation between the energy of the spectrum and the electrical effect. It is probable that the electrical action is of the nature of a resonance effect in which case the energy would not be the primary factor though it would of course contribute. Except at the maximum there is no similarity between the energy curve of Pfliiger and fio. 4. A final point of interest is that hitherto it was supposed that there was no electrical action in the visible spectrum, but with the sensitive methods of measurement used in this research a small effect was eae in the visible spectrum (which extended from n= 1°54 to n=1°56) though it was only two per cent. of the maximum effect. To make certain of the presence of this, a plate of glass was inserted 300 Method of Reading Torsional Angle of Rotating Shaft. to cut off any possible stray ultra-violet light but the small electrical effect still persisted. In conclusion I should like to express my warmest thanks to Professor J. 8S. Townsend, in whose laboratory this research was conducted, and at whose suggestion it was undertaken, for assistance and encouragement always most readily given. XXIX. An Optical Method of Reading the Torsional Angle of a Rotating Shaft. By F. J. JERvis-Suirn, 1.A., Oxon., TERT ER taal |e ane certain experimental work in which it was necessary that the energy transmitted by a rotating shaft should be known, I devised the following method of reading the angle of twist of the shaft, and found that it gave good results. It is easily applied, and might be used for other similar purposes in the mechanical and physical laboratory. Clear readings can be taken from a pointer, moving over a circular dial which reflects light, while it rotates about a diameter. The optical principle involved is similar to that of the thaumatrope of Dr. Paris, in which on one side of a card the head of a man was painted, and on the other side a hat. When the card was rotated by means of twisted strings attached to the opposite edges of the card the head and hat appeared as one picture. Tbe rationale of the experiment being that the picture of the head is retained by the organs of vision until the hat appears, the two separate impressions thus making one picture. In my apparatus the reflecting dial, made of mirror glass, is fixed so that its plane is parallel with the axis of the shatt, or spiral spring, the angle of twist of which is to be measured. The mirror is perforated in the centre, and through the perforation the pivot which carries the pointer passes. The pointer is de- flected through an angle proportional to that of the twist of the shaft by means of connecting links attached to the shaft and also to a sleeve fixed to the shaft. A parallel beam of light is projected on to the mirror dial by means of two plano-convex lenses, as used in a projection lantern. The image of the pointer moving over the divided scale can either be viewed direct by reflexion, or, which is far more convenient, the image can be projected on to a screen by means of two achromatic lenses, and then it may be viewed simultaneously by several observers. ‘This latter method of viewing the pointer is preferable to the * Communicated by the Author. T heory of the Structure of the Electric Field. 301 former one, which is rather fatiguing to the eve, owing to the intermittent flashes. Even when the speed of the shaft is slow, about 300 revolutions per minute, and the flashes occur at intervals of 4+ second, the image of the pointer is clear and well defined. I have applied this optical method of reading a moving dial to a torsion work-measuring machine placed aes an “electric motor and a dynamo feeding are lamps. The best way of reading the torsional angle ‘of a shaft is, undoubtedly, by means of a scribing point ‘actuated by differential gear ‘which makes the movement of the point directly proportional to the angle of torsion. The earliest application of such a device is due to Hirn (Les Pandyna- mometres, par G. A. Hirn, Paris, Gauthier-Villars, 1876). In it eight cog-wheels are required to meve the pointer and show the torsional angle. Subsequent inventors have recently arrived at the same result with a reduction of the number of cog-wheels. In a recording torsional ergometer, by the author of this com- munication, four cog-wheels were employed and two flexible joints. (Shown at the Royal Society, May 2, 1894.) In some cases such an elaborate method of reading the torsional angle is not required, and the optical method I have described is sufficient for those cases in which only rather small powers are dealt with, as for example, in aeroplane engines, and motor-launch internal combustion engines. December 29, 1909. at its Gali sasion to Rontgen Ridinsion and to Light. By Sir J. d. THOMSON, Professor of Experimental Physies, Cambridgye*. Gar theory considered in this paper—that the electric field is made up of a number of discrete units—is one which naturally suggests itself, if we use the conception of tubes of electric force for representing the state of the electric field. T have in several papers and also in my * Recent Researches on Electricity and Magnetism,’ and in * Electricity and Matter,’ shown how the phenomena of the electric’ field can be regarded as the method of working of a mechanical system, the parts of which are the tubes of electric force, which on this view are endowed with mass, momentum, and energy. The properties of the tubes of force are determined “by the charge at their ends, hence if that charge is the charge * Communicated by the Author. 302 Sir J. J. Thomson on a Theory of carried by a corpuscle, or a particle in the Canalstrahlen, which as tar as we know is incapable of further sub-division, the corresponding tube of force will be incapable of further subdivision and will form a natural unit. Now the ques- tion at once arises does the tube of force attached to a corpuscle spread out uniformly in all directions, like the lines of force round a charged sphere, or is the tube confined within a cone of small vertical angle. We usually regard the electric field round an electric charge as spreading out uniformly in all directions, so that the force remains unaltered in magnitude as long as we keep at the same distance from the electric charge. We must remember, however, that the charges used in the experiments by which this result has been established are many million times the charge on a corpuscle, and it is evident that when we superpose the fields due to millions of corpuscles the result will be the same whether each individual field is uniformly distributed in all directions or is confined within a small solid angle. Hach would give a field uniform in all directions: on the first supposition the field would be continuous, while on the second it would have a structure. We are now, however, able, in cathode rays, to observe the behaviour of particles carrying the indivisible unit, so that the nature of the field round the unit charge is an important subject, and one which we might hope to be able to settle by experiment. The properties of a corpuscle, on the supposition that the electric field around it is uniformly distributed, have been known for sometime. In this paper I shall consider what the properties would be if the electric force due to a corpuscle was practically exerted in only one direction. Let us suppose then that the tube of force attached to a corpuscle is a double cone of small semi- vertical angle, so that it is only within this cone that the corpusele exerts any electric force. There is discontinuity between the electric force inside and outside this cone: to reconcile this with the principle of the Conservation of Energy we may suppose that a finite amount of work is spent or gained when one corpuscle crosses the boundary of a tube of force due to another. On this view the electric field due to a number of corpuscles is a mosaic, as it were, made up of a number of detached fields. The electric field itself, as well as the electric charges in it, being molecular in constitution. In this paper I shall consider the properties of the field due to a single corpuscle: in another paper I propose to consider the action of two or more such fields upon each other. We assume that the properties of the isolated tubes are —— Se it ip he the Structure of the Electric Field. 303 the same as those possessed by a tube forming part of a con- tinuous electric field. (1) That the product of the electric force due to the charge and the cross-section of the tube is constant and equal to 47 times the charge at the end of the tube. (2) That when the tube is moving relatively to axes which are fixed with reference to the instrument used to measure the physical quantities, there is a magnetic force at each point in the tuhe, the direction of the magnetic force being at right angles to the plane containing the electric force, and the direction of motion of the tube at the point, the magnitude of the magnetic force being the product of the electric force, and the component of the velocity of the tube at right angles to this force. (3) That there is momentum with reference to these axes throughout the tube equal per unit volume to the vector product of the electric and magnetic forces divided by the square of the velocity of light. (4) That if R is the resultant electric and H the resultant magnetic force at a point in the tube measured in electro- static units, c the velocity of light, the energy per unit volume Fig. 1. of the tube is EY pase fo ( Re + ce aah, Si Ce From these principles we shall calculate the values of certain quantities which are required in the discussion of the properties of these tubes. The energy in a tube at resi. Let the corpuscle be regarded as a sphere of radius a, the tube of force being the portion outside the corpuscle of a double cone whose solid angle is w with its vertex at the centre of the corpuscle. Let A B, A’ B’ be two adjacent cross-sections of the cone, 8 the area of AB, x the distance of AB from the vertex of the cone, and dr the distance between AB and A’ B’ ; then if R is the electric force at AB, the energy per unit volume is R*/87, so that the energy included between AB and A'B/ is equal to i 8 dr R? Sir ° If e is the charge on the corpuscle the charge at the base of each half of the cone is $e, thus RS=2ze and the energy 304 Sir J. J. Thomson on a Theory of between AB and A! B’ is lredr _1me'dr Dior 2@r° Thus the energy in one half of the cone is equal to oy Te" * dr pices Fy peat Dee a And the energy in the double cone is 7re?/wa. Thisis greater than if the lines of force were uniformly distributed round the corpuscle in the proportion of 27 too. nergy ina tube moving with uniform velocity. Suppose the tube is moving with a velocity w in a direction making an angle @ with the axis of the tube. The velocity of the tube at right angles to itself is wu sin @: there is therefore a magnetic force at right angles to the tube equal toRsin@u. The energy due to this magnetic field is equal to 1 R°w? sin? 6 S73 c per unit volume. ‘Thus the energy due to the magnetic 9 “ field is 4 sin? 6 times the energy due to the electrostatic field. The expression for the electrostatic energy obtained in the last paragraph requires modification, as when the lines of electrostatic force are moving they suffer a compression parallel to the direction of motion equal to 4 / 1— — m which we shall denote by g:; we can easily show that this i SENS 4 ; cos? 6\= contraction diminishes the solid angle to g (sin? 6+ ) @, and since the energy is inversely proportional to the solid angle, the total energy, electrostatic and magnetic, is equal to Ne T : : & tate sia 2 é | : og (sin? $+ co eye ‘ The kinetic energy thus depends on @, and the motion will not be steady unless @ has the value which makes the kinetic energy a maximum, 7. ¢., unless = =, so that the bo} the Structure of the Electric Field. 305 axis of the tube is at right angles to the direction in which it is moving. Thus when the field due toa corpuscle is limited to a small cone, the electric field will always set itself at right angles to the direction of motion of the tube, whatever be the value of w, and not merely when w=c, which is the only case where the lines of force become at right angles to the direction of motion when they are uniformly distributed in all directions when the particle is at rest. Momentum inthe tube. Since the electric and magnetic forces are at right angles, the momentum per unit volume is equal to the product of the electric and magnetic forces divided by the square of the velocity of light ; the direction of the momentum is at right angles to both the electric and magnetic force, and hence, if the angle of the cone is small, will be approximately at right angles to the axis, and in the plane containing the axis of the cone and the direction of motion. Since the magnetic force is uw sin @ times the electric force, the moment per unit . usin @ R? volume is -—~—; Aare tration of the tubes we find that the momentum in the tube is equal to : taking into consideration the concen- 2a u e z — . —- ————————_, 55 sin 0 @ Ca bik, cos? @\3 g (sin? 6+ ) 9 Q° and is at right angles to the axis of the cone. When the motion is steady 6=7/2 and the momentum is parallel to the direction of motion of the corpuscle and equal to Ir ue @ aeg Thus the mass of the corpuscle is Par Nig ha wae,’ when considered as a function of the velocity it varies in- versely as q or 4/ ee This is the result given by the Principle of Relativity when the lines of force are supposed to be distributed all round the corpuscle. The experiments made by Bucherer on the mass of @ particles moving with different velocities show that this law agrees well with his observation. Phil. Mag. 8. 6. Vol. 19. No. 110. Feb. 1910. D.¢ 306 Sir J. J. Thomson on a Theory of For slowly moving corpuscles the expression for the mass 9 9 9 e a ‘Dim ° * Z e e e e is —— instead of = =, which is the value when the lines @ Ca 3 7a of force are supposed to be uniformly distributed. Since 2a/w is a large quantity the value of a founded on the sup- position that the lines of force are concentrated in a cone will be much larger than the value calculated on the basis that the lines of force are uniformly distributed. On the effects produced when a moving corpusele is suddenly stopped. Let O represent the corpuscle moving forward horizontally, and suppose when it is at A its velocity is reduced by u ina short time t. Let us consider the configu- Fie. 2 ration of the tube after a time ¢ measured a from the instant when the stoppage began. Before the corpuscle was checked the tube | was at right angles to its direction of motion, i. e. it was vertical. Now any disturbance is propagated with a finite velocity ¢, ¢ being the velocity of light. Hence, if we describe a sphere whose radius is ct and whose centre is at the centre of the corpuscle, the region outside this sphere will be unaffected by the stoppage of the particle, and the tube of force in this region will occupy the position it would have done if the particle had gone on moving uniformly. Again, the fact that the velo- city of the corpuscle has been reduced will have been telegraphed to a distance c(t—7) from the centre of the corpuscle, so that inside a sphere of this radius the tube of force will have the reduced velocity, hence since the tube preserves its continuity it will have a configuration similar to that repre- sented in fig. 2, where the portion inside a sphere whose radius OA is c(—7) is vertical, and the portion outside a sphere whose radius OA’ is equal to ct is also vertical, but separated by a distance u(t—7) or, since we take 7 to be small compared with ¢, ut approximately from the portion inside the smaller sphere. The portion of the tube of force connecting the two vertical portions will be constricted, and hence the electric force in this portion will be greater than it would have been at the same distance from the corpuscle if the latter had not been stopped. a ae ee Ps the Structure of the Electric Field. 307 Tf S is the area of the cross-section of the tube at AB the area of the constricted portion is Ssin @, where @ is the angle between AB and AC, aoa AA! CT me AO att oa CT es: approximately when ¢ is large. Thus the area of the constricted portion of the tube is ae so that if F is the value of the electric force at AB its value in the portion AC will be F<. If m is the solid angle of the tube and v=OA, F7?@=27re, hence the electric force 2tre ut . = . . in AC equals —— — » or since ct =1" approximately, the force ro ¢ in AC is 29ré u ro oT” The lines of electric force in this portion are moving approximately at right angles to themselves with velocity ¢, hence they give rise to a magnetic force at right angles to the pJane of the paper, and equal to 2Te u @r CT Let us now calculate the energy in the portion AC. The energy per unit volume is equal to e- wu? eS Re Or CT The length of AC is approximately equal to ut and the CT cross-section is 7°w x 3 hence the volume of the portion u AC is +*@ x eT, and the energy is equal to and there is an equal amount transmitted along the other half of the tube, hence the amount of energy travelling out as radiation is equal to 508 Sir J. J. Thomson on a Theory of If the corpuscle is reduced to rest, the energy possessed by it before it was stopped was equal to 2ae° wv @ ea 2a? hence the fraction a/et of the energy is radiated away. I regard this radiation as constituting the Réntgen rays. I gave a similar theory when the electric force was supposed uniformly distr ibuted round the cor puscle. In that case the energy of the corpuscle is radiated in all directions, and is diffused throughout a very large volume. In the theory just given, however, the energy is not diffused but is concentrated ina kink ina sin gle tube of force. Thus when a number of cathode par ticles. bombard an anti-cathode the resulting Réntgen radiation is concentrated into small patches which possess momentum and energy, in fact we have a condition of things much more closely represented in many respects by the old emission theor be of light than by the wave theory in the form in which it is usually represented. A corpuscle when it strikes the anti-cathode is not reduced to rest by the first collision it makes with a molecule, but rebounds from one molecule to another. The tube of force attached to it is jerked spasmodically by the collisions and a series of small discrete transverse pulses travel outwards along it with the velocity of light ; the condition of the tube is very analogous to that of a stretched stri ing the end of which is spasmodically jerked backwards and forwards. I pointed out several years ago that the properties of the Rontgen rays, especially those concerned with the ionization of gases through which they pass, suggested that the electric and magnetic forces constituting these rays are not uniformly distributed through space, but are concentrated in regions whose volume is a very small fraction of the space through which the rays are passing. If we suppose, as in the usual explanation of the forces in the electric field by the tensions along the lines of force and the pressures at right angles to them, that there is a repulsion between tubes of force running in the same direction ; we can see that the energy radiating out into space along tubes of force may be communicated to charged bodies in the region traversed by the tubes, and that when two pulses come close together the energy in one may increase at the expense of that in the other. The energy radiated resides, as we have seen, in the kink in the tube of force produced by the stoppage of the corpuscle, and remains constant as the kink travels outwards into space | ; yw aS the Structure of the Electrie Freid. 309 following the tube of force which is supposed, until the dis- turbance reaches it, to retain the velocity it had before the stoppage of the corpuscle. The constancy of the energy is due to the length of the kink being proportional to its dis- tance from the cor puscle from which it arose, and this again is due to the uniform motion forward of the part of the tube not yet reached by the disturbance. If the motion of this part of the tube is interfered with after the pulse is started, then the length of the kink may be no longer proportional to the distance from the corpuscle, and then the energy in the pulse will no longer remain constant as it moves away from the corpuscle. If for example we retard by any means the further portions of the tube, the length of the kink when it reaches them will be reduced, and the energy in it less than if the motion had not been retarded. The ener ey lost will either appear as the energy of a reflected pulse or ‘it will be communicated to the system which retards the motion of the tube of force. One way by which the motion might be retarded is by the repulsion exerted on it by a similar tube of force which it is overtaking, or it might be pushed forward by the repulsion of a similar tube from behind which was overtaking it. In the first case the energy in the kink would be reduced, and in the second it would be increased. We can in this way picture to ourselves interchanges between the energy in the pulses and that of charged bodies in their path, of a some- what similar character to those which occur between colliding molecules, and which might be expected to lead to statistical equilibrium between the ener ey in the pulses and the kinetic energy of the charged bodies. We can see, too, in a similar way that there might be interchange of enere gy between two pulses ¢ enerated by corpuscles so that one might overtake the other. And thus if we had an assemblage of such pulses travelling backwards and forwards in an enclosure with perfectly reflecting walls we should expect that the interchange of energy between them would produce, when the sy stem was in a steady state, a distribution of energy analogous to that which holds for the distribution of energy among the molecules of a gas in a steady state. When the field round the corpuscle is supposed tniformly distributed the Réntgen rays produced by a stoppage of a slowly moving corpuscle are symmetrically distributed with respect to a plane through the cor puscle, at right angles to the direction in which it was moving. \W hen the velocity of the corpuscle approaches that of “light there is sensibly 310 Sir J. J. Thomson on a Theory of more energy radiated in front of the plane than behind. When the electric field of the corpuscle is confined to a small cone we see that all the energy radiated is in front of the plane, whatever may be the velocity of the corpuscle. When the rays pass through a region in which there are free corpuscles they tend to make the corpuscles move in the same direction as those which produced the rays Radiation from a corpuscle whose motion is being accelerated. We can easily adapt the preceding investigation to the case when the velocity of the cor pusele i is changing gradually. If the velocity changes by Sv in the time 7, then the investi- gation just given shows that there will be a kink in the tube of force carrying an amount of energy equal to adh OP QO Ga | Now 6u/7 is equal to the acceleration of the corpuscle ; hence the energy travelling outwards along the tube of force is equal to 27e" , rie ak when fis the acceleration of the corpuscle. This energy takes a time t to leave the corpuscle. Hence the rate at which the energy is radiated from the corpuscle is equal to WC This expression is of the same form as when the lines of force are uniformly distributed round the corpuscle. In the latter case, however, the radiation spreads out in all directions in space, while in the case we are considering it is all concen- trated along the tube of force attached to the corpuscle. In the former case we have the uniform distribution of energy usually associated with the wave theory of light, while in the case we are considering the energy is concentrated in a manner more comparable “with the emission theory, for the energy is concentrated in a single system—the tube of force attached to the corpuscle which is thrown into transverse vibrations by the movement of the corpuscle. Regarding light as arising from the vibrations of corpuscles and charges of positive electricity, we see that on this view we should have transverse vibrations travelling along some, but only some, of the tubes of electric force which pass through the space surrounding the luminous body. The a ee ee the Structure of the Electric Field. dll tubes which are transmitting the energy are those which are terminated by the moving charged particles which are the source of the light. The energy would thus be localized along these tubes, and would not be distributed continuously throughout the space within sight of the luminous body. The theory would have some of the characteristics of both the emission and the usual form of the undulatory theory. It would agree with the emission theory in supposing that the energy of light is concentrated in small specks in the space round the luminous body, while it would be in agree- ment with the undulatory theory in supposing that the disturbance whose propagation constitutes light is a vector quantity. Isketched a theory of this kindin my ‘ Electricity and Matter,’ and pointed out there that the electrical pro- perties of light, suchas the ionization of gases by ultra-violet light and photo-electric effects, could be more easily explained by a theory of this kind than by one requiring a continuous distribution of energy throughout the ether. With respect to purely optical effects, the explanation of most of these would be the same on either theory ; the case of interference, however, requires special consideration. Inter- ference implies that over the slit, or obstacle, or whatever system is used to produce the interference, the disturbance producing the light should be in the same phase, or at any rate that the disturbances at different parts should have their phases definitely related. It is not necessary that this disturbance should be absolutely continuous over the slit, if, for example, the disturbances travelling along a number of discrete lines of force passing through the slit reached the slit in the same phase, we should get much the same set of fringes as if the disturbance had been uniformly distributed over the slit. If, however, there were no phase relations between the vibrations of the tubes of force as the disturbance passed through the slit, there would not be interference unless the amplitude of the vibration extended from one side of the slit to the other, so that the electric force in the disturbance would excite secondary vibrations in the cor- puscles in the metal on the two sides of the slit: as these secondary vibrations would be in the same phase they would be able to produce interference effects. In any realizable case it would seem, however, that the vibrations along a considerable number of tubes coming from a luminous body must have their phases related in a definite way. For consider a corpuscle vibrating in a definite period ; in its neighbourhood there will be many other systems having the same time of vibration, and the vibrations of these will ol2 Sir J. J. Thomson on a Theory ot he excited by resonance and will be in phase relation with the primary vibration. Again, as the vibration from the corpuscle travels through the luminous body it strikes against other corpuscles, and excites in them vibrations which are in definite phase relations with the primary vibra- tions. It appears therefore that any vibration excited in a single tube of force will, after it has travelled through the parts of the luminous body surrounding the corpuscle at its extremity, be accompanied by a number of secondary vibra- tions in other tubes of force in phase relationship with the primary vibration. So that even on the view that the vibrations which constitute light travel along discrete lines of force, the phases of the vibrations along the different tubes of force which pass through any area under considera- tion, such as the slit used to produce the diffraction-fringes, will not all be independent. There will be groups of lines of force in which the vibrations are connected by definite phase relations, and the secondary waves started when the vibrations of the different members of one of these groups strike against the boundary of the siit, will also be in phase relation, and will therefore be able to interfere. The existence of interference even when, as in experiments such as those made by Mr. Taylor (Proceedings of the Cambridge Phil. Soe. xv, p. 114), in which sharply defined interference-fringes were obtained when the intensity of the light was reduced so much that it required an exposure of days or weeks to photograph the interference-fringes, is thus not incompatible with the view that light consists of vibrations travelling along discrete tubes of electric force. Hxperi- ments with very feeble light do, however, show, unless we assume that the amplitude of the transverse vibrations of a tube of electric force can be greater than the width of the slit, that the energy in the vibrations running along the different tubes of force cannot all be equal unless we suppose that this constant value is exceedingly small, much smaller than the unit of radiant energy for light in the visible part of the spectrum given by Planck’s theory : for in Mr. Taylor’s experiments I estimate that the light was so feeble that there was less than one of Planck’s units per litre : unless the units came very irregularly from the luminous body it is difficult to imagine that they could be sufficiently crowded together to interfere. On the effect of a magnetic field on the set of the tube of electric force attached to a corpuscle. We have seen that the axis of a tube of force attached to a single corpuscle moving uniformly along a straight line, , (wm—vn) + R(lu + mv + nw) {usin 6+ cos O(v cos 6+ wsin d)} = 0, the Structure of the Electric Field. 313 sets itself at right angles to the direction of motion of the corpuscle; we shall now investigate what effect a magnetic field will have on the direction of the axis of the tube of force. We shall take the case when the magnetic field is uniform and parallel to the axis of x. Let 1, m,n be the direction cosines of the axis of the tube of force ; u, v, w the components of the velocity of the corpuscle. If the angle of the tube is small, the components of the magnetic force due to the moving tube at a place where the electric force is I, are respectively R(mw—nv), Riwu—lw), R(lv—mu), and if ais the external magnetic force, the components of the magnetic force at this part of the tube are respectively a+R(mw—nv), R(imu—lw), R(lvo—mu). The energy per unit volume is therefore aa {(a+R(mw—nv)) + R?(nu— lw)? + Re — mu)? i, if the axis of the tube makes an angle @ with the axis of «, and if the plane containing the axis and the axis of # makes an angle @ with the plane of ay l=cos#, m=sin@cosd, =sin@sin ¢. When things are in a steady state, if T is the energy per unit volume, aT wT Bers Gem The first of these is equivalent to (). and the second to alu— (lu + mv + nw)(a + ROivw — nv) ) = (), From these equations we get ba Oy lutme+nwo=0. The first of these shows that the axis of the tube sets itself at right angles to the magnetic force, and the second that it is also at right angles to the direction of motion of the corpuscle : thus, where there is a magnetic field, the direction of the axis of a uniformly moving tube is determinate. — ee 2 aie On the Rate of Evolution of Heat by Pitchblende. as 1 By Horace H. Poon *. Outline of Method. HE following method was adopted to directly measure the rate of evolution of heat by pitchblende. The powdered pitchblende (Joachimsthal) is carefully dried and placed in a spherical Dewar calorimeter. This is placed in a large vessel of planed ice, and the difference of tempe- rature between the surface layer of pitchblende in contact with the bottom of the calorimeter and the ice is measured by means of a sensitive thermo-couple. ‘The pitchblende, initially at air temperature, cools comparatively rapidly at first, but eventually attains a temperature which remains approximately constant. In this state the rate of generation of heat by the pitchblende is equal to the rate of loss. To find this rate the calorimeter is filled with water and the rate of cooling noted. The cooling was in accordance with Newton’s law, so that the rate of loss of heat is pro- portional to the difference of temperature between the inside and the outside of the calorimeter. Now as it is the tempe- rature of the surface layer of pitchblende that is measured, the conductivity of the latter does not affect the result; so that the heat evolved per hour by the pitchblende equals ‘the constant temperature difference between the surface layer and the ice multiplied by the thermal conductance of the calorimeter (7. e. the number of calories which escape per hour when the inside is 1° C. hotter than the outside). Calorimeter and Ice Vessel. The calorimeter employed is approximately spherical with aneck about 8 cm. long and 1 em. internal diameter. Its capacity is about 160 ¢.cs., and the walls are silvered in the usual way. The calorimeter is supported ona wooden tripod inside a cylindrical zine vessel 12 inches high and 10 inches diameter which is filled with planed ice. ‘This is supported on a wooden stand inside a larger galvanized iron vessel, also filled with ice, which in turn is placed inside a barrel, the space between the walls being filled with cork dust, and a bag half full of the same being placed over the lid. The inner * Communicated by the Author. tate of Evolution of Heat by Pitchblende. 315 vessel not being water-tight drains into the outer, from which the water is siphoned off at intervals. By re-filling the outer vessel with ice every ten days or so the ice in the inner vessel could be kept unmelted almost indefinitely. It was found necessary to soak the ice in water for an hour or more after planing in order to bring it to zero, as it was initially much below it. Even when all possible precautions were taken small temperature-differences seemed to occur, possibly due to small variations in the purity of the ice or to regelation effects. Ice frozen from distilled water not being available, commercial ice frozen from the Dublin City supply had to be employed. This point is discussed later. Couples and Galvanometer Arrangements. The thermo-couples employed consisted of five pairs of iron and nickel silk insulated wires of diameters (bare) 0°12 mm. and 0°15 mm. respectively. The use of these fine wires minimized the conduction of heat along the couple. Two couples were used. In one of them the stem inside the calorimeter was cased in a glass tube and reached down through the centre of the calorimeter to the bottom. This couple was used in all the water-cooling experiments, and also in the first pitchblende experiment. The second couple, which was employed in all the later pitchblende experiments, had a slightly longer stem. The part of the stem inside the calorimeter not being enclosed in a glass tube was flexible. The couple was inserted in the calorimeter before the piteh- blende and pressed down so that the stem curved round the side of the calorimeter and avoided the centre, where the temperature must be slightly higher than at the surface, — The neck of the calorimeter was plugged with cotton-wool above which wasa thick layer of vaseline to exclude moisture. Over all was a sheet rubber apron bound above round the stem of the couple, and below round the neck of the calorimeter. The outer junctions of the couple are encased in a thin glass tube and buried in the ice. Between the glass tube and the calorimeter the wires are protected by a thick seamless rubber tube. Flexible conductors, as used in electric lighting, are employedas leads. To lessen conduction of heat down these a length of over five feet is buried in the ice. Tor about two feet nearest the couple the leads are encased in a rubber tube and lie inside the inner ice vessel. The next two feet or so of leads are enclosed in a copper tube SEG Mr. H. H. Poole on the Rate of laid on the lid of the inner ice vessel and completely buried in ice. For about two feet more the leads are encased in rubber ; most of this portion is also in the ice. Between the couple and the galvanometer is placed a reversing-key. It consists of two blocks of wood each about 3 inches square by 2 inches deep. In the centre of the lower face of one block and of the upper face of the other were bored holes about one inch in diameter and one inch deep into which molten lead was run. The ends of the lead cylinders so formed were finished off flush with the surfaces of the blocks. The bared ends of the leads from the couple are stuck down to one lead cylinder, being insulated from it by paper soaked in molten paraffin-wax. The leads to the galvanometer are similarly stuck to the other block in such a way that by laying one block on the other the bared wires come into contact and close the circuit. By turning the upper block through 90° the circuit is opened, and a further turn through 90° reverses the original connexions. The whole is placed under a bell-jar and the upper block lifted and turned by means of a glass rod passing through the stopper. In this key thermal eftects are reduced to a minimum, the lead blocks serving to equalize the tempe- rature of the points of contact ; the resistance of the key was found to be negligible. A suspended coil mirror-galvanometer is employed. Itis enclosed ina fireclay cylinder to minimize thermal effects at its terminals. ven so such effects are never quite absent, but are eliminated by the system of reading employed. The circuit is always left open, and beforemaking an observation the open circuit reading of the galvanometer is taken. This changes from day to day, and seems to depend on the tempe- rature of the cellar in which the apparatus is set up. The circuit is then closed and a reading taken; the circuit is reversed and «another reading taken. The difference in reading produced by reversal is entered as the deflexion, and from it the temperature of the calorimeter is obtained. Thermal effects at the galvanometer only render the de- flexions on each side of zero unequal and do not affect the total deflexion produced by reversal. Calibration of Lhermo-couples. The couples when used with this galvanometer are too sensitive to be conveniently compared with a mercury thermometer. Two methods of calibration were employed. One method consisted in calibrating the galvancmeter by Evolution of Heat by Pitchblende. B17 means of a cell of known voltage, using a standard 0:02 ohm manganin resistance, and measuring the H.M.F’. of the couple when there was a known difference of temperature (a few degrees) between the junctions. Hence, knowing the resistances of the galvanometer and the couple, we at once obtain the fraction of a degree which corresponds to one division of the galvanometer-scale. It was found that the galvanometer was deflected (on reversal) 13°32 scale-divisions per microyolt applied to its terminals, the arrangement being such that thermo-electric effects were as far as possible eliminated. The H.M.I. of the couple used in the pitchblende experiments was found to be 159-7 microvolts per degree. This is the open circuit E.M.F. as the arrangement adopted was a nul method such that when the H.M.I’. was measured no current was being taken from the couple. ‘The resistance of the galvanometer being 30°0 ohms and that of the couple 25:4 ohms, 1° C. corre- F 92.299 y 1). sponds to eee ey 1152 scale-divisions. oor4 The other method consists in placing one terminal of the couple, encased in a glass tube, in a vessel of planed ice and water which can be subjected to varying pressures by means of a mercury reservoir and a flexible tube. The reservoir can be raised or lowered by a cord passing overa pulley and the level of the mercury in it read on a vertical scale. As the pressure vessel is completely filled with ice. and water, the variation of the level of the mercury in it is negligible. The vessel is buried in ice in the zine ice-vessel. The stem of the couple passes through a rubber stopper which is firmly tied down and the outer junctions are buried in the ice. A reading of the gaivanometer deflexion having been made with the reservoir at a certain height, the latter is raised a known distance and the variation in the galvano- meter deflexion noted. Jor every 76 cm. that the reservoir is raised the temperature of the ice and water is lowered 0°-00748 C.; and hence by noting the change of the galvano- nomeier deflexion the sensitivity of the apparatus is found, since the temperature of tlie outer junction remains constant. After taking a reading with the reservoir raised, the latter is again lowered to the standard position. The galvanometer reading generally did not return exactly to its original value. Accordingly the mean of the galvanometer readings, with mercury at standard height, obtained immediately before and after each reading with mercury raised, is subtracted from the latter, and the ditference entered as the deflexion produced 318 Mr. H. H. Poole on the Rate of by the change of pressure. The following is a typical set of readings :— | . Height of Mercury Deflexion Means | Met ee beer above Standard. | (on reversal). (see above). y ah : em. Scale-divisions. | Scale-divisions. Senlediviices 0:0 46 132°0 19-4 4°65 14:75 1200 182 | 4°85 13°35 0-0 50 110°0 17:05 | 5:05 12:0 0-0 515 139°5 208 | ol 157 0:0 d'1 | 130:0 19°55 5:2 14°35 0-0 53 | It will be seen that the numbers in the last column are sensibly proportional to those in the first. The mean of all the determinations made by this method is a deflexion of 11:21 scale-divisions per metre of mercury. Taking the density of ice as 0°91674 (Bunsen) and latent heat of water 80°16, we find that the lowering of freezing- point is 0°-00748 C. per atmosphere. Hence 1° C. corre- 11°21 x 76 100 x 00-0748 method gave 1152; so we may take as a mean that 1° C. corresponds to 1145 scale-divisions. sponds to =1138 scale-divisions. The other Tuperiments with Water in the Calorimeter. Three experiments were made with water to determine the thermal conductance of the calorimeter. The straight stem couple was used. As this is slightly shorter than the one whose calibration is referred to above its resistance is less, so that it is rather more sensitive. With this couple 1° C. corresponds to about 1220 scale-divisions. In every experiment the water in the calorimeter eventually attained a lower temperature than the junction in the ice outside. This could hardly be due to the conduction of heat down the leads to the external junction considering the precautions taken to prevent this source of error. It is probably due to the ice not being at zero even after all the precautions taken. The ice will accordingly rise in temperature, and the lag of the calorimeter will cause the water to become colder than ——- Evolution of Heat by Pitchblende. 319 theice. This is borne out by the fact that the greater the precautions taken to bring the ice to zero the smaller was this effect. In the first experiment the ice was only finely broken and then soaked in water fora couple of hours. In this case the water attained an apparently steady temperature of 10-0 scale-divisions, 7. e. about 0°:008 C. below that of the ice. In the second experiment the ice was planed to a fine powder but not soaked, and in the third experiment it was soaked for an hour after planing. The temperatures attained in these cases were —4°0 scale-divisions and —1°9 scale- divisions respectively. This effect could hardly persist long enough to greatly affect the pitchblende experiments, and would be greatly reduced by the smallness of the specific heat of pitehblende, but of course would tend to make the results too low. In each case the temperature-difference approached its final steady value along a true exponential curve. This is what we would expect from Newton’s law of cooling if we assume as an approximation that the ice is rising in tempe- rature at a constant rate. For if a be the rate of rise of temperature of ice in scale-divisions per hour, 6 differ- ence of temperature between junctions in scale-divisions, K thermal capacity of calorimeter and contents, C thermal conductance of calorimeter, K (-@ —2) =, or K dé If the calorimeter finally attains an apparently steady tem- perature 6), Cé,+ Ke=0 “= 0 KY =0(0-6), If 6’ be temperature measured from final steady value as zero so that d'=0—6), then , dg’ —K ae the same eauation as we should obtain for @ if « were zero, so =C6', 320 Mr. H. H. Poole on the Rate of that the effect of the rise of temperature of the ice is only to depress the zero, leaving the form of the curve unaltered. The solution of this equation is cr 6’ = Oye where @ is aconstant and e the base of the Neperian system \ of logarithms. We can give @ and : such values that the experimental and exponential curves coincide at any two points, and observe how they agree elsewhere. The table given shows readings obtained in the died experiment. The figures in the third column are obtained N n= 0:0375; these values are chosen so as to make the observed and calculated values coincide at the two points marked with an asterisk. by making 0)=192°8 and Time, @ (observed), | 0 (calculated), Differences, hours. scale-divisions. visions. | scale-ti scale-divisions. scale-divisions. 0-0 189-5 5 A, OS 190-9 ae). B15 | 157°3 157°0 +03 | 15°77 104-2 104-7 —~05 17°62 97-6 97-6 00x 24-67 75:0 | 745 +0°5 28-98 63-2 | 63-1 +01 39°75 41-7 | AL‘5 +02 52°52 250 24-9 +01 63:77 15-4 | 15-7 —0'3> 64:88 | 149 150 —O1 67-08 13-6 13°6 0-0 | 69°47 1 yh 12:3 00x | 73-70 10:0 10-2 —0-2 | 76°55 89 | 9:0 —O-1 : 87°12 53 54 —01 | 90:02 4-6 4-7 —O1 | 100-42 25 2-5 0-0 | 112-87 0:8 08 00: | 124-28 —0-2 —0O1 —O0'1 | 135°65 —0°8 =F —01 148°37 3 —1-2 —O1 15883 —15 | —1-4 —01 | 172:07 Ley | —16 —01 183-70 —1-9 | ong —0-2 209-20 —19 —1:8 —Oi 231°73 | 27 —19 +02 236:28 —1:8 = 1:9 +01 240°85 suey a +0°2 In this experiment the calorimeter contained 158-2 grms. of water and about halfac.c. of oil. The thermal capacity of Evolution of Heat by Pitchblende. 321 the calorimeter was estimated at about 6°0, making K=164°4 about. This gives C=6:165. The two earlier experiments | a. C=6-31 and C=6 175, so we may take as a mean ic 62, 7. e. when the calorimeter is 1° ©. hotter than the ice 6°2 calories escape per hour. ‘The oil used was intro- duced in the last experiment only, as it was feared that distillation up the neck of the calorimeter might increase the loss of heat. It was obtained by distilling common petroleum with a pear still-head and collecting a small quantity of the distillate which came over at 205° C.: about half a cubic centimetre was poured into the calorimeter after the intro- duction of the water. Since in the experiment to which the ‘ above table refers 5 = 00375 and @;=—1'9, we have a='071 scale-division per hour. Hence as 1220 scale- divisions are equivaient to 1° C. a rise of 0°-0014 C. per day in the temperature of the ice would account for the negative temperatures observed. Euperiments with Pitchblende in the Calorimeter. Five experiments were made with pitchblende. The preliminary one gave an extraordinarily high result. This was probably due to chemical action, as, though the powdered pitchblende was in a desiccator for several days, it may not have been perfectly dry, and was in an atmosphere of air in the calorimeter. This experiment was only continued for about a month altogether. In this experiment the short stem couple was used. In the next experiment the pitchblende was placed under an air-pump over H.SQ, for several days before insertion in the calorimeter. After insertion of the flexible couple and pitchblende the calorimeter was placed under the receiver of an air-pump which was exhausted and then filled with dry atmospheric nitrogen. This was repeated four or five times and then the neck of the calorimeter was plugged with vaseline. About ten days after packing the temperature became approximately steady and remained so for over a fortnight. The mean deflexion on reversal was 9°2 scale-divisions. This corresponds to 0°0080 C. On this occasion the calori- meter contained 560°7 germs. of pitchblende, so Heat Evolved 2 Go X10, | me OO pitchblende. In this experiment the ice was only finely broken and soaked, but not planed. Before the third experiment the pitchblende was kept in a Phil. Mag. 8.6. Vol. 19. No. 110. Fed. 1910. x ‘9x 10-° calorie per hour per gram of 322 Mr. H. H. Poole on the Rate of desiccator for a month, while water-cooling experiments were being made. It was then inserted in the calorimeter and a slow stream of nitrogen, prepared from NH,Cl and NaNOs, and dried by passing through two P.O; tubes, was passed through the calorimeter to expel the air and remove any free water which might still remain in the powder. This was continued for five or six hours and then the neck was plugged and the calorimeter buried in planed water-soaked ice. After about a fortnignt the temperature became approximately steady and continued so for two months, at the end of which time the experiment was discontinued. The variations of tempe- rature during this interval are plotted on the chart (No. 1). ¢ \ “ue, SCALE OMN/S/C 4 1 : mI A 7 ‘ = 4 z ! a h | | Maes. | |- te 3] 10 20 50 40 0 60 70 DAS. The mean reading was 7:1 scale-divisions, and as there were 539°8 grms. of pitchblende in the calorimeter we find the Heat HKvolution=7'1 x 10-° calorie per hour per gram of piteh- blende. In order to ascertain whether the presence of air had much effect the couple was removed from the calorimeter and the nitrogen expelled by blowing in undried air with a eycle-pump. In order to make the couple stiff enough to be again inserted and to penetrate the pitchblende it had to be enclosed in a glass tube. Otherwise no change was made in the arrangements. At the end of a week the temperature became steady and reinained so for a month. The mean deflexion was 9°4 scale-divisions. Evolution of Heat by Pitchblende. 323 After the conclusion of this experiment some water un- fortunately got into the calorimeter owing to the failure of a rubber tube. This caused the powder to cake so that it could only be removed by washing it out with water. The wet powder was placed in a steam oven, where it was left for a fortnight, being occasionally stirred to expose fresh surface. The Calender was carefully dried out, absolute alcohol being used, and the couple (which was not encased in class) and the pitchblende were inserted. A stream of carefully dried air was then passed through the calorimeter. This air before entering the calorimeter was bubbled through H,SO, and then passed through four tubes containing solid NaOH, CaCl,, and P.O; (two tubes). The current was passed for about eight hours a day for four days. Finally, the air was displaced by nitrogen passed through the same drying tubes and the calorimeter closed and buried in theice. The calorimeter contained 538°5 grms. of pitchblende. As will be seen from the chart (No. 2) the temperature at first showed a tendency to become steady at about the same value as before, but after some rather large fluctuations eventually fell Pa ne quickly to about 4°7, w herd it remained for a fortnight. The experiment was then discontinued. The mean temperature for the last 35 days of the experi- ment is 5°45 scale-divisions, but the mean of the last 18 days is only 4°7. This looked as if some chemical heating was occurring at the commencement which subsequently ceased. To test this conclusion the calorimeter, which had never been opened, was again buried in the ice three weeks after the conclusion of the last experiment. ‘The temperature variations are shown on the chart (No. 3). The mean for the last 36 days is 5°85, so that the low readings at the end of the previous experiment are not confirmed, and must probably be ascribed to some irregularity in the temperature of the ice in the neighbourhood of the outer junction of the couple. As all these experiments give values for the rate of evolution of heat considerably g ereater than was to be expected we must consider the most likely sources of error. Sources of Hrror—Chemical Action. The most obvious source of error is the possibility of chemical action which would cause too high a result. It is, however, difficult to explain the high results obtained in the foregoing pages by this means. One experiment lasted ¥2 324 Mr. H. H. Poole on the Rate of eleven weeks, and in the last two combined the calorimeter was closed for almost four months, yet there was no evidence of any diminution in the rate of heat-generation. Hence it would seem in these two cases that any chemical action occurring must be maintained by the continuous or inter- mittent entrance of air into the calorimeter, as otherwise the action would have diminished considerably towards the end. Some estimate can be made of the possible effects of such a cause. When the calorimeter was opened at the close of the last experiment it was inferred from the appearance of the vaseline plug that air might possibly have been forced into the calorimeter by barometric changes and also during the initial cooling, but that absolutely no liquid water could have got in. Also, owing to the plugging arrangements before described, diffusion would seem to:be negligible. B examination of the barograph charts for the period of the last experiment (August 31st-October 19th, 1909) it was found that the total rise of the barometer amounted to about 5 inches. The volume of the calorimeter is about 160 c.es., and of this the pitchblende must occupy at least half, so the gas in the calorimeter may be taken as occupying about 80 c.es. Hence barometric changes would have introduced about 13:3 c.cs. The initial cooling from 15° C. to 0° C. would draw in about 4:4 c.cs., making a total of 17°7 c.es. of air to account for the excessive heat generation. Now, as will be shown further on, the theoretical value for the rate of evolu- tion of heat is 4°4x10~° calorie per hour per gram of pitchblende, i. e. 23°7 x 10~* calorie per hour for the whole of the contents of the calorimeter. The rate of evolution : 6°2 x 5°85 observed in the last experiment was —~—~—~ —31-7x 10-3 calorie per hour. 1145 The difference to be accounted for is therefore 8 x 10~° calorie per hour, or a total of 9°4 calories during the 49 days that the experiment lasted. Hydration and oxidation seem to be the only likely actions. Now one cubic metre of saturated air at 0° C. contains 4°835 grams of water. Hence weight of water-vapour entering calorimeter during the last experiment amounts to A835 x 17°7 x 10-6 or 85°5 x 10-° gram ; even if each oram of water yielded 840 calories, as much as it would liberate by combining with CaQ, the total heat generated would only amount to ‘072 calorie, so heating due to hydration seems to be negligible. As regards oxidation, 3°72 c.cs. or 5°3x 10-* orm. ae oxygen entered the calorimeter during the experiment. Let Sa se a ee Evolution of Heat by Pitchblende. 325 us assume that all the oxygen entering the calorimeter is absorbed ; then to generate the required 9:4 calories, the evolution would have to be nearly 1800 calories per gram of oxygen. This is almost as much as is generated by the union of one gram of oxygen with one gram of sulphur. As uraninite is essentially an oxide the formation of a higher oxide is the only likely occurrence, hence this evolution seems excessive; moreover, our assumption that all the oxygen enters into combination is an extreme one. A rough analysis of air which had lain over some powdered pitch- blende in a closed jar for two months indicated that the absorption of oxygen was very small, and indeed could not be detected with certainty. It is worth noting in this connexion that in the long ex- periment (chart No. 1) which gave a higher result than either of the last two, there was no evidence of the entrance of air as the vaseline plug was absolutely intact at the end. Irregularities in the Temperature of the Ice. The ice is frozen from the Dublin water-supply. The total solids in this water amount to about one part in 23,000, but the greater part consists of colloidal substances which would not much affect the freezing-point. The chief salts present are chlorides, mostly of Na, Ca, and Mg, there being 0°70 part of Cl per 100,000 by weight. If we assume that NaCl is alone present in amount equivalent to the total Cl we shall over-estimate the lowering of freezing-point, which, on this assumption, will be 0°-00074 C. corresponding to a deflexion of 0°85 scale-division, The concentration of salts in the last parts frozen might cause larger variations. This may explain some of the irregularities observed, as the purity and hence the temperature of the water in contact with the outer junction may vary from time to time, but it is exceedingly unlikely that the outer junction was in every experiment in a region considerably below the average temperature. Difference of Temperature beween Centre and Surface of Pitchblende. If » be internal radius of calorimeter, k be thermal conductivity of pitchblende, q be quantity of heat evolved per second per unit volume, 6 be difference of temperature between centre and surface when a steady state has been reached, we readily find that » 4 ~, 326 Rate of Evolution of Heat by Pitchblende. From the known volume of calorimeter we obtain 7=3'4 cms. nearly. Using the mean value found for the heat evolution per gram of pitchblende we find g=5°7x 10-*. The con- ductivity of powdered pitchblende is probably not less than that of dry sand, about 910-4. Putting in these values we obtain @=1'2 x 10-4 of a degree or about 0°15 scale-division. Hence, even if the junction were at the centre of the pitchblende, the error would be very small. Conclusion. The three experiments for which the temperature charts are given seem much the most reliable, as in the earlier experiments the importance of having the powder thoroughly dry had not been realized, nor were the experiments continued long enough to ensure that heating caused by initial dampness and oxygen had ceased. The same remark applies to the experiment in which damp air was intentionally introduced. It will be observed that of the three charts No. 1 is much more regular than Nos. 2 and 3. From chart No. 1, Heat evolution 7:1 x 10-5 calorie per hour per grm. ” » No.2,» ” 5°45 x 10-5 ” . Oe. Be “4 5°85 x 10-5 % We may take as the mean of these three values that each gram of pitchblende evolves 6°1 x 10~° calorie per hour. This result is surprisingly high. An analysis of a sample 7 2? ” 7) of the pitchblende used indicated the presence of 64 per cent. of uranium. According to Boltwood each gram of uranium in equilibrium is associated with 3:4x 107" of a gram of radium. The accepted value for the rate of evolution of heat by the uranium and products in equilibrium with one gram of radium is 5°6x10-? calorie per second. According to these figures each gram of pitchblende should only evolve 4-4.x 10-° calorie per hour. However, the figure 5:6 x 10-? calorie per second for total heat associated with each gram of radium is based on the assumption that one gram of radium in equilibrium with its immediate products evolves 110 calories per hour; the latest determination of this quantity by von Schweidler and Hess (Le Radium, Feb. 1909) gives 118 calories per hour, which would somewhat reduce the discrepancy. It is hoped to repeat the experiment in a slightly modified form, using a solid block of pitchblende. In conclusion, I wish to express my gratitude to Dr. Joly, at whose suggestion the work was carried out, for his kindness and valuable advice during the progress of the research. Physical Laboratory, Trinity College, Dublin, October, 1909. Sere } XXXII. Pleochroic Halos. By J. Joty, P.RS* 7 HEN recently examining a section of a greisen from Altenburg, Saxony (supplied to me by Krantz of Bonn) I noticed that some of tbe pleochroic halos scattered through the Muscovite presented an appearence which, so far as J am aware, has not hitherto been referred to by petrologists. It might, indeed, escape the notice of any one unaware of its significance from the point of view of the theory which. ascribes to these halos a radioactive origin. I may add that I have several times sought for the structural feature now to be described, but hitherto in vain. The appearance referred to is that of an inner and an outer halo of very different densities; so that an inner darker sphere and an outer less deeply coloured shell are clearly indicated. In the field the central dark area is fairly uniform save along the outer margin, where it diffuses rapidly, being immediately succeeded by an area, fading somewhat—almost imperceptibly—outwards, but wonderfully sharply defined at the extreme boundary. The radial width of this corona is about one half the radius of the inner disk. It is considerably fainter in colour. Its diameter is easily measured by a mi- crometer eyepiece. The diameter of the inner sphere is not so well defined, but by careful focussing very consistent measurements are obtainable. There is no doubt as to the closely approximate truth of these measurements. ‘Trials were made to see by how much the travelling line of the micrometer could be displaced on the margin without sensible error; it was found that a very few scale-divisions limited the permissible displacements. There is unfortunately great difficulty in reproducing photographically the appearance presented. This is because the halo is imbedded in a medium far from flawless. When under observation it is easy to focus through these flaws ; but in the photographs they bring in lights which confuse the margins. The following are the readings taken on three such halos. The readings are taken in different azimuths, which are defined by colons; and are the diametrical lengths in small divisions of the micrometer head. A. Halo seen on a cleavage section. Corona 84, 84, 83 : 83, 85,85. Mean 84, Halo o1, 92: 53, 54,52. Mean 52°4, * Communicated by the Author. 328 Prof. J. Joly on Pleochroic Halos. B. Halo on a basal section. | Corona 83, 81, 80: 84, 83 : 85, 83: 84, 86, 84. Mean 83°3. Halo 50, 54,52 : 54,56,56°: 59,56,50 +: .5955,00. Mean aa ae C. Halo on an approximately basal section. Corona 80, 81 : 82,81 : 80,80, 79. Mean 80:4. Halo 52,49: 50,48, 51: 49,51, 52. Mean 50-3. That no unconscious bias affected these readings will be evident when I add that the whole of them were obtained before the calculations given below were made. With the same optical arrangements it was found as the mean of several readings that an engraved one-tenth of a millimetre was traversed by 122 small divisions of the head. Accordingly, we have in millimetres for the radial measure- ments of the halos and their coronas :—- Corona. Halo. Calc. radius of Halo. ate EARS 0:034 0:022 0°023 1 Eikeg lan Das er NA 0:034 0°023 0:023 Oe ae A et 0-033 0-021 0:022 The inference that the complex structure of these halos finds an explanation in the varying ionization ranges of the a rays of the uranium-radium family is based on the following considerations. The effective ranges of these rays,as measured by Ruther- ford and later by others, will account for the measurements made above. The most accurate determinations of the ranges are probably those of Bragg (Phil. Mag. Sept. 1905). Brage’s results are as follows :— Radium css, Pear ei LIEN 3°50 ems. Emanation or Radium A. 4°23 ems. Radium A or Emanation. 4°83 ems. Rada 2 a eee 7-06 cms. To these measurements may be added those of Levin (Phys. Zeit. 1906, p. 461) for Ral, 3°85 ems. and by Allen (Phys. Rev. 1908) for uranium 3°5 ems. It is apparent that a particle of radium in equilibrium will, in air, be surrounded by a sphere of very intense ionization which will be limited by a radius of 4°83 cms., and beyond this for a further distance of 2°23 cms. there will be a shell within which the ionization is comparatively feeble, pro- bably only of one-sixth or one-seventh the intensity of that prevailing within the inner sphere, and of course still further diminished by the divergence of the rays. If we calculate from the extreme radial dimensions of the Prof. J. Joly on Pleochroic Halos. 329 corona the radius of the inner darker region, according to the ranges given by Bragg for Ra C and Ra A or emanation, we find for the inner halo the dimensions given in the third column, The value of this test is of course independent of the absorption or stopping-power of the muscovite, the figures being simply comparative. It would seem probable, from the agreement of the figures, that the corona is accounted for by the very co onsiderable difference between the ranges of 2 rays from Ra C and from the other members of the radium family. We may also estimate the theoretic radii of the outer and inner halos by applying Brage’s law, that the ranges are proportional to the square roots of the atomic weights. Taking the formula of muscovite as 2H,0, K,O, 3A1,03, 6Si0,, I find 4:2 is the average square root of the atomic weight. or air the corresponding value is 3°79. From these data, and assuming the density of muscovite to be 2°8, we get for the rays of 7:06 ems. in air, a range in muscovite of 0°0337 mm., and for rays of 4°83 ems. in air a range in muscovite of 0°0230 mm. It will be seen that the agreement is very exact. It would be even closer if a slightly higher density was assumed for muscovite ; which would be quite permissible. Measurements made in other rock-sections have given me higher values for the limiting radial dimensions of halos. The possibility that thorium may be responsible for the halo, in some cases, must be borne in mind. I have had cordierite exposed under specks of thorianite now for nearly four months, as yet without definite coloration, but taking into account the feebleness of the radiation compared with what can be commanded in similar experiments with radium, this result is not surprising. I find that the a rays of ThC, which have been found by Hahn (Phys. Zeit. 1906, p. 461) to possess a range in air of 8-6 ems., would, on Bragg’ s law, account for a halo in muscovite of 0-042 mm. radius. It may be asked why all pleochroic halos due to radium do not show the differentiated structure described above. The answer is, I think, that in very old halos, or in halos due to very active nuclei, this structure becomes obliter ated ; for somewhat the same reason as an over-exposed photo- graphic plate loses contrast and detail. The effects of the more intense inner ionization probably reach a maximum, or at least do not continue to visibly increase at the initial rate, Finally, the effects in the corona must accumulate till there is sensible equality in density. In the greisen in which the complex halos oceur there are also found halos in the final 330 Prof. J. Joly on Pleochroie Halos. stages, intensely dark to their extreme radial limits ; and again others, surrounding very minute specks, where the earliest development as an undersized border of brown coloration is apparent. Both the age of the rock and the intensity of the source of radiation affect the development. A greisen is an alteration-product and halos in it may be much younger than the original granitic mass, for not only the profound mineral changes but any excessive heating will destroy original structures of the kind. The complex halos described above evidently contain no effective quantity of thorium. Thus Hahn (loc. cit.) gives for the ranges in air of the e rays of thorium derivatives the values in centimetres: Radio-thorium 3°9, Th X 5:7, Emana- tion 5°5, ThB 5:0, ThC 86. Such rays if present would obliterate the structure observed ; nor would they agree with the measured maximum radius considered in connexion with Brage’s law. In studying the ordinary pleochroic halos, petrologists would seem possessed of a means of determining the average atomic weight of the containing mineral: the alternatives, in fully-developed halos, being limited to calculations based on the ranges of the rays of RaC and The. One other point remains to be noticed. It would be expected, I think, that a spherical structure derived in the manner of a halo and viewed in plan should not be very definitely defined at its extreme boundaries. The boundary of the corona is, however, when carefully focussed found to be very perfect. (f am not sure that it is not even some- what intensified.) Might not this definition be ascribed to the known fact, appearing so clearly in Bragg’s curves, that the ionization of the « ray increases in intensity just before it becomes finally ineffective ? The halo, in short, approxi- mately represents a projection, on the spherical radius, of a Brage curve for the uranium-radium family of @ rays ; intensity of colouring replacing the measured effects in the ionization chamber. That the effects visible in the halo are due to ionization and not to mere storage of helium appears to be set beyond doubt by the observed limit of the rays in the mica agreeing with the observations on the ionization ranges as obtained in the laboratory. A mere accumulation of helium should exhibit considerably larger radial dimensions. The beautiful artificial halos obtained by Rutherford (Phil. Mag. Jan. 1910) would, probably, reveal structural particulars similar to those described above, if examined at a sufficiently early stage of their development. [i B3Mb ] XXXIII. Notices respecting New Books. Cours de Physique. Par H. Bovassn. Siath Part: Ltude des Symétries. Paris: Librairie Ch. Delagrave. ee sixth part of this important text-book of Physics follows somewhat unfamiliar lines. Beginning with a geometrical study of kinds of symmetry, including the main facts of crystallo- graphic forms (142 pp.), it proceeds to a detailed study of the physical properties which are linked with them. These include the elastic properties, thermal dilatation, electric and magnetic relations, the Hall phenomenon, piezo- and pyro-electricity, phe- nomena of double refraction, pleochroism, absorption, the pheno- mena of Kerr, the anisotropy of liquids, magnetic rotary polari- zation, and liquid crystals (pp. 143-408). The treatment throughout is mathematical, and no use is made of modern conceptions based on electronic theory. The matter dealt with loses in consequence a great deal of the vividness which a more physical treatment would have givenit. Though we regret this, yet it must at the same time be asserted that Professor Bouasse is producing a masterly treatise which supplements the less mathe- matical text-books of physics. Results of Observations made at the Coast and Geodetic Survey Mag- netic Observatory at Cheltenham, Maryland, 1905 and 1906, Washington : Government Printing Office. 1909. T'HIs gives in 110 quarto pages the results derived during 1905 and 1906 from the magnetographs at the central magnetic station of the U.S. Coast and Geodetic Survey. The monthly tables give hourly values from the declination, horizontal intensity and vertical intensity curves, with the daily maxima and minimaand their times of occurrence, also the mean value and the range for each day. Inequalities are derived from the ten quietest days of each month. A. table is given of the beginning, duration, and relative intensity of the principal disturbances, numbering 40 in 1905 and 29 in 1906. Only three storms, commencing respectively on March 1, March 31, and November 14, 1905, reached the highest grade, 4, of disturbance. Some of the disturbed curves are reproduced. The Observatory is also provided with Bosch-Omori seismographs recording N-—S and E-W earth movements. A table is given including particulars of 39 earthquakes recorded in 1905 and 43 in 1906. — An Introduction to the Science of Radioactivity. By C.W .Rarrery. London: Longmans, Green, & Co. 1909. Tus claims to be aconcise and popular account of the properties of the radioactive elements. We consider that the author is justified in his claim, and that his book will be found useful to those who do not profess a first-hand acquaintance with the subject. Many readers no doubt will find the treatment too 332 Geological Society :— slight for their purpose ; if so, the book is not intended for them, but for those only who, while wishing to gain a general idea of the important work done during the last dozen years, find the more authoritative popular accounts too advanced for them. XXXIV. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 208. ] May 12th, 1909,—Prof. W. J. Sollas, LL.D., Sce.D., F.R.S., President, and afterwards Dr. J. J. H. Teall, M.A., F.R.S., Vice-President, in the Chair. ’ i ‘HE following communications were read :— 1. ‘ The Hartfell-Valentian Succession around Plynlimon and Pont Erwyd (North Cardiganshire). By Owen Thomas Jones, A RIN ths OME UALG cP In this paper the author deals with the stratigraphical suc- cession and the geological structure of an area of about 40 square miles, lying in the hilly district about 12 to 16 miles east of Aberystwyth. In an historical introduction the work of previous observers, including Sedgwick, Ramsay, and Walter Keeping, is dealt with. The rocks within the district are divided into three Stages, which are further subdivided into Groups and Zones, as follows :-— ( (ec) Rhuddnant A Rhuddnant Grits. | Group. 1. Rhuddnant Shales. C. YstwyTH } (6) Myherin Te Blaen Myherin Mudstones. STAGE. | Group. 1. Dolwen Mudstones. E (a) ree Sure : Mudstones with thin grit-bands. ( (3. Flags and shales. (Zone of Monograptus sedgwicki.) Castell | 2. Shales and mudstones. (Zone of Cephalograptus (ec) Caste : Pp grap 1 | | | : Group. cometa.) rs . Mudstones and shales. (Zone of Monograptus = L convolutus.) = (4. Black shales and mud- ( ¢. Leptotheca-band. A IR. Pont : stones. J y- Magnus-band. = : | (Zone of Monograptus | B. Triangulatus-band. > Erwrp 4 (5) Rheidol communis.) La. Triangulatus-var. band. Group. a S. Flags and black shales. (Zone of Monograptus STAGE. é cyphus, s. 8.) 2. Flags with thin shales. (Zone of Monograptus 2 ray Sp. nov. U1. Flags and shales. (Zone of Monograptus sp. nov.) (2. Flags, shales, and grits. (Zone of Cephalograptus (a) Eisteddfa 2 ? acuminatus.) Group. | 1. Flags with thin shales. (Zone of Glyptograptus "4 [ C persculptus.) 5 ( (c) Brynglas Group. Mudstones. 2 A. PLYNLIMON | (6) Drosgol Group. Grits, conglomerates, and mudstones. a | STAGE. 9 (a) Nant-y-moch Flags with thin shales. (Zone of Dicellograptus <\ L Group. anceps.) on flartyell-Valentian Succession around Plynlimon. 338 The Plynlimon Stage is developed in the northern part of the district, between Plynlimon and Pont Erwyd; the Pont Erwyd Stage along the two valleys of the Rheidol and the Castell, which converge near Pont Erwyd village; while the Ystwyth Stage is developed on the plateau-like tract extending from the Castell Valley to the Ystwyth Valley. The stratigraphical succession is demonstrated by a series of sections and traverses across various parts of the district, and lists of fossils (mainly graptolites) collected from the various zones are given. The paleontological evidence is in entire accord with the stratigraphical evidence. The structure shows many points of interest, and is clearly brought out by the mapping. Three types are dealt with— (1) folding, (2) strike-faulting, and (3) normal faulting; but the first is predominant. The rocks are fulded into a primary anticlinal fold or anti- clinorium, with a southerly ‘pitch’ of 10° to 15°. This primary fold is composed of a number of secondary folds of a symmetrical type, with axes ranging nearly due north and south and pitching as above. Occasionally, the secondary folds are complicated by smaller folds or crumplings. The effect of the plication-structure on the rock-outcrops, and especially on the topographical features, is dealt with in some detail. The latter are shown to be dependent in an important degree upon the pitch, and to a much smaller degree upon the dip of the rocks. Strike-faults play a minor role in the structure, and their effects are of little importance ; they have not been observed to carry mineral deposits. The normal faults of the district are of greater interest. They range in an east-north-easterly direction, and nearly always carry sulphidic ores of lead, zinc, or copper. They appear to be quite independent of the folding, and behave generally as if they were of later date. A brief account is given of the district lying on the western limb of the anticlinorium between Pont Erwyd, Devil’s Bridge, and Aberystwyth. Evidence is given for assigning to the ‘Aberystwyth Grits’ of earlier observers a position much higher in the geological sequence than has hitherto been attributed to these ‘ Grits.’ The paper concludes with a tabular list of fossils, correlation- tables, and a description of two species of graptolites of zonal importance. 2, ‘The Geology of the Neighbourhood of Seaford (Sussex).’ By James Vincent Elsden, B.Se., F.G.S. This paper illustrates the application of zonal methods to field- geology in a Chalk area. It deals with a portion of the South Downs lying between Eastbourne and Newhaven. The inland outcrops ‘of ‘the uppermost zones of the Chalk are mapped. In tracing the boundary-lines, fossil evidence is alone relied upon, 304 Geological Society :— lithological characters being found untrustworthy. Certain struc- tural features of this area are thus brought to light and discussed. On the east of the Cuckmere River, the beds examined are found to be nearly horizontal. On the west side they are bent into a sharp uniclinal fold, striking east and west. Scaford Head repre- sents a remnant of this fold, the westerly extension of which is destroyed by marine erosion. The low ground between Seaford and Chyngton occupies the trough of the fold, from which dip- slopes rise gently to the north and sharply to the south. There is thus formed a true synclinal depression striking westwards, with a low pitch, into the English Channel. The complete disappearance of the fold on crossing the Cuckmere cannot be satisfactorily explained by the normal process of dying-out, the distance being too short for so rapid a recovery. It is suggested, therefore, that a transverse fault may exist beneath the alluvium of that river, thus, perhaps, accounting for the narrow gorge at Cuckmere Haven. The top of the zone of AMarsupites testudinarius, which lies in an approximately horizontal position at about the level of the 300-foot contour on one side of the river, sinks below sea-level in the trough of the fold on the other side. The fault, if it exists, seems to die away rapidly northwards, since no trace of it has been detected higher up the valley towards Alfriston. This feature would be expected from the nature of the movement. The relation of the Seaford fold to the main fiexures of the South Coast is considered. and it is suggested that this fold represents the eastern termination of the structural area known as the Hampshire Basin——-being, in fact, a continuation, en échelon, of the Purbeck—Isle-of- Wight system. Certain existing physiographical features are ascribed to the influence of this flexure, which facilitated the retention of the Eocene cover in the synclinal hollow thus formed. A result of mapping the outcrops, also, has been to prove that the numerous dry valleys in this area are not of the nature of ‘sinks’ or ‘dolinas,’ but are true valleys of surface-erosion. Attention is drawn to certain features of these valleys. A brief comparison is made between the fossils of the inland exposures and those of the cliff-section, the most notable difference being the evidence in the former of a Conwlus-band at the top of the zone of Micraster cor-anguimum. May 26th.—Prof. W. J. Sollas, LL.D., Sc.D., F.R.S., President, in the Chair. The following communications were read :— 1. ‘The Cauldron Subsidence of Glen Coe and the Associated Igneous Phenomena.’ By Charles Thomas Clough, M.A., F.GS., Herbert Brantwood Muff, B.A., F.G.S., and Edward Battersby Bailey, B.A., F.G.S. The succession of volcanic rocks in Glen Coe is mainly a series of lava-flows, of which there are three types—augite-andesite, The Cauldron Subsidence of Glen Coe. aa0 hornblende-andesite, and rhyolite. Agglomerates, tufts, and sedi- ‘ments form but a small portion of the sequence. The Lower Old Red Sandstone age of the rocks is proved by the occurrence of plant-remains in shales at the base. The sequence is divisible into groups, which are not, however, persistent over the whole area. Each group may contain different types of lava, which interdigitate one with the other. It is probable that the district was supplied from more than one centre, the foci being independent as regards type of material erupted, although their periods of activity overlapped. The volcanic pile with patches of conglomerate and breccia at the base rests upon an uneven floor, evidently a land-surface, of the Highland Schists; and further, the eruptions appear to have been subaérial. The cauldron subsidence, which let down the volcanic rocks and the underlying schists some thousands of feet, affected an area roughly oval in shape and measuring 8 miles by 5. It is delimited by a fault, the hade of which is sometimes normal, sometimes reversed. ‘The lavas abut against the fault-plane, and are frequently tilted up into a vertical or even overturned position nearit. Further evidence for the boundary-fault is afforded by the displacement of the outcrops of several members of the Highland Schists. A zone about a mile wide lying immediately outside the boundary-fault has been invaded by a number of masses of granite and porphyrite, spoken of collectively as the ‘fault-intrusion.’ Where the fault-intrusion comes against the boundary-fault, it is chilled, and its margin is a smooth even plane, while its junction with the schists is highly irregular. The fault-intrusion causes contact-alteration in the schists outside the fault; but the voleanic rocks and schists inside the fault are scarcely affected by it. The movement along the fault-plane has caused intense shearing and crushing, leading finally to the production of ‘flinty crush-rock.’ The latter owes its characters to extreme trituration, probably accompanied by incipient: fusion due to frictionally generated heat. In certain places, flanking faults older than the main boundary- fault, and accompanied in each case by a mass of igneous rock on their outer walls, are found near the main boundary-fault, and parallel to it. The subsidence therefore took place in at least two stages. After subsidence in the cauldron had ceased, a multitude of dykes, mainly porphyrites, but including quartz-porphyries and lamprophyres also, were intruded along lines trending north-north- eastwards and south-south-westwards. It is shown that the dykes add their width to that of the country traversed, and that they have their focus within the Etive granite-mass. They have a parallel, not a radial, arrangement ; and a vast majority are concen- trated into two swarms, which extend north-north-eastwards and south-south-westwards from the granite. The authors discuss several questions arising out of their con- clusions. With reference to the relative age of the faulting and 336 Geological Society. the fault-intrusion, it is concluded that they are contemporaneous, and that the uprise of the magma may be considered as comple- mentary to the subsidence. The Glen Coe subsidence is compared with the subsidence which took place in the Askja caldera in Iceland in 1875. It is considered probable that the lobe of the Cruachan granite, which invaded the sunken area ot Glen Coe, was admitted by a further subsidence of part of the rock-mass within the cauldron, and that the granite occupied the-cavity thus formed. A theory is advanced that the Cruachan granite-mass also originated in a subsidence of the schists in place of which the granite is now found, the magma welling up the sides of the sinking mass and filling in the subterranean cauldron. Evidence for this is adduced in the form of the intrusion ; in the presence of a curved, flanking fault; and in the shearing of carly consolidated parts of the mass. The dykes point to the operation of regional tensional stresses which, co-operating with the pressure of the magma, opened parallel north-north-east and south-south-west fissures. It is suggested that the concentration of the dykes was due to the pasty condition of the internal parts of the granite-mass, which yielded to the stresses and caused a localization of fissures in the surrounding solid rocks. After the intrusion of the majority of the dykes, a further sub- sidence within the Cruachan granite-mass admitted the central . core of the Starav granite. The principle underlying the interpretation of the phenomena described is the upward movement of igneous magmas in correlation with complementary subsidence of portions of the earth’s crust. 2. ‘The Pitting of Flint-Surfaces.” By Cecil Carus-Wilson, F.R.S.E., F.G.S. Regular pittings of uniform size are occasionally seen on flints which have been exposed to the weather. ‘They have been referred to by various authors, but no satisfactory explanation of their origin has been given. ‘The author procured some interesting examples occurring in a recent deposit near Folkestone. This deposit is formed of materials which appear to have been washed down from the adjacent chalk-hills. The flints appear to have been derived from the sandpipes in the Chalk: their surfaces are much decomposed. The removal of the colloid silica has rendered them very porous, and they absorb a good deal of water. It is believed that the pittings are due to mechanical action. Obser- vations and experiments carried out by the author indicate that such markings cannot have been produced by blows, nor by any process of desiccation, and that the freezing of the absorbed water seems to be the only satisfactory explanation to account for the various details of the phenomenon. Phil. Mag. Ser. 6, Vol. 19, Pl. IIL. 60° 50° TON cd LL. SY, Tess LOR 40° SS 50° One ee FOR WESTONCELL. al ite ig ae aba ee ag i wai” mi Ea 10° | ee wc ie PEPCENTAGE OF CAOMIU, : [ie ie 16 20 Phil, Mag. Ser, 6, Vol. 19, Pl. III. 100840 SMITH. Fie. 2. in a 103340 = + a =e a ae a 15° 1.03840 | \ | Td H ° | rca =] "| 25 1.03440 ESS | . =r ! ia 102340 . Res l | r SJ] | 40” = L bey th | Y 45° 102640 aoe oni f FOR rv) ‘ | WAlsTon ckzL. | Ky ala S 10/840 a Alea: & G rozeaa FS LI (Gee ES | | STON CF LL o || ) 2 v & 101840 50° }— +} Taal I] 101540 We v 9 — S NS }h LS 4 en 101940 = ; Yh? supper ie § WESTONCELL 20° = 1-01040. = || 100640 - ERPRORE HSS | Z 0.99840 SJ] =| 0° _|PERCANTAGE OF CHOMIL, © Ty 20° 30° 40° 50° 60° (eae 5 = ere eal 0 4 8 le 16 20 100340 099840 PERCENTAGE OF CAOMIUM. asc ROYDS. Fig. 1.—Sn. Capacity ='0306 mfd. Period=3:0x 107° sec. A=initial air-discharge. B=a second air-discharge of short duraticn. C=air-discharge of long duration. Lower electrode initially positive. Phil, Mag. Ser. 6, Vol, 19, Pl. IV. F1q. 2.—Lower electrode Hg, upper electrode Pb. Capacity =-0306 mfd. Period =3:0X 107° see. Upper electrode initially positive. Fie. 4.—(Positive photograph) Mg. Capacity =0206 mfd. Period=9°5X10~° sec. Lower electrode initially positive. Fic, 3.—(Positive photograph) Sn. Self-induction = "0337 henry. Fie. 5.—Ca. Capacity ='0306 mfd. Period =1°65 x 107° see. Upper electrode initially positive. é AY on, 7 s iy " S Prmeg ew a f ’ ¥ : 7] pe td F cin r Fo sevay . THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES. ] MARCH 1910. XXXV. Homogeneous Corpuscular Radiation. By CHARLES A. SapDuer, M.Sc., Oliver Lodge Fellow, University of Liverpool*. [Plate V.] ae investigations upon the nature of Réntgen rays various experimenters observed that when these rays were allowed to fall upon metal plates secondary radiations were emitted. From an element of low atomic weight this radiation was of a type indistinguishable from the exciting beam ; it carried no charge and had the same penetrating power as the primary. From an element of higher atomic weight a radiation carrying no charge, but considerably less penetrating than the primary beam, was produced{. But in addition to the secondary radiation of these types there was found to exist in some cases a radiation which was completely absorbed in a few millimetres of airt. This radiation was found to consist of negatively charged particles identical in character with the negative corpuscles which carry the current in a discharge-tube. : ‘ It was soon found § that the intensity of this corpuscular radiation depended both upon the “ hardness” of the primary beam employed and upon the nature of the metal plate upon which it fell, but the results obtained by different investi- * Communicated by the Author. 1 Barkla, Phil. Mag. June 1906, pp. 812-828. } Townsend, Proc. Camb. Phil. Soc. x. p. 217 (1899). § See publications by Perrin, Sagnac, Langevin and Townsend, and others on corpuscular radiation excited by X rays. Phil. Mag. 8. 6. Vol. 19. No. 111. March 1910. Z 338 Mr. C, A. Sadler on gators were very inconsistent. This evidently is only to be expected, since the primary beams used in all these investi- gations were necessarily heterogeneous, and it is impossible to determine which constituents of these various beams were chiefly concerned in producing the corpuscular radiation measured in any particular instance. While investigating the secondary radiation of the Rontgen type emitted by metals exposed to a primary beam from a Rontgen tube f, it was found that the radiation from each element whose atomic weight lies between those of chromium (52) and tin (119) was quite characteristic of that element, and moreover was very homogeneous. The coefficients of absorption in aluminium of these homo- geneous radiations have a wide range of values as will be seen by reference to the following Table. The coefficient » is defined by the equation fi, Ca where I, is the initial intensity of the radiation and I the intensity after passing through a plate of aluminium of thickness # cm. TABLE [. Hlement from which Atomic weight \, ie oe _ the secondary homogeneous of ia sia mieten: radiation is obtained. element. “ie Sey aluminium f. Chiromuwiy, §, 8.54 chc. canes 52 367 MOM Dc sc ce mats BANS gil Ss: 559 239 Comal MEI A eee 59:0 193°2 MWieice! Seas raion oe Buse 61:3 * 159°5 COPEL io. wesnsnictes spews pote 63°6 1289 TAN ge aa lat SERRE re Ne Nar. 65°4 1063 PANISERUC Hii fo08 fo ccoeceteancaesees 750 60°7 D@leAVUEA: ibis a daielee due cw eigcls dbame 79:2 51:0 SUE RC HCCI C00 | Bae a RR aR 876 35'°2 DMolybdenum , ....0...ceccsns0e- 96-0 12% thodum set siects dee eeeaee 103°0 8:44 SS LIGRCEN reese er acre ren MPa TO 1079 6°75 TT ee ae eae RA oe et 119°0 4°33 * Note the atomic weight of nickel is here given as 61'3 instead of the usual value 58°7, for reasons explained in a paper on the atomic weight of nickel (Barkla and Sadler, Phil. Mag. September 1907, and in later papers). t Some of these values are quoted from a paper by Barkla and Sadler, Phil, Mag. May 1909. Perhaps itis hardly necessary to point out the enormous + Barkla and Sadler, Phil. Mag. Oct. 1908, pp. 550-584. Homogeneous Corpuscular Radiation. 339 advantages to be gained by the substitution of these homo- geneous secondary beams for heterogeneous primary beams. In a course of some previous experiments *, in which these homogeneous radiations were used to excite tertiary radia- tions in metals belonging to the chromium-tin group, the emission of the tertiary Réntgen radiation was found to be governed by the following laws :— (1) With a given substance as radiator, its characteristic radiation is only excited by those secondary beams which are more penetrating than the tertiary radia- tion characteristic of the substance. (2) When the secondary beam is only just more penetrating than the tertiary, the intensity of the latter is small, but, as the secondary beam becomes more penetrating, a very rapid increase in the intensity of the tertiary radiation to a maximum takes place. (3) As the secondary beam becomes more penetrating still the intensity of the tertiary radiation decays as a linear function of the ionization produced in a given volume of air by the secondary beam. (4) When these secondary beams are absorbed by thin sheets of metals from the group chromium-tin, a large increase in the absorption takes place when the secondary beams become more penetrating than the radiation characteristic of the absorber,. this increase in the absorption being intimately connected _ with the emission of tertiary radiation by the absorber. The fraction of this increase in the absorption of the energy of the secondary beam which is re-emitted as tertiary radiation is not constant, but decreases.as the secondary beam becomes more penetrating, slowly at first and then more rapidly when a relatively ‘penetrating secondary beam is used. eae The results obtained with these homogeneous beams ap- peared to be sufficiently regular to warrant an attempt being made to find in what manner the emission of corpuscular radiation by a substance depended upon the exciting radia- tion. It was thought probable that when the intensity of the tertiary Réntgen radiation decreased as the secondary beam became more penetrating, an increasing fraction of the energy of the secondary beam absorbed by the tertiary radiator would be expended in corpuscular radiation. : — To investigate this point was the main object of the present research. : * Sadler, Phil. Mag. July 1909. Z 2 340 Mr. C. A. Sadler on ' ¢ It was desirable that the homogeneous beams used should be of sufficient intensity to produce an easily measured ionization ; for this purpose an arrangement of the apparatus somewhat similar to that employed while investigating the tertiary Réntgen radiation was adopted *. Fig. 1. Lead screen. The experimental arrangements are shown in plan in fig. 1 drawn to scale. A rectangular tube of lead 5 cm. long lined with aluminium 2 mm. thick, was fitted into the walls of a lead screen which surrounded the Roéntgen ray bulb. ‘The rays proceeding from the anticathode K were allowed to fall upon the secondary radiator placed at R,. The secondary rays from R, entered the aperture A of the ionization-chamber J, passed through the thin aluminium window B, and fell upon the tertiary radiator R,. This plate was insulated from the itonization-chamber and connected to Es, a sensitive electroscope of the type used in the experiments on the tertiary Rontgen radiation. A and B were circular apertures in the ends of a tube sliding with a telescopic fit within another tube which formed the outer casing of the ionization-chamber J. This chamber was mounted upon an insulating stand of ebonite. The diameter of the inner tube was 4°8 em., while the diameters of the apertures A and B were each 2‘7 cm. The end B of the inner tube was ground accurately perpendicular to its axis. Thin brass bars in the form of a cross were soldered * Sadler, Phil. Mag. July 1909, Homogeneous Corpuscular Radiation. 341 into the aperture B flush with the surface of the end plate. A thin sheet of aluminium (003 cm.) stretched as tightly as possible was fastened over this aperture with Chatterton compound. The pressure inside the ionization-chamber was kept a little higher than atmospheric, and under these con- ditions the aluminium window was as nearly as possible flush with the plane of the end plate, even when the window was only 1 mm.away from the plate R, and subject to its electro- static attraction. The position of the inner sliding tube relative to the outer case could be determined by means of a millimetre scale engraved upon the surface of the inner tube, while the outer casing carried a suitable vernier scale. The diameter of the circular tertiary radiator was approximately 4°2cm. The ionization-chamber was charged to a poten- tial of +240 volts, a potential high enough to ensure a saturation current from the case to the insulated plate in all Fig. 2. Fig. 3. co) os Ba bs, Ga “>gnitar ii ir Noma YY, j WMA YLUMW: LN cases. A portion of the secondary rays from the radiator R, passed through the window of E,, an electroscope of the ordinary Wilson type. ‘The ionization produced in this electroscope served as a measure of the intensity of the secondary beam. In figs. 2 and 3 are given views, in plan and elevation respectively, of the various parts of the ionization-chamber I. The wires leading from the radiator R, to the electroscope Hi, were surrounded by earthed tubes. A key fitted to the electroscope E, made it possible to earth or insulate the radiator R, at will. When R, was earthed the gold-leaf of the electroscope EH; to which it was connected took up its zero position, and this could be observed by means of a microscope fitted with a scale in the eyepiece. On insulating 342 Mr. C. A. Sadler on R,. the gold-leaf retained its zero position so long as no ionization was produced in the chamber I. When a secon- dary beam passing through A and B entered I a steady movement of the gold-leaf from its zero position took place, and on cutting off the secondary beam the new position of the gold-leaf, as viewed in the microscope, served as a measure of the ionization which had occurred in J. A chamber, con- sisting of .two tubes, one end of each being closed, the one being made to slide within the other, was charged to a potential of —240 volts. Into one end of this chamber passed an insulated wire connected at the other end to the gold- leaf of E3. This chamber served as an adjustable compen- sator by means of which the normal ionization in I could be approximately eliminated. The compensator when once adjusted remained unaltered while a complete series of readings were taken. In making measurements upon the corpuscular radiation it became necessary at each reading to adjust the position of the aluminium window attached to the inner sliding tube of the ionization-chamber IJ relative to the fixed tertiary radiator R,. These adjustments involved changes in the electrical capacity of the system. It thus became necessary to calibrate the capacity of the whole system—consisting of the ionization-chamber I, the compensator C, the electroscope E;, together with the capacities formed by the wires con- necting the chambers I and C to the electroscope E; and the surrounding earthed tubes-—for each position of the aluminium window relative to the radiator R, occurring in the course of subsequent readings. The method adopted was one suggested to the author by Dr. Barkla, and was carried out as follows. A tube containing 5 milligrammes of radium bromide was placed in a tunnel arilled into a block of lead; the mouth of the tunnel was then placed opposite to the side wall of the compensator C. The radium produced an ionization in 0 which soon reached a steady state. The ionization-chamber I was connected to earth. The gold-leaf of the electroscope H; was first earthed and then insulated. Owing to the ionization produced by the radium in C the gold-leaf at once began to deflect and the time taken for it to move over thirty scale- divisions was observed. This was repeated for each position of the aluminium window relative to the radiator R, required in the subsequent experiments. In the following table are given relative values of the capacity obtained in such a series of readings; the capacity when the surfaces were 3 cm. apart being taken as unity. Plotting the value of the See ae ee ee ee | Homogeneous Corpuscular Radiation. 343 Tase II. | Distance apart of Ratio of capacity of system in aluminium window this position to that in which | and radiator R,. the surfaces are 3 cm, apart. | ee i aN eM | 1-427 | OT CWB | a8 .220- 1-291 | PRO eile Seeress.: 1-200 MEV Cie? To cek see 1-137 SON CM Fete. an 1-097 it a ee 1:069 ay ae a 1-050 PMIEMER! Si san | 1-033 OEMS ane santa | 1-023 5 LE ee ae 1-018 i es 1-012 | BEMPOM, bsiisk ide. | 1-004 =| UC ne oe 1-002 ' = Ie ae 1:00 | capacity of the system against the distance apart of the aluminium window and the radiator R,, the points were found to lie upon a smooth curve, so that where necessary intermediate values could be obtained by interpolation (see Pl. V. fig. 4). With each new radiator R, this calibration was repeated. When the radiation from R, passed into the ionization- chamber I the ionization produced was due to the following sources :— (a) The corpuscular radiation from the face of the tertiary radiator upon which the Réntgen radiation from R, was incident. (b) The corpuscular radiation from the interior surface of the aluminium window through which the radiation passed. (c) The exciting Réntgen radiation from R, in its passage across the space between the aluminium window and the tertiary radiator Ry. (d) The tertiary Roéntgen radiation from the tertiary radiator R,. (e) The tertiary Réntgen radiation from the aluminium window. From some results obtained during the investigation upon the tertiary Rontgen radiation* it was evident that the contribution of (d) and (e) would be small. It was, however, tested for separately, and a correction made for it. During some early tests the effect of (6) was shown in * Sadler, Phil. Mag. July 1909. 344 Mr. C. A. Sadler on general to be small compared with that of (a), but later ex- periments showed that this was not universally true. In fact, in some cases the correction due to it became of grave importance. Details of these results will be given later. Preliminary experiments showed that in no case did the corpuscular radiation penetrate to a greater distance than a few millimetres in air at atmospheric pressure, so that no ionization due to the corpuscular radiation could be detected at a distance of 1 cm. from the radiating surface. If the inner sliding tube was so adjusted that the distance between the aluminium window and the radiator R; was 1 mm. and the ionization measured, then the inner tube withdrawn a small distance at a time until the surfaces were about two centimetres apart, the ionization being measured at each stage, it was found that the values of the total ionization increased rapidly at first, and finally at a slower and uniform rate. These readings were corrected in each case for the effect due to the change of electrical capacity of the system involved in the various relative positions of the two surfaces. If the distances apart of the plates be taken as abscissee and the total ionization produced in the ionization-chamber as ordi- nates we obtain the curves shown in Pl. V. fig. 5, which represent the total ionization curves with iron as tertiary radiator, Fe, Sr, Mo, and Ag acting respectively as sources of secondary homogeneous radiation. ‘The final straight portion of the curve corresponds to the region in which no corpuscular ionization is produced. It is readily seen from the figure that the corpuscular radiation from iron excited by the homogeneous radiation from silver is of a more penetrating type than that excited by strontium. In practice, a constant deflexion of the gold-leaf in the electroscope E3 was used, and the corresponding deflexion upon the electroscope H, observed. The ratio of the readings of E, to that of E,, corrected for changes of capacity, gave a measure of the relative ioniza- tions. But the difference in the values of the total ionization when the bounding surfaces were 2 cm. and 1 cm. apart respectively, corresponded to the ionization produced in 1 cm. of air by the effects (c), (d), and (e). If a proportional amount of this ionization be subtracted from each of the series of readings previously obtained, we get a measure in each case of the ionization due to the corpuscular radiation from sources (a) and (6). Plotting the ionizations due to the corpuscular radiation as ordinates against the distance apart of the bounding surfaces as abscissze, we obtain the curves shown in Pl. Y. Homogeneous Corpuscular Radiation. 345 fig. 6, which show the corpuscular radiation from iron excited by Sr, Mo, and Ag respectively. The horizontal portion represents the space in which no further ionization is produced by the corpuscular radiation. It is easily seen from the curves that the corpuscular radia- tions from iron excited by the various homogeneous Roéntgen radiations penetrate to different distances in air. Homogeneity of the Corpuscular Radiation. In an account of some experiments upon the corpuscular radiation from metals Cooksey showed that in some special eases the corpuscular radiation was absorbed by thin sheets of Al according to an exponential law™. In these experiments, using an ordinary primary beam, he found that the values of the absorption coefficients deduced from his readings depended upon the degree of “hardness ”’ of his bulb. The corpuscular radiation excited by a “ soft” primary beam was in some cases very heterogeneous, while that excited by a “hard” beam was more homogeneous, and also of a more penetrating type. But since a heterogeneous primary beam was used to excite these corpuscular radiations no definite values of their absorption coefficients in aluminium were available. It was desirable to obtain detailed knowledge of the cor- puscular radiations excited by the various homogeneous beams, and if possible determinations of their absorbability. For this purpose air at atmospheric pressure was used as the medium in which the absorptions were measured. The method of obtaining the absorption coefficients was similar to that described above. As an example, the following observations were made to determine the character of the corpuscular radiation from iron excited by the secondary homogeneous radiation from silver. The inner sliding tube of the ionization-chamber I was adjusted until the distance between the aluminium window B and the radiator R, was 1 mm. (the radiator R, had been ground plane and adjusted as accurately as possible in a plane perpendicular to the axis of the sliding tube, so that the plate was parallel to the plane of the aluminium window); the adjustment being made by means of the scale on the sliding tube and the vernier on the outer case. The deflexion on the electroscope E, corresponding to a deflexion of 20 divisions upon the electroscope E; was noted. The sliding tube was now withdrawn through 0°5 mm. and a * Cooksey, American Journal of Science, October 1907, pp. 285-304. 346 Mr. C. A. Sadler on reading of the deflexion of the electroscope E, for 20 divisions on E3; again noted. This process was repeated for the following distances between the two parallel surfaces: 1°5,)2°0, 2°5, 3°0,.3°5, 4°0, 4°5, 8°0,.16°0 mm. The: ratio var the deflexions of the electrescopes E; and E, gave a measure of the total ionization at each distance. Preliminary readings had shown that the corpuscular radiation in this case was not perceptible at a distance greater than 6 mm. from the tertiary radiator. The difference between the last two readings of this series gave a measure of the ionization produced by the secondary and tertiary Roéntgen radiations in 8 mm. of air. If from the ratio of the deflexions of the electroscopes E; and EH, for all distances from 1 to 8 mm., corrected in each case for any change in the electrical capacity of the system, we subtract a proportional part of the ionization due to the radiation of the Réntgen type, we obtain a measure of the ionization due to the corpuscular radiation alone in successive half milli- metres of air; all distances being reckoned from the radiating plate. If from the total ionization produced by the cor- puscular radiation we subtract the ionization produced by the corpuscles when the bounding surfaces are separated by a distance of « cm., the number so obtained is a measure of the intensity of the corpuscular radiation at a distance of « em. from the radiator. Ii the corpuscular radiation be absorbed by air according to an exponential law, then plotting as ordinates the natural logarithms of the numbers representing the intensity of the corpuscular radiation at different distances from the radiator R, against the distances from Ry, as abscissee, the points so obtained should lie upon a straight line. In Pl. V. fig. 7 are shown the curves obtained when the corpuscular radiation from iron was excited by the secondary homogeneous radia- tiens from tin, silver, molybdenum, strontium, and arsenic respectively. It will be seen from the figure that within the limits of experimental error the corpuscular radiation is absorbed in each case according to an exponential law. In all cases investigated the exponential law was found to hold very closely, even though corpuscular radiation of widely different absorbability was excited. If we define £8 the coefficient of absorption of this cor- puscular radiation in air at 0° C. under a pressure of 760 mm. of mercury by the equation [= I) e—®*, where I, is the initial intensity of the radiation issuing from the metal surface and I the intensity at a distance of « cm. from the surface ; then Homogeneous Corpuscular Radiation. 347 if the intensity at two distances v,; and «, be I, and I, respectively we deduce the expression for § in the form logL, —log.[, B= of Xs ae | vy In the figure the tangents of the angles which the straight lines make with the horizontal axis were taken as the values of 8 in the various cases. The absorption coefficients for the corpuscular radiation from different metals excited by the several homogeneous beams used are given in the following table. TABLE III. Radiators which act as the source of homogeneous secondary radiation. Tertiary radiators. ee | ie wey Sik | | es : Ni. | Cu. | Zn. | As. | Se. | Sr.) Mo.|Rho.| Ag. | Sn. a 38:9 | 37:0 | 35°8 | 30:2 |26-4 /21-5 |15°5 |10°9 |8:84 |6-41 Copper ......... ms ss | OOD | S04 | ... (206/152 [10's hia 6°67 I fap ssnaess. sis Wave | BBR | G02 |).06 (21°2 |15°4:|10'3 (8-78 16-63 Aliminium ...| ... | ... | ... | 29° |... [200 (15-2! ... |8-90 6-54 It should be stated in passing that with aluminium as tertiary radiator the graphs from which the values of 8 were deduced were not quite so regular as in the case of the other tertiary radiators. ‘The means of several determinations were finally adopted for inclusion in the table. In the case of the other tertiary radiators at least two determinations for each value of 8 were made and the mean taken ; but these values agreed among themselves very closely. It will be seen that within the limits of experimental error the absorption coefficients of the corpuscular radiation ex- cited by any particular homogeneous radiation from metals which differ so greatly in atomic weight, density, and other properties as aluminium, iron, copper, and silver, are identical, This is more clearly brought out in the following table. 348 Mr C. A. Sadler on Tas_eE LV. Absorption | Coefficient in Aluminium | Absorption Coefficient in Air Densiey Seen of the of the corpuscular radiation Tertiar of J Weicht of Homogeneous) excited by the Secondary | Ra cents Tertiar Te eae _ | Secondary | Homogeneous Radiator from "| Radia ae Ra ee Radiator the following metals. i | * characteristic of the | |————_________ Terti | Ru ho As. | Sr. | Mo. Rho.| Ag. | Sn. Alumiuium .. ATE 27 29'6| 200) La2e 8°90) 6:54 BRON Picci. sic 779 559 239 30°2) 21°5| 15:5) 10°9) 8°84) 6°41 Copper ...... 8°95 63°6 1289 304) 20°8) 15-2) 10-8) 8:81) 6°67 STVER ace. 10°57 107°9 675 | 80-2; 21-2) 15-4 10:3) 8-78 ae * The absorption coefficient of the homogeneous radiation from aluminium has not yet been determined. Since the absorption coefficients of the radiation from aluminium when both radiating surfaces are of aluminium are the same as the corresponding coefficients when one of the surfaces is replaced by iron, copper, and silver, respec- tively, no correction is necessary in the determination of the absorption coefficients of the radiation from these substances due to the superposition of the corpuscular radiation from the remaining aluminium surface. However, this point was tested for separately; the secondary rays from silver were allowed to fall upon a copper ter- tiary radiator, after passing through a thin copper window substituted for the aluminium window used in the previous experiments. No difference in the value of the absorption coefficient could be detected. These results are shown in Table V. TABLE V. 3, Absorption Coefficient in Air. Secondary Radiator. Al Window. | Cu Window, SLLVER. Sei adeeces 8-81 | 8:84 | Homogeneous Corpuscular Radiation. 349 The very easily absorbed corpuscular radiation excited by the softer homogeneous beams is almost completely absorbed in the millimetre of air nearest the radiating surface. It was extremely difficult to work with a much smaller distance between the surfaces than 1 mm., for it was impossible to altogether avoid a direct leak from the aluminium window to the insulated plate owing to the presence of dust particles. Moreover, the leakage was very irregular, and so could not be allowed for. Under these circumstances, the device was adopted of using a mixture of air and hydrogen as the ab- sorbing medium. With this mixture, readings for the absorption of the soft corpuscular radiation could be taken over a range of about 3 mm., since hydrogen proved a much less efficient absorber than air. Control experiments in which harder secondary beams exciting more penetrating corpuscular radiation were used made it possible to deduce the absorption coefficients of the softer corpuscular radiation in air at atmo- ‘spheric pressure. A strikiny relationship was found to exist between the absorption coefficients in air of the corpuscular radiation excited by the several homogeneous secondary beams in any one substance used as tertiary radiator. If the absorption coefficient of the corpuscular radiation excited by each homogeneous secondary beam be plotted as ordinate against the atomic weight of the metal used asa source of the secondary exciting beam as abscissa, it is found that over a wide range the points so obtained lie upon a straight line. This is shown in fig. 8, where iron is used*as a tertiary radiator. It will be seen that for the more pene- trating secondary beams there is a departure from this relationship. It has been shown *, however, that from sub- stances whose atomic weights are greater than that of silver, in addition to the emission of a very penetrating homogeneous beam, a softer type of Réntgen rays are produced. From a substance like lead (207) this softer type appears to pre- dominate. In the present experiments the exciting radia- tions from silver and tin were first passed through aluminium screens to sift out the soft radiation. Connexion between the Corpuscular and the Exciting Réntgen Radiations. But turning now to the main object of the present inves- tigation 1 1s important to learn in what manner the intensity of the corpuscular radiation depends upon the metal emitting * Barkla, Phil. Mag. Jan. 1906, pp. 812-828 ; Barkla and Sadler, Phil. Mag. Oct, 1908, pp. 550-584. 350 Mr. C. A. Sadler on it, and upon the secondary homogeneous radiation that excites it. . Using the same form of apparatus as that previously used for the determination of the absorption coefficients, the method of experimenting was as follows. , A given metal to be used as a tertiary radiator was placed in the position R, with its plane perpendicular to the axis of the sliding tube. The aperture B was covered with aluminium sheet (-003 cm.), and through this passed a portion of the secondary homogeneous beam from R, exciting corpuscular radiation from the inner face of the aluminium window and from the face of the tertiary radiator upon which the beam fell. (In all cases the tertiary radiator was sufficiently thick to completely absorb the secondary homogeneous beam.) A measure of the total ionization in the space between the aluminium window and the tertiary radiator when the sur- faces were separated by a distance of 1 cm. and 2 cm. respectively was determined by observing the deflexion on the electroscope E, for a constant deflexion of 20 divisions on the electroscope E;. nt Let the ratio of the deflexion on the electroscope H; to that on the electroscope H, be d, and d, in these two cases respectively. Then d,—d, is a measure of the ionization produced by the exciting Réntgen-ray beam and the tertiary Réntgen radiations in a layer of air 1 em. thick. The ioni- zation due to the tertiary radiation was determined in- dependently. Let its value be denoted by a. Then d,—d,—a is a measure of the ionization produced in a layer of air 1 cm. thick by the secondary beam alone. With the same notation d,— (d,—d,) =2d,—d, is a measure of the total ionization due to the corpuscular radiation from the aluminium window and the tertiary radiator. We thus obtain in the expression 2d,— dy d y— ad, =<, =R, the ratio of the total ionization due to the corpuscular radiation, to the ionization produced in a layer of air 1 cm. thick by the secondary exciting beam. If the radiator R, be made of aluminium, and if the intensity of the corpuscular radiation does not depend upon whether corpuscles are produced by the Réntgen rays at inci- dence or on emergence, then one half of the value of the expression R in this case may be taken as the correction to be subtracted from the values of R obtained with other metals as tertiary radiators. Other investigators had made this Homogeneous Corpuscular Radiation. 351 assumption in correcting for this effect, but it occurred to the writer to test the matter by direct experiment. With the apparatus previously employed values were obtained of the total ionization due to the corpuscular radiation produced at incidence and emergence respectively. The latter was found to be roughly twice as great as the former. More elaborate precautions would, however, be necessary to secure satis- factory data. Experiments are now in progress with this end in view. In many cases the ionization produced by the corpuscular radiation from the aluminium window was small compared with that produced by the corpuscular radiation from the tertiary radiator, and the necessary correction was thus very small. In all cases, however, two-thirds of the value of R obtained when two aluminium surfaces were employed was subtracted from the value of Ik when any other metal was substituted for aluminium as the tertiary radiator to obtain the value of R for the tertiary radiator alone. These cor- rected values are tabulated below. TABLE V. Values of R. Metals which serve as the source of Homogeneous Secondary Rontgen Radiation. Tertiary Radiators. l ei | l i, | Cu. | Zn.| As. | Se. | Sr. | Mo. | Rho. Ag. | Sn. ‘010 | 013 | 028) 038 -071 | +264 | 400 | -500 | -680 eh aketes es he (1 013 | -245 | 250 | 255 | -272|-290. -807 | 362 | -500 | -590 | -700 ET sapere) stnacs] coeeee| ceceee O18 | ‘059 | °363 | 377 -387 | -458 | 550 | 630 | -730 Biveh) dies 079 098} 119 |-148 178 | -383|-410 -672| -972/ 1-08! 1:15 2-19 The smaller values of R could not be determined with very great accuracy, while in some cases it proved too small to be measured at all, as, for instance, when the secondary radiation from iron fell upon copper as the tertiary radiator. It was not to be expected that the sign of the potential of the ionization-chamber I would influence the speed of the corpuscules, since the field in which the corpuscules exist on leaving the atom must be very much greater than that pro- duced by the voltage employed in these experiments. However, to test this point the outer case of the ionization- chamber was first charged to +240 volts, and subsequently 352 Mr. C. A. Sadler on to —240 volts. The corpuscular radiation from copper excited by Sr and Ag respectively was usea for measurement purposes, but no appreciable change in the absorption co- efficients in air could be detected. Itis evident then that the repulsion of the corpuscules by the plate R, in the one case and their attraction by the plate in the other is too small to affect their velocity to a measurable extent. It was reasonable to suppose, however, that the value of R might possibly depend, in part, upon the sign of the voltage applied to the ionization-chamber I. For consider first the case when the ionization-chamber is charged to + 240 volts. When the plate R, is earthed it acquires a negative charge. On insulation it stiil remains at zero potential, and possesses the same charge. If now a secondary beam of Rontgen rays of given intensity pass into the ionization-chamber for a given length of time, a certain amount of ionization will be produced. The positive and negative ions will be equal in number. The positive ions will be attracted to the plate and will diminish its negative charge. This will be true whether the ions are produced by the corpuscular radiation or by the incident secondary Réntgen radiation in its passage across the ioni- zation-chamber. But the emission of negative corpuscules by the plate will of itself tend to diminish the negative charge on the plate, so that from both causes the negative charge will diminish and the potential of the plate and its con- nected system will rise correspondingly. This change of potential of the system causes a motion of the gold-leaf in the electroscope E;, and by observing the deflexion of this leaf we obtain a measure of the change of potential of the system. This gives us a measure of the charge received by the system on some purely arbitrary scale. Consider now the case when the ionization-chamber is charged to — 240 volts. When the plate R, is earthed and then insulated it possesses a positive charge equal in mag- nitude to its previous negative charge. If we allow the same secondary beam to enter the ionization-chamber for the same length of time as in the previous case, an ionization will be produced by the radiation of the Réntgen type equal to that obtained in the former case. We have seen that altering the sign of the potential of the ionization-chamber produces no measurable change in the velocity of the cor- puscles, and it is reasonable to suppose that the ionization which they produce will be the same as before. Thus the ratio of the total ionization due to the corpuscular radiation, to that produced by the secondary exciting beam in 1 em. of Homogeneous Corpuscular Radiation. 393 air, would be unaltered by the change in sign of the potential of the ionization-chamber. The negative ions produced by these two sources of radiation will thus communicate a negative charge to the plate R, equal to the positive charge imparted in the former case. But the negative corpuscles emitted by the plate R, carry away a negative charge, and this is equivalent to giving an equal positive charge to it. Thus the total loss of charge by the plate will be less in this case than in the former. Let g, be the charge carried to the plate by the ions pro- duced by the secondary exciting beam in lem. of air. Let gz be the charge carried to the plate by the ions produced by the corpuscular radiation. Let g3 be the charge carried away from the plate by the corpuscles themselves. Then the true measure of the ratio which we have called Reis ol 41- The ratio experimentally determined from the readings of the electroscopes E, and H3, as previously described, is really Jat 93 gt when the ionization-chamber is charged to + 240 volts, and G2— qi— Qs when the chamber is charged to — 240 volts. Below are given the values of these ratios when copper is used as the tertiary radiator and silver as a source of the secondary homogeneous Réntgen beam. Taste V1. Voltage on Ionization-Chamber. Value of Ratio. +240 ‘547 — 240 545 +240 540 It will be seen that within the limits of experimental error the ratios are identical in value, and cannot appreciably differ {rom the theoretical ratio R. It appears from this result that g; must be small compared Phil. Mag. 8. 6. Vol. 19. No. 111. Mareh 1910. 2A 354 Mr. C. A. Sadler on with either 9; or g 2. ‘[his will be the case if the number of ions produced by a corpuscle before it ceases to act as an lonizing agent be a comparatively large number. The fact * that the corpuscular radiation is absorbed by air according to an exponential law would lead one to suppose that the absorption of the corpuscles is due to their diffusion by the air molecules, and not to any appreciable extent toa diminution of velocity. One striking result which appears from an examination of the figures given in Table V. is the following. With iron as tertiary radiator, a large increase in the value of R occurs when the secondary radiation from nickel is used. With copper as tertiary radiator, a large increase in the value of R occurs when the secondary radiation from arsenic is used ; with silver as tertiary radiator, a large increase in the value of R occurs when the secondary radiation from tin is used. These values are underlined in the table. These results are well indicated in fig. 9 (Pl. V.) where the ordinates represent values of R while the abscissee are the atomic weights of the secondary radiators. Now in each of these cases this large increase in the emission of corpuscular radiation coincides with the corre- sponding increase in the production of tertiary Rontgen radiation characteristic of the tertiary radiator previously mentioned in this paper f. It would appear then that some intimate connexion exists between the production of Réntgen radiation in a substance and the emission of corpuscular radiation by that substance. The writer has not met witha single instance in which homo- geneous Rontgen radiation characteristic of a substance is produced without the simultaneous special emission of cor- puscular radiation. But the converse is by no means proved. For an examination of Table V. shows that a considerable emission of corpuscular radiation from silver used as a tertiary radiator occurs even when none of the homogeneous radiation characteristic of silver is produced. For instance, when the secondary exciting beam from silver itself falls upon silver as tertiary radiator, no homogeneous Réntgen radiation is produced, and yet a considerable emission of corpuscular radiation occurs. But in the cases { where none of the homogeneous Rontgen radiation characteristic of a metal is produced, a portion of the exciting beam falling upon the metal is scattered. It is * ¢Conduction through Gases,’ J. J. Thomson, p. 379. + Sadler, Phil. Mag. July 1909. } Barkla and Sadler, Phil. Mag. Oct. 1908, pp. 550-584. iN 4 y ‘ " ' f, 7! Homogeneous Corpuscular Radiation. 305 probable that the emission of corpuscular radiation in these cases is connected with the scattering of the homogeneous exciting radiation. In this connexion it is interesting to note that during some experiments carried out in conjunction with Dr. Barkla*, upon the seattering of an ordinary primary beam by different substances, it was found that silver behaved as a much more efficient scatterer than light elements such as carbon or aluminium. For equal quantities of matter traversed, the silver proved to be about 6°5 times as efficient a scatterer as carbon or aluminium. No determination has, however, yet been made of the relative scatterig of homogeneous beams of different penetrating powers. It is, however, significant that in the cases where there is no production of the Rontgen radiation characteristic of either aluminium or silver, the corpuscular radiation from the latter is invariably greater ‘than from theformer. The ratio of the corpuscular radiation from silver to that from aluminium diminishes from about 15 for very soft secondary exciting beams to a little over 2 for the hard secondary beam from siiver. Experiments are at present in progress to determine in what manner the scattering of homogeneous Rontgen radiation by different substances takes place. It is hoped that the results of these experiments will throw additional light upon this part of the subject. Perhaps the most striking result of the present investigation is the entire dependence of the corpuscles for their velocity upon the “ hardness” of the Réntgen radiation used to excite them, and their independence of the kind of metal from which they are produced. This is in marked contrast to the Roéntgen radiation characteristic of a metal. This latter is of constant penetrating power, and this penetrating power depends solely upon the metal from which it is produced, and not at all upon the hardness of the Rontgen radiation producing it. Thus as far as the absorbability of these two types of radiation is concerned, we see that in one case the exciting radiation is the controlling factor, while in the other the radiator emitting the radiation itself furnishes the control. The variations in the intensity of the corpuscular and Roéntgen radiations appear to follow similar laws, but the evidence is far from being complete. A fuller discussion of the foregoing results, including those on the scattered radiation, will be attempted in a further paper. * Barkla and Sadler, Phil. Mag. Oct. 1908, pp. 550-584. 2A 2 356 Mr. S. D. Chalmers on the Sine Condition Summary of Results. (1) The corpuscular radiation excited in different metals by homogeneous beams possessing a wide range of penetrating power is absorbed in every case according to an exponential law. (2) The absorbability of the homogeneous eorpuscular radiation from a tertiary radiator depends solely upon the degree of “ hardness ” of the exciting homogeneous secondary beam, and not at all upon the nature of the tertiary radiator. (3) The absorption coefficient of the corpuscular radiation from any metal is a linear function of the atomic weight of the metal which acts as the source of the homogeneous exciting beam. (4) In the case of those metals which emit a characteristic homogeneous Réntgen radiation when subjected to a suitable exciting beam of Réntgen rays, it is found that whenever the homogeneous Roéntgen radiation is produced it is always accompanied by a special increase in the corpuscular type of radiation. Where the homogeneous type of radiation is not produced, we still have a portion of the exciting radiation scattered by the metal, while the intensity of the corpuscular radiation is quite considerable. It is probable that the cor- puscular radiation emitted in such cases is intimately con- nected with this scattering. In conclusion I wish to express my indebtedness to Pro- fessor Wilberforce for the interest he has shown throughout this research, and for the ready help and advice he has accorded me. I wish also to thank Dr. Barkla for his advice in the ~ early stages of the work. George Holt Physics Laboratory, University of Liverpool. 29th January, 1910. XXXVI. The Sine Condition in Relation to the Coma of Optical Systems. By S. D. Cuaumers, M.A.* HE condition for the correction of coma in a centred optical system is the well-known Sine Condition. This has been proved by Clausius, Helmholtz, Hockin, and others, and the importance of this condition in the design of optical systems has been pointed out by Abbe, Steinheil, Conrady, and others ; but so far as J am aware no discussion of the effects of failure to satisfy this condition has been published. * Communicated by the Physical Society: read June 25 1909. in Relation to the Coma of Optical Systems. 357 Steinheil appears to have arrived at a correct estimate of the errors in telescope systems, on the basis of a number of trigonometrical calculations. I propose to obtain the relation between the Coma of a system and the errors in the Sine Condition. If O and A be the angles which the initial and final rays make with the axis, and g the theoretical magnification of the system, the Sine Condition may be stated 1 po sin O Seaan A —1 =0 for all values of the angle O, fo and mw being the refractive indices of the first and last media. We shall consider centred optical systems only and take the axis as the axis of <, and the object may be assumed in the plane y=0. The length of the theoretical image will be denoted by z and the errors, in the directions « and y, for the actual Lv ray, will be denoted by Sx and 6y, and — and ey will specify the Coma for the ray considered. From approximate theories we can see that these terms depend on the particular ray considered and may be represented as follows :— = = 1,(2 cos*r+1) sin°’A+i,(4 cos*r+1) sintA + terms of higher order in sin A, = 1,(2 sin cosy) sin?A +%,(4 sin cosy) sin*A + terms of higher order, where is the angle between the « z plane and the radius Fig. 1. to the point of the stop through which the ray passes (fig. 1). 358 Mr. S. D. Chalmers on the Sine Condition If we take w=0 or 180°, ae = 37, sin?A, + 5%, sin*Ay- oy = 0 if ar == 90°, ee = 7, si Ay) eI ee Oye Refraction of Rays not in one plane. If the direction cosines of any ray be ly) mp before and 1, M9 Nq after refraction at the surface (1), the normals being L,M,N,, we have : Holo — Pals Oe LAI cr Pe CON ra (I ) 1A | = 1 N, 9 e ° oe e ° where fp and py are the refractive indices of the two media. Then po(/o)M,—mol;) is invariant on refraction. We may choose the coordinate axes and origin so that v Ly, Myr, and Nyr,4+ 7, represents the point on the surface at which the refraction takes place. To specify the ray independently of these co- ordinates we can take the points where the ray cuts the planes v=0 and y=0. ~ Tet these points be 0, mopo, do 3 loo, 0, Do, then ag Do\ Po (= Lape J is the distance along the ray between these points. We have Ly, ee My" —™Mp Po Nit (8 —d) ra pS vane ok Peele ‘ lo Mo No > iD giving (Lym — foMq) 71 = — Lomopo; OY fMolpMoPo 1S Invariant on refraction. The condition that L,7,, M,7,, and Ny,+; represents the point on the surface is satisfied, with different values of 74 and 3, for all points on a surface of revolution about the axis of z; and since flpmpPp 1s independent of the position of the origin it is nvariant throughout any system having a common axis. This invariant may be written u(2yn—yj,!), where a, and y, in Relation to the Coma of Optical Systems. 359 are the coordinates of the point in which the ray meets any plane at right angles to the axis. Considering a complete system we have [o%oMg = Me(A2Mz—Yola), OF Moog = Molly + Oxy) Mg— fydY/alo, if the ideal values of the coordinates on the image plane be yp. Thus uy mM Modv l ) 9 HoXoINo 1 n Lg0.V2— lo 2) fess dv ly by Mody MoXsMy Lo) Io is, It remains to obtain the value of Ba rly By fy «My Be in suitable form. The aberrations of any centred system may be expressed as besa ee ae OW, = aa, (pat; P2°5 X12) 'z dN are = {F(e/, p2”, X12) }> where ay, represents the point at which the ray meets the stop plane, Xyq the point in which the ray should meet the theo- retical image plane, praeptyy, p22 Har +yo’, Xi2= + (ry%2+yY2), and IF is a function of the second and higher orders in Pi» po, and xp. oF oF Oils — 22 ~ 9 - re Opi “OX Then or oF MyGWg—lydy/o => 2 (mya — ly7/;) = + (mg2'y — [52 ) Speen a aie i Obs pi e's OX12 n oF 0 0. flo(My02ag— lady) = Zugxymy ——s + poMoty —— $ of go bg Y2) ovo Spe BK Ox Hotoy _ _ yHotomy OF | OE MoetyMe fiyit’gMy Op. Ox OE > 2 2 AN ae Moro f’ Opry Oxi, MoksNg sy or 360 Mr. 8. D. Chalmers on the Sine Condition The denominator is generally nearly unity. When the object is small and nearly on the axis we need only consider terms of zero and first orders in 22, y, that is, F = f,(p1”) + X12,fo(p1”), OE ysiy? 5h) igen Ofs Oxi Pher) 3 Op: Opry” i Opi” These three terms may be considered separately :— (1) The lateral spherical aberration is 2p, oh : 1 (2) The comatic effects are made up of ox = Xofo( px") 3 >/ end) (3) 62 = 225 oe = 2020s = ‘ Ofe fs) oY = Dumas = 2yX Xo oe Thus we have lat. spherical i C, a Cs BotoNo tah 1 iF MoVoMog P 1 i) Uy where C, is the x displacement due to Coma when X,.=0. C, is the additional x displacement due to Coma when 73.540. Except when a, is very small, the last term = becomes 1 unimportant ; if «, be very small the left-hand side becomes indeterminate and dependent on the actual value of «x, as chosen. If we take y,.=0 we have Mm, = sin A, MoXo sin B if Me®o sin A Vy Vs (b) Lo? ne Pong tt smn: ss Coma _ 6v a b22 and B and A may be obtained from the axial ray if the image be small and v be the distance from the plane (1) to the point at which the axial ray meets the axis. This result may also be demonstrated as follows :-— Consider the ray OPQI (fig. 2) in the plane yz passing through the system and cutting the axis in I and the theore- tical image plane in J, O,P a neighbouring ray through O, oe -— ~ = eww in Relation to the Coma of Optical Systems. 361 where OO, is at right angles to the plane yz. The ray OO, meets the wz plane in I, and the image plane in Jj. Fig. 2. OO,PN IT,QN, (equation IL. above); Ko OP == Fo QI é but when OO, is not large, PN PN A O,P = OP = SIN B, or = hy = sin A. 1 Hence a Hy sin B But if there be spherical aberration present, JJ; QJ NK, fs Ol” NL’ and we have Mo SU sin B N,K, psn A’ Ni ; if KK, be the ideal value of the image JJ; —KK,=Coma ; and we have Coma =(z OO, 4,smnB NK -1) — ( OO; po sin B ) RiP ARK, «sin A“ Ni = (aK, wsinA KI / 00, _ po sin B Nil (ack w sin x) JJ, = OO; Coma KI _ ( OO; wo sinB _ ) KK, —Ni=(kK pwsinA ? provided the product of the KS and a may be neglected. 362 Mr. 8. D. Chalmers on the Sine Condition Thus (III.) v i) 67 , bv sin B a o=(¢ aa Even though the value of II, be large, the relation p00, sin PO|N=pll, sin QT,N, holds good ; but the point Q is no longer the theoretical image of P or even on the ray OPI. The relation (II.) above, expressed in the form w.m.II, is invariant furnishes a convenient check upon the results of trigono- metrical calculations for oblique rays. Rays in one Plane. For the case of rays in one plane we may adapt the method of Hockin to give further information. In this method the optical length, 2. e., 2x actual length, is stationary for the actual path. We cannot, since aberrations are present, assume that all optical paths between the same points are of the same length, merely that the difference between two neighbouring rays is zero. Fig. 3. All the rays are in one plane. O,, Og are two neighbouring object points ; P,, P, two neighbouring points on the stop plane ; N, the centre of the stop ; f O,P;, O,P., and O,N, pass through the system and meet O,P, O.P., and O.N, in the points Q,, Q., and Q3 and intersect the image plane in L, I,, and I, ; O,P,, O2P,, and O.N, meet the image plane in Jy, Jy, andJ3; Q,1,, and Q.I, meet in EK, ; QJ, and QJ. meet in Ff). in Relation to the Coma of Optical Systems. 363 Considering the optical paths of neighbouring rays, we have the optical lengths O,P,Q:E, = 0,P.Q.K,, O2P,Q:F; = 0,P.Q.F,, 2.€., OPy—O2P14+(Qh—QWJ1)— (Hi, — BL) + Fi — Fi) = (0,P,—02P2) + (QoIl.—Q,J2), or (O,P,—O2P,)—(O,P2--02P2) + (Qil, — QiJ1) — (Q2l2— Qe 2) = (E,I,—E,I,)-(¥,J,—FJ,). We denote the angles between the axis and the rays O,P;, O.P2, O,N and O.P;, O2P2, O2N by a, a, a3 and 81, Bo, B3 respectively ; and the angles the axis makes with Gili, Qele, Q;I3, and Qi, QeJ2, Q3J; by L, kL, I; and Ji, Ja, Js respectively. Then O.P, — Oub y= —O 10, eae By = — s — 0,0, sin a1, provided O,O, be small. Similarly Q1,—-Q.J, = —LJ, sin I, QeI,—Q.J., = —I,J, sin I,, Ht: cos I,—cos I, E,,—E,I, = Bee) cos J.—COse 1 sin(J» »— J) cos I, + Q, (cos(I, + Q)) ) "sin {(1,—L) +(Qs— Qi) t If now we can assume the angles Q, and Q, small,—that s, I,J, and I,J. small, we have cos sI; — cos a) _ Q, sinI,— +5 sin], V cosI,—cosI, J FJ, —F,J,= —J,J.- cos cos(I,—I) sin(I,—1[,) ie Q, cos I, + Q. cos I, cosl,—cos I,) (1+cos(I,—I,)). cos I,—cos I, = Jy ith 7" sin(l,—=1,) 7 oe ( 364 Mr. S. D. Chalmers on the Sine Condition Thus cos 1,—cos I (E,], — E,I,) = (FJ, Fe F,J>) = (JiJo— I,I,) sare —I,) : Q, cos I,+Q, cos I, . +JiJ> 1+cos(1,—]J,) ak | or cos I,—cos — Q0,0,(sin 2,;—sin #2) — (LJ ; sin I, —I,J_ sin I.) = (J,Jo—L],) —— Sa (L—I,)e Ly BE) 4 los cos I, cos I gem 7 : t ee 1 2 Osa + cos (L,—I, yy On summing throughout the aperture for O, on the axis —O,0, sin a,—I,J,; sin 1, = > (Jy o— I,I,) sin 1,4 23a cos I, 9 IJ NY where the value of J,J3 is assumed small as compared with NI. Thus 0,0, sin ay I,J; sin I, + 3(J,J.—LI_) sin +3(JJ2 cost 57). But L,J,-1,J 3 = Comatic error for rays in one plane ; and = (J; J 2015 sinI, is the Coma x the average value of sin],: this average value depends on the way in which the Coma varies with | the aperture. If we assume et ee a 1) 2a tian = 37, sin?’A + 57%, sin*A, the mean value in the first term is in the second # of 52, sintA. Thus we have, since A= —I,, 0,0, sin ay ) i) f == es ——— a ‘ 4 a ' I,J, sin Ay Ae Coma of first order +1 Coma of second order } | JJ_1 IN sinI = — 13 Coma of first order ++ Coma of second order t Bess | long oO. .g. spher. aberr. INH in Relation to the Coma of Optical Systems. 365 This result is equivalent to that proved above for the rays passing through x#,=0, on the stop, provided that for 2,=0 the Coma may be denoted by 2%, sin’?A +72 sintA, and for 2; =p, by 32, sin?A + 5i. sin’A. This shows that the proper expansion for the Coma is in the form 7, sin?A +72, sintA rather than 7,A?+7,A4,since the latter would not separate the terms which are in the ratio of 3:1 from those in the ratio of 5:1,and thus would lead to a more complicated expression for the Coma. The summation above may also be applied to the case of an oblique ray. We then have = O0,0,(sin a,;—sin ot3)) —(I,J, sin 1,—J,J; sin I;) = >(JJ,—L,I,) sin I LJ +2J,Jo cos*I NL’ : —0,02(sin «—sin 3) _ ie ne 1 : \ +1,J,(sin 1, —sin J) ~ J,J3(sin L, sin 15) : aah Tee + *( (Jhb) sin 1,)+ ¥( (ids cos *I = é | — (IJi— I3J3) sin L, The summation may be effected provided we know the approximate expressions for the quantity J,J,, but in the ordinary way these expressions are too complicated to admit of simple expression. The condition for the absence of Coma for a small object not on the axis is easily found provided one point be perfectly defined : 0 1) 0 throug ci it is (7 yi aia |, 1) =0( throughout the aperture, where & refers to the ray through the centre of the stop, In conclusion, it will be seen that the Comatic effects asso- ciated with any departure from the sine condition may be calculated, even though spherical aberration be present, from the relation (III.) above. Pa seg ni) XXXVII. Reversion of Power Series. By C. EH. Van ORsTRAND *. _ LARGE number of the series employed in pure and applied mathematics are special cases of the integral power series, Y= aot + Oye? +Gle t... , sie In the numerous applications of this series, it is oftentimes necessary to express w as a function of y by means of the integral power series e=Agy+Ay?+ Ag+... ony ic - ° . (2) The usual method of procedure is to substitute the value of y in the second equation, equate coefficients, and then solve for A,, Aj, As,... in terms Of a, @;, @o,... | Deen three or four coefficients may be determined in this way without much difficulty, but the coefficients of the higher powers of y are so complicated that this method is almost useless for the determination of their values. To obviate this difficulty, Professor McMahon + bases the development of the second equation on Lagrange’s series. He puts ao Ao Ao a=2tb,0? + boa +...=2+d(a), n'=n(n+1)(n+2)...(n+r—1), and obtains eerg ntl oat anaes a | create (a) as the coefficient of 2”~" in the reverted equation. The exponents and subscripts of the 0’s in this expression satisfy the conditions ptgt...=r+l]1 pttgqjt ...=n—2. Another method of obtaining the general term of a reverse series has been suggested by Professors Harkness and Morley. They differentiate (2) with respect to « and divide by y”. * Read before the Philosophical Society of Washington, D.C., Oct. 9, 1909. Communicated by the Author. + “On the General Term in the Reversion of Series,” Bull. Am. Math. Soc. p. 170, April 1894. Reversion of Power Series. 367 These operations give a [= Pes Pe ae y + (n+1)A,+ (nt+2)Ansiy + Lay diate (8) It may be shown by substitution of y from (1) that nA,-1y~1y' is the only term in the right-hand member which contains z~!. That this is true is shown also by means of the equation y 1 d =) y —(n—1)da\y" if d payee a a ae == | (age) OP (14+4—e+ tat...) ] —(n—1)dex a df a a | = ae ae geet << s tantanaet «.], a series which after differentiation contains no terms in 27! for integral values of other than unity. Hence, by equating the coefficients of «-! in (3), we obtain * Prd ': PE . Fan ° s Pe ‘ ‘ 4 (4) where the expression in the brackets means the coefficient of w—! in the development of y~” as a function of «. This coefficient may be found without much difficulty. Performing the indicated expansion y= (age + aye? + agit? + ee ahi —(a.n\-" RO oe "i = (age) (1+ ad a Hb vais ) = (a2) "(1 —b,a— bya? + ...)™ =(a je)” E +n(byu + boa? +...) +... n(rn+1)...(n4+r—1 - pe eeu hs Le, (bya + bow? +...) +.. ]-6) ene and Morley, ‘Introduction to Analytic Functions,’ p. 144. 368 _ Mr. C. E. Van Orstrand on The polynomial theorem gives r! (ba +boa? + .. eerie ae Ts ee 2 the exponents being subject to the conditions ptqotr.-.-=7, 7 pt2qg+ ...=n—1L (7) The first condition arises from the homogeneity of the b terms in the expanded equation, and the second is imposed by the condition that the terms in 2 must be of degree (n—1) in order that the complete expansion of y~” contain a term in wz. These conditions therefore require that the 6 terms are of order r and weight (n—1) instead of order (r+1) and weight (n—2) as in expression (a). This dif- ference in the order and weight is due to the fact that (a) is the coefficient of 2”~* instead of 2”. Finally, by substituting (6) in (5) and (5) in (4), we obtain Lae S(n+1)(n+2)...(n+r—1) ee ca. as the coefficient of y"a) "=<". Formulas (7) and (8) hold for all positive integral values of rand n. The seemingly exceptional case, r=1, is readily seen from (4) and (5) to give a numerical coefficient unity for all values of x. Since the 6 terms are of a given order and weight, they may be taken in part from tables, such as Bruno’s table, “Symmetrische Funktionen der Wurzeln einer Gleichung,” contained in his treatise, ‘ Bintire Formen,’ which contains all terms of successive orders and weights from 1 to 11 inclusive. Terms of weight 12 may be deter- mined with the assistance of the same table, for evidently the expression, 6, x terms of weight 11+, x terms of weight 12+..., contains all terms of weight 12, including dupli- cations. The process may be continued so as to determine all terms of any given order and weight *. Having determined p, gq... in the manner indicated above for any particular value of n, the corresponding coefficient is readily determined by substitution in either of the preceding He * For a precise method of determining order and weight, see a paper by R. A. Harris, “ On the Expansion of Sn 2,” Annals of Mathematics, 1888, p. 87. P j Reversion of Power Series. 369 formulas. By proceeding in this way, the first 13 terms of the reverse series of (1) are found to be C=2+ 02? + [by + 2b)71 23 + [b3+ 501d, + 50,? | <4 + [by + 6)1b3 + 3b.” + 216,75, + 140,"] 2° + |b; + 7(b,b4 + b253) + 28(b,2b3 + bib”) + 84b,b, + 42b,° |2° + [bg +4( 2,0; + 2b.b4 + 63”) + 12(3b,7b, + 6b,bob3 + b?) + 60(26,3b, + 3b;2b27) + 330b)4b. + 1326,° Jz + [67+ 9(byb, + babs + 6364) + 45(b,7b; + byb3? + b57b3 + 2b ,bob4) + 165(b,2b4 + b,b.3 + 3b 7bobs) + 495(by4b3 + 2b13bo?) + 1287b)°b, + 429,73 + | bg + 5 (2b; + 2babg + 2b3b5 + 64) + 55(Ds2l5 + baby + bods? + Qdybabs + Qbybsb,) + 55(46,7b5 + 6b,2b5? + 12b,2bob, + 12,b.7b3 + 04) + 715 (b,4b, + 26,7b.? + 4b,°bab3) + 1001 (26,°b3 + 5b, 4b57) + 500568. + 14300,5]2° + [by -+ 11(byg + bab, + bgbg + b4bs) + 22(3b.2b7 + 3b.2b; + 3b,b,? + 6b babe + 6b1b3b5 + 6bob3b4 + 55°) + 286 (b,2bg + b3b3 + 3072 b9b5 + 3b Pb gb4 + 3bpb.7b4 + 3b 1b9b5°) + 1001 (6,405 + bybo¢ + 26,3b5? + 4b 2264 + 602973) + 1001(36,°b, + 106,3b.3 + 156,*b9b3) + 8008(6,%3 + 3b1°bo7) + 19448b,7b, + 4862b,°] 2! + [O19 + 6(2b,b9 + 2babg + 2bgh; + 2b yhg + 057) + 78(b7bs + b97bg + bab? + b32bg + 2b bab + 2b bgh¢ + 2bybybs + 2bgb3b5) + 182(2b,°b7 + 227d, + 2b,b3° + 3b,2b,? + 36,76, + 6b7babg + 6,7 b3b5 + 6b1bo7b5 + 120, b2b504) + 273(5by'bg + 206326; + 200, %b3b4 + 206, 53b5 + 306 1727b4 + 306;7b2b5? + 55°) + 2184(26)75 + 50,03? + 5b,7bo* + 100,426, + 206 ;7027)5) + 12376(b,8b4 + 6,%bob3 + 5b40,5) + 15912 (20,73 + 7b1%bo”) + 755820 ;8b. + 1679 66,2 |U Phil. Mag. 8. 6. Vol. 19. No. 111. March 1910. 2B 370 Mr. C. E. Van Orstrand on + [by + 13(0,b19 + dob + b3bq + Bid, + D5D¢) : + 91(b42D, + Ds2b7 + bbs? + b32D5 + Dgh,2 + QD ybabg-+ Qdybaby } 4D dabe + 2bodabg + 2bababs) © , + 455(b2bg + by9Ds + bobs? + BD 2baby + 3b:2b3bg + 3d :)270g | 4+ 3b 2babs + BD bob22 + BD 52D, + 3b02b3b, + 6B1boba0s) + 1820 (by4b, + bots + 20132 + 204233 + AD12babg + 4d Dabs | 44D Do8by + 6129205 + 6D 1052052 + 120,2bob3b4) + 6188(b,5g + bybo? + 5by4b9b5 + 50,43, + 1005.20, + 100,%bob3? + 10b,7b23b3) + 18564(d,%; + 3D,5D32 + 51205! + 6D Baby + 1514.25) + 50388(b,"b,-+ 70;ob3+ Thpbs°) + 125970(b,8b3 + 4bby?) + 2939300,%, + 58786),2] 2 + [dyg + 7 (201011 + 269049 + 2b3b9 + 2b4bg + 2057 + bg?) + 35(30,7b19 + 3bo7bg + Bbob5? + B37bg + Obybaby + 6b, b3b3 + 6b,b4b7 + 6b,b5bg + 6b2b3b; + 6bab4bg + 6b3b4b5 + 04°) + 140 (4)%bg + 4b5%b, + 6b17b5? + 60224? + 12b,7babg + 1207b3h7 + 126,b.7b, + 12b2bibg + 120 1b37b; + 1261364? + 12b,2b3b5 + 12b9b37b, + 24b,bob3bg + 246 ,b5b4b5 + 0") + 2380(by2bg + bobg + 2b2%b5? + 4032), + 40;2b3b, + 4b7b405 + 4b,b.2b; + 4bybob33 + 6b17b52b, + 6b,7b9b2 + 6b77b37b, + 12b,2bob3h; + 126,b.7b3b4) + 1428 (66,°b; + 150,40, + 206,753 + 306;*b2b, + 30b;*b30; + 306 ,b54bs + 606,3b57b; + 600 17b0°b, + 900,7b57b," + 120b,?b2b3b, + 6.8) + 27132(68b, + 3b)2b2° + 6b;?babs + 667364 + 15040574 + 15b,4b2b3? + 206,°b.°b3) + 19380(40)'b5 + 14b,°b3? + 280,%bobs + 35b,4bo* + 840,573) + 67830 (30,80, + 24b,'bob3 + 28b,5b,°) + 248710(2,°3 + 9b18b5) + 11440666,1%, + 2080120,” |2¥ ieee : Reversion of Power Series. 371 In order to obtain a complete check, the numerical co- efficients in the above series have been computed twice with Professor McMahon’s formula and once with formula (8). Use has also been made of the partial check obtained by putting for in this case e=y-yPty—..., ‘as 1s otherwise made evident by writing the original and the reverted equations in the respective forms, y=(#)(l—2)-! and s=y(1+4+y)-. ‘This result suffices to establish a theorem in regard to the coefficients of terms of all orders and of a given weight, viz.: the sum of the numerical coefficients of the terms of even order is greater or less by unity than the sum of the nume- rical coefficients of the terms of odd order according as the weight is even or odd. There are a number of special series reducible to the form (1) and therefore capable of reversion in the usual manner. Such are for example equations containing an absolute term c. It is then sufficient to replace y in (1) by y»=y —c. If the power series contains even powers only, it may be written y=aX? + a, X*+a,X5+..., and this series is reduced to the form (1) by the substitution v=X?*, If the coefficients of the first successive powers of « vanish, Y=CHAm- 18" Fay t™ tit ..., and the required transformation is Yr =(Y—-) + Ani =a2"(1 + aya + agx? + ...) yy = 2 + By2? + Boa? + .... There are m reversions in this case corresponding to the m roots of yj. Other series containing zero coefficients are reversed by substitution in the complete expansion. Thus, if the series 2B2 372 Mr. C. E. Van Orstrand on contains odd powers only, and the reversed series 1s wot bez? + [b, + 3b.7 Je? + [bg + 8bab4 + 120.3] 27 + [bg +10bab6 + 5b,? + 55b—2b, + 55b9° |Z” + [ Byy + 12bobs + 12b4bg + 7897, + 78b2b,? + 364b,%b, + 2736," |2™ + [ byg + 146.19 + 140403 + 76,” + 105b,7bs + 210b2b4b, + 390," + 5600,°b, + 840b97b,? + 238002", + 14286,°]<¥ Fase Bieler . Again, if the series proceeds by alternate odd powers beginning with the first, b=), =b; =); = be=b;=lbo= ... =0, and the preceding series reduces to eset hye? + (bg +5b7)2° + (by + 140 ibs + 3504? )2? 4- 0. An important case arises when the number of coefficients. which do not vanish is finite. The reversed series is then an expression in terms of an infinite series for one real root of a polynomial of the mth degree*. The first terms in the solution of the quadratic, cubic, and biquadratic are given below. (a) Solution of quadratic. Y= At + ay” c= a—l,2 wee tbi2? + 26,72? + 5b 324 + 146,42° + 426,226 + 1326,827 + 429,728 + 14300829 + 486262210 + 167966, 24 + 58786)b,U-" + 20801206,2-" + |. _ * Joseph B. Mott, “On the Solution of Equations,” The Analyst, vol. ix. 1882, p. 104. Merriman and Woodward, ‘ Higher Mathematics, Melee , Reversion of Power Series. 373 - (b) Solution of cubic. (1) Y = Apt + aya c= a—lya® SS ee ee B=L toe? + 322° + 120.72! + 5dbot29 + 273d72" + PES we (2) y=agv+ Aye? + Ayn” cma —hy a" —bya* ga=ithet+ [bet 2b? 2? + [5b,b,+ i a + [30,7 + 21b,7b2 + 14,* \2° + [2801.7 + 840;%b2 + 42b,° 28 + [120.7 + 180,205? + 33004) + 1320,° Jz’ Pe.) + [1650,0,° + 990050," + 1287,"bs + 4290," Jz (ec) Solution of biquadratic. (1) y=agr + a3a*. 2=a—bzn". w= 2+ byet + 40227 + 2232" + 1405342 + ...- (2) y=agr + age +a. =o — boa? la, Laz tbo? + gz" + 3d72° + Thobsz® — [ 4b, _ 126," | 2! “+ 45b_7b32* = [ 55dqb3° + 55d,* |2° + [2263 + 2860,°bs |2!°+ [ 5460,7b3? + 27307 Jz" + [455b.b35 + 182052*b; | =!° + [ 1408,‘ +4760,%b3" + 14286,°]z° + ..,, (3) y= aoa + ayx? + aga + age". c= a—),a? — by2* — ba". wHZtd 2+ [bgt 2b) ]2> + [bs + 50d. + 5b,*] =". + [6,b5 + 3b.? + 215°. + 14,* 2° + [7bob3 + 28(5,°b3 + byb,") + 846°b. + 426,°|2° + [8b.b, + 4:2 + 726, b.b; + 12b,° + 1206;°b; + 180b,2b.2 + 3306,%b. + 1326,° |=" + [455,b,? + 455,°b3 + 1650,b,° + 495b,7b.b3 + 4956;'bs + 9900,°bo2 + 12875,°b, +4290;"|=° 374 Mr. C. E. Van Orstrand on The preceding expressions are not always sufficiently conver-- gent* to be useful in practice. They may be made convergent, however, by substituting an approximate value of x in the original equation. It thus becomes possible to obtain all of the real roots of any polynomial. After one or more of the roots a, &, a3,... have been obtained, use may be made of the relations f Ay = — (% +4, +a3+ Arte etn) Q,= (aya tayas+ ... &n—1n) a= — (21 toes + Oy h3ty PF ... An 2m —10n) An (—1)”(ayetgetg 2.4 On —-1%n)5 in the evaluation of the remaining roots. A method for determining all of the real and imaginary roots from a single series has been given by McClintock ¢. Following are some examples : Bian) OSP pene (1) y= cosh X-1=5, +75 + G 4+ -- =ytntht c= oy h=— 4: b.= = ae i a 90 2 se"t sif0 2 @Qye za) dena gee = 0°8862 2693, +0:2320 1367y°+0:1275 5618y +0°0865 5213y7+0-0649 5962y?+0-0517 3128y!"4..... * For a proof that the reverse series converges in the same domain as the original series, see Harkness and Morley’s ‘Treatise on the Theory of Functions,’ p. 116. + Burnside and Panton’s ‘ Theory of Equations,’ Chapter III. ¢ Bulletin Am. Math. Soc. i. 1894, p.3; Am. Jour. of Math. xvii. ae pp. 89-110. Merriman and Woodward’s ‘Higher Mathematics,” p- 27. Reversion of Power Series. aTd (3) 2—2e?+20c=2 1 di < We On i i 1 ok Ueki! Ghee =0°1+0:001 —0°00003 —0°000002 =0°100968. | &=4,+2 2 + 823 + 240¢/°+322,=1 1 me 3 1 fo Peas tat + 39° 1 2 1 1 aq 393 A= “ b= (3 9s=— 35° &=0°0312500 —0:0007324 + 0:0000266 —0-0000011 =0°030543 x= 2°030543. Substituting the original letters y, a, @, 4, shah) SE arranging the terms with respect to the successive powers of the a’s, the general equation is *— a? 3 ay a ~ Ay a2 alos y* an _ BY, [2% -2 0 3+ BSF ag ME rt be wade Ao Ay Ao 7 a Ay Ay Ao a, ae ly 414% —21%1, © 4.6% oly gM ane] ay dp oi Ay” pa ae, 5 +[ 2%, 4.84% @_ 93% % ly Ag° Ao Ag” My 2 6 ay ag” 4 Ag Az as; a +t ta) A — = Fe Ay Ay da. + [132% ~ 330%. @ 4.190 Apt Ag a ri Ay? ( - 9” - a2 bay as a ag” “) ay a Ay a2 a Ay” 4 3 ag |y' acs. +2 ie i a : do Ay ao ao 7 ay! * See also “Inverse Interpolation by Means of a Reversed Series,” this Journal, May 1908. 376 Mr. J. L. Hogg on Friction in 7 5) 4 og) | 1ae7 ee L a’ Ay’ ao Ay* Ay 3 Oey oe 047 Ay G3 "Oe +1655, (—6% A) 45 (11% a) a De Nie A, Ao a A ~ Ay Gs. a Ay As A4 Ag + 15%(11@ 32 oe ao ag Ao ag Ao Ao ao 8 A> Ag As As As A, Az} + 98(— 52% 4M), 90 Se a1y a aN ° The last equation is quite sufficient for relations involving experimental data. In many other cases, however, such for example as the construction of mathematical tables, it is con- venient to have the equation in the more extended form, and thereby save the labour of making many tedious transforma- tions, or of devising new methods for the development of the inverse (or anti) functions. The second of the preceding problems is a case of this kind. Another important application consists in being able to estimate, without reversion, the error due to neglecting terms of a certain order in the reverse series. For example, the error due to neglecting the term in z” in problem (1) is of the order of magnitude of 98786 x 12-1 x (2y)", or roughly 3 x 10-*y", while the same error for problem (3) is approximately 6x 10-®xy”. Since in the first case y may exceed unity, the series will not always suffice for computation, but in the second case y=0°1 and inspection shows that the remaining terms in the co- efficient of 2!” will not greatly exceed 58786 x 6", consequently. the error arising from the omission of he Ah. 7 TO. Lo hy 92 hh Pe -987 1°56 | 2:0 27 2°65 | 10°28 40°4 1515 6 x10 | x10°° | x10" Sulphur. \- Time Dane an | 6m aoe 50m.|25h.| 19h. | 27h gerne 816 “818 83 84 06 >). 39. | tae 3°54 x10°° 10° Indiarubber. Time a 5b m. | 1 in. 2 m. 4°5 m. | 12°5 mj S22 ae 60 m. | ‘Tir saat hk ae ORGS LL? ——— ns eo ened es Der ccrs | 057 ‘084 "105 "136 ‘29 (2 1:66 | | 16° | | | x10°° x10 | Fused Quartz. | | Time ...| ‘5 m. | 1 m. 5m. 15 m. 20 m. 25 m. | Bee Ses ‘O71 112 “179 24 39 1:15 x10°° 10° 7. NATURE OF CURRENT DURING SLOW POLARIZATION. The electrical movement within the substance during slow polarization may be a translation of free electrons, as in metallic conduction, a transfer of charge, as in electrolysis, or a separation of the internal charges of the molecule, so that its electrical moment increases. In order to determine which of these occurs in the present Dielectrics in a Steady Field of Force. 405 case, a flint-glass ellipsoid was suspended in a glass cell (fig. 3), the upper part having pole plates, so that a field Fig. 3.—Testing Cell for exposure to Radium. Ibe) Wii “IN. NZ =e 4 Ny N F AY | AN SS SSS Mri itr NM NJ % Z Z cS Ms oe RSS N N Be ‘B = N : N | N N \ IN N N ee N acind f N ic -Pole plates. of) N N N ' IN N N eeeeress oN By i [er i N N ——- Radius eS : N [| enaearen y- mae Hes. aoe {_ ee ee Saree smears * i a ii i a ass could be maintained there, and the lower being filled with disulphide of carbon. Outside the cell a massive iron cylinder was placed, having a small central hole pointing towards the line of suspension. The ellipsoid was exposed to the field for several days, and its steady period noted. It was then lowered into the liquid and exposed to the radiation from a strong tube of radium bromide placed in the end of the iron cylinder. After five minutes’ exposure the ellipsoid was raised, allowed to stand in the field for the same time that it had been removed from it, and its period then found to be unchanged. The liquid was used to shield the specimen from any ions which might possibly gather on it from the poles, and to prevent any change in the gradient between the poles by the diffusion of ions from the line of the radium discharge. If the charge on the ellipsoid had been in any sense free, whether residing on the surface or inside, it would have been rapidly removed and the period lengthened by exposure to 406 Polarization of Dielectrics in a Steady Field of Force. thé: radium. The experiment was repeated with the con- ditions varied in several ways with always the same result. It would appear from this that the electrical movement in a dielectric when isolated in the field is entirely confined to the molecule, and is therefore neither metallic nor electrolytic in type, but is the continued displacement of the atomic charges to a greater degree of separation than has been hitherto recognized. It may be remarked here that the high values of the dielectric constants obtained in the paper are confirmed by measurements on electric cables: Ashton* found that an ozokeritized rubber cable charged at 206°5 volts for 7200 seconds, absorbed 13°8xX10-5 coulomb. From the given dimensions the dielectric constant corresponding to this is 13‘1. In this case also the slow polarization was elastic in type, as shown by the fact that the quantity discharged on short circuit was just twice that absorbed at constant potential. It was early suggested by Mossotti that the behaviour of a dielectric can be explained on the assumption that it is made up of isolated conducting spheres. On any of the present electronic theories of matter the separation of charge under the influence of the field gives an arrangement similar to this. It is now suggested that the first stage of polarization pro- duces such a charged system, but that it differs from a system of conductors in that the charges undergo further separation automatically by reason of the attraction and quasi-elastic approach of opposite charges on adjacent molecules, The limit of the first stage of “rapid polarization is that of the re-arrangement of atone charge within the molecule, that of later stages the establishment of electrical strain throughout the mass under new conditions of internal stress set up by the first polarization. The range of the dielectric constant in the first stage ey be taken as the intercept on the vertical axis in fig. 2. Writing this Ko, it is of interest to compare it with the values fetid for the same substances in alternating fields, asin Table III]. The influence of frequency may from this be anticipated. For those substances in which the ratio is nearly unity it will be at low frequencies small, but for the first few in the list a greater change may be expected * A, W. Ashton, Phil. Mag. no. 52, 1901, p. 501, “On the Resistance of Dielectrics.” +t Vide Roy. Soc. Proc. A. vol. Ixxxii. p, 422 et seg. On Saturation Specific Heats, &c. 407 between zero and 80 alternations a second than from 80 to the frequency of light. TABLE III, Ratio of the Initial Values of Dielectric Constants in Steady Fields to those in Alternating Fields. ae | Ky. gee | Co | || a ey eee | 4:6 | 13 2°8 Mimt-class A 41 ...............-.- 8:5 25 2°9 Dako oe 3°09 7:0 2:3 OCC CE 2°72 | 19°5 6:9 | Paraffin-Wax ...........ccccceses. 2-32 | 5-6 a | RPIRT ONE, | vusevcnasseceeeroeene 4:43 | 70 1:58 BPP GIIE, 202020020. ..0Jsusiedeeesscce. 403 | 4°38 1:07 | U4 |) 3°08 3-4 e. 11 Oo er eenee 2-79 42 PHLT MN i Se Ee 4°54 4°55 10 Pint, Gilags A 3°30... 2.00.00 ss000. 6°98 70 10 PND Nair cndae dakve wo ssn secs gators 28 2°8 10 BEM WAS 2 .c0ccscuneseseescescse 4°56 ce. 46 1-0 =e —— — . = ate = de — a ee ————_————— = - a i TE XL. Saturation Specific Heats, §e., with van der Waals’ and Clausius’ Characteristics. By Rosary EB. Baynus, IA = [Plate VI.} VER twenty years ago, and shortly after reading Planck’s 1881 Memoir “ Die Theorie des Siittigungs- gesetzes ” in vol. xiii. of Wiedemann’s Annalen, I calculated in the following manner the values for different temperatures of the latent heat of vaporization and of the saturation specific heats on the assumption of van der Waals’ equation, as also the values of other magnitudes. I did not publish my results as the equation does not represent actuality ; but such of them as have not since been published by others I desire now to record, since they contrast very greatly with the results of similar calculations which I have lately made on the assumption of Clausius’ characteristic. : * Communicated by the Physical Society : read January 21, 1910. 408 Mr. R.E. Baynes on Saturation Specific Heats, &c., The thermal capacities of a fluid with van der Waals’ characteristic (p+av-*)(v—b) = Ret being defined by ‘dH = kdt+ldv = Kdt+Tdp, their values are Rt R te | Se ee Bee 1—2a(v—b)?/Rtv” where & is in the general case a function of ¢. Their further calculation is simplified by using the charac- teristic in its ‘reduced’ form (7 + 3v-7)(38v—1) = 8r, in which the units employed are the critical values P, V, T of the pressure, volume, and temperature respectively, in terms of which we have i ake. d= a R:=36PN /a8 and if, to simplify our calculations, we introduce * a new variable wu defined by = +,/{(3v—1)?—4771'*}, which will be real in all the cases we need consider, and for shortness of expression put L=—- A =3v-l, B =468v—1—p), C—43B8r—1+yp), H=i(4r-1+n), D=H3rt1—p), B=iGv+l4p), F=18-3—p), G=13—3r-4p), we have m7 = BC/r’*, Te ADH ey, /P = DE/’, (K—&)/R=DE/FG, L/V = —ADE/3¥G. To determine the variations of the thermal capacities we may plot their values for given values of 7 against the values of zw orv. We easily see that the Iz, Lr, Ly curves may present singularities while the /y curves do not; that the value of K— &, which is positive except when v lies between 1+4y and 1—4y, i.e. except when v lies between the largest pair of the roots of the equation 47v? = (3v—1)? which are * This variable was also employed by Ritter, Wien. Sitz.-Ber. July 1902. with van der Waals’ and Clausius’ Characteristics. 409 all real when <1, is always a maximum when v=1 what- ever the value of 7, the maximum value for 7 being Rr/(r—1) and the corresponding reduced pressure being 4r—3; that K—& has also a minimum value R when v=d, this being also its value for any value of tr when v=~. If the isopiestic for 7 cuts the isothermal for 7 in three points and y, v”’, v’ denote the corresponding volumes in ascending order of magnitude, then, if yu is defined as above strictly with reference to y, vy’ = v/U, ¥ = 0/5, and the heat-capacities corresponding to v’, expressed in terms of vy, are given by ’/P = ABD/r’, (K’—k&)/R = AD/uG, L'/V = —ADE/3yuBG. Case of Saturation. Now, if a is the saturation-pressure at t, we have also the relation 3(7 +3/w’)(v’—v) = 8r log {(3v’—1)/(3v—1)}, so that, in the case of saturation, v and yw are connected by the relation log (E/AB) = 3vGH/ADE, which may be looked upon us the equation of the liquid side of the connode or boundary curve. [ The equation of the vapour side of the connode is of the same form but with the sign of « changed ; for we similarly obtain hs bv +1—p im dv'(3—3v'—p’) (9v'—1—p’) ® GV —NG@/—1+p) (@Bv—1)Br'+14+p) 8 41—p) if w= + V7{(3r—1)?—4rv*}.] From the above equation we may determine the value of y for any value of yp, or vice versé, and then at once obtain the corresponding values of the saturation pressure, temperature, and vapour volume 7, 7, v/, as well as of v'', so that we can easily plot the curves connecting these magnitudes with p, or indeed with v or 7, &e. We easily find that ~=0 for both y=2 and v=1 and that it has a maximum value *3065658, to which correspond yv="5430962, v' = 3°365713, w="4713545, r=°8371466. 410 Mr. R. E. Baynes on Saturation, Specific Heats, &¢., We further find, by somewhat labori ious work, for the case of saturation sd shone a — —EF/pAB, — = DEF/8r' ; dv and consequently / dm _ sBH/ADE, = — —8r/uABD. adr The saturation specific heats s=— K+ lLdp/dt, s' == K’+ L'dp/dt, are thus given by ($s —k)/R = (DE—BH)/FG = 3r/F (k—s")/R = (HAD) w= 3y/n, whence alsa ($—s')/R = 3vG/uF. Furthermore, the latent heat of vaporization » is given by r/PV == (v'—v)t dr/dt = GH/r’ ; the aul: of vaporization w by w/PV = a(v'—v) = CG/r ; and the real latent heat »! by NM/PV = 2r/PV—7('—1v) = 3G/v. We may hence tabulate the values (p. 411) of, and plot curves for, all the above magnitudes with either vy, 7, or 7 for abscissee (PI. VI.), and we at once see that : (.) 3—k is always positive, increasing from R tow ast increases from 0 to 1: Gi.) 8’—& is always negative and has a maximum value —4°95824 R when p= “288953, to which correspond y ="477565, v'= 6°64469, ow = 238058, a 72433 5s at ae —« when Tis either 0 or 1: (il.) inversion in the sien of s’ (which is negative for the highest and lowest temperatur we will therefore take place if &/R>4: 99824, or, on the assumption that &/R=N-+ 4 for an N-atomie gas, if N>4: 45824, so that inversion will occur if the gas has at least jive atoms in its molecule: ta van der Waals’ lately published Lehrbiich der Thermo- dynamik, edited by Kohnstamm, it is found by an approxi- mate calculation that for s’ to be positive k == 1+ R/& must All x as ~~ oes S =) a S =~ Ss — S “x = “—_— % = is) La) SS AS) = ices a) mw 3 Ss = D SS) = i] > — “=> = S 05 ‘07399 ‘08524 12418 15441 18568 "18824 ‘23018 26254 ‘26682 ‘28199 28895 29955 30306 30657 30641 30564 ‘29642 ‘29089 28864 ‘28345 "27353 26795 26013 "24156 21437 | "16922 0957 1: Vv. ao 833835 5) 358 36176 10 38602 “B9197 4 ‘42191 44596. 45 467 24 47757 dD 512 54310 ‘DD 56033 60340 *619538 62538 ‘63785 ‘65903 ‘66988 - 68412. 71494 "75514 81459 ‘9 | | | a3 975 « 107 1°34 x 10° 2°59 x 104 911-12 212-00 T7759 68°026 23°706 11-836 10°820 78054 66447 49888 43960 3°3697 32015 2°9856 2°3488 21826 21289 20246 18734 18064 17271 1°5830 '1°4369 1-2785 1°1220 1: ‘OO000 ‘00000 00002 ‘00097 ‘00495 ‘01533 ‘O1784 ‘O5875 “12753 14076 “20062 °23810 32054 *36405 “47135 -49378 | -52627 64700 “68609 69946 72659 ‘76851 ‘78806 ‘81188 "85670 ‘90298 - 95075 ‘OS7T94- I: 8')/R, ‘0 | i * L 15306 || 11053 | 21- 22°1058 "2165 11598 145146 15:6744 ‘2443 || «11-1864 «| 12°7317 | 13/9181 ‘33834 || 11-2851 90597 103448 ‘39785 || 1:3725 7D 88725 ‘45950 | 41-4718 65 79718 ‘46920-14891 6:3748 78589 ‘56175 || 16880 55 71830 ‘64249 =) 19119 50960 70079 65482 19520 50596 70117 70011 2-1298 4-9708 7-1006 "T2433: 92-2415 4-958 7:1997 ‘77003 2-4991 50075 79066 79115 || 26461 50683 path 1 83715 || 3-0622 53146 8:3768 84590 | 3:1622 58849 85471 ‘85816 || 33177 5-5 88177 ‘9 | 40524 61070 101593 9125 | 43705 | 63894 — 10-7599 ‘91668 44926 65 10-9926 925 47661 | 67509 | 11:5169 ‘9375 52766 | 72281 | 12:5047 ‘94319 || 55636 75 13-0636 ‘95 | 59705 78899 138604 9625 in 69909 88790 15-8699 ‘975 | 87095 105675 | 19-2771 ‘9875 || 12-6061 | 144378 | 27-0439 Rye apse Ms seas 646449 ; oO Zan der Waals’ characteristic. ; = | (@—kY/R. | (k—-8")/R. | s— d/PV. )- 89796 8:°9572 89442 88825 88041 86856 $6620 8°3520 79262 78491 7°DO85 72988 8375 55909 5°9630 58267 5624 48239 4°5402 4°4396 4-229] 3°8841 Slee 3°4949 30447 2°5006 1:7786 “8788 ‘0 m(p’—y). ‘() ‘4082 ‘D774 "6515 "S856 1°0465 11859 1:2061 1°3680 1°4526 1°4597 14722 14684 1°4388 1°3305 1:3093 1:2763 1°1293 1:0724 10517 1:0076 9333 "8956 “8467 “74387 “6156 “4413 "2193 ‘0 APY. 9° 85714 8:3799 82928 cond 79967 Ne Ada A - 7 {910 74996 74559 69840 64736 63894 60364 58304 53987 51769 4°6325 4°5175 43491 3°6946 534678 33879 32215 - 2°9508 2°8177 2°6481 23010 1:8850 13373 6595 0 412 Mr. R. E. Baynes on Saturation Specific Heats, &c., be less than 19/17, which is equivalent to &/R>8°5. Dalton (Phil. Mag. April 1907, p. 537) has given the condition «<1:202 by an interpolation method, “and the foregoing calculation gives «<5°95824/4°95824 — 1:20168. (iv.) g—s' is always positive and bas a minimum value 7:00789 R when #=*262535, which correspondstov=*445963, 0636477 — 12730. r= 642488 : (v.) the work of vaporization has a maximum value_ 1:47215 PV =:55206 RT when w=:281990 (as also found by Ritter), to which correspond v=*467237, v’=7°80538, a =°200617, r=°700110; it is 0 when 7 is either O or 1: (vi.) the greatest value of both and 2/ is 9 PV = 27 RT/8, this occurring for w4==0, v=4, w=0, r=0, and both decrease continuously to the value 0 at the critical point. Inversion of the sign of s' occurs where 3'=0, that is, where 3v/u=k/R, and we can easily see, as was pointed out by Raveau* and earlier by Duhem, that this is at the points where the entropy along the vapour side of the connode has critical values. On the assumption that k/R=N+4 we may therefore determine the temperatures of inversion for different values of N either graphically G.) from the diagram of the values of (&k—s')/R in terms of 7 or (ii.) from the diagram giving the values of v,7, 7 in terms of mw, by finding the inter- sections of the lines 3v/w=N +4 with the vp-curve, or thus by calculation. If we put n =3(N—4)/(N +4), the desired points on the connode correspond to w=(3—n)v and are therefore given by he (6—n)v+1 av(3—nv)}(12—n)v—1} SP (ip ad )(rnv—1) ~ (8v—1) (nv +1) {(6 —n)v+1f? and for these we have 7= (nv—1)}(6—n)v—-1} / 4 t= (3v—1) (mv + 1) {(6—n)v +1} / 320%. We also have for these points vy’ = 2v/(8v—1—p) = 2v/(nv—1), i.e. v=v'/ (nv'—2), or pepe) ev 7) bv’ (nv! ne yi tT} (83—n)v +2 ~ (8v'—1)(nv’—1) {(8—n)v! + 2} w=(nv’—2){(8—n)r +1} /v" T= (8v’~1)(nv!—1) {(3—n)v' +2} /8r?. * Journ, de Phys. 1822, p. 461. with van der Waals’ and Clausius’ Characteristics. 413 We thus get for the points where inversion of s' occurs U Vv. T. | y N. “p' T. 7 5 2°9855 52627 85816 6 2°1289 ‘69946 ‘91668 | 7 18064 ‘78806 94319 8 9 23-706 | 05875 | “56175 | 77-759 | 01533 | -45950 21200 | 00495 =|: “39785 543°88 | -00172 | 35353 | | | ' 163 | 840 | -958 152 | 873 | -967 13628 | 000623 | -31918 10 1°45 “899 os 33894 | 000229 | -29136 7 1-40 ‘918 ‘979 | 8430-1 000s 26818 12 1:34 933 | 983 21036: | 0000315 | -24849 I have lately been interested to examine by this method the values of the heat-capacities, &e. of a saturated fluid in the neighbourhood of the critical point. For though v. d. Waals’ equation is not a correct representation of the behaviour of gases, yet it represents so near an approxima- tion that its indications are of value. If we put « =1—y, then in the neighbourhood of the critical point w and w are very small so that we may expand the terms in the equation of the connode in ascending powers of w and w; we then obtain 5, 4-382 +p _ __ d(1— 4) (82+ w)(8—9x + p) ~ ©? (2—32)(2—30—p)~ (2— 3)(4—30—p)(4—3e + p) 9 wel: =— 1091 (3+ p)?4a—pwt 3 (172? —172p—2y*) + +(312°—31e2*5— Tap? —p*) bi, rie + 55 (83704 —837a%e— 275.27 u?—79.1y3—6y!) +... whence 2 16 229 Bes wy Ee ted or L= pt 5H + 55H + 75 +... or and thence we deduce LO RBS og 129 2, + B9BT 4” ~ 90” ~ 300 414 Mr. R. E. Baynes on Saturation Specific Heats, §c., If now we put 2? =1—T, we get 26) een a teen = De 2 ee oo TE: OEE OO TE 1—9e4. 18,2_ 1475, 1992 . ET eee ae 875 © yA oe GS Cae Ge 7w=—1—Az ie Tae ie Wee o Pel soemn tie ($—%&)/R= 5¢ (l— x24 50° gree Fs ) + Lis tal a are da (&—s!/R= 5271+ get et eet ) NEV ede ) = 1oz( Be te nears D3, T™(v —v) = 41. — 552 tale Thus in the neighbourhood of the critical point we may write 7=47—3 or, with greater accuracy, a =1—*8(6r—1)(1—7), and also A/RT=6V7(1—7) or, with very great accuracy, A/RT="12(27 + 237). /(1—7); further, the work of vapor- ization is one-quarter of the latent heat. | We likewise see that the mean of the saturation-densities near the critical point is *1(11—7) if :04(1—7)? is negligible, so that within this limit only the law of the straight diameter is exact with v. d. Waals’ characteristic. Saturation with Clausius’ characteristic. With Clausius’ characteristic Rt c ye oe t(v +B)? the critical state is given by V=3a+2@, P?=cR/216y’, T?=8c/27yR, where y=a+8; and if we _ write v=(v+ )/(V +8), the reduced characteristic becomes ST EA a 2 ee 3v—17 cw?" % : J ; with van der Waals’ and Clausius’ Characteristics. 415 In the case of saturation we obtain from Planck’s mémoir (loc. cit. ) . =3(r+1l—rcos¢), v=" +1+rcos¢) | _ 3/38(r+1)er sind ~ 99? sin? db +2741) 6\/3(7? sin? d —1) ~ (r +-1)2(7? sin? 6 + 27 + 1)r sin a A/t 8r*sin? 6+ 4r+1 | RT~ (7 +1)rsin ¢ tan db’ where cos d cot” d ri = ; “Tog cot 46— cos’ our previous mode of calculation‘then gives for the saturation specific heats g—k J k—gs' 2(r? sin? d6-+27r4 1) eek rem (7? sin’ 6— 1)?—3(7 tht 2" adh Se 1)rcos¢ (7? sin? d—1)(2r+2—7° sin? d)—2(r4+1) ” and the work of vaporization w is {RT zr cos gs Calculation for different values of ¢@ gives the following table (p. 416), whence it appears that (i.) s—%& is always positive, is infinite for 7=0 and r=1, and has a minimum value 15°333 R a) T2"foead: (ii.) s’—f& is always negative, is —« for r=0 and r=1, and has a maximum value —11°355 R for t= "80579: (iii.) $—s' is always positive, is © for T=0 and r=1, and has a minimum value 26°741 R for r="81962: (iv,) the latent heat > increases continuously from 0 to «© as T falls from 1 to 0: (v.) the work of vaporization has a maximum value °68567 RT when r=°76610, being 0 for r=0 and r=1. With this characteristic too inversion may occur in the sign of s’, but now when «<1+088 or, on the former assump- tion that */R= N+4 for an N-atomic gas, only if the gas has at least eleven pie in its molecule. The contrast between these conditions and the corresponding ones for v. d. Waals’ characteristic, especially in regard to s-k and A, is very marked and is very clearly shown by the curves that have been plotted. This contrast subsists further in the neighbourhood of the critical point; for, if 9 as before we write no= Vin(Vr aa AP at LL (5) adbE+ bint +82 =VIE(Vr—r) cand from these equations values for a, 6, c can be at once written down. 7. Before discussing the choice of axes, it will be convenient to obtain general expressions for the probable errors of a, b, and c. The residuals Vr—r may be regarded as the sole source of error, since all the other quantities involved in (5) are known with ample accuracy for our purpose. In dealing with the various errors which may affect the values adopted for the quantities Vr—v, let the suffix 4 distinguish true values, while the suffix , indicates those quantities which are variable from one Vr —7 to another. Thus let V=Vo+2, r= 4- hrs i Warenternt GB) so that v is the error in the accepted value V for the velocity -of light, and & is the proportional error in the value assigned to the solar parallax. (The angular magnitudes involved in our estimate of » can be so accurately ascertained that no appreciable error can arise from them.) 8. Again, let (T»); denote the time actually taken by light to travel from the satellite at mid-eclipse to the observer. Then, (79); being the difference between the apparent and the -actual time of mid-eclipse, the value ts which is available to us will be liable to differ from (79), owing to three sources of error :— Gi.) The irregularity of Jupiter’s surface causes the actual eclipses to succeed one another at intervals of time which are sensibly discrepant from those calculated theoretically. Thus Ts is affected by an error of prediction which is periodic or quasi-periodic ™ in character, and which we denote by y,. * Though Prof. Sampson tells me that no period has been recognized -in the residual errors. 2H 2 420 Dr. C. V. Burton on the Sun's Motion Gi.) There will also necessarily be some slight uncertainty as to the epoch of the actual eclipses, and tr, will on this. account be liable to a small constant error (y). (iii.) In determining the apparent time of mid-eclipse, am error of observation 1s “incurred: let the error thus introduced. into the value of 7, be called v,; we have accordingly T:=(To)stHs ty + vs- e ° e . ° (7)) 9. For any single eclipse-observation, to a sufficient approximation, Vr—r=VoTo—1+ Viurstxsty)+tv—kr.. . (8) Without, in the first place, taking account of the motion of the solar system through the sether, we may make y very small by making use of a number of eclipse-observations. extending over at least one Jovian year. Moreover, y only enters into our results in association with factors which tend approximately to disappear in the long run ; its effect may therefore be disregarded. [or similar reasons our final results will not be appreciably affected by the small errors v and k. (When the numerical values of a, b, ¢ have been found from (5), it will be possible, without much additional labour, to determine more definitely the influence of given errors y, v, k on the values found for a, b, c, as well as the probable errors in a, 6, ¢ arising from an assumed “all round” value for the probable observational error corre-. sponding to a single eclipse. For the present we must be content with a more tentative estimate of the accuracy to be: expected. ) LO. Finally, since the residual effects for which we are searching depend on the direction of the line drawn from the carth to Jupiter, and have thus a principal period of about 12 years, it is especially important for our purpose that the Jovian system should be free from any influence having an approximately equal period, and capable of affect- ing the times of eclipse of the first satellite. Prof. Sampson kindly went into this question with me, and found that no disturbing influence of the period in question was to be. feared. We may accordingly confine our attention (in this preliminary discussion at least) to the errors typified by Ns Us. 11. Referring now to equations (5), we find that the error Vs+us in a single observation gives rise to an error in a equal to Be ae te St 2 SE Sep Ste i a3 ey Day ane | ; where D= | Em Sy? Zl & 2g 2S" aCe 2G Se with respect to the Aither. 421 Thus if ¢ represents the all-round probable value of the error ¥,+u; for a single eclipse, we shall have ae (__ | & 2En B82)! p.e. in Oey: Si) gy Za?) SHG | Cs Ig 20? =WeD-i(Sy? Xl? — Syl. SnS)i3 - . - (9) as is found on expanding and simplifying. The probable errors in 6 and ¢ can of course be similarly expressed. 12. In order to obtain a preliminary notion of the con- ditions of the problem without undue labour, I have considered a simplified system which is not strictly speaking a dynami- cally possible one, but which represents the actual ‘system sufficiently nearly for the purpose in view. The observer is supposed to be carried uniformly round the sun in a circular orbit whose radius R is equal to the mean radius of the earth’s orbit, the time of revolution being one year. The Jovian system is supposed to move in a very slightly elliptic orbit, with the sun at the centre of the ellipse, the inclination, de of this orbit to that of the observer being 1° 18' 41, which is the actual inclination of Jupiter’s orbit to the plane of the ecliptic. The excentricity of the modified orbit of Jupiter is such that its projection on the plane of the observer’s orbit is a circle, this circle being uniformly described in 11 years 315 days, and having a radius R! equal to the mean radius of Jupiter’s actual orbit. To take account of the fact that eclipse observations are impracticable for some time before and after Jupiter is in conjunction, an eclipse is considered as possibly observable only when the : angle earth-sun-J upiter lies between the limits + (yr=102°, say). Between these limits, eclipse observations are taken to be uniformly frequent and uniformly weighty. With the axis of « through the node of Jupiter’s (modified) orbit, the axis of y in the per- pendicular direction in the plane of the ecliptic, and the axis of 2 perpendicular to the ecliptic, the results found are ey oe 2V'e? By | Poe OLS REIRIR sin apt R” IV 4-2 (p.e. in 6)?= as ME eg ad 8 be (10) nm K?(L— sin? yr/+p") eae 2V%e? R?—2R’Rsin f/>p+R? is ~ ‘sin? @R2R?(L— sin? W/p?) 13. In these expressions R=148 x 10° kilom., R//R=5:2, V=3 x 10° kilom. per second, 6=1° 18’ 41"; x, the number 422 Dr. C. V. Burton on the Sun’s ALotion of available observations, is about 330; and e, the “ probable ~ discrepancy between the observed and calculated time of am eclipse, is estimated by Prof. Sampson at about 4°5 seconds. The numerical values thus derived from (10) are p-c. in a=44°0 kilom. per second by) Ming tO od ee ? .- «5 99 C=10,000 7, ey) 14. The determination of the velocity-component c (per- pendicular to the plane of the ecliptic) is so badly conditioned that the investigation can hardly be considered to aftord any light on that point. It may be admitted as probable: that the velocity of the solar system through the eether is. very far below 10,000 kilom. per second ; otherwise we should have to suppose (for example) that practicall y all stars. whose radial velocities have been measured are moving through the ether with velocities of thousands of kilometres per second. ‘This certainly appears unlikely, though perhaps the possibility ought not to be too lightly denied. An in- direct argument against a very high velocity of the solar system may be derived from other considerations. For we know that the velocity components in the plane of the ecliptic must be relatively moderate, otherwise there would be a marked anomaly, having a period of about 12 years, in the observed times of the ‘eclipses of Jupiter’s satellites*, and a priore it is unlikely that the velocity of the solar sy ‘stent should be nearly in some arbitrarily assigned direction, such: as that perpendicular to the plane of the ecliptic. 15. On the other hand, if we begin by admitting complete ignorance regarding the velocity-components to be deter- e “only our assumption that they are small com- pared ea the velocity of light—then 10,000 kilom. per second will represent approximately the probable error in the value of the component ¢c ; and zn general the determi- nation of the component of v elocity resolved in any direction in space will be affected by a large probable error arising from the uncertainty in the value ‘of ¢. On this view, the exact choice of coordinate axes becomes of importance, the: relatively large probable error in } given by (11) being due to the influence of the error in ¢. So long as the axes remain as specified in § 12, this unduly large error in b can only be avoided (if at all) by an excessive arbitrary weighting of the observations, so designed as to secure the vanishins: * It may be hoped that the investigation now proposed will indicate these components more definitely, or at least as superior limit to them. with respect to the Avther. 423. of the coefficient of c in each of the first two ot equations (5); these two equations being then solved for a and b. 16. In the simplified problem of § 12, it appears that, without special weighting of the observations, the probable error in the velocity-component b can be greatly reduced by a slight change of axes. Theaxis of # remaining unchanged, the axes of y and < are to be rotated in their own plane through a small angle; and if the velocity-components along the new axes are called a, b,, c, it is found that the probable error in 6; is a minimum when the axis of y lies very nearly in the plane of Jupiter’s (modified) orbit. If we take that plane as the plane of xy, the probable errors, corresponding to our simplified problem, are: p-e. in a =44:0 kilom. per second y 1 9999 b= 49°95 ” ” i o : (12) bb) ) ¢,= 10,000 Sb) oe) using 330 observations, each with a probable error of 4°5: seconds. 17. In the absence of definite reasons to the contrary, it would be natural to take the plane of the ecliptic as one of the coordinate planes, but it appears that, by taking the plane of Jupiter's orbit instead, we can avoid increasing the pro- bable error of one velocity-component (6 or b,) by an unknown amount, and this without adding seriously to the labour of computation. In the actual problem, therefore, it will pro- bably be advisable to determine the values of two velocity- components in the plane of Jupiter’s orbit, using for this purpose equations corresponding to the first two of (5), and causing the coefficients {f£, {nf to disappear as nearly as may seem necessary by a moderate special weighting of the observations. We may then proceed more tentatively, on the assumption that ¢ is moderate, to identify the velocity- component ); so found with the corresponding component 6 in the plane of the ecliptic. If, for example, on general grounds, and in the absence of any positive information, we assume ¢=0+500 kilometres per second, the probable error in } thus arising will be approximately 12 kilom. per second, and the total probable error in } will be about ¥ (45°5?+ 12?) = 47 kilom. per second. I hope very shortly to consider this problem more in detail ; meanwhile I wish to thank Prof. Sampson for his kind and most helpful advice. Boar’s Hill, near Oxford, 9 October, 1909. Th DA al XLII. Rays of Positive Electricety. By Sir J. J. Thomson, WA., PLS.* i the experiments described in my previous papers on this subject (Phil. Mag. Aug. 1907, Oct. 1908, ea 1909) the streams of positive electricity were produced by means of a large induction-coil. When the discharges are produced in this w ay the potential-difference between the terminals varies considerably during the discharge, and the method is not suitable for those exper riments in which accurate measurements of the potential-difference during the discharge are essential. Tor example, I showed in my paper in the Phil. Mag. for Dec. 1909, that the velocity of the most rapidly moving particles i in the positive rays is independent of the potential- difference between the terminals and is equal to about 2x 108 cm./sec. A particle of the kind found in the positive rays, if it moved with this velocity, would possess an amount of kinetic ener oy equal to that acquired by the fall of the charge through a potential-difference of about 20,000 volts. It is clearly a matter of importance to investigate whether or not positive rays moving with this velocity can be produced when the maximum difference of potential between the terminals of the discharge-tube falls below this value. To test this point satisfactorily fairly accurate measure- ments of the potential-difterence between the electrodes are required. JI have, therefore, obtained a Wehrsen (Mercedes) electrostatic induction machine with two movable plates, this furnishes a very constant discharge through the vacuum- tube; the index of a Braun’s electroscope whose terminals are connected with the terminals of the discharge-tube remains quite steady. lL repeated with this machine ‘the ex- periments for which I had previously used an induction-coil, and found exactly the same resujts. In particular I repeated the experiments on the effect of the potential-difference on the velocity of the Canalstrahlen, and again found that the velocity of the swiftest rays is independent of the potential- difference between the electrodes. With the Wehrsen machine I was able to measure the potential-difference between the terminals of the discharge- tube with an electrometer; a method less open to objection than that of measuring the length of the equivalent air-gap placed in parallel with the discharge-tube. As a matter of fact, however, I found by measuring the potential-difference first by an electrometer and then by the air-gap method, using the electrostatic machine to produce the discharge, * Communicated by the Author. Rays of Positive Electricity. 425 that if suitable precautions are taken the air-gap method may be made to give very satisfactory results. Using the electrostatic machine I was able to get observa- tions on the Canalstrahlen with potential-differences between the electrodes varying from 40,000 to 3000 volts; the voltage depends upon the pressure in the discharge-tube, being much greater when the pressure is very low than when it is con- siderably higher. Tor this range of potential-differences and pressures the maximum values of v the velocity and e/in the ratio of the electric charge to the mass remained unchanged. As might be expected, the appearance of the phosphor- escent patch produced on the Willemite screen by the pencil of Canalstrahlen, when these were deflected by electric and magnetic forces in the way described in my former papers, changes considerably as the pressure varies. At very low pressures the phosphorescent patch was very bright and sharply defined. It consisted of a comparatively faint straight portion OA (fig. 1), ending in a well-defined spot A, joining on to this is a tail AB, in Vig. 1. shape approximately a parabola, touching the 0 vertical line of no electrostatic deflexion at Wi O the position of the undeflected spot. In many cases another bright spot could be ere observed between O and A. A a As the pressure of the gas in the tube increased the phosphorescent patch became more diffuse but the position of the left- hand boundary remained unaltered, the spread of the phosphorescent patch being to the right. ‘This left-hand boundary remained unaltered up to the highest pressures at which the phosphorescence was visible; at this pressure the potential-difference between the terminals of the discharge- tube had fallen to 3000 volts. The appearance of the phos- phorescent path at this pressure is represented in fig. 2. Since the left-hand boundary remains unaltered there are among the Canalstrahlen produced by a potential-difference of 3000 volts some particles moving as fast as the fastest of those produced when the pressure was very low and the _ potential-difference 40,000 volts, these particles too have the same value of e/m at all pressures. The appearance of the phosphorescence at the higher pressures shows that while the velocity of the majority of the particles is less than the maximum velocity, there are some which possess this maximum velocity 426 Sir J. J. Thomson on which is about 2 x 108 cm./sec.; for some of these e/m=10".. Hence if V is the potential-difference required to give a char ge e when carried by a mass m, this velocity Ve=tmne or Ved x10) <4 xa = 20,000 volts, whereas the actual potential-difference between the terminals: was only 3000 volts; hence the positive particles acquire a much greater velocity than they could get by falling through the whole potential-difference between “the terminals of the discharge-tube. This result has such important relations with the question as to how the positive rays originate, that it is necessary to: consider very carefully any sources of error that may lurk in the method by which it was obtained. The method involves. the assumption that the magnetic and electric forces acting on the positive rays after they have passed through the cathode remain unaltered, though the pressure in the discharge-tube- increases from the value it had when the potential-difference- was 40,000 volts, to the value it had when this difference was reduced to 3000 volts. There seems no reason to suppose that the magnetic force should be affected by the pressure :. the case of the electric force, however, is not so simple. In the experiments a fixed potential-difference was established between two parallel plates, and the positive rays were de-. flected as they passed through the space between the plates;. the electric force acting on them was assumed to be the potential-difference between the plates divided by the distance: between them. Now the positive rays as they pass through the gas between the plates: ionize it; and if the ions were sufficiently numerous, the aggregation of positive ions in one place and negative ones in another might appreciably modify the force between the plates, and make the force midway between them less than the value which it is assumed to- possess when deducing the values of e/m and v. Thus it might be possible for the electric force on the positively charged particles to be appreciably less at the higher pres-. sures than the assumed value. If this were the case, the values of v and e/m deduced on this assumption would be too: large. It seems very unlikely a priori that if this change in the electric force occurs, it should adjust itself with such: nicety as to leave the boundary of the phosphorescent patch: Rays of Positive Electricity. 427 unaffected by changes in the pressure. For assuming that the velocity of the positive rays in this boundary does not remain constant, then if v is the deflexion due to the magnetic force H, x that due to the electric force X, then y is pro-- portional to He/mv and w to Xe/mv?: thus for a particle on the unaltered boundary e/mv must be constant as the pressure changes, and X must be proportional to v: it would thus require a very elaborate adjustment of variable quan- tities to keep the boundary fixed. I thought it, however, desirable to test this point by direct experiment, and tried several methods of doing so. In one way I used a tube in which the only connexion between the part of the tube where e took place and the part traversed by the rays. the discharge =) after passing through the cathode, was through the long, hollow, narrow hypodermic needle through which the rays. passed. With this arrangement, it took some time for the pressures in the two tubes to get equalized if a difference between them was produced in any way. Thus it was. possible to have for a short time the pressure considerably higher in the discharge-tube than in the region traversed by the rays after passing through the cathode. The phosphorescence was first observed when both the discharge- tube and the space through which the rays passed was ex- hausted to a very low pressure; a little air was then suddenly let into the discharge part of the tube, so that for a short time the positive rays were produced in a gas at compara- tively high pressure and then passed through a gas at very much haan pressure on their w ay to the willemite screen. In this case there was no displacement of the left-hand boundary of the phosphorescent patch, though now the electric force acting on the charged particles must have been the same as it was ‘helare the pressure in the discharge-tube was increased. Another method I tried was to reverse the direction of the discharge, and use the perforated electrode as the anode: in this case a stream of cathode 1 rays passed through the needle. and fell on the willemite plate. The deflexion oe these rays was measured under the same electric field with different pressures in the discharge-tube; and it was found that the deflexion of these cathode rays increased enormously as the pressure in the discharge-tube was increased. It was quite easy to get at the higher pressures deflexions ten times as great as those at lower ones; the magnitude of the deflexions varied approximately inver sely as the potential-difference between the electrodes, which is the result we should expect 428 Sir J. J. Thomson on if the electric force between the plates remained constant. Thus we may conclude that the cathode rays at any rate do not produce enough ionization to affect the electric field ‘between the plates. The most direct proof that the positive rays do not affect this field is given by the following method: The charged ions in a gas do not modify the electric field when this is strong enough to send the saturation current through the gas, fo in see case the lons are removed as fast as they are tormed and have not time to congregate and affect the electric field. Hence, if we prove that the potential-difference between the plates used to produce the electrostatic deflexion was sufficient to produce the saturation current through the gas, we are safe in assuming that the electric field between the plates is uniform. I therefore measured the current sent through the gas ionized by the positive rays when the potential- -difference applied to the plates was the same, 100 volts and upwards, as in the experiments made to determine the electrostatic deflexion of the rays. To do this, one of the parallel plates was connected with the leaf in a gold-leaf electroscope, and initially was con- nected with the earth, the other plate was connected with one of the terminals of a large battery of storage-cells the other terminal of which was earthed. The current between the plates was measured by the movement of the gold-leaf when its connexion with the earth was broken. It was found that there was a very considerable current through the gas ionized by the positive rays, so that to get measurable de- Hexions it was necessary to connect a small leyden-jar with the electroscope so as to increase its capacity. The current varied very greatly with the pressure: it was, which is im- portant, very small when the pressure was so high as to reduce the potential-difference between the terminals in the discharge-tube to 3000 volts, it reached a maximum when the pressure was quite low, and rapidly fell off as the pressure was still further reduced. ‘The results given in the following table show that even when the ionization was greatest the current through the gas was practically the same when the potential-difference between the plates was 100 volts as when it was 200; so that the current was saturated by the field used to deflect the C:nalstrahlen : hence the electric field between the plates must have been uniform and the electric force equal to the potential-difference divided by the distance between the plates. 0 a SS Sa ee ee ee ee ee ee ee ee = ‘- Rays of Positive Electricity. 42% Deflexion of Electroscope Potential Difference in volts per minute. between the plates. Cia wea al ee hs Bi bat 200 | CE PARR SARA EER ae 200 | Pressure so high that Ditch cate eet ot 100 $+ spot was only just CUOMO aa oe NOS 200 |; _ visible on the screen. Gree: erstatnncie tera Ue 100 } 1.16) Sa oe ee cari 200 | a. Wer we a - Pressure a little lower. aa MA defeat aad on at 100 | HRA eosin iia ws boven 200 | ol 66 ET too Still owerZpressure BO deiweke nae tana 200 MP ita bess ans Cae 200} ROSE Sea Sec ne sae ee 100 | | ES Bes ee 200 Much lower pressure. BE choy actagicin aan 100 ee eee nies tae can 100) The fact that the current between the plates was saturated. by the potential-difference used to deflect the Canalstrahlen. is confirmed by the fact that the electrostatic deflexion of the Canalstrahlen is proportional to the potential-difference between the plates ; this would not be the case unless the current were saturated. It follows from these results that the electric force deflect-. ing the Canalstrahlen was the same when the pressure was very low and the potential-ditference between the electrodes 40,000 volts, as it was when the pressure was higher and the potential-ditterence only 3000 volts; and therefore that the. constancy of the detlexion in the two cases implies that the velocity of the particles was the same. It thus appears that when the pressure is high the velocity of some of the positively charged particles is greater than that which could be imparted by the electric field as the. particles move from the place where they are liberated up to the cathode. The place where the positively charged particles are pro- duced can, by observation of the shadows thrown by obstacles placed in front of the hole in the cathode, be shown to be the region near the boundary of the dark space ; thus, if the particles when liberated were positively charged, the velocity they would acquire from the electric field would be that due to a fall through the dark space; hence if this were the source of their velocity, a passage back through the dark space from the cathode to the anode would bring them to rest at the boundary of the dark space, and they would not travel any further down the tube. 430 Sir J. J. Thomson on We can test this point in the following way :—A perfo- rated cathode C (fig. 3) is fixed in the middle of a sym- metrical tube, and anodes A and B are placed symmetrically Fig. to) Cathode | © -on opposite sides of it. If one anode, say B, is insulated the Canalstrahlen can be seen travelling ‘down the B side of the tube, while if A is insulated and 'B used as an anode they travel down the A side of the tube. If, now, A and B are connected together and used simultaneously as anodes, dis- charges take place on both sides of the cathode, the Crookes dark space is developed on both sides, and extending through and far beyond these are the two pencils of Canalstrahlen, These are now somewhat thicker close to the cathode than they were when the discharge took place in only one-half of the tube, as if some had been stopped close to the cathode ; but these so stopped form only a small fraction of the whole, the great majority travelling on right through the dark space on the side of the cathode opposite to that in which they were generated. This may be due in part to many of the ‘Canalstrahlen being uncharged for a time after they have passed through the cathode ; but if this were the complete explanation, the fall of potential in the dark space must be concentrated quite close to the cathode, so that the velocity of the Canalstrahlen, if this were due to the attraction by the cathode of the positive particles, would be very small ‘until the particles got quite close to the cathode. Now it can be shown that the Canalstrahlen possess their characteristic properties long before they reach the cathode. One way of doing this is as follows :—One of the most charac- teristic properties of the Canalstrahlen is their power of exciting the sodium radiation in suitable sodium salts. I find that one of the best ways of getting a surface sensitive to these rays is to use a flat perforated cathode and plug up the whole with a short capillary tube made of soda-glass. When a vigorous discharge is sent from this cathode the tip of the tube. shines with a bright yellow light, and some salt of sodium geek off from the glass and is deposited on the adjacent parts of the cathode. This deposit is exceedingly -sensitive to Canalstrahlen, and the places where these strike Rays of Positive Electricity. A31 the cathode are indicated by a bright yellowish glow. The alteration of the distribution of ie eae ivahlen as the pressure changes can be followed with great ease by this method. To trace the distribution of Canalstrahlen at a distance from the cathode, I took an insulated aluminium ring (fig. 4) Fig, 4 mounted on a piece of glass tubing, to which a piece of iron was attached, so that by means of a magnet the ring could be moved up or down the tube and its distance from the cathode varied. The ring was first made sensitive to Canal- strahlen: this was done by moving it up against the cathode which had a short glass tube fixed in the middle. A vigorous discharge was then sent from the cathode, and one side of the ring got coated with the sensitive deposit. ‘The ring was then withdrawn from the cathode and moved up and “down the tube ; the yellowish glow due to the impact of the Canal- strahlen on the face of the ring turned away from the cathode could be detected at a distance of more than a centimetre and a half from the cathode, showing that even at this distance from the cathode the Canalstrahlen had sufficient energy to produce their characteristic effects. These and the other effects already described all point to the conclusion that the great speed of the Canalstrahlen is not due to the direct attraction of the cathode on their positive charges. And since we can get some positively charged particles with energy represented by 20,000 volts when the potential-difference between the electrodes is only 3000 volts, itis evident thatif the energy of the Canalstrahlen is derived from the cathode rays, then the energy of several of the cathode rays must have been drawn upon for the production of one particle in the Canalstrahlen, tor the energy of each cathode-ray particle cannot meer that due to the fall of potential between the terminals of the discharge- tube. Thus, unless we suppose that there is some chemical change in the molecules from which the C analstrahlen arise, resulting in the liberation of energy, we must regard these molecules as capable of storing up the ener oy communicated to it by cathode particles w hen they strike a against it, so that 432 Sir J. J. Thomson on the molecule has not lost the energy acquired by one collision before it is again in collision with the cathode particle when the fresh energy it acquires is added to the store it already possesses. Thus the energy of a molecule exposed to the bombardment of the cathode rays goes on increasing until it reaches a critical value, when the molecule explodes and positively charged particle, or it may be a neutral doublet, is shot out with a velocity greater than 210° cm./sec. On the theory of the Ganalstrahlen given in my paper (Phil. Mag. Dec. 1909) the velocity of the positive particles when they have travelled some distance from the cathode is deter- mined, not by the strength of the electric field nor even by the pro perties of the molecules from which they are ejected, but by the properties of a system of dissociating neutral doubléts. That this is the nature of the Canalstrahlen, or at any rate ot those of them which are deflected by electric or magnetic forces, is suggested by many experiments. We know that a particle with a positive charge does not remain in this state indefinitely : after a time it unites with a negatively electrified corpuscle and forms a neutral doublet ; : and again we know that constituents of the “ positive rays” may be at one time without a char ge, and then subsequently become charged, presumably by breaking up into a positively charged particle and a corpuscle. Thus a stream of positive rays may be regarded as a gas. which is continually dissociating and recombining. ‘The molecules of the gas correspond to neutral doublets, the atoms into which the molecules dissociate to the positively charged particles and negatively charged corpuscles. Now it is only within certain limits of temperature, that is it is only when the kinetic energy of the gas is between certain limits, that there can be appreciable amounts of the gas in both the dissociated and undissociated states. For if the temperature is above a certain value practically all the gas will be dissociated, while if it is below another value there will practically be no dissociation. The temperature of the gas corresponds to the kinetic energy of the particles in the positive rays: hence we see that it is only when the velocity of these particles is within certain limits, which do not depend at all upon the strength of the electric field in the discharge-tube, that the positive rays could possess the properties disclosed by the experiments made upon them. We could not measure the velocity of much slower rays, for these would not dissociate and could not therefore be deflected either by electric or magnetic forces ; and if the rays whilst approaching the cathode could not exist as neutral doublets, Rays of Positive Electricity. 433 it seems probable that they would be so deflected by the intense electric forces in the neighbourhood of the cathode, that they could not travel down the long narrow tube inter- posed between them and the phosphorescent screen without striking against the sides and so getting absorbed. I have measured also the velocity of the “ retrograde rays ”’—those positively electrified particles which travel away from the cathode in the same direction as the cathode rays—and find that, as in the case of the Canalstrahlen, this velocity does not change when the potential-difference between the electrodes undergoes wide variations: the velocity of the “retrograde rays” is the same as that of the Canalstrahlen. I was not able to study the “retrograde rays’ when the potential-difference between the electrodes was anything like as small as 3000 volts. These rays are not nearly so bright as the Canalstrahlen, and become too faint to be observed when the pressure in the tube is high enough to make the potential-difference comparable with 3000 volts. The direction in which the retrograde rays come off is much the same as the direction of the cathode rays ; they are much denser along the normal to the cathode than in any other direction, and though they can be detected in directions making an angle of a few degrees with the normal, they rapidly fall off in intensity as the angle increases and soon become too faint to be observed. It would thus seem that the electric field produces some kind of polarization in the molecules which makes them eject the uncharged doublets along the line of motion of the cathode rays in their neighbourhood. I have tried a good many experiments to find the place from which the Canalstrahlen and the retrograde rays start. One of these was as follows :—The cathode was an aluminium disk 2°5 cm. in diameter. A narrow thin piece of mica coated with fused lithium chloride was fixed to a piece of iron, and could be moved towards or away from the cathode by a magnet. The chloride when struck by Canalstrahlen or retrograde rays gives outa bright red light, and is thus a good detector of the rays. Suppose we start with the mica at a distance from the cathode less than about half the thickness of the dark space, we find then that there is no red light to be seen on the side of the mica next the cathode, showing that no retrograde rays strike against the mica. The anode side of the mica, however, is a brilliant red, showing that plenty of Canalstrahlen are striking against it. Let us now pull the mica further away from the cathode. Phil. Mag. 8. 6. Vol. 19. No. 111. March 1910. 2 F 434 On Rays of Positive Electricaty. When the distance from the cathode becomes equal to about one-half the thickness of the dark space, red light appears on the cathode as well as upon the anode side of the mica, showing that the mica is now struck both by Canalstrahlen and retrograde rays. This continues until the mica reaches the limit of the dark space, when the red glow disappears from the anode sile, showing that it is no longer struck by the Canalstrahlen ; the cathode side continues to glow, showing that it is still struck by the retrograde rays. We can also get the same effect by altering the pressure without moving the mica, if we put the mica at, say, about 1:5 cm. from the cathode and start with the pressure fairly high, so that the mica is outside the dark space: the cathode side of the mica is red, but not the anode. If now the pressure is gradually reduced so that the dark space increases, then, when it reaches the mica, the red light appears on the anode side as well as the cathode, and as the pressure is reduced the red light remains on both sides until the pressure falls so low that the mica is about in the middle of the dark space. The red light disappears now on the cathode side, and this side remains dark when the pressure is still further reduced, the other side being covered with a brilliant red light. There can, I think, be no question that the Canalstrahlen originate in the region near the outer boundary of the dark space. The place of origin of the retrograde rays is more difficult to fix. The experiment just described might suggest that these rays arose from the portion of the dark space beyond (2. e., further from the cathode) the place where the red luminosity on the cathode side of the mica screen dis- appears. Other experiments, however, have led me to the conclusion that most of the retrograde rays originate close to the cathode, and that the absence of them when the mica is brought close to the cathode is due to the shadow cast by the mica on the cathode. The experiments were as follows :— If the cathode is in a large bulb and the mica screen is put a little on one side of the cathode, the red lithium light can be observed on the cathode side quite close to the cathode even though the dark space may be 5 or 6 cm. deep. Again, when, as in the first experiment, the tube is of one bore throughout, as the mica is moved towards the cathode across into the dark space the luminosity on both sides is very bright, and remains so until the mica enters Goldstein’s first layer, then the luminosity becomes suddenly very much fainter and in some cases disappears on the cathode side, at the same time the shadow thrown on the cathode by the mica becomes much more marked and suddenly increases in size. Bending of Electric Waves round the Earth. 435 ‘This is, I think, due to the screen becoming positively elec- ‘trified as soon as it enters the first layer, where, as we know, there is a great accumulation of positive electricity. The eeerely electrified screen repels the positively electrified Janalstrahlen which pass it on their way to the cathode, and deflects them from their course so that they now strike the -cathode outside the projection of the screen on the cathode ; in this way a considerably larger area of the cathode is screened from the Canalstrablen, the portion so screened no longer emits cathode rays, and the region in front of it is traversed by little if any current, and ‘the bombardment by retrograde rays almost ceases. The same effect can also easily be shown if the mica screen be replaced by a very fine platinum wire. When this wire is moved across into the dark space it becomes red hot, and wemains so until it enters the first layer. It then immedi- ately becomes cool, and the shadow which before could hardly be detected now becomes prominent and much thicker than the wire ; the change takes place very abruptly. In some cases, before the wire entered this layer I have observed a reversal of the shadow, i. ¢., the projection of the wire on the cathode instead of being ‘darker was brighter than the rest of the cathode. ‘This, I think, indicates that the wire in this position got negatively electrified and attracted the Canalstrahlen, instead of repelling tem. The retrograde rays are formed quite well with gauze electrodes, indicating, I think, that some of them start from the positively electrified first layer, and not from the metal of .the cathode. XLII. On the Bending of Electric Waves round the Earth. I. By J.W. Nicuotson, M.A., D.Se.* i a former note? two criticisms were directed against an investigation, by M. Poinearé, of the extent to which diffraction can account for the remarkable results achieved by experimenters in wireless telegraphy. It was indicated that the results of M. Poincaré were at variance with those of a paper by the writer, completed some time ago, but only now in course of printing. The difference was traced to two points in M. Poincaré’s inv estigation which were mathema- tically unsound. In the meantime M. Poincaré has also * Communicated by the Author. +t Phil. Mag, Feb. 1910. 2 2 436 Dr. J. W. Nicholson on the Bending of noticed these points, and has published, in the current number of the endiconti del Circolo Matematico di Palermo*, an exhaustive investigation of the problem amended in each of these respects. The general conclusions. to which he is now led are substantially in agreement with those of the writer’s paper, which were expressed in the previous note. Instead of, at any point of the earth’s sur-- face, an intensity due to diffraction of magnitude sufficiently large to furnish an explanation of the observed phenomena,. M. Poincaré now finds an intensity containing an exponential factor of large negative argument propor tional to mi 50, where: @ is the orientation of the point from the oscillator, and m is. a magnitude roughly of order 10°, depending on the ratio of the wave-length of the oscillation to the diameter of the: earth. But this agreement is only general in character, and does not extend to the actual quantitative results. The object of the present note is to point out that M. Poincaré’s expo- nential formula dues not appear to be of such general appli-- cation as his investigation would indicate. From my own investigation, which proceeds on different lines to that of M. Poincaré at many stages, the exponential factor may be- obtained readily for points whose orientation, measured from. the oscillator, is not great. But for points at a greater dis- tance, such that the argument of the exponential exceeds a cer tii moderately large value, it ceases to be the important term in the dimen effect. The rate of decay of the intensity round the surface, as given by M. Poincaré’s. formula, would therefore be much too rapid, although, as stated in the previous note, it is certainly sufficiently rapid to eliminate diffraction as a possible explanation of the. phenomena. M. Poincaré’s mode of solution is briefly as follows. The electric force at any point on the surface of the earth is first determined in the form of an harmonic series, the coefficient of the zonal harmonic of order n being a meromorphic function of n+4 derived from the Bessel functions of this. order. This series ceases to converge when the oscillator is placed on the earth’s surface, thus necessitating some trouble- some mathematical processes for its evaluation in this limiting case. In the writer’s paper, the magnetic vector has been dealt with instead, chiefly on account of the fact that, by its use, a convergent series may be obtained. M. Poincaré then proceeds to an examination of the infinities of the mero-. morphic function, with a view to its expression in a more * Marzo-Aprile, 1910, pp. 169-261, Electric Waves round the Earth. 437 “appropriate form by the use of Cauchy’s theorem. He finds that one infinity is of greater importance than the others, and corresponds to a value of n, whose imaginary portion is very great, and of order ms. The function is then replaced by the term corresponding to this infinity in the ordinary way. Certain formule of summation of series of zonal harmonics are then developed, in their first appreximations only, and finally, the particular series in question, with the coefficient of the harmonic replaced by the formula corresponding to its chief infinity. The result stated above is the con- ‘sequence. Now the terms neglected during the summation do not necessarily all lead to exponentials, in their final contributions to the result, in the same manner, and may therefore lead to a sum which, for a large orientation from the oscillator, does actually transcend that due to the terms retained. For however small their sum may be, if its decrease with orien- tation is not exponential, it may soon be of greater importance than M. Poincaré’s exponential effect, as the point moves round from the oscillator. This is undoubtedly the origin of the discrepancy stili existing between M. Poincare’s results and those of the writer, who has found, by a different process of summation, that the effect is certainly exponential at first, but only for a few hundred miles. As soon as this expo- nential becomes smaller than a certain (not very well defined) limit, its magnitude has really fallen below that of certain other non-exponential portions of the sum, which although decreasing as the orientation increases, do not decrease so rapidly. “These portions are determined to a great extent by Jow order harmonics in the series. For the great distances mainly in question, therefore, another formula would appear to be necessary, and this will be found in the investigation appearing shortly In conclusion, it may be pointed out that M. Poincaré’s modified expressions for the Bessel functions whose argument and order do not differ widely, are in accord with the values deduced by the writer in earlier papers, and mentioned in the previous note. Trinity College, Cambridge, February 10, 1910. potas XLIV. On the Charge of the Electron. By Juan PERRIN,. Professeur de Chimie Physique a 0 Université de Paris”. ROFESSOR MILLIKAN has recently published in this: volume (pp. 210-228) a careful determination of the: charge on an electron. He has employed the well-known method of condensation by sudden expansion invented by J. J. Thomson and afterwards improved upon by H. A.. Wilson and himself, and he has obtained the value 4°65 x 10-12, exactly equal to that which Rutherford and Geiger obtained in 1908 from the study of the « rays. The new method! which I based on the observation of the Brownian movement (May 1908) gave the value 4:1 x 10-° (Oct. 1908). The difference does not reach 12 per cent., and there is. thus a striking agreement between these very different methods. There remains to be examined the degree of pre-. cision which each of them admits of. In regard to this I am: not of the same opinion as Prof. Millikan, and the objections which he takes to my method lead me to think that he has. not yet seen two at least of the Notes which I have published! in the Comptes Rendus or the detailed discussion which I have given in the Annales de Ch. et de Phys. (Sept. 1908,. pp. 1-114). I may be permitted to refer to these communi- cations which are not open to the objections cited by Prof.. Millikan. In effect: (1) I have been able to determine the radius of the grains used by me without having recourse to Stokes” law; at the same time, far from assuming this law a priori, | have been able to establish experimentally that it applies exactly to the grains showing the Brownian move- ment (C. #&. vol. exlvi. p. 475 ; and Ann. pp. 45-52). I may- even say that I have there brought forward an indirect contribution to the method of condensation by sudden ex- pansion, where this law is applied, since in every case, om account of the feeble viscosity of the air, the droplets formed have a notable Brownian movement which a strong magni-— fication would reveal, i (2) Lhave not had to assume, and I have proved experi- mentally, that the particles follow Maxwell’s law (Ann. pp. 81-86). . (3) I have not assumed, in the final measurements, that the density of the particles was that of gum gamboge in mass, and I have determined this density by two different methods (C. £&. vol. exlvii. p. 531, and Ann. pp. 38-40). * Communicated by the Author. a ra On Pirani’s Method of Measuring Self-Inductance. 439 (4) The value 5-410, given indeed in the Comptes Stendus for the constant N of Avogadro, resulted from an error (the omission of a factor #) in the calculation of ex- periments which gave exactly 7-1 x 107. I must apologize for this error in calculation, which was rectified some days later, and refer for the exact figures to my publications subsequent to September 1908. It is clear that the difference between 5°4 and 7:1, which would indeed be inadmissible, has no bearing on the precision of the method employed, a precision which is I believe entirely a question of patience and of time. XLV. On Pirani’s Method of Measuring Self-Inductance. By J. P. Kuenrn, Ph.D., Professor of Physics, Leiden University ™. N the September number of this Magazine Professor Lees+ gives a simplified proof of the formula for Pirani’s method of comparing an inductance and a capacity. The object of this note is to show that the result may be obtained in a much simpler way, owing to the fact that the conditions prevailing in the method are much simpler than appears to be recognized. First consider the effect of breaking the main current and assume that the current in the main branch is instan- taneously reduced to zero. Using the notation in Professor Lees’s paper and reckoning the integral transient current from the moment that the current is opened, we have «=0. The condition is that the galvanometer branch must be free of integral current, i.e. <=0. Applying Kirchhoft’s second law adapted to impulsive currents to the mesh GPR (com- pare the diagram which is taken from Professor Lees’s paper) and remembering that G contains no electromotive impulse, whatever the inductance may be, it follows that the current in PR is zero, i. e. y=0, hence likewise by Kirchhoff’s first law the currents in the conductors Q and L, the only wire therefore which conveys electricity being the resistance which completes the condenser circuit. Finally, we have from either of the meshes SGQ or SRPQ Ly=y7rKxr or L=Kr. Q.£.d. * Communicated by the Author. tT C. H. Lees, Phil. Mag. [6] xviii, p. 482 (1909), 1 440 Pirani’s Method of Measuring Self-Inductance. Hividently there is a complete balance between the in- ductance L which provides the electromotive impulse and the current from the condenser. This balance has nothing to do with the bridge; obviously in a case like this where all BE 2) P Q. le : FDQD : S7 ~\ iin Oe “a | & K YL the inductances and capacities are confined to one of the branches of the network, the rest of the network may be left out of account altogether. The only part which the bridge plays in the method is to provide a galvanometer which is free from current in the steady state, in order that it may serve to indicate any transient current during the discharge. The network might, therefore, be very much more complex ; as long as the galvanometer were placed in a currentless branch (conjugate with the battery branch) the result would be the same. On the other hand, if it were possible to switch off the current and putin a galvanometer simultaneously, the bridge arrangement could be entirely dispensed with. In order to bring out the real significance of the induction phenomenon, we assumed the transient current in the battery branch 2 to be zero. The result is, however, independent of this assumption. Without further calculation we may observe that an additional current « in the main line must distribute itself over the network as in the steady state, and cannot therefore affect the galvanometer. For confirmation we may repeat the above calculation, dropping the assumption that «z=(0. From the mesh GPR we now have («—y) P=yR, and Notices respecting New Books. 44] from the mesh SGQ Ly,=97,K7? +yS—(a@—y)Q=7, Kr", by the condition PS=QR for the steady balance. The other case of the current being made is rendered equally simple by the observation that the impulse \E di can in itself have no effect on the galvanometer, if PS=QR, or in general if it is placed in a link which is conjugate with the battery branch. Or using Professor Lees’s equations (12) to (14), we may imagine them first solved leaving out the second side of (13), 7. e. with L=0 and K or r=0; this gives PS=QR. Then subtract from the actual equations when the second side of (12) disappears ; it is then seen that the condition for the additional z being 0 is Kr?=L, that this condition is independent of PS=QR being satisfied, and finally that it involves the disappearance of the additional « and y. The additional current is thus confined to 7 as in the previous case, L providing the necessary impulse. The principle that E cannot have any influence in the induction experiment after the steady balance has been obtained is of course of more general application ; in the present instance it makes the writing down of the general equations unnecessary, and the result follows immediately by Kirchhoff’s laws as in the first case. XLVI. Notices respecting New Books. The Scientific Papers of Sir Witi1aM Hvaerns. Edited by Sir Wirr1am and Lady Hueerns. London: William Wesley & Son. Price £1 11s. 6d. net. HE book under review is the second volume of the Publications of Sir William Huggins’s Observatory at Tulse Hill, and con- tains a reprint of the published Papers on the work done in the observatory since its foundation by Sir William Huggins in 1856. Itis complementary to the first volume, ‘An Atlas of Representative Stellar Spectra,’ published in 1899, which contains the later original work of the observatory. The present volume is of extreme interest containing, as it does, the record in contemporaneous documents by the pioneer himself -of the development of astronomical research in an entirely new direction through the application of the Spectroscope to the heavenly bodies other than the Sun, and supplying what is prac- tically a history of the birth and development of the science of Astrophysics. It is just fifty years since Kirchhoff’s great discovery of the true significance of the Fraunhofer lines in the Solar Spectrum, -and the deduction therefrom of the chemical coustitution of the 442 Notices respecting New Books. Sun, suggested to Sir William Huggins the possibility that this- novel method of research might be extended to the other heavenly bodies. He would find out whether, as required by Laplace’s. nebular hypothesis, the unity of the law governing their motion had not its counterpart in the unity of their composition. Hager to explore this virgin field, Sir William yet did not underestimate- the great difficulties involved, and in the ‘ Papers’ before us we- find with what patience, perseverance, and fertility of resource they were severally met and overcome. The first section relating to the observatory and instrument is deserving of careful study, and illustrates the perfect adap-- tation of means to the end by many ingenious contrivances. It must be remembered that at this time a Star Spectroscope was. a thing unknown, and all the necessary details had to be supplied by the inventive genius of the explorer. Amongst these contri-- vances may be mentioned the cylindrical lens for widening the linear star spectrum into a band on which the lines might be seen ;: the polished slit of speculum metal whereby the slit itself was made to serve in keeping the star image steadily on any part of it ;. and the method of applying the comparison spectrum so as to avoid a spurious shift of the lines. The advantage of a mirror of speculum metal together with Iceland spar for the prisms and: quartz for the lenses in photographing the more refrangible part of the spectrum was also perceived and utilized. The first-fruits of Sir William’s work were contained in a note- to the Royal Society in 1863, ‘On the Lines of some of the Fixed Stars,” in which diagrams were given of the spectra of Sirius, Betelgeux, and Aldebaran, showing the principal lines and. their position relatively to the chief solar lines. This was followed by a paper which appeared in the Philosophical Transactions for 1864, ‘* On the Spectra of some of the Fixed Stars,” in which are- given the results of a most careful comparison of the spectra of Aldebaran and Betelgeux with the spectra of sixteen terrestrial elements, as well as the result of the examination of several other stars, from which the author draws the inference that “a similar unity of operation extends through the universe, as far as light enables us to have cognizance of material objects. The importance of supplementing eye observations of the- spectra of stars by photographs was so obvious, that as far back as the year 1863 an attempt was made to obtain ‘such photographs, but it was not until the introduction of the gelatine dry plate in 1875 that success in this direction was attained. Very notable was the photograph of the spectrum of Vega taken in 1876,,. showing seven dark lines. A later photograph showed twelve- lines, and from their similarity and the symmetry of their arrange- ment ‘‘ the suggestion presents itself whether these lines are not intimately connected with each other and present the spectrum of one substance.” Sir William observed that -two of the lines coincided with two known hydrogen lines in the solar spectrum, but it was not till much later that so many as twelve of the series. - Notices respecting New Books. AAS. were observed in the spectrum of hydrogen in the laboratory. Reproductions of these historical photographs are given in the ‘Atlas.’ In 1864 the nature of Nebule was still an unread riddle.. Were they composed of fiery mist or were they merely stellar galaxies too remote to be separated into their component stars ?’ The problem was solved on the evening of August 29, 1864, when for the first time a planetary nebula was examined in the spectro- scope. Three bright lines! ‘“ The answer which had come to us in the light itself read: Not an aggregation of stars, but a luminous. gas.” Further observations identified one of these lines with the all-pervading hydrogen, but the principal line, although subjected to careful examination, still preserved its encognito as belonging to: a substance as yet undiscovered on the Earth. Details are also. given of the spectroscopic examination of a large number of nebul, one third of which proved to be truly gaseous. The determination of the motion of stars in the line of sight by the application of Doppler’s principle, suggested itself to Sir William some time in 1862-3, soon after the commencement of his work. He quickly saw how the change of refrangibility might be detected; for as soon as the observations had shown that certain terrestrial substances were present in the stars, the: original wave-lengths of their lines became known and any small. want of coincidence of the stellar lines with the same lines pro- duced in the laboratory might be safely interpreted as revealing the velocity of approach or recession between the star and the Earth. In 1868, after many toilsome disappointments, he was able to announce, ina paper published in the Philosophical Trans- actions, the foundation of this new line of research. The method was devised before adequate instrumental power was available for it, and Sir Wilham’s observations can be considered as experimental only ; but time has proved how correctly he had forecasted the possibilities and value of the method which has since become so: fruitful in other directions. Itgives the power of distinguishing not a few double stars, the components of which are too close to be separated in the telescope, by revealing the ditference of their motions in the line of sight as they revolve about their common centre of gravity. The motion of the gases constituting sun-spots has just been determined by the same method, but perhaps most remarkable of all is the recently published Memoir from the Cape Observatory on the spectroscopic determination of the value of the Solar Parailax from the observed changes during the course of the year in the radial velocity of the Earth, relative to a large number of stars. New or variable stars did not escape Sir William’s attention, and iu 1866 he was the first to discover the compound nature of their spectrum in his observations of T Corone, whilst his work on Nova Aurigz in 1892 was of extreme importance. The spectra of Comets are very fully dealt with. Here again it was Sir William Huggins who first in 1868 showed that in addition 444 Notices respecting New Books. to reflected light certain comets emitted light characteristic of carbon vapour, and who later by photography in 1881 detected the bands of cyanogen, showing the presence of both carbon and nitrogen. From this important discovery he deduced an intimate relationship between comets and meteors. In the section dealing with the Sun and corona Sir William gives one of the earliest detailed descriptions of the spectrum of a sun-spot. The method proposed by Sir William for viewing the spectrum of the solar prominences without an eclipse being quite correct in principle, it was a disappointment that non-success attended his efforts to see it until after the identity of the principal lines had been recognized at the eclipse of the Sun in 1868. But he was the first to observe and draw the complete form of a pro- minence in full sunshine by the spectroscopic method. The papers on the form and nature of the corona are especially full of profound reflexion and insight; and the attempt to photograph it in full sunshine is astriking instance of patience and ingenuity, though the task proved too difficult for accomplishment. Nor were the Moon and planets neglected, and much careful attention was devoted to the spectrum of Mars and the examination of the absorption lines in it, indicating the presence of water vapour—although this latter is still in doubt. One of the difficulties which Sir William had to surmount before he could even begin his work on the chemical analysis of the stars is brought home to the reader in the section on Chemical Spectra. ‘“‘At that time (1863) no convenient maps of the chemical spectra were available, and to meet this want I devoted a great part of 1863 to mapping the spectra of twenty-six of the elements.” ‘The results are shown in two plates and a table. For the greater convenience of the reader, the Papers have been arranged primarily in order of subject matter, secondarily in each division in the order of their publication, although in a few cases this arrangement necessitates the occasional dividing up of a paper. In conclusion, attention should be drawn to the many scientific illustrations accompanying the text, and the two fine portraits of the distinguished author and the gracious lady who since 1875 has shared his labours. Legons sur les Alliages métalliques. Par J. CAVALIER. Paris: Vuibert et Nony, 1909. THE aim of the author of this book has not been to bring together the known facts of all metallic alloys; but to write a treatise, suitable for teaching purposes, which explains the principal methods of investigation, and illustrates them by examples. The volume is divided naturally into two parts. The former deals with generalities such as the modes of preparation of alloys and the study of them in their chemical, metallographic, and —— ——— Notices respecting New Books. 445, thermal relations, together with their other physical and mecha- nical properties (200 pp.). Inthe second part (230 pp.) a study is. made of particular alloys. In spite of the very modest pretace,. we have not come across any other text-book in which so much information of such varied kinds is given. Whether the reader wishes for a description of experimental methods or of the modern data in regard to coexistent phases, or data in regard to com- mercial applications (including relative cost of materials), all is set forth clearly here. There are numerous diagrams representing the physico-chemical properties of the alloys selected for description. We think that this book will be found to be a most useful one. Vectors and Vector Diagrams applied to the Alternating Current Circmt. By Q.Crame andC. F.Smrra. London: Longmans, Green, & Co. 1909. Tis should prove a valuable book to the student of electrical engineering. With a slightly modified notation the authors discuss clearly and systematically the methods of vectorial graphics. which have been developed during the last twenty years. These methods have grown naturally out of Maxwell’s theory of the sinusoidal current and the geometrical representation of the complex variable. ‘The first three chapters contain a presentation of the foundations of the method, after which follow important chapters on self and mutual induction, the transformer, motors of the induction type, and alternating current commutator motors. There is, then, a mathematical chapter on the product of two vectors leading up to the two concluding chapters, which deal at considerable length with locus diagrams and examples of the appli- cation of locus diagrams. The book is well illustrated by numerous. vectorial diagrams, which are all-important in werk of this kind. The Elements of Non-Euclidean Geometry. By Jurtan LOwWELu Cootipen. Oxford: at the Clarendon Press. 1909. THe Harvard Professor has in this book given a well-planned exposition of non-Euclidian geometry of three dimensions. As explained in the Preface, he approaches the subject from three different points of view, namely: (1) The elementary geometry of point, line, and distance ; (2) Projective geometry and the theory of transformation groups; (8) Differential geometry with the concepts of distance-element, extremal, and space constant. Chapters XVIII. and XIX. treat respectively of the two last methods. The other sixteen chapters contain an instructive de- velopment of the first method. Professor Coolidge builds the system on XIX Axioms, whose consistency is discussed in Chapter VI. for the three types of geometry—the Euclidean, the hyperbolic, and the elliptic. Thereafter follows the geometrical superstructure of which the foundations have been laid in the- 446 Notices respecting New Books. preceding chapters. Successive chapters are devoted to such ‘subjects as congruent transformations, higher line-geometry, conic sections, quadrics, &c., the principle of duality playing a most important role throughout. The theory of curvature and torsion of curves is developed in Chapter XV.; and in Chapter XVI. -on differential line geometry the author includes some of his own researches in isotropic congruences. The book is eminently readable, and finishes with a picturesque comparison of the merits -of the three methods of approach. We are told that ‘“ there is no answer to the question which method of approach is the best. The determining choice among the three will, in the end, bea matter of personal esthetic preference. And this iswell. Let us ‘not forget that, in large measure, we study pure mathematics to satisfy an esthetic need. We are fortunate when, as in the present ease, we are free at the outset to choose our line of approach.” We have noticed two or three unimportant slips, such as that in Theorem 32, p. 22, where some words are evidently omitted. Lehrbuch der Practischen Physik. Dr. Frreprich KoniRavscn. Eleventh edition. (Leipzig and Berlin, B. G. Teubner, 1910.) M. 11. Wuite this book is under review comes the news of the death -of its renowned author. In its present form it contains, therefore, the last additions to be made by his own hand. It is a monv- ment to a seeker after precision in physical measurement ; and, especially in its latest form, it will serve to perpetuate the aims of ‘its author. . The additions made in this eleventh edition are very considerable. Important modifications have been made in many sections. For example in connexion with recent work on absolute thermal radiation and its practical applications, with the phenomena of the -cathode stream and the measurement of the ratio of charge to mass ; with the phenomena of electric waves. But the most important is an extra section of thirty-three pages in radioactivity. This is the best summary we have yet seen of this subject from the point of view of physical measurements and it should be of -immense service to the large number of new recuits who are con- tinually being brought to this branch of study. Useful changes have also been made in the tables at the end so as to bring them -completely up to date. There is no need to recommend the book for it has already taken up an established position as a book of sreference in every physical laboratory. [ 447 | XLVII. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 336. | ‘June 16th, 1909.—Prof. W. J. Sollas, LL.D., Se.D., F.R.S., President, in the Chair. ee following communications were read :— 1. ‘The Carboniferous Limestone of County Clare.’ By James Archibald Douglas, M.A., B.Sc., F.G.S. 2. ‘The Howgill Fells and their Topography. By John Edward Marr, Sc.D., F.R.S., F.G.S., and William George Fearnsides, M.A., F.G.S. The Howgill Fells form a monoclinal block, from which the “Carboniferous rocks have been denuded. ‘he gentle northern slope probably corresponds very closely with the sloping plane of unconformity between the Carboniferous rocks and the under- lying Lower Paleozoic strata. On the south side, the steep slope ‘to the Rawthey is along a block-fault which has several minor parallel step-faults to the north. The chief drainage was originally north and south from the watershed at the summit of the bloek, ‘but the swifter south-flowing streams have in several cases cap- ‘tured the headwaters of those flowing northwards, and thus the watershed has been largely shifted to the north. Some of these captures occurred probably in pre-Glacial times, but others un- -doubtedly took place in the Glacial Period, and others again are still proceeding. The tract was glaciated by its own ice, but ‘ foreign’ ice was conterminous with the local ice on all sides. The rocks are, from the point of view of erosive effects, nearly homogeneous, and a rounded form of feature was produced by weathering and stream-erosion in pre-Glacial times. The chief erosive effects of glaciation were the truncation of spurs; the formation of conchoidal scoops in the concavities of the valleys ; a general widening of the valleys, and but slight deepening, as marked by the slight difference of grade (amounting to but .a few feet) at the junction of tributaries with larger streams. A feature of interest is the contrast in this small area between these glaciated valleys and others of V-shaped cross-section, which -are typical water-carved valleys unaffected by glacial erosion. The two great hanging valleys of Uldale and Cautley, where the streams plunge scores of feet down waterfalls, are due to river-capture, and not to the deepening by ice of the main valleys. 3. ‘A New Species of Sthenurus.’ By Ludwig Glauert, F.G.S. 4. ‘Some Reptilian Remains from the Trias of Lossiemouth.’ By D. M. 8. Watson, B.Sc. 5. ‘Some Reptilian Tracks from the Trias of Runcorn (Cheshire).’ By D. M. S. Watson, B.Sc. 6. ‘The Anatomy of Lepidephloios laricinus, Sternb.’? By D, M. 8. Watson, B.Sc. 448 Intelligence and Miscellaneous Articles. November 3rd.—Prof. W. J. Sollas, LL.D., Se.D., F.R.S., President, in the Chair. The following communications were read :— 1. ‘Certain Jurassic (Lias-Oolite) Strata of South Dorset, and their Correlation. By 8. 8. Buckman, F.G.S8. 2. ‘Certain Jurassic (‘‘ Inferior Oolite ”) Ammonites and Brachio-- poda.’ By 8. 8. Buckman, F.G.S. 3. ‘ The Granite-Ridges of Kharga Oasis : Intrusive or Tectonic ?” By William Fraser Hume, D.Sc., A.R.S.M., F.G.S. The author quotes the detailed records given by Mr. Beadnell in his paper published in the Quarterly Journal for February 1909 (vol. lxy. p. 41), and although in entire agreement with the facts. there stated, differs with regard to the interpretation of those facts. Whereas Mr. Beadnell regards the granite as intrusive, on account of the high dip of the sedimentaries, and the changes which they exhibit as regards colour and hardness, near the granite, the author considers that the dips are due to fold-movements almost at right angles to one another, since they lie on the same line as the crater- like basins, the rims of which are formed of the compact and steeply- dipping limestones of the Lower Kocene, and he adduces as further evidence the fact that dykes and quartz-veins penetrating the erystalline rocks cease abruptly at the edge of the sandstone. He attributes the bright hues and silicification found near the contact to gases and heated waters acting along the boundary between the dissimilar rock-masses. 4. ‘The Cretaceous and Hocene Strata of Egypt’ By William Fraser Hume, D.Sc., A.R.S.M., F.G.S., Superintendent, Geological Survey of Egypt. XLVI. Intelligence and Miscellaneous Articles. CORRECTION TO MR. E. GOLD'S PAPER ON “‘THE RELATION BETWEEN PERIODIC VARIATIONS OF PRESSURE, TEMPERA TURE, AND WIND IN THE ATMOSPHERE. i fees denominator in the expression for tan J, on p. 33 of my paper on Periodic Variations in the January number of the Ei losaphical Magazine, is wrongly given as n(3 cos’ ¢ —2)(4n* sin® @ +0). The coefficient of n should be (3 cos’ ¢—2) ~4n* sin? ¢(3 cos’ ¢+ 2). The values of W, in the table on p. 34 are unaffected for 7=0, but for the other two cases they are as follows :— Latitude...... 0° 15° 30° 45°—s OPTS (i= 19% «= 197 08- at A Be OS ps tae 207 Qll 224 240 «257 270 O73 ~ The phase in the theoretical value of w,, at the bottom of p. 34, becomes 344° instead of 374°, and a comparison shows that the theoretical phases of u,and v, for J=jn are in almost perfect agreement with the phases found by Hann from the observations on Siintis. H. Gorn. Phil Mag. Ser. 6, Vol. 19, Pl. V. 9. Fig. yy 10S39NTHf SY. Ma Fro. AG. 90 100 110 120 Aromic WE/GHT. 80 70 Phil Mag. Ser. 6, Vol. 19, Pl Y, SADIMUM, Fia. 4, Via. 5. Lae = | | Ss Fig, 9. _| | 7oTAtL /ON/SATION. Vaives or PR. i ror. | 110) | Ls, ANS 16 Zan 8 9 10 tl 2 3a 5 6 rm. DISTANCE BETWEEN BOUNDING SURFACES. 1:00) 4 r HM bet) PSS 16 ize og DISTANCES BETWEEN Ab. WINDOW ® fy | ABSORPTION COEFFICIENTS. Pia. 6. Ma. 7. | 3 = ] al FE_AS TERTIARY FAQIATOR 9 7 | “BI- | 7 7] | “6h 8 f 3. ; 4 : ; = 5 S| | 2 . ae N | > SS ee ee | : ra fice Cs a vot 75: SW 90. a) ee S 90 10 120 130 OA 3 Aromic WeiGhT. | SAMA EROTAG: | 90 100 it) 120 Aromic Wesanr a 2 a rr a) 6 cm | 60 70 8090 100 vam tel The curves shown above were picked from among those determined DISTANCE BETWEEN BOUIVDING SURFACES experimentally, and are corrected for Capacity changes and for | variations in the temperature and pressure of the air used as absorbing medium, S— + a Ol ees Teenie, Dead (eo Phil. Mag, Ser. 6, Vol. 19, Pl. VI. Fie. 4. Latent and Specific Heats, &c. for Saturation in terms of T. Clausius’ characteristic. ian eae eee aN a JERR oe a SSRN lh ees ee | | BAYNES, Corresponding Values of ¥, 7, 7 in terms of » for Saturation with yan der Waals’ characteristic. VT Corresponding Values of 7, v, v' in terms of 7 for Saturation with van der Waals’ characteristic. Latent and Specific Heats for Saturation in terms of r with yan der Waals’ characteristic. Phil. Mag, Ser. 6, Vol. 19, Pl. VI. Tie, 4. Latent and Specific TTeats, &e. for Saturation in terms of 7. Clausius’ characteristic. ; ——— al LL ' (S-/A CURVE By i+h g 3y(9y 1+ W(3—3y +1) Lane} — | 1 : Jay tM B22 (Gos 8 = ae (So) (Sve =a)(ev-ale= A) pe (3y —1—p)(8v—14)) =D +1 p(B) + 14H) ae tae THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] APRIL 1910. XLIX. On the Pressure of the Electric Wind in Hydrogen containing traces of Oxygen. By A. P. Caarrocn, Pro- fessor of Physics, and A. M. TYNDALL, B.Se., Lecturer in Physics, in the University of Bristol *. ag a recent paperf dealing with the influence of oxygen on point-discharge in hydrogen at atmospheric pressure the present authors showed that if the amount of oxygen in the hydrogen be reduced to a fraction of one per cent., certain phenomena accompanying the negative discharge are modified in a marked way. — Among these it seemed possible that the velocity of the negative ions was to be classed, its value being apparently raised as the hydrogen approached a state of purity (loc. cit. pp. 44-47). As, however, the principle upon which this statement was based is not above criticism (see below p. 456) it seemed desirable to measure the ionic velocities again in some other way. This we have now done, using the wind-pressure method t for the purpose, and the results obtained are discussed in what follows. * Communicated by the Authors, t Phil. Mag. 1908 [6] vol. xvi. p. 24. ¢ Phil. Mag, 1899 [5] vol. xlviul. p. 401. Phil. Mag. 8. 6. Vol. 19. No. 112. April 1910. 2G 450 Prof. Chattock and Mr. Tyndall on Pressure of the Method of the Experiments. The hydrogen and its admixture of oxygen were prepared and introduced in the. manner described in our paper of 1908 (p. 30), the identical apparatus being used for the purpose. As the arrangement of the discharge-vessels and manometer required considerable alteration they were con- structed afresh. Fig. 1 gives a perspective view (diagrammatic) of the new design. T is a tilting-table, supported on three points 8; S: 83, and S, is the end of a spherometer screw, on the large divided head of which (not shown) it is possible to estimate a travel of S3 of 1/40000 inch. Upon ‘I is fixed the glass discharge apparatus. The four bulbs M, each 10 cm. in diameter, contain mercury in their Electric Wind in Hydrogen containing traces of Oxygen. 451 lower halves. Above M, M, are the discharge-tubes D, D, containing the gas to be experimented on. Above M; M, is water W, W.. _ W, is continued upwards to form a bubble of water, B, in benzene (shaded) at the top of an inverted glass funnel sealed into the tube which contains it. The upper surface of the bubble is focussed on to the cross-wire of the micro- scope A, and forms a very delicate indicator of the small differences of pressure applied between the surfaces of M, and M,. The diameter of the bubble is about 4 mm. In making a measurement the bubble is kept on the cross- wire by tilting T with 83, and the tilt then affords a measure of the difference of pressure applied *. When working well the movement of the bubble is just visible under a pressure difference of 10-? dyne em.~? E E are taps for locking the manometer during the pumping out of the discharge-tubes. At the bottom of each discharge-tube is a plate of thin rolled sheet zinc covered with perforations (about 500 of them) and connected with a sensitive galvanometer. This plate, shown by a dotted line, is the surface against which the point P discharges. It is more convenient than the ring used in the earlier wind-pressure work, as it does not entail so long a discharge-tube. The perforations must be such that the pressure below the plate is the average of the pres- sures on its uppersurface. For this to occur the perforations must all be equal and equally spaced, and the flow of gas (if any) through the plate at each perforation must be pro- portional to the difference of pressure producing it. This is not an easy condition to fulfil in general; but in our case the differences of pressure that could occur between the two sides of the plate were so extremely small that we were able to obtain the requisite proportionality with sufficient accuracy by using the point of a fine sewing-needle suitably supported to make the perforations. * This manometer is a modification of the one designed twelve years ago for the original measurements of the electric wind (Chattock, Phil. Mag. 1899, loc. cit.). The latter was subsequently improved by Mr. J. D. Fry, and in this form used and described by Dr. T. E. Stanton (Proc. Inst. ©. E. vol. elvi. 1903-4, pt. ii.). In the instrument just described benzene replaces oil as it is less sluggish, and its surface remains clean for much longer periods, provided that the water in contact with the benzene is air free. When, as in our case, it is necessary to trap the water with mercury the effect of the latter on the working of the manometer can be practically got rid of by using large mercury surfaces and putting the mercury tubes at right angles to those containing the water, as indicated in the diagram. The manometer must then be calibrated in terms of a second manometer containing water only. 2G 2 452 Prof. Chattock and Mr. Tyndall on Pressure of the The difficulty that the plate could not be perforated right up to the side of the discharge-tube was met by putting extra holes as near the edge as possible. D, and D, are connected by the tube G; und the gas is in- troduced through G by way of a glass spiral (not shown) sufficiently slender to admit of the tilting of the manometer. The discharging point P, of finest platinum wire cut obliquely with sharp scissors, and sheathed almost to its end with glass, is supported by a thin steel rod sliding in a mercury-tight stuffng-box. This rod is so protected by a column of mercury, C, that no part of its surface which has been exposed to the air passes into the discharge-vessel when the point is lowered. Stufting-box and connecting tube, G, are attached to an ebonite plug which closes the top of D; the joint with D being made gas tight by an indiarubber band (not shown) protected from the air by mercury. Below the perforated plate hangs a small tube of phos- phorus pentoxide. The internal diameter of D,, the tube we chiefly used, is. 4-47 cm. One turn of the tilting-screw corresponds with a pressure of 9:07 dyne-cm.-? The pressure of the gas was. adjusted to 75 cm. The temperature was that of the room. The principle of the wind-pressure method may thus be. described. If a poimt discharges a current of strength c through a uniform tube of cross-section a against a plate at a distance z from the point, the specific velocity of the ions. V is given by the expression ce [dp ae where p is the average pressure of the electric wind per unit area of the plate; it being assumed that the plate fills the tube, and that ions of one sign only are present in the dis- tance dz. The measurements were made in the same way as those described in the 1899 paper (loc. c7t.). Discussion of Results. Harlier measurements by the wind-pressure method * led to values of V, which are in satisfactory agreement with those obtained by other methods. For dry hydrogen the values of V+ and V— were respectively 5-4 and 7°43 cm.- sec. in a field of 1 volt-cm.-!, the hydrogen containing a * Phil. Mag. 1901 [6] vol. i. p. 79. Electric Wind in Hydrogen containing traces of Oxygen. 453 small but unknown quantity of air. Other methods gave for dry hydrogen 5°67 and 7°18 as the most probable values of V+ and V—*, In the present experiments we found that when more than, say, 0°5 per cent. of oxygen is present in the hydrogen, the values of V by the wind-pressure method are about the same as before (V+ =5'8, V—=7'6); but in the case of negative discharge alone, a very striking decrease in the values of dp/dz, i.e. increase in the apparent values of V—, is observed as the hydrogen is made purer, the increase being more than forty-fold in the purest gas we could at first obtain. This is shown in the Table of Results, where are collected the bulk of our determinations; and graphically in Curves I. (p. 454), in which the first and third columns of the Table are plotted together. Kacept where otherwise stated all measurements are given in C.G.S. units. Table of Results. Negative Point. Positive Point. 1a t A | P t A t ercentage p- arent) _ ercentage Ai parent) _ of ome dpide. ve. ve GE Capa: apite. Y ha ¢ aes us 9 0-0 0:22x10°"| 299 |19 || oo ll1x107"| 58 |03 30 210 O07 | 11°5 56 105 4 290 1-1 11°2 57 104 32 200 2°2 0:0017 10°5 Gl +01 "Sl 210 18 “0052 11°0 58 |U-1 27 240 i+} "116 10°3 6:2 0:3 39 160 1°8 33 11°3 wr on PO | 0:0002 ‘87 74 | -94 10°9 595° 102 | “0003 1:29 50 21 1:77 11'8 ha) |0'5 0010 86 75 2°8 3°73 10-7 60 |03 ‘0027 12} 53 + 5 | il: eae a aa —_—} —— 0039 115 56 19 Mean 58 103 "0052 1:44 44 2°4 | ‘0063 1°59 40 2°5 0124 1 30 2:0 0150 2°4 27 1:3 “0209 2-5 26 19 029 a 24 2-0 116 56 MD Tis || 33 6:9 93 |04 94 89 72 108 177 79 <2) 10:2 373 8:7 74 103 Mean of last three ... 76 | 04 * Winkelmann, Handbuch der Physik, iv. p. 595. 454 Prof. Chattock and Mr. Tyndall on Pressure of the dp/dz is the slope of pressure for a current of 1 micro- ampere from the point, and V is the ionic velocity in a field of 1 volt-em.—} CurvEs I, | The Curve marked by crreles t$ Che same ag that marked by dots wrth the percentage scale. mognified one hundred fold — V (appar ent), fa) | 7 2Q 3 Ter centage Oxygen It was found that the values of dp/dz were not independent of the current used to obtain them, and that they increased as the current was reduced—the negatives considerably. The current used for all but the very low percentages was about 2 microamperes. For these, in the case of negative discharge only, it was increased about three-fold, as the pressures to be measured were then so very low. : The numbers in the second column have all been corrected to zero current. This meant an increase in the pressure slope per microampere obtained from a current of 2 micro- amperes of about 20 per cent. for negative and 5 per cent. for positive discharge, in the case of the higher percentages. Now it is most unlikely that the negative ion travels at anything like the speeds shown here--speeds which are moreover out of all proportion greater than those suggested by our 1908 work; and, in fact, we found later that it was possible to obtain values of dp/dz which were actually though very slightly negative for discharge in what was perhaps still purer hydrogen ”*. * A little paraffin-wax had been discovered between the two sets of experiments upon the glass under one of the mercury seals; and this may have given rise to a very slow leak of air into the apparatus between the wax and the glass in the first experiments. Electric Wind in Hydrogen containing traces of Oxygen. 455 This result proves that here at any rate we have a case of discharge which cannot consist of the simple stream trom point to plate of ions of like sign to the point postulated for the formula. It strongly suggests the presence of positive ions travelling against the wind, ions which, even if present in smaller numbers than those from the point, might still reverse the pressure as they travel slower; and it is then of course only reasonable to regard these ions as the cause also of the lowering of dp/dz in those samples of hydrogen which contained oxygen, and for which dp/dz was not negative but only very small. The view that back discharge is the cause of the reduction of the wind is further borne out by the appearance of the plate when examined in the dark. In the purest hydrogen, and for a negative point, the whole plate glows brightly over the surface presented to the point; but the glow fades gradually as small quantities of oxygen are added to the gas. Ata current of 2 microamperes, for instance, the plate was still faintly luminous with 0:1 per cent. of oxygen present, but became quite dark before the percentage reached 0°9. With a positive point no such effect is observed. At 2 microamperes the plate was quite dark, but a very faint luminosity was always obtainable with much stronger currents which was apparently litile if at all affected by the amount of oxygen present. It thus appears that for negative discharge the glow is a maximum when dp/dz is lowest, viz. for the purest hydrogen, and that as oxygen is added the glow fades and dp/dz in- creases, until at about the point where dp/dz gives’ normal values of V— the glow vanishes. If, then, we may take the glow to mean ionization at the plate, we have in these facts strong support for the view that the lowering of dp/dz is due to back discharge. It is interesting that under certain conditions the glow seems to become unstable, tending to contract suddenly often to a point which glows brightly, and is probably coincident with some slight projection on the plate. For if at any region on the plate the back discharge accidentally increased, the lines of force from the point would converge towards that region and concentrate the current there, thus tending to increase the back discharge still further, and so to intensify the concentration of the lines. We have seen this effect on several occasions, and there can be little doubt that it has been responsible for a certain irregularity which characterizes the wind of negative discharge. And this suggests another consideration. As the point is moved further from the plate 4356 Prof. Chattock and Mr. Tyndall on Pressure of the the field at the surface of the plate is increased, owing to the increased quantity of free electricity in the discharge-tube, It may well be that this entails an increase in the back discharge when the latter is present, and a consequent further decrease in the wind pressure. If this is so the apparent increase in V —, while directly dependent upon back discharge alone, may be indirectly dependent through it upon two dis- tinct causes: one the cause of the back discharge itself, and the other the cause, just referred to, which renders the back discharge greater at large than at small values of z. In the 1901 work on hydrogen large variations were met with in the calculated values of V— which could not be attributed to errors of observation, and which seemed to imply real changesin V—. At the time it was suggested that the intermittent escape from the point of occluded gases other than hydrogen might be responsible for the effect. In the light of the present experiments, however, it is clear that changes in the back discharge due to very small changes in the amount of oxygen present are capable of having pro- duced even larger changes in the value of V— than those which were actually found. As such small changes in the purity of the hydrogen must certainly have occurred in this early work, it seems safe to replace occluded gas by variation of back discharge in explaining the variations of V—. Heating Effect of Discharge. In our 1908 paper it was shown that if point discharge occurs in a closed vessel the rise of temperature of the gas depends on the specific velocity of the ions; the rise per watt of electrical energy supplied being a simple function of V and increasing with it. The meaning of this is that if V be increased from any cause, the ions by passing through the gas more rapidly will exist at any moment insmaller numbers between the point and the plate. The drag on the gas and consequently the wind will be less. Hence the cooling will be less, and a higher final temperature will be reached. Abnormally high temperature rises were recorded in the case of negative discharge through nearly pure hydrogen (loc. cit. pp. 44-47); and this was the reason we thought, as mentioned above, that V— might show a corresponding increase above its value in less pure gas. But back discharge will also lead to rise of temperature by reducing the wind, and we now find that a reduction of the oxygen present promotes back discharge when the point is Electric Wind in Hydrogen containing traces of Oxygen. 457 negative. It is therefore much more likely that the high temperatures in question are due to increase in the back discharge rather than to any change in V—. ~ Zoe There remains one other point to be referred to—the curious shifting of the negative wind-pressure curve parallel to itself—an effect which was first observed in the 1901 experiments. In the present work it has been again met with, and now turns out to be another of the effects which a small amount of oxygen has the property of destroying. An examination of the wind-pressure curves themselves will make the matter clearer. f Curves II. + / 0-947 > 3 ; —/ 3-73% 3 LY) = | } yi : 2 Fe 0-947, v3 Ms = 003% Oxygen. ah a ft ad we ° 1 2 3 4 5 iS) 7 8 a In Curves II. are given typical wind-pressure curves for positive and negative discharge, and a current of 2 micro- . amperes in hydrogen containing various percentages of ; oxygen. The wind pressures for different distances, z, : between point and plate are here plotted with z, the origin corresponding with the position of the plate. As it is the 458 Prof. Chattock and Mr. Tyndall on Pressure of the shape and position of each curve which is now of importance, we have arbitrarily chosen a different scale of ordinates for each so as to bring out these characteristics most clearly. The scale of abscissee is in centimetres, and is the same for all. For positive discharge the curve is a straight line cutting the axis of z very near the origin, and very little affected as. regards shape or position by the presence of the moderate. quantities of oxygen we used. The distance from the origin to the point of cutting is z (see Table), and its value lies between 1 and 5 mm. For negative discharge the curve is also straight in its upper part, but in the case of low percentages of oxygen it bends towards the origin, so that z) which is now the cutting point of the straight part produced may have much higher values than for positive. Those we have observed have lain between 1 and 30 mm. In the curves the continuous lines are ruled straight, the dotted portions following the experimental points where necessary. : In the earlier experiments on hydrogen by the point and ring method (loc. cit. p. 87) the same sort of effects were met with; z) for negative varying between 8 and 13 mm., and for positive between 0 and 2mm. The high value for negative was there attributed to a back discharge of positive ions from the ring, these being supposed to meet and com- bine near the ring with some of those from the point. In this way the small values of dp/dz near the ring were explained, and the fact that beyond a certain distance from the ring the curve became straight was taken as proving that the combination all occurred within that distance. This view is probably wrong. In the present experiments, for instance, the electricity in each cub. centim. of the dis- charye-tube was always considerably less than 1 H.S. unit, which means that for every ion there were more than 10° molecules of hydrogen. The chances of back discharge ions combining with those from the point must thus have been negligibly small except quite close to the point, and they must be pictured as traversing the tube practically from end to end. ‘Their effect on the wind-pressure curve will thus be to reduce its slope without altering its straightness ; and so, unless we arbitrarily assume the existence of two kinds of positive ions, one of which combines with the negatives while the other does not, we find in back discharge a simple explanation of the abnormal ionic velocities, but not. of the large values of 2. ——- Electrie Wind in Hydrogen containing traces of Oxygen. 459 There are two other ways in which these large values may possibly arise :— (i.) The ions from the point may travel an appreciable distance before growing large enough to produce much wind. (i.) The gas may be ionized for an appreciable distance from the point—this being equivalent to a lengthening of the point so far as wind production is concerned. It is possible that both these processes may be involved in the explanation of z). There is at any rate definite evidence available for (i.), since Franck has shown* that when discharge occurs in air from the surface of a fine wire in a strong field, it is extremely probable that the ions, whether positive or negative, do not reach their full size while travelling a distance comparable with 7 mm. when driven by the strong fields he used. Nowit 29 is to be due to (i.), we should expect its value in air to be less than 7 mm. for the weaker fields which accompany discharge at a sharp point ; and in fact we find it to be about 3 mm. for both positive and negative f. This is consistent with Franck’s result, and makes it likely that for air Z) 1s what it is owing to the growth of the ions after leaving the point. For negative discharge in hydrogen containing more than 1 per cent. of oxygen, and for positive discharge at all percentages, the mean value of z, is 3 to 4 mm. It is possible that these numbers may be a little too low as the point (in the case of the hydrogen experiments only) unfortunately projected from its glass sheath rather further than it ought (about 3 mm.), and there may have been a little discharge from its sides. The error cannot, however, amount to much more than a millimetre. Taking z) as 4 or 5 mm., it is com- parable with the value for air, and thus presumably explainable in the same way. But for lower percentages of oxygen and negative dis- charge the case is different. 9 is so many times larger that it appears necessary to invoke some special process to account for it. The emission of corpuscles from the point seems to us to furnish such a process. It is known that oxygen tends to prevent this emission, and we now find that zo is only large when the amount of oxygen is small. If the removal of the * J. Franck, Ann. der Phys. Vierte Folge, Bd. xxi. p. 984. + This determination was made with the present apparatus, in which the substitution of a plate for the ring renders dependable measurements of z possible for the first time. 460 Pressure of the Electric Wind in Hydrogen. oxygen is followed by a stream of corpuscles travelling at very high speeds from the negatively charged point, they may travel considerable distances before settling down into the state of ordinary negative wind-producing ions. Moreover they will ionize the gas at these considerable distances. Hence by the principles of both G.) and (ii.) we may expect large values of zp» on this view. Experiments now in contemplation on the effect on 2 of changes in the discharging point should throw light on this question. In this connexion it is worth while to recall the fact that the state of the point has a very marked influence on the amount of combination which occurs between oxygen and hydrogen when the percentage of oxygen is low. For the four phe- nomena of glow, fall of wind pressure, shift of negative curve, and abnormally rapid combination all take place within about the same narrow limits of oxygen percentage; and it is there- fore at least tempting to think that they may all ultimately prove traceable to a single source. Summary. 1. The gradual removal of the last one per cent. of oxygen from hydrogen gives rise toa marked fall in the electric wind when a negative point discharges through the gas against a plate. | 2. This fall is probably due to discharge of positive elec- tricity from the plate, and may even amount to a slight reversal of the wind in very pure hydrogen. 3. The fall is accompanied by a glow upon the plate ’ which may be regarded as further evidence for such back discharge. 4, Within the same limits of percentage change of oxygen the straight part of the negative wind pressure curves becomes shifted parallel to itself through a distance which may amount to 3 centimetres. It is suggested that this shift may be con- nected with the growth of the ions from the corpuscular magnitude to that of the ordinary negative ion. 5. None of the above effects occur with positive discharge in hydrogen, which is practically uninfluenced by the presence of oxygen up to the highest percentage (3°7) used. In conclusion we wish to acknowledge our indebtedness to the Royal Society for a grant in aid of this research. F461 J L. On the Electrostatic Effect of a Changing Magnetic Field. By J. M. Kurwne*. [* the development of Maxwell’s Electromagnetic Theory there is implied, if not directly expressed, the existence of four distinct mutual effects between electrostatic and magnetic phenomena, some of which have to this day eluded experimental verification. While it may not be said that the value of the theory depends to any very serious extent upon the demonstrability of these assumed effects, yet a great theoretical interest attaches to the question whether the foundations of this theory, which has shown such pheno- menal fruitfulness in directing research, can be experimentally established. ‘he four assumed effects are :— 1. A moving electric charge produces a magnetic field. 2. A moving magnetic pole produces an electric field. 3. A changing electric field produces a magnetic field. 4. A changing magnetic field produces an electric field. In all four cases the difficulty in the way of experimental verification lies in the extreme smallness of the force which is finally to be observed and measured, together with the comparatively very large disturbing forces which must come into play when electric charges and magnetic tiglds, of. suf= ficient magnitude are used to make the eftect:scught for at all observable. This is shown at once by the fact that in the denominator of the formula in each case there enteva the well-known v of Maxwell. Of these four effects the first was demonstrated by Rowland in 1876, the third by Hichenwald in 1963. The ecnverse effects (2) and (4) have, so far, resisted ail attempts.-at, ex- perimental proof. ‘The electrostatic effect of a changing magnetic field may be considered as derivable from Row- land’s experiment, if the assumption is made that motion of the electrostatic lines of force relative to the ether is not an essential condition of a mutual magnetie and electrostatic effect ; but the justification of such an assumption is by no means clear. It was the hope of giving a direct and un- equivocal proof of the existence or non-existence of the last- named effect that prompted the present research. Several previous attempts at the solution of this problem have been published, the results, however, being either non- committal or apparently directly contradictory. The first of these attempts was made by Sir Oliver Lodgef in 1889. * Communicated by the Author. + Phil. Mag. [5} xxvii. p. 469. 462 Mr. J. M. Kuehne on the Electrostatic The experiment consisted of suspending two charged mica or gelatine vanes fastened to the ends of a light rod, after the manner of a very delicate torsion balance, inside and in the plane of a large magnetized ring placed vertically, and then reversing the magnetization of the ring synchronously with the natural period of oscillation of the vanes. The experiment was believed to have yielded positive results ; but the fact that it was purely qualitative and that no data are available for deciding whether the observed deflexions were even remotely of the order of magnitude demanded hy theory makes it at least very doubtful if one or more of the very numerous disturbing forces was not entirely responsible for the effects observed. Crémieu* in 1900 published an experiment in which an attempt at a quantitative determination was made. His apparatus consisted of a charged disk of aluminium suspended horizontally by a light glass frame between the flat ends of two spools of wire, thus constituting a parallel plate con- denser. farads: Moment of torsion of upper suspension ¢=58"2 dyne cm. per radian. Scale distance D=388 cm. A B A—B Phasing 6°32 4°43 1 y4 dots x4 a 6°42 4-76 1:49 enti 608 | 4-90 1:29 6 30 a'40 114 Electro- 6:78 510 1°43 magnetic 6°28 5°30 1:12 deflexions. 6°56 | 540 113 6°51 / 5 40 1:00 Rabon 629 | 494 1:32 Ae pas | 6-24 4-765 1-44 bisa | 6 ee 1:44 5:92 47 1:12 Electro- 5°80 5-00 1:00 magnetic 6°20 512 113 deflexions. 6 30 | 5°16 1°20 6:43 | 5:25 1:08 pe 28 | 4-96 1-41 Torsion 6°51 5°67 ‘Q2 deflexions. 6°67 561 115 ATA Mr. J. M. Kuehne on the Electrostatic In the groups ‘electromagnetic deflexions” column A gives the scale readings with both charging and magnetizing circuits closed, and column B the readings taken alternately with the A readings, but with the magnetizing current re- versed. A—B therefore gives twice the electromagnetic deflexion. Hach “reading” is the point of rest determined from three snecessive turning-points. The differences A—B are obtained by subtracting each B reading from the mean of the preceding and. following A readings. In the groups “torsional deflexions” the A readings are taken under the same conditions as the A readings in the other groups, and the B readings are obtained by turning the upper torsion- head over 20 cm. of the scale, so as to produce a deflexion in the same direction, and of nearly the same magnitude, as that due to reversing the magnetic field. Both the electric and magnetic fields were left unchanged while the torsion deflexions were made. If now 2d is the mean A—B for the electromagnetic, and 2d’ the mean for the torsional deflexions, D the scale distance, and @ the moment of torsion of the suspension wire, then the observed moment due to the electromagnetic effect is Hence from the above set the observed moment is pa LS x5 x 58:2 = 7-34 x 388 =°63 dyne cm. while from theory 823 XO 9X AON L =—>S x10 C= 64 dyne cm. In the following table of results : Ei, =the “ effective” charging H.M.F. in volts. E,=the “effective”? H.M.F. in volts induced in one turn of a conductor encircling the magnetic field. 2d=the (double) deflexion due to the eiectromagnetic effect, in cm. 2d' =the deflexion due to turning the torsion-head through an angle 20/2D. L=the moment in dyne cm. calculated from theory. L’/=the measured moment in dyne cm. The last column gives the percentage deviation of the observed from the theoretical value of the moment. The Effect of a Changing Magnetic Field. 475 capacity, scale distance, and moment of torsion of the upper suspension are the same as in the foregoing specimen Set. ; ; Per cent. ie E,. | 2d. 2s L. L’. aa 600 *345 1:16 3°34 33 26 —2] 606 *B42 2°70 572 Oo 15) +6 606 *490 1:29 2A "47 45 —4 600 “499 ‘73 1:24 47 44 —6 956 *346 "69 1:02 "*H2 48 —8 956 348 2:06 3°36 "525 “46 —13 806 “444 1°59 O24 "565 03 —7 772 "485 TOL 1-33 59 D7 —4 828 "490 1:13 134 "64 63 —2 840 “494 Dp 62 "659d 64 —2 848 495 65 ‘73 66 67 +] 864 "485 27 13 66 73 +10 916 490 2°39 2°70 ey | ‘06 —7 956 47 1°43 dy a ‘716 ‘70 —2 956 “490 87 1:02 ‘74 7 he —3 956 492 1:78 1:86 74 “72 --3 1294 ‘480 45 32 98 1-05 +7 1314 498 40 29 1:03 1:04 +1 Ay. L' 3 per cent. too small. Conclusion. Considering the experimental difficulties involved, the form of apparatus used, and the smallness of the force measured, the accuracy of the foregoing results is all that could be expected. ‘The observed moment shows a decided tendency towards lower values than the calculated amount. This is possibly due partly to the effect of the magnetic field on the charging current, partly to a phase difference other than 90° between the electric and the magnetic field. These are the only two sources of error not eliminated by the method of observation, and since both act in the same direction of diminishing the observed deflexion, the sum of their effects is shown by the results to be only about 3 per cent. of the quantity measured, which is not far from the probable ex- perimental error. This can hardly be considered as casting any doubt upon the result. However, an attempt will be made to determine the phase difference, so that it can be taken into account in the calculation. The experiment as it stands shows that a charged body in a changing magnetic field, or perhaps more exactly in a field swept by magnetic lines of force, is subjected to a mechanical 476 Dr. J. Robinson on Kénig’s Theory of the force, whose magnitude and direction are in agreement with that demanded by the electromagnetic theory of Maxwell. The first part of the work was done in the laboratories of the Universiiy of Chicago, the second part at the University of Texas. To the members of the Physics Faculty of the University of Chicago I wish to express my thanks for the appreciative interest they have continually shown in the progress of my work. , Austin, Texas, August 15, 1909. LI. On Kénig’s Theory of the Ripple Formation in Kundt’s Tube. By James Rosinson, (.8¢., PhA.D., late Pemberton Feliow of Armstrong College (University of Durham), Newcastle-on- Tyne *. eta time ago on the suggestion of Prof. Eduard Riecke, hD I performed some experiments to test Koénig’s theory of the ripple formation in Kundt’s tube experiment. ” 33 R, and R, are the radii of ei ehedie 7) is the distance between the centres of the two spheres. Now Kirchhoff did not integrate these equations generally for special values given to the forces, but he calculated the forces necessary to produce uniform motion. He assumed Uy, Vj, Wy, Uy, Vg, We Constants, and so obtained the potential of the forces a Y,, Z; to be 1 1 Oe) REY oe RRA ut =" sq +," "4+ w,? Fe C Vi YE Pa | 1 a Ce ide ; + 20010) 10 + 2u,w; = alk + 2v,u ia 56 ; Konig * makes this the starting-point for his theory of the formation of the ripples in the Kundt tube experiment. * Wied. Ann. xlii. p. 549 (1891). 478 Dr. J. Robinson on Kénig’s Theory of the The sound-wave travels along the tube. He lets Uy = 0) =U =), = 0, and Ya and so finds the potential of the forces between two small spheres inside the tube to be al O= —a7lPR.2w? Qc’ if the density of the fluid be taken as unity. Now letting w vary periodically with the time as it does in the Kundt tube, according to the law t w=wy) cos 27 -,; T QO also varies with the time. K6nig finds the mean value of © to be The components of the force between the spheres are immediately found. Thus x= 2, y= 28, = 08 Deduction of the Theory on Different Grounds. The theory can be obtained also in the following way:— In the Kundt tube experiment the sound-waves travel parallel to the axis of the tube. Then if we put == = p= 0), and W, = We in Kirchhoff’s general equations, we get for the kinetic energy, Ripple Formation in Kundt’s Tube. 479 considering as before between two spheres, | 1 : 1 On T=K.E. of the spheres + 3 Rywit+ 3 Ry?w? — rRYR Pw = or putting R,= R,=R, 3° T=K.E. of the spheres + = Riw?— 7 Rw? = Now if we suppose the period of the sound-wave to be 7, then the velocity of the air at any point of the tube, at any instant ¢, is oy i ale ee where w =the velocity at that point at the times =0, 7, 2r,... Now to find the force between the two small spheres we have rae 0 Wa Wd dt Ow de +Z for the Z component of the force on one of the spheres. Now T is expressed explicitly in terms of wy, wy, ¢,, 2, and so by using the equation for T, the equation of motion, and the equation w=w, cos 2 : = Wy Cos 2a, we ean find Z. Thus ae d or 6 ol dt Qw Oc! ai 2 o> Pa. se ae + —,- R’w? —r Row? —?, v 0c 9 T=mw where m is the mass of each sphere. ol OF = (m+ RoR 2) 2 a} 27 5 OL Fs, t =(m+ —h?—7k*— 2w, cos 2ar-.« 5 0c" T 480 Dr. J. Robinson on Kénig’s Theory of the oa “cre Qa Po AM 3 “= Ni (m+ a Rea R42") = Wy sin 2a os if fo or Ay ED ee eC) and al Og Z=—(m+ 53 aly PMN ol 2 wy sin 2r- Colom 34 ee t 7 4+ Row? cos? ar — ——* Troe Calculating the mean value of this force over the period 7 é : : t ; we see that the term involving sin 27— vanishes, T as Di Wisk t = : ee S| sin am Ob), wae) 2 Py Retiree a 3 Wo! Bil 1s” dar — dt 0 it 2. Pere WO igee: 2 ee 3 For the X component we have d Ob Wolke dt Du. naan Here again the only term remaining after taking mean values is vel ou’ and thus ae 5 7 RE v0 Sa Sade" If we with Konig suppose the motion to be in the zz plane, ehawev 0. These, values of XX, Y, Zare the same as those given by Konig. Ripple Formation in Kundt’s Tube. 481 Letension of Kénig’s Theory. This method of deducing the forces from Kirchhoft’s general equations admits of an extension which may be of some use. K6nig’s investigation was limited to the case when the two spheres experienced the same velocity. A ease has arisen* when it would be of value to remove this restriction, and this has been done in the following. In Kirchhoff’s general equations we will assume as before that w;=v,=u,.=v,=0, but that w, is different from w,. Then the equations for the Z forces are OT IGR ida ST (LOT dt Ou, dy a Haus aa. 2.3 and the energy equation is Mey rites T ? eRe Fe Rew. aRowioy —-— +4mwe +h mw’. Now as T is ex +e%8... explicitly in terms of w,, we, ¢,, C2, we have ae oF us wa oT mR®w,w2——," OC olay Further, if w, and w. are simple periodic functions of the : t ; time ( cos an”), and we wish to find the mean value of Z, Wy the only terms which need be retained are those involving et wee sin? 2m or cos’ 27r-. Hence che Z, = — Mean value of | — rR®w, wy ——™. Cy" | If we now suppose that the velocities w,; and w, vary along the tube as well as with the time according to the law, TC 21° 9 t w= Wy Cos 27- cos %G * Robinson, Joc. cit. Phil. Mag. S. 6. Vol. 19. No. 112. April 1910. 21 482 Dr. J. Robinson on Kénig’s Theory of the we have gl Ly= a7 Wo R® cos a Ee5, a 1 pee § —"0 — — (3 cos O—5 cos? 6), CCE oh F where @ is the angle between the Z axis and the direction of ». Therefore 2 3a R®w yw, cos 8 (3 —5 cos? A) (== = =aage de The same expression is got for Zp. This only differs from the expression when the velocities are the same in the terms w, and wy, 2. e. we have w, w, in place of w,?. We can thus apply the treatment given by the author*, if we make this substitution of ww, w, for wo. Then we find that the force exerted by one whole ripple A; on the next one A, is Ae 2nyra R8w, we j a2" o being the density of the gas in the tube and ay, the distance apart of the ripples A, and Ay. If the distance apart of the ? = ripples A, and Aj is a3, the force exerted by the ripple A; on A, is re 2n3s77 7 RSwe Ws 2 3 A23 P] if m, is the number of smali spheres per unit length of A), ke. * Loe. cit. p. 182. Ripple Formation in Kundt’s Tube. 483 Equating F, and F; for the equilibrium of the ripple A, we get NyW We N3loW2 ; Gos at Ay3 To a first approximation, putting ny=n 3 we get : (2) = WgW3 Ayo) WyWe For the forces between the other ripples we get similar equations :— \3 =) pede ae A23 WolU3 Pere) rt) te i ae is Wr 1,” Forming the product of the terms on the left, and of those on the right-hand side of these equations, we get Gr. r+) 3 Wy, Wr +1 a2 Wy We are ac approximately. If a. is the distance apart of two ripples at an antinode, and a,.-41 this distance at a distance & from an antinode, we have We COS (= oa) ae ( 3 2 r) » 7 k: Ae Wy 21 and Up r+) 27 k —_— COS3 D 7 e Qy92 i l This is a more rigid investigation than that in the Phil. Mag. for July 1999, where it was found that There the investigation was based on the following assump- tion :— Konig’s theory was limited to the case where both spheres 212 484 Dr. J. Robinson on Kénig’s Theory of the experienced the same velocity. In applying this to the case when they do not have the same velocity, it was assumed that the force of one sphere on the other was a function of the velocity at the former sphere. The present investigation shows that this assumption was not well ae The law cos$~ = Esher ed only that the experimental results Bf given in the paper quoted agreed qualitatively with Kénig’s theory. The law here found gives also what may be taken as a quantitative agreement, and so the experimental results quoted give another verification of KGnig’s theory. The following calculation shows how well the law just developed agrees with the experimental results. In the paper quoted above, the mean distance between two ripples was given at different places between two nodes. This latter was divided into three equal parts, left, middle, and right, and the mean distance between two consecutive ripples in each part measured. Hence the distance between two ripples was measured at the antinodes and near the nodes. ‘The ratio of these distances was found as follows :— Using fine iron powder and a frequency of 3208, the distances between the consecutive ripples in the three parts were found to be 1:01, 1:25, and 1:07 mm. respectively. The mean distance at the sides was thus 1:04 mm., and at the centre 1°25 mm. The ratio of the mean distance at the sides to that at the centre thus was ae =a es In this way this ratio was found, for the different powders used, to be 0: eh for fine iron powder. oo coarse 0-70 (0 ne 0°78 va 0-39 f » emery pow der. 0°66 0-70 f sand. The two values for each powder are for two different frequencies. The mean ratio is 0°78. Ripple Formation in Kundt’s Tube. 485 This :ratio can be calculated for the law cos? as follows: lf 21 is the distance between two nodes then the middle vs 2 part of the distance between the two nodes extends 5 to each side of an antinode, and the sides extend from 3 to l. Thus we need to plot the curve y=acosi@, and then the ratio required is the ratio of the mean ordinate of this curve from 30° to 90° to its mean ordinate from 0° to 30°. The curves y =a coss0 and y=acoss0 were drawn, and from them the required ratios were calcu- lated. The results are shown in the following table. Now as the ripples generally do not extend as far as the nodes it is not correct to take the mean ordinate from 30° to 90°. Thus in the following table this has been calculated for three different cases, 30° to 90°, 30° to 80°, and 30° to 70°. mycin jae cos’ @ | cos*@, Mean ordinate from 80°—90° aa Mean . 5» O°—80° hk te Mean ordinate from 30°—80° 0-65 O-45 Mean, 0° —30° Mean ordinate from 80° —70° come aed Meni, “yn uy, 0°—308 ike hee Now the mean value of this ratio from the experimental results is 0°79. Hence considering the ripples to extend only from 0° to about 70°, which is frequently the case, we see that there is a very good agreement between the experi- mental results and the law cosi@ deduced from Kénig’s theory; in fact these results furnish another quantitative agreement between experiment and this theory. Arnistrong College, Newcastle-on-Tyne, December 17, 1909. r 486 J LIl. Lgnoration Problem. Explicit form of Result. By R. HARGREAVES*. FIND that the coefficient of each term in the result of the elimination required for the ignoration problem can be expressed by a simple determinant. In respect to one of three types of coefficient this is already known; the present note deals with all types, and I think the use of the values given here adds something to the symmetry and complete- ness of the solution of this important problem. § 1. A kinetic energy T> involving x velocities is given in the form 21 = ene! + 2a 00 We+ 5: cos ee and the variables are divided into two groups, 2; to 2m and : or ’m41 to x, If we write == for a momentum, the Or problem is to express each of the sums & xpé, and & &mirEmer r in terms of @, and &,,4,, that is in terms of velocities of the first group and momenta of the second group. Half the sum of these expressions is the energy T,, and half the dif- ference is the kinetic potential L, in the modified or reduced system. We shall find that > Lpep= om == I, > ee Eh 2K — E p r : Z so that Tp=T+K, and L=T—K+I; “) where, if A denotes the determinant (@n+41,m41...@n,2) and An+r,m+s 4 first minor, the forms of T, K, and I are QAT=3% bppitg +2 > Bp Byker | \ Pp Pd 2 ZAK > Ande eee ain 2 = Lee fits Cones M-+S> f . (3) rs r,s Al = S Cp, mien ae nm pr The position as regards K is well known; it will appear that 6,,, is the determinant A with a border added, and Cp,m+r 18 the same determinant with a row altered. §2. To establish this we must solve n equations of which the first is Ay, + Gee eta = &, iad VC (4) * Communicated by the Author. ae r Cf. Thomson and Tait, vol. i. p. 320 sqq. Mr. R. Hargreaves on the Ignoration Problem. 487 viewed as a system in which # and &,4, are treated as known quantities, while E,and %m+, are to be found. The value of w+, is given by Te EN cece 1 mtr, m+s( Em-+s— 2 Am-+s, p cay : 2 (5) 8 Pp and therefore : 2 A 2 nee pas Amtr, m+r Bate ae 25 Bante nes m+rom+s r r 7, s —> Leora pi Am-+s, p a an rates Dt s The interpretation of the summation 2 an4sAm+rm+s OF $s > dy, m+sAm+r,m+s is that in the determinant A is to be sub- stituted a row beginning with apn, in lieu of the row beginning with dnir,m41, V1Z- Cp, m+r = Am+1,m+l1 +. Ap,mt+1leee+- + Joe hee. (6) The value of &, is given by hase by, gitg + Cy, maecneee ye ° (7) where ava = Up, 99 Up, m+1 “er oS S An+1,q) Am+1,m+1 ee im 30 ates To obtain (7) the multipliers applied to the rows of (4) are zero for the first m rows excepting the pth; for the pth _and the rows after the mth they are the first minors of 6y,, corresponding to the members of its first column. The multi- plier for £4, is minor corresponding to the r+ 1'th member of the column, and is (—1)" times the value with @pm4 as top row; while ¢p,m4r is (—1)"~1 times the same value, that is the multiplier is —¢p,m4,, the negative sign corresponding to the fact that &, and &,4, are on opposite sides of equa- tion (7), on the same side in equations (4). § 3. In a recent communication* it was shown that this transformation of a homogeneous to a new heterogeneous form, in which one group is represented by velocities, the * Phil. Mag. July 1908, 488 Dr. P. Lowell on Photographs of Jupiter other by momenta, is valid apart from any question of ignored coordinates. Thus generally the force-equations being PML. Oli - kaa i Gaston: Ai F(2,)= dt Oip ie Oe’ Re.) = Dea SQaaume OL and @m4, being — aS there is exact agreement between m+r these forces and those derived from the formula dew Va: att di 62 "oe When the coordinates 2,4, do not appear in the coefficients e e . d of Ty, and there is no corresponding external force, dE mtr = (0) dt : i. e. these momenta are constant. Thus what for the general case is an admissible modification of the expression for energy, and of the form of the force-equations, becomes in the case of ignorable coordinates the reduced form of ‘the system. In this case K has the true character of a potential energy, while the existence of I marks the interaction between the two groups originally shown by the presence of product terms of the form &p%m4+. In conclusion it may be roted that T is the minimum value of Ty which can be obtained by giving real values tO @m41...%n, because the minimum problem leads to the __ group of equations (4) with &,4,=0. LI. Photographs of Jupiter taken at the Lowell Obser- vatory, April 1909, by Mr. E. C. Slipher and the Director, Dr. Percival Lowell. By Dr. Lowxu.*. ae photographs of Jupiter taken here this spring may ° be considered under two distinct heads :—1, researches on what measures of them yield ; 2, study of the markings represented. The first investigation was rendered possible by the great size to which it proved the images could be enlarged without loss of detail. The results are interesting, for they have something to say about the planet’s physical condition. 1. We will begin with the ellipticity of Jupiter’s globe. Measures of the equatorial and polar diameters in five * Communicated by the Author. taken at the Lowell Observatory. 489 images on a plate of mine taken April 14 gave as a mean for the ratio of the two :— polar diameter —s_— 1 equatorial diameter 1:0677° This makes the oblateness = paw = : : a 15°8 i a—b 1 and for the ellipticity = hay ee At the time of the photographs the phase amounted to = of the equatorial diameter. This was partially offset by the irradiation from the limb. Also the poles of the planet are notably darker than the rest of the disk, causing a less visible extension there, and this effect is increased by the photographic plate. As the corrections thus work opposite ways we may, to a sufficient degree of approximation, consider them as neutralizing one another. And in this connexion a caveat should be entered against measures made with a colour screen by daylight, especially when the planet shows a phase, unless these are subsequently corrected for relative irradiation and loss by faintness on the ter- minator. For in such cases measures may show less than reality, thus erring in the opposite direction from what the earlier measures, where irradiation was neglected, did. We will now compare this photographic value with what micrometric and heliometer measures of the planet have yielded. From the various determinations which have been made of Jupiter’s oblateness, we may perhaps select the following as the best :— Date. Oblateness. ee th, Cassini), a wayels « « LEOL. bails Interesting as being the first ever made. 2 SA Ae 1833 1:15:73 MHeliometer cor- rected for phase. ee hie © eee 1856 1:16:06 Micrometer. PROD. a Sataaaee OF 1657. L.s.ker6a E ES yaw apne aba aaee abd 1sgs , 13.1580 Ee Pe noes 1880 1:16-76) ES sg EA 1892-4 1: 15-99 pe W. Hz. Pickering, Lowell 1894 1:16-11 id Obs. Sy a ge wee hea 1894-5 1: 16-7) EPS ocia cote a eee 1900 - ¥:15°53 3 _—_—_. Mean of the last unbracketed seven. 1 : 15°83 —-——- - 490 Dr. P. Lowell on Photographs of Jupiter From the motion of the perijove of the Vth satellite have been deduced :— Date. Oblateness. LID eee ee 1896... 1: 16°50 PVE SPeNGaMS fas eo ae 1899 > besos For some reason the oblateness deduced from measures of the orbits of the satellites seems to come out too large. Thus La Place got from those of the other satellites —o : if and De Damoiseau 13-40 The oblateness has now been so well measured that we are certain that these denominators are too small. We see, then, that the mean of the visual measures and of the photographic agree thus :— it ‘oe oe ~ = photographic deter minations ——< Oblateness, mean of visual determinations . aa A concordance so striking shows how serviceable these photographs may be for future measures of the rotation periods of different parts of the disk. or the present ones can be compared with those taken at any later date. The actual ellipticity of Jupiter combined with its rate of rotation gives us an insight into the planet’s present physical condition. By Clairaut’s theorem the limits between which the ellip- ticity of a rotating body must lie, no matter what the law of its distribution of density be, are }¢ and 3¢ where ¢ is the ratio of the centrifugal force to gravity at its equator when the body is homogeneous. Gravity is the attraction there less the action of the centrifugal force, therefore : i ora fh w2a eee Wer os, a where @ = angular rate of rotation, T = its period, a = the equatorial radius, g = gravity, j = the attraction of the bedy on a point at its equator. The limit $¢, as the equation by which it is determined shows, corresponds to a distribution of density where all the taken at the Lowell Observatory. 491 matter composing the body is concentrated at its centre ; the limit 3, to that in which the body is homogeneous. Now, if we know the body’s equatorial radius and its ellipticity, we can determine its volume, and from its mass its mean density ; and if in addition we know its rate of rotation, we can from its density and its rotation rate deduce g’ by the formula where the unaccented quantities refer to the Earth and the accented to Jupiter. 2 Por. V'= er may thus be found, V being 0°00115, and k being the constant of attraction ; and ¢’ may be expressed in terms of V’ by a process which the reader will find given in Tisserand’s Mécanique Celeste. Tisserand’s data for the densities &c. are not in accord- ance with modern values. It is necessary, therefore, to recalculate ¢. Using, now, for Jupiter the most modern determination of the equatorial radius and the ellipticity which, as we have seen, the measures of the photographs bear out, to wit :— a = 38" at Jupiter’s mean distance, the solar parallax at 8'"80, = 89040 miles or 143300 km., T = 9"-842 at Jupiter’s equator, and c= Bis: or 7= El ~ Peer 158" where e=the ellipticity E am and 7 the oblateness om) 1 we find o 10°2° The ellipticity is the more philosophic quantity to use, although the oblateness is more commonly employed, because the ellipticity associates itself directly with ¢', and thus with the planet’s physical state. The limits then between which the ellipticity could lie are 1 E 20-3 and ah the former corresponding to total concen- tration at the centre, the latter to homogeneity *. We thus * For anew and more exact method of determining these limits and ¢, see paper by the avthor appearing in a following number. 492 Dr. P. Lowell on Photographs of Jupiter see that Jupiter with an ellipticity of ae fulfils Clairaut’s law, and furthermore that the matter composing him comes nearer to central condensation than to homogeneity. Going into this more carefully and comparing the cor- responding data for the Earth, Mars, and Saturn, we get the following table :— p 5 a—b a—b Ratio 2 4? ae ig ee PBA Hs 5 os tee = kia a = =(0'97o Milas es kc = = = a = iter Jupiter .... a a a S = 0-632 Saturn. s,./) = ; = = == (7575 These are the only planets for which we have the complete data. Mercury and Venus rotate so slowly, in 88 and 225 days respectively, that their oblateness is too small to be measured, so that nothing can be deduced of their laws of density. Of Uranus and Neptune we lack as yet the rotation periods, and in the case of the last even the flattening, so that their distribution of matter remains equally hid. In considering the table we see that the Harth and Mars lie nearer homogeneity than central condensation, since the conditions for these states are e=0:5d and e=1°25¢. Jupiter and Saturn, on the other hand, approach central condensation, Saturn indeed almost reaching it. Now, when we reflect upon the great pressures to which the matter composing the major planets must be subjected owing to their enormous masses, we perceive that such slight densities as they present can only be maintained through great internal heat. The force of attraction at Jupiter’s equator, for instance, would be 2°61 times the Earth’s were his substance homogeneous, and 2°51 times were all his matter concentrated at his centre, being given by _ ,Ampb 3 9 PHISE (1+ 5-35 where 0 is the polar radius, in the one case, and by e? + ete. ), m J= k ei in the other. taken at the Lowell Observatory. 493 As we have seen that he more nearly approaches the central concentrated than the homogeneous state we may take 2°54 as the value. His centrifugal force, so called, at his equator is Cif gh eae | ou, whence we finally get or °23 times the Harth’s attraction, 2°31 for the force of gravity at his equator compared with that at the Harth’s. If Jupiter were cold this would cause a density greatly exceeding the Harth’s, while the facts are the other way, Jupiter’s mean density being to the Harth’s as 1°32 to 5°527. It is evident, therefore, that he must be expanded by heat. Furthermore, the gradation in density from core to cuticle must be gradual. There can be no sudden change from a dense kernel of Jupiter to a vast gaseous husk. For the great pressure in such a case would act to thin the air envelope relatively more than in the case on Earth. Nor would any permissible increase in the relative amount of air held by the planet come within the figures needed. For if the density of the nucleus were only that of the Harth, a modest estimate, the volume of the air outside it would be about 9 times its own, an amount of air with which we have no warrant for crediting it. 2. The next point is the position of the several belts. The measures of the images on the plates of April 14 give for their apparent places, t taking the polar radius as “the radius of the sphere of comparison, which, since the tilt was only 1°3, may be taken as the apparent minor axis of the ellipse :-— tamer ec Sa, a I ER a BNYD South Temperate belt. . 5 Vis ew tage, ae eG South edge of 8. Tropical Bictbee uset Ohad 3 —19°5 dark line in a a ee North edge of South Tropical bali i be - — 6°8 North Tropic Bart. 43.8% bie pr tf | GP ad. North stripe of 8. Tropical Gabe Edt AY ne + 17°83 North Temperate belt. . . + WaT EO) At the time the photographs were taken, April 14, the latitude of the planet’s equator with regard to the Earth was — 1°3. 494 Intr oducing this correction we have: Dr. P. Lowell on Photographs of Jupiter Antarctic belt «.. —38°°5 South Temperate belt. —28°°9 South edge of S. Tropic belt. —20°°8 dark line in . "8 —17°2 North edge OF pee i — 81 North Tropic belt . . + 81 North stripe of 8. Tropic belt +16°5 North Temperate belt. . . . Mah +33°7 From this we see that the belts near the equator are here symmetrically disposed with regard to that line and regard- less of the declination of the Sun at the time. Hence - they cannot be caused by him but by Jupiter himself. Jupiter’s cloud envelope is therefore totally unlike our own, and isa self-raised, not a sun-raised phenomenon. This state of things can only be due to great internal heat setting up currents which are persistent in their main features for ‘long intervals of time. At other longitudes their latitudes differ; a feature, interesting as it is, into which I do not propose hare to enter, and unaffectin g apparently the present conclusion. The Jatitudes we have deduced are those on a sphere of radius equal to the polar radius. To correct these to the apparent latitude on the planet we proceed as follows. In an oblate spheroid any meridianal plane cuts out an ellipse in which all lines perpendicular to the polar axis are in- creased over those of the sphere with that axis as diameter in the proportion of < to b where b a = equatorial diameter, b = polar ‘9 Tf, then, a = colatitude measured on the sphere, ellipse, a 9 ” ” a we have tan a’! = tan a. This gives for the apparent latitude on the planet for the several belts :— Antarctic . . a laced’ vid id Gree ky achiral South Temper es MR ee Ne South edge of S8. Tr oe ele’ he ee dak ene Gark line im . 5; Me eS North edge of __,, ee North Tropic . OME AS eta North stripe of N. Tropic Me eS) North Temperate te + 32°-0 taken at the Lowell Observatory. 495 3. Turning now to the features themselves, we note that the great red spot, as a spot, practically no longer exists—it is now only faintly suffused with rose—but what was its oval cradle is well defined in the photographs, shown much as it has shown for the last thirty years. In fact in this connexion it is interesting to note that this cradle proves to be very much older than most people think. Long before the red spot made its sensational appearance the cradle it was to occupy was there. [t is distinctly traceable in the drawings of Sir William Huggins made in 1859-1860, which he kindly sent me. My eye detected it the moment I examined them. Here then we have evidence of a feature which in its general outline has been stable and persistent for fifty years, a marvellous length of time for a cloud form as we know such things to continue to exist. This alone would suffice to demonstrate that Jupiter’s meteorologic conditions owe nothing to the Sun. That this cradle then became the centre of a vast ruddy mass, which after a time disappeared to leave it in its former condition, indicated it as the seat of some violent outburst from below, over which the cloud veil, rent at the height of the explosion, settled down again, covering the furnace from sight. That the feature is so permanent hints at a certain plasticity as op- posed to what we call gaseous in the constitution, and likens it to those permanent seats of disturbance with whose vents we are familiar on Harth, A volcano in embryo is what its behaviour points it out to be. It probably forms a con- necting link between volcanoes on the one hand and sun- spots on the other. 4. Perhaps the most interesting feature of the photographs is their exhibition of the wisps lacing the equatorial belt. These wisps were first detected and studied, I believe, by Mr. Scriven Bolton, who communicated them here through Prof. Turner. They were then visually observed at Flag- staff, and many drawings made of them. Lastly, they appeared in the photographs, being well shown in the prints. They are very curious markings, being shreds or filaments of a dusky character traversing transversely the bright equa- torial belt. They proceed always from black spots of a triangular shape set on the edge of the north or south tropical dark belts as the case may be. They join such to one another and to the medial dark line which with more or less interruption threads the planet’s equator. Their course is usually inclined some forty-five degrees to the dark belts ; but sometimes they go straight across. When tilted they are as often right-handed as left-handed. ‘They are widest 496 Photographs of Jupiter taken at the Lowell Observatory. at their sides, where they leave the belts, and tenuous in mid-career. What they betoken is difficult to decide, because from their dark tint they would appear of a negative rather than of a positive character, gaps, that is, in the cloud envelop, rather than entities in themselves. That they give us a clue to the meteorological circulation of the planet is clear; but it is not at all clear what that clue is. That they are an essential feature of a planet in the early semi- chaotic state is indicated by a recent discovery at this obser- vatory: that Saturn’s belt is laced, though much more faintly, in precisely the same manner. 5. The colour of the dark markings of Jupiter is everywhere a cherry-red. As the physics of his condition show that he must be very hot, his density not being compatible with his gravity under any other supposition, the dark markings must be where the lower and hotter layers of the body show through. For that red heat is the maximum possible for his surface glow is evidenced by his albedo °78 of pure reflexion ; no perceptible light being emitted by him. We are thus warranted in concluding that the lighter, yellowish belts lie at a higher altitude, and are what they seem to be, clouds. From Dr. Slipher’s spectrograms these clouds appear to be made up in part at least of water-vapour, in part probably of other substance, though about this we have no positive evidence, as the spectroscope informs us only that unknown gases are certainly present in the planet’s atmo- sphere, but gives no assurance that these are condensed to cloud, possible and even probable as this may be. That the bright belts are cloud and show no self-emission of light appears also from their albedo being almost exactly that of cloud, 0°78 as against 0°72. | The various lines of evidence thus point to one conclusion : that Jupiter is still in a fiery chaotic state, that the materials that went to make him are still in disconnected fluid form, that they are much condensed toward his centre, but are kept from absolute solidification by his intense heat; in short, that he is in a midway stage between Sun and Earth. | 6. To the great use to which these photographs and their successors may be put I may here draw attention. Per- mitting, as they do, of measurement of the planet’s features it will be possible in the future to get a record of alterations in the appearance of the belts which no one may dispute and any one may verify by inspection himself of reprints from the negatives, and which in addition allow of quanti- tative valuation. Thus the rotation time of the several belts : | } On the Use of Mutual Inductometers. 497 or of special details of them may be got with greater ease than at present, and any changes in rate from time to time may similarly be detected. Unatfected by any personal equation, the photographs have the further advantage of being original documents which can be used at first hand by any investigator. Flagstaff, Arizona, Nov. 29, 1909. LIV. On the Use of Mutual Inductometers. By ALBERT CAMPBELL, B.A.* (From the National Physical Laboratory.) . Introductory. . Modified Mutual Inductance Bridge. . Measurement of Effective Inductance. . Null Method in Iron Testing. . Tests of Current Transformers. WPA OAC Or CO DO ee §1. Introductory. i. a former paper f I have described arrangements, which we may call mutual inductance bridges, by which self inductances, even of very small amount, can be directly measured {. J wish here to discuss further these and other methods in which use is made of variable mutual inductances or, to give them the more convenient name, mutual inducto- meters. § 2. Modified Mutual Inductance Bridge. In the equal arm bridge already discussed there is con- siderable loss of sensitivity resulting from the insertion of an auxiliary balancing self inductance in one of the arms. By a modification of the arrangement of the bridge, however, the use of the auxiliary coil is rendered unnecessary and the whole apparatus becomes simpler and more efficient. Let us first consider the more general case shown in fig. 1, where the ratio arms are not equal. * Communicated by the Physical Society: read January 21, 1910. + Phil. Mag. Jan. 1908, p. 155. { I would mention here that many years ago Dr. Oliver Heaviside investigated a very general case of inductance bridges (Phil. Mag. p. 173, vol, xxiii. 1887). Most of the possible combinations are included in his paper. Phil, Mag. 8. 6. Vol. 19. No. 112. April 1910. 2K 498 Mr. A. Campbell on the Let N be the self inductance of the coil to be measured, ry a constant inductance rheostat, L, L, self inductances, Fig. 1. WL 7g, OC é Ce ae a 2, M 0 Sa Ce A B P Q Sy 6p) while the current 7 acts inductively on both sides of the bridge as shown, the mutual inductances being M and m. The coils L, L, may be the upper and lower fixed coils in the inductometer which I have before described (loc. cit.), the inducing coils carrying the currentz. Thus L, and L, will usually have mutual inductance (call it y) between themselves. Let 2, and 2, be the instantaneous values of the currents (of sine wave form) in the arms L, and Ly, respectively, the resistances of these arms being P and Q. Let a=o/—1, where w=27n, n being the frequency, and let R/S=c. Then, when there is no current through the galvanometer, we have Re, = Se On agp—25 and (P+ Lye)t, + Matt yar, =(Q+ Loa + Na)tg— mat + yar. Since i=7,+72, we have Pa, + Lyaty + Ma(t, +25) + yards = Qi, + Lyat,+ Naty—me (ty +79) + yori. Separating the real and imaginary parts, we obtain Po and o(L,+N)—L,=(m+M)(eo4+1)—v(e—1) . . (1) Use of Mutual Inductometers. 499 The most useful case of this is when the ratio arms are equal, 7. e. c=1, and then these equations reduce to Oe te ae eis sl (2) and Le—LIy+tN=2(m+M).. . . . . (3) The best arrangement is to make L,=L, permanently in the inductometer, and then the unknown inductance is given directly by vg an nam © 2 2 When L, and Ly, are the upper and lower fixed coils in the inductometer, the reading of the instrument is m+ M and thus N=2x Reading, and thus is read directly. The sensitivity of the bridge is here much greater than when L, consists of a separate balancing coil. §3. Measurement of Effective Resistance. If an alternating current at ~ per second having ap effective value I in flowing through a circuit wastes energy in it at the rate of R’ I? watts (including losses due to edd currents, magnetic and dielectric hysteresis, &e.), then R’ is called the Effective Resistance of the circuit (for frequency n). In general it increases with the frequency, and when tele- phonic frequencies (500 to 2000 ~ per sec.) are reached, it may often become very much larger than the ohmic resistance. As its value governs the loss of energy, it is of the utmost importance in telephone work to be able to measure it with accuracy. In former papers already cited I have shown how it can be directly measured by a mutual inductance bridge simultaneously with the effective self inductance of the circuit. When the bridge has equal ratio arms which can be interchanged to ensure that they are identical, the method is free from serious error ; but if unequal ratio arms have to be employed, large errors may arise from the very small but unavoidable self inductances of these arms. The importance of this in the simple self-inductance bridge was pointed out lately by Giebe *, and upon his mathematical result he based his ingenious method of measuring very small inductances. I have applied similar treatment to the somewhat more complicated case of the mutual inductance bridge as follows. * Ann. der Physik, p. 941 (24), 1907. 2K 2 500 Mr. A. Campbell on the In fig. 2 let the resistances of the four arms be P, Q, R, and 8, and their self inductances L, N, /, and X respectively. Fig. 2. Let M be the mutual inductance as shown. By procedure similar to that in §2 it is easy to show that PS —QR=o7[((L—M)A-(N+M)l]_ . . () and SL-RN=(S+R)M—Pa+Qil . . . . (6) Several cases are important. Case (1). When M=0 we have PS+QR=e@7CAN=N)) 2.) a and SL—RN=QI-PA. ... .. 3 ‘These are Giebe’s equations for the ordinary self-inductance bridge. Case (2). If R=S, then (P—Q)R=o*[(L—M)a—(N+ Mi] and (L—N—2M)R=Q/—Pa. If also A=1, then (P—Q)R=o7l(L— N—2M) and (Q—P)l=R(L—N—2M). Hence either R? = —’/?, which is impossible, or P=Q ...',) . 2 and L—-N=2M.. °°). | ae i ! | Use of Mutual Inductometers. 501 Thus we see that in an equal arm bridge the simplest con- ditions are got by making sure that the ratio arms have equal self inductances. They can be adjusted by interchanging them and altering until the interchange does not alter the balance of the bridge. Case (3). If Rand S are unequal, let R/S=o as before. The arrangement is best used as follows. Let Q be an auxiliary balancing coil, set to give a balance in the bridge initially when M=0 and P=P,. Thus we have for the preliminary conditions Poo ea (LANL) ayia ey ee (ED) and Sota Qiao wh, te thas ts, ¢ tee) Now let a coil to be tested, having resistance T and self inductance X, be introduced into the arm P, and !et the balance be restored by reducing P, to P, and by setting M. Then we have (P,+T)S—QR=o?| (L+ X—M)rA—(N+M)/] and §S(L+X)—RN=(8+R)M—(P,+T)r+4+ Ql. Subtracting from these (11) and (12) respectively, (P},—P,+T)S=o7?[(X—M)rA— M7] =o [XrA— M(\+)) | and SX=(8+R)M—(P,—P,+T)r. In the most useful practical case 7 and X are small com- pared with X and M, so we may take as a first approximation SX = (8+R)M or Pilea Lt OLN ces Ria ic Vy, aah ED Hence Bae) Tg oh —t ie or T=P,—Pi+@*(oA-1)M/S. . (14) and a closer approximation then is X = M[1+oe-@*(orA—I)A/S] . . . (18) Thus the unknown T and X are obtained from the change made in P, the reading of M, and the corrections due to X and 7. It will be found that the correction is usually almost negligible in the expression for the inductance X. On the my Mr. A. Campbell on the contrary, for the effective resistance T the correction may become very important, since w? is large even for moderate frequencies. For example, let R=99, S=1, P,—Pi=20, M=1 millihenry, ]=10 microhenries, X=1 microhenry, and n= 1000 ~ per sec., giving w°=40 x 10°. Then we have T+P,—P,+3°6 = 23'6 ohms, while X=100M—3-6x10-° =0°1—3°6 x 10-° henry. Thus the small inductances 7 and X affect the measurement of the effective resistance T by as much as 15 per cent., while the self inductance X is only affected by 3°6 parts in 100,000. This example shows how much more difficult it usually is to measure effective resistance than self inductance. The difficulty is got over, however, if we can make RA=S/ (c.e. oA=1), for then the terms involving / and A disappear and we have A=(ls+o)M tc. eS and T=Py—Pj.. wy sp 4 To ensure that RA=S/, or in other words that the pro- portional arms have self inductances in the ratio of their resistances, is not avery easy matter. ‘The following is the best method that I have tried. A coil (A) is constructed of highly stranded wire to give moderately high self inductance (say 01 henry) with as high effective insulation resistance as possible *. Its distributed capacity 4, which should be as small as possible, may be measured by connecting it through a thermoammeter to a small variable condenser and adjusting the latter to give resonance with an alternating current of known frequency (2000 to 10,000 ~ per sec.) in a loosely coupled neighbouring coil. If K be the reading of the variable condenser in mfds., and L the inductance in henries of the coil (A), then m/(K+4)L=159°3, and so & can be found. It is well known that the effective resistance T’ and induc- tance X' are given by Dolezalek’s formulas, Xt X20. «2 ee and TY T+ 20h),. oe. where X and T are the values if & were absent (in this case the values for very low frequency). * See ‘ Electrician,’ Dec. 10, 1909. Use of Mutual Inductometers. 503 The coil (A) is then tested in the bridge, and if the values obtained for X! and T’ are not those given by equations (18) and (19), a small amount of inductance is added to Ror S until X’ and T’ are read correctly. When this adjustment has been made once for all, it will be found that any other effective resistance will be measured correctly. It may be mentioned that a well proportioned 0:1 henry coil of 7-strand wire, each strand being of 0°2 mm. diameter and separately insulated, should show practically no variation in efiective inductance and resistance due to skin effect at 1000 ~ per sec.*; its distributed capacity should be less than 0:0001 mfd. If a coil when tested at various frequencies shows percentage variations in its effective resistance which are double those in its inductance, then these variations can be accounted for by distributed capacity; but if the variation of the effective resistance is at a greater relative rate, then we must look for additional causes such as leakage or skin effect (or errors in the bridge). When the self inductances of the inductometer coils are high, their distributed capacities may cause errors at the higher frequencies. The effects of the capacities of the coils L, and Ly (fig. 1) are got rid of by the method of preliminary balancing ; unfortunately this does not eliminate the effect of the capacity & of the external inducing coil. If its resistance and self inductance are p and z respectively, then for an equal arm bridge, instead of equations (2) and (4) we have Q=P+20*%khpM—wkM? . . . . (20) and Saat aria 8 ory oe aD) where & is in farads. 7 The error in Q due to the second term in (20) is usually the most serious. Both errors can be approximately elimin- ated by connecting a small capacity of suitable amount across .the points ¢, d (fig. 1). § 4. Null Method in Iron Testing. The use of a mutual inductometer affords a null method for the magnetic testing of iron; this is analogous to the self inductance method which Max Wien investigated very thoroughly some years ago +, and there are cases in which it * See M. Wien, Ann. der Physik, p. 1,(6), 1904, and Dolezalek, /. ¢. p- 1142 (12) 1908. 7 Ann. der Physik, p. 859 (66), 1898. 504 Mr. A. Campbell on the may prove of distinct value. The connexions are shown in fig. 3, Fig. 3. An iron ring is wound with superimposed primary and secondary coils of turns N, and N, respectively, m being the mutual inductance between them. ‘These coils are connected, as shown, through the coils of a mutual inductometer D toa source of alternating current and a vibration galvanometer (or tuned telephone), a sliding contact E allowing a part Q to be selected from a resistance connected with the junction of the coils of D. The galvanometer circuit is best made highly inductive by a coil J. Let the hysteresis * and eddy current loss in the ring be represented by a tertiary closed winding, evenly distributed, of resistance R and self induc- tance L, having mutual inductances F and G to the primary and secondary coils respectively. If N,/N,=6, then G=O6F. Let 2, 22, 23 be the instantaneous values of the currents in the primary, secondary, and tertiary coils respectively, I,, I, and I; being their effective values. Also let the current 2, be of sine wave form, which may be attained by electrical tuning or otherwise, the period of the galvanometer being also tuned to that of 7,. The galvanometer deflexion can now be reduced to zero by adjusting D and E to values M and Q respectively. * If this be not considered rigorous enough for the hysteresis loss, the part relating to it can be proved by another method. Use of Mutual Inductometers. 505 Then we have (R + La)23 = Fai, and therefore [Ry Tet) =Po’, . . . 2) Also (m— M)ai;= — Qt; — Garis, ast A FGo?, —— (2, + R. te La y) lot FGo?Ri, FGLe?«i, | Sat RT! B+ Le Hence nptes ea avid FG Lo? mM te Lh and FGRo? FPR a? | Bs? Q=R Le? b(R?+ L?w*) 51?" Therefore the total iron loss RI,’ mah ay i an el? e N,/N.. e ° ° ° ° (23) Also m—M=—QOL/R. 2.4 Ge) I have assumed that the conditions are not sensibly affected by the small harmonic currents of higher frequency which pass through the galvanometer without any consider- able effect on its deflexion. The object of the inductance coil J is to check these, and this could be done even more effectively by adding also a condenser of suitable capacity. Since the magnetizing current 7, is of sine wave form, the flux density 4 and the total flux ® will in general not be sinoidal. It is usually necessary to reduce the results of iron tests to a standard value of Ajax with induced secondary voltage of sine wave form. When this is required the Bnax and the form factor of the secondary voltage are observed simultaneously with the above measurement of power. This is done in the usual way by the help of a synchronous com- mutator *. By making tests at two different frequencies the hysteresis and eddy current losses may be separated. It should be noticed that the work done in the iron by the current 7, is got by integrating 1d4; if % has no harmonics, the harmonics in .# disappear in the integration. * See Lloyd, ‘Bulletin Bureau of Standards,’ p. 467, vol. iv. 1908; also the Author, Proc. Inst. El. Engs, p. 553, vol. xliii. 1909. 506 Mr. A. Campbell on the I have tried the method with a ring of stampings from ordinary transformer sheet, and obtained a result in very fair agreement with the ordinary wattmeter method; but further experiments are desirable. I have found the method convenient for testing small samples of iron for telephonic work where the tests have to be made for very low values of H (say about 0:01), and in such cases the method is so sensitive that it is easy to test very small rings weighing only a gram or two. When the eddy currents are very small, equation (24) becomes m=M; and since ~ m=Ar x 10-°N,N.ys/(circumf. of ring), where s=section of ring, we can at once find the permea- bility « for various values of I, (and hence of #%). In such cases I have found the values obtained at moderate fre- quencies in agreement with the results of ballistic tests. When the eddy currents are large enough to affect the phase of ®, but do not alter its magnitude appreciably, as Mr. T. L. Eckersley has pointed out to me, we can still find p; in the galvanometer circuit in fig. 3 the vectors Ql, oMh, and wN.® form a right-angled triangle (® being the effective value), and hence ~ eNO? =(Q2+oM)L2 . . . . (5) Thus we can find ® and wp for any value of I, by observing Q, M, and the frequency. -§5. Tests of Current Transformers. In current transformers used with ammeters, wattmeters, &c., it is necessary to be able to test accurately the ratio of current transformation (I,/I,) and the angle of lag (¢) between the secondary and the reversed: primary current. The above method may be used for this as shown in fig. 4, where the transformer has any desired load; but part of the load is the known resistance 8. By considering the galvano- meter circuit, when a balance is obtained, the vectors oMI,, QI,, and SI, form a right-angled triangle, and hence oM tan i = O and J,?/I,?=8?/(Q?+ M’w?). . . (26) 2 2,2 When ¢ is small, this =a(1-“gr Use of Mutual Inductometers. 507 or Ty/Toe= G (13 tan? 6) S P? =a(1-s} Usually ¢ is so small that we may take ey GO a Ca Fig. 4. zy | 0 00209 0 0900090 <= O00 hp With regard to the actual working of the method, the inductometer should be a low-reading one whose primary coils can carry the large primary current. The resistance Q may be a standard low resistance shunted by a slide wire along which the slider runs. Tests by this method on a commercial transformer gave results in practical agreement with those obtained by Mr. C. C. Paterson by a wattmeter method, the source of current being a sine-wave alternator. This agreement is interesting as-showing that the harmonics in the secondary current (which are not taken account of by the vibration galvanometer) have very slight effect on the results. The method should only be used where the primary current can be made approximately sinoidal. In conclusion, I would express my best thanks to Dr. Glazebrook for his kind interest in the work, and to Mr. T. L. Eckersley for valued and suggestive criticism. [p08 J LY. On the Laws regarding the Direction of T. hermo- blocs Currents enunciated by M. Thomas. By CHarues H. Less, DSe., F.RS., Professor of Physics in the East London College, University of London*. N number 8 of the Bulletin de la Classe des Sciences of the Académie Royale de Belgique for 1909, M. Bruno- Joseph Thomas enunciates the following laws as the result of his examination of the direction of the electric current pro- duced in more than 100 thermo-couples :-— “1, When a difference of temperature is established be- tween the two junctions of a thermo-electric couple, the current produced flows from the hotter to the colder junction through the metal which is electrically the better conductor, if the product of the coefficient of electrical conductivity of that metal by the coefficient of thermal conductivity of the other metal exceeds the product of the coefficient of thermal conductivity of the first metal by the coefficient of electrical conductivity of the second. “2. If the former product is less than the second, the current flows from the hotter to the colder junction through the metal electrically the worse conductor. “3. If the two products are equal no current is produced.” + The author states further that at different temperatures “for a given couple the strength of the current produced increases with the ratio (“rapport”) of these two products, and other things remaining equal, the strength will be pro- portional to that ratio” f. M. Thomas supports his theory by 14 pages of tables, in which the values of the products for each of the 105 couples which can be obtained by combining together in pairs the 15 metals of Becquerel’s thermo-electric series, are com- pared. Nota single exception to the theory is found, and at first sight it appears as if M. Thomas had succeeded in establishing a most important and hitherto unsuspected law of connexion between the thermo-electric properties of a metal and its conductivity for heat and for electricity. On more careful examination, however, it is seen that M. Thomas in his tabular proof has based his argument entirely on the order of the metals in Becquerel’s thermo-electric series, on the heat conductivities of 7 of the metals according to * Communicated by the Physical Society : read February 25, 1910. + Bulletin, Classe des Sciences, Roy. Acad. Belg. 1909, p. 903. ¢ Ibid. p. 925, Laws regarding Direction of Thermo-electric Currents. 509 Berget, of 2 others according to Wiedemann and Franz, and on the electrical conductivities of 12 of them according to Latimer Clark, and of 1 other according to Becquerel. Of the heat conductivities of the 9 metals thus obtained M. Thomas states himself that he modifies 4 by amounts varying from 2 to 10 per cent. in order to make the values of the products fit in with his laws*. As he had no values for the heat conductivities of the remaining 6 metals, nor for the electrical conductivities of 3 of the metals, he states himself that he assumes values for these quantities which fit in with his lawt. It cannot be claimed that such a method of treatment pro- vides a firm support for the theory advanced by M. Thomas, and it is necessary to examine it more critically in the light of the more accurate values of the constants which are available. In the first place, it is possible to put the theory into a more compact form than that adopted by M. Thomas. According to him, if electrical conductivity of the better electrical conductor x heat conductivity of the worse electrical conductor is greater than electrical conductivity of the worse electrical conductor x heat conductivity of better electrical conductor the thermo-current flows from hot to cold in the better electrical conductor. If, on the other hand, electrical conductivity of better electrical conductor x heat conductivity of worse electrical conductor is less than electrical conductivity of worse elec- trical conductor x heat conductivity of better electrical con- ductor the thermo-current flows from bot to cold in the worse electrical conductor. Dividing the inequalities through by the product of the heat conductivities of the two metals, the statements become :— electrical conductivity kt ——_____——— heat conductivity >electrical conductivity <= heat conductivity of better electrical conductor of worse electrical conductor the * With regard to the values for these four metals—copper, gold, iron, and lead—-M. Thomas says (p. 904) :—‘‘Si je les acceptais tels qu’ils sont, j’aurais nécessairement un petit nombre de couples dont le sens du courant ne répondrait pas a mes lois. Je les ai done modifiés légérement ainsi qu’il suit :—” + Of the latter quantities M. Thomas says (p. 906) :—“ Je ne connais ni les coefficients de conductibilité électrique, ni les coefficients de con- ductibilité calorifique du cobalt, du manganése et de l’arsenic,.... Je prendrai donc par hypothése pour coefficients du conductibilité électrique du cobalt, du manganése, et de l’arsenic Jes nombres 1580, 1000, et 200.” 510 Prof. C. H. Lees on the Laws regarding the thermo-current flows from hot to cold in the a worse | electrical conductor. That is, the current flows from hot to ea electrical conductivity heat conductivity cold in the conductor for is the greater. In the original form the law was tested by a comparison of the 210 products of the electrical and heat conductivities of pairs of the 15 metals; in its present form it may be tested by a comparison of the 15 quotients of the electrical by the heat conductivities of the 15 metals. It may be noted that the electronic theories of conduction of heat and electricity in metals make this quotient the same for all metals, and that these theories and that of M. Thomas are therefore incompatible. In selecting the experimental material with which to test M. Thomas’s theory it is important to remember that both the electrical and the thermal conductivities and the thermo- electric properties of metals are influenced by the presence of impurities, and that it is advisable to use only results obtained with pure materials, If, however, results for abso- lutely pure materials are not available, those for approximately pure materials may be used, if in the case of any particular metal the impurities in the samples used in the different sets of observations were the same in amount and in material, or better if the whole of the measurements have been made on the same material. Fortunately in the present case we have the experiments of Jager and Diesselhorst *, who determined the quotient of the thermal by the electrical conductivity and the thermo-electric power of each of a number of rods of very nearly pure metals. Their results for the quotients of the two conductivities have been con- firmed by more recent observations made by other methods +, while their results for the thermo-electric powers of the metals agree with those obtained previously for approximately pure metals f. They may be used therefore without hesitation as the best results available at the present time for the comparison of M. Thomas’s theory with experiment. * W. Jager and H. Diesselhorst, Wissenschaftl. Abhandl. der Phys.- Techn.-Reichsanstalt, ii. p. 270 (1900). + C. H. Lees, Phil. Trans. Roy. Soc. Lond. A. 208, p. 381 (1908). t K. Noll, Ann, der Physik, lili. p. 874 (1894); W.H. Steele, Phil. Mag. xxxvil. p. 218 (1894). Direction of Thermo-electric Currents. OW Me The following table contains their results at 18° C. for each of the metals tested. Where they give more than one result that one has been used to which they attach most weight. The unit of heat is the watt second ; electrical conductivity is expressed in reciprocal ohms per centimetre cube :-— Conductivities, Thermo-electric powers in micro-volts Metal. per degree ; thermal electrical with respect to electrical, thermal. cone ia Aluminium ...... ‘63610—°| 1:57x10° +32 —03 UO 665 1°50 —0°3 +3:2 CL 672 1°49 —O1 +30 BUNGE ads vos0cccs. 686 1:46 0 +2°9 Lo) 699 1°43 eae Malice: Cadmium ......... ‘706 1°42 —0 +3'5 i) ‘709 1°41 0 +2°9 Lic ls eae “Gib 1°40 +2:9 0 LT a "735 1°36 +26 | +03 Platinum’ ........: "753 1:33 +6'1 —32 PARTUM .2 2.005. 754. 1:326 +115 | —86 Lc ee 838 1:19 —8-4 +11:3 raninbly, foes conics "964 1:04 +733 —70°4 The numbers in the second and fourth columns are those given by Jager and Diesselhorst, the third column contains the reciprocals of those in the second, the fourth gives the thermo- electric powers of the metals with respect to one of the copper bars, the fifth has been calculated from the fourth and gives the thermo-electric powers with respect to lead. The signs have been reversed to comply with the convention which makes the thermo-electric power of a metal positive if the current flows from lead to the metal through the hot junction, i. e. if the current flows from hot to cold through the metal. If the laws stated by M. Thomas are true the numbers in the third and fifth columns of the table should increase and decrease together. It is seen that as the numbers in the third column decrease those in the fifth change irregularly in magnitude and are sometimes positive, sometimes negative. They aftord therefore no confirmation of the laws, but on the contrary disprove them. The following diagram, in which each metal is represented by a point, with abscissa the quotient of the electrical by the thermal conductivity and ordinate the thermo-electric power 512 Laws regarding Direction of Thermo-electric Currents. of the metal, shows perhaps more clearly than the table the want of agreement of the theory with fact. On the theory the current flows from hot to cold in that metal of a pair whose abscissa is the greater. According to the observations ABSCISS : QUOTIENTS OF CONDUCTIVITIES. ORDINATES: THERMO-ELECTRIC POWERS. the current flows from hot to cold in that metal of a pair whose ordinate is the greater. The two are only in agree- ment when the line on the diagram Joining the two points corresponding to any two metals slopes upward to the right. It is obvious that the number of exceptions to this rule, even for the metals dealt with, must be almost as large as the number of agreements, and that the laws enunciated by M. Thomas are not supported by the observed facts. ars LVI. The Variation of Disk Resistance with Temperature im Water. By Professor A. H. Gipson, D.Sc., University College, Dundee *. POSSIBLE law of the variation of disk resistance with temperature may be deduced from purely theoretical considerations, the only assumptions made being that resist- ance of each element of the rotating surface is proportional to the same power ‘7?’ of its velocity, and to some power of the viscosity mw, and of the density w of the fluid, both of which vary with temperature T. On these assumptions it may be shown that the resistance, other things being equal, is probably proportional to ?-”. w"—?: and to test the validity of the reasoning a series of expe- riments has recently been carried out by the author on disks 12 inches in diameter rotating in a closed casing, with different values of n, and with a range of temperatures from 60° F. to 160° F. The values of n were determined in each case from a preliminary series of experiments carried out at as nearly as possible uniform temperature, all these results being corrected to 65° F. by an application of the foregoing hypothetical formula. This involves the method of successive approximations for finding the true value of ‘n’; but as the temperature corrections were always small (usually much less than 2 per cent.), the first approximation was in general sufficiently accurate. In this way the following values were obtained. SERIES A, n 12-inch brass disk in smooth casing with +} inch side clearance ,, 1°785 SERIEs B. 12-inch brass disk in rough cast iron ,, 32 re ; .. 1800 Series C. 12-inch brass disk with radial vanes ,, 2 - - .. 1950 The results of the temperature variation experiments are given in the following tables. * Communicated by the Author. | By the theory of dimensions. The method is applied to pipe-flow by the author in a paper in the Phil. Mag. ser. 6, vol. xvii. p. 389 (1909). # is calculated from Poiseuille’s formula, ¢ being in ° Fahr. ue 3716X10-8 P~ 0-4712+0-01435 t+-0°0000682 # Phil. Mag. 8. 6. Vol. 19. No. 112. April 1910. 2 L lb. per sq. ft. 514 Prof. A. H. Gibson on the Variation of Series A.—1400 revs. per minute. Temperature ° F.| 65. | 72. | 74. | 80. | 95. | 100.} 109. | 113, 128. | 140. | ee Resisting moment | | 3.979 | 3.903 | 3:194 | 3-117 | 3-002 | 2:986 | 2°920 | 2-894 | 2-803 | 2-701 (foot lbs.) ...... Rte ES Be | |1:000 |0:977 |0-973 | 0-950 /0-915 | 0-910 | 0-890 | 0-883 | 0855 | 0825 Series B.—1175 revs. per minute. Temperature ° F....... 65. 78. | 98. 104. 115. 133. 159. ee | | OO Moment (foot lbs.)........ 2-480 | 2394 | 2°249 2°257 2°175 2°105 2:010 See noe 1:00 | 0965 | 0908 | 0910 | 0-877 | 0850 | O-810 SeRIES C.—-1050 revs. per minute. iemperiearers Fe, oo uo tae 60. 765. 100. | 132'5, Miemour (nat lbs) ote as 6-460 6377 6215 6-138 Ratio to moment at 65° F.......... 1:003 0:989 0:963 0°950 In addition to these a number of results of experiments by Dr. W. C. Unwin on a disk having n=1°'85 are available * These are as follows:— Temperature °F, ,..5....05s0.0.00 41-2. 53:0. 70°4. 130°5. ertetawen Pde. 01215 0°1149 01112 0-1003 For the sake of comparison the whole of the foregoing results have been plotted in the diagram against the curves representing the relationship 2—n Resistance at ¢t° F.= Resistance at 65° F. x { He \ {2 ae M65 65 for corresponding values of n. From these it appears that * Minutes of Proc. Inst. C. E. vol. liii. p. 221. Disk Resistance with Temperature in Water. 515 Temperature Degrees Fahr. J J r t Relative Resistances. 212 516 Dr. J. W. Nicholson on the Bending of the theoretical curves fit the experimental results remarkably closely, quite sufficiently closely indeed to justify the adoption of this formula. To render the series more complete the theoretical curves for n=1°9 and n=2:0 have also been added. The results show that the resistance diminishes with an increase in temperature, the amount of the variation with a given temperature-difference increasing as the temperature diminishes and also as diminishes. Its value in the neigh- bourhood of 65° F. with a polished brass disk (n=1°8) is about one-third of one per cent. per degree Fahr. When n=1°9 this falls to one-seventh of one per cent. per degree Fahr., and when n=2:0 it becomes inappreciably small. LVI. On the Bending of Electric Waves round a Large Sphere: I. By J. W. Nicwouson, M.A., D.Sc.* fea effect of an obstructing sphere upon incident waves has been examined by Lord Rayleigh, more especially when the waves are those of sound. When the character of the obstacle differs only in a small degree from that of the medium around it, a solution may be obtained, whatever its size and form. But when the difference of character is very marked, as for example in the case of electric waves, when the permeability and dielectric constant are arbitrary, the solutions of value relate only to the case of small spheres. In the general cases the entire motion outside the sphere may be regarded as consisting of two parts, (a) the undis- turbed motion which would exist in the absence of the sphere, and (b) a secondary disturbance zero at infinity radiating outward from the sphere. The motion may thus be readily expressed in the form of an infinite series of zonal harmonics, whose coefficients involve Bessel functions and their derivates in a complex manner. This general series cannot be summed, and it is comparatively of little utility. But the character of the secondary disturbance is deter- mined by the ratio of the radius of the obstacle to the wave- length of the incident vibration, and when this ratio is very small, the terms of the series decrease rapidly in order of magnitude, and any desired approximation to the solution may be effected. ‘This series therefore leads to the physical solution of the problem of the small sphere. A case of equal, if not greater, importance is that in which * Communicated by the Author. + ‘Theory of'Sound,’ 1896, § 334. Electric Waves round a Large Sphere. 517 this ratio is very large. This problem has been attacked by more than one investigator, but no quite satisfactory solution has been obtained. Lorenz*, in a discussion of the problem of the incidence of light on a transparent sphere, indicated a mode of obtaining a first approximation to the solution. But his method is not sufficient for the treatment of many important problems, including that of the bending of electric waves round the earth. This special problem has given rise to some discussion. Macdonaldt deduced, for a source situated on the surface of a perfectly conducting sphere, an effect at a great orientation from the source, of the same magnitude as though the sphere were absent. But Lord Rayleigh t pointed out that this result was not in accord with optical experience, for the ratio of diameter to wave-length is about the same in Marconi’s experiments as for visible light incident on an opaque sphere of an inch diameter. Even for a highly polished spherical obstacle, the light does not sensibly creep into the dark space in this case. Lord Rayleigh further suggested that the important terms of the harmonic series would probably have an order nearly equal to ke, where 2cr/k and ¢ are respectively the wave-length of the vibration, and the radius of the sphere. Taking account of this flaw, which was also independently pointed out by Poincaré §, and employing some results from Lorenz's paper, Macdonald subsequently || investigated the effect close to the surface of the conduetor, and found that the first order effect was zero at a finite angular distance from the oscillator, in harmony with experimental fact. But the problem. still remains whose object is to determine the actual amount of diffraction taking place. A recent investigation by M. Poincaré{ has been the subject of previous notes*”*. The corresponding acoustic problem has been investigated by Lord Rayleight+ when ke is as large as 10, but, as he suggests, although this case clearly indicates the formation of the sound shadow, it probably throws little light upon what happens when ic is really large. The object of this communication is to indicate a method capable of finding the solution of all problems of this class for which an harmonic series can be obtained, and to examine * Cuvres Scientifiques, i. p. 405. + Proc. Roy. Soc. 1903, Ixxii. t Proc. Roy. Soc. 1904, lxxiii. § Ibid. || Ibid. | Comptes Rendus, April 29, 1909, and other dates. Rendiconti del Cire. Mat. di Palermo, Marzo-Aprile 1910. ** Phil. Mag. Feb. and March 1910. +t Phil. Trans. 1903. 518 Dr. J. W. Nicholson on the Bending of further the special case of diffraction round the surface of a largesphere. The type of analysis is suitable for all problems of diffraction by large obstructing spheres or circular cylin- ders, with other forms of less importance, Many portions of the analysis are applicable without change to a large class of problems. The harmonic series for an oscillator in the presence of a sphere. Let C be a radial oscillator in the presence of a sphere of centre O, where OC=7,. OC may be taken as axis of z in a system of cylindrical polar coordinates, the distance of a point P from the axis being p, and its spherical polar co- ordinates (7, 6,6). Let CP=R. By considerations of symmetry, the magnetic force in the combined system is distributed in circles having OC for their common axis. Let be its value at P. The magnetic force y, due to the oscillator alone is it 3 e—ikR = Op R _ ahs ° . . (1) where the strength of the oscillator is unity, and the time factor is ignored. But y must be a solution of the equation a! else 2 , ag et ee =O" ae = pop Ze + Kk") yp (2) where 27r/« is the wave-length of the oscillation. Writing p= cos 0, p=rsind, |‘ z=reos 0. Then 2% at ope (<, + 2 ae +x? )yp=0. Thus yp=r? sin? 6 s { And (xr) mh Bud i 2m (rry)—? eK, (Ary) J,,(kr) Pa(u). ee tae all But py = (1-# ee ns), and after some reduction, if gn(7)) Ne np=rt sin? 8S 9,(r,)Jn(br) Ge, (5) a formula given by seid But there is an important point to be noticed at this stage, which formed the real subject of Poincaré’s criticism of Macdonald’s investigation. For the oscillator is afterwards placed on the sphere, and when the effect at a point on the sphere is sought, we are involved in the case r=r,._ It requires to be proved, therefore, that the formule for r>7; and rr,, is found to be —itkR 2 Sai (17,73 x 2mJ,,(kr1) K,,.(kr) P(e), * Loc. cit. 520 Dr. J. W. Nicholson on the Bending of and leads after some reduction to dbs dm’ (5a) and ig obviously continuous with (5) at r=7,, except perhaps in derivates. But by separating off the portion of yp which is infinite at the pole, a mode of determination can be found which leads necessarily to continuous series. The oscillator, having an external magnetic force yp=— 5 BP mrt fF (= PT ned lr) br Bdn (em) (1 =p) fy Ps) eo kK Te Op R behaves, in its own neighbourhood, like a simple doublet, since we may then write k=0, a time factor being present in the strength of the doublet. The expansion of this simplified function in an harmonic series is, on reduction, [oe ue mene, ee orto te Pa ie as nt eee GA so that in the actual case, with (5), Aw, eee ea emai nP=P >," R ae ae { ga(rs)i2J,, (kt) + dp » (6) where the series vanishes with &, and is finite everywhere, including the neighbourhood of the oscillator. The disturbance y,9 produced by the sphere must, near the oscillator, be that due to its electrostatic image in the sphere, which consists of (1) a doublet of strength —a?/r,? (that of the oscillator being.unity), and (2) a residual charge a/7,2, both situated at a distance a?/r, from the centre of the sphere. These are referred to the original doublet as a unit. The magnetic effect of the image, the strengths having a time factor, becomes on reduction, if R’ denote distance from the image, | AP he © ee ate TP ne Gpeeon ae R’ y) and its harmonic expression for external points r>a’/r, is on A eet? AP to) eee a — sin? 6 > 4 and the complete disturbance due to the obstacle is, being Electric Waves round a Large Sphere. 521 finite at infinity, of the form he ated a rh—T, ae ot L Yop? = re PA R’ _ re OR + sin 7 >, Dae K,,(kr) neh arty ab, n gry nr? dp - de (7) D, is determined by the surface condition that when r=a, Dare 00501 4 ete (8) and therefore with (5, 7), since the portions not in the series satisfy the condition at once, it follows that n+1.a"+1 Regn = —Jn(71) oO az) (ka) — 9 0a and this gives a series in y.0 which is finite everywhere. Finally, all conditions are satisfied by a total magnetic force given by me! 6a G1 D, 2 abKy(ka) = or Dn = —gn(r}) 2 83 (a) | je Cees yo 09) Op R— re? dp Rae OR : 0 / Imre ry '\adP mee sin? @> NT D ; aie k n (= te e™ Ka(kr) T(r) — ) - in? @ of(n+1l Task AIK ys ee + sin? 0S” (“= serra tDyrtKm(ir) JO", (9a) where r is equal to or greater than 71, m=n+4, and D, is given by (9). The expression for y,p appropriate to a diverging wave has been used. The series are convergent at all points. If the oscillator is on the surface, so that 7;=a, _ a Fs t707r ‘ dP, yp = 7 — sin? @ > { = Tm( ka) — D, j?Kn( kr) a o 2 n-1 | + sin’ @ >, at fz = 9 the summations together vanishing with k. But the second is at all points equal to the first term of yp, and finally, as a series appropriate to all external points, representing divergent n+1l.a” Oa gt? ° 922 Dr. J. W. Nicholson on the Bending of waves only, and convergent at all points on or outside the sphere, dP yp= — sin? dS" (“em In( at) —Dy )7?Kn (kr) Ta 29) Reduction of yp. Let the oscillator be now placed on the surface of the sphere, so that 7,;=a. If new functions be introduced, defined by Jem(2)=(+)" ea Ge + ae in terms of the Bessel functions of order m or n+4, then from the relation defining K,,,(z) it appears that 2K,,(2)= (=) ornate d a 2 — UnT—t £aK.@=(gh) GR), the accent denoting differentiation with respect to z. Thus 2K,,(2) } = 22K ,,(z)=tk'R,, cos Xp e'Xn, where tan ¥,= —iR,’. ° ie SU ee (13); abi with some reduction, £ ab In(e)= (=) 008 (ie See as In all further work, z will denote ka, and thus R,, dn are functions of argument ka. The corresponding functions of kr will be denoted by Ray, nr» In accordance with this, Ca (At) “9, after some reduction. Thus = nto $28 (TEs Ber) dP YP resin 9s = a \ 2k awkr/] du x [sin b,—¢ cos (fat xno Jee an and therefore t ied a 2° m(R,Ry)'(Eh For 4 drt) BP (46) (aa Electric Waves round a Large Sphere. 523 In particular, on the surface of the obstacle, where r=a, isin? 6 ax,) Pn - = = mR(1+e laa relay The values of Ru, bas Xn It has been shown by Lorenz* that, if z—m is of higher order than 2 and positive, R,(z) =2/(2?—m*)# \ 18) bn(=) =(2?— m?): +m sin-!m/z— nT, and that if z—m is of lower order than 2%, and positive or negative, 3-3 24\3 R,(z)=m33 te HY m(—§) +m—z.0(—¥/ =) | ‘ Se »( 24) +... : rs (19) . ar ; R,’ (z-—my gu(Z)= & +Rale—m)— 2R,2 2 ants j where, in the latter formula, Rn», R,’ correspond to z=m. m(s) denotes Gauss’ 7 function, and is identical with [(s+1). An extension of the first result, required for diffraction problems in which a_ second approximation is desired, has been given by the writer f. Tf n+$=m=Zzsina, . «| 's «.'-,,(20) thus defining a subsidiary angle a, As Ra(e)= see at oo, gee? a+ o,f Iya jsec’ at . where Ag= —4F, As =h (27 — 24m”), A;= 51, (1160m? —1125—40m’), and st3.dAczat (+2) rAszet2m?s.st1.s+2.r,+m's.s?—4.A,2=0. (21) i al icing * Loe. cit. + Phil. Mag. Dec. 1907. 524 Dr. J. W. Nicholson on the Bending of Moreover, T gn(z) =i + (cosa—F —a. sina) — + (m-* me + a — #tana+ 2 \,(tan a—$ tan® a), .. 0, 0S where there exists the identity Lp pt tpt? + 0... SCL 4+ Age +527 + ...)7% (23) The next approximation to that of Lorenz becomes, on selection of the proper terms, R,=2/(2?—m*)3 b,=77 + 2(cos a— 5 —asin a) + as ~ (sec a+ ? tanta sec x). (24) Lorenz has also shown that if m_is greater than z, and m—z of higher order than 2°, Fae) =(+)"( aoe tS where 2Tn=2/(m?—2”)2, eh 24 =— log 2+(m?—2’)2 +m log ~ Lae 2 } Cae The range of variation of y, may now be investigated. It was defined by OR, Oz Now when m and z are not too nearly equal, to the second order tan Yn +3 = Uh oR, "02 where m= sin «. Thus to the same order, X,= sin? a/22 cosea, . 4 5 eS = —m?/(z?—m’)?= — sin? «/z cos? a, and is very small within the range of application of this formula. ee Electric Waves round a Large Sphere. 525 When m and z are nearly equal, of = —7(—4)242/B8at + ... =—2//73 when m=z. Thus at the critical point m=z, y,=17. This value exhibits no discontinuity on passing the point. When m is much greater than z, 2T, =2/(m?—z’)?, 2T,! =m?/(m?—z")2, and by a comparison of substitutions for the Bessel functions, Na 2l,, cosh 2¢,, OR, Oz since by an identical relation, | Oana: = 2T;,! cosh 2t,+ 2 sinh 2t,, But ¢, is large and negative, and O0R,/dz is therefore very great for a great value of n. Accordingly, y, tends to the value 3a when n is infinite. Thus as n increases from zero to infinity, x, is never of the order of ¢, in z. It ranges from zero to dz, taking the value 47 when n+4=z2, Within the range of application of the first type of expansion of the Bessel functions, it is very small and of order z—1,) Harmonic terms of infinite order. When n is very great, a substitution n+3=m=z cosh 8 may be made, where z denotes ka as before. Thus from the appropriate expansions of the Bessel functions, 2T,»= cosech B tra= —4 log 2+2(sinh B—fcosh B), f ° (29) a quantity ultimately very large and negative. Comparing the two types of expansion of the Bessel functions, tan dra= erin, 526 Dr. J. W. Nicholson on the Bending of Tf n+i=kr cosh Ai=% cosh Bi; so that cosh B= ~ cosh 8, and #, is never greater than £, 2t tan drr=e"™, where tne= —4} log 2+2,(sinh 8; —/, cosh 8)). Then tan ( hn om; dnr) a ezin erinr since tn is also very large and negative. For a large value of n this is evanescent, and dn— Prr= 0. The typical term of yp therefore depends on But 14e7%=2/ (1—v tan yn] =2/ (145%), and R, = 2T, cosh 2t, = ius e—2tn, under the present circumstances, ORn __ 02 —1) etn _, cosh? 8 a By eS —1)e 2tn ad The typical term therefore contains zcosh B(TrT»,)? e—”—™ dP, L+he—%m dy.” which is not greater than < cosh B cosech 8 cosech as? ; to the same order. (30) Electric Waves round a Large Sphere. 527 Employing the asymptotic expansion for the zonal harmonic, a2 is found to be of order m3 cos (mo— ip and therefore the typical term contains cosech 8, cosh?@, . . . . (31) or e?9-%, which is ultimately evanescent, however great r may be. This compensates any power of z present, and the sum of the harmonics of order n much greater than z cannot contribute to the value of the magnetic force. Formule of summation. It has been shown by the writer* that the series n=nN92 n n S = > (2) eXp. ZV (), AP et 6 tae) where 2 is great, and where differentiation with respect to n/z=« does not increase the order of u or v in z, is equivalent to the integral “€, «zu V V. 2("e (vo+ HF t n)de, 6. (83) oe eo & where n/z in the functions has been replaced by a continuous variable « and where M, N+1l=<2(e, €2), \ uv wv | Vo=uby = *, ar a, ee. | 2 sin eS 2 e Vv; = uly + teu Wo, | Vo= dul py t dela! + du) bs — fu! by, / ds @ Y= de® S| es | : 3 . . ° . (35) api soul ei, the accents denoting differentiations with respect to w. If v’ has no zero or very small value at any point in the range an asymptotic value for the sum may be obtained by a series of integrations by parts. In this case, the result becomes, to the second significant order (Vg eee va BV yy yt == — ei” 4 2 af 4 —t — 8 S es af z\v' v? vy? * Mess. of Math. Oct. 1907. where so that (36) | 528 Dr. J. W. Nicholson on the Bending of Vanishing of the derivate of an exponent. It has been shown that esin? @ 2» PE eras 5 emir en . = a m(RaRnr )?(e $n—Pnr +e n—Pnr +2%q) a. This may be reduced to a dependence on series of the above type for all harmonic terms of high order, by the use of the asymptotic expansion for the zonal harmonic, valid when 6 is not small. But this process is not justifiable until it is proved that the harmonic terms of low order do not contri- bute substantially to the magnetic force. Some other expression must therefore be used, and the Mehler Dirichlet integral formula is at once suggested. Writing hide ] cos mo dd P= 24 V2 (cos 6 —cos 0)’ - ae and defining the action of an operation g (@) upon any function w by 2esin?@ d (°? aodp No = end a oe F: H@) ka*r du), »/2 /(cos 6—cos 6) ee Then yp = g(0)S mi TE, Fn, )2( e'Pn— nr 4. oton— Par + an) cos m@, il and yp = (OE u(er+e4 er +e), 1 where u = 4m(R,Rny)2 " y= Pr— nar + mh | Vg = gn— Pnr — bn — mp r ® : Fs maetine ; : 5 (39) v3 = hn —= Dur = 2Xn + mo | a= Dn — Dar + 2Xn— Mh 3) ; It is now necessary to determine all possible cases in which the exponents of these series can have zero derivates’ with respect to w (the case v'=0 of the summation formule). Let n+4=m=czsina as before. Then when z—m is of higher order than z:, « being between zero and 47, bn = 2cosa—tn7r+ ma, 0¢,/d02= a—-t, taking the main terms of ¢, only. Electric Waves round a Large Sphere. 529 Thus 0¢,/07 is negative. As n increases, it tends to the order m-3. If z—m is of lower order than 23, and is therefore again negative, and of order m-? with 1/R,, on both sides of the critical point m=z. When 2 is greater, writing m=z cosh £, 0t,/On=—B by (29). But tan b,=e"", and therefore 0¢,/dn is again negative. Moreover, it decreases rapidjy as 7 increases, on account of tn. Finally, 0¢,/dn is always negative, as must also be OPn/On. Since, moreover, they are of order not greater than m—% except in the first region of expansion of the Bessel functions, it appears that the four functions v cannot have zero derivates except in this region. Thus in finding such derivates, we may write, if z=n+4/z Pn = -{ (1-2) asin x— Fo} af ; _ (40) Pp = £4 (eta hte sin~! ca— Fa} + — 2 4 where ¢ denotes a/v, and the four derivates become, rejecting Xn Which is very small, i vy = 2(sin7' «—sin—! ca+ 9), = 2(sin“'w—sin—! ca—), v3) = 2(sin7'w—sin-! ca + 9), vy = 2(sin~* w«—sin-! ca—¢). Ss wo | Since ¢ is not greater than unity, sin-! ca cannot exceed sin—' w for a positive value of « (corresponding to an harmonic of positive order), and therefore v,! and vs! can never be zero, er ! p 7 iahq 7 vy and v,' have the same vanishing point, given by sin~'w = sin-!ea+q, or x= cxecosd+sin d V1—cx?, leading finally to w=sind/(l—2ccosd+e). 1. . (41) Phil. Mag. 8. 6. Vol. 19. No. 112. April 1910. 2M 530 Dr. J. W. Nicholson on the Bending of This value of 2 tends to unity as sin? d@ and 1—2ccos +e? tend to equality, so that c=cosq. The value of ¢, used above then ceases to be valid. Now it is clear from the form of the integral in the operation g(@) that the important part of the range of inte- gration is near the upper limit ¢=0@. The important harmonics are therefore clustered about the value of n given by a2 = sin 0/(1—2ecos0+c?)?, . . . (42) This is unity when ¢ = 60s 6, Of VarCcosio — a, The plane r cos 6 = a is therefore inside a region of space in which the important harmonics have an order nearly equal to z. It has already appeared that v' cannot be zero for any of the four component series of yp within such a region. The real effect of this value of w is to make the equal functions v,’ and v,' very small, although not zero. It represents the point at which they attain their minimum values, and will be considered later. This region will henceforth be called the “region of transition,” between, as will afterwards appear, brightness and very complete shadow. When @ is not very small, and « is defined by (20), the value of n corresponding is of the same order of magnitude as z. In this case, the important harmonics are of very high order, and the zonal harmonic may be replaced by its asymptotic expansion. Writing therefore Poe (——,) éosi(md—ia), - sen so that the leading term of dP,,/dw is Ami? « 1 ( ane) sim (m@— 7), it is found that yp =X, u(eon—Frr 4 ga Suet Mn) sin (mO—Iar) where um — sin a y= == ka? 1 Again rejecting two series which cannot have a vanishing Electric Waves round a Large Sphere. 531 derivate of an exponent, j ieee Sar an ae ame) ae eee = a een eter) (44) where in terms of c=m/z, u = ia@(22e sin ORaR,,)?/a72, athagnss X45) and where the exponents have zero derivate when xe = sin 6/(1—2cecos0+0?)?, c=a/lr. But when @ is obtuse, sin~!cxv and sin-!z cannot have a difference equal to 6. In fact at v=1, the solution ceases, so that on the left of the plane rcos@=c, there is no point leading to a vanishing derivate of an exponent. The value of yp would then, except in the region of transition as previously indicated, be deduced from the integrated sum- mation formula. It would consist of limit terms, of which those at the upper limit would vanish in accordance with previous theory. Since therefore the sum is to be deduced from integrated terms at a lower limit, it is obvious that the use of the asymptotic expansion of the zonal harmonic may not be legitimate. The expression for yp given in (39) must therefore be evaluated instead for points beyond the left of the region of transition. But as written, the calculation would present difficulty, as @ may in its range take such a value as to cause a zero derivate of the exponent. This may be avoided by the use of the alternative expansion rit eed sin md dp } deus poe = s/2/(cos8—cosd) * * 6) If dar—d, cannot be so great as +0, a fortiori it cannot attain the value ¢, which ranges from @ to 7. Defining in this case an operation G(@) by _ Qisin?@ d (7* wil = kaa = d0\g /2/ (cos 0—cos d)’ * (47) then nk yp = G(A)=, m(RnRnp)(e'?a— Par 4 etn Pnr +2hn) sin mod = G(A)S, u(elr + e's — els —el), where = 5. (R,R.,, 3, | vy = DPn— Por Ti md, % = $a—Pnar+mh+t 2x, Ge 2) V3 = da— ar —Md, | U4 = bn—Par—Mh + 2H, J bo & bo 032 Dr. J. W. Nicholson on the Bending of This form must be used on the left of the transitional region. Moreover, the derivates of the functions v are never zero. Magnetic force in the region of brightness. Proceeding first to the consideration of this special portion of the problem, we note that the value of « defining the zero point is given by e=rsin0/R or in the figure, g@ = sin OCP =‘sin @j, 9s) 20 See \ thus giving a simple law followed by the important harmonics corresponding to any point P in the region. For the present, only a first approximation to the solution | is desired. More accurate values of the zonal harmonic and of @, would be needed if the calculation were to proceed further. The second approximation will be given later. By a slight change of notation in (44), for this region, | ype ewe a | where | wt wu = —Le(1 +2) (Qe sin ORnRny/ aa)? 0 = —dbyth,—mé. i . (50) In the neighbourhood of the zero point, it has already appeared that X, is negligible in comparison with unity, so that e2n»=1, and we may write more briefly, y = —a(22n sin OR, Ri /o*mr). | ee By (83), rejecting terms of order 2~' in comparison with those retained, if accents denote differentiations with respect. Electric Waves round a Large Sphere. 533» to w, the sum of the series (50) is ue = ZU yp = e4 A) Ure” dz, é where wer a@=(n+4)/z, and Up =(uv'/2sindu’)e 2, . (52) e being 3/2z, or zero to the order contemplated. This integral contains one, and only one point, in the range at which v'=0, and it is well known* that in such a case, the important part of the integral is, if v9” is positive as in the present problem, (2ar/2v9'')3 Uo, 0 et2%o— der A Ge Le (53) the zero suffix referring to the zero point. The ultimate value may be independently shown to be of a higher order of magnitude in z than the greatest term of the harmonic series for yp. Substituting therefore for the integral, yp = —u(2tr2/v9’)8U,, oe'%. Now since vw’ =0, Uo, = % by (52). Thus writing x = sin 8, in (50, 51), Uo,o = —sin 9,(2¢ sin 6 tan 0, sec(O,—0)/aar)} , Vy = — Par t+ drn—md = —zR/a = —kR, after some reduction, remembering that sin—! (csin 0,) = 0,—@ and using the values of Rn, dn, dar given in (18, 40). Now so that v' = sin~!a—sin— ca—8@, vy! = (1—a?)-§-—e(1— ea) -3 V/ = sec 6,—c sec (0,—0) = = sec 0, sec (0,—8), and after final reduction 2 yp = —2uk - sin? @e—*R, or the magnetic force is given by Y Bas — 20k Fasin fear. ws (54) _ * Cf. a paper by Lord Kelvin, Rey, Soc, Proc. xlii. p, 80, and other investigators. i 534 Dr. J. W. Nicholson on the Bending of For the oscillator alone, : 0 e—kR == —whs3 sin ll SR taking the important term. The intensity at any point, and consequently the mean energy per unit volume, is therefore quadrupled in this region, following the inverse square law from the centre of the sphere. This result was to be expected, for in calculating the first approximation, the sphere may be regarded as an infinite plane. The effect of a plane (perfectly opaque) in quad- rupling the intensity is well known. But although it is only a first approximation, the formula is very accurate even for a large orientation from the axis of the oscillator. For example, in the numerical case typical of Marconi’s experiments, the error is of relative order 10—'? for all points possessing this type of solution. In such a case, the necessity for a second approximation disappears. The result is very useful also as a justification of the mode of summation of the harmonic series, and as a very compre- hensive check on the accuracy of the work. In a later paper, the mode of approximation to higher orders will be given. Points near the axis. For points in the neighbourhood of the axis through the oscillator and its antipodes, @ is small, and it has already been shown that the formula (39) must be employed. Points very close to the oscillator are again excluded. Thus if _ Msn? Oo ad (° odd Bes kaa cl /2/(cosd—cosO)> * oe then : yp = g(A)>, ue? 4 eles a. ei2%s 4 pizrs) u=dn(BR,) Os AV, V2) = grn— Prrtkmp 203, U4) = dn— Par + 2K%n tM, in which y, may be neglected as before, in the neighbourhood of a vanishing derivate of an exponent. The only series which can have such a vanishing derivate where Electric Waves round a Large Sphere. 9395 are the second and last. Neglecting therefore the first and third, as having sums of a lower order of magnitude, oi g(O)E, Quel?" Mieiaaes ¢ (9S) since v, = v, when yx, is ignored. Thus if v denote vz or vy, so that zv = dr—dyr— md, then at the zero point, after some reductions, w# = sin d/(1—2c cos d + ¢”)?, U = —c(1—2ccos d+ c?)?, = (1—2c cos d + c*)2/(cos 6—c)(1—c cos dh), and is positive, for cos ¢ is never less than cos 0, and there- fore a fortior: not less than ec. R,R,, = (1—2¢ cos 6 + ¢*)/(cos 6—c)(1—c cos $), whence Up = $2 sin d/(cos 6 —c)2?(1—c cos p)?, and by use of the summation formule, v,! being positive, yp = g(0)4 (az) dug(2vg!")-2e'%o-4", and on reduction 1 2 sind = (% 23\5 = ae x rae 1 {iced "216i doeas ee Cxp. “(47 +kr(l—2ccosp+c?)?}, (59) or yp = —4usin 0(2k/am)s Sa, (60) provided that a Psin pd exp.—t{4ia—kr(1—2¢ cos p+ ¢*)t} (61) Jo «= 2(cosp—cos 6)2(1—2ecosp+e?)t * The important elements of this integral are those near the upper limit. Writing therefore bd = 0—u, and retaining no powers of « higher than the first, cosh = cos A+ usin 8, 2 1—2ccosd+c’ = a —2cusin 8, and th (7) Pin Odu exp. —u( dart kR—hay, w sin ) ee ee ee 0 to the most significant order, by the usual theory of integrals of this type. 536 Dr. J. W. Nicholson on the Bending of If u=o’, 2r° sin O p 1 (7-02 kar . | ! = (=) i} d@ exp.t ( —jar—kR+ koe Gu), (62) and I is expressed as a linear combination of Fresnel’s \ integrals. This result holds for all points within the region 4 of brightness, for @ has not been assumed small. d1/d@ vanishes with 6, so that the axis of the oscillator is a line of no magnetic force. This could have been foreseen by considerations of symmetry, for the oscillator alone can produce no magnetic force at points in its axis. { Except when @ is practically zero (in which case the | magnetic force becomes evanescent), the upper limit of the integral may be treated as infinite, on account of the rapid oscillation of the integrand. : Now by well known results, ice) { cos Aw*dw = (71/8A)3 = +f sinrw’ dw, 0 0 so that ia kar . ah \e Loe Bi 95 OP RN Piece ‘| deo exp.i( dor —kR R sin Oo )= ie : ; Thus SO ange ON Se a : | I =_ (sens) é AR, e e 5 e (63) i and since BR agen cei the leading term of 01/08 is ee (wkar*/2R4) sin Oe—*8, leading to yp =—2ikgs sin? Ge—"B, 5 |. (64) as for the points more remote from the axis. There is thu no change of type in the solution near the axis. Extent of the region of transition. An estimate of the extent of the region hitherto called transitional may be obtained from the consideration that it is the region in which the expressions, previously used for the Bessel functions in the calculation of the significant harmonics, cease to be valid. Electric Waves round a Large Sphere. 537 Thus in the transition, the value of n given by n+4/2e= « =sin O/(1—2ce cosO+¢?)2 must make z—n—}4 of higher order than 23. In the case of electric waves and the earth, or of visible light and a sphere of an inch diameter, z is about 10°, and one order higher than 23 is sufficiently represented by TO* “or ’22. Now a e—n—-$ = &, 1—sin 6/(1—2c¢ cos 0+ ¢”)? = 2-3, leading to es cos Gee ein 2.) oc OB) This defines the exterior of a double cone whose axisis along the oscillator, and whose semi-vertical angle is the complement of tan-'2z2-+. For the numerical case in question, this angle is about two degrees. Deep Brigh C785. ~ Transition Thus the transitional region does not extend more than about two degrees on either side of its axial plane through the oscillator. It will be shown in a subsequent communication that this region consists of a series of true diffraction bands determined by the maxima and minima of an Airy’s integral, and that behind it, there is a very dense shadow. Trinity College, Cambridge. i Wegind LVIII. Electrical Recording Thermometers for Clinical Work. By H. L. Cattenpar, V.A., LO.D., FRS., Professor of Physics at the Imperial College of Science and Technology, rl a In Memoriam Dr. A. GAMGEE, F.R.S., obtet Mar. 29, 1909. L ae scientific interest and importance of a method of continuously recording the temperature of the body, whether in health or disease, is now being generally recognized, and I have been persuaded that my experiments in the construction and testing of suitable apparatus for the purpose may be of some value at the present juncture, although I have no results of physiological importance to produce. | My attention was first directed to the subject by Professor J. G. Adami, F.R.S., of McGill College, Montreal, at whose suggestion I constructed some electrical resistance thermo- meters specially adapted for recording the temperature of different parts of the suface of the body. Unfortunately I had to leave Montreal early in 1898, before the apparatus. had taken its final shape, and the results obtained at that time, consisting merely of experimental records of my own normal temperature, did not seem to be of sufficient interest. to merit publication. I did not return to the subject until November 1908, when I undertook to make some thermometers. for Dr. A. Gamgee, F.R.S., and to collaborate with him in evolving a practical method suitable for general use in clinical work. His paper, then recently published, “ On Methods for the Continuous (Photographic) and Quasi-Continuous- Registration of the Diurnal Curve of the Temperature of the Animal Body” +, dealt exclusively with thermoelectric methods of temperature measurement: but he had become convinced that the thermoelectric method was unsuitable for general use, on account of the delicacy of the apparatus, and the necessity for employing an elaborate thermostat. The Copper-Constantan thermocouples which he employed gave an H.M.F. of 40 microvolts per 1° C., which was ample for delicate photographic methods of registration, but insufficient: for the more robust type of instrument required for general use with ink records. A more serious difficulty in practice was the necessity of keeping one of the junctions of the. couple at a constant temperature in the neighbourhood of * Communicated by the Physical Society: Presidential Address, February 11, 1910. + Phil. Trans. Roy. Soc. B. vol. 200, p. 219 (1908). Electrical Recording Thermometers for Clinical Work. 539 37° C. for long periods. In the application of the resistance method no thermostat was required, and it was easy to obtain a tenfold larger deflexion with the same galvanometer. The objections that have commonly been urged against electrical resistance thermometers for this kind of work are (1) that resistance thermometers are more difficult to con- struct and to insulate satisfactorily than thermocouples, and (2) that their indications are liable to be disturbed by the heating effect of the current employed. These objections undoubtedly exist, and have frequently proved fatal to the employment of resistance thermometers: but they have arisen chiefly from faulty application or construction, and not from defects inherent in the method. Objection (1) is readily surmounted by proper methods of construction, and objection (2) by a proper consideration of the conditions of sensitiveness. Conditions of Sensitiveness. 2. The conditions of sensitiveness in measuring a resistance by the Wheatstone bridge method have been discussed by Maxwell (‘ Electricity and Magnetism,’ vol. i. p. 437) and Heaviside (Phil. Mag. Feb. 1873), whose results have generally been quoted and applied to the problem under consideration. They start with the assumption that the battery power available is limited by the internal resistance of the cells, and give rules for obtaining the maximum current through the galvanometer with this implied limit- ation. The limitation of battery power was no doubt an important consideration in many kinds of telegraph testing thirty years ago, but it is rarely applicable in modern laboratory practice, and never in dealing with electrical resistance thermometers. In the majority of resistance mea- surements, and more particularly in electrical thermometry, the limiting condition is imposed by the heating effect of the current on the resistance to be measured, and the resistances, etc., should be chosen and arranged to give the greatest sensitiveness for a given limiting value of the current through the resistance to be measured. The resistance of the battery circuit is quite immaterial provided that the battery can be arranged to give the required limiting current. The problem has been discussed from this point of view by Guye (Arch. Sci. Phys. Nat. Geneva, 1892), and by Schuster (Phil. Mag. xxxix. p. 175, 1893), and in several of my own papers, but it will facilitate discussion to reproduce here the investigation itself in a simplified form. 540 Prof. H. L. Callendar on Electrical Let the annexed diagram, fig. 1, represent the arrangement of resistances in a Wheatstone bridge, in which C is the Fig. Diagram of Wheatstone Bridge. current through the resistance to be measured R, and ¢ is the current through the galvanometer of resistance G, when the bridge is not balanced. The resistance in series with R is nR’ traversed by a current C +c; the resistance in parallel with R on the same side of the galvanometer circuit is mR/ traversed by a current C!+c. The resistance in the opposite arm of the bridge to R is nmR’, traversed by a current CO’. The currents are assumed to flow in the directions indicated by the arrows, and it is evident that they satisfy the condition of continuity. The numbers n and m, representing the ratios of the arms of the bridge, may have any positive values. The bridge is balanced when R=R’, in which case c=0, and CU im: Since the difference of potential between the ends of the galvanometer circuit is Ge, we have by Ohm’s law, RC—mR'(C'+¢)=Ge=nmR’/C'—nR’'(C +c). BHliminating C', we obtain for the ratio ¢/(, c/C=(R—R)/(G(1+n)/n++m)R’). © . (A) It is obvious that this ratio, which may be regarded as a measure of the sensitiveness of the arrangement, is quite independent of the resistance or H.M.F’. in the battery circuit. It is also at once evident that, for a given defect of balance as measured by R—R’, the value of the ratio c/C will be a maximum when n is as large as possible, and m as small as ey the limiting value of the ratio ¢/C in this case being, when n=o, and m=0, c/C=(R—R’)/(G+R’). Recording Thermometers for Clinical Work. DAL This point has been emphasized by both Guye and Schuster, i and comes out clearly in Heaviside’s investigation. The advantage which can be gained by making x large and m small is not very great, because when n=m=1, ¢/C=(R—R/)/2(G +R), or the sensitiveness is still half as great as in the limiting case. For practical purposes it is more important to observe that ! if n is small or m large, the sensitiveness may be indefinitely reduced. This arrangement should always be avoided if possible, as shown by the following example. Maxwell’s Rule. 3. Maxwell has given a rule which is very often quoted and applied (‘Hlectricity and Magnetism,’ vol. i. section 348). ‘Of the two resistances, that of the battery and that of the galvanometer, connect the greater resistance so as to join the two greatest to the two least of the four other resistances.” This rule is seldom applicable to accurate resistance measurements, and when applied in platinum thermometry has often led to disastrous results. A simple numerical example will make this clearer. Suppose that it is required to measure the resistance of a platinum thermometer, R=10 ohms, with a post-office box having ratio arms 10 and 1000 ohms. Suppose that the galvanometer has a resistance G=10 ohms, and that the battery is a 2 volt cell of negligible resistance. According to Maxwell’s rule we should connect the galvanometer so as to join the two resistances of 1000 ohms to the two resistances of 10 ohms, in which ease we should have n=1, and m=100 in fig. 1. Suppose that the balance is slightly disturbed by a change dR=R—R’. Substituting in equation (1) we obtain : Case (A), Maxwell’s Rule, n=1, m=100, e/C=dR/1030, C=0:1 amp. If, however, we interchange the connexions of the battery and galvanometer, which is simply equivalent to inter- changing the values of n and m in the equation, we obtain for the Anti-Maxweil arrangement: Case (B), Anti-Maxwell, n=100, m=1, e/C=dR/30, C=:002 amp. | 542 Prof. H. L. Callendar on Electrical The actual value of the current through the galvanometer in case (A) is greater than in case (B) in the proportion of 3 to 2. Maxwell’s rule is so far justified. But the current through the resistance to be measured is 50 times greater in (A) than in (B), and the heating effect of the current is 2500 times greater. With an ordinary platinum thermo- meter arrangement (A) might give an elevation of tem- perature of about 5° C., which would be fatal for any purpose of measurement. Whereas arrangement (B), with nearly the same sensitiveness, would give an elevation of *002° C. only, which is sufficiently small for the most accurate work. The essential point is to observe that.the ratio c/C of the galvanometer current to that through the thermometer, which is the true measure of sensitiveness in this case, is 34 times greater in (B) than in (A). (B) is therefore by far the better arrangement in the case where the heating effect of the current on the resistance to be measured is the primary consideration. The rule to replace Maxwell’s in this case, if there is any choice as to the arrangement of the battery and galvanometer connexions, is: “‘ Connect the battery so as to make the resistance in series with the thermometer greater than the resistance in parallel.””? This will make n greater than m in fig. 1, and it is evident that the ratio c/C will be diminished or the sensitiveness reduced, if n and m are interchanged. In the practical use of platinum thermometers, we have the further restriction that n must be equal to unity, because it is necessary to compensate the changes of resistance of the leads by the equal changes of resistance of a pair of compensating leads on the opposite side of the galvanometer contact on the bridge-wire. The loss of sensitiveness, as compared with the case of n very large, does not amount to more than 20 or 30 per cent., and is of no consequence as compared with the trouble and uncertainty involved in measuring the resistance of the leads separately at each observation. Moreover, if n is made large, the heating effect of the current C on the resistance nR’ may become serious. As a matter of fact this is a common and very insidious source of error in the use of a P.O. box with n=100, especially if the coils are of german silver or platinoid. With maganin resistances the effect is much less marked. It is still possible to gain some advantage in point of sensitiveness by making m smaller than unity, provided that it is not made so small that the heating effect of the current C/m on the resistance mR’ becomes appreciable. For the most accurate work I have generally employed thermometers Recording Thermometers for Clinical Work. 543 having a resistance about 25°6 ohms at 0° C., with a funda- mental interval of 10 ohms, and have made the ratio coils about 6°4 ohms, giving m=1/4*, which gives an advantage of about 20 per cent. in point of sensitiveness as compared with m=1. A more important consideration in this case is that the same box can be used with thermometers of lower resistance, e.g. Ry=2°56 ohms, F.J.=1 ohm, without serious loss of sensitiveness. Resistance of Galvanometer. 4. It is generally possible to vary the resistance of a galvanometer through a wide range, by altering the con- nexions of the coils or by substituting one coil for another, without altering the mass of the coils. In this case the sensitiveness will vary approximately as the square root of the resistance G. We see from equation (1) that the de- flexion of the galvanometer, which is proportional to c,/G, will be a maximum when G=n(1+m)R'/(1+7). This result agrees with that given by Maxwell, and shows that in the majority of cases which occur in practice, G should be of the same order of magnitude as R, being restricted to the limits 2R and R/2, if n is never less than 1, or m greater than 1. If we are restricted, as in platinum thermometry, to the case n=1, we should take G=(1+m)R’/2. The best resistance for the galvanometer is restricted to the same limits, 2R and R/2, provided that m does not exceed 3. For example, in the case of the box already quoted, with ratio coils 6°4 ohms, when used with a thermometer Ry=25°6 ohms, the best resistance for the galvanometer was 16 ohms. When the same box was used with a thermometer ten times * The reason for choosing this particular value of m in the case of the compensated box with platinum-silver and platinum coils, which was exhibited at the Royal Society in May, 1893, and fully described later (Phil. Trans. R. 8S. A., vol. 199, p. 92), was that the change of resistance of the platinum-silver coils due to the heating effect of the current might be of the same order of magnitude as that of the platinum coils, and might take effect at the same rate. The thermal capacities and radiating surfaces of the coils being nearly equal, but the temper- ature coefficient of the platinum-silver nearly 16 times less than that of the platinum, the platinum-silver coils would carry about 4 times as great a current as the platinum for the same change of resistance. This was found to give a fairly safe limit for the value of m, but the effect was so small as to be of little importance. D44 Prof. H. L. Callendar on Electrical smaller, Rp =2°56 ohms, the best resistance for the galyano- meter would be 4°5 ohms. This could be secured approxi- mately by putting the two coils of the 16 ohm galvanometer in parallel, but the advantage gained thereby would be only 10 per cent. If, on the other hand, the ratio coils had been made equal to 25°6 ohms each, according to the rule given by Maxwell and generally followed, there would have been a loss of sensitiveness of 20 per cent. with the 25°6 ohm thermometers, and 50 per cent. with the 2°56 ohm thermo- meters, which could not so well be neglected. Resistance of the Thermometer. 5. For a platinum thermometer of resistance R, at 0° C., the change of resistance per 1° C. is approximately ‘004 Ro, which must be substituted in equation (1) for the value of dR or R—R’, in order to find the deflexion of the galvano- meter per degree change of temperature. Making this substitution, and remembering that n=1, we find that, when the resistance of the thermometer is changed, the deflexion per degree, which is proportional to c\/G, varies as ROV/G/(2G-+ (14+ m)R). | The rise of temperature produced in the thermometer by the measuring current C is directly proportional to C?R, and inversely proportional to the rate of dissipation of heat per degree rise of temperature of the thermometer above its surroundings. ‘The rate of dissipation of heat, depends on the form and surface of the thermometer, and on the con- ditions of exposure. For similar thermometers of different resistances, but of the same size, under similar conditions of exposure, we must have C?R the same in order to secure the same degree of accuracy as limited by the heating effect of the current. The permissible current C will therefore vary inversely as the square root of the resistance of the thermo- meter. Making this substitution, we have o/G varies as \/GR/(2G+(1+m)R). We see from this result that, if it is possible to choose the best resistance for the galvanometer, namely (1+m)R/2, and if m is the same for ali, the sensitiveness and accuracy of the thermometer, so far as the heating effect of the current is concerned, will be independent of its resistance. In accurate laboratory work, where a delicate galvanometer is available, the heating effect of the current is so small as to be relatively unimportant, and the resistance of the thermometer is chosen chiefly with a view to minimise errors -_ Recording Thermometers for Clinical Work. 545 due to defects of insulation or imperfect contacts, or small differences in the resistance of the thermometer and com- pensator leads. On the other hand, for practical applications, or for ink recording, it is necessary to employ a fairly robust and portable galvanometer of the suspended or pivotted coil type ; the current employed must be considerably increased if a large scale record is required, and it may be important. to choose the resistance of the thermometer to suit the galvanometer or vice versa, in order to minimise the heating effect of the current. The symmetry: of the above expression in respect of G and R shows that the condition governing the choice of the resistance of the thermometer, if the galvanometer is given and m is constant, is the same as that already given for the galvanometer in terms of R, namely, If, however, the value of mR=S is constant, being fixed by the resistance of the ratio coils, each equal to S, of the bridge employed, the maximum sensitiveness is obtained. when R=2G+S. The latter condition gives the limit beyond which no ad-- vantage can be gained by increasing R and diminishing m, even if the heating effect of the current on S can be entirely neglected. If the heating effect of the current in the thermometer is kept constant, as already assumed, the heating effect in S increases in proportion to R/S. On the same assumption the total current to be supplied by the battery (which is a consideration in records of long duration) is a minimum when R=S§, but is only increased in the ratio 0/4 when R=48. The maximum sensitiveness is obtained when S=0 and R= 2G, but the sensitiveness will be reduced by little more than 10 per cent. if we make S=G/2, keeping R=2G. It would be reduced a further 20 per cent. if we made R=G=S, according to Maxwell’s investigation *, In employing the deflexion method described below, the advantage of making R larger and S smaller than G is really greater than would appear at first sight, because, if the sensitiveness is reduced 20 per cent. for the same heating effect of the current, it means in practice that the current must be increased 20 per cent. to give the required deflexion, * Maxwell, ‘Electricity and Magnetism,’ vol. i. § 349. This is the right arrangement if all the four arms of the bridge are similar, and equally affected by the current. Phil. Mag. 8. 6. Vol. 19. No. 112, April 1910. 2N 546 Prof. H. lL. Callendar on Electrical or that the heating effect of the current must be increased more than 40 per cent. The most important error to avoid is to make R small compared with G or 8. The sensitive- ness may in this way be very greatly reduced, and the heating effect of the current increased five or ten times. Thus, a pyrometer of low resistance designed for work at high temperatures (where the resistance is high and an open scale is not required) would not be a suitable instrument to employ for recording small variations of temperature on an open scale at low temperatures, unless G and 8 were made inconveniently small, and the pyrometer itself specially wound with extra thick wire to give a large cooling surface. Construction of Thermometers. 6. The construction of the thermometers depends on the situation in which they are to be used. I have made three principal types:—(1) For insertion in the mouth; (2) for the rectum ; (3) for the surface of the skin or for the axilla (armpit). (1) The mouth is not a suitable position for records of long duration, but it is of interest to be able to insert an electrical thermometer in this way in order to investigate the conditions of lag, which cannot be observed so accurately with a mercury clinical thermometer. It is most essential that the thermometer should be of small thermal capacity, in order to minimise the effect of inserting a cold thermo- meter, since the tissues of the body are not very good con- ductors of heat. In any case, if the mouth has not been kept shut for some time previous to the insertion of the thermometer, there will be a large apparent lag due to the recovery of temperature of the mouth itself. A so-called half-minute clinical thermometer may take upwards of five minutes to get within a halfa degree of the true temperature under these conditions. I have tested several recording thermometers from this point of view, and the results are of some interest. | Flat Glass Bulb Thermometer.—The ordinary type of platinum thermometer has the wire wound on a mica cross with compensated leads insulated by mica disks. Thermo- meter and leads are slipped into a containing tube about 10 mm. diam., which is removable, and can easily be replaced if damaged or broken, and which permits withdrawal of the thermometer for adjustment or repair, if required. This type is most convenient and suitable for ordinary laboratory work with sensitive galvanometers ; but is insufficiently Ee Wr le mg On es Recording Thermometers for Clinical Work. SAT sensitive for insertion in the mouth. Moreover, since the wire is surrounded by air, the heating effect of the current may be excessive when such a thermometer is used with recording instruments on an open scale, especially if its resistance is low. A great improvement on this type, in point of quickness of action and diminution of the heating effect of the current, is readily effected by winding the wire on a flat plate of mica in place of a cross, and melting the lower part of the con- taining tube (which is preferably of lead glass) down on to the wire so as to form a flat bulb, as illustrated in fig. 2. If the glass is thin, the sensitiveness may be increased nearly five times, as compared with a thermometer of the ordinary type, and the heating effect of the current reduced in nearly the same proportion. When inserted in a water- bath at 37° ©. this thermometer takes less than a minute in arriving within a hundredth of a degree of the final tem- perature ; but when inserted cold in the mouth, it may take 4 or 5 minutes to get within a tenth of a degree, because the tissues of the mouth take time to recover from the cooling effect of inserting the thermometer, as further illustrated below. Where quickness of action is an essential condition, as in some kinds of calorimetric work, the flat- bulb thermometer is a great improvement on the ordinary type with a round tube, but it shares with the mercury thermometer the disadvantage that, if broken or damaged, it cannot easily be repaired. It is essential that thermometers of this type should be provided with compensated leads, otherwise there will be a variable immersion error, and an apparent lag due to slow ‘conduction of heat along the leads. It is also most important for accurate work to avoid screw terminals in the head of the thermometer, and to provide each instrument with flexible leads two or three metres long, permanently soldered Fig. 2. WT SS SS UITIANTI ALT S/DE-VIEW ENLARCED Flat Glass Bulb Thermometer. +o the thermometer and compensator leads, and securely attached to the head of the thermometer, as indicated in fig, 2, 2N 2 548 Prof. H. L. Callendar on Electrical Hereus Quartz-Glass Thermometer.—The smallest size of quartz-glass resistance thermometer constructed by Herzeus after the experiments of Dr. Haagn has a bulb 25 mm. long and 3 mm. in diameter, and is therefore very similar in dimensions to an ordinary clinical thermometer. The bulb contains a coil of fine platinum wire wound on a rod of quartz-glass, and protected by a very thin tube of quartz- glass fused over it. The bulb is fused on to a larger quartz tube, containing the two leading wires, which are of gold. The resistance of the coil is 25 ohms nearly at 0° C., but, as there are no compensating leads, the resistance cannot be measured very accurately. This type of thermometer is very suitable for insertion in the mouth, but is unsuitable for records of long duration owing to its shape and the risk of breakage. It does not appear to be quite so sensitive as a mercury thermometer of similar dimensions, partly owing to the low conductivity of the central rod of quartz-glass, and partly to the fact that the leads are not compensated. When inserted in a water-bath at 37° C. this thermometer starts very quickly to rise, but takes about a minute in arriving within a tenth of a degree ©. of its final reading, and continues to change appreciably for three minutes, possibly owing to slow conduction along the leads, one of which passes through the centre of the coil. When placed under the tongue, after keeping the mouth closed for ten minutes previously, it arrives within a degree of the final temperature in one minute, but takes seven or eight minutes to get within a tenth of a degree. A typical record obtained with this thermometer is given in fig. 3. Flat Platinum Tube Thermometer.—This is a type of thermometer which I specially designed for calorimetric work, where quickness of action and small mass are im- portant. It is similar to the flat glass bulb thermometer, except that the containing tube is of platinum about 0°05 mm. thick, which permits the thermal capacity of the bulb to be greatly reduced. The coil is wound on a flat plate of very thin mica, and insulated from the flattened platinum tube by thin strips of mica on either side. The lag of this thermo- meter when placed in a water-bath at 37° C. was so much less than that of the galvanometer that it could not be recorded satisfactorily. Owing to its size and shape, it was not very suitable for insertion in the mouth, as it could not be placed in the usual position under the tongue. When placed along the side of the mouth (after closure for ten minutes) between the tongue and the teeth, it arrived within 0°:3 of the final temperature in less than a minute, but Recording Thermometers for Clinical Work. 549 continued to rise appreciably for nearly ten minutes, apparently indicating that the mouth had not been kept closed long enough beforehand to reach a practically steady temperature. A comparison of the records of the platinum tube thermo- meter (Pt) and the Herzeus quartz tube thermometer (He) when placed in the mouth under similar conditions is shown in fig. 3, which illustrates the importance of extreme quick- ness and small thermal capacity in such tests. Fig. 3. [Zc poor | “4 a 4 6 8 /0 Comparison of Quartz and Platinum Tube Thermometers in Mouth after closure for 10 minutes. It might appear at first sight as though the slow and long continued rise indicated by both thermometers in different degrees, were due to the heating effect of the measuring current, which might easily produce a result of this nature if the current were excessive or the mass considerable. But the current employed in taking these records scarcely exceeded one hundredth of an ampere in the case of either thermometer, and the heating effect of the current was measured in both cases and found to be very small. With the Herzeus thermometer, when placed in a water-bath and traversed by the same current, the rise of temperature due to the passage of the current was found to be only 0°085 C. and to reach a steady value in about two minutes. When enclosed in its metal protecting tube, so that the bulb of the thermometer itself was surrounded by air instead of being in direct contact with the water, the heating effect of the current was increased to 0°31 C., and became steady in less than three minutes. The comparatively high value of the heating effect of the current with this thermometer is evidently due to its small size and surface. When placed in the mouth the heating effect of the current could not be 550 Prof. H. L. Callendar on Electrical determined accurately, as the temperature was not perfectly steady, but it was probably less than a tenth of a degree,. and therefore insufficient to account for the observed lag. With the flat platinum tube thermometer, owing to its much larger surface, the heating effect of the current was far too. small to be appreciable on the scale of the records. That the apparent lag of the thermometer is really due to. the slowness of recovery of the temperature of the mouth itself after being cooled in any manner, is furtker shown by the following records taken with the platinum tube thermo- meter under different conditions. The first curve, marked (1) in fig. 4 was taken at the conclusion of an ordinary Fig. 4, Cesee| romans | @ | | tt wi liidcth8) 3s eee eee ne ee Platinum Tube Thermometer Records in Mouth after different periods of closure. conversation, without any precautions as to keeping the mouth closed or refraining from speaking, until the thermo- meter had been inserted. The thermometer did not reach 32° ©. for more than a minute, and was still nearly 1° C. below the normal temperature after a lapse of seven minutes. Curve No. (2) represents the record obtained after keeping Recording Thermometers for Clinical Work. dol the mouth closed for three minutes previous to the insertion of the thermometer. Curve No. (3), which is practically a reproduction of the record already given in fig. 3, represents the result of keeping the mouth closed for ten minutes previously. There is a further improvement with this thermometer after closure for fifteen minutes, but unless the mass of the thermometer is very small, or the thermometer itself is previously warmed, little advantage is gained by keeping the mouth closed more than ten minutes, because in any case the mouth takes some time to recover from the cooling effect of inserting the thermometer. In all the records given in figs. 3 and 4, the thermometer was inserted cold, being taken direct from a water-bath at 15° C. The difference between the records in fig. 4 is due almost entirely to an actual difference in the temperature of the mouth at starting. The difference between the quartz and platinum tube thermometers in fig. 3 is due chiefly to the greater cooling effect of the quartz tube, owing to its greater mass in proportion to its surface. (2) For records of long duration, the best results are undoubtedly to he obtained by insertion in the rectum. The thermometer bulb should be of small dimensions and the leading wires should be enclosed ina flexible but inextensible tube of small diameter, such that it can be attached to the bulb in a perfectly secure and water-tight manner. Such a thermometer may be made in a variety of ways. The following is a description of a method of construction which I have found very satisfactory. The fine wire forming the bulb is wound on a thin celluloid tube 5 mm. in diameter, and is protected by another thin celluloid tube which closely fits the first. One end of the protecting tube is closed by a small celluloid stopper, the other is cemented with celluloid cement to the cut end of a Porges catheter 5 mm. in external diameter, and 32 cm. long, through which the flexible thermometer and compensator leads are passed. This type of catheter has a smooth surface, and is very flexible, but practically inextensible, thus protecting the leading wires and joints from possible strain. It also makes a very good joint with the celluloid tube protecting the bulb. I have tried rubber tubes and metal bulbs, but these appear to be in every respect less suitable. A rubber stopper fixed on the catheter at a distance of 3 to 5 cm. from the near end of the bulb, is a convenient method of limiting the depth of insertion. The thermometer is smeared with vaseline before insertion. This prevents irritation, and does not appear to have any injurious effect on the celluloid or catheter. 552 Prof. H. L. Callendar on Electrical The curves shown in figs. 5 and 6 are typical records taken with this thermometer on consecutive nights under similar conditions. In the first case there was an unusually rapid rise in the external atmospheric temperature during the night which occasioned some feeling of discomfort. In the Fig. 5. 16 12 2 4. 6 g Rectal Thermometer on Thread Recorder. Fig. 6. 32° 1 A se | el ee 0) 12 ! 2 A 6 38 Rectal Thermometer on Thread Recorder. second case there was a sharp frost, and the patient awoke feeling appreciably chilled. From a comparison of similar records it would appear that the normal temperature of the body may be influenced to some extent, as one would naturally expect, by the external conditions prevailing at the time. There is often a fall of nearly half a degree C. during the night corresponding more or less with the diurnal change of atmospheric temperature. : The changes here observed cannot have been due, as might appear at first sight, to the effect of change of temperature on the resistance coils of the box or measuring apparatus, because increase of resistance of the balancing coils in the bridge would act in the opposite direction, producing an apparent fall of the thermometer. Moreover, the balancing coli in this particular thermometer was made of fine man- ganin wire having a resistance equal to that of the thermo- meter coil at 37°C., and was placed inside the celluloid tube ‘| ‘ ii aa > Recording Thermometers for Clinical Work. 553 on which the thermometer coil itself was wound, so that any changes in its resistance were due to change of temperature of the thermometer, and were allowed for when the thermo- meter was calibrated. This construction was adopted in order to avoid the necessity of applying any correction for change of temperature of the resistance box ; but it is open to the objection that it makes the thermometer rather less sensitive, and doubles the heating effect of the current. The sensitiveness of the celluloid rectal thermometer, though only about half as great as that of the Herzeus thermometer, was ample for records of long duration. The heating effect of the current was only a tenth of a degree, and did not intro- duce any appreciable error, since it remained practically constant, and was allowed for in the calibration, in which the same current was employed. A more important source of systematic error would lie in variation of E.M.F. of the battery during a long continued run. SBut unless the storage cells employed were nearly run down, this would not be likely to amount to more than 1 per cent. of the galvanometer deflexion, or two or three hundredths of a degree, since the galvanometer deflexion ‘corresponds only to the small difference from the balance point which is in the neighbourhood of 37°C. The zero of the galvanometer and the deflexion per degree would of course be tested, and, if necessary, adjusted daily. (3) Surface or Aaillary Type-—A tube form of thermo- meter is not very suitable for this purpose, since the contact between the thermometer and the skin is necessarily imper- fect. It is desirable to give the thermometer as large a surface as possible, in order to diminish the heating effect of the current, and at the same time to make the bulb flexible and thin, so as to conform approximately to the contour of the surface to which it is applied. The first thermometer of this type which I constructed for Prof. Adami of Montreal in 1897, was made by sticking a thin bolometer grid of platinum foil on to a thin sheet of celluloid photographic film. This satisfied the required conditions very perfectly, but the bolometer grid was excessively fragile (unless its resistance were made unduly low), and was also difficult to attach securely to the leading wires. The substitution of fine wire for the foil greatly facilitated the construction and adjustment, and was found to make little, if any, difference to the sensitiveness. In any case it is most important that the thermometer should be fairly robust, as it has to undergo somewhat rough usage. In my experience nothing answers so well as celluloid film for the insulating material. The 554 Prof. H. L. Callendar on Electrical wire may be protected by cementing over ita thin film of celluloid ; but a better plan is to insert the thermometer in a thin flat celluloid sheath, as illustrated in fig. 7, which can be changed at any time if it becomes soiledi The flexible leads are flattened and insulated between celluloid films at the end where they join the thermometer coil. In applying the thermometer to the surface of the body, a truss, similar to that employed by Dr. Gamgee (loc. cit.) for thermo- couples, may be used; but, in my experience, a simpler and more generally applicable method is to back the thermometer with a pad of cotton-wool to secure good contact, and to keep it in place by means of an elastic band to which the thermometer and leads are pinned or sewn. When the thermometer is placed in the left axilla, which is generally the most convenient location, the elastic band should pass under the left arm, and over the right shoulder near the neck, =| i il | TN HN i “i Axilla or Surface Thermometer (Celluloid). A surface thermometer of this type is extremely sensitive, and shows no appreciable lag when placed in a water-bath. But when placed in the axilla, or elsewhere on the surface of the skin, it may take fifteen minutes or more to reach a steady temperature. The reason of this is partly that the skin is slightly chilled by exposure during the insertion of the thermometer, and partly that the application of the thermo- meter with its wool pad tends to raise the temperature of the skin locally by preventing evaporation. The temperature of the skin under normal conditions with free evaporation is generally lower than that of the body. If the patient is. already in bed and well covered up, the initial lag may be greatly reduced, but is in any ease of little or no consequence - for records of long duration. A thermometer of this type should not be heated to 100° C. or directly exposed to water at temperatures as high as 50° C., because this procedure is liable to buckle the film. If the temperature coefficient of the wire is known, it can be cali- Recording Thermometers for Clinical Work. DDD brated with sufficient accuracy by a single observation in a water-bath at the ordinary room temperature with a reliable mercury thermometer. If it is to be tested in a water-bath at 40° or 50° C., it should be protected by a thin copper sheath. I have generally employed thermometers adjusted to a fundamental interval of 10 ohms, giving a change of one-tenth of an ohm per 1°C. Each thermometer is provided with a balancing coil, the resistance of which is adjusted to be equal to that of the thermometer at or near 37° C. The chief difficulty in obtaining reliable records of the body temperature by means of a thermometer inserted in the axilla, is that the indications are liable to be disturbed by movements of the patient, more particularly by putting the left arm out of bed, or by raising the bed-clothes and admitting cold air in turning over from one side to the other. Fig. 9. Axillary Thermometer B, with elastic band. These effects are illustrated by the records shown in figs. 8 and 9, which were taken with different thermometers in the 556 Prof. H. L. Callendar on Electrical axilla on consecutive nights. The curve is often very smooth during sleep, but becomes irregular owing to greater rest- lessness on awaking. These accidental excursions seldom exceed a few tenths of a degree, unless the patient gets out of bed, and are chiefly due, in all probability, to slight dis- placements permitting intrusion of air betwen the thermo- meter and the skin. In the record shown in fig. 10 a very Fig. 10. 39° L 6 Sam. Axillary Thermometer C, fastened in place with adhesive wax. smooth curve was obtained by sticking the thermometer to the skin with adhesive wax. Unfortunately this record was of shorter duration, only six hours, and, as the patient did not get to bed until 2.30 a.m., it is possible that excessive fatigue may have induced unusual soundness of sleep. Recording Instruments. 7. Two principal types of recording instruments are avail- able, corresponding respectively to the Deflexion Method, and the Balance Method of measuring variations of electrical resistance. In the deflexion method, the greater part of the resistance to be measured is balanced on a Wheatstone bridge by a balancing coil, and the small variations are observed by the deflexion of the galvanometer. In the balance method, the galvanometer contact is moved along a bridge-wire until balance is obtained. This method is not so quick for small variations, but is generally most suitable tor work with resistance thermometers, and has the advantage that the scale is uniform and independent of the E.M.F. of the battery. In clinical thermometry, the range of temperature to be covered is so small that the deflexion method is generally applicable and the scale practically uniform. The scale is readily adjusted to read in degrees of temperature by means of a suitable rheostat in the battery circuit. If storage cells of sufficient capacity are employed there is Recording Thermometers for Clinical Work. 557 little risk of error from variation of the E.M.F. of the battery. Thus, for clinical thermometry, there is little to choose between the balance and deflexion methods, if the instru- ments employed in either case are equally sensitive. In the installation of underground thermometers, buried at various depths in the soil at McGill College in 1894, the deflexion method was employed, as being the quickest, for the daily readings of each thermometer ; but the balance method was adopted, as being the most accurate, for taking continuous records of the diurnal variations when occasion required. In order to obtain a strictly continuous record by the de- flexion method it is necessary to employ photography, but a practically continuous record may be obtained, as in the Thread Recorder made by the Instrument Co., Cambridge, by depressing the pointer attached to the galvanometer coil so as to mark its position on the record sheet by means of an inked thread at regular intervals of a minute, the index being free for the greater part of the time. Tor records of long duration the separate dots coalesce into a continuous curve if the variations are not too rapid. Adjustment and Use of the Thread Recorder. 8. The standard type of Thread Recorder is a fairly robust instrument, having a resistance of about 10-12 ohms, and a scale of 80 mm. to one millivolt. With a Copper-Constantan thermocouple, this instrument gives a scale of about 3 mm. to 1° C., which is rather too small for clinical records. With a resistance thermometer, it is easy to get a scale of 50 mm. to 1°C. if desired, but a scale of 10 to 20 mm. is usually sufficient. In using the thread recorder with a resistance thermo- meter, some form of platinum thermometer bridge with a slide-wire is required, and a rheostat or resistance box capable of fine adjustment between 30 and 50 ohms. This rheostat is connected in series with a 2 volt storage cell to the battery terminals (marked B on the bridge), and serves to adjust the scale of the record. The thread recorder is connected to the terminals marked G on the bridge, one of which is con- nected between the equal ratio coils of 10 ohms each, and the other to the sliding contact on the bridge-wire, as shown on the diagram, fig. 11 (p. 558). The two thermometer leads are not connected to the terminals marked P on the bridge (which is the usual arrangement when the balance method is used), but to the 558 Prof. H. L. Callendar on Electrical terminals marked C, C, (for compensator) so as to be in series with the adjustable plug resistances denoted by R in the annexed diagram of connexions. ‘The compensator leads are connected in series with the Balancing Coil, denoted by S$ Fig. 11. Rheostat B; Wes e Si Therm Diagram of Connexions for Thread Recorder. in the diagram, to the terminals marked P, P,. With this arrangement the resistance S+C is nearly constant, so that if the galvanometer deflexion is adjusted to be correct when the thermometer is at 0°C. or 20°C., or any convenient temperature, the scale will be very nearly correct when the thermometer is at any other temperature, the plugs R being suitably adjusted. As an example of the adjustment of the galvanometer scale, we will suppuse that the thread recorder is properly set up and levelled, and the zero adjusted to the right hand side of the scale, which reads from right to left. The standard type of record sheet is divided into 25 scale-divisions, with a heavy line at each fifth division. With this sheet it is most convenient to adjust the scale to read 5 divisions to 1° C., giving a range of 5° C. on the record sheet. “Suppose that the thermometer has a resistance of 26 ohms at 0°C., and is adjusted to have a fundamental interval of exactly 10 ohms, giving a change of 0:1 ohm per 1°C. Suppose that the balancing resistance 8 is 30 ohms and is adjusted to be equal to the thermometer at 40°C. Ten centimetres Recording Thermometers tor Clinical Work. 559 of the bridge-wire, or ten units of the plug resistances R, will ‘correspond to a tenth of an ohm, or 1°C., on the thermometer. To adjust the galvanometer scale, the thermometer may be placed in a vessel of water at the temperature of the room, and the bridge balanced. If the temperature is 15°C. (which is 25° C. below 40° C.) it will be necessary to unplug 250 units (equivalent to 25°) in the box at R to balance 8, when the sliding contact is at zero in the centre of the bridge- wire. If balance were obtained with 230 units in R, and the sliding contact at 7°3 cm. to the right of zero, it would mean that the temperature of the water was 40—23+4+°73 =17°73C. In any case, to obtain a fine adjustment for the balance, the sliding contact must be shifted till the galvano- meter shows no deflexion, and then clamped in position. The galvanometer being thus balanced at a steady tempe- rature, increase the resistance unplugged in R by 50 units (0°5 ohm), corresponding to a rise of temperature of 5° CO. on the thermometer. If the adjustment of the rheostat is correct the galvanometer should deflect 5° to the left, 7. e. to the extreme left of the record sheet. If the galvanometer deflects to the right the connexions of its terminals must be reversed. If the deflexion is less than 5°, the resistance of the sliding rheostat must be reduced, and vice versa, until the desired deflexion is obtained. The adjustment will then be correct for any temperature within the range of the instrument up to 40°C. To get a range on the record sheet from 35° to 40° C., for records of normal body temperature, it would then be necessary to unplug 50 units in R and to set the bridge-wire contact at zero, so that the zero of the galvanometer scale may correspond to 35° C. Similarly for a range from 40° to 45°C., the resistance unplugged in R should be zero. Temperatures above 45° could be measured by increasing the balancing resistance S, but it is not advisable to go above 45° C. with celluloid thermometers. The accuracy of adjustment of the thermometer and its balancing coil S can be tested at any time by immersing the thermometer either in melting ice, or in water at the room temperature if a reliable mercury thermometer is available. If, for instance, the galvanometer reads zero when the ther- mometer is in water at 17°67 C. with 230 units unplugged, and the contact at 73 mm. (reading 17°73 C.), the thermo- meter reads 6 mm. or ‘06° C. too high at this temperature, and the error may be corrected by setting the bridge-wire contact 6 mm. to the right of zero for the record, instead of at zero. Several different thermometers may be used with 560 Prof. H. L. Callendar on Electrical the same balancing coil, provided that allowance is made in this way for the small differences between them. Testing for Defects. 9. To test the thermometer and compensator leads for defective insulation disconnect one of the thermometer leads, say Co, from the battery, and the opposite compensator lead, say P,, from the bridge-wire (or vice versa, C, from the bridge-wire and P, from the battery). If, now, the galva- nometer circuit is made, there should be no deflexion if the insulation is perfect, since the bridge-wire and galvanometer contact are effectively isolated from the rest of the circuit, unless there is defective insulation between the thermometer and compensator. Before disconnecting the wires for this test the galvanometer circuit should be broken, otherwise the instrument will be violently agitated, and its zero may be affected. To measure the heating effect of the working current on the thermometer, place the thermometer in water at a steady temperature, balance the bridge as already explained, and adjust the galvanometer scale if this has not been done. With the galvanometer balanced at zero, and the current adjusted to its proper value, change the battery from one storage cell to two cells of the same E.M.F. Since the resistance of a storage cell is negligible compared with the other resistances in the circuit, this change will have the effect of doubling the current through the thermometer. The rise of temperature of the thermometer due to the passage of the current will be quadrupled, since it varies as the square of the current. The change of resistance of the thermometer due to changing from one cell to two, will be three times the change of resistance due to the heating effect of the current with one cell. The deflexion of the galvano- meter for a given change of resistance will also be doubled, since the current is doubled. The deflexion of the galvano- meter produced by changing trom one cell to two will therefore be six times the heating effect with one cell measured in degrees of temperature. Hence observe the galvanometer deflexion in degrees of temperature and divide by six to find the heating effect for the normal current with one cell. If the heating effect so deduced does not exceed a tenth of a degree, it will not introduce any appreciable error, since it remains practically constant, so long as the scale adjustment is the same. With a given thermometer and galvanometer the heating effect varies as the square of the number of scale divisions per degree. Thus if the ite ee ee ee ee ee es ——— —— Se . Recording Thermometers for Clinical Work. 561 heating effect were found to be a tenth of a degree with the galvanometer adjusted for a scale of 2 cm. to 1° C., it would be four tenths if the scale were increased to 4 cm. per degree, and the zero would be raised three tenths. Caltbrat'on of the Scale of the Galvanometer. The angular deflexion of the galvanometer for one milli- volt is rather large, and the scale may not be accurately one of equal parts. This should be tested for accurate work. With the aid of the bridge it is easy to make the test, either by placing the thermometer in water at a steady temperature, or by substituting a fixed resistance, equal to the balancing coil, in place of the thermometer. Balance the galvanometer at zero by adjusting the bridge-wire contact. Adjust the scale to the required value. Unplug successively resistances in R corresponding to 1, 2, 3, 4, 5°, and observe the galvano- meter reading for each on the record sheet. Repeat in descending order to test the galvanometer for change of zero due to imperfect elasticity of the suspension. The errors should not as a rule be appreciable on the scale of the record unless there is something wrong with the levelling or sus- pension. It is of course extremely important that the galvanometer coil should be perfectly free in all positions of the boom, and it must be remembered that a variation of level may cause a variation of scale, as well as of zero. Balance Method with Slide-wire Recorder. 10. This is the type of recorder generally employed with resistance thermometers, especially where a large range of temperature is to be covered. It is equally applicable for clinical work, or for small ranges of temperature, provided that a suitable thermometer is employed, and that the current is properly adjusted to avoid excessive heating of the thermo- meter. It gives a strictly continuous ink record, with a pen attached to the sliding contact on the bridge-wire, which is automatically maintained at the balance point by means of a pair of motor clocks actuated by the deflexion of the galvanometer. The galvanometer coil carries a light arm terminating in a contact fork consisting of two short platinum wires close together. A wheel with a platinum edge rotates between these two wires, one or other of which makes contact with the wheel (according to the direction of de- flexion of the galvanometer), and completes a relay circuit actuating one or other of the motor clocks. This form of Pat, Mag. 8. 6. Vol. 19. Ne, 112, April 1910. 20 562 Prof. H. L. Callendar on Electrical relay is incredibly delicate, and has since been adapted by Brown for the purpose of submarine telegraphy. The con- tacts keep themselves clean automatically, and never require adjustment. Since the deflexion of the galvanometer is limited to a fraction of a millimetre, there is no chance of accidental disturbance of the zero by excessive deflexion. The scale of the instrument is independent of the E.M.F. of the battery, being determined by the resistance of the bridge-wire in relation to that of the thermometer. The bridge-wires are usually made with a scale of 2:0, 1:0, or 0°5 ohm toa length of 20 cm., which is the length of the record sheet. With a thermometer having a zero resistance of 26 ohms, and a fundamental interval of 10 ohms, these bridge-wires give ranges of 20° ©., 10° C., and 5° C., respec- tively on the record sheet, or scales of 1, 2, and 4 cm. to 1°C. With thermometers of smaller resistances the scales are proportionately smaller and the ranges larger. For clinical work the most convenient range is from 35° to 45° C., with a scaleof 2cm.to1°C. This is obtained with a 26 ohm thermometer and a 1 ohm bridge-wire. A recorder of this type is more expensive than the thread recorder, but is complete in itself, with the exception of the battery, and dves not require either an auxiliary resistance box, or a rheostat for adjusting the scale. It is a great advantage in practice that the scale never requires adjust- ment, but is always correct to about 1 in 1000, provided that the bridge-wire is correct and uniform. It is easy to change frum one scale to another when required by changing the bridge-wire, which gives a range of 1 to 4, or by changing the thermometer, which gives a range of 1 to 20, from a fundamental interval of 10 ohms to a fundamental interval of 0°5 ohm. For ordinary work thermometers are generally provided with an “ice-bobbin ” or balancing coil, equal in resistance to the thermometer at 0° C. The ice-bobbin for each thermo- meter is connected to its appropriate terminals when the thermometer to which it belongs isin use. If the thermo- meter is required to cover an extensive range of temperature, a series of auxiliary resistances, generally ranging from U to 20 ohms is provided, which enables the range to be extended to 20 times the range of the 1 ohm bridge-wire. With a 26 ohm thermometer the range thus obtained would be 200° C., or 2000° C. with a 2°6 ohm pyrometer. For clinical work these auxiliary resistances may be dispensed with if the balancing resistance is adjusted to 30 ohms to balance the thermometer at 40° C., and the zero fixed at the centre: Recording Thermometers for Clinical Work. 563 of the bridge-wire so as to give a range of from 35° to 45° C., with a 1 ohm bridge-wire. Some observers have experienced trouble with this type of recorder owing to the heating effect of the measuring current. Such troubles have arisen chiefly from the employ- ment of unsuitable thermometers with excessive currents. With a 26 ohm thermometer the instrument can be adjusted to work perfectly on a scale of 4 cm. to 1° C. with a current of one hundredth of an ampere through the thermometer. The record can be read easily to less than a hundredth of a degree, or 1in 30,000 ; and the heating effect of the current, with a celluloid thermometer of the surface type above described, is only two or three thousandths of a degree, or practically inappreciable on the most open scale of the record. Testing and Adjustment of Slide-wire Recorder. 11. The most important points to test in a slide-wire re- corder are the adjustment of the zero of the galvanometer, and the zero of the slide-wire. When the galvanometer coil is free and the instrument levelled, the two prongs of the contact fork should be just clear of the contact wheel on either side. If there is any torsion in the suspension, one of the prongs will bear against the wheel, and the pen will travel continuously in one direction when the circuit of the relay magnets is made, the galvanometer circuit being open. The torsion head should be adjusted until the pen does not travel either way under these conditions. This is a very delicate test. If there is much torsion in the suspension, the apparent zero on the slide-wire will vary to some extent with the resistance in circuit when the galvanometer circuit is closed. ‘This adjustment is made before the instrument is sent out, but itis desirable to test it occasionally, especially if the suspension has been strained or damaged, or if a new suspension is fitted. To adjust the zero of the slide-wire, short circuit the pyrometer and compensator terminals, PP and CC, with short equal pieces of copper wire. When the battery is switched on the pen will quickly come to rest at the zero. If the position is not quite correct in reference to the record sheet, the slide-wire may be shifted in the direction of its length through a distance equal to the observed deviation, For many purposes it is most convenient to have the zero in the centre of the slide-wire. The wire can then be changed, if desired, for another of different resistance and scale, without altering the zero. [f it is desired to have the € ao a —— | i tl S64 © | Prof. H. L. Callendar on Electrical zero at one end of the scale, it is necessary-to provide a small auxiliary slide for correcting the adjustment when the slide- wire is changed. The freedom of the galvanometer sus- pension from torsion may conveniently be tested at the same time as the zero of the slide-wire, by doubling the resistance in the battery circuit, so as to halve the current through the slide-wire. If this does not produce any appreciable shift of zero, the suspension must be very nearly free from torsion. It is important that the external resistance added should be only in the bridge circuit, and not in that of the relay magnets, which will not work satisfactorily if the Pag on the relay circuit is much reduced. This mistake has often been made in using this type of recorder. Slide-wire Records. 12. It is possible to obtain records on a more open scale with the slide-wire recorder than with the thread recorder, owing to the greater width of the record sheet, which is 20 em. for the slide-wire recorder, as compared with 8 em. for the thread recorder. The latter is limited by the range of the galvanometer boom, which cannot be indefinitely elongated without making the deflexion too sluggish. The record reproduced in fig. 12 was taken with an axillary thermo- meter differing in a few details of construction only from those employed for records 8and 9. The scale of the original record was 4cm. per 1°C., which is 2°5 times the scale adopted with the thread recorder. The scale has been ALA CCS a bee ls ei Axillary Thermometer D, on Slide-wire Recorder, 4 cm./deg. reduced to one third, to facilitate reproduction and com- parison. It will be observed that the record shows the same characteristic dip, culminating between + and 5 a.m, as records 8 and 9. When first observed, it seemed as though this dip of temperature might be due to some instrumental error, but it has been observed on several occasions with Recording Thermometers for Clinical Work. 565 this particular patient, with different thermometers and different recording instruments. The dip is not always present. The record reproduced in fig. 13, which shows an extremely steady temperature, was taken on the same patient, Axillary Thermometer D, on Slide-wire Recorder, 2 cm./deg. with the same recorder and thermometer, but on a scale of 2 cm. per 1°C., similarly reduced in reproduction. The current employed in both cases was the same as with the thread recorder for records 8 and 9, and the thermometer was attached in precisely the same way. The only effect of reducing the scale with the slide-wire recorder would be to make the instrument proportionately quicker in responding, which cannot account for the greater smoothness of the record. A record taken simultaneously on the same patient with the rectal thermometer and thread recorder, is repro- duced in fig. 14. The variations shown are very slight, and the differences between the two thermometers are such as would be likely to exist in different parts of the body. 10 TX Lam. Ly 6 Sg Rectal on Thread Recorder (simultaneous with record 14 i The records above given are sufficient to show, if proof were needed, that accurate clinical records of temperature on a fairly open scale under practical conditions may readily be obtained by the aid of electrical resistance thermometers, with either of the recording instruments above described. The recording instruments are standard types, which have been thoroughly tested by the experience of a great variety of users for many years, and are not likely to giye trouble. LIX. The Absorption Spectra of certain Uranous and Uranyl Compounds. By Harry OC. Jones and W. W. Srrone *. (This is part of an investigation that is being carried out with the aid of a Grant from the Carnegie Institution of Washington. | (Twenty-ninth Communication.) EVERAL investigators have already studied the uranyl bands, but only Deussent and the authors} have noticed the marked changes in the frequencies of the uranyl bands when the solvent in which the uranyl salt was dissolved is changed. No one, so far as we know, has hitherto observed the gradual change in the uranyl bands when one uranyl or uranous salt is changed into another salt. Prac- tically no work § has been done on the absorption spectra of uranous salts, and very little upon “solvent”? bands. Jones and Anderson || have shown that neodymium possesses “water” and “alcohol ” absorption bands when the salt is dissolved in mixtures of water and alcohol. The “water” bands gradually decrease in intensity as the amount of water is decreased, while the “ alcohol ” bands increase in intensity. The frequencies of the “water” and the “alcohol” bands were found to remain constant and both sets of bands may appear in the same solution. The purpose of the present investigation was to photograph the absorption spectra of uranous and uranyl salts dissolved in different pure solvents ; to find the relationship between the uranous and the uranyl bands of the same salt in different solvents by changing the solvent; to find the relationship between the uranous and uranyl bands of different salts by changing the acid radical of the salt, and to find the effect of the presence of acids and foreign salts upon the bands. The general method of preparing the solutions of the uranous salts was to add a little acid (corresponding to the anion of the salt) and zine to a solution of the uranyl salt. The nascent hydrogen reduced the yellow uranyl salt to the very green uranous salt. Most of the uranous salts keep for a very long time when not exposed to the air. Uranous bromide and uranous chloride ure stable in all the solvents * Communicated by the Authors. Tt Wied. Ann. lxvi. p. 1128 (1898). t Phys. Zeit. x. p. 499 (1909). § Formanek, ‘ Die qualitative Spectralanalyse anorganischer Kérper, Berlin, 1900. || Phys. Rev. xxvi. p. 520 (1908). Carregie Institution of Washington, Publication 110. | Absorption Spectra of Uranous and Uranyl Compounds. 567 that we tried. Uranous sulphate and uranous acetate are not us stabie as the chloride but may be kept in most solvents. Uranous nitrate is very unstable and no photographic work could be done upon its absorption spectra. When nitric acid is added to a solution of uranous acetate the salt is changed into a yellow uranyl salt. Photographs have been taken of the absorption spectra at the various stages of these changes. A spectroscope with a Rowland grating was used, and the method of work is that described by Jones and Strong *. The uranyl absorption bands are about twelve in number, running from % 5000 out into the violet, their width and dis- tance apart decreasing in general with the wave-length. The uranyl bands often appear in the absorption spectra of uranous salts, and probably are due to the presence of unreduced uranyl salt. Starting with the red end of the series, the uranyl bands will be designated by the letters a, b,c, d, ke. It has been found that the width, intensity and wave-length of each one of the uranyl bands is ereatly modified by changing the solvent, by changing the salt, by the presence of free acid, by a foreign salt or by a change in temperature. [ew if any other series of absorp- tion bands can be so greatly modified by changing external conditions as the uranyl bands, and any theory of absorption spectra to be at all adequate must explain these changes. The absorption spectra of uranous acetate, bromide, chloride and sulphate have been photographed in acetone, methyl and ethyl alcohols. glycerol and water. The absorption spectra of uranous bromide and uranous chloride are very similar to each other and differ considerably from uranous acetate. Acetone solutions of uranous salts give absorption spectra that are quite different from the absorption spectra of these same salts in other solvents. Even in the case of uranyl salts the absorption spectra in acetone is characteristic of that solvent. For example, uranyl chloride in acetone gives six bands at AX 4980, 5000, 5030, 5240, 5270, and 5295. The uranyl bands themselves are broken up into several compo- nents. Ifa small amount of free hydrochloric acid is pre- sent the d, e, # uranyl bands are broken up into triplets, each triplet consisting of a strong central component and weaker satellites about 20 A. U. wide. d. é. K A470 4340 4205 4430 4290 4160 A385 4250 4120 * Amer. Chem. Journ. xlii. p. 37 (1910), 568 Messrs. Jones and Strong on the Absorption Uranous chloride in acetone to which free hydrochloric acid is added has absorption bands, each from 10 to 20 A. U. wide, at AA 5195, 5210, 5220, 5910, 5960, 6000, 6040, 6090, 6340, 6365, 6390, 6470, 6490, 6555, 6600, 6625, 6690 6740, 6780. These bands have been discovered in no other spectrum and are as sharp as the erbium and neodymium bands. Hitherto it has been found that by lowering the temperature the uranyl bands are broken up into fine com- ponents. In the case of uranyl chloride the resolution of the bands is accomplished at ordinary temperatures. When no free hydrochloric acid is present the groups of fine bands consist simply of broad hazy bands. The absorption spectra of the various uranous salts in methyl alcohol is very similar to that in ethyl aleohol. The absorption of uranous chloride in methy] alcohol. for example, is almost the same as in ethyl aleohol, and this is practically the same as the absorption of uranous bromide in methyl alcohol. The absorption consists of rather diffuse bands at AX 4150, 4300, 4450, 4680 (very strong), 4950, 5100, 5250, 5900, 5650, 6250, and 6650. The relative intensities of the uranyl bands are very considerably modified by the amount of the reduction of the uranium salt present.in solution. In several cases the uranyl bands are broken up into several components, and in other cases several uranyl ‘bands may merge into a single wide band. Uranousacetate in methyl alcohol possesses a series of bands in the red that are very similar in character to the uranyl series, except that it runs in the opposite direction. Four such bands have been measured at AX 6700 6800, 6870, and 6920. The absorption spectrum of uranous bromide in glycerol is almost identical with that in methyl alcohol. The absorption bands of uranous salts in glycerol are very wide and hazy. The presence of free acid does not in general increase very greatly the sharpness of the bands. Glycerol solutions of uranons bromide and chloride consist of a strong band at A> 6250, a hazy hand at X 5300 and at x 5000. ‘The intensities of the uranyl bands depend largely upon the extent of the reduction of the uranium salt. Several uranyl bands of the partly reduced uranyl acetate are broken up into two components. The absorption spectra of the various uranous salts in water are very much the same. Uranous bromide and chloride have bands at AXA 6750, 6500, 6340, and 5500 ; and uranous acetate at AX 6850, 6700, 5600, and 5550. In aqueous solutions the uranous and uranyl bands are diffuse and in many cases extremely weak. ee eS ee eee ee eee eee ee ee eee ee ee ee ee ee Spectra of certain Uranous and Uranyl Compounds. 569 The general effect of the presence of free acid is to make the uranyl bands narrower and muci sharper. ‘The bands of uranyl nitrate in aqueous solution * are of shorter wave- Jengths than those of the other uranyl salts. The effect of sufficient free nitric acid is to shift the b, ¢, d, e, and 7 bands about 30 A. U. to the violet, and to make all the bands much sharper. The absorption of a solution of uranyl nitrate with water of erystallization in strong nitric acid consists of bands shifted slightly to the violet, as compared with the absorption of anhy - drous uranyl nitrate in nitric acid. The former solution when freshly prepared gave the fine band absorption spectra of nitrogen dioxide. The addition of sulphuric acid to an aqueous solution of uranyl sulphate causes the b. c, 2,7, k, 4, and m bands to be shifted towards the red, theg and h bands to come together, while the remaining uranyl bands are apparently shifted to the violet; the d band doubling. Free acetic acid has very little effect on the position of the uranyl acetate bands. The effect of free hydrochloric acid on uranyl chloride is to shove the uranyl chloride bands into the red, and also to change very greatly the relative intensities of these bands. The study of the effect of the presence of foreign salts has been limited almost entirely to the chlorides. Strontium, zine, calcium, and aluminium chlorides cause the uranyl and uranous bands of aqueous solutions to appear much stronger; and in general the presence of these foreign salts causes the bands to be shifted to the red. This shift is a gradual one, being greater the more foreign salt present. In the case of the addition of aluminium chloridef to an aqueous solution of uranyl chloride the e (A4460) and /(\ 4315) bands merge into one broad band. The absorption of concentrated aqueous solutions of mixtures of uranyl chloride with aluminium chloride, or zine chloride, or calcium chloride, or hydrochloric acid is practi- cally identical, and it seems quite possible that the effect upon the bands is due largely to the presence of the chlorine ion. Very interesting changes in the uranous and urany] bands are caused by adding acids to solutions of uranium salts. Acetic, hydrobromic, hydrochloric, nitric, and sulphuric acids have been added to aqueous solutions of uranous * Phys, Rev. xxix. p. 555 (1909). f+ Amer. Chem. Journ. xliii, p. 87 (1910). 570 Messrs. Jones and Strong on the Absorption acetate. In the pure acetate solutions the a and b bands are very weak. The c¢ band, on the other hand, is quite the strongest one of the series, the shorter bands gradually decreasing in intensity. The addition of sulphuric acid does not break up the acetate bands. On the other hand, the addition of nitric acid apparently causes the g-h and the j-k acetate bands to become a simple band. The addition of acetic acid to uranous chloride in water causes the b, c, and d bands to double and the e, f, g, and h bands to shift to the red. Sulphuric acid added to a nitric acid solution of uranyl nitrate does not greatly affect the e (A 4370) and f(A 4230) nitrate bands. The g nitrate band disappears, however, and the h, 7,and 7 bands are each shifted about 50 A.U. to the red. The & (43740) and 1 (A 3660) bands are unaffected. In a spectrogram showing the ab- sorption of uranyl nitrate in nitric acid to which sulphuric acid is added, the d nitrate band is gradually shifted about 70 A.U. to the red by the addition of sulphuric acid, until it joins the ¢ band which remains stationary. Nitric acid has very little effect upon the uranyl bands of uranyl! sulphate in strong sulphuric acid. Sulphuric acid causes the uranous and uranyl bands of an aqueous solution of a mixture of uranium chloride and aluminium chloride to shift to the violet. Hydrobromic acid, like sulphuric acid, does not greatly modify the uranyl acetate bands. The addition of hydrochloric acid to a solution of uranyl nitrate and nitric acid increases very greatly the ultra-violet absorption. Even a drop or two of hydrochloric acid pro- duces a very marked effect upon the uranyl bands. Thea band of the pure nitric acid solution is very weak and narrow. Apparently it fades out and the a band of the chloride becomes stronger and stronger. The 0 nitrate band does not shift when hydrochloric acid is added, but a chloride band farther to the red takes its place. The ¢ nitrate band is very slightly shifted. On the other hand, the 6 nitrate band is shifted very greatly towards the red by the addition of hydrochloric acid, and finally unites with the ¢ band to form a single broad band. The other nitrate bands are shifted to the red when hydrochloric acid is added. A given uranous or a given urany] salt often has different absorption bands in different solvents. In some solvents the differences are not great. For instance, the uranyl bands of uranous chloride in glycerol and in methyl and ethyl alcohols are very similar. In other cases the absorption spectra are very dissimilar. Uranous chloride in acetone has the b band 4 7 . _ Spectra of certain Uranous and Uranyl Compounds. 571 at 4920, ¢ at 4750, and d at 4590. On adding water the “acetone” bands gradually decrease in intensity, but are not shifted, and the “water” bands b (X4980), ¢ (~ 4700), and d (X 4570) increase in intensity; both bands existing at the same time. . Uranous chloride in methyl alcohol has the distinct “alcohol”? bands ¢ (A 4770 and 4670) and d (A 4600). A very typical example is that of the red “water” bands of uranous chloride, bromide, and sulphate. There exists a rather wide “‘ water,”’ band at > 6500, and another band at % 6750 about 30 A.U. wide. As the amount of uranous chloride is increased these bands widen into one broad band. Glycerol sclutions of uranous salts give a broad band in this region, but in no case does this ever break up into two bands, except when free hydrochloric acid is present. In marked contrast with these solutions those in acetone, methyl and ethyl alcohols show transmission throughout this region. In mixtures of these solvents with water the ‘“ water” bands gradually decrease in intensity as the amount of water decreases, but the wave-lengths are unchanged. When hydrochloric acid is added to an aqueous solution of uranous chloride the > 6500 band becomes much narrower and the dX 6750 band widens. There are few if any salts that show such characteristic ‘‘solvent”? bands better than uranous salts. The explanation of “solvent”? bands seems to lie in the fact that the salt forms a more or less definite compound with the solvent. The persistence of the “solvent” bands may be taken as a rough measure of the stability of these compounds, The explanation of the gradual shift of the bands is much more difficult. Larmor™ suggests that loose aggregates, on account of their mutual influence, would vibrate in longer periods. In the case of uranyl bands, however, the different bands are shifted by very ditterent amounts, and apparently the different bands are occasionally shifted in opposite direc- tions. The study of the changes of the uranyl and uranous bands during chemical changes should throw considerable light upon the nature of the latter. A very good example is the addition of nitric acid to uranous acetate in water. The uranous acetate is gradually changed to uranyl uitrate. * Astrophys. Journ, xxyi. p. 120 (1907). 572 Absorption Spectra of Uranous and Uranyl Compounds. During this change the uranous acetate bands gradually dis- appear, while very y marked changes occur in the uranyl! bands. The resulting spectrum is not ‘the same as that of uranyl nitrate dissolved in water containing free nitric acid, but is probably due to a different compound. The shifting bands may correspond to the formation of a number of intermediate compounds. It seems highly probable that the uranyl and uranous: bands are due to systems of absorbers in the uranium atom, and that the effect of change of salt or solvent is to change the electromagnetic field in which these absorbers vibrate. Kach frequency should thus be traced while external changes about the uranium atom are taking place, and in many cases this has been done. For instance, the vibrator producing the d band consists of a system which can absorb three dif- ferent frequencies, as is shown in the case of uranous chloride in acetone and hydrochloric acid. Thee band in this case is doubled. Uranyl nitrate in nitric acid, to which hydro- chloric acid is added, gives the « and d bands asa ‘single band. It would be very interesting to know how this e-d band breaks up at low temperature. To summarize the results thus far obtained with uranous and uranyl salts, we find that when the solvent or the salt is changed, or acids or foreign salts added, the absorption spectra are greatly modified. This modification is of two kinds: The existence of characteristic bands for each state of the uranium salt; or a gradual shift of the bands. When one salt of uranium is transformed into another salt there is usually a gradual shifting of the one series of bands into the series corresponding to the second salt, and in this way a number of relationships hetween the uranous and the uranyl bands have been discovered. Uranous salts in different solvents show very different absorption bands, and the solvents in question either have practically no absorption or absorb only in the ultra-violet. We regard this as evidence for the solvate theory proposed in this laboratory, which says that there is some kind of union between the solvent and the dissolved substance. It is difficult to see how the different solvents could affect the absorbers unless they combine in some way with them. Physical-Chemical Laboratory, Johns Hopkins University. January 1910. LX. Note on the Energy of a “Double-layer © Condenser of . Electronic. Origin. By Wiuttam C. McC, Lewis, M.A., 2.30.” NE of the most characteristic properties of colloidal solutions and emulsions is that the colloid particles and emulsion particles show a movement in one direction or the other when the solution is placed in an electric field. These particles are therefore electrically charged. To the class carrying negative charges in water belong colloidal metat!s prepared electrically (yold, platinum, silver, mercury), some sulphides (suchasarsenic trisulpbide and: antimony trisulphide. i suspensions (such as selenium and sulphur) and emu'sions such as rubber-latex and hydrocarbon oil particles. To the class carrying positive charges in water belong colloidal hydroxides in general. The s ls of common metals such as Jead, bismuth, and iron, are also charged positively, but it is practically certain that in these cases we are really dealing with the hydroxides. From a determination of the velocity of migration of these substances under a known potential gre adient one may calcu- late the p.p. which exists between each particle and the liquid medium surrounding it from considerations first put forward by Lamb, and founded on the Helmholtz double-layer theory Lamb’s expression as modified and employed by Burton f (to whom we owe a great deal of our knowledge of the subject) is where V= the p.p. between the particle and the medium. K = the dielectric constant of the medium. n= the viscosity of the medium. = the migration velocity of the particles in em./see. under a potential gradient of X units per cm. The following table summarizes the results obtained with some typical colloids, emulsions, and suspensions. ‘The par- ticles in such cases are usually designated by the term ‘the disperse phase.” As regards the size of these particles we tind great variation even in the same prepiration. Emulsion particles (oil, for example, in suspension in water) have an average diameter of 4x10-° cm. Burton obtained values * Communicated by the Author. + Burton, Phil. Mag. xi. p. 425 (1906). 574 Dr. W. C. McC. Lewis on the Energy of a of the same order for his colloidal metal preparations, though these more frequently belong to the order 10~® to 10-7 em. Liquid Medium— Distilled Water. Velocity in cm./sec. | P.D. of the Disperse phase. under a gradient | particles Observed by 1 volt per cm. in volts. Arsenic sulphide ...... 22x io” —0°'032 |Linder & Picton*, Quartz (suspended) ...| 30x10°° —0-044 |Whitney & Blaket Hydrocarbon oil ......| 48x107° —0:05 — |Lewist. Prussian blue .......:. 40x 10” —0:058 |Whitney & Blake. ail See ae 40x10 —0 058 , Platinum ...sessesees. 30x10" —0-044 i olathe aint as 21-6x10 ° —0:032 |Burton§. Platina fed 20°3x107” ~0030 | ,, Siiver ee ee 23610” ~0-034 i json PRC ae ee. 30x 10" 40-044 |Whitney & Blake. Bismuth (hydroxide) .. 11x10 +0016 /|Burton. Rera(hedreside) ae 12x10” 40018 ‘ Iron (hydroxide) ...... 19x10” +.0:028 . * Linder & Picton, Journ. of the Chem. Soe. Ixxi. p. 568 (1897). +t Whitney & Blake, Journ. Amer. Chem. Soc. xxvi. p. 1839 (1904). t Lewis, Zeitschrift fiir Kolloide, iv. p. 211 (1909). § Burton, Phil. Mag. xi. p. 425; xii. p. 472 (1906). The most important fact about the above values for the p.p. of the particle is that they are all of the same order of magnitude. ‘There appears to be an entire absence of any specitic effect due to the chemical composition of the disperse phase. This is brought out more clearly when instead of altering the nature of the disperse phase we alter the medium, as was done by Burton (I. ¢.). Medium. iis — KF ‘ Fe Ethyl Ethyl Methyl Disperse phase. Water. matonate. alcohol. alcohol. ea BINARY ok evs otis sues ve —-0:03 volt —0 054 EtSiCE nqse seen seoeseneneor —0:032 —0:033 SHER, Ure Peace sds sajnbes ans —0:034 —0 040 Lead (hydroxide) ...... -OOUS eases +0023 +0:044 | Bismuth (hydroxide)... +0°016 \.. Soult became cnn nT cata areaes +0°022 I think one is Justified, in view of the above figures, in saying that the p.D. of particles in colloidal solution is in- dependent of the chemical nature of both the disperse phase and medium. *Double-Layer”’ Condenser of Electronic Origin. 575 Now as to the mechanism whereby the P.D. is set up, we find two views held—first, that it is an electrolytic ion effect; secondly, that it is electronic. For the first view it is evident that we must assume different electrolytic ions coming into play as we alter the medium. The chief objection is that colloidal solutions can be prepared from substances with which it is extremely unlikely to suppose electrolytic ions associated—such, for example, as colloidal platinum in chlo- roform. The electron view, on the other hand, is applicable in all cases, and the absence of specific effect noted above is very strong evidence in its favour. It must be confessed that it is by no means easy to form a mental picture of the mechanism whereby a double layer can be maintained by the emission of electrons. Perhaps the equilibrium is a dynamic one, the potential finally reached being determined by equality between the number of electrons emitted by the particle and those returning to it. Whatever the mechanism may be it may be of interest to point out a relationship which appears to hold between the electrostatic energy of the double layer (regarded as equiva- lent to a small condenser, the plates of which are molecular distance apart) and the kinetic energy of the electrons. According to the prevailing theory of metallic conduction metals contain a certain number of free electrons in tempe- rature equilibrium with their surroundings. At 0° C. the average velocity of an electron is 10’ em. per second (‘ Cor- puscular Theory of Matter,’ p. 52). Now suppose such an electron escapes from a colloidal particle and is brought to rest after traversing the distance corresponding to the thick- ness of the double layer. The decrease in kinetic energy is 3mv*. The increase in electrostatic energy is }e, where e is the charge on an electron, and if we equate these expressions and solve for 7, mv PE M= spon Of the hydrogen atom=5 x 10-* gram, e= 10-* electromagnetic unit =3 x 10-?° electrostatic unit, and therefore a= 1'6 x 10-° electrostatic unit = (0048 volt, which is in fair agreement with the observed Y.v. of colloidal metals. Chemical Laboratory, University College, Gower Street, London, W.C. yaa | LXI. The Solution of the Integral Equation connecting the Velocity of Propagation of an Earthquake- Wave in the Interior of the Earth with the Times which the Disturbance takes to travel to the different Stations on the Earth's Surface. By H. BarEman™*. 1. FFNHE records of instruments at different places on the Barth’s surface indicate that the seismic disturb- ances produced by an earthquake are of several distinct kinds}. The disturbances that are first recorded are culled by Milne the “ preliminary tremors.” They are supposed to be propagated through the body of the Earth with a speed depending on the elastic properties of the material through which they pass. In reality there are two speeds, for the seismic waves in which the vibrations take place in a longi- tudinal direction generally travel faster than those in which the vibrations are transverse to the direction of motion. The speed of propagation in either case is given by the formula | where D is the density of the material and E the appropriate elastic modulus. It has been suggested that this formula may provide us with a means of estimating the elastic constants of the materials of which the Earth is composed. The problem to be solved is the inverse one of calculating the speed of propa- gation at different points inside the Harth from the observed times of transit to different places on the Harth’s surface. Radzkit and V. Kévesligethy§ treated this problem mathematically by regarding the rays along which the energy is propagated as brachistochronic paths; their ma- thematical assumptions, however, were made so as to get a soluble case, and had no dynamical harmony with known seismological facts. Recently Prof. ©. G. Knott || has discovered a law of velocity which gives results agreeing closely with the obser- vations, while Wiechert and Zoeppritz have worked out * Communicated by the Author. + A good account of the subject is given in Prof. Knott's book, ‘The Physics of Karthquake Phenomena,’ Clarendon Press (1908). { Beitrdge zu Geophysik, 111. (1908). § Mathem. u. Naturw. Berichte aus Ungarn, xiii. (1897); xxiii. (1905). | Proceedings of the Royal Society of Edinburgh, vol. xxviii. Part 3 1907). " Géttinger Nachrichten (1907), Ieft 4. Velocity of Propagation of an Euarthquake- Wave. 577 an interesting case in which it is assumed that the rays are circles. For theoretical purposes it is desirable to have a general method of solving the inverse problem, and so an attempt has been made in this paper to study the properties of the integral equation to which the problem is reduced when it is assumed that the earth may be divided into a series of concentric spherical shells within each of which the material is uniform. In this ideal case the elastic con- stants, and consequently the velocities of the seismic waves, are functions only of the distance from the centre of the earth. The integral equation may be easily identified with one which was partially solved by Abel* in 1823, and later by Liouville t in 1832. By a few transtormations we may obtain a formula con- necting the speed of propagation with the times to different points on the Earth’s surface. A simple case has been worked out for purposes of illustration. The problem has been solved on the assumption that the seat of the disturb- ance can be regarded as a point on the surface of the Earth. This condition is only approximately satisfied in actual cases, for the hypocentre may be several miles below the surface, and the disturbances may radiate trom a crack instead of a point. This may be allowed for roughly by assigning the time T=0 to the points of a small circle surrounding the epicentre. 1 wish to express my gratitude to Dr. Schuster for sug- gesting the investigation and for the interest which he has taken in it. 2. In order to simplify the mathematical analysis it is convenient to treat the Harth as a sphere of radius R, and to suppose that the elastic constants of the substance of which it is composed are the same at all points of a concentric sphere. ‘The velocity of propagation of a disturbance at any point is then a function of the distance of the point from tle centre of the Harth. Let v be the speed at distance r from the centre,and T the time the disturbance has taken to travel from the seat of the earthquake to the point in question. Then by Hamilton’s * Collected Works (Sylow and Lie edition), vol.i.p 11. | + Liouville’s Journal, vol. iv. (1889) p. 238. Journal de l’Ecole Poly- technique, cahier 21 (1832) p. 1. Phil. Mag. 8. 6. Vol. 19. No. 112. April 1910. 2 P 078 Mr. H. Bateman on the Velocity of an general method * of treating brachistochronic problems we have (V+ a écvit- ow he where @ is the angle which the radius r makes with the radius through the origin of the disturbance. Putting r=7h, T=SR, the equation becomes We get a solution of this by putting oe (2) =a, a quantity independent of # and @. This gives 12 Vee kt ROE The negative sign must be chosen for the first half of the path because a begins to decrease as S increases. This holds until a point on the path is reached at which the radius is cut at right angles. This radius is then a line of symmetry of the path and the point in which it meets the path is a vertex of the path. It is clear that after passing the vertex « increases with 8, and so the positive sign must be attributed to the square root. - Substituting the values of a and — in the equation fol fol) dss ry] — dO + = =— da, we get Q : 1 ee dS=ad0+ -4/% ~a?. da When @=0, x=1; hence for the first half of the path we have S=ai— \'SV3-4. —e S The equation of the path is obtained by Sitbs oe equal * Transactions of the Royal Irish Academy, 1833. I am following here the analysis given in Prof. Knott’s paper. Earthquake- Wave in the Interior of the Earth. 579 to an arbitrary constant. Thus the equation of the first half of the path is Since 6=0 when «=1 the constant is zero *. Let y be the value of @ for the line of symmetry and z the corresponding value of x, then (z, y) are the polar coordinates of the vertex and 2y is the angle which the radius through the point of emergence makes with the radius through the origin. These quantities are connected by the equation 1 dz + —— ee ate Os oa: Ate o x \ =a (9) a mG a ee v Let wf be the angle at which the radius cuts the path at any point, then or PEE oe Thus y= 4 when a=av; the value of zis therefore given by the equation glad a be em ws AY If eis the angle which the ray makes with the surface when it emerges and U the value of v at the surface of the sphere, 7. e. when «=1, we have CW: nas ae rea dat: Sele dat C The value of a for any particular ray may be determined from the equation _ oS 08” where 8’ and @ now refer to the point on the surface at which the ray emerges. Now 8’ may be supposed to have been determined as a function of 0’ by observation: hence the equations os! T Or * We assume here that the disturbances radiate from the hypocentre, 7. €. the point 6=0, r=1. i's Pi aie a (9) a 0 = 2, 580 Mr. H. Bateman on the Velocity of an may be used to express vy as a known function of a. Putting xy=f(a), B=a, we have the integral equation fe) = Seer (10) 0/5 Ta from which to determine v(w). In this equation the value of f(a) is given from a=0 to e=8; and since y vanishes for the infinitesimal path which is a tangent to the sphere at the origin, 7. e. when e=0, =. =, we have /(8)=0. This equation may be simplified by putting we) 73 it then takes the form This may be identified with a well-known equation solved by Abel by putting 1 i if Cio n=, ca Ss t a Slog e)= 2d), Ka) =F (9) We then get “s h(ty)dt EG )e= ~~, ..,} (s) -a /s—t and the condition that F(s)=0 when s=a. Now a necessary and sufficient condition that this equation should have a solution ¢(¢) which is continuous in the interyal (a 7 = O yon) therefore ldx__ 2 @ (8 fla)da wv dn 7 a V a?—n This was the method by which the result was first obtained. Earthquake- Wave in the Interior of the Earth. 5&3 The value of S may be calculated most conveniently by eliminating @ from the equations | $$ $$ u , AD S=a0— | - = —2’, d t B “igh O+a a Pion Ton: —(), b 2 wh on / ana 1) 1 gu vd 1 UV av? This gives The time of describing the whole of a path is just twice the time up to the vertex, therefore si=2( & da = wy UN eae where z is the value of « for which a?=av?(x). If 8 is regarded as a known function of 6 the solution of this integral equation may be made to depend on that of the preceding equation. The number of paths which reach a given point on the surface may be determined geometrically by drawing the curve which is the locus of the vertices of the paths*. Ita radius making an angle 46! with the radius to the epicentre is cut by this curve in n points, n distinct rays of the type under consideration will reach a point on the surface specitied by the angle @. Since the equation ! s 7, (log wr) dy flalme (Hoe vV/ 7? — a" d is linear as regards f («) and —— (log «), it is clear that if the dy i oan values of a (log x) corresponding to particular forms of f(a) are known, the value corresponding to a function f (2) which is a linear combination of these particular forms may be written down at once. In the following table is given the solution of the integral equation in a number of cases. * This method waa used by P. G. Tait in the theory of mirage to determine the number of visible images. Transactions of se Royal Society of Edinburgh, vol. xxx. (1881) ; Scientitic Papers, 1. p. 427. 584 Mr. H. Bateman on the Velocity of an } To apply these results to a practical case we must endeavour to obtain an approximate representation of the results of observation by means of some linear combination of these : particular forms of 7 (2). The constant 8 which determines x the velocity at the surface is found from the value of a i which makes 7 («) vanish. . Table of Solutions. cos! < 1 1 8 7) B ——— 3 atan— (up “B?—a") mu’ [ 1412622") | oh V/ a? —a? n 2 | nee b 7?—a? V +R a Vite ep ea VP apa? 1 _-_- ‘aa pa Ja? =) / 1-+-p?a? V (a2 — 1)(1+p2a*) n(d + p2n?) Lh V a2 1 A 1 pte — pata 1 | Sap ua 5a2—1 nw 1+44a2p2n? p(a? — 22), a ae p(B—a@) 7 Pie The solution given by Prof. Knott is fourth on the list, the fifth solution corresponds with the law of velocity con- sidered by Wiechert and Zeeppritz. The other solutions by themselves give irrelevant laws of velocity, but by combining them with the others we may obtain possible laws of variation of the velocity. If there is any difficulty in expressing the results of obser- vation by means of the function given in the table, it may be convenient to assume different expressions for § for different ranges of values of @: for instance, we may assume S=a+b0+cé? for 0< 0 800° £6°48 009-1 000-T = OOO" sit te dinteeaiall me BE Se eae “OUILT, sauily, Jo saivnbg s9oUe4SI(T ee eee “SOUL, a —- —s __s srs e* gungden oe Og S588 a But Ws 6 ES s° 4h, 2 oe 9 Sag Cees SNUB.) uangey dojidn ¢ yey | STLUd A SAN00.19 JX 604 A new Binary Progression of the Planetary Distances. of the distance of Neptune is the outermost position of the planet in relation to the other members of the system, with the consequent conjoint attractions of all the planets, acting through every part of their orbits, to contract continuously and permanently his radius vector to the amount shown in the observations. The large amount of this contraction is strong presumptive evidence against the existence of a planetary body beyond the orbit of Neptune. 25. A further consequence of the outermost position of Neptune is the small amount of the eccentricity of his orbit, 0009, or nearly six times less than the eccentricities of Uranus, Saturn, and Jupiter, which, excepting Venus, 0-007, is the nearest approach to a circular orbit of any member of the system. 26. It is not a little remarkable that the inevitable effect of the outermost position of a planet, to contract continuously its radius vector, has never presented itself to Lagrange, Laplace, and other writers on celestial mechanics, who have elaborated the doctrine of the absolute stability of the solar system. The effect of the conjoint attractions of all the planets upon Neptune is clearly demonstrated by the diagram, whereon, from the exigencies of space, the intra-Jovian planets are not included. 27. Reverting to the small amount of the difference between the sums of the binary progression in column 4, Table I., and the observation distances in column 5, it will be seen that the latter is a plus quantity, as 104°162— 103°25=0°912. Now as the amount of the contraction of the radius vector of Neptune is 19°410 Mercurian units (696,000,000 miles), as shown in column 6, the plus difference, 0-912, between the two sums of the binary progression and the observation distances may well be accounted for as being the amount of the reciprocal attractions of all the planets upon Neptune in accordance with Newton’s third law of motion, acting through periods of time too immense for calculation in the present state of our knowledge. 28. Assuming the future contraction of the orbit of Neptune to be continuous, his radius vector will ultimately coincide with that of Uranus, when the two bodies would either revolve together about their common centre of gravity in the same orbit, or coalesce to form a single self-luminous planet, when the same operation would be repeated in succession with other members of the system. 29. It is further postulated that all the planets would ultimately coalesce to form one or more self-luminous bodies revolving round the Sun, as one of the binary or ternary ee eee aa Absorption of B Rays prom Radium by Solutions. 605 systems of stars, of which upwards of ten thousand have been discovered and catalogued during the last century. 30. The probability that the ultimate transtormation of the solar system will be brought about by the means, and in the order herein set forth, derives further support from the fact that one of the stars of long recognised binary systems is itself a close double star, revolving about its common centre of gravity, as instanced in w Herculis and y Andromede. 31. Recapitulation—(1) That the exact binary progression of the planetary distances is the primordial and fundamental law (shining forth alone in the formless void) from ‘which the principal elements of the planetary orbits have been derived ; (2) that the apparently irregular differences from the law are the direct consequence of the mutual attractions of the planetary bodies amongst themselves, but without affecting the validity of Kepler’s laws, as the distances and periodic times are necessarily correlated; (3) that as planetary systems have been evolved in regular order from a nebular substance, -o the transformation of these systems will proceed in like order to form the numerous binary and other revolving systems observed in the immensity of the stellar universe. LXIV. Absorption of B Rays from Radium by Solutions and Liquids. By W. A. Boropowsky, M.A., Privat-docent of Chemistry, Tourjew (Dorpat) University, Russia*.- Introduction. LARGE amount of work has been done by various observers in determining the absorption of @ rays from radioactive matter by metals and solids, but little attention hus been given to the allied problem of absorption of the rays by solutions and liquids. N. Campbell f examined the absorption of the 8 rays from uranium by liquids and solutions, but his method of obtaining uniform layers of liquid by means of filter paper is open to objection. The results obtained by him were somewhat irregular, but he concluded that “it is possible that the value are (where > is the coefficient of absorption and p the density of the solution) for a solution should be greater than either of the values for the solvent or the solute, or should be less than either of these values.” While the present work was in progress a paper was * Communicated by Professor E. Rutherford, F.R.S. + Phil. Mag. 1909, vol. xvii. p. 180. 606 Mr. W. A. Borodowsky on Absorption of — published by 8.J.Allen*. This deals chiefly with the secondary radiation emitted by solids, solutions, and pure liquids when 8 rays from radium fall on them, but also gives some results of the absorption of B rays by liquids. He obtained some remarkable results with organic liquids, and attempts to explain them on the view that the arrangement of the mole- cules in the substance has some effect on the secondary radiation and absorption. J. A. Crowther+, on the other hand, found that for solid bodies the absorption of @ rays from uranium for any particular clement is quite independent of its state of chemical coim- bination. In the present paper the following questions are con- sidered :— 1. Does the absorption of 8 rays depend upon the physical or chemical state of the solution ? 2. Does the absorption in liquids take place according to the same laws as in solids ? 3. Is the absorption of 8 rays by complex substances additive ? Haperimental Arrangement. *52 mgr. of radium bromide in radioactive equilibrium was used as a constant source of @ rays. It was contained in a copper capsule with an air-tight mica window. All the @ rays Pisods were stopped by the mica, but the 8 and y rays readily passed through into a small 8 ray electroscope, which was placed above the radium at a distance of about one centi- metre from it. In some experiments the solutions and liquids examined were contained in a glass cell and in others in a glass wedge (figs. 1 and 2). + Phys. Rev. 1909, vol. xxix. p. 177. + Phil. Mag. 1906, vol. xii. p. 379. “s Rg wa B Rays from Radium by Solutions and Liquids. 607 A round ring (fig. 1) with well polished edges was fastened by means of Canada balsam or by means of water-glass to a very thin cover-glass, which formed the bottom of the cell. The area of the cells used was about 4 cm.? and the depth between 2and 3mm. The area of the bottom of each cell was accurately determined by measuring the diameter of the ring, and the depth of the cell by weighing the cell filled with mercury. The cell was completely filled with a liquid and a thin flat cover-glass placed on the top, the exces: of the liquid being withdrawn by means of filter paper. The mass of liquid was determined by weighing the cell full and empty. The absorption of @ rays by different powders was also examined in the glass cells. The powdered salt was put in a sifter and dropped into a glass cell through fine muslix. The thickness of the powder was determined from a knowledge of the weight and density of the salt. The glass wedge was constructed as follows (fig. 2). Strips of thin glass (140 x 40x°2 mm.) were fastened along their edges by suitable cement to accurately planed steel plates (150 x 5x 3 mm.) which were stiff enough to prevent any bending when the wedge was filled with the liquid. Liquids with high surface tension would stay in the wedge of their own accord, but those of low surface tension were kept in by means of glass sides. At one end of the wedge the glass plates (strips) were in contact, at the other end they were kept apart by a piece of glass 2-3 mm. thick, but in each case the thickness of the glass was determined exactly. The parts of the wedge were held together by means of steel clips at the ends of the steel plates. During the experiments the gliss wedge filled with solution was supported by a piece of wood with a scale on either side so arranged that the zero mark corresponded with the centre of the source of 8 rays; thus the position of the wedge could be read off directly from the scale, and the thickness of the layer of the liquids over the centre of the source of @ rays could easily be calculated. The thickness of the liquid in the wedge obviously increased in direct proportion with the distance from the end at which the plates are in contact. The small 6 ray electroscope used had a base 3x3 em. and a height 4em. It was provided with sulphur insulation and with a leaf of thin aluminium foil in the usual manner. Method of Procedure. Preliminary experiments were made to find how far the y. ray effect would interfere with the main experiments. This question was settled by studying the absorption of the 608 Mr. W. A. Borodowsky on Absorption of 8 and y rays by different metals. The ionization current in the electroscope was measured when screens of different thickness were inserted between the radium and electroscope. Thin screens of aluminium, copper, tin, and lead were used. When the logarithms of the activity as ordinate were plotted against the thickness of screens as abscissee, it was observed that the curve after a rapid initial fall became nearly a straight line parallel to the axis of abscissa. The point where this line, when continued backward, cuts the axis of ordinates was taken as the initial value of the y ray activity. This value was found to be nearly the same for the curves obtained with the different metals. The mean initial activity of the y rays alone found from these experiments was 7 per cent. of the total initial activiry. When the empty glass cell or wedge was placed in its position between the 8 ray electroscope and the source of radiation the emergent rays consisted of the same quantity of y rays, but of a smaller quantity of @ rays. Consequently the relative activity of the y rays was increased. The thin plates, of which the glass cells or the glass wedge were made, absorbed a considerable fraction of the 8 par- ticles, and the value of the y ray activity was about 12 per cent. of the total activity for the glass cells and about 15 per cent. for the glass wedge. The constancy of the y rays over the range of experiments was clearly shown by allowing the radiation to pass through a sheet of lead 1:65 mm. thick covered by a glass wedge with varying thicknesses of concentrated sulphuric acid. It was found that the absorption of the y rays, even in the thickest layer of acid, was too small to detect. The activity of the radium measured by the ionization in the electroscope remained nearly constant over the whole time of the experiments. There was a small variation from day to day which was due to variations of barometric pressure and temperature. All the readings of the aluminium leaf were taken between the same points of the scale. The limits of error of several readings did not exceed one per cent. The correction for the small natural leak lay between ‘1 per cent. for the highest observed activities and ‘6 per cent. for the lowest. The Relation between Absorption and the Chemical and Physical State of Bodies. Two cells with thin glass bottoms were filled with different liquids. One was placed in inverted position over the other in such a manner that their edges exactly coincided, and the B Rays from Radium by Solutions and Liquids. 609 liquids in the two cells were separated by a sheet of mica ‘01 mm. thick. The amount of liquid in each cell was determined by weighing. The cells arranged as described above were placed under the electroscope over the radium, and the activity of the beam of £ particles passing through both liquids was measured by the ionization in the electroscope. After ten concordant observations had been taken, the thin sheet of mica was carefully removed without changing the position of the cells, and laid on the top of the upper cell. Then the liquids were mixed and the activity of the B rays passing through the mixture was again measured. In no case was air allowed to enter the cells or any liquid allowed to escape. The results of several experiments of this kind are given below in Table A. The activity (1) of the pencil of B and y rays passing through both empty glass cells with mica between them was taken as 100; the y ray activity was 12. In Table A given below, the upper line gives the weight of the liquid in the upper cell and the lower the weight of the liquid in the lower cell. On the right-hand side are given the physical or chemical results of the mixture. The percentage activity observed in each case after the 8 rays had passed through the solution is represented by I. TABLE A. Not mixed. Mixed. The mixture contained : 92 gr. H,SO, (12°4 per cent.) I. 151 gr. H,O = 4°5 per cent. H,SO4 I = 20°67 per cent. I = 20°51 per cent. 1030 gr. H,SO, (20°4 p.c.) Lis 1-050 gr. H,O 10°1 p.c. HySOx I = 20114 p.c, I = 20°09 p.c. 1:160 gr. H,SO, (39°2 p.c.) én. HII. Sea btster. HO. 20°6 p.c. H,S0, I = 19:09 pic. L = 1913 p.c. 1:150 gr. sugar solution 63:1 p.c. _ ae i ; IV. ; 1-040 gr. Reh hi GGA 33°2 p.c. sugar solution. I = 19°25 p.e. I = 19°11 p.c; Difference =—-75 p.c. 1:00 gr. CaCl, solution 3 ; E Wi. 1-14 gr. H,SO, == precip. CaSO4+2H Cl+aq. I = 19°42 p.c. L = 19°48 p.c. Ay lut. VI, Equiv.—_ as ae = precip. 2AgCl+ prec. BaSO,+aq. f= 21°43 p.ec. f= 2P47 p.c. Phil. Mag. 8. 6. Vol. 19. No. 112. April 1910. 2h 610 Mr. W. A. Borodowsky on Absorption of Experiments I. to III. give the effect of dilution of an electrolytic solution, and IV. of a non-electrolytic solution. It will be seen that there is practically no change in the activity after mixing. With a viscous solution like sugar, a small amount of liquid sticks to the mica and is removed with it. This is spread over a large area, and so the effective mass in producing absorption is a little less. This accounts for the slight diminution in activity observed in case IV. Experiment V. gives the effect when one substance is precipitated from the solution on mixing, and VI. the effect when two substances are precipitated. In the latter case equivalent amounts of solution were taken, so that after mixture the silver and barium were both completely preci- pitated. In all these experiments the variation of acta lies within the limits of error of observation. If the cells are not full to the top and covered with mica, the @ ray activity before and after mixing of solutions is not the same, e.g. for a mixture of H,SO, and H,O a 12 per cent. change in the activity was observed, while for a mixture of sugar and H,O the change was about 21 per cent. This is due to the fact that the surface tension alters on mixing; the meniscus becomes flatter and the effective thickness of matter in the path of the @ particles becomes greater. The results of the above experiments show conclusively that the absorption of @ rays is independent of the physical or chemical state of the solution. Method of comparing Absorption in terms of Aluminium. Since the 8 particles from radium bromide in a state of radioactive equilibrium are not absorbed according to an exponentiai law, it is advisable to express the results of absorption by different liquids in terms of the absorption by a definite metal, for example, by aluminium. In order to compare the absorption of the 8 rays by liquids with that by aluminium, thin sheets of aluminium (‘078 mm. thick) were put between the strips of the glass wedge and the variation of activity with increase of thickness was plotted. The initial activity, taken as 100 per cent., repre- sents, as in the case of solutions, the effect of ihe, 8 and y rays passing through the empty wedge. ‘The decrease of the activity with increase of thickness of the layer of the solution was plotted ona large scale on the same paper on which a similar curve for aluminium was drawn. Hxamples of typical curves obtained in this way are given in figs. 3 and 4. B Rays from Radium by Solutions and Liquids. 611 Fig. 3 gives the variations of activity for solutions of CaCl, of different thickness and concentrations. Curves 3, 3, re and 11 give the activity observed for concentration of calcium Fig. 3. 100 —_ SIS 1 60 oi : Hed : 3 40 4 ; Co ck, 4 8 a : 20 thickness O40cem 7 ” 72" Té - 20- chloride corresponding to 3,5, 7 and 11°75 gram-molecules of 4CaCl, per litre of water respectively. For comparison, the curves obtained for pure water and aluminium respec- tively are added. Fig. 4 gives the variation of activity for solutions of cane- sugar of different thickness and concentration. Curves ], 3, Fig. 4. Cx Hs 0, /cane Sugar) He 3 a] CizHeed, > ' thickness OY crn ‘O8» S22 bem 90. and 5 give the activity observed for concentrations of cane- sugar corresponding to 1, 3, and 5 gram-molecules of sugar per litre of water. The thickness of aluminium which cuts down the activity to the same extent as a definite depth of solution can be easily read off from the curved paper. Rs = al — Pave ee 612 Mr. W. A. Borodowsky on Absorption of A new series of curves (figs. 3 a and 4a), corresponding to figs. 3 and 4, were then drawn, in which the depths of the solution were taken as abscisse, and the corresponding thicknesses of aluminium to give equal absorption were taken Fig. 3a. Ml fare) Sw x ~~» & eo CeHee 0, a SESacilicnucn link 6 thickness O4em "08. /Q- /6- ‘90. as ordinates. Supposing in the case of aluminium that the abscissee correspond to the thickness of aluminium, the curve for aluminium is obviously a straight line bisecting the angle between the axes. For each concentration the curve obtained is a straight line passing very nearly through the origin. This shows that, for a given concentration, the absorption in terms of aluminium is directly proportional to the depth of the solution. A slight divergence was observed for thin layers of the solutions. This results from the bulging of the glass strips forming the wedge in consequence of the forces of surface tension. ; 8B Rays from Radium by Solutions and Liquids. 613 These results are clearly seen from Tables I. and II. Table III. gives similar results for different thicknesses of sodium chloride in the form of powder. Corresponding to each concentration, the first column gives the thickness, ¢, of the layer in cm., the second column the observed absorption, a, in terms of ratio of a/t. em. of aluminium, and the third column the TABLE I. CaCl, Solution. Absorption in terms of Aluminium (cm.) for various thicknesses and concentrations. Concentration... 3 gram-mol. 5 g.-mol. 7 g.-mol, 11°75 g.-mol. Thick f ; om ne “a aa a solution " Beet alt a. | aft. a. alt. ges alt. in cm. G40 ‘ aluminium. 12 cm. 0490 em. of Al.) 41 || -0517 | -43 | 0552 | -46 | 0635 | 53 14 0569 +41 ||-0604 | -43 || 0658 | -47 | -0740 | -53 ‘16 0648 40 ||-0686 |-43 | 0741-46 | 0845 | 53 18 ‘0726 '-40 || 0770 |-43 || 0824 |-46 | -0950 | °53 20 ‘0810 40 || 0855 |-43 | 0919 -46 | “1055 | 53 21 ‘0854 41 ||-0905 |-43 | 0970 | -46 | +1106 | 53 Mean ... -405 43 | 46 | 53 f TABLE II. C2H.0;; (Cane-Sugar) Solution. Concentration ... 1 Mol. 3 Mol. 5 Mol. ts a. alt. a. aft. | a. alt. ‘10cm. | -0888cm.of Al.) -39 ‘0427 43 0451 “45 12 ‘0467 39 0516 43 || -0546 ‘46 14 0540 -39 0604 43 0643 -46 16 0616 39 0688 43 ‘0732 ‘46 18 ‘0688 ‘38 0775 43 ‘0822 46 20 ‘0765 38 0859 43 ‘0918 ‘46 ‘21 -0806 ‘38 0906 43 ‘0970 ‘46 39 43 46 614 Mr. W. A. Borodowsky on Absorption of TaBLeE III. NaCl (powder). a. alt. °O1 cm. 0075 cm. of Al. af) 02 0152 “76 03 0229 16 04 0308 ait 05 0384 are 06 0455 “76 ‘OT 0529 “76 08 "0610 ae 09 0688 ‘76 10 ‘O765 sid “hi 0834 “76 "12 "0895 “7D Mean ... ‘76 These tables show clearly that the absorption is directly proportional to the thickness of the matter traversed. Since the cross-section of the @B ray beam is the same for each layer it follows that the absorption in terms of aluminium is directly proportional to the mass of matter of one kind traversed. Relation between Absorption and Concentration. In order to compare the absorption of solutions of different concentrations we must know the density of the solution, its concentration, and the thickness of the layer exposed to the beam of @ particles. The quantity of salt in many solutions was exactly determined by chemical methods, and likewise the densities at different dilutions. The results of the above determinations agreed well with the values given in Landolt- Bornstein’s tables. Let the percentage of salt and water in a given solution be respectively M, and M,,, and let D be the density of the solution. Then in a layer of solution of thickness h the amounts of salt and water in the path of the @ rays per sq. cm. cross-section are respectively : M, M,, 100° ~D.h and T00° Wn 8 NY If A is the absorption in terms of aluminium for pure B Rays from Radium by Solutions and Liquids, 615 water for a layer of thickness h, then the absorption of the layer of water in the solution is M, a Too * Di since the absorption is proportional to the mass of matter traversed. If A is the observed absorption of a given layer of the same thickness, the absorption for the salt itself is A, x ML xD fh on or per unit mass: A x100—A, x M,, x D M,xDxh If the absorption of the 8 rays is an additive property of matter, then the above ratio must be constant for different concentrations, 2. é., Ax 100—A,, x M,,x D M,xDxh For pure substances M,,=0 and M,= 100, obviously the above equation becomes: SFR ga io | CB A ; Wem Be re) 1) If the absorption in terms of aluminium is multiplied by the density of aluminium, 2°70, then the two equations give the ratio between the mass of aluminium and the mass of any substance which absorbs the same quantity of the 8 particles. This ratio may be called the relative absorption of the substance compared with aluminium. The values of K calculated in this way from equations (1) and (2) are given in Table IV. (p. 616) for a number of solutions. The relative absorption per unit weight compared with aluminium is also given in the third column. By the method of comparison of absorption in terms of aluminium the results are quite independent of the actual coeflicient of absorption “2X” of the rays employed, 7. e. the results are independent of the penetrating power or lack of homogeneity of the rays. This conclusion was confirmed by a direct experiment. In the earlier observations the radium was covered by a sheet of aluminium 1°60 mm. thick, and the constant of absorption K for sodium chloride solutions was °35. This constant was unaltered in later experiments, when the thick sheet of aluminium was removed. 616 Mr. W. A. Borodowsky on Absorption of TaBLE LV. Densities. | Constant. Relatiy. ie absorption. D K Kx 2°70 5: Wp nme eae 2 Batty 2°70 "370 1-000 O73, 2A0 Gee ee ‘9981 354 "956 Glycerine solutions : 25 per cent. /cac cco ee 10595 33 DO 5,0) Dene ee 1°1249 33 TD. 9) 4 Gee eee 11927 35 Glycerine pure ............... 1°2604 “35 2 (OF: oe AlSieee “92 Cane-Sugar solutions: 1 Mol. ii. ds tegee eee Oe 1:1070 33 Dl ay woaeenne caer ee 1:2351 "34 Dn igg | Vnccee odeemea tes eeeeeeee 1-3095 "35 Cane-sugar (powder) ...... 1-61 "35 CE haestwas ‘92 NaCl solutions : it MOL: od enseemem seca eee ees 10384 "34 ) 2b isch elueeseaee agin eee 1:0748 ‘33 ae ata ass db aR Sls oo 1-1085 "33 Assy ieatheceaceeceneneaear mes 1:1402 33 } Do antl | waded cane cate me ceemeee 1:1704 “34 GlA Mol. .shauee acer 1:2017 35 NaCl(powder)).pc.ctesnse 217 "35 ) Bo AA ae “92 CaCl, solutions : 2 MMOL. 5.03 cxscpee eee 1:1249 “39 Di bye) seneneseeee cen eee 1:1955 38 1 Sel MALE gE 1:2627 "39 VFS. Moly vy icoa. dest eee eens 14267 “39 "oO rece see 1:04 SrCl, solutions : pA OM OM tach cotnc bee ees 10296 "44 BOB Tess feces sce, aire eens 1-0592 "44 7242 1 a a ARMS EI, 2 11614 “46 ; Gio. ih oe 1:3698 “44 | Ae poseeaee 1:18 | BaCl, solutions: : As Mol i. Aes ke See 1:039 "59 | SRO) VN deeds oery rgeikheate 10775 “61 | MN ee te aes act cat heer 1:1180 "61 > ESTEE ATES « coccce's Lee ccces 1/1750 ‘61 | 23 0 RAL a Ree oe 1:2790 58 ) BO cshlacs ep 1°62 B Rays from Radium by Solutions and Liquids. 617 The results obtained should thus hold for any source of B rays. It must not, however, be forgotten that for each experimental arrangement, a curve of absorption for alu- minium must be obtained for purposes of comparison. We see from Table IV. that there is a good agreement between the constants of pure substances and the constants deduced for the same substances when dissolved in water. Both electrolytes and non-electrolytes were used, but the _ value of the constant for each particular substance remained the same throughout, no matter whether in the form of dilute or concentrated solutions or even pure substance. These results show that the absorption of 6 rays is propor- tional to the amount of salt present, and is independent of its physical state. The absorption per unit of mass is further found to be greater for heavy atoms (Ca, Sr, Ba), than for light atoms (Na)—a result noted by previous observers. The internal constitution of the atoms therefore plays some part in the mechanism of absorption. Absorption as additive property of Matter. The absorption by carbon (graphite and charcoal), sulphur, and bromine were determined separately. In the same cell the absorptions by CS, and CBr, were also found. If the absorption does not depend on chemical change we can calculate the absorption by these compounds from the ab- sorption by their constituent elements. The absorptions per unit mass per sq. cm. of surface of graphite and charcoal were found to be identical—a result previously found by Mr. Crowther. The absorption of 1 grm. C = -2066 grm. of aluminium. is i eho T EL) By. x Ie " “ In one gr. of C8, there is *1575 gr. of ©, and the absorp- tion of this amount ='1575 x -2066=:0325 ; in one gr. of CS, there is =*8425 gr. of 8, and its absorption is =*8425 x "2598 = °2189. .. Total absorption according to additive law for one gr. of C+ S.="0325 + °2189=°2514. In a direct experiment with CS, it was found that 1°130 er. of CS, gave an absorption +2822; consequently 1 gr. CS, gives an absorption *2497. The difference between the values calculated and found, viz. °2514 and *2497 is 0°7 per cent., which is well within the limits of experimental error. For CBr, the calculated result was *2598 per gr., and that found experimentally *2668, These again are in good if uy 618 Absorption of 8B Rays from Radium by Solutions. agreement, since there are obvious difficulties in the use of volatile substances like bromine and carbon tetrabromide. These experiments as well as those with the solutions show, as far as they go, that the absorption is an additive property of matter, and that the absorption in liquids takes place according to the same law as in solids. The internal struc- ture of the atom has its influence on the absorption, but the arrangement of atoms in the molecule does not affect the absorption at all. This result is contradictory to the view expressed by Mr. Allen *. He found that the absorption of the B rays from radium by ethylene chloride is three times as great as that by ethylene bromide. I have repeated these experi- ments using the cell method. In the cel] °239 cm. high the ethylene chloride absorbs as much as *2813 grm. of alu- minium of the same cross-section. In the same cell the ethylene bromide absorbs as much as ‘5684 grm. of aluminium. Thus the absorption of 8 rays by ethylene chloride was one half as great as that by ethylene bromide—a result which fits in exactly with the additive law. Further, Mr. Allen found equal absorptions for CHCl; and CCl,, while the result found in the present experiments is less for CHCl, than for CC],. I think the difference between my values and those found by Mr. Allen result from the imperfect experimental method adopted by the latter. He did not take into account the — errors arising from the shape of the meniscus and the mass of the vapour in the upper part of the cell. We have seen earlier (p. 610) that serious errors may easily arise unless these disturbing factors are eliminated. The experiments described in this paper are free from these sources of error, since the liquids were always contained between two sheets of thin glass. Summary. 1. The activity of the @ particles after passing through layers of solutions is independent of the chemical or physical state of the solutions. 2. The method of comparing absorption in terms of alu- minium may be applied to homogeneous or non-homogeneous rays and consequently gives comparative results quite inde- pendent of the coefficient of absorption “2X.” | 3. The absorption is directly proportional to the mass of matter traversed. * Loe. cit. | ; : Anomalous Effects on First Loading a Wire. 619 4. The absorption of the salt is directly proportional to the amount of salt present in solutions. 5. Solutions absorb @ rays according to the same laws as solids. 6. The absorption by compounds follows an additive law. 7. Relative absorption depends upon the internal structure of atoms but is independent of the arrangement of the atoms in the molecules. I have much pleasure in expressing my gratitude to Prof. E. Rutherford for his permission to work in his labo- ratory and for his kind interest in this work. Manchester, January, 1910. ——- SS eS ee Ee eee SS SS ® LXV. Anomalous Effects on First Loading a Wire, and some Effects of Bending Overstrain in Soft Iron Wires. By A. I. Steven, WA., B.Sc., Assistant Lecturer and Demonstrator in Physics, University of Liverpool*. iG the recently published text-book on ‘ Properties of Matter,” by Poynting and Thomson, there is a para- graph under the title ‘“‘ Anomalous Effects on First Loading a Wire” (p. 58), in which notice is taken of the fact that the extension produced by a given load is, in general, different on the first test from that obtained in subsequent ones. It does not. seem, however, to have been noticed that the “‘anomalous”’ stress-strain diagrams obtained under these circumstances bear a close resemblance to those obtained, say, in the case of an iron bar immediately after a severe longi- tudinal overstrain. Thjs apparent defect of Elasticity has been noticed by others, and notably by Muir (Phil. Trans., A. excili. p. 1), who investigated the effect of temperature on the elastic recovery of iron. If the resemblance had been noticed, it seems probable that the determining factor in the production of such anomalies would have been discovered, for every wire after manufacture is subjected to a non-homo- geneous overstrain by being wound on a circular bobbin of dimensions comparatively small. If the wire remain on the bobbin, probably part of the internal stress disappears before the wire is tested, in the same way as iron gradually recovers its elasticity if left to itself, but when the wire is wound off the bobbin for testing, another set of stresses is superposed on the previous residual internal stress. These internal stresses * Communicated by Prof. L. R. Wilberforce. 620 Mr. A. I. Steven on Anomalous seem to be what give rise to the so-called anomalous effects,. for even if the two opposite stresses immediately follow one another, the supposition that the processes of bending and unbending should exactly neutralise each other could scarcely be justified. If they did, then there would require to be some relation between the position of the neutral axis, and the yield-points for extension and compression ; for instance, if the neutral axis were in the centre (as does not seem to be the case when there is a large overstrain) then the yield-points for compression and extension would require: to lie symmetrically about the point of zero load, and if a change took place in the position of either, then the alteration, of one would require an equal change in the other. The most suitable material for an investigation of these effects experimentally is some kind of soft metal, and in the following experiments a soft iron wire was used. It had been intended for electrical purposes, and after the insulation had been removed with sandpaper, had an average diameter of 0364 cm. The apparatus used was of quite a simple nature. The wire hung horizontally from a rigid support at one end of a table and from a spring balance at the other end. ‘The spring balance was attached to a spiral screw working horizontally in a fixed nut, so that the tension might be varied continuously. The index of the balance was sharp, and by observing its image in the well burnished scale and thus avoiding parallax, the increase in load could be measured accurately to at least 1 per cent. The extension of a length of 150 cm. was measured by means of two microscopes :—the one which measured the extension nearer the balance travelled hori- zontally and was actuated by a micrometer screw graduated to ‘005 mm.; while the other, which observed any motion of the fixed support, was provided with an eyepiece scale, and was carefully calibrated beforehand. The best index-marks were found to be two small pendulums of about 5 cm. length consisting of a single silk fibre with a little bob of soft wax, the bob being allowed to hang in a small vessel of water to damp their oscillations. The fibres were easily attached to the wire at their proper places by a thin film of melted wax, and when viewed through the microscope against a dark background were immensely superior to the scratch on the wire advocated by many writers of text-books on practical Physics. A certain amount of initial tension (corresponding to the + 1b. mark on the spring balance) was required to keep the wire straight, and this tension is the zero of tension on the diagrams illustrative of this paper. Hffects on First Loading a Wire. 621 In fig. 1 are shown the results obtained in three successive tests on a wire immediately after being unwound from a Fig. 1. fo} 2 3 4 Se ame! (1) Behaviour of wire on first test.' (2) Fe ps immediately following (1). (3) ” 9 ” ? (2). bobbin of about 8 cm. diameter. The first curve indicates the anomaly of a wide separation in the upper part from the two following ones. It does not seem to show any definite yield-point as they do, but a continuously increasing extension for the same increase of tension. A certain amount of per- manent extension remains after the load has been removed. For the sake of making the comparison of the curves easier, this permanent extension has not been shown in the diagrams. All the curves have been moved along the axis, so that the position of the zero of tension might be the same. This gradual increase of extension with the same increase of load could scarcely be due to kinks in the wire, as the process of removing the insulation was carried out with the wire in 622 Mr. A. I. Steven on Anomalous tension and any kink would be pulled out or flattened out during the operation. | | A series of experiments were now carried out to see whether the same type of curve always resulted when a wire suffered a circular deformation. A number of cylinders were turned of different diameters, and the wire, after it had shown behaviour identical with fig. 1 (2), was rolled up on a definite cylinder, being kept taut by a definite mass hanging at its lower end. The following table illustrates the consistency of behaviour of such a wire during four such tests in the same circumstances—a cylinder of 3 em. diameter and a tension of 300 grams weight. Tner PASC of Extensions in ‘01 mm. Tension. Ne 1p ITI. TV? Average. lbs. oz. || -——-———|- ——— —— = 12 34 34 35 36 Be) 1 4 63 62 62 65 63 1 12 89 87 89 92 89 2 + 121 119 122 128 122 2 12 In57( 156 162 160 159 33 ' 187 ire? 181 183 182 33 4 221 207 218 208 212 3 8 245 231 244 — 237 239 3 12 282 265 283 271 275 4 Pad 325 306 a All 323 319 4 4 425 382 398 393 399 4 8 536 549 529 500 529 Different wires, however, do not agree so well with each other, even though they are all off the same bobbin. The effect of coiling the same wire on a succession of different cylinders was also tried, and after each test for this. purpose an intermediate test was applied to see how far the wire had recovered its normal state as in fig. 1 (2). The results are exhibited in fig. 2. The diagram shows the effect of cylinders of 6, 4°5, 3, 1:5 cm. diameter on the wire respectively, the tests being made in that order. The straight line indicates the normal behaviour of the specimen, and is the average of the four tests which immediately pre- ceded the four other curves to see whether the wire had completely recovered. It is obvious that the effect increases with increased curvature, moreover the longitudinal stress. applied during the test seems to hasten the recovery of the material to a more homogeneous state. On relieving the load a permanent extension was found to have taken place. Effects on First Loading a Wire. 623 — Similar sets of curves have been obtained with copper wires. Fig. 2. q N & “ S N EXTENSION IN MM. l é 3 4 5 (1) Normal behaviour of specimen. (2) After being on a cylinder of 6 cm. diameter. (3) 39 re +P] 4 5 7 99 (4) ” ”? ”? 3 9 3 (5) 9) ?? 3) 15 ”? ? The amount of tension applied to the wire in the process of winding it, at least if it remains within the elastic limit, does not seem to affect the form of the curve very appreciably. A very large number of specimens have been examined, and if there is any effect, it is almost of the order of experimental error and is in the direction of increasing extension with increasing tension. That these effects are of the same nature as the apparent elastic loss observed in the case of iron or steel rods after overstrain is further shown by the fact that the effect dis- appears slowly with lapse of time, if the wire is kept suspended in a straight condition under very small tension. Fig. 3 (p. 624) illustrates this recovery. In two months’ time, a specimen had recovered half-way to its normal condition as a straight wire, while even in a week considerable progress had been made. Similar results have been obtained with other sizes of cylinders as well. 624 Mr. A. I. Steven on Anomalous The recovery of iron from elastic overstrain on warming to a temperature of 100° C. for a short time, investigated by Fig. 3. % Q N = ~~ S Q N EXTENSION IV MM. l (3 3 4 5 (1) Normal behaviour of specimen. (2) After coiling on cylinder of 4°5 cm. diameter. (3) Six days after coiling on a cylinder 4°5 cm. (4) Two months after coiling on a 4°5 em. cylinder. Muir, suggested that the same recovery might be found in this particular type of bending overstrain. Tig. 4 is illus- trative of this effect. (1) showsas usual the normal behaviour of a straight homogeneous wire, (2) the effect of winding on a 2 cm. cylinder. After again obtaining curve (1) the specimen was wound on the cylinder, and unrolled so that the wire was pulled straight up a long straight glass tube. Steam was then passed along the tube for 15 minutes, when, after cooling, the wire showed the behaviour indicated by (8). A second experiment showed (4) for the effect of the cylinder again without stcam-heating. None of the samples showed complete recovery, even though the steam-heating was con- tinued for as long as an hour, but all showed some recovery, this diagram (fig. 4) being a fair example. These effects suggest that the condition of internal stress in the material alter bending overstrain might affect the period of a spiral spring, so that its period immediately after Effects on First Loading a Wire. 625 being coiled would differ from that after a lapse of time sufficient for recovery of its normal elastic properties. To Fig. 4. 5 ee eee =e (i yeetoe (4) (2) a +8 Ny = ‘~* S ae] iL EXTENS/ON IN MM. \ e aS a (1) Normal behaviour of specimen. (2) After being on a cylinder of 2 cm. diameter. (3) After being on cylinder of 2 cm. diameter and heated in steam fur 15 minutes. (4) After being on cylinder again. investigate this, a long spiral of about 600 em. length of iron wire was wound on a cylinder of 1°5 cm., and its period found with a small load. After being heated in steam for some time, so that it might recover considerably, its period was found to be still the same, so that the effect of the internal stress was either relieved by the first few oscillations or did not affect its torsional properties. It does not seem probable that the internal stress was relieved, as a wire after being coiled on a cylinder, unwound, and then maintained in transverse vibration under small tension with a tuning-fork for an hour, showed similar behaviour to one which had not been so vibrated. Poynting and Thomson suggest that microscopic investi- gation may shed further light on the problem of these ano- malous effects. With wires of such small section as have been experimented on, there is considerable difficulty in Phil. Mag. 8. 6. Vol. 19. No. 112. April 1910. 28 626 Dr. J. W. Nicholson on the obtaining the desirable polish for the ready observation of the effects of strain, and as the wires are of circular section, the whole specimen is not in focus at one time, so that in- vestigations of wires microscopically will be laborious and troublesome. The main object of this paper is to point out that the so-called anomalies are only special cases of the loss of elasticity consequent on severe overstrain, and any theory which explains oue type of effect will be equally applicable to this one. The effect of the first load is clearly to aid in the equalization of the internal stresses in this particular case. In conclusion, I should like to express my indebtedness to Prof. Wilberforce for his interest and helpful suggestion at various stages in the work. George Holt Physics Laboratory, 21 July, 1909. LXVI. On the Size of the Tuil-particles of Comets, and their Scattering Effect on Sunlight. By J. W. NicHoLson, M.A., D.Se.* Vee opinions have been held as to the probable degree of aggregation of the molecules in the tails of comets. That the majority of the particles are not of molecular size, in spite of the probable tenuity of the gases in the tail, now seems to be fairly certain. The theory of radiation pressure proposed in general terms by Arrhenius f, and examined later by Schwarzschild ft from a. strictly mathematical standpoint, demands, if it is to give a descrip- tion of the phenomena associated with the tails of the most difficult type, the existence of particles of such a size that the pressure of solar radiation on a single particle must be about nineteen times as great as the solar gravitation. This value of the repulsive force from the sun, which is undoubtedly present, has been shown to be necessary, apart from any theory as to its origin, by the exhaustive work of Bredichin, Belopolsky, and others, in a classical series of papers in the Annales de l Observatoire de Moscou §. The validity of a radiation pressure theory of comets’ tails thus depends upon its capacity to supply a force of this mag- nitude on a single particle, and also upon an experimental indication that particles of the size for which this force may * Communicated by the Author. + Phys. Zeit. i. Jahr. Hett vi-vii. ~. Setzungs. der Math.-Phys. Classe zu Miinchen, 1901-2, p. 293. § Vide e.g. Series ii. vols. 1-11. Size of the Tuil-particles of Comets. 627 he obtained are present in large numbers in the tail, so that the effect of radiation upon these large particles ‘shall be predominant. In Schwarzschild’s paper, the pressure ptbdieed by a plane wave train impinging on a perfectly reflecting sphere placed in its path is examined, in order to obtain an upper limit to the possible pressure. Tor it is evident that in this case the pressure must be a maximum for a given dimension of obstacle. When the diameter of the sphere is not greater than about a quarter of the wave-length of light, assumed to be monochromatic, Schwarzschild finds that the pressure is given by the formula P = 22477°a7E/3\4 within an accuracy of 20 per cent., where E is the energy per unit volume of the incident beam, and a and 2 the radius of obstacle and the wave-length respectively. Adopting the specific weight of unity for the material, and taking the energy density of the solar radiation at the surface of the sun * to be 27°5 10-4 erm. cm.~*, with a gravitation constant at the sun’s surface of 27°5 times that of the earth, the ratio of pressure to solar gravitation becomes 3/4a, when a is in wp, and equal to X/8. The two forces are thus of the same order of magnitude. Their comparison at the surface of the sun instead of in the comet, is legitimate, since both obey the inverse square law. But when the diameter of the obstacle increases, Schwarz- schild finds that the pressure rises rapidly to a maximum, and becomes nearly twenty times the weight for a particle whose diameter is about one-third of the mean wave-length of visible light. Account has now been taken of the distribution of energy among the different wave-lengths in the spectrum, by the use of Wien’s law. When the diameter increases still further, the pressure sinks to the value wa*E valid for very large obstacles. The writer has recently repeated these calculations of radiation pressure by a shorter analysis, which may be readily adapted for rapid numerical calculation. The arithmetical results are not in complete accordance with those of Schwarzschild, but as regards the maximum value of the radiation pressure, and the size of particle corresponding to * Arrhenius, Joc. cit. + These results will shortly be presented to the Royal Astronomical Society. 282 628 Dr. J. W. Nicholson on the it, which are the most important points in the theory, there 1s very complete agreement. This agreement from two inde- pendent modes of calculation appears to place these two special points on a firm foundation. The analysis in each case is somewhat intricate, and independent confirmation therefore very desirable. It thus appears that on a radiation pressure theory of the tails, particles must be present whose dimensions are com- parable with the wave-length of light, and these must consist of aggregates of a very large number of molecules. In view of the low pressure at any point of the tail, a necessity appears to arise for the presence of a large number of particles of much greater size, which by continuous disintegration give rise to a sufficient number of the dimensions proper for i maximum radiation pressure. For an explanation of hydrogen tails, this maximum must be attained very nearly for a large number. The question of the origin of these large particles, whether expelled from the nucleus of the comet or not, may be left open for the present purpose. Now it seems possible to obtain certain experimental indications of the distribution of particles of various sizes in the tails, and the object of this note is to suggest, with necessary precautions, some lines along which inquiry might prove fruitful. If the particles were all small in comparison with the wave-length, and with properties not greatly different from transparency, the light scattered in any direction would be partially polarized, and completely so in a plane perpendicular to the incident rays. In such a case, Lord Rayleigh’s investigations in connexion with the blue light of the sky would be applicable. The intensity of the light scattered in any direction would be, at a distance 7, proportional to ma®/r°A*, where a is the radius of a particle and m the number of such particles. But when larger particles are present, in the number required for Schwarz- schild’s theory, the scattered light loses its character, and ceases to be polarized, in the absence of polarization of the incident light. The writer has recently worked out in detail the case of incidence of a plane beam on a perfectly reflecting sphere, which may be taken as an illustration of the scattering effect to be expected. The results necessary for the present pur- poses may be obtained at once by elementary considerations. For if E be the mean energy per unit volume of the incident beam, and if the particle be sufficiently large to throw a well- defined shadow, the energy stopped by its presence in the path of the incident beam is wa?H. Since the scattered wave at a great distance must be spherical, and this amount Size of the Tail-particles of Comets. 629 of energy is radiated off, the amount crossing unit area at distance » must be wa?H/4ar? or Ea?/4r?, in a first order approximation. The corresponding result for a number of such particles follows at once if, as seems certain, their distances apart are large multiples of the mean diameter. In other words, the shadow cast by one particle is supposed not to fall on the others. This assumption is borne out by such observations as have hitherto been made. For example, Rosenberg *, from a series of observations of the surface brightness of Comet Morehouse (1908 C), has concluded that the material is in so extreme a state of tenuity that there can be no question of the casting of shadows. ‘The Lommel- Seeliger law of diffuse reflexion was used in the deter- mination. Moreover, it has long been a matter of visual experience that comets do not greatly darken a bright star when passing over it. In addition to its indication of an absence of occultation of one particle by another, this fact affords valuable evidence of the smallness of the absorption. But in the presence of sodium vapour, or of other vapours which may possess similar properties, great care must be taken in the interpretation of results of experiments on brightness, after the remarkable researches of Wood tf on the extent to which absorption is affected by fluorescence. The need for a determination of the polarization of the scattered sunlight is therefore evident. In the case of Comet Morehouse, Rosenberg has recorded that no visual effect of polarization could be detected, but, as he remarks, the reflected light was in this case only a very small fraction of the total light, at any rate within the limits of the visible spectrum. As exact a determination as possible of the proportion of scattered light which is polarized would go far towards a determination of the average size of the particles, and experiments on these lines do not appear to have been per- formed as yet f. The absence of any great amount of polarization in ihe total light affords no strong indication that the proportion of reflected light is small. Rosenberg § appears at one point to regard this as a possible corollary, but, as we have seen, this can only be the case in the absence of a large number of reflecting particles too great in magnitude to polarize the scattered light. If, when the intrinsic light of the comet and * Astrophysical Journal, Nov. 1909. t Phil. Mag. Dec. 1908, Oct. 1909, and other Papers. t I find.that Prof. H F. Newall has designed a series of experiments on these lines, in view of the approach of Halley’s Comet. § Loc. cit. p. 268. 630 Prof. J. Joly and Mr. A. L. Fletcher : the reflected sunlight are separated, there is not a large portion of unpolarized light in the latter, a serious objection to a radiation pressure theory of the tails appears at once. When a comet has a dense bright nucleus, a certain small proportion of its intrinsic light must show evidence of polarization, due to scattering by the tail, of light emanating from the nucleus. But this effect would probably be tco small to be observed in any case. Radiation from the nucleus may also, in some cases, give a radiation pressure assisting that of the sun, although the gravitational action of the nucleus can never be important. All observations relating to polarization would be liable to be affected by the presence of fluorescence, and in view of the results obtained by Wood *, deductions from them demand as much caution as those from the observed brightness. LXVIL. Pleochroic Halos. By J. Jory, F.R.S., ont ARNOLD L. FLEtcHER, B.A.L.T [Plates VIII. & IX.] ee the date of an earlier paper on the subject of Pleochroic Halos{ the examination of many halos of special interest has elicited points of sufficient importance to justify fuller consideration of the subject. In the paper referred to it was shown from the measure- ment of halos in a Greisen that the radial dimension of the corona, or outer halo, when compared with that of the darker central area was such as to support the conclusion that the corona represents a shell of less complete ionization (or other effect) due to the « rays of radium C only ; while the more intensely darkened centre was ascribable to the influence of all those rays which range no further than those of RaA. It was also found that calculation of these radii, according to Brage’s law connecting the ionization range with the square root of the molecular weight, gave distances closely corresponding with those observed. The mica in which these halos occurred was subsequently found to be a lithia-bearing variety—the somewhat variable species classed as zinnwaldite. But, as will be seen further on, this emen- dation introduces no contradiction to the views originally expressed. In a letter to ‘Nature’ of February 10th the fact of the existence, in some cases, of an accentuated outer border to the corona, in accordance with the increased effect * Phil. Mag. July 1908, p. 184. + Communicated by the Authors. t Joly, Phil. Mag. February 1910. Pleochroic Talos. | 631 of the 2 ray towards the end of its range, as shown in the Bragg curves, was confirmed; and the identification of thorium halos recorded. _As the number of measurements multiplied it appeared as if the calculated or theoretic values tended to range a very little above those determined by observation. The source of this has been found to lie in the mode in which Bragg’s law was being applied. The method of applying this law had in- vited the ¢ comparison of the average square root of the atomic weight of the mineral containing the halo with that of air. Now Bragg and Kleeman’s figures aud results do not bring air quite harmoniously into line with the solid substances examined (Phil. Mag, Sept. 1905), and it seemed a more correct procedure to calculate the range in the mineral by comparison with the square root of the atomic weight of a substance more generally in agreement with the law and similar to the mineral in physical state. Choosing aluminium for the purpose, the table given further on for the ranges in some important halo-bearing minerals, has been calculated both for the « rays of RaC and RaA and for ThC and Thx. The calculation is most readily made as follows. Bragg and Kleeman found that if the products of range and density in a number of different substances are compared with the corresponding products in the case of air, the several quotients obtained stand in the same relation one with another as the square roots of the atomic weights of the several sub- stances. The quotient for aluminium bas the value 1°23. If we assume the range in air to be one centimetre, then the : : Lee ; A x 1°23 corresponding range in aluminium is r= hier eae, where A and 6 are the densities of air and aluminium. Also we have Se =a: where 8’, 7’ are the density and range in any particular n mineral; a’ is the average square root of the atomic weight of thie mineral, and a ‘the square root of tlie atomic weight of aluminium. Substituting the expression for » and taking the density of sap as 0°0012 ; and writing 5-15 for a; we have 7/=0: 000287 6 ; which gives the range in the mineral corresponding to one centimetre in air. Thus for certain biotites u’ is found to be 4°5 and & 2°8, hence r’'=0°000461. Then for the rays of RaC, having a range in air of 7:06 cm., we find the range in this biotite to be 0:0326 mm. The quantity a’ may be determined either from 632 Prof. J. Joly and Mr. A. L. Fletcher : a molecular formula or from a chemical analysis. In the first case we multiply the square root of each atomic weight by the number of such atoms present; sum the values so found ; and finally divide by the number of atoms entering into the formula. In the second case we deal with each molecule separately, adding the square roots of the component atomic weights and dividing by the number of atoms in the molecule. The number obtained is the average square root of the atomic weights entering into the molecule. It is to be multiplied by the percentage quantity of the molecule present, as given in the analysis. ‘The numbers obtained are to be added and their sum divided by 100. The following are the ranges in air of the « rays of the uranium, actinium, and thorium families of elements :— Lo aie Tbs | Iv. | em. | om. | eins 4 | en, os Uranium ...| 35 28 | Radioactinium., 4°8 | Thorium ...... 3°5 | Tonium ...... | 2°8 Op | ill lieben eee de 6°55 | Radiothorium .| 3°9 | Radium ...... | 3:54 | 2°84 || Emanation...... (ore | Wh Xa eee 57 | | Emanation...| 4:23 | 3-60 || ActB...........- | 550 | Hmanation...... 5'd | 1 i ee | 483 | 4-25 | Th B iscis ee | 50 BaO p20) 04. 706 635 | Th 3 eee sg | 172 as 386 | 316 | Column I. gives the limiting ionization ranges in air at 760 mm. pressure, and Col. II. the approximate distances in air at which the ionization effect is a maximum in the case of the uranium family. The values in Col. II. are deduced partly by scaling directly from the Bragg and Kleeman curves, partly by assuming that a deduction of 7 mm. from the limiting range gives the approximate position of most ‘intense ionization. These values we shall require later. The distances in Cols. III. and IV. are the limiting ionization ranges in air in the case of the actinium and thorium families*. In a pleochroic halo we may have either the uranium derivatives acting or those of thorium ; or, again, a mixture of both. In the first case the radius of the halo is limited * See Bragg and Kleeman, Joc. cit. and Phil. Mag. June 1906; Allen, Phys. Rey. xxvii. 1908 ; Le Radium, Jan. 1910, p. 2; Levin, Phys. Zeit. 1906, p. 521; Kuéera and Masek, Phys. Zeit. 1906, p. 8389; Hahn, Phil. Mag. Sept. 1906. Pleochroic Halos. 633 by the ionization range of RaC in the particular mineral in which the halo is formed. In the second the radius is limited by the range of ThC. In the third it will again be limited by the range of ThC. It might, in general, be difficult to distinguish the last two cases from one another ; but if the halo is not over exposed we would expect in the Ra-Th halo a corona with inner and outer radii corresponding to the radii of RaC and ThC. Thusif formed in biotite the radius of the pupil would be about 0:033 mm. and of the corona or iris 0-040. There is no difficulty in distinguishing uranium trom thorium halos, however blackened up they may be. In biotite the outer radius of'a good uranium halo is never very different from 0°033 mm. and that of a thorium halo from 0-040. The difference, unless the originating radioactive particle is large enough to exaggerate the dimensions of the former, is such as cannot escape detection. In these cases if the halo is not over exposed the radius of the pupil will be respec- tively about 0°022 and 0-026; the former being due to Raa, the latter to ThX. The table which follows gives the radii (in millimetres) in the case of the commoner halo-bearing minerals. Only some of the figures in the above table are based on an accurate knowledge of the density and the chemical composition. Both quantities require to be known for the particular mineral. The more reliable values will be recognized by the accurate figures given for the specific gravity. Uranium. Thorium. é. RaC. RaA. | C. x: _. Eo Ee =o | | ! Giomen ae '0-0346 0-0237) 0:0422 0-0280 (MgFe),Si,0,(AlFe),O,. Hornblende} 31 | 00336 0-0230 0:0409 00272 r Diopside ... 3492 | 00281 0:0193) 00343 0°0227 Hedenbergite. } i Biotite ...... 28 0:0826 0:0222) 0:0397, 0:0263 (HK),(MgFe),Al,(SiO,),. | Anouite...... 2°846 0°0321 0°0219 0:0390 0:0259 Analysis selected by Tschermak. Haughtonite. 2-934 0-0334 0:0229'0:0407 00270 ~—,,_—s see p. 638. | Muscovite ... 28 —0:0304 0:0208 0:0370 0-024" A1,KH,Si,0,,. - 2830 0:0310 ae 0°0378 0°0250 Analysis selected by Tschermak. Lepidolite ...| 2°885 0:0307 0:0210 0-0374 0-0248 Analysis selected by Tschermak. pI | 0-0275 00188 0:0335 0:0222) KLiAl, Fe, HSi,0,,. | | Zinnwaldite.. | 0:0322 0:0221 00391 0:0260 Analysis selected by Tschermak. 00829 0:0225 0:0400 0:0265 20 Li, 20 Al, 348i, 99 O, 20 Fe. 63+ Prof. J. Joly and Mr. A. L. Fretcher : In selecting halos for investigatiun several sources of error must be borne in mind, the neglect of which has doubtless been largely responsible for the general failure of petro- logists to recognize their wonderful obedience to law. In the first place it is apparent that onlv a central section of the sphere can give the true radius or, if there is differentia-- tion, a correct ratio of the radii. Thus in complex halos a section off the centre tends to exaggerate the dimensions of the iris compared with that of the pupil; and in simple or structureless halos the radius of an excentric section of the halo must be misleading. With respect to this source of error the only safeguard is the certain presence of the origi- nating radioactive particle. When this is plainly defined we cannot be far from a central section. A second source of error arises when the originating radiouctive mineral has dimensions which are not neglivibly small compared with the radius of the halo. When the radius of the nucleus is, relatively, not negligible, its mean diameter should, in general, be deducted from the diameter of the halo. This is so because in many cases it is demon- strable that rays leaving the surface—cr from points near the surface—of the nucleus have determined the boundary. Thus a large square zircon-section may show a sub-circular halo enclosing its sides ; the measurements from the boun- daries of the zircon being correct for radium C. In general halos occasioned by large nuclei are badly defined and hazy upon the edge owing to rays coming from different distances in the nucleus spoiling the boundary definition. In very small nuclei there is but little error due to the passage of central rays through the nucleus. Thus for zircon the re 7 ° . ht ; x e ° V/ atomic weig Uta 1:1; in the case of thorite it density is 1:2; in uraninite about 1; and in the cases ot hornblende and biotite about 1°8 and 1°6 respectively. There is, there- fore, some retardation in nuclei formed of the radioactive minerals ; but it is easy to see that the error introduced by quantity neglecting it is of quite a different order from that which: arises when the ranges in air of rays proceeding from beneath the surface ofa solid film are being observed. ‘The correction, such as it is, seems to be in most cases subtractive. Thus in the case of a nucleus having a radius of, say, 0°0021 mm. we cannot well assume that the outer boundaries are defined either by rays proceeding from the very surface, nor yet from the very centre of the radioactive particle. If we ' é . | ty | a 4 | Pleochroic ITalos. 635 assume that their average point of dejarture is about 2/3 the radius measured from the centre, we find that the distance, 0°0007 mm., traversed in the denser ‘mineral (we will suppose uraninite) is equivalent to 00011 in biotite, say. So far there should be an additive correction of 0:0004 mm. But the point of departure not being at the centre, but 0°0014 mm. away from it, the subtractive correction exceeds the additive, leaving finally a subtractive allowance of 0°0010 mm. The correction is, probably, of approximately this amount. The effect on measurements of outer radii is small. On inner and much lesser radii it may be more serious. In certain minerals the nucleus itself takes a roughly circular form and is opaque, and when the whole is blackened up quite erroneous conclusions may be arrived at. Hxami- nation by reflected light will, in such cases, generally reveal the central nucleus. “Indeed, it is well to examine all much darkened halos by reflected light before forming conclusions as to their radius. So far as our experience goes halos in cordierite or in andalusite cannot be used for accurate measurements. Ap- proximate measurements quite in accord with calculated values can be obtained, but the boundary of the pale yellow halos is too indefinite to permit of accuracy. Structural features we have not traced in such halos. Cordierite is much more sensitive to the « radiation than biotite*. It is possible that even a lesser quantity of a radioactive substance could be detected in this medium than in biotite. The greater sensitiveness probably leads to more rapid over exposure and loss of detail. As an instance of the necessity of making allowance for a relatively large nucleus we may refer to the photograph of a halo in biotite illustrating an earlier paper on this subject ft. The measured diameter of this halo is0°096 mm. The radius, 0-048, however, will not agree with any calculated radius in biotite, but it is found that the central mineral, in this case zircon, is too large to be neglected. A number of Observations in different azimuths from the boundary of the zircon to the periphery gives the radius as 0°039. This is quite in harmony with the view that this is a thorium halo. In the next table we have collected some measurements of halos in various minerals. * Miigee, Centrbl. f. Min. 1909, p. 6d. t Joly, Phil. Mae. March 1907. 636 Prof. J. Joly and Mr. A. L. Fletcher : | | Containing | R. | *. {Radioactive Rate | mineral. mm. | mm. | substance. ; Bictite © .....- 0:023 a Ra | Durbachite, Schwartzwald. | + ae eee 005950. % Th | S Rae tub gs 0:039 | 0:023 Th Diorite, Redwitz. a ae 0:032 1) (ca. Ra LA Se TA Ls 0:033 | 0°021 Ra Granite, Freiberg. te Re 0:0393) 0:0244 Th » Ochsenkopf. 4 ee 0:0319} 0:0191 Ra a * oes ye aaae 0:033 | 0:022 Ra », Leinster. Zinnwaldite..|0°0312| ... Ra _—|: Greisen, Altenberg. A ...|0°0305) 0:0195 Rav ds be Ms “a ...|0°0321| 0°0211 Ra | ot is o ...|0°0323) 0:0202 Race 2 | a se Corona detached. x HO OS26 aes a Ga A ...|0°0317| 0:0198 ase Ee . is Corona detached. a ...|0°0301| 0:0191 Ria. aye " i a ” i ...|0°0316) 0:0190 Ray ys | rm % fs Lepidolite ...;0°028 |... Ra _—|,s Lourdalite, Langesundfiord. Cordierite ...|0°031 e Ra Hornblende ..| 0:040 | 0°028 Th | Syenite, Knorre. Aa | O38 5 ae Rr st. Maurice. One of the thorium halos in the durbachite is that of which we have just given details, and is illustrated in the paper referred to. We reproduce here (Pl. VIII.) a photograph of a thorium and a uranium halo (contained in the one crystal of mica from a granite of Ochsenkopf in the Fichtelgebirge) the dimensions of which are given in the table. The uranium halo (in lower part of figure) is too dense to permit of clear photographic reproduction of the pupil; but the latter can be seen and measured in the microscope. The corona of the thorium halo is not strongly developed, but is quite distinct. The nuclei in the case of both these halos are small and practically negligible. From the measurements given in the table it will be seen that halos developed in biotite are somewhat greater in radius than those in zinnwaldite or lepidolite. This agrees with the theoretic radii in these minerals as already deduced. The fact that the radius of the pupil is not in every case quite up to the theoretic size is, we think, explicable on the view that these halos are not always fully developed. It will be seen further on that the corona appears before the pupil reaches its limits. Thorium halos are not so frequenily met with as uranium halos. The thorium halo observed in the syenitic hornblende of Knorre is a very beautifully coloured object when seen between crossed nicols. The nucleus in this case is pre- sumably thorite. It is black and nearly opaque and almost Pleochroic Halos. 637 square in cross-section. Thorite is isomorphous with zircon, and its presence in syenites has been demonstrated before now. The inner radius is not sharply defined in this-case ; the measurement given is approximate only. Reference to the table of calculated radii will show that the agreement between observed and calculated values is in general very striking. Slages in the Development of Pleochroic Halos. We have included in the preceding table of observations only those halos whose development appears to be nearly or quite perfect. In some cases these halos may be described as “‘over-exposed”’; that is, the detail within the halo is obliterated by excessive darkening of the medium. In others we see the halo at an earlier stage; corona and pupil being distinct, or the former even detached from the latter and encircling it as a delicate ring. Plainly this last is an earlier stage of development. But halos may be observed in still earlier stages. We have been so fortunate as to find ina single large crystal of biotite (var. haughtonite) from the Leinster granite, uranium halos in every stage of develop- ment. Those stages we shall presently describe ; but, first, some account of the conditions attending the occurrence of these halos will be of interest. The crystal of biotite, which measures about 2°5 em. in its greatest dimension, is enclosed within a fringe of silvery white mica, a relation not uncommon in the Leinster granite. This appears to be an original association, the muscovite erystallizing after and in continuation of the biotite. But the same association is believed to occur as the result of meta- morphic action in certain rocks. The boundary is abrupt an|, even in the microscope, the demarcation is sharply de- fined. Halos are developed only in the biotite, and principally in the vicinity of the junction of the micas. The central parts of the biotite crystal are ha!o-free and clear of all in- clusions. The radioactive substance is often found extending along cracks in the biotite, which then are intensely dark, and may carry strings of halos, their nuclei contained in the cracks, their spherical shells extending into the clear biotite. In other places the halos are crowded in clusters or, again, scittered without apparent connexion. Much radio- active blackening has also occurred around elongated or shapeless inclusions, or in stains and blotches. The radio- active darkening never extends into the limpid muscovite; a halo often appearing as bisected where it meets the colour- 638 Prof. J. Joly and Mr. A. L. Fletcher : less mica. The diverging arrangement of the darkened eracks from large areas of blackened biotite situated near the margin of the crystal strongly suggests the idea that a radioactive solution has at some time penetrated the biotite. An occasional yellowish staining of the muscovite may pos- sibly indicate the presence of this solution.. Zircons, not present in the muscovite, often act as radioactive nuclei in the biotite, but some of the smallest nuclei are, apparently, not zircon. ‘They are, possibly, uraninite or some allied mineral. The following analysis of this mica shows it to be of the variety called haughtonite. The iron is nearly all in the ferrous state and is calculated as such. SiOste nee. eee SG 6h eat 2A 1 AIGOS EO, Zoro2 Me eee 0°48 IRGOMA Ct 13°40 OR (4 The small quantity of magnesia is remarkable. The mica is of a deep reddish brown colour with small optic-axial angle. A careful experiment gave the density as 2°934. From these data we find the average square root of the molecular weight to be 4°84, and the quotient : to be 1:65. Hence 7’ has the value 0000473; 2. e. the range in this mica corresponding to one centimetre in air. A determination of the radioactivity of the mica was made ata sacrifice of one gram of material. The experiment was made in the new Physical Laboratory (by kind permission of Professor W. E. Thrift) under circumstances precluding any probability of contamination. The result arrived at, on two identical determinations, was 11°87x10-¥ gram per gram. This result gives, on the face of it, an idea of the extremely small quantity of radioactive material sufficing to originate ahalo. It is not too much to say that some ” thousands of halos, or equivalent rad oactive staining, occur in a gram of this mica. An endeavour to observe any local radioactive intensity by placing halo-rich parts of the mica beneath and in contact with a screen of sensitive zincblende gave a negative result. Using the value obtained above for the r ange-equivalent of J cm. in air, and the measurements given in the table p- 632, the following are the limiting ranges and the ranges ge eS es ee Pleochroic Halos. 639 of maximum ionization of the various # rays of the uranium series in this biotite. Maximum Limit mil, mm. mite: Ge a. ees 0:0301 0:0334 mW ee 2) 2 vig 0°0201 0°0229 Emanation ......... 00-0170 0:0200 Reaper Geert yh 0:0150° = O-OLSS Remar) 0. 0°0135 0:0168 Monit soe 0-0100 0:0133 These ranges appear to be concerned in the genesis of the halo, but in a manner which can only be completely under- stood by reference to the Bragg curves showing the ionization at every point of the path of an @ particle in air. For better convenience we reproduce here the curves (p. 640) as given by Bragg and Kleeman in the paper which has been ‘the starting point of all subsequent work of the kind™ (Joe. ctt.). Toniza- tion being plotted as abscissze, and distance from the origin as ordinates, the complete curve for the a rays expelled by RaC (curve cbad) shows that the ionization due to the « particle increases slowly at first, but augments very rapidly towards the end of the ionization range. Ata distance of about 6°3 cm. the intensity of ionization isa maximum, and at 7:06 em. the ray no longer i ionizes. Its presence beyond this distance— if it penetr: ates further—could only be detected by a cumula- tive experiment and spectroscopic analysis; for it will neither affect a photographic plate nor a sensitive zine sulphide screen, nor will it ionize a gas. The recent experiments of Geiger (‘Nature, Feb. 24, 1910, p. 508) support the view that the a ray then comes to rest, and, probably, losing its charge, takes on the character of an atom of helium (Ruther- ford and Geiger, Proc. R. 8, Ixxxi. A. p. 162). The form of the ionization curve of RaC! is found to apply to the other rays, so that the effects attending the motion of each particle in the medium are defined by transporting the curve for the rays of RaC parallel to itself, and so as to suit the ranges of other rays. Each ray has therefore the same intensification of its effects just before it ceases to ionize: the initial velocities and ranges of the rays alone vary. A radioactive nucleus giving out rays in all directions from the several #-ray-producing constituents is, therefore, sur- rounded by successive shells of maximum ionization. That * We desire to thank Professor Bragg for his permission to reproduce these curves. 640 Prof. J. Joly and Mr. A. L. Fletcher : due to ionium lies nearest to the centre, in air at about 2°1 em., in biotite at about 0°Ol mm. After this the shells of the several maxima crowd together, merging one into the other, as the table of ranges, given above, shows. The com- plex curve, given by Bragg and Kleeman for radium which after 28 days, ” ” ” TD. Tonization curve of Ra after 140 hours. alter 64 hours. after 90 hours. ’ ”? ” Experimental readings marked by crosses. ” ” has developed emanation and the ensuing products of rapid change, illustrates this effect; the line of ionization sloping, with perturbations, upwards from right to left, the slope ending with the effects due to RaA. In the halo we must Calculated readings marked by dots in circles. si ee he Pleochroie Halos. 641 imagine additional rays at work which would prolong the Bragg-Kleeman curve further to the right, and also bring it downwards nearer to the origin. Here we may recall Rutherford’s suggestion, based on Boltwood’s work, that actinium is probably a collateral member of the uranium-radium series; certain of the unstable atoms disintegrating according to a different plan and determining a separate line of descent. As such atoms are known to be subordinate in number it is probable that halos generated by actiniuin, or even definitely revealing its presence, will never be observed; its effects being always associated with and, as it were, submerged in, those of the preponderating radium series. There is som» reason to betieve that the fastest moving « ray products of actinium, ranging from 5°5 to 6°5 cm. in air, are concerned in finally blackening up the uranium-radium halo; but so far as our observations have gone no halos referable solely to this radioactive series occur. The extension of the work of Bragg and Kleeman has re- sulted in showing that the laws discovered by these investi- gators apply to « rays however originating, whether from actinium or thorium. In the Leinster haughtonite we find that a considerable number of small halos exist, possessing a radius of about 0-013 mm. These are simple, generally structureless, halos, but occasionally showing a darkened peripheral border. They may vary much in intensity, the fainter being associated usually with the smallest nuclei. In most cases the nucleus is clearly and sharply defined as a minute black or nearly opaque speck, which may look square or angular. The colour of these halos varies from that of a faint smoky grey toa deep brown or even black. Their form is perfectly circular. The following are some radial dimensions in millimetres read in cleavage flakes of the mica :—0-0129, 0:0147, 0°0125, 0-0132, 0:0128, 0:0128, Otho, 0-0137, 0°0137, 0-0111, 0°0141, 0°0135, 0:0132, 00138, 0°0122. We might add many scores of such deter- minations. These halos only very rarely reveal any trace of an outer corona or ring. Thus two’of the above, having radii in each ease of 0°0137 mm., showed a first very faint trace of the corona. Such coronas lie at the correct radial distance for RaC as given in measurements cited further on. It is re- markable that halos at this stage may be much darkened and yet fail to show any trace of the action either of RaC or other of the more penetrating rays, whiie those showing traces of a corona may be only fairly well defined. As we may assume that the time of accumulation of the radioactive Phil. Mag. 8.6. Vol. 19. No. 112. April 1910. 2T G42 Prof. J. Joly and Mr. A. L. Fletchier : effect (whatever its nature may be) is the same for all, the differences in development must simply depend upon the radioactivity of the central originating substance. The larger nuclei are, accordingly, generally found to be attended with the best dev eloped halos. It seems certain that these embryonic halos owe their radial dimensions to the effect of the slower moving @ rays; those of ionium having their maximum effect at 0:°010 mm. defining the smallest, and the maxima of radium and uranium lying at 0°013 mm. defining the larger. Some allowance for the radius of the nucleus, which very often has been found to be about 0:0022 mm., would refer some of these halos solely to the effects of ionium. But none are ever found of radial dimensions less than can be accounted for by tonium. This seems additional evidence that ionizing « rays of less velocity than those given out by ionium do not exist in the uranium-radium series. The influence of the radium and uranium a rays is, pro- bably, an early one; the latter element being known to give two « rays upon disintegration. At the radial distance of 0-0135, therefore, three a rays out of the total eight which are evolved in the entire sequence of changes, exert their greatest effect. It is, therefore, not to be wondered at that the number of embryonic halos having a radius ascribable to these influences is very great, but owing to the uncertain allowance for the nucleus and the gradation of the observed dimensions it is impossible to clearly define those due to any particular ray at this stage of development. In agreement with this, simple halos of varying dimensions somewhat greater than those cited above are found. ‘This, too, is quite in accordance with the Bragg curve. If this curve were prolonged to the right so as to bring in the uranium, and the slower moving ionium, ray, it is ‘evident that aaale halos would not be expected to show a lesser radius than is due to the ionium ray, the enlarging of the radius under the intense ionization of uranium and radium should be rapid. It should be noted as generally true of the initiation of halos that the first beginning is not from within outwards. The halo suddenly presents itself as a very faint spherical darkening of the mica, which if differentiated at all is darkest at or near the surface. Any greater darkening occasionally ob- served near the nucleus is more or less irregular and fuzzy, is confined generally to small radial distances, and may be due either to the convergent concentration of the rays or to some escape of emanation. This mode of initiation is, of course, in keeping with Brago’s observations. — Se — es a SS | : j | \ Pleochroic Halos. ° 643 Two shells of maximum ionization succeed closely those due to the radium and uranium rays; that due to RaF at 0°0150 mm., that due to emanation at 0:0170 and extending to 00200 mm., about. Just beyond this limit should lie the maximum effect of RaA. The following are typical measure- ments of halos whose limitations may be ascribed to Ral or to emanation :—0°0158, 0:0167, 0°0165, 0°0170, 0:0181, 00182, 0°0184, 0°0188. We might quote very large numbers of such measurements. Such halos are very often, if not most generally, still without a corona. These simple halos may even be quite blackened up and yet show no signs of the effects of RaA or RaC. The dimensions of the pupil at which RaC first makes its appearance are, typically, somewhat greater than the above, but not such as can well be ascribed to RaA. This is one of the most remarkable points about the development of these uranium-radium halos. RaC seems to create an effect in advance of RaA, although the density of the former is less owing to the greater radius, and although RaA is somewhat strengthened by a coincident actinium ray. The radius of the faint, early, corona is never less than can be ascribed to RaC, as will be seen from the list of dimensions which follows. R is measured to external radius of corona, R’ to its inner radius. | | } % R’. R. Nuclea oy radius, ae i ee } 00165 a 0:0297 0:0169 a 0:0300 0-0170 a4 00300 0-0171 a ' 0-0318 00174 eS | 00328 00020 00176 i | 00314 0 0029 0:0174 ee 0:0319 | 00019 0:0177 a 0:0306 | 0:0179 Af 0:0323 | 0:0180 Me 0:0328 0-0183 00290 00323 0:0025 0-0184 00267 00323 0:0029 0:0185 00286 0:0323 0:0185 0 0284 0:0317 Such halos as the above are not, of course, fully developed. The corona changes indeed but little in its subsequent dimen- sions, but the pupil expands considerably. The next feature observed on close examination is sometimes the appearance of a very fine and faint ring lying outside and close to the pupil. It is, indeed, Peale ae all cases connected to the 212 644 Prof. J. Joly and Mr. A. Ji. Fletcher : pupil by intermediate shading less intense than that which makes the ring visible. This ring is very probably repre- sentative of the effects of RaA. The following are some measurements :— Be i RaA. R', R. 0:0103 0 0169 00228 0-0295 00333 00187 0:0227 0:0327 0 0186 00229 0.0284 0:0539 0:0192 00233 00280 0:0338 | 90-0201 | 00219 | 0:0283 | 0:0323 In the case of the first halo the inner radius, 7’, of a dark band edging the pupil is recorded. It will be seen that there is a very close agreement amon the measured radii of the ring ascribed to RaA. The radii are measured to the outside of the ring. Allowing for some effects due to the central nucleus the coincidence between the measured and calculated ranges (the calculated maximum is 0°0201 and limit 0°0229) is sufficient to leave hardly any doubt as to the influence of RaA in forming this ring. In Pl. IX. fig. 2 we reproduce photographically a field of halos in various stages of development, one showing the pupil well developed and three others in various stages of incom- pieteness. In the more rudimentary ones the appearance of the pupil will serve to convey the form of the simple halos which precede the advent of the corona. In Pl. IX. fig. 3, a very much enlarged halo is shown bearing the ring due to the rays of RaA. This halo is 0°033 mm. in diameter, and as it scales here about 3 cm. the linear magnification is about 450 times. It is difficult to offer any explanation of the relatively backward effects of RaA. If there was “initial recombi- nation” progressing and such recombination was stimulated by the shock attending the passage of the fast rays of RaC, such an effect might be more marked here then elsewhere, and so the absence of the effects of RaA be explained upon their elimination. It must be borne in mind that the changes are progressing in a crystalline medium, and there may be forces tending to restore the original molecular grouping which may be stimulated into action where they are not Pleochroic Halos. 645 permanently overcome. ‘The true explanation may, on the other hand, lie in effects not yet suspected. The further development of halos is marked by widening of the corona and advance outwards of the pupil. In general there seems to be a pause after the effects of RaA have brought the pupil out to such dimensions as are given in the foregoing table. In many old halos, as described in the first part of this paper, the differentiation is limited to the pupil and corona—the latter a more lightly shaded continuation of the former. But for some reason the development does not always progress as if leading to this result. Thus in the mica we are dealing with we have found halos with a fairly uniform dark pupil brought out to a radial distance of 0°0278, and still separate from the corona. That this is not due to a large central nucleus is shown by the outside radius of the corona—0°0335 mm.—and can perhaps best be explained on the activity of several nearly coincident actinium rays ranging over 5cm.inair. If this is the normal process of develop- ment—that is by the gradual advance of the pupil—the state of simple ditferentiation into pupil and iris must be explained on the subsequent greater accumulation of effects within the limits reached by RaA. But it must be admitted that while the general features of the development scem free from any obscurity, the relative intensity and order of the events are not wholly explicable. We are, however, in total ignorance of what really takes place in the crystalline medium. The highest magnification applied to halos does not appear to throw light upon the latter point. A faintly radiate structure is sometimes seen, in particular lighting, in halos which are under-exposed. This may sometimes appear to be confined to the corona. Its significance, if it has any, is unknown. Doubtless helium is actually stored in the suc- cessive shells which mark the limits of each ray. Under what conditions, whether free or possibly in synthetic com- bination with atoms into which it has penetrated, is not known. Strutt’s results on the helium stored in various rocks and minerals point to a mere mechanical storage of the helium. It seems very improbable that the presence of the helium occasions the darkening observed. There is really very little of it. As will be seen later there is, probably, not sufficient to create an atmosphere pressure if distributed throughout the halo. Again, if the helium is the cause of the darkening, why are halos confined to certain minerals? The restrictions are not due to want of radioactive nuclei, for zircon and apatite, which in some minerals give rise to halos, in others in the same rock-section are without effect; and, as observed, 646 Prof. J. Joly and Mr. A. L. Fletcher : the halos in this Leinster biotite may be abruptly bisected where they abut against the muscovite. This, too, is a frequently observed appearance in granites, syenites, &e. The halo is not continued inte quartz or felspar adjacent to the biotite or hornblende, in which it is developed. If the appearance is due to a mere storage of helium this should not be. On the other hand, the view that the molecular structure is broken down by the passage of the positively charged ray seems in harmony with the photographic and other chemical effects which have been shown to attend the passage of the ray. The nature of the changes produced is as yet unknown, or whether the crystallographic structure influences then. There might be phenomena resembling “ reversal ” in photo- graphic films. We are really in very complete ignorance of the nature of the wonderfully graphic yet minute phenomena of the halo, as the number of speculative possibilities associated therewith demonstrates. It is of interest to note the extremely minute quantity of radioactive material which suffices to make a halo. We have already suggested that the nuclei are in some cases possibly uraninite or some related substance. For they may be black and opaque, and, when their lustre can be observed by reflected light, show the coaly appearance of pitchblende. The assump- tion does not involve an amount of radioactive matter greater thin the ascertained gross amount per gram of mica might indicate. The diameter of a nucleus sufficing to evolve all the characters of the halo, pupil and corona, is very often less than 5x10-*em. Thecorresponding volume is about 65 x 10-Yem. If composed of uraninite having the density 8 the mass would be approximately 5x 10-1° gram, and the associated radium in each nucleus about 10-16 gram. As one gram of the mica contained nearly 12x 10-! gram radium there is sufficient to build 100,000 such halos per gram. We may, therefore, without unduly taxing the resources available, sup- pose uraninite to be one of the radioactive substances present. The quantity of helium stored in a halo must be small on any probable estimate of the time which has elapsed since its initiation. On Rutherford and Geiger’s estimate that 3:4 x 101° helium atoms are expelled per second per gram of radium, the quantity 10-!§ gram radium and the 8 «@ ray pro- ducing elements in equilibrium with it evolve only about 850 helium atoms in a year. If the time was 108 years the number of atoms is 85x 10%. Rutherford and Geiger have ealeulated that there are in one cubic centimetre of a gas at standard pressure and temperature 2°72 x10!" molecules. . 3 | } Pleochroic Hatos. 647 The volume of helium is, therefore, only aLout 3x 107 c.c. If this is distributed uniformly throughout a halo having the radius 0°033 mm. the pressure is less than the forty-sixth part of an atmosphere. A nucleus of ten times the mass would still yield a much less amount of helium than would fill the halo-sphere at atmospheric pressure. Nothing is so surprising in connexion with the pleochroic halo than the extremely minute effects which suffice to give rise to it. The calculation given above shows that nuclei of the size repeatedly measured contain but 10-' of a gram of radium, even if we suppose them composed of the most radioactive of naturally occurring minerals. This quantity gives off but one « ray inabout ten hours. But even smaller nuclei suffice to produce small simple halos whose radioactive character is still quite unmistakable. A series of measure- ments, using a high magnification, gave the following readings in millimetres for the nuclear diameters of embryonic halos previously described as averaging 0°013 mm. in radial dimension :—0°00195, 0°00243, 0°00144, 0°00243, and for two such embryos showing very faint coronas 0°00292, 000316. Calculating the volume corresponding to the diameter 0:002 mm. we have, finally, a quantity of uraninite less than one-sixteenth the amount giving rise to the more developed halos. Weare recognizing, therefore, by mere inspection, the presence of considerably less than 10-' of a gram of radium ; a quantity expelling about 80 helium atoms in a year. It is to be added that many of the smallest nuclei are probably zircon, which would lead to still lower estimates of the amount of radium involved. We are in some hope that experiments now in progress upon the rate of growth of artificial halos in this biotite may enable us to arrive at an estimate of the time required for the formation of these halos. Here we would remark upon an aspect of the study of pleochroic halos which may, in the future, prove of im- portance. We are dealing with quantities of radioactive sub- stances many thousands of times less than can be measured by any known method, and not only so, but the means of discriminating between one radioactive family and another is given tous. Hven more, a certain number of the specific a@ rays can, under favourable conditions, be inferred, and the ranges of some accurately measured. Thus in the uranium-radium halos of the Leinster biotite we might, if starting the research in ignorance of the specific results arrived at by Bragg and his successors, have inferred the presence of the ionium ray: of that very vigorous and pro- bably, therefore, complex group of rays which enlarge the 648 Mr. G. H. Berry on the primary halo and darken its border: of the ray having the definite range of RaA; and, finally, of the very penetrating ray giving rise to the corona. It would seem, therefore, that in the pleochroic halo we possess a method of search for new and rare radioactive substances as yet unapproached for sensitiveness.1 This is due plainly to conditions analogous to those which enable the photographic plate to reveal stars which are too faint to be visually detected, even in the most powerful telescopes. ‘The halo, however, not only integrates the radio- active effects of ages, but presents them to us sorted out according to the laws which govern its formation. But there is another aspect of these conditions. The fact that mica, sensitive to the « radiation and capable of inte- grating its effects over geological time, is found unaffected and unaltered in association with many elements in the rocks —elements present in quantities which are enormous relatively to those we have been considering—appears to shew con- clusively that these elements do not expel any ionizing a rays even at very prolonged intervals of time. This we appear entitled to accept as testimony of the high stability of many rare and common elements throughout geological time. LAVIL. Lhe Striking Point of Pianojorte Strings. By G. TW. Berry *. [Plates X. & XI_] iG is the practice of all pianoforte makers to construct their instruments so that the hammers shall strike the strings near the fixed bridge. The well-known explanation of this, first given by Helmholtz, is that striking the string near its end tends to eliminate the upper partials which are dissonant with the fundamental. Experiments carried out by Hipkins + showed that these upper partials, having a node at the point struck, could still be plainly heard, and it would seem that this explanation of Helmholtz is nota sufficient one. The object of this paper is to show that an entirely different reason can be given for this preference of striking- place. Apparatus. Dr. Barton has published} the results of research work on the vibrations of the various parts of the monochord and * Communicated by Prof. E. H. Barton, D.Sc., F.R.S.E. ’ + ‘Sensations of Tone,’ translated by Ellis, 8rd ed. p. 77. t Phil. Mag. July 1905, p. 149: Dec. 1906, P 576 ; icin abel 4s6; Aug. 1909, p. 933, Striking Point of Pianoforte Strings. 649 violin. The writer has used apparatus of a similar nature, the principle of the arrangement being that a beam of light is set vibrating by the sounding-board or other part of the musical instrument and this beam of light traces a wavy line on a moving photographic plate or film and the result is in the nature of a time-displacement curve. A photograph of the apparatus is shown in Pl. X. fig. 1. Ky, Ky, Ki are sections of sounding-boards of a pianoforte. In this paper the section K, only was used. The sections are glued and screwed to a strong wooden frame, the construction of a pianoforte being followed as closely as possible, and the whole securely bolted to the main wall of the building. The steel string used weighed 0°062 grm. per cm., had a vibrating length of 66°5 cm., and was tuned to c' 261 vbns./sec. The hammer H forms part of a section of a “ tape check” action, and is caused to strike the string by allowing the weight W to fall on the top of the key. _ The weight falls 2°7 em. vertically and weighs 128 grm. _ A concave galvanometer mirror M is mounted on a table on the “hole, slot and plane” system as described by Dr. Barton. In Pl. X. fig. 2, L is a straw forming the connexion between the sounding-board and the table carrying the mirror. The straw has a small wooden base block, and a needle, driven into the block and into the sounding-board, holds it in its place. Two small aluminium nuts grip the other end of the straw. The mirror turns on a vertical axis passing through the hole and slot. A beam of light from an electric are passes through a pin-hole (these are not shown in the photo- graph), falls on the mirror M, and after reflexion passes through the shutter S and is brought to a focus upon the cylindrical drum of the phonograph G. The latter is turned to face the mirror when in use, The weight and shutter are released simultaneously by pressing the bulb B. The films used were the “ Kodak Brownie No. 1”; a strip of the film being fastened round the drum of the phonograph. The mirror and table are carried by an adjustable cross-bar which is not directly connected with the sounding-board except by the straw L. Results. Plate XI. shows nine photographs taken with the apparatus as described. With the exception of No. 9, the only alteration was in the position of the striking-point. From straw to axis of mirror was 0°6 cm. 650 The Striking Point of Pianoforte Strings. From mirror to film was 164:0 em. The magnification is therefore 2 x 164:0+0°6 =547. In No. 9 the film was only 77 cm. from the mirror but the blow was somewhat greater. On examining the photographs it is at once apparent that there are two distinct vibrations recorded on each film, one vibration being about five times the pitch of the other. The vibrations of the string are transmitted through the bridge (shown immediately above the cross-bar carrying the mirror in fig. 1) to the sounding-board, forcing the latter to vibrate with the same period as that of the string. Now the sounding-board K, is fixed only at its two ends, and together with the whole length of the string it must have a natural period of its own. Any blow upon the string or bridge will set the sounding-board vibrating with its natural period, and this vibration together with the forced vibration due to the string appear upon all the films. A regular variation in the intensity of the light will be noticed. This is due to the alternating current passing through the are. The pitch of this oscillation was about 30 per second, but is stated to vary somewhat. The pitch of the string being 261 it is at once apparent that the longer wave is the one due to the natural period of the sounding- board. The straight white line shows the spot of light as the weight is falling, and the effect of the blow on the string is to drive the spot of light in many cases off the film. The film being on a cylinder, when exposed the wave passes immediately from the right-hand edge of the film to the left-hand edge. Taking an average of the respective amplitudes of the vibrations the results have been plotted in the form of a curve in the accompanying diagram. The curve A gives the amplitude of the forced vibration of the sounding-board produced by the vibration of the string. The curve B gives the amplitude of the natural vibration of the sounding- board. Itis the aim of the pianoforte maker to obtain the maximum effect from the string and the minimum effect from the natural period of the sounding-board. It will be seen that this point is reached when the string is struck by the hammer at ; of its length from the fixed bridge. The pianoforte maker has reached the same result by empirical methods. For this particular string, measurement of the striking-point on two English pianos gave z; and ;5 respectively as the point chosen, and on a German piano x. The natural vibration of the sounding-board falls off very i aa a we a ed = Notices respecting New Books. 651 quickly and the sound produced may be described as a thump rather than a musical note. 3 ! 52 a6 iz 7 The writer takes this eed of acknowledging the kindness of Prof. V. A. Mundella when he (the writer) began research work in 1902, and also of Dr. Barton, whose sug- gestions have rendered this paper possible. 14 City Road, London, I.C. March 11, 1910. LANIX. Notices respecting New Books. Physical Science in the time of Nero: being a translation of the Questiones naturales of Seneca. By Jonn Crark, M.A. With Notes on the Treatise by Sir ArcaiBatp Geiktp, K.C.B., D.C.L. etc. (London: MacMillan & Co., 1910.) 10s. net. VEN those who are most absorbed in the methods and aims of modern science must, sometimes at any rate, feel their interest aroused in the scientific world of the ancients. To such this book will be specially welcome because the Querstiones naturales q = SRS ES 652 Notices respecting New Books. of Seneca was the latest statement on the subject of physical specu- lation emanating from classival times. No translation has appeared since (or before) that of Thomas Lodge in 1614; and although it may to some extent be true that further progress has widened still further the breach between ancient and modern times, and thas made translation more difficult, yet it is acknow- ledged that Lodge’s translation is inadequate and the present is an attempt to render more satisfactorily into modern English the equivalent of the arguments and beliefs of Seneca. When these ideas themselves are examined it must be acknow- ledged that the sentence that comes uppermost in our mind is one of Seneca’s own: ‘“ We must, theretore, listen indulgently to the ancients. No subject is perfected while it is but beginning.” Only a little way had been travelled along the path of discovery ; and though, here and there throughout this book, occur definite trains of reasoning which even now possess cogency, yet, most of the arguments may be dismissed, often as irrelevant, generally as non- probative of the assertions made. It is specially difficult to be certain in making a translation, when the original is a deid language, that the precise equivalent of the original ideas is reproduced. Every word which may be used in the translation has a connotation which in part at least must be foreign to the author of the original. All we can say is that the present translator has judiciously chosen his modern equivalents. The translation reads well. We are very seldom pulled up by anything which sounds impossible. One such case may be cited, referring to the artificial production of rainbows. “Tf you will at any time watch a fuller at work, you will observe the same appearance: when he has filled his mouth with water and spirts it lightly on the clothes stretched on pegs, the air thus besprinkled exhibits plainly the various colours that shine in the bow.” We believe that this is a correct literal rendering but we understand neither the use of the operation described in the cleansing of clothes nor the skill necessary to make it yield effective bows. We hazard the guess that the ‘ mouth’ is a technical term of lost significance differing from its usual one. Amidst all this medley of perverse arguments and curious theories we yet detect the spirit of one who could rise above the sordid things of his courtly life and take the keenest interest in physical enquiries. ‘‘ Life would have been a useless gift were I not admitted to the study of such theorems. What cause for joy would it be to be set merely in the number of those who es ae aaa Away with the priceless boon! Life is not worth the heat and the sweat ...... The full consummation of human felicity is attained when, all vice trampled under foot, the soul seeks the heights and reaches the inner recesses of nature.” i eae LXX. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 448. } November 17th, 1909.—Prof. W. J. Sollas, LU.D., Sc.D., F.R.S., President, in the Chair. 5 aa following communications were read :— 1. ‘The Geology of Nyasaland.’ By Arthur R. Andrew, F.G.S., and T. Esmond Geoffrey Bailey, B.A., F.G.8S. With a Description of the Fossil Flora, by E. A. Newell Arber, M.A., F.G.S.; Notes on the Non-Marine Fossil Mollusca, by Richard Bullen Newton, F.G.S. ; and a Description of the Fish-Scales of Colobodus, etc. by Ramsay Heatley Traquair, M.D., F.R.S., F.G.S8. 1. The greater part of Nyasaland consists of crystalline rocks, which comprise :— (a) Highly metamorphosed sedimentary beds, including graphitic gneisses with lmestones, and muscoyite-scliste. (6) Foliated igneous rocks, especially augen-gneiss, derived from granite or syenite. (c) Plutonic intrusions,usually granite or syenite, more rarely gabbro. In two localities nepheline and sodalite- syenites are found; these are perhaps of the same age as the similar post-Waterberg and pre-Karoo syenites of the Transvaal. 2. In the north-western corner of Nyasaland is a somewhat altered sedimentary series, which forms the Mafingi Hills. It consists of a thick accumulation of quartzites, grits, and sandstones of pre- Karoo age. 3. The Karoo System is represented both in the north and in the south of Nyasaland ; in the north it occurs in patches, which owe their preservation to faulting, It has afforded remains of freshwater lamellibranchs (Paleomutela), fish-scales (Colobodus), and species of G'lossopterts. 4. Recent lacustrine marls and sands are found at great heights above the present level of the lake, and as much as 15 miles away from its margin. 5. Pumiceous tufts, associated with recent gravels containing pebbles of Tertiary lava, are found in the extreme north of the country ; across the border, in German East Africa, fertiary and recent lavas and tuffs are widely distributed. 6. Nyasaland consists of high plateaux rising irregularly one above the other. The Nyika and Vipya plateaux were doubtless at one time continuous as ‘a platform of erosion,’ which originated after the main faulting of the Karoo in Northern Nyasaiand, and before the formation of the great Nyasa fault-trough. 2. ‘ The Faunal Succession of the Upper Bernician.’ By Stanley Smith, M.Se., F.G.S. 654 Geological Society :— 3. ‘Notes on the Dyke at Crookdene (Northumberland), and its Relations to the Collywell, Morpeth, and Tynemouth Dykes.’ By Miss M. K. Heslop, M.Sc., and Dr. J. A. Smythe. The dyke at Crookdene is exposed in the bed and banks of the Wansbeck about 15 miles above Morpeth. It is intruded along a fault-fissure in beds of Bernician age, and apparently comes to a natural head. ‘The basalt is characterized, microscopically, by narrow lath-shaped felspars and curved augites. Macroscopically, its most interesting feature is the occurrence of large inclusions of a felspar, which is shown by chemical analysis to be closely allied to anorthite. ‘Che exterior of the inclusions in contact with the cround-mass is strongly zoned, the latter showing a slightly chilled edge; the individual crystals are intergrown and are cracked, faulted, and in places completely shattered. In no case is the dislocation great, and, in fact, the crystals seem to have burst in situ. These phenomena point to a plutonic origin of the felspathic inclusions and connect them with the porphyritic felspars of the Tynemouth Dyke, for which a similar origin has already been suggested by Dr. Teall ‘he dyke which comes to a head in the coast-section at Collywell, about 24 miles distant, shows almost precisely the same peculiarities. Chemical and microscopical examination of the two basalts and their felspathic inclusions show them to be practically identical. Considering these facts. and the general field-relationships of the dykes, it appears probable that they belong to the same intrusion. The work of Dr. Teall upon the dykes at Tynemouth and Morpeth has been amplified by further observations. The re- semblances among the four dykes are so strong as to render it probable that they are derived from a common source. ‘The observed differences are such as could be readily accounted for by differences of physical condition operating during the period of consolidation of the dykes. December Ist, 1909.—Prof. W. J. Sollas, LL.D., Sce.D., F.R.S., President, in the Chair. The following communications were read :— 1. ‘The Tremadoe Slates and Associated Rocks of South-East Carnarvonshire. By William George Fearnsides; M.A., F.G.S., Fellow of, and Lecturer in Natural Sciences at, Sidney Sussex College, Cambridge. 2. ‘On some Small Trilobites from the Cambrian Rocks of Comley (Shropshire).’ By Edgar Sterling Cobbold, F.G.8. 3. ‘The Rocks of Pulau Ubin and Pulau Nanas (Singapore), By John Brooke Scrivenor, M.A., F.G.S. Pulau Ubin and Pulau Nanas are islands set in the eastern entrance to the Straits of Johore, and consist of igneous rocks ——— The Rocks of Pulau Ubin and Puluu Nanas. G55 of consilerable interest. Pulau Ubin is composed mainly of horn- blende-granite, but a pyroxene-bearing microgranite is found also; while the hornblende-granite is cut by rhombic-pyroxene bearing * veins and also contains angular masses of rock resembling the veins. The following grouping of the Pulau Ubin rocks (with which is included a rock found in the granite of Changi) is adopted. (1.) Normal hornblende-granite with a little monoclinic pyroxene. (IL.) Pyroxene-microgranite with dark masses resembling III (i). CLL.) (i) Porphyry, with peculiar spongy masses of hornblende. (ii) Masses of a rock at Changi having the mineral constitution of an amphibole-vogesite. GEY..) (i) Veins of quartz-norite in the normal granite. (ii) Veins and masses of enstatite-spessartite in the normal granite. (iii) Masses of quartz-biotite-gabbro in the normal granite. Pulau Nanas consists of dacite-tuffs and dacite which are referred to the Pahang Volcanic Series, of Carboniferous or Permo- Carboniferous age. The tuffs and lavas have been altered by the adjacent granite of Pulau Ubin, and contain much secondary biotite and hornblende. They also contain some fragments that appear to be altered chert, but their most remarkable feature is the presence of fragments of altered granite. The author discusses the mutual relations of the different rocks, and arrives at the following conclusions :— (1) The normal granite of Pulau Ubin is hornblende-granite, the age of which is certainly post-Triassic and pre-Eocene, perhaps post-Inferior-Oolite and pre-Cretaceous. (2) Veins of quartz-norite and masses of quartz-biotite-gabbro, and veins and masses of a fine-grained rock which may be described as enstatite-spessartite, are found in the normal granite of Pulau Ubin. These point to an early differentiation of a granite and a gabbroid magma, perhaps in pre-Cretaceous times, and they are. referable to rocks in Borneo and Amboyna. , | (3) A pyroxene-microgranite and porphyry on Pulau Ubin, and a rock at Changi, having the mineral constitution of an amphibole- vogesite, are described. ‘heir relations to the other rocks are not clear. (4) The dacite-tuffs of Pulau Nanas contain fragments of granite which must be of pre-Carboniferous age, and are referable to the granite of Amboyna. (5) The fragments of granite, and perhaps certain pebbles of schorl-reck, are the only evidence found us yet in the Malay Peninsula of pre-Carboniferous rocks. 4. ‘The Tourmaline-Corundum Rocks of Kinta (Federated Malay States).’ By John Brooke Scrivenor, M.A., F.G.S. Overlying the limestone on the west side of the Kinta Valley is 656 Geological Society. a thin cap of schists, with which are found. certain rocks the two chief constituents of which are tourmaline and corundum, ‘They are often carbonaccous; and, in the many variations found, white mica, brown mica, pleonaste, rutile, and metallic sulphides oceur, The tourmaline-corundum rocks contain certain structures which are described in detail. ‘hese consist of round and oval cavities and bodies, the largest of which are about 6 millimetres in greatest width. Nothing can be proved regarding their origin, but the description of the rocks is summarized and an hypothesis adopted regarding their history, as follows :— (1) The tourmaline-corundum rocks of Kinta consist of varying amounts of tourmaline, corundum, carbon, white mica, spinel, and other minerals. (2) They contain cavities about 6 millimetres in greatest: width, generally bordeied by a layer of corundum grains, with tourmaline grains on the inside of this border. Sometimes solid bodies similar in size and shape to the cavities occur. They are composed of tourmaline and corundum: the former mineral, gencrally speaking, being more abundant towards the centre. Such bodies also show concentric structure. (3) Smaller bodies occur, sometimes, but not always, accompanied by the larger cavities and bodies. They consist of tourmaline, of corundum, and of tourmaline and corundum. When both minerals are present, the corundum forms a shell to a nucleus of tourmaline. ‘The corundum bodies frequently show concentric structure. (4) The tourmaline-corundum rocks are associated with other rocks, which lead to the conclusion that the structures described in 2 & 3 are the result of replacement of the materials of pre-existing bodies at the time of extensive granitic intrusions. (5) They also are associated with rocks which point to the original beds having been laid down under conditions similar to those that obtained when the Pahang Chert Series was deposited. (6) As tourmaline-bearing partings in the limestone at Changkat Pari constitute a case of selective metamorphism, so it is thought that the tourmaline-corundum rocks as a whole mark a process of selective and intense metamorphism in beds associated with schists overlying the Kinta Limestone. (7) These beds were probably chert and silicified limestone, both being In many cases carbonaceous. (8) The larger cavities and bodies mentioned in 2 are believed to be the result of replacement or partial replacement of oolitic grains. (9) The smaller bodies may be, in part, the result of replacement of the materials forming casts of radiolarian structures ; in part, the result of the further development or replacement of spots scen in soft partings in the iimestone at Changkat Pari; and, in part, the result of the replacement of small oolitic grains. eee ee = — - —< — —_—_——— s — = See = SS SS Fe ae = —— = == - — : = ar - ~ SS ES ee ome - ~ 120 4) D g oshfews igh 90 Phil. Mag. Ser. 6, Vol. 19, Pl. VIL. 60 EG. i2* S VATS: 30 DISTANCES (rrcitinlied by) *»—- 0 NETTLETON. ——<—« FYILYY FAW LZ JOLY & FLETCHER. Phil. Mag. Ser. 6, Vol. 19, Pl. VIII. Riek Thorium and Radium Halos in Biotite. x 150 diams. soLy & FLETCHER. Phil, Mag. Ser. 6, Vol. 19, Pl. IX. Big, 2: Group of four developing Halos in Biotite. x 75 diams. Halo in Biotite. x 450 diams. Sbowing ring due to Ra A, Phil. Mag. Ser. 6, Vol. 19, Pl. X. BERRY. Fia. 1. 3 ~ BERRY. Phil. Mag. Ser. 6, Vol. 19, Pl. XI. = ie } , of : CL Ae ace A THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] MAY 1910. LXXI. The Effect of Dust and Smoke on the Ionization of Air, By A. 8. HEvs, WA. D.Sc., McGill University, Montreal *. LARGE number of observers, using the instrument, or ion-counter, devised by Ebert, have tound a considerable excess of positive over negative ions in the atmosphere. The ratio of the number of positive to the number of negative ions has been found to vary much with meteorological and local conditions, but the mean value 1:17, obtained by Simpson f in Lapland during a vear’s continuous observa- tions, may be taken as a fair average result. The number of ions of each kind per cm.? in the atmosphere is of the order 1000 to 2000, so that an excess of 17 per cent. corresponds to a numerical excess of the order 200 per cm. Now, if an excess of this magnitude were maintained for a height of 4 kilometres, the positive charge in the air would be 4x 10° x 200 x 4°6x10-” or 87x10-? E.S.U, per cm. of the earth’s surface, This would be far more than sufficient, indeed 75 times too great, to balance the corresponding negative charge on the earth’s surface, which has been found by C. T. R. Wilson to be of the order 5x10-* E.S.U. per cm.? Doubtless, as the results of balloon ascents prove, the distribution of electricity is different at various altitudes, and * Communicated by the Author. + Phil. Trans. A, cev, (1905), Phil. Mag. Ser. 6, Vol. 19. No. 113. May 1910. 2U 658 Dr. A. 8. Eve on the Lifect of the excess of positive electricity in the air near the earth’s surface may be expected to be greater than at a considerable altitude. On the other hand, since equal quantities of positive and negative electricity are produced by ionization. such an excess as 17 per cent. of positive ions is surprisingly large and requires explanation. It has been found by Smirnov* that the average density of charge (4 km. to the ground) is 9x 10-” E.S.U. per cm.*, and this is equivalent to the charge on about 2 ions. Also at 700 m. he found p to be 3:4 x 10-°, and this is equivalent to the charge on about 8 ions. Hence we see that the nega- tive charge on the earth, and the positive charge in the atmosphere, may be accounted for by a small excess of positive over negative ions, something of the order of one _ per cent. of the total positive or total negative ions present. Inasmuch as negative ions diffuse more rapidly than positive, it may be urged that the large observed excess (17 per cent.) of positive ions may be due to the greater loss by diffusion of negative ions to the sides and top of the testing vessel in the Hbert apparatus; but the arrangement is such as to reduce any such diffusion loss to a minimum. Also it will be remembered that there are.in the atmo- sphere large ions, first detected by Langevin t, which pass through the Ebert machine mainly without detection by the testing vessel and electroscope. These have a mobility of about 1/3000 cm./sec. per volt/em. As to the number of these large ions there seems to be much discrepancy among observers, for Langevin found them about fifty times as numerous as the small ions, Gockel{ 1000 per em.’%, Daunderer § 6000-7000, and Pollock 600-5500. Valuable work has been done at Sydney by Pollock ||, who finds three types of ions present, small, intermediate, and large. His results may be summarized thus :— Type. Number of Ions. Mobility. OTE earn ae 0 to. fd 7 About 1:0-1°8 Intermediate ... 200 to 1000 1/15 to 1/150 Thapoe voices 40 600 to 5500 1/1280 to 1/3370 The conditions at Sydney appear somewhat singular, but it is possible that obseryers elsewhere have grouped most of * Acad. Set. Pétersbourg, Bull. ix. p. 759, 10th May, 1908. + Comptes Rendus, exl. p. 232 (1905). t Phys. Zeit. x. p. 396 (1909,. § Phys. Zeit. x. p. 113 (1909). || Science, 11th June, 1909. Dust and Smoke on the Ionization of Air. 659 the intermediate ions with the small ions. Pollock finds that © the number of intermediate ions appears to have in the main a linear relation with the amount of water vapour present in the atmosphere. It is also important to note that Pollock finds more negative than positive large ions, the mean values of their numbers being n+, 1914; n—, 2228; n—/n+, 1°16. This, of course, suggests that more negative than positive small ious transform into large ions, a result which might he ooo from the known physical properties of the two kinds of ions. The ionization of the atmosphere is difficult to investigate on account of the number of variable and unknown conditions, mainly meteorological. Laboratory experiments have been made with much trequency and accuracy, but in these care has been taken to avoid as far as possible all the disturbing elements, such as dust, smoke, water vapour and mist. It seemed then desirable to make some experiments on the ionization of air as it occurs, and without removal of impurities, using a strong and constant ionizing source. A large paper or metal cone (fig. 1) was, therefore, placed Bie TT. above the Ebert instrument, and the y rays from 14 mg. of radiam bromide, about 4 to 2 m. distant, were made to traverse the upper part of the cone, so as to ionize the air on its passage to the testing vessel, which was, of course, screened by thick lead from the rays from the radium. The results obtained by Bates and the writer were published in a letter to ‘Nature,’ llth March, 1909. The observations were made in a large class-room in the Arts Building at McGill University, under the very dry conditions of a Canadian winter. The temperature was about 18° C., and smoke and dust were avoided as far as possible, apart from 2U 2 ~ 660 Dr. A. 8. Eve on the Effect of any special precautions, so that doubtless there was much dust in the air. Taking the ionic charge to be 4°6 x 10-19, and assuming that there were no multiple-charged ions present, then the number of ions detected in the testing vessel of the Ebert instrument for various distances of the radium were as follows :-— Series. m+ - n— Ratio. ee One 51000 A6A0N0 1°10 7 LANs 27000 13700 (197) Be Te 30200 22800 1°33 ene TE 18000 16000 j toil b= Eye aa 9900 7850 1:26 Mean 1:20 Without radium... 1740 1420 1°23 Thus, with a wide variation of conditions there appears to be an excess of positive over negative ions equal to about 20 per cent., whereas we know that equal quantities of positive and negative electricity must have been produced under all conditions by the ionization. Moreover, when an earthed, wide-meshed gauze cylinder was placed in the cone and over the testing vessel so as to increase the loss by diffusion, the number of ions of each charge was certainly diminished considerably, but the ratio n+ /n— was not much altered. Hence the inequality was not due to diffusion alone. This deduction then appears inevitable, that more negative than positive ions rapidly become either attached to dust or otberwise transformed into large ions, and pass without detection through the testing vessel of the Ebert apparatus. It might. be expected that, because the small negative ion is more mobile than the small positive ion, an excess of the negative would he shown by the electroscope. But it is not so, because ail small ions are stopped in the testing vessel, and previous to the arrival there of the air examined there has been a greater loss of negative than positive small ions in virtue of the greater mobility of the former. And this loss has taken place in two or three ways: by diffusion to parts of the apparatus employed ; by diffusion to centres such as dust, mist, and smoke in the air: and, as Pollock has shown, by the apparent loading due to water vapour, even when far from saturation. The fact that these three operations act in the same direction, all causing an excess of small Dust and Smoke on the Ionization of Air. 661 positive ions, makes the measurement of their relative mag- nitudes a dithcult and intricate matter. In order to investigate this point further, apparatus of the following type was used (fig. 2). Fourteen mg. of pure radinm bromide in two test-tubes and a cylinder of lead about 2 mm. thick, were placed about 10 cm. beneath the timned iron cylinder V (17 cm. long, 17 cm. diameter), from which the ionized air passed along the brass tubes (3 cm. diameter) and round an insulated brass central cylinder T (0°7 cm. diameter, 12-3 cm long), which could be charged to any desired potential so as to remove some or all of the small ions in the stream of air. Three testing vessels, A, B, C (C, which was just below B, is not shown in the figure), with their corresponding electro- scopes, could be used either separately or together. A had an axial rod, 8 cm. long, 0°2 cm. in diameter. B had a comparatively large cylinder, 8 cm. long, 2°2 em. in diameter, whilst the outer tube was 3:1 cm. The third testing vessel, C, and electroscope were those of the Elert apparatus. ‘The hand-wound spring turbine of the instrument was used to eause the current of air, xzbout a litre per second, moving with a velocity of about 120 cm. per seeond along the tubes, so that the air passed from the ionizing chamber V to the testing vessels in about one second. It was found that the ionized air on arrival at A caused the electroscope leaf to move at the rate of 130 divisions a minute when T was earthed, but a moderate voltage on T, about 70 volts, reduced this to about 1 div./min. A larger voltage, say 400 volts on T, produced little further decrease, so that the ions which move to the axial rod of A were mainly small ions (see curve 2, fig. 3, p. 662). In the same way, if the central rod of A was kept at 230 volts, B caught but one per cent. of the ions detained when A was earthed. 662 Dr. A. 8S. Eve on the Effect of The mobility of the ions was also calculated to be about 1:2 cm./sec. per volt/em. This was done by raising T to a few volts, when the current measured in terms of divisions a minute with electroscope B was— Voltage on T. -+ ions to centre. —ionsto centre. Ratio n+/n— UP es Cage 21:1 ILS Lik Bigs eek Mae 181 16°5 1:10 ‘OPtificial| Fog peciat|aeaeai l IG |_| 200 Volts onT 600 800 When two balls of loosely wound wire were placed in the tube near V, the current measured at the electroscope B was reduced by diffusion,— + ions to centre. —ionstocentre. Ratio n+/n—. No wire balls... 20°3 18:2 1A Wire balls ...... 6:3 5°0 1°26 This indicates that more negative than positive small ions are lost by diffusion to the wire balls. When the lengths of the tubes between V and A were varied, it was found that the ionization current was less for long than for short tubes. The loss is, of course, due to recombination and to diffusion to the sides of the tubes. Some experiments were made to test the effect of altering the length of the tube between V and A,— Length. 10 cm. 200 cm. n+ /n— 1:10 1°35 ee ee a oe ee Dust and Smoke on the Ionization of Atr. 663 This shows that diffusion is an important factor in the value of the ratio. This has long been recognized. With the longer tube (2 metres) a loosely wound ball of fine wire was inserted midway along the tube. Humidity + tocentre — to centre (relative). D/m. D/m. eae: Marwire ball... . 50- 90°5 66 Loe Wire ball...... 50 56 36 1°56 Wire ball...... 90 bi 45 E27 The wire ball increased the loss due to diffusion, especially that of the negative ions. An increase of the humidity of the room, however, while affecting the positive ions little, reduced the velocities of the negative ions, and therelore their loss by diffusion was lessened, and the ratio n+ /n— was decreased. Tubes of zine, aluminium, and brass were tried, but the material made no difference in the ionization current. Variations due to Atmospheric Conditions. Observations were next made on different days to find the influence of various atmospheric conditions on the ionization eurrent at B. ‘The room was well ventilated with open windows, the velocity of the air current was maintained con- stant at each reading, and the radium was left zn situ. There was no variant except the type of air examined. The voltage on B’s central system was about six hundred volts, and the results obtained for the ionization curreut expressed in divisions a minute were :— + ions to centre. —ions to centre. Ratio n+./n— Weather. Liss Lea 1:00 Very clear. 16°0 £5'0 1:07 Ciear. 13'5 12°8 1:05 Smoke and mist. 12°3 sie a 1:09 rh iO With high voltages the ratio n+ /n— is generally nearer to unity than with low voltage, due to increased approxi- mation towards saturation. The above results are remarkable, because with an electric force of the order 1000 volts/em. in the testing vessel B, not only was the ratio n+/n— not unity, but the number of ions caught when the air was clear was about 50 per cent. greater than when the air was smoky or misty. altitaenieal 664 Dr. A.S. Eve on the Efect of A systematic series of similar measurements was, therefore, begun and continued for more than a month, whence the results in the table following :— TABLE I. Divisions a minute. + to — to Ratio centre. | centre. | 2+/n—. A au Sept, 25... 20-0 20°0? 1:00? 66 N., clear after rain. Ss ver Te ae RED 14:0 113 95 N.E., rain, misty. A 2B asa Dos 158 1°15 95 W., heavy rain. Octs 9 cit 124 11°2 1:08 76 S. wind, smoky, misty. he ee he: Monn 16:0 1:04 65 S. wind, clearer. Sy) RAO eRe: 146 115 72 Morning S.W., misty. sis 186 17-9 1-04 ft. Atternoon, clearing. x id..4 188 16:9 ime dD S.W. le eiet y Lacd 14:5 1:20 54 S.W. PG cee) eel, 185 1-06 86 W., clear. BS ech 20 14 1-04 17 S.W., clear. aes (18:0 16:0 1°12 ase Smoking in room.) (13°6 12-2 1-11 sie Bunsen flame also.) ae (128 Ill Tete ae Kettle boiling freely also.) pi loreeel an bone 179 1:07 67 N.W. Ree Opene mim be) 17-4 1:08 65 S.W. aes (9:0 70 1-28 ane Kettle boiling over Bunsen.) Steal A Mere 11:8 1:13 85 Very smoky and misty. see ant: ek dsb 13:8 1:06 95 8.W., clearer after rain. Aye eee pee 5) 17-0 Til 82 : Be ec SIE ie Lee 11°5 1:24 88 S.W. wind, smoky, misty. OO e ne! peel 19:0 1-11 70 S.W., very clear. Nov.2 ...| 15:0 14:2 1:06 92 S.W., misty, a little rain. SOS NES, ae! fs 17:2 1:06 85 S.W. Mean... 1°10 Nach reading is the mean of about six readings, taken in a room about 7x 4x5 m.? with the windows open. The ratio n+ /n— is a very variable and complex number, and its magnitude may be dependent on humidity, dust, smoke, electric force, intensity of ionization, diffusion, and type of apparatus employed. Moreover, itis hard to measure, because it is the ratio of two quantities each of which is variable. It is clear, however, from the results in Table I., that with a constant ionizing source at V and with a constant air current, the ionization current in B varies greatly with the type of atmospheric conditions. Dust and Smoke on the Ionization of Air. 665 The maximum values on very clear days were :— -+ to centre. —tocentre. Ratio n+/n—. 80) ag cane 21:1 19-1 11 Sept. 25...... 20:0 20:0 ? 1-00 ? and the minimum values, occurring on smoky or misty days, were :— Rb. see 1271 LEZ 1:08 G4 ae 13°3 11°8 PLZ | ae 14°4 115 1:24 Hence, the number of ions detected in very clear weather was nearly twice as great as on a dull, smoky, misty day. Large values were always obtained with “clear shining after rain,” when the air had been largely freed from impurities. Indeed, apparatus of the type used might prove efficient for the purposes of testing the degree of purity of the air, at any time or place, as regards the amount of smoke, dust, or mist present. It should be explained that the position of the Physics Building at McGill University is such that a slow moving wind from the city brings with it much smoke from the soft coal used in the fires of the ever increasing factories, whilst a wind from the west or north-west is of exceptional purity. The maxima and minima recorded above indicate this difference well. . If the window was shut, an artificial fog of tobacco-smoke, steam from a boiling kettle, and vapours from the flame of a two-ringed Bunsen burner were found to lower the ionization current still further to about one-third of the maximum. By blowing tobacco-smoke directly into the cylinder V, the ionization current at B may be reduced to a very small value. This is merely a repetition of the experiment of Owens*, who first showed the marked diminution of the ionization current between two plates when tobacco-smoke was introduced between them. What then becomes of the ions? The explanation usually given is not altogether satisfactory, namely, that “in dusty air, the rate of recombination is much more rapid than in dust-free air as the ions diffuse rapidly to the comparatively large dust particles distributed throughout the gas”? (Rutherford’s ‘ Radioactivity,’ 2nd edition, p. 42). And again, “ The effect produced by dust is easily explained, * Phil. Mag. Oct. 1899, i eee ee 656 Dr. A. 8. Eve on the Lifect of as the dust particles are in all probability very large compared with the ions ; thus, if a positive ion strikes against a dust particle and sticks to it it forms a large system which is much more likely to be struck by a negative ion and neutralized than if the positive ion had remained free ;. in this way the presence of dust will facilitate recombination of the ions ” (Sir Joseph Thomson, ‘Conduction of Electricity through Gases,’ 2nd edition, p. 20). It may be objected to this explanation that pairs of small mobile ions will recombine more readily than pairs of ions, one large and one small. In fact, if the rate of production of ions is constant, the rate of recombination « may be expected to decrease, and, therefore, the total number of ions will increase, if dust or smoke is present. It is true that the small ions tend to disappear, but large ions replace them and recombine more slowly, so that there is on the whole an increase. Further experiments on this point will be described later in this paper ; for the present it is sufficient to state that the variations recorded from day to day in Table I. are attributed to the relative presence or absence in the atmosphere of dust, smoke, or other centres, which cause the many small ions rapidly to become large ions and thus pass the testing vessel without detection. Moreover, it is probable that the small negative ions, owing to their greater mobility, pass in larger numbers to the particles of dust or other centres than do the positive ions, and on this account an excess of small positive ions is indicated by the electroscope, whilst an excess of large negative ions passes through the testing vessel. The degree of humidity, and diffusion to the parts of the apparatus may be ecntributory to the observed excess. Ions from Flames. In order to ascertain if humidity affects the ratio of the nuniber of positive and negative ions, a kettle was boiled vigorously in the room (7x4x3m.°) over a two-ringed Bunsen burner, at a distance of 3 metres from the apparatus. With the radium in position it was found that the ionization current was diminished about 50 to 70 per cent., but this diminution was produced also when the kettle of water was absent. The effect was due to the ions from the flame alone. A remarkable result may be recorded. If the ionization current was measured at B, with 1adium near V (fig. 2), with the Bunsen flame burning in the room, and if an increasing potential was given to T, then the readings Dust and Smoke on the Ionization of Air. 667 obtained are shown in curve L., fig. 3. It will be seen that a very large potential on T does not withdraw the ions, which are, however, detected at B, although the potential is lower. Without the flame curve II. was obtained. When the radium was removed from the room, similar results were obtained when the gas was burning, but the initial value for zero potential on T was less. Potential on T (volts). Div./min. at B. ) | 4-0 1500 3°0 2000 2°4 3000 2*L 4000 bY 5000 kee The whole room contained a large number of the slow- moving ions due to the flame, only a few minutes after shutting the windows and lighting the Bunsen burner. Over a large city the contribution to the supply of large ions, but slightly mobile and slow to recombine, by fires, furnaces, flames, and smoke, must be considerable, and this fact may help to account for the discrepancies among different observers in their determinations of the number of large ions. With the windows open, without flame, smoke, or radium in the room, the “ natural ionization” in the Physics Building gave about 0-4 D/m, and it has been found that this is equivalent to about 3000 ions per c.c.; we can, therefore, determine approximately the number of ions detected at B under the conditions of the experiments. Div./min. No. of ions per em’. Natural ionization ...... Or4. 3,000 1 Nal ERS OS a a 4-() 30,000 Radium, clear weather... 20°0 150,000 fe smoke and mist 11:0 80,000 He artificial fog ... 9:0 70,000 There was an essential difference between the natural smoke and mist entering the window and that made artifi- cially by flame, in that the ions due to the natural fog were mainly removable by a moderate potential on T; not so for the artificial fog. I attribute this to the very large persistent ions due to the Hame. It may be worth recording that, in order to determine the qa oS : pe — DP hal vaw os" * Po = y bf ig a — 7 668 Dr. A. 8. Eve on the Lffect of effect of dust, an old coat was struck smarily with the hand, — and the cloud of dust was found to be strongly electrified with large ions giving 30 to 60 Div./min. at B, corresponding to 500,000 ions per cm’. The charge observed was, doubtless, due to friction, as with the dust in Lichtenberg’s figures. Some experiments were made with various fine dusts, but these soon affected the insulation, and after recleaning the instrument these tests were for the time abandoned. Again, a positively charged wire from a Wimshurst machine was suspended at the mouth of a funnel which replaced the cylinder V. No negative ions were detected at B because they were drawn to the positive wire, but the number of positive ions was very large and the ionization current at B was great, giving about 50 Div./min. Smoke and Ionization Current. To investigate the effect of smoke on ionization, on the recommendation of Professor H. A. Wilson, arrangements were made to work at a much higher potential. The single leaf on electroscope B was replaced with a leaf of paper with Dutch-metal foil pasted on each side; this was suspended freely by a short strip of single Dutch-metal foil. The cylinder and leaf were first connected to a leyden-jar and Kelvin Telephone Circuits. 675 have only a few entries. But calculation is easy enough. Write r(@) in the shape a+. Then cosh r(@) = cosha.cos@+isinha.sin8 and sinh +(@) = sinh a.cos8+icosh a. sin £, so that the calculation is easy for any values of 2 and @. I will first make a few statements, the foundation of my reasoning. lL. It is well known that any looped line or cable con- taining a receiving instrument at the end of the loop has its equivalent in a cable whose far end goes to earth through ihe receiving instrument, the outer coating of the cable being everywhere to earth or of zero potential. The mathe- matical proof of this is very easy and I shall not give it, although I do not know that it has been published. 2. In all difficult problems cables are so long that for all practical purposes we may suppose them to be infinitely long. Thus it is well known to practical people that when lines of 30, 40, &c. miles of the same cable are experimented upon, the rate of diminution of current per mile is the same except for regions near the ends, and these end effects are much the same for all lengths of line except in short lines. 3. In any long uniform cable at equidistant points A, B, (, &e., when a simple periodic current is flowing, its value at Fidae points being Ca, Cp. Co, &e., the ratio of Cy to Cg is the same as that of Cg to Co, &e. This is true except near the ends. 4. In a long cable with perfectly similar contrivances, A,A,, B,B,, C,C,, &e., at equidistant places A, B, C, &e. ; when a simple periodic current is flowing, its value at corre- sponding points being Aj, Ay; By, B,,; C;, C,, &e., the ratio ” A, to B, is the same as that of B, to C,, &c., and these are also the same as the ratio of A, 40,39, 43g )t0- Ci f&e. We may say the same about the voltages at peneepioadine points. It is seen that I assume an infin.te line, but even in the case of an infinite line my assumption is that there is some diminution in the amplitude of the current from A, to B,. If the circuit behaved like a mere Ohmic resistance _the current would be the same everywhere but the voltage would not. 5. Neither telephone nor telegraph signalling can be studied correctly through the behaviour of periodic currents, but it has been found in practice that in telephony, if we assume that all currents are periodic functions of the time, the frequencies being not much less than 600 nor much 2X2 | it I 676 Prof. J. Perry on greater than 1000 per second, the most important being 800, good transmission of such currents is found to mean good transmission of speech. 6. It is quite a usual thing to find that experimenters are using transmitting and receiving devices which are not the best for a particular cable. For a receiving instrument, for example, there is a best resistance and inductance (or com- bination of inductance, capacity, back E.M.F., &e.). In every case yet studied by me (but I can imagine others) the most suitable receiver causes the current and voltage near that end of the line to be different from what they would be if the line were infinitely long. Thus, for example, with simply periodic currents in the standard cable it will be found that at any place far from the ends, voltage divided by current is of the form 7(—45°) or e—ai. If the telephone or other receiving instrument has a resistance R and an inductance L each of which is proportional to the square of the number of turns of wire upon it, and if the effect on the instrument (such as the deflexion of a coil, &c.) is propor- tional to the current turns received, this effect is a maximum if R=a and if L=a/q, where q is 27 times the frequency. It is easy to show that if we wished the receiving instrument to produce the same effect as an infinite prolongation of the cable, it ought to have a resistance R and a negative induct- ance L=—a/q, which is a capacity l/ag. It is then not merely when unsuitable instruments are used, but also when the most suitable instruments are used that we have end effects. 7. A line with equidistant similar contrivances when they are not too far apart may be replaced (mathematically) by a line whose properties (as to resistance, inductance, capacity, leakance, &c.) are continuous, for many purposes of calculation. The simplest case of uniform distribution of properties in an infinite line was studied in my paper read before the Physical Society in 1893. Let v and ¢ be the voltage and current at any place whose distance from the sending end is w; let vwysingt be the voltage when x is 0. If 7, 1, s, and k are the resistance, inductance, leakance, and capacity per unit length, we have v= ve7™ sin. (gt ~ ga)y.\. =». where kgr CG eae 3 lene VAIN Leela . ae @) y Rhy ie x “ . -f. 4 a ia ng sf % ; . b Telephone Circutts. 677 is the value of h if the minus sign be taken and the value of gif the plus sign be taken. The following formula tor the current ¢ is useful— l = gas SSI (g¢—gie+ tan~'y ~ tan-17), (3) I showed in tables how good signalling was affected by changes in the values of the constants and specially by alterations in! and s. There was nothing very new in my paper, but it enabled the non-mathematical man to under- stand some of the discoveries of Mr. Heaviside. At that time I saw no easy way of calculating how the introduction of detached equidistant contrivances caused discrepance from the idea of continuously distributed properties. In much that follows, lg will be so large compared with r that, taking s=0, we have k us na=taft, g=q Niel. If s is small but not negligible, g is as before, but il sl k h= s(r+ lt: As for other uniform distribution of properties, it is only necessary here to speak of those which may immediately be put in the above shape. For example, besides having merely y and / in series in the line, we may have something of the nature (mathematically) of 4; farads per mile, so that instead of r+lqi we have ; AL By de r+ lgut+ La, or r+gi(i-za) 1971 1 This isas if we had merely the resistance 7 and an inductance i! = [——,. kyq’ Again, instead of the leakance s which is the reciprocal of a resistance R, let us have also something of the nature of an inductance L, say, and assume R to be small relatively to Lg. Then instead of s we have pi Rite SeYN/ i4 2 « R+ Lgi = £2,?~ Lg’ 678 Prof. J. Perry on Thus the common expression s+/qi becomes R a (k= pa 1293 qt ny For preliminary calculations it is generally sufficiently correct to assume R/L?q? to be zero. We may now consider that the telephone line has, per mile, : : i a resistance 7, an inductance //= /— 73 a leakance s=0, . 1 e zi s ° and a capacity k’= k— Tee It will be found desirable in 4 almost all cases to have /’ so large that we may take r hk! in ea and’ g= 9 Vhs Assuming these conditions to exist we have, very nearly, ki 1 [rye os TTI C= 7 voc VET sin (gt —ge/k'l'). For a line of, say, L miles in length the amplitude of the a7 pa¥l. In fact the arriving current is the sending current divided by e. It will be found that in practice we never have so small a value of h as this gives, and we generally try to make hf as small as possible. We can approximate to the above conditions by inserting contrivances at equal distances in the line. I will take a few examples. IJ prefer just now to consider a cable in which I have recently taken an interest ; »=18 ohins per mile, k=°055 x 10-§ farad pér mile, s=0, J=0. I take g to be 5000 (that is, a frequency of 800). arrivin eg current is a maximum if 1. This cable with no contrivances, Ip == j= /kqr/|2 = "D497: So that if @ is 1 mile, e-*=.0:9513. That is; thetetigtam attenuation of 5 per cent per mile. 2. Let l/’=l=0. That is, there is neither inductance nor capacity inserted in the line. Now let there be an inductance leak to earth which causes k’ to be 0°01375 x 10-®. Then h=q=Wk'qr/2 = -024875, p se? ———— lf." ——— a. 4 Telephone Crreurts. 679 so that e~*= ‘97513, or there is an attenuation of 2°55 per cent. per mile. The approximation to this is by an induct- ance to earth of L henries per mile, where 055 x 107° ~ is = "DISS */ 1D", Lig or L=‘972 henry. When I use an inductance to earth I generally prefer to put it in series with a condenser. Thus, instead of an inductance of 0°972 henry, I would use an inductance of 1:012 henries in series with a condenser of 1 microfarad. This enables the line to be tested more easily. Instead of an inductance leak of 0°972 henry per mile we may insert 0°972/m henry every m miles. As m gets greater the discrepance from the continuous conditions gets greater. The problem to be solved, later, is in all cases, to find the discrepance for a particular value of m and what therefore is the most convenient distance by which con- trivances ought to be separated. 3. Suppose l/= —:09112 and #=—-:0125x10-* These values will cause c to be a maximum if the distance is 300 miles. It will be found that they mean, a capacity of 0-439 x 10-® tarad per mile in series with the line and an inductance leak of 0°5925 henry per mile. Here g=0°16775 and h=3AV/j'/l/= 0:00333. This means an attenuation of 0°333 per cent. per mile when g is 5000. Now when q is taken as 3000 or 4000 the attenuation is 0°64 or 0°53 per cent. per mile. The approximation to these conditions is (a) a contrivance every m miles consisting of a capacity of 0°439/m microfarad in series with the line and an inductance leak of 0°5925/m henry ; or (}) a contrivance every m miles consisting of a capacity of 0°439/m microfarad in series with the line between two inductauce leaks each of 1:185/m henries ;- or (c) a contrivance every m miles consisting of two condensers In series, each of the capacity 0°878/m microfarad, the point of junction being put to earth by an inductance of 0-5925/m henry. It will be found that as m is smaller and smaller the line with detached contrivances approximates more and more to the line with continuous properties. Detached Contrivances. Suppose we have contrivances at the equidistant places A, B, &., m miles apart. There is a contrivance whose terminals are A and Ag, another whose terminals are B and 680 Prof. J. Perry on B,, &e. Between A, and B we have the standard cable: Let the currents in the line at A, Aj, and B be e, ¢, and C. » Let the voltages at these points be v, vj, and V. Let V/C=v/c=p. Knowing the nature of the contrivance we can calculate vy and ¢) from v and ec. It is also known that r+ le 2 hs V = wy cosh mn+ / ¢o sinh mn, a: n : C = ey cosh mn+ -Up sinh mn, r+lgi where n= V(r+lqi)(stkqt). Putting V/C or p equal to v/c, we have a quadratic to calculate p and therefore V and C. I am in the habit of writing c=1, meaning c = sin qé, and then v=p. Whatever the contrivance may be, Vi a+Bp — G7, P mraaipe where a, 8, a and b are given in value; they are usually complex unreal quantities. Solving for p and finding © we have two answers, ©, and Cy say. In general C,C,=1, and if (a+) be called P, C=P+(/P?—1. Now P is very easily evaluated, and therefore C. Example :—(1) If our contrivance is a resistance 7, in series with the line q¢=c=1, »=r14+ p. Thus if 7 is lan Ohmic resistance R in series with a capacity K and an inductance J, : iL T= NE MIN es This isa very simple case, the simplest form of which is that of Mr. Pupin, a mere inductance coil inthe line. The trouble in such a case is that the whole telephonic current passes through the coil and there is considerable Ohmic loss of energy. It is to be remembered that the value of R to be taken in any such case may be many times the ordinary resistance, because c’R istaken to be the total loss of power, and this is due to hysteresis, eddy-currents, &c., as well as mere Ohmic loss. (2) Again taking a case already mentioned; the con- trivance consists of two equal resistances 7; in series with the line and a resistance leak to earth 7; from the point of junction of the other two. LL a a Telephone Cireuits. 681 7. Tis easy to show that and 3 ; Vy = 27, + a +p(1+ =) and it follows that P= (1 -- 2) cosh mn 4+- tHe + 5] ae i a sinh mn. It may be well to work out one example. In the above- mentioned cable r=18, k=0°055x 10-§, s=0, 1=0; take g= 5000, n = Nrkyi = 07036 (45°), -= ‘003909 (45°), ” = 955°8 (—45° ‘ee 255°8 (—45°). Let us place contrivances at m=4'263 miles apart. I choose this distance because mn=0"3 (45°), and this happens to be given in Prof. Kennelly’s table, which tells me that cosh mn = 1:0007 (2°. 35’), sinh mn = 0°29994 (45°. 51’). To imitate the continuous case given above we need an inductance leak of 0°5925 henry per mile or L=*5925~4:263 every 4°263 miles. Hence r, = Lagi = 694-71. pli Na l . ~ -4°263 = Kai’ 7,=—971:137. Inserting these values in the above formula, I find P = —0°05384—-068078 7 7, Is a condenser of capacity K= 10-6, so that as and Y= 1:057 (183°. 54’) or 0°946 (—183°. 54’). That is, there is an attenuation or increase of 5°5 per cent. in 4°263 miles or 1:3 per cent. per mile. Observe how different this is from the eftect of the continuous distribution. As another case, a most important one, consider the standard subterranean cable r= 88, k='05x 10-8, s=0, r=0. I. Without contrivances h=g=,/rkg/2. If ¢=5000 (a frequency of 800), h=‘1049=g9. There is an attenuation er + eg ——_ aaa ae eeiien Zottieiadll e Pe * — — ae ——12 — Ee 682 Prof. J. Perry on of 11 per cent, per mile. It is to be observed that notes of frequencies 1:2 are attenuated in the proportion 1: 7/2. _ This is probably good, because it is known that the graver notes of speech need to be of greater intensity for distinct hearing. But the changes in phase are also as 1: 72, whereas for good hearing they ought to be as 4/ 22s Using this cable with no contrivances it is found in practice that there is good telephonic speech to a distance of 40 miles. When g=5000 a current 1 is reduced to 0-015 or ge of itself in 40 miles. If g=6000 (a frequency of 960) a current of 1 is reduced to 0°01 with a lag of 263°; whereas if g=4000 (a frequency of 640) a current 1 is reduced only to 0°0238, but it has a lag of 215°. In fact the two lags in actual time are as 22 to 29. If, then, we find such attenua- tions and lags in any case we may inter that good telephone speaking is pessible. I]. I will first give examples of a tuned circuit, that is, very perfect transmission of one musical note of frequency 800. Keeping to the contrivance which has two condensers in series, each of capacity K in the line and an inductance leak of L henries to earth from the point between the condensers, it is easy to write out the general expression tor P in terms of mand 7, and 7’; (at all events when mnis small), and it may be studied. Now for any particular m find the values of r, and 73 which will cause P to be nearly 1 (0°) and there- fore C to be nearly 1 (0°). I have already explained that we have no right with the above formule to get a really constant current like this, but we have a right to approach it very nearly. If m=1 mile, then K=°5917 microfarad and L='8324 henry. If m= 2:755 miles, K =-2273 microfarad and L='3692 henry. Hven when m is as much as 5°51 miles, a very considerable distance for such a cable, I find that I can use values of K and L which cause the attenuation to be comparatively small. We see that by means of contrivances placed at con- siderable distances apart we can tune a long circuit so that it will transmit a current of one frequency, just as if the line were a mere Ohmic resistance. | I do not know whether practical men sce great value in having this power. It may be important to inventors. It is well, however, to say that when we have such a line, unless the contrivances are rather close together, the circuit is really a tuned one, tuned to a particular frequency and not good for the transmission of other frequencies. — Telephone Circuits. 683 III. Isolated contrivances. When m is so great that cosh mn and sinh mn are practically equal, calculations are particularly easy. Thus, taking the case in which ¢g=5000, r= 88, k='05 x 10-%. We have — p = Vr/kgi= 593 (—45°) = 419°3 (1—2) = a—ai = a—PqI, say. (1) If in the middle of a very long line we merely insert an inductance leak to earth of L henry; L=~=§='11725 gives the best eftect. This multiplies the received current by 1°414 and causes a lead of 45°. (2) An inductance of L=*‘11725 henry in series with the line and a leak to earth which is a condenser of capacity 1+ ¢? farads (in the present case 0°532 microfarad) ; this causes the current to be multiplied by 1414 and to get a lead of 225°. (3) An inductance of 0°11725 henry in series with the line and no leak to earth. ‘his multiplies the received current by 1*414 and it creates a lag of 45°, IV. Returning to the contrivance consisting of two equal condensers in series with the line and an inductance to earth from the point between the condensers. For good telephonic speech it is evident that the inductance ought to be small, otherwise there is too much dependence on frequency ; there is some tuning. We want k'/!’ to be small and nearly constant and k'l’ to be small also. We cannot effect both these objects, but we can obtain fairly good speaking by compromise. Thus, let k’ = —:05 x 10-8 and l’/= —:128, so that K = is microfarad and L=0°4 henry. Thus if m is 1 mile, 71, =—83207 and r3= 2000 2. . I find that if gq =5000, C= 0:9704 (24°.3), if q¢= 4000, C= 0:9647 (35°). That is, with a frequency 800 there is an attenuation of 3 per cent. and a lag of 24° per mile, whereas with a fre- quency of 640 there is an attenuation of 33 per cent. per mile and a lag of 35°. We get practically the same results by the continuous formule. 684 Prof. J. Perry on Telephone Circuits. Practical men will now see their way to easy calculation of the effect of placing any kind of contrivance in a circuit, and I need not give any more examples. It is to be remem- bered that condensers suitable for telephonic purposes are very cheap, a tew shillings per microfarad, whereas induct- ances are expensive. Also, it has to be remembered that when the whole telephonic current passes through an induction coil, its resistance becomes of very serious importance, for the cost of coils of similar construction is proportional to the 3/2 power of L/R if L is the inductance and R the resistance of the coil. With the permission of the Society I here append a table of the values of senh r (45°) and coshr (45°) to supplement that published by Kennelly. His 7 is 0,071, 0°2, &c., and we need closer entries for the above kind of calculation. My table has been computed by Mr. J. J. Brookes. r. sinh 7 (45°), cosh r (45°). “O01 ‘010000 (45°-00) 100000 (-0C°) 02 ‘020000 (45°-00) 1:00000 (40°) 03 "030000 (45°-00) 100000 (02°) 04 "040000 (45°01) 100000 (-04°) 05 "050000 (45°:02) 100000 (07°) 06 “O6U000 = (45°°04) 100000 (10°) 07 "070000 (45°-05) 100000 (-14°) 08 “O80000 (45°06) 1-000U0 (18°) ‘09 "090000 (45°-08) 100000 (22°) "10 ‘10000 =(45°-09) 100001 (°29°) "15 "15000 = (45°21) 100004 (64°) ‘20 20000 = (45°58) 1:00013 (1°:15) 2) 25000 (45°-60) 100032 (1°79) 30 30C0L — (45°°86) 100067 (2°56) "35 35003 (46°17) 100125 (3°°50) “40 40005 (46°53) 10021 (49°58) "45 45010 (46°°93) 1:0034 (5°°79) “50 50016 = (47°°39) | 10052 (79°14) 55 55027 = (47°89) 10076 = (8°63) “60 60042 = (48°44) LOIO7 (10:25) 65 °65063 (49°03) 10148 =(12°-01) "70 ‘70093 = (49°-68) 10200 © (13783) “715 OLS]. (Ol ean) pee 10263 (15°89) "80 “80181 = (51°10) 103386 = (18°°01) "85 *85245 = (51°89) 1:0426 (209-24) “90 90327 (52°°72) 1:0583,. (22957) "95 "95428 (53°°60) 1:0659 (24° 99) 1:00 | 100553 = (54°53) 10803 (279-49) e [685-4] LXXIII. The Change of Resistance of Metals in a Magnetic Field at Different Temperatures. ByS. C. Laws, M.A., D6. XPERIMENTS on the change of resistance of metals in a magnetic field have been made by Patterson J, Grumnach ¢ and others. These observers have found that all metals show a change of resistance in the magnetic field, all except the magnetic metals showing an increase ot resistance in all fields; except for Bismuth, however, this increase is very small, being usually less than +} ,th of 1 per cent. for a field of 10,000 c.a.s. units. Fleming and Dewar §, working at temperatures varying from 19° to— 203° CU. have made measurements on the change of resistance of Bismuth, and have found a very considerable increase at low temperatures ; while Righi || has made similar experiments at temperatures ranging from 19° to 108° C. with similar results. From the point of view of the electron theory as at present understood, a change of this nature is to be expected if a lowering of temperature produces an increase in the mean free path of the corpuscle. This change is required from considerations of the thermoelectric effect and the change of resistance with temperature. The experiments described in this paper were undertaken in order to get some further evidence on this point. Cadmium and Zine were selected as suitable metals for the experiments, since in these the effect to be measured is larger than in any others among those examined by Patterson or Grumnach. Experiments have also been made with pure graphite, this substance being interesting theoretically from the tact of its negative coefficient of increase of resistance. It has also been found to show a comparatively large change of resistance in the magnetic field. : The apparatus with which experiments were made con- sisted of a Wheatstone bridge arrangement in which the metal under examination formed one of the four arms of the bridge. The adjacent arm consisted of two sets of standard resistances arranged in parallel, one of these, Y, serving to provide a first approximate balance, the second, X, which was of higher resistance—1000 to 2000 ohms—serving * Communicated by Sir J. J. Thomson, F.R.S. + Patterson, Phil. Mag., June 1902. } Grumnach, Ann. der Phys. xxii. 1, 1907. § Fleming and Dewar, Proc. Roy. Soe. lx., 1897. || Righi, Att: della R. Acead dei Lincei, xix., June 1884, 686 Mr. 8S. C. Laws on the Change of Resistance of Metals to adjust the balance accurately. The ratio arms of manganin were 51 and 5°1 ohms (or in the case of carbon 51 and 0°48 ohms) respectively. These, together with the battery series resistance, were immersed in an oil-bath. Fig. 1. m4 x The method of procedure was to first adjust Y and then X until the balance was nearly obtained and then, keeping X and Y constant, to observe the detlexion produced by reversing the battery current through the bridge. ['inally, the readings were standardized by observing the deflexion due to a reversal of the battery-current when a small alteration of the resistance X was made. In this way, the resistance r of the experi- mental wire was obtained both when the field was “ off” and © On. In the case of the metals Cadmium and Zinc, these were obtained in the form of silk-covered wires °*003 inch diameter, and a piece about one yard in length was wound lightly round a small rectangular mica frame so that the coil would lie easily in the space between the poles of an electro- magnet. Conducting leads were fastened to a strip of vulcanized fibre and the ends of the experimental wire attuched with soft solder to these. Fig. 2. 0 The temperatures were obtained by surrounding the coil with a rectangular bath, a section through the centre of which is here represented, with recesses in opposite faces to bring the pole-pieces of the electromagnet as nearly as possible to the coil. The pole-pieces were maintained at a constant distance ina Magnetic Field at Different Temperatures. 687 apart by means of two adjustable brass pieces cast so as to fit over the pole-pieces and held by means of nuts and bolts at the desired distance. This bath was connected to a boiler and condenser, and so could be filled with vapour of liquids boiling at suitable temperatures. The results given were obtained when the bath was empty and when filled with the vapour of acetone or steam. The use of this steam-jacket surrounding the coil com- pelled a gap between the pole-pieces nearly a centimetre in width, so that the fields ovtainable never reached 14,000 C.G.s. units. Hxperiments were also made at the tempera- ture of liquid air, for which purpose a vacuum vessel was made by Baumbach by flattening the lower portion of a glass cylinder 2 inches in diameter until the opposite faces were about 4 mm. apart, and surrounding this with a jacket of similar shape. The experimental coil was then enclosed in mica and immersed in liquid air contained in the vessel. The use of this vessel required a gap between the pole-pieces of about 1-8 cms., so that experiments at this temperature were made with fields which did not exceed 4000 c.a.s. units. The metals used were supplied by Messrs. W.G. Pye & Co., while for the Graphite I am indebted to the Morgan Crucible Co. Experiments with Cadmium. Experiments were made with currents through the electro- magnet varying from 1°8 to 10 amperes. ‘Lhese currents were measured by means of a Kelvin Standard Ammeter. The following table shows a few readings from which the results given are obtained. December 13th, 1908. Bath heated by acetone. oo oa. Y=210. X=1520. r=185 ohms. Current through | Deflexion on| Deflexiondue | Electromagnet. reversal, to fieid. | / 0 19 4-4 | Ll 23 | 0 29 | 44 45 22:7 0 25°5 4°4 3 20 7 0 22 | | As previously explained an approximate balance was found by making Y=210 and then adjusting X to 1520 ohms. 688 Mr. 8. C. Laws on the Change of Resistance of Metals Keeping these resistances fixed the deflexion due to a reversal of the current (about 5}5 ampere) through the bridge arrangement was observed. ‘This observation was made first with no field on, then with a current of 4:4 amperes through the coils of the magnet; it was then repeated with the field zero, and so on as shown in the above table. The deflexions on reversal were obtained by observing the position of the image of an incandescent-lamp fiiament on a { scale some 3 metres distant from the galvanometer, first with i the current through the bridge arrangement direct, then reversed, and finally in its original direction, the deflexion on reversal being obtained by subtracting the second reading i from the mean of the first and third. These successive readings of the position of the image of the lamp filament 1 were taken at intervals of 15 seconds. The galvanometer was a D’Arsonval instrument, whose resistance was 60 ohms | and sensitiveness 2°5 x 10-9 ampere per scale-division. The : average of the above readings together with those made | under similar circumstances with other currents are tabulated naa dinedemee standardized bat Ghee eee beiow. The readings were standardized by observing in the same way the deflexions on reversal when the resistance X | was changed from 1520 to 1530 ohmns, there being no current in the magnet coils. The deflexions on reversal obtained in this experiment are as follows :— x Deflexion on Defiexion due to : reversal. change of resistance. 1530 75 1520 —125 O17 1530 83:5 95°5 1520 —115 96 7 : 1630 87 97°5 ] 1520 ~ 95 95 1530 84 92°5 ‘ 1520 — 75 915 fi 1530 84 — 94°6=mean. P ) ‘| With X=1530 the equivalent resistance of X and Y was ; I 184-655. | With X=1520 the equivalent resistance of X and Y was i 184°509. i So that a change of :146 in the resistance of this arm pro- duced a galvanometer deflexion of 95 divisions. ina Magnetic Field at Different Temperatures. 689 The change in resistance of the wire in the magnetic field was then found from the relation Pach Dp gy) SEB a ( illann Ly or= x 95 iain e 95 x ¢: where @ is the mean deflexion on reversal due to the field in any case. In this way the following results were obtained. Deflexion on or 1 5 =-7 C. H. reversal, orx 10". - x10", | H’x10 18 8450 146 22°5 1°22 7714 2:5 9300 Wel 26'3 1°42 8:65 3°4 10200 19:1 29-4 1-59 10 4 4:4 PLEOOOM: | 22°1 34:0 1°85 13°] 56 11800 27°6 42-5 231 13-9 Toe 12600 29°7 45°7 2-48 159 10:0 13300 322 49-6 2°70 17-7 The strengths of the field for each current used were measured by snatching a coil from between the pole-pieces and observing the throw of the galvanometer (the same instrument as for other measurements). To reduce damping, the coil and galvanometer were connected through a standard resistance varying from 800 to 1000 ohms. Interposed in this circuit was also the secondary of a standardizing apparatus. The latter consisted of a primary coil—a long solenoid— through which a known current could be passed, and a short secondary placed at its centre having 322 turns of fine silk- covered wire in a single layer. The throws of the galvanometer were standardized by reversing the currents in the primary. If, then, 6, be the throws of the exploring coil and 6, those of the standardizing coil, while C, and C, are the corr sepohd- ing currents, we have He 22. Ny 4rCon a oo 8 Te Gy” where N, and N, are the number of turns in the exploring and standardizing secondary coils respectively, d,; and d, their respective diameters, while n represents the number of turns per unit length of the primary of the standardizing - solenoid. In these experiments N;=8 : No=322: n=31‘7: di. ="500 CM.'s @4==3°90 cm. Phil. Mag. S. 6. Vol. 19. No. 113. May 1910. a ¥ 690 Mr. 8. C. Laws on the Change of Resistance of Metals With the apparatus described above, readings of the change of resistance were also made at the temperatures 16° C. and 97°-7, that is, with the jacket empty and filled with steam. The results obtained are tabulated below :— December 11th, 1908. t= 16° CL YY =160 SAO) Fs Defiexion on| 4. or 4 ail C. H. AE or X 104, es 10°. | H2x 1OGe 18 8450 22°1 30°5 191 7:14 2°5 9309 278 38°3 2°39 8°65 34 10200 32°4 44-7 279 10°4 4:4 11000 37°8 52:1 3°26 12:1 56 11800 42:1 58:1 3°63 139 (is) 12600 48:1 66:3 4:14 159 10:0 13300 51:8 714 4°46 Lia December 10th, 1908. C= 991 VE 2407 X= T1S00 Ge ie Defiexion on or 4 —7 C. Hi, pee sie orx 10+. ar x10*. | H2x10 7S. 1:8 8450 8:3 15°4 0-73 7:14 2°5 9300 100 18:5 0°88 8°65 34 10200 11°6 21°4 101 10°4 4:4 11000 13'3 24°5 1:16 12-1 56 11800 iy) 29°3 1°39 13-9 io 12600 18:3 33°8 1:60 for 10:0 13300 19°7 36°4 173 ule 7¢ It will be noticed that the value of the magnetic field produced by a given current .in the magnet coils is not affected by the changes of temperature of the bath. | This fact was ascertained by a special apparatus erected to test the point. This apparatus consisted of a few turns of fine insulated copper wire wound on a very small ebonite disk with the ends of the wire connected to a small split tube commutator. The disk and commutator were attached to a light brass spindle so that the former revolved within the steam-jacket and in a line with the pole-pieces of the magnet—occupying in fact the same position as the wire under examination in the above experiments. in a Magnetic Field at Different Temperatures. 691 This small armature was connected in series with an exactly similar but larger one revolving about a horizontal axis at the centre of a horizontal standard solenoid. The armatures were driven from a countershaft to which were attached pulleys of the requisite diameter, the countershaft itself being driven by a small motor. The currents generated in the armatures were arranged to neutralize one another in a cireuit in which the sensitive galvanometer used throughout these experiments was included, and by adjusting the field of the solenoid a complete balance was obtained between the armature currents. It was then found that the current in the solenoid necessary to produce this neutralization of those in the armature was quite inde- pendent of the temperature of the bath surrounding the poles of the magnet. } This method of measuring fields was therefore abandoned in favour of the practically simpler one of snatching a small exploring coil from the space between the pole- ieces. This latter operation was effected mechanically by attach- ing the exploring coil to one end of a lever and fixing the coil (by means of a thread attached to the other end of the lever and passing over a pulley) between the pole- pieces against the elasticity of a rubber cord. On burning the thread the eontraction of the rubber eord eftectively removed the coil from the field. ARIES or It will be noticed that the values of ~ at the temperature of steam are only about 2 of those obtained at ordinary 2) *. atmospheric temperatures, a fact which gave some reason to hope that the changes occurring at the temperature of liquid air might be measurable in spite of the much smaller fields that must necessarily be employed in this case and of the difficulty of preserving these temperatures to any extra- ordinary degree of constancy. Accordingly, the experiments were tried with the apparatus already described, which allowed the pole-pieces to be brought within 1°8 cm. of one another. In working with liquid air the difficulty of obtaining a constant temperature and therefore a constant resistance was very pronounced, the resistance always rising during an experiment—in the course of experi- ments tabulated below from »=4°12 to r=4°20—and the aecuracy of the resalts suffers accordingly. There can be no doubt, however, that these results are quite of the right order. A characteristic set of readings obtained is given below, | 22 692 Mr. S. C. Laws on the Change of Resistance of Metals April (th, V9G9: t=—186°C. Y=44. X=800. r=4°17. Current through | Deflexion on Deflexion due magnet coils. reversal, to field. 0 140°5 4-4 83 33 0 91 4-4 43°5 27 0 50'5 4-4 2:5 29 0 12°5 4°4 — 41°5 30 0 — 33 30= mean. The following table shows the actual results obtained. Deflexion on i or 4 -7 C. H. reversal, or X 104, ao 10", | H?x10 ". 3°4 3230 26°5 21 5:0 1:05 Ad 3460 30 24 oy 1:20 56 3680 34 27 66 1:35 75 3800 42°5 34 8:2 1:44 | 10:0 4060 26°5 37 9:0 1°69 The above results are collected in the following table, which shows that :— (i.) PR is practically proportional to H? at all temperatures; (ii.) fora given field oP increases considerably as the tem- perature falls. Va H 7 {=98. t=5) t=16 t= — 186. 25; 0 ina Ma 3 Nl | ER les ac" UI Cals ood vega 478 SAGO! MN PE UE Ried cb Rene See ee 47°5 SOSO, flit Geeledec'l este eee eds eee ce 48°9 2101010 an ee nti BM SR Ua Wer Ose 5S 56'8 4401610 ae RN eS a ARR SPN si isa ip dik Sous 54°5 8450 1:02 1-71 2°68 9306 1:02 1:64 2°76 10200 ‘97 1:53 2°69 11000 96 1:53 2°69 11809 1:00 1:67 2°62 12600 1:00 1:57 2°61 13300 98 1:52 2°52 "99 1:60 | 2°65 51=mean. ht se in a Magnetic Field at Different Temperatures. 693 Experiments with Zine. The method of making experiments with zine wire differed in no respect from that employed with cadmium. The resistance changes being, however, only about one-sixth of those obtaining with cadmium, no measurements were taken at the temperature of steam. Below is a summary of the results obtained. May 22nd, 1909. pe Go. Y= 960) R—1120. += 8'85. Deflexion on t or t 2 —7 C. Hi; ne or X 104. ig 10") ><¢ 10 j 18 8220 127 5°56 63 6:76 2°5 9110 15°5 679 “Ti 8:30 34 10200 178 780 "88 10°4 44 10900 20°0 8:76 92 IL-9 5°6 11700 22°6 9°90 112 13-7 75 12100 25:1 11-0 1-24 14-7 100 12700 28°9 12-7 1-43 16-1 May 24th, 1909. moet, Y=H=1IO. KR=—1150. r=16°04. re} H Deflexion on | or x 104 or x 10! H? x 107! : reversal, 7 r ; 25 9110 9°5 4°53 45 8°30 3-4 10200 12°3 5°87 *b8 10°4 4-4 10900 13:5 644 “64 11°9 56 11700 15°4 y's) oF 13°7 TD 12100 167 797 “79 14°7 10-0 12700 tT! 8°30 83 1671 Liquid-air Experiments. April 7th, 1909. t= — 186°. Y=29. X=900. r=2°81. C. H, | Deflexionon) sige | ox 10% | Hex 1077, reversal. a 56 "3580 8-2 53 2-0 1:35 75 38800 10°6 | 25 1°44 10-0 4060 15:2 10:2 3°6 1°65 694 Mr. 8. C. Laws on the Change of Resistance of Metuls | Be The proportionality between a4 and H? is again apparent as in the case of cadmium. 4 1 Sr The following table shows also the value of Ee 3t ° U different temperatures, 1 12 > «10 i. 650" t= 185, t= —186 PLGA MCs gl RE Geet 4 Rt Sa tee i 150 ToC VERSE ok Riot HAP ee any Bag SESS SRS 170 4060 SSN slo vie SiS o8 22:0 8200 Ae 93 9110 D4 “92 10200 56 85 10900 54 83 11700 53 82 12100 D4. 8D 12700 Di ‘B9 54 87 18°O=mean. Heperiments with Carbon. A considerable portion of the time devoted to the experi- ments described in this paper was occupied in the examina- tion of the change of resistance of carbon in the magnetic field. The work was unsatisfactory in the first place because of the very wide range of value observed for different specimens of carbon, and secondly because in the case of graphite, which alone gave anything like consistent results at atmospheric temperature, the resistance was altogether too unsteady for the purposes of this experiment as soon as the temperature was raised. This difficulty seems to be inherent in the nature of the material. In liquid air, however, the resistances were quite steady, The earlier experiments were made with the carbon fila- ments from incandescent lamps, but it was found that whilst some filaments showed practically no change of resistance in the magnetic field, others again showed as large a change as O°D per cent. in a field of 10,000 c.a.s. units. This change of resistance was found to take place in the outside layer of deposited carbon, for when this was removed by burning or in a Magnetic Field at Different Temperatures. 695 scraping all filaments agreed in showing no change of resistance. _ The filament from an ordinary Hdiswan 16 c.p. 110 V filament gives a value of the order of 0°5 per cent. for the change of resistance in a field of 10,000 c.a.s. units. The high voltage filaments show very little change of resistance in the magnetic field. Also unflashed filaments specially supplied by the Robert- son Company and ordinary filaments from their lamps similarly showed no change, but some tbat they flashed for me showed a change which was larger the greater the amount of flashing. | Though this layer to which the change of resistance was due could be easily removed, it was not obtainable again by the process of flashing in coal-gas or various liquid hydro- carbons which I tried. Owing to this lack of uniformity in the incandescent lamp filaments, [ made experiments with graphite, which was found to show a considerable change of resistance in the magnetic field. Experiments have been made on the impure graphite from numerous specimens of lead pencils as well as with some specimens of pure graphite supplied by the Morgan Crucible Company. These all agree in showing a change of the order of 1 per cent. in a field of 10,000 units (and also in having an un- certain resistance at temperatures higher than atmospheric). Below are tabulated the readings taken with a specimen of pure graphite, at 15°°5 C and in liquid air. July 8th, 1909. ato is. Meals, SO ool. Sear Ung § tee nae i : Deflexion on 4 or 4\ cre —7 1 or 12 tp H. et jorx 10 ~ 10! H? x 10 | iP 7 *10 | 18 | 2580 32°5 2:33 543 665 81:7 2°5 | 2900 401 3°48 668 ‘84 T4 3:4 | 3220 47-5 4°13 7-92 1-04 T7'5 4:4 | 3360 541 470 | 9-02 1:23 167 56 | 3660 62-1 539 =| 10:4 1°34 768 75 | 3940 740 | 640 | 12 inh | 79 10:0 | 4140 73:3 | 679 134 | 171 761 | 78:2=mean. ! 696 Mr. 8. C. Laws on the Change of Resistance of Metals July 8th, 1909. re='— 186°.” Y=102.) X=—90).) 70 cow Deflexion on eee 4\eye tif) le or 12 C H. bE A or x 104. ; «10451 HA LOG » ee >< TOs 8 42580 112 10°6 16°3 665 249 2°5 | 2900 138 Lyi 19:8 *84 236 3°4 | 3220 160 15:2 23:0 1:04 225 4-4 | 3360 182 17°3 26:2 1-13 22 56 | 3660 203 19°3 29:2 1:34 PALL 7-5 | 3940 230 21:8 33°2 1:56 TAS) 10:0 | 4140 256 24°3 36°8 al. 215 225= mean. These results indicate that for graphite the change of resistance varies approximately inversely as the absolute temperature: the change of resistance in a given field increases threefold between the temperatures 18°C. and —186° C., whereas in the case of the metals examined the change of resistance at the temperature of liquid air is about 20 times that at atmospheric temperature. In endeavouring to maintain the resistance of graphite constant at higher temperatures, [ made some experiments with fairly satisfactory results with the substance in the form of powder. The powdered graphite was enclosed in the form of a cylinder about 1 mm. diameter and 6 mm. in length, along the axis of a short wooden cylinder fitting into a hole drilled in a small piece of brass, and was maintained in a state of compression by means of a vertical brass plunger. This plunger was forced down upon the graphite by the pressure exerted by the short arm of a lever, to the long arm of which an upward force was applied by means of a suitable weight suspended over a pulley. These experiments, which were made at temperatures of 17° and 98° with fields ranging from 4300 to 6930 ¢c.q.s. units, indicated that between these 7 2 diminished in the ratio of 1:2 to 1. As the absolute temperatures are in the ratio of 1 to 1°28 these experiments serve to confirm the results previously given for graphite rod. As already mentioned, any explanation of electrical con- duction in metals in terms of the original corpuscular theory as developed by Sir J. J. Thomson, Lorentz, and others, and according to which conduction is due to a system of freely limits of temperature the values of ina Magnetic Field at Different Temperatures. 697 moving negatively charged corpuscles in statistical equi- librium with the surrounding atoms or clusters of atoms, requires that the mean free path of these corpuscles should increase as the temperature is reduced. Sir J. J. Thomson * points out that for most pure metals X should vary almost inversely as the absolute temperature, though the fact that the resistance of cadmium and zinc increases rather faster than the absolute temperature as well as that the Thomson effect is in each case positive, would cause the mean free path to show a somewhat larger variation with temperature than this inverse first power law requires. By the aid of the expression dia) i UE CD Ee a, ta Se ania Tio lecplisp given some years ago by Prof. Thomson f for the change of resistance in a magnetic field, we can determine » for the different temperatures at which _ has been found. It will be seen from this equation that if > varies as : 1 or 1 Ip should vary as ae The experiments with cadmium and zine described in this paper indicate that at temperatures between 15° and 100° ©. 1 or ; : — varies somewhat more rapidly than I H? ¢ ha than a whilst at the temperature of liquid air the variation though less is rather less than be . Onthe whole, therefore, the variations 63 of X as found from the experiments here described agree fairly well with those required on the same hypothesis by con- sidering the variation of resistance with temperature and the Thomson effect. The absolute values of X at a temperature of 16°C. obtained from the above equation by putting be ae POT Tit and v = 107 ¢ are m for cadmium Ack a meee Te and for zine r= 19x 10-6, * ‘Corpuscular Theory of Matter, p. 80. + Rapports piésentés au Conyrés Internationale de Phys‘que, iii. p. 138 (1900). { ‘ Corpuscular Theory of Matter,’ pp. 10 and 52. 698 Mr. 8. C. Laws on the Change of Resistance of Metals From his experiments, Patterson*, using the same for- mula, found x for cadmium = 4 sce 0-° and _ for zine = 2°3 x 10-6. The difference in these numbers is a consequence rather of the different values assumed for v and e/m than of any differences in the values found for es ; For, working with fields varying between 17,000 and 29,000 c.a.s. units, Patterson found for cadmium ee 28°2 x 10-38, while my experiments for fields between 8000 and 13,000 give 26°5 x 101*: in the case of zinc both sets of experiments give the value 8°7 x 10-} for this quantity fF. The mean of the values obtained by Grunmach { with fields varying from 8000 to 16,000 c.a.s. units are 27°3 and 11:0 respectively. 1 or H? + varies approximately inversely as the absolute temperature, so that the same formula for this quantity shows that A is approximately independent of temperature. The same result is again arrived at by considering the effect of temperature on resistance ; for, since the resistance is practically inde- pendent of temperature, the equation 1 1 en The experiments with graphite indicate that here makes A independent of temperature, provided that here, as in the case of metals, n is proportional to 4/0, that is, if the specific heat of electricity is not abnormally large in the case of graphite §. Actually, the resistance of graphite decreases somewhat with rise of temperature so that X should increase slightly at higher temperatures. The experiments here described point to the same conclusion, for these indicate that the change in ee is rather less than is required by the inverse first power * Patterson, loc. cit. p. 695. + By an obvious slip the values given by Patterson are 282 and 87 respectively, doc. cit, p. 652. t Grunmach, loc, cit. p. 170, § This point has been recently verified in some experiments, as yet unpublished. vr ( ina Magnetic Field at Different Temperatures. 699 law. The above expression therefore indicates that » increases somewhat with rise of temperature. As far then as the variation of mean free path with tem- perature is concerned, these experiments may be satisfactorily explained on the older electron theory. Sir J. J. Thomson* accounts for this change of A» by supposing that collisicn occurs in metals between the cor- puscles and clusters of atoms, these clusters undergoing a process of disintegration at higher temperatures. The same hypothesis would enable an explanation to be given of the relatively large values of X which experiments on change of resistance in a magnetic field require on this theory. Sp The values of ie obtained at the temperature of liquid air: in the case of the metals cadmium and zine might be taken to indicate that at very low temperatures » tends to become constant, but any further speculation should rather be reserved until more complete data are obtained. The results obtained in this paper may be summarized as follows :— (i.) In the case of the substances examined cadmium, : ate zine, and graphite—the change of resistance, —, in a mag- netic field H, is proportional to H® for the range of fields used. (i.) This change of resistance in all cases increases at : . 1 ér lower temperatures; for the metals cadmium and zine He e at the temperature of liquid air is about 20 times as large as , Sy) eae at atmospheric temperatures, while for graphite li; shows Sy _ only a threefold change over this range of temperature. (iii.) The increase of resistance, = in the case of graphite is about 1 per cent. in a field of 11,000 c.a.s. units at atmo- spheric temperatures ; this is about equal to the decrease of resistance of nickel in the same field, and is many times larger than the change of resistance occurring in any of the non-magnetic metals hitherto examined except bismuth. The above experiments were carried out in the Physics Laboratory of the Blackburn Municipal Technical School. Technical Institute, Loughborough, January 51, 1910. * “Corpuscular Theory of Matter,’ p. 72. [.. 700: J LXXIV. On the Limits of the Oblateness of a Rotating Planet and the Physical Deductions from them. By Prot. PERcIVAL LOWELL*. i a Clairaut we owe the first important work on the relation of the oblateness, the rotation spin, and the matter-distribution of a planetary body. By taking into account only terms of the first order in the ellipticity he showed that, whatever the law of density from surface to centre, the oblateness must. be comprised between the values = and 3 of the ratio, d, between the centrifugal force so-called and gravity at the equator when the body is homogeneous. In the present paper it will be shown that if we extend the investigation to higher orders we can prove not only that the upper limit is very approximately 3? ¢, but that the ‘lower limit is not dependent on a series but may be expressed exactly in terms of the mean density and the rotation. 2. To find the conditions for equilibrium in a fluid mass, consider the little parallelopiped dx, dy, dz. The increase of pressure between its two faces in the direction of x is dp 7.00 OY, Ae da ch tat the minus sign being used because the pressure decreases a. The resolved part of the force acting on it in the direction of w is pXtda dy\cdz. Similarly for the other coordinates. For equilibrium these must balance. Whence dp _ ep ( OPN geod or dp=p(Xde+ Ydy+ Zdz). This is the differential equation for fluid equilibrium. 3. To find the integral equation for the pressure at a point, suppose the fluid in equilibrium and canals drawn from any point within to a point on the surface. The pressures pro- duced by the fiuid in the several canals must all be equal at ihe point since otherwise there be a flow to or from it. The pressure due to any canal must therefore be independent of the form of the canal. * Communicated by the Author. Limits of Oblateness of a Rotating Planet. 701 The pressure due to any canal is \dp=| p(Xdw+Ydy+Zdz)... . (1) taken between the ends of the canal. Now if this cannot be integrated without expressing p, X, Y, and Z in terms of z, by the equation to the canal, we shall have {=P =f. feo 2 de ene (2) in which f will differ for each canal and the pressure be dependent on the form of the canal. But if p( Xdx + Ydy + Zdz) be an exact differential, then its integral between limits can be expressed solely as a function of those limits; or Paani 4}. Y15 £1) — r(x, Yoo Zo) aa) var? sa (3) and will be the same whatever the form of the canal. In this case equilibrium will be possible. Secondly, the values of the integral for all canals drawn from the point within to different points of the surface must be the same, since all these must exert the same pressure on the point. That is Wy, Y15 21) — (xo; Yos 2) = W(x, J Z9) — (xo, Yoo 0) or W(x, Y25 2) — (21, 1; 2) ° Therefore the variation of dP=0 at the surface, or RXda+Vdy+Zdzthere=0, . 2°. , (4) which indicates that the whole force must be perpendicular to that surface. These two conditions completely satisfy the problem. 4. Now if the force be any function of the distance, say or, then . © = aa Y : Zz = Y ar approx. ike g=ayn + ayn? +. a3n?, Ke., then if the series be convergent n= bgt beg? +)3q° Ke. and we get | (ab; —1)¢ + (abe + aby”) g? + (aybs + 2a2b1b2 + a3),°) 9°, Ge. =0. Since gq may have any value within the limits of con- vergence, the coefficients of this series must be identically Oblateness of a fotating Planet. 705 zero. As the first equation ajb,-1=0 determines 4, all the other 4’s are determined seriatim. In this manner we find at aes +2:489? approx. . . (10) or =? a is Mig+i11 96g? approx. 11. The ratio ¢ of the centrifugal force to gravity at the equator is 2mpay , sin7"e as hot V1—e. 2mpa( a M16) _ompay substituting for g and e their values in 7 or € we get 4 ed P= =— 77RD Ke. ep Biss (11) | {P+ 554? ) ooo». Sa ule ie also 3 : 12. For the Earth For any other pianet its y, or 7,, can be found from the equation 9 ~ n= Teh Baa wszom Sacer t ,€12) in which the unaccented quantities refer to the Earth, the accented to the planet. Plal. Mag. 8. 6. Vol. 19. No. 113. May 1910. Ply f 706 Prof. P. Lowell on the Limits of the In this manner we get for homogeneity the following table :— | g. 1. ¢. in seta aes } | ot Sater en hours. | 1 1 Barta, s.oseces 00230 2317 5R0-G 5527 23°933 | IVE HESS 28 cut coe ‘00306 it : 3°924 24-623 pane 174 217-5 | Jinpiterc..s. seen 05686 mene id 1322 9:842 | 9°13 11:4 Saburh) ceceeeee 10177 a a 0:685 * 10:22 4-97 6°23 * Deduced from the mean of H. Struve’s two determinations of the diameter 1894, 17/52 at dist, 9:539 and of his oblateness. <=. using the latest determinations of the rotation spins at their equators, and of their several mean densities. The decimals of the second and third places in the values of p are given merely for the computation. 13. Comparing with these values of 7 the actual observed oblateness 7, of the same planets, which are 11 1 m I e ° e e Earth 5975 1 Mars x ‘ i. 190 Jupite oe Pp Ie J e e e 7 . 15:8 it S t fi e ° e e e e Seater aturn 9:5 we see that none of these planets are homogeneous, but that the inner and smaller come nearer to it than the larger and outer ones. 14, Turning now to the other limit for the law of density, to wit when its rate of change is infinitely great from surface to centre, we note in the first place that this condition corre- sponds to where all the matter composing the body is con- -ccentrated at the centre. There is in this case a kernel of a gl aes Oblateness of a Rotating Planet. 707 infinitely dense matter of infinitely small volume constituting a finite mass. The curve expressing this distribution, instead of being a straight line parallel to the axis of abscisse, as for homogeneity, is asymptotic to the two axes of coordinate. Here again the forces are all functions of the distance, and p is constant till it becomes infinite at the centre, a mere point, being throughout a function of the distance. Throughout the volume, therefore, the conditions for equilibrium hold. 15. To find the resulting oblateness iu this case we shall proceed as follows :— Suppose as before two canals from surface to centre, the one polar, the other equatorial. That equilibrium should exist between the two it is necessary and sufficient that the pressures produced by the fluid in these two canals should be equal. This will give us the polar and equatorial diameters and consequently the oblateness, whether the figure of equi- librium be a spheroid or not. The pressure produced by a portion dp of the polar caual at the point p is Poop, where P is the accelerative force, the section of the canal being supposed unity. 16. Let uw denote the whole pressure at the point p. Then, since p is reckoned from the centre, an increase of distance dp decreases the pressure by P.dp. Whence du=—P.dp. Since all the matter is supposed condensed at the centre, : an are 09 p and du=— = dp, whence Re M ae P or eae ho N bo mm Se oe ii 708 Prof. P. Lowell on the Limits of the At the centre, therefore, Mos NE u= cl an where p=0. Similarly the pressure from an equatorial eanal is > qe P IMD ares sa ee wat ot qa’ +O, and the pressure at the centre OMAN SAME) age Poy atare ve ae a’, where p=0; M M_M M, 2r? , 2a? , = 3 =_— __ zai erecta: eas Mae > p b p a ee pe: hs ae (13) / It might seem at first as if the term uM vitiated any solution. For becoming infinite when p=0 we have in our equation infinite terms side by side with finite ones from which no conclusion could be drawn. 17. Butlet us look into this more closely. As the radius p of the kernel contracts to its limit zero, the ratio of the centri- fugal force to the equatorial attraction itself decreases to zero, since the one force depends on p directly, the other on its inverse square. In consequence the polar and the equatorial forces tend to equality as they severally tend to infinity. At the limit the terms so introduced therefore cancel each other. The kernel in fact becomes perfectly spherical. Transferring these terms ee to the same side of the equation, we haye 1 MME a ye aes Tee Fa nears trace Po pan aan ae i oz 5 at er ie since the term 72 p* at the limit =0. As p tends to zero, the left side eventually becomes and our equation becomes Me 2a oe: oath |i.) aaa eee) a 2 | 18. Let g= 4S as before, p denoting what the mean density of the mass M would be if that mass were spread uniformly over the spheroid. Rs aa? Then rany rely “ise ae $ra'b | Sarna l* = MT” and 27? , 3 M.¢ Our equation thus becomes b Be ae or fie ae g ps b which, since b=a(l—7,), where 7, is the oblateness of the _ spheroid in this case, gives m= 2q. e ° . . . e . (15) The simplicity of this result is very remarkable. 19. As Hamy has shown, the figure of equilibrium in the general case is not composed of concentric spheroids. In the actual cases of the planets it is not so for another reason, to wit: that in the Sun, Jupiter, and Saturn, different latitudes rotate at different rates. As their equatorial spins are the fastest, this tends to flatten them more in mid- latitudes than would otherwise be the case. But the want of a spheroidal figure does not affect the limits of the oblateness, for in any case the spin must decrease to nothing at the poles. 20. Evaluating now 7p, for the several planets we find for their oblateness at the lower limit, corresponding to central 710 Prof. P. Lowell on the Limits of the condensation, the following amounts which are set opposite their values of 7 for homogeneity and their actual observed flattenings 7. n- 2» M+ =. if I 1 1 Barth - 393g eq 2 os ae . 1 1 1 Mere 60 ea Tse 150° aay Jupiter. . pal vee neo e weds Dik 15'3 22°38 | 1 1 1 1 Saturn . 97 res 5 123 21. To see how this value of 7, compares with Clairaut’s approximation to it to terms of the first order in the ellipticity, we shall express 7) in terms of ¢. Since AD NOT os PO: ed g I Bog" : ) de, i ie g Wet 3 2" very approx.; whence, combining this with equation (11), we get 209 p=2y2+ a1 + = 2 +%,", very approx.; and reversing, ees 1 AG tells, i a= 5 p 1680” AC. = 9 os) 3? , very approx... (16) We thus find accordance between the two results to the first order of approximation, and also how to evaluate the neglected terms in his result. | The two limiting values of the oblateness corresponding, the one to homogeneity, the other to central concentration, in the matter constituting the planet, give us an insight into the present distribution of that matter. The upper limit is very closely : d, and therefore depends upon the ratio ¢ borne by the centrifugal force to gravity at the equator in the homo- geneous state. ‘The lower limit, on the other hand, depends directly on q, that is on the rotation spin and the mean density Oblateness of a Rotating Planet. (jth of the body, or what would be such were the mass distributed; in other words, on the planet’s spin and mean density only. Itis somewhat less than y and increasingly, so as the spin increases and the density diminishes. If now we compare the observed oblateness with its possible extreme values, we can get a criterion of how the matter is distributed. For each planet respectively the limiting values and the actual oblateness in terms of ¢ are:— Ys oF i i Hath os. 1:25 and -50¢ 973 pb mats 4 @ LZ 44, | 30h 1145 ¢ Jupiter. . 1256 ,, ‘“487¢ 72 pare f 256 ',, “AT6$ ‘66> Thus the percentages of 7, in terms of 7—7, are: Harth,< ,.*| )» "63 Jupiter). «88 Wes og Oe OE Sb veg ee: Here a marked distinction shows itself between the major planets and the terrestrial ones. The major are from aquarter to three-eighths way from central concentration to homogeneity, while the terrestrial ones show the opposite relation. Now when we consider the small mean density of both Jupiter and Saturn we see that this means that tenuous matter occupies « large part of their volumes ; that they consist of a more or less dense fluid kernel surrounded by a vast shell of gas. Also that we never see what may by courtesy be called the surface of their globes. For the dark cherry-red portions of what we perceive in the telescope lie very nearly as high as the brighter portions, and cannot therefore be of much denser material. A secondary consideration from the table implies that the observed flattening of Mars is probably a little too large im = 190 and that pee or even is nearer the truth. This seems to 200 210 be corroborated by the fact that its oblateness as deduced by H. Struve from the motions of the apsides of the satellites i } ' 1S 790" and for some unexplained reason the values of the oblateness of planets so deduced seem always to be a little too 712 Mr. 8. H. Burbury on large. That of Jupiter from the actions of his satellites is an example of it. For these range from for the earlier to iL —~~ for the later values; while the flattening observed is —, =. 15°5 15°8 Thus theory helps us to a determination of the distribution of the matter composing a planet, which, since we cannot get inside it, would otherwise remain unknown. Boston, Mass. January 15, 1910, LXXV. Boltzmann's Law of Probabitity e~™. By 8. H. Burstury, F.R.S.* IVEN a system of molecules between which mutual forces of attraction or repulsion act, such forces having a potential y, the chance, or comparative frequency, of the molecules being in a configuration in which the potential of the mutual forces is x, is proportional to e—"%, h being defined as follows: ¢, the mean kinetic energy of a 3 e e . . molecule, =F: This is Boltzmann’s law of probability, proved by him to hold for gas molecules. May it not be extended to any material system in stationary motion ? Assuming this law to hold, the second law of thermo- dynamics can be deduced from it. (See a paper of mine in Phil. Mag. for January 1876; also Watson’s ‘ Kinetic Theory of Gases.’) It seems to me now that there are strong reasons for holding that the converse is also true, namely, that assuming the second law to hold, the truth of Boltzmann’s law can be deduced therefrom. For, the system of n molecules being in stationary motion, Clausius’ equation must be true, namely, . 3 pu=dt+45> Rr, in which ¢ is the kinetic energy of a molecule, L¢ the sum of the kinetic energies of all the molecules, p is the external pressure on the vessel of volume v in which the system is moving. Also, R is the repulsive force, 7 the distance, between a pair of molecules, and the double summation includes every pair. Now we may suppose v to become v+ 01, while ¢ and also p remains constant, and every 7 is increased * Communicated by the Author. Boltzmann's Law of Probability e-*x, (ees in the same ratio. Then r becomes 7 7, or ara; v ov and if x, y, are the rectangular components of 7, Ov=2 5-, Ov with similar values for Oy and dz. Multiply Clausius’ equa- tion by ano”. That gives | hpov= | re +h3E Ro, =no logu+n > ror =no logu—h axa», afk) MoS Canc aah oe EP ea dv where x denotes the mean value of that function. Next let the system receive a small accession of heat, QQ. That will be spent (1) in work done in increasing v against the external pressure p, yes is pOv3 eee in increase of the mean kinetic energy ¢, or 5 , that is — Sah (3) 1 in increase of mean potential x: ae leads to 9Q=pdv— sa dh+dy, and multiplying by - ho * ce) NaQq=° =m ak x é pegs ——— + =3 (hy) ame = 5nd log »— Xav— ee ie " O(hix) = xh, by (A). Of these five terms, 9 log v, 2 and “3 (hy) are complete 0Q differentials. In order therefore that - ; may be a complete differential, it is sufficient and necessary that the sum of the 2 2,d two remaining terms, namely — 5 xoh— 3 bh” On, be a complete differential of some function, u, of h and v. That is, 9 as we may omit the factor 5 du du = KONA MX Br= Tht T' dr, 7i4 Boltzmann’s Law of Probability e-*x., and therefore since Of and Qv are oS independent, dh ener dv =— 1 ; a.du, i ad an And since Jc dh ee have ax _ d dy 6 Fah Ge. ro) (B) Let now 7 be the function of probability, so that x=| Tudo, dv . 2 e e se . d the integration for ds including all configurations, also a { 7 Kaa, Then (B) gives io) ete {7 Mao + \x @ ao, } x ix df dx at ail (1 72 dc) Jafry Ye + fag 2 id d a ixte= ia Ope ac) becomes dx af t5—( 72x df dx (7S Naot | xihde= (7? do (ct dv ug ( df df dx or hares = ha ay do. and since y is a function of v only, we may write ws ices a! au dy dv’ and we have \ Bees tax (in Kae, and the equation which leads to Lae =n That is, 7 is a function of the product hy. Also afte), df ey eddoo ws clog f Menge Tah pO Sang onan and log J is a function of ix. The simplest solution of which is fae", epaera ls Oya ~LXXVI. Earth-Air Electric Currents. By GeEorGE C. Simpson, D.Sc., Government of India Meteorological Department *. ee PERIN G the present state of our knowledge of atmospheric electricity, perhaps the most important factor requiring study is the current of electricity which is constantly passing between the earth and the air. This eurrent has already been investigated to some extent, and two methods have been used for the purpose. (GerdienT determined the current by measuring the potential gradient and the conductivity of the air; and Wilsont has investigated it by observing the actual loss of electricity from a small plate exposed to the atmospheric field and kept at zero potential. The measurements made by these two methods have only been few; and in view of the large variations which the current undergoes in the course of a few minutes, the need has been strongly felt for some method by means of which the current could be automatically recorded. Neither Wilson’s nor Gerdien’s methods lend themselves for automatic registration, but an instrument has_ been developed in the Meteorological Office, Simla, which has proved entirely satisfactory in use. If we confine ourselves to the consideration of normal conditions during fine weather, the problem may be stated thus :— Consider a small area of the earth’s surface. On account of the electrical field in the atmosphere there is an induced surface charge of negative electricity on the area. This charge suffers change on account of two causes: (1) Every change in the potential gradient is accompanied by a change in the induced charge—the greater the potential gradient the greater the charge, and vice versa. (2) The electrical field in the atmosphere is constantly driving free positive ions from the air on to the surface ; or, as it is generally expressed, the conductivity of the air results in a current of negative electricity from the earth into the air. As the area under consideration forms part of the surface of the earth, its potential remains zero ; hence each change of charge from either of these causes gives rise to a flow of electricity between the area and the earth. If, on the other hand, the area were insulated its potential would be constantly undergoing change unless steps were taken to keep it zero, by either adding or withdrawing a charge similar in amount to that which would have flowed to or from it if it had been * Communicated by the Physical Society: read March 11, 1910. + Gerdien, Gotting. Nachr. 1907, p. 77. ¢ C. T. R. Wilson, Proc. Roy. Soc. A. Ixxx. p. 587 (1908). 716 Dr. G. C. Simpson on earth connected. The problem therefore consists in finding a method for keeping an insulated plate placed on the ground at zero potential, and for measuring the change of charge which this entails. The solution obtained was the following :—Water was made to flow from an insulated vessel in fine jets, and the points at which the jets broke into drops were surrounded by an earth-connected cylinder. The vessel was connected by a wire to a large plate placed in the open as near to the ground as was consistent with efficient insulation. By the well-known ‘‘collector” action the jets prevented the potential of the insulated system from varying appreciably from zero, and the drops as they formed carried with them the charge necessary to effect this. In other words, the drops carried away all the charge set free on the insulated plate which remained at zero potential. The drops as they fell were caught in an insulated vessel connected to an electrometer, and the change in the deflexion of the electrometer recorded the charge carried away by the drops. The details of the apparatus as used in Simla are shown diagrammatically in fig. 1. The nozzle from which the water issued is shown at A, surrounded by the earth- connected cylinder B. In order to get a large discharging capacity with a small expense of water, it was necessary to have the water issuing from the nozzle in fine jets which broke up into very small drops. For this reason, the nozzle was attached to the lower end of a glass tube 150 cms. long which was usually kept full of water, so that the jets issued under a considerable pressure. There were three jets each having a diameter of about*2 mm. To the top of the glass tube a vessel C was attached into which a regulated supply of water dropped from the funnel D. The point at which the water detached itself from the funnel D was well within the surrounding cylinder E, so that the drops were entirely free from charge when they fell into the vessel C. At first considerable trouble was experienced on account of the fine holes in the nozzle becoming choked up. To prevent this filter-papers were introduced into the vessel C and also just above the nozzle itself; but it was found that owing to some form of impurity in ‘the water, the papers became impervious to water after being in use for a day or so. After trying several methods of filtering the water without success, the following satisfactory method was hit upon. A piece of sponge was placed in the funnel D to remove all large impurities, and then the water was caused to pass through the trap F in the vessel C before it reached Earth-Air Electric Currents. - T17 the glass tube. The trap consisted of a closed brass tube fitting over the pipe which communicated from the upper to the lower parts of the vessel C. A hole was made in the Bip. y, TO EXPOZED PLATE TO EARTH TO ELECTROMETER covering tube somewhat below the level of the top of the communicating tube, so that the dust which floated on the top of the water in the upper part of C, and the grit which sank to the bottom, could not pass into the giass tube. 718 Dr. G. C. Simpson on Usually the water-supply was regulated so that it just provided the flow through the nozzle when the olass tube was full; if, however, the supply fell short of this amount, the head of water sank in the tube until the flow from the nozzle just carried away the supply; if the supply was too great, the vessel C gradually filled with water until the surplus overflowed from the pipe G. The water which escaped here carried away no charge as the drops detached themselves within the cylinder H. The water which left the nozzle A fell into the insulated funnel I and left this free of charge by dropping from the end of the small pipe K well within the cylinder L. Thus all the electricity which the drops from the nozzle carried into the funnel I remained in the funnel and its connexions. The capacity of this funnel was made as small as possible in order to obtain a large deflexion of the electrometer to which it was conuected. As the drops of water from the nozzle entered the re- ceiver I with considerable velocity, it was deemed necessary to investigate whether any error was produced by the “Lenard effect.” A careful investigation proved that no error of this kind was present, and this agreed with previous experiments, which had shown that, owing to some impurity in the Simla tap-water, it is incapable of developing the Lenard ettect. If this work were repeated elsewhere it would be necessary to test for this source of error, and if found it would no doubt be possible to remove it by adding a little salt to the water. The electrometer used was one of the pattern designed by Benndorf for automatically recording the potential gradient. In this instrument an electrical circuit containing a magnet is closed every two minutes, and a light boom attached to the needle of the electrometer is thereby pressed into contact with the paper taking the record. Thus the deflexion of the freely swinging electrometer-needle is automatically recorded at the end of each two minutes. A slight change in the construction of the electrometer made it possible to auto- matically earth the receiver as soon as its potential had been recorded. Hach deflexion, therefore, was proportional to the charge obtained by the receiver in the previous two minutes. The potential gradient was recorded by a second Benndorf electrometer, and the recording mechanisms of both electro- meters were actuated by the same electrical current. Thus simultaneous records were made on the two instruments. It was desirable that the exposed plate-should be placed ee ee Pl eee ee ee Fe, ee oo hk * —ie Earth-Air Electric Currents. 719 where the earth’s normal electrical field was undisturbed. This, however, was quite impossible in Simla, for the town is situated on a ridge of the Himalayas, and there is no singie plot of oround within many miles of the town on which the field is not disturbed by the configuration of the surrounding hills and valleys. ‘The Simla Meteorological Office is at present located in a house originally built for a dwelling on a spur branching off the main ridge on which the town is built. The house is situated on the extremity of the spur, and the ground slopes down for many hundreds of feet at a very steep incline on three sides of it. There is, however, between the house and the main ridge a small piece of level ground (about 100 ft. x 50 ft.) on which a tennis court has been made. ‘The tennis court is surrounded by stop-nets about ten feet high and several trees 30 or 40 feet high are growing against them. ‘he branches of the trees which were growing over the court have been cut away so that the court itself is fully exposed to the sky. The middle of this court was the best site which could be obtained, and a small tea-house built close to the stop-nets on one side afforded a suitable shelter for the instruments. Thus the plate was exposed in the middle of a small open space, the electrical field of which was considerably attected by the near trees and stop-nets as well as by the irregularities of the surrounding hills. The exposed plate itself consisted of a wooden frame (285 x 570 cms.=17 metre”) on which canvas was tightly stretched. ‘The canvas was covered by a sheet of brown paper made conducting by means of a thick coating of blacklead. In this way a large conducting surface was obtained which was at the same time light and strong. The sheet was insu- lated on six sulphur insulators which raised the upper surface of the sheet about 15 cms. above the general level of the ground. It would have been much better if the sheet could have been actually at the ground level, but as this would have necessitated cutting up “the surface of. the tennis court it was not possible to doit. It will be seen from the above that the exposure of the plate was far from that of a piece cf the earth’s surface exposed to the normal potential gradient ; but it was the best which could be obtained ander fia, circumstances. Results. Let each cm.’ of the exposed plate be receiving a charge from the air at the mean rate of w els. units per second. Then in a given time ¢ the whole plate will receive from 720 DroGoe Simpson on the air Adt els. units. If the potential gradient imme- diately over the plate changes from P, to P, volts/meire during the interval, there will be also a quantity of electricity AGP } - els. units set free on the plate. Both these charges are transferred to the receiver; hence, if the mean charge per second recorded by the electrometer is Q els. units, we have A(P,- P.2) Am30000 ” or Q hon) v= —-- er a x ihe <8 UU Qti= Aat— > ae + Pa els. units. If it is assumed that the whole charge x reaches the plate by means of the ordinary conductivity of the air (i. e., neglecting the charge carried by dust, &c. which settled on the plate against the electrostatic field), the conductivity can be calculated from the relationship 5) naa satay els. units, in which P is the mean potential gradient expressed in volts/metres during the interval ¢. This mean potential gradient will in general be different from $(P,;+P,). This treatment applies to a plate in a uniform electrical field, and therefore will not hold strictly for the plate exposed in the disturbed field described above. It has, however, been applied, and in discussing the results it must be remembered that while the absolute values obtained are likely to be affected, the daily range of the different factors are no deubt near approximations to the truth. | The data obtained were treated thus :—The trace printed by the electrometer consisted of a series of dots, each repre- senting the deflexion at the end of a two minute interval. These dots were first joined by a line, and then a freehand curve was drawn to take out the smaller irregularities. The approximately smooth curve thus obtained was measured at five equidistant points in each hour, and from these five measurements the mean deflexion of the electrometer during the hour determined. The mean deflexion was then con- verted into volts and the quantity Q/A determined. The part of this value due to the change in the potential gradient was then obtained from the actual value of the potential gradient at the beginning and the end of the hour. The Earth-Air [electric Currents. pea Jatter charge was subtracted from Q/A, and the remainder gave the mean value of w during the hour. _ The following gives a typical example :—The smoothed eurve for the hour 20 to 21 Indian standard time on the 19th November, 1909, was measured at five equidistant points, giving a mean value of 30°83 mm. As the sensitive- ness of the electrometer was 5°9 volts per mm. deflexion, the mean voltage recorded at the end of each two minutes during the hour was 30°8 x 5°9 = 181°7 volts, 2. e. *606 els. unit. The combined capacity of the receiver and electrometer was 35 em.; hence, during this hour the average quantity of electricity removed from the plate during each two minute interval was *606 X35 = 21:21 els. unit, or during each second 21°21/120=°177 els. unit. The area of the plate was 17x 10* cm.”; hence Q/A=:177/17 x 10°=104 x 10-8 els. unit. At the commencement of the interval the potential gradient over the plate was 20°8 volt/metre and at the end 49°3 volt/metre, hence rite 28°5 ble oie Reai<10° 3600xa-77x10° 7% 10 ts. amit. Hence wv, the mean charge received from the air each second by each em.’ of the exposed plate, was (104 —2)10-§=102 x 10-® els. unit. The mean potential gradient over the plate during the hour under consideration was 34°8 volt/metre, hence / 34°8 / 30000 The values obtained during ten days in November are shown in Table I. Only those days on which the weather was fine, without a cloud in the sky, and on which an unin- terrupted record was obtained from midnight to midnight, have been utilized. The lines marked z give the mean charge received from the air by each cm.” of the exposed plate in a second during the hour stated at the head of the column. The lines marked P.G. give the mean potential gradient over the exposed plate during the hour. The lines marked % give the mean conductivity of the air during the hour. The time entered at the head of the table is Indian Standard Time, which is 21 minutes ahead of mean local time. Phil. Mag. 8. 6. Vol. 19. No. 113. May 1910. 3A r~=102«10-§ =°7 x 10- els. unit. “qlun's[a -—N] X "9.19 UT/}[O A ‘qun's[a g—O] X “quun'sya p-OT X "a.t7oUI/9[0 A ‘qtam'st9 g—OT X *ylun's[o -—(OT X oT INS “qIun's[T9 g—(Q[ X ‘qlan'sye 7- AT X *a.13UT/J[0 A “Tun'sTe g—O1 X SruIesls P—()] Xx "A.AGOUT/4[O A “quum's[® g—QT X “}Un'syo F—()T X “14 OUU/4[O A ‘qlun's[a g—OT X 1G ee #9 _—_ VG 04 Uvop, vA +N LY Ee ze 68 1 he Og L-¢ |e | 19 | F-G | 2 | T-¢ | 9. CF (2-F | Gb 168 |8-F 166 |e G |¢-F (GF (88 |6-6 | 9-6 | oP © | Wy | OE | cs | ee | Ss | 8c SF iO | co 2h 68" 26. 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IG GL i! 074 0 "AON TUG “AON PUSS “AO NT IS1G "AON WG “AO NT GEL “6061 i y AON UILT | pavpuryg ueIpUty ‘qqun's[> p—OT X| 8.G rorqout/9[O A “}run's[s g—OT X *}TUN'STO F—OT X *a.tjgouu/4TO A “jun'sya g—OT X *j1uN'sTo p—OT X *OAQOUL/A[O A “yun'sTa g—OT X |'qrun'sye F-OT X "O.1OU/}TO A ‘yrun'sya g-OT X TEUN'STS 4-OT X "OIJOUT/J]O A “ran's]9 s—OT X “un'syo p—-OTX *AL19UT/}]O A “WUN'STO g—O] X 6-9 G& —S 10} poj}VItlOo SIN|VA UBS] poate ‘apuryo d1polsed-uol IOF SONTVA uvaw [BNIOV S49 ‘sXep uaq ~ —_—_—_—+ s 1:4, which is about three times the value for the radium y-rays, In our previous paper (EF. Soddy and A. 8. Russell, Phil. Mag. 1909, xvi. p. 620) the y-radiation of uranium X pre- parations from about 45 kilograms of uranyl nitrate was exhaustively examined. The y-rays were naturally far more powerful than those at Eve’s disposal, and in addition the rays had not first passed through a considerable thickness of material as in Eve’s experiments with uranyl nitrate. We found the absorption did not proceed exponentially in any substance until a thickness equivalent to 1 em. of lead had been penetrated and then the absorption coefficient remained constant (and, except for lead, mercury, and very light sub- stances, proportional to the density) at least until 5 cm. of lead or the equivalent had been penetrated. The value of r for substances of density between copper and aluminium was only 1:18 times the corresponding value for radium y-rays, while lead was somewhat anomalous with the ratio 1-465. The value of X found for lead for the radium y-rays was 0'495 from 1 cm. to 9 em. The results for the initial part of the absorption curves up toa thickness equivalent to 1 cm. of lead were reserved for the present communication. Tt was obvious that the whole question of the absorption of the cnn tas —a Question of the Homogeneity of 4-Rays. 727 y-rays needed careful re-examination before any conclusion as to their heterogeneity or homogeneity could be formed, and that the shape of the curves and values of the absorption coefiicients depended greatly on then unknown disturbing influences. A quantitative comparison of the ratio of the intensity of the B- to y-radiation for uranium and radium showed that the y-rays were relatively about 60 times less intense for uranium than for radium, and on account of the similarity of the penetrating power of the radiations of the two elements it was pointed out that the @- and y-rays could hardly be regarded any longer as interdependent phenomena. In Part I. of the present communication the absorption curves of uranium y-rays for various substances up to thick- ness equivalent to 1 cm. of lead are given and discussed. It is concluded that although for certain metals, for example zinc, the curve can be represented as the sum of two positive exponential terms, one due to the “hard” y-rays previously studied and another due to a “soft” y-ray with absorption coefficient about 14 times greater, this explanation does not hold good for other substances. Jn any case the supposed soft y-radiation is relatively feeble and unimportant. In Part II. it is shown that with certain new experimental dispositions the y-rays of radium are absorbed by lead as a homogeneous non- scattered radiation with a constant value for the absorption coefficient > (=0°50) in the strict sense of the equation dI/dT=—AlI over the whole range up to 7:7 cm. In these experiments the ionization vessel takes the form of two hemispheres with the radium at the centre and the absorbing plates are in the form of truncated hemispheres, this being a disposition for which the theoretical expressions can be evaluated. or zinc hemispheres again some evidence of a secondary penetrating radiation generated in the zine, with a value for A about 4°5 times greater than for the primary, has been obtained. The uranium y-ray curves show, however, small departures from the theoretical over the first part of the ran ge. In Part IIL. further evidence of the homogeneity of the y-rays of radium is given. The departures of the curve from the simple exponential type both at the beginning and the end of the range in lead are shown to be due to disturbing factors. With regard to the departures at the end of the range (Tuomikoski) a simple exponential law has been found, with suitable means of measurement, to hold up to 22 cm. of lead with a value for X, 0°498, practically identical with that given in the earlier paper. These results and others in this paper show that the y-rays are capable of measurement 728 «Mr. & Mrs, Soddy and Mr, A.'S. Russell on the to a very high degree of accuracy under proper conditions. The whole laboratory in which the measurements have been carried out has been preserved scrupulously from contamin- ation, and no radium, except in sealed tubes, has ever been brought in. With regard to the departures at the initial part of the range, it is shown that these are in opposite senses for different materials and vary greatly with the disposition employed, Although these departures have not yet been fully cleared up it is probable that they are due to equilibrium between the primary and secondary radiations, probably as regards distribution in space, not having yet been attained for the initial thickness of absorbing material, | Confining attention to the y-rays of radium and lead for which most evidence is available it is difficult to escape from the conclusion that the rays are homogeneous and are expo- nentially absorbed, without scattering of the primary beam. In metals other than lead, softening and scattering of the primary y-rays may, itis true, occur. But this does not affect the important conclusion that initially the primary y-rays are probably homogeneous. Now there is considerable evidence that the B-rays of radium are not homoge: eous. Hvenit the very soft 8-rays due to radium itself and to radium B (O. Hahn and L. Meitner, Phys. Zeit, 1909, x, p. 741) are lett out of consideration as possibly too feebly penetrating to produce a detectable ry-radiation, there is evidence that the 6-radiation of radium C itself is also complex (compare ibid. p. 697). The view that the y-rays of radium are homogeneous thus carries with it very strong support of our earlier view that the 8- and y-rays are not interdependent, It is not advyo- cated that this point of view is as yet established but merely that a considerable body of evidence exists in its favour, We are now at work to see if Prof. Brago’s theory that some metals like zine soften and scatter the y-rays before converting them into @-rays while others like lead trans- form them into $-rays at one step, will prove helpful in unravelling some of the intricate phenomena observed. It appears, however, if this is so, that zine will be found tu behave to the ry-rays of uranium like lead to the y-rays of radium, ny Part I.—Jnitial part of Absorption Curves of the y-Rays of Uranium X. The disposition employed was identical with that used in Series I. of the earlier measurements (loc. cit. p. 632), all the absorbing plates being clamped up to form the base of the Question of the Homogeneity of y-Rays. 629 electroscope, the uranium X preparations being 14°63 cm. below the upper surface of the base. The curves are shown in fig.1 (Pl. XII.). The ordinates represent logarithms of the ionization and the abscisse the thickness multiplied by density of material, i. e. what may be termed equivalent thickness. The ordinates have been corrected for the decay of the activity of the preparation during measurements and so the curves show the approximate relative magnitude of the ionization when equivalent thicknesses of various materials form the base of the electroscope. The actual results are plotted and no attempt has been made to smooth the curves. In the case of lead part of the irregularities are doubtless due to irregu- larities in the thickness of the only foil then available. It will be seen that the curves arrange themselves one above the other in oider of the density of the material, the hghter substances giving for equivalent thicknesses far more ionization than the denser. The secondary radiation pro- duced by the primary y-rays of uranium X from the emergent surface of the absorbing plates obviously increases as * the density of the material diminishes. Accor ding to the neutral- pair theory of y-rays the emergent radiations should all be equal it the density law of absorption holds true for both B- and y-rays, and if the latter are homogeneous (Bragg and Madsen, ‘Phil. Mag. 1908, xv. p. 670). The dotted lines in the curve represent the prolongation back to the zero axis of the straight part of the absorption curves over the range at which absor ption is exponential. The lead and iron dotted Jines are from actual observations taken at the same time as the others with greater thicknesses of metal than can be shown on the curve. Unfortunately in other cases it was not possible to do this. The zine dotted line is drawn with an arbitrary ordinate, from the value of A, 0°37, before found. It will be seen that the curves differ from each other progressively as the density of the material decreases, [or zinc ‘the curve is not very different from what would be the case if it consisted of two superimposed curves, one due to the hard y-rays and a second due to a solter ty pe with a coefficient of absorptionabout 14 times greater. Butfor the lighter materials, for example cardboard and aluminium, this is not the case. With diminishing thickness the slope of the curve at first is greater than that of the bard y-rays, then diminishes and runs for a little practically paten to it, and then increases again. But just before the 8-rays begin to enter the electroscope a well-marked diminution of slope is again noticeable in the card curve. Attention may first be directed to the dotted lines. These 730 = =Mr. & Mrs. Soddy and Mr. A. 8. Russell on the represent the primary hard y-rays capable of penetrating 1 cm. of lead, together with the equilibrium amount of secondary radiation they produce in the absorbing material. If only hard y-rays were present, on the simple theory (dvd. p- 641), assuming the ionization produced by secondary and primary to be directly proportional to their absorption coefficients, one would expect the following relation to hold:— Iy T—-A, —A,T Kz —_— = —_— 2 & = =a te . ° ° 1 T; (1+). Be “", where 8 Tee (1) Here Ir and Jy refer to the ionizations, A, and A, to the coefficients of absorption of the primary and secondary radiations respectively and « is the coefficient of transfor- mation, or fraction of the energy of absorbed primary transformed into secondary. For great thicknesses the second negative exponential term becomes negligible if Xe is greater than A,. The two radiations are in equilibrium and the relation becomes :— la if - 9 This is the equation of the dotted-line curves. Since for thicknesses greater than 1 em. of lead the curves are exponential it may be assumed that A, is always much greater than Ay. The relation indicates that when the radi- ations are in equilibrium the ionization is due to the two parts, j KXo ar one, ei", due to the primary and the other, ~~ \-e~™", 2— due to the secondary. Now for different substances « may possess any value between 0 and 1, and since on not very much greater than unity, it follows that for equivalent thicknesses of various materials the ionization cannot vary very much more than in the ratio of 1 to2. So faras can be seen the results certainly bear this out. At equivalent thicknesses 5'6—the greatest examined for card—the ratio of the ionizations of lead and card are 2°15, and, at 2°6, of pine- wood and lead are 2°5. A similar 2:1 ratio in the emergence radiation from different elements for the y-rays of radium is to be seen in Bragg and Madsen’s curves (figs. 2 and 3, Phil. Mag. 1908, xvi. p. 925). It was not possible with the lighter substances to extend the curve sufficiently far to give all the information required, for the source of radiation was too weak to be removed to a greater distance to allow of greater thicknesses of absorbing material being used. A fF Question of the Homogeneity of y-Rays. tai prolonged attempt to interpret the curves obtained led to no very definite conclusion, partly perhaps on this account. If the ionizations shown by the dotted curves are subtracted from the observed values one would expect to obtain a curve having the relation I, S = —Be—rt4 Ce-Ast , Pn ts WON (3) ss where C and A; refer to the soft primary y-radiation, if such exists. ‘The zinc curve and to lesser extent the lead and iron curves so obtained are nearly straight lines. Hither «, and hence B, must be very small (which cannot be true at least for iron as the dotted-line curve shows) or A, must be relatively very large. In other words the secondary radiations are wholly @-rays and secondary y-rays play no appreciable part. The zinc curve can be approximately represented by the equation Ty = Ae—Ait+ Ce-A,T 1, where A, is 0°37, A, 5°15, A0°3 and C 0:7. The following table shows that the agreement between the observed and calculated values is fairly close. The deviations are not large but they are greater than the errors of measurement. L, eu T cm. — observed.| — (calc). | 0 I) } 1065 [184 (B-rays)) 692 142 627 622 1775 ‘541 | 560 | 213 “BVO ames tetany 2485 ‘446 f? "Sage Ps 2840 | 418 | 432 | 355 ‘| (375) | (375) 426 | 342 | “334 | ‘497 | “314 | 303 | 568 |: -284. | asl wid 639 269 | 263 745 “255 243 "852 238 | 227 The ratio A,/A, is 14, and if the energies of the two types of rays are proportional to the ionizations divided by the absorption coefficients it follows that the energy of the soft type initially is only one-sixth of that of the bard type. Even if such a soft type exists, therefore, in the y-radiation of 782 Mr. & Mrs. Soddy and Mr. A. 8. Russell on the uranium X, it must be unimportant relatively to the hard type. he curves on the whole, however, do not bear out the view of the existence of a soft primary radiation. It is possible by choosing the values of A, and A; near together to get a more or less close approximation of the curves for the lighter substances with the theoretical equation consisting of the sum of three exponential terms, two positive and one negative, but only by using values of B inconsistent with the dotted-line curves. Experiments were performed with the uncovered uranium X preparation between the poles of a powerful electromagnet to see if B-rays had any influence on the results and to explore the curves over smaller values of the equivalent thickness in the hope of getting evidence of the negative exponential term. The electroscope rested on a thick cast-iron plate supported as a table above the magnet, over a circular hole cut in the plate, to the under side of which the absorbing plates were clamped up. Using just sufficient card or lead completely to absorb @-rays it was found there was no difference in the value of the leak whether the magnet was excited or not. The same was true when card of equivalent thickness 4°4 was used. Hence the 6-rays do not affect the results, or produce any appreciable secondary y-radiation. The dotted part on the curve (fig. 1) indicates the reading obtained for card over the region usually masked by the 6-rays. The ordinate of the curve is naturally arbitrary. It must be remembered that §-rays are never completely, deviated by a magnet (Soddy, Phil. Mag. 1909, xvi. p. 860), although with the arrangement used “they were aainond probably to less than one per cent. of their initial value. The results are therefore not of great value. For the metals sufficiently accurate results to be of service could not be obtained owing to the heat of the magnet disturbing the readings of the electro- scope. In light of the results to be given in the succeeding sections it seems that the distribution of the emergent radiation as regards direction in space must be taken ‘into account in the interpretation of such y-ray absorption curves. Bragg and Madsen (Phil. Mag. 1908, xvi. p. 926) have suggested that the differences between light and heavy metals are due to the soft type of y-radiation, supposed to be present initially, being more readily absorbed, relatively to the hard type, by ence) tien by light eulasranigest the absorption causing a proportional production of soft @- radi- ation. The soft y-radiation is thus used up rapidly in lead, Question of the Llomogeneity of y-lays. 733 whereas in aluminium it is not used up so fast and it produces Jess secondary radiation. But such‘a soft §-radiation will always be in equilibrium with the soft y-radiation, except over the part of the curve not investigable owing to primary #-rays. The logarithmic curves obtained by subtracting the effect due to the hard type from the observed effects should be straight for aluminium no less than for lead, whereas it is obvious that they are not. All that the explanation, if we understand it aright, would seem to involve is that the slopes of the curves, plotted against equivalent thickuess, should be greater for dense than for light materials. Part 1l.— Absorption of y Rays in truncated Hemispheres. H. W. Schmidt (Ann. Phys. 1907, xxiii. p. 689) has given the solution of the problem of the absorption of a homo- geneous radiation from a point source uniformly distributed in all directions, in a plate of uniform thickness placed directly over the source, assuming that the absorption proceeds exponentially as with light. His paper and the theoretical absorption curve he gives suggested at first sight that the departure of the experimental curves from the exponential form might be due, not to any want of homo- geneity of the rays, but to the obliquity of part of the beam. This, however, as will be shown, is not the case. The error so introduced is trifling with the disposition employed, the found absorption coefhicient being about 2°5 per cent. greater throughout the whole range than the true theoretical coeffi- cient for a parallel beam of rays. Schmidt solved the problem in connexion with his study of §-rays, to which it will be shown the solution does not apply owing to scattering. Experiments, however, on the absorption of the y-rays with some new dispositions for which the absorp- tion according to the theoretical expressions can be calculated, have strongly confirmed the view that the y-rays of radium are homogeneous and that a soft primary type does not exist, although still much remains to be cleared up. Unfortunately, i in Schmidt’s paper the accidental omission of a factor in the final expression makes the solution ¢ appear false, and we are indebted to Sir Joseph Larmor for solving the problem for us independently. A plate of infinite area and thickness ¢ is in contact with a point source sending out radiation uniformly in all directions which are absorbed 734 Mr. & Mrs. Soddy and Mr. A. 8. Russell on the exponentially by the plate, the absorption coefficient being represented by X in the sense of the equation dl apa Diagram A. It is required to find the quantity of radiation after it has passed through the plate. let E be the intensity of the radiation at any point. Let I, and Iv, respectively, be the measure of the total radiation over the angle of 180°, with the absorbing plate absent and present. Within a solid angle dw the intensity of the radiation is Hd with no plate present, and EH. go. e— \t/008 With the plate. do = 6(—27 cos 8), | w/2 Ty =| Kd(—27 cos @) = 27H, 0 ae L= ( E(e 7! °°) d(—27 cos 8), e0 w/2 i =( Bes ee 27 cos 0). a jo Let AT/cos 6 = y, ae The integral ie .dy is known as the Exponential Inteeral. It is expressed by the symbol Ei(z). Values both for Hz(z) and Ei(—.) have been tabulated by J. W. L. Glaisher (Phil. Trans, 1870, clx p. 367). The result may re Question of the Homogeneity of y-Rays. 735 thus be written :— is =r = Tans AE Be NE tte, yx. that CS) This is Schmidt’s result, the factor AT in the second term having been omitted through an oversight. The graph of the function is shown by the lower curve in fig. 2 (Pl. XII), the lower curve representing log” . The upper curve re- 0 presents the value of logio i according to the simple expo- 0 nential law. It will be seen that the curve somewhat resembles the experimental curve, the slope at first decreasing until a certain initial thickness has been penetrated, after which it remains nearly constant. As Schmidt has pointed out for values of AT greater than 5 the slope is about 1°16 times greater than for the simple exponential curve. As drawn, the slope over the range from AT = 2 to AT = 4,2. ¢., for a thickness of lead from 4 to 8 cm., is very nearly 1°25 times the slope of the exponential curve. The above solution, however, applies only to the case where the cone of rays included subtends an angle of 180°. The expression holding for a cone of any semi-angle @ is, however, readily obtained. If Io! and I,’ represent respec- tively the quantities of radiation in a cone of semi-angle @, before and after passage through the absorbing plate, iho: ~° —1—cos 0, 0 At/cos 0 0 a) Py —Xt/cos 0 a n ie a =| Z d(—eos 0) = ar am . dy, AT / T= {<9 (oon ye" 4a BH(—AD) 0 —AT| Ei(—AT/cos 6) | } +(l-cos@). . (5) In the experimental disposition described with cylindrical electroscope 12°8 cm. high and 10°5 cm. in diameter, and the preparation 8°25 em. below the upper surface of the base, the base and top of the electroscope subtend respectively cones of semi-angle about 22° and 14°. The mean semi- angle may be taken as about 18°. A particular case was 736 Mrs. & Mr. Soddy and Mr. A. S. Russell on the 21 worked out for cos @ = = (6 = 17° 45’). In the following / table the values of a and = the latter calculated from the 0 0 simple exponential formula, are contrasted for various values ot AT :— Ar, | ie Ir Bae Ute. | 1 Ip | | 00 | 1-000 / 100000 = | | 20 ne gisra (RCL crane A001). Jegbaage Olio) megoRa Te») lee GUT ‘54075 54881 | es 80 | -44058 44933 hice 2-0 12896 13534 ie 1-02145 40 016674 018316 In the last column 2//A represents the relation of the apparent to the true absorption coefficient, 4’ being the value that would be obtained for the absorption coefficient if calculated for the two values of AT indicated in the table according to the simple exponential law instead of from equation (5). It will be seen, therefore, that the general effect of using a cone of rays of semi-angle 18° is that the absorption coefficient experimentally obtained is from 24 to 2 per cent. too high throughout the whole of the range. Thus the obliquity of part of the beam is completely insufficient to account for the initial irregularities of the absorption curves. Diagram B. —— Rr ———_ > Equation (4) assumes the plate to be of infinite area, and it is scarcely possible working with the y-rays of radium to Question of the Homogeneity of y-Rays. 137 use plates so large that no radiation gets through the edges. If instead of plates of infinite area we use truncated hemi- spheres, the part of the radiation escaping from the sides of the truncated hemisphere can be allowed for. The total initial radiation I, over the whole 180° is divided into two parts, S and P, to represent the portions escaping: through the sides and top of the truncated hemisphere. Then §, = I, (T/R) where T is the thickness of a truncated hemisphere of radius R. Then S: = Io (q)e™ : cos 9 = T/R | | —— e—At/cos 8 J _ cog @) 1, cos9 = 1 7 AR eee Ve ey 3 =e (3.)¢ anf ey dy ; AT - = eM $T[Ei(—AT)]-AT[BW AR). . (6) Equation (6) has been tested experimentally with hemi- spheres of zinc and lead built up of circular plates of diminishing radius from base to top, the radioactive prepara-- tion being placed at the centre of the base. Fig. 3 shows: the apparatus employed. The ionization chamber consisted. of two copper concentric hemispherical bowls, A and B, of diameter 40 cm. and 20 cm. respectively with the absorbing hemisphere C concentrically placed within the inner bowl. The wall thickness of B wasabout 0°5 mm. The radium was placed at the centre of C at the point marked D. Thus the paths of all rays over the whole 180° within the ionization chamber were equal. The ionization chamber was fixed to the under side of a table HE, carrying the electroscope F and reading microscope. The base of the electroscope was protected by a circular block of lead, G, 5 em. thick, of the same diameter as the electroscope, through central holes in which was placed an insulating plug H bearing the electrode K. ‘The latter consisted of a construction of wires similar to an open umbrella frame at its lower end, which carried on its upper end a gold leaf. J was an additional lead plug. The absorbing hemisphere rested on a platform which could be raised or lowered at will. The dotted lines show the hemisphere in its lowered position. The platform consisted Phil. Mag. 8. 6. Vol. 19. No. 113. May 1910. 3B 738 Mr. & Mrs. Soddy and Mr. A.S. Russell on the of a block of lead L, 10 cm. diameter and 5 em. thick, ina small hole in which the radium tube was placed at D, so that the radium was in the centre and as near the surface as possible. For experiments with uranium X another similar Fig. 3. N Notz /nsulation thus Lead in Section thus Other Sections thus KS — 5 Z—- ss eed hiks 4 Phe oe ie pees Pe Pa 7) PAR a RE CSS Sey ey cas peer ree eee tie ea See ue pee SB SSS Se E24 FaseSaSSSse Sasa aRere SSS : peas toc a Sectional Plan foken through ZZ Approximate Scale e los platform was used, with a rectangular depression 8 cm. xX 4cm.x 5 mm. deep, capable of taking the three platinum trays on which the uranium X was spread in the form of thin films. The apparatus was designed so that there should be as little material as possible round the base and sides of the ionization chamber, to produce secondary y-rays (Kleeman, Phil. Mag. 1908, xv. p. 639). But owing to : 1. == a M\ << Question of the Homogeneity of Y-Rays. fag the weight of the absorbing hemispheres it was found necessary to support the platform by four T rods of iron MM running horizontally close beneath the base of the ionization chamber (which consisted of } inch sheet brass), and these probably produced a little secondary radiation capable of penetrating the base of the ionization chamber. Any such secondary radiation would, however, be constant, and would merely increase the value to be subtracted from the readings as “natural leak.” The results with the lead hemisphere and the y-rays of radium bear out, in a remarkably close manner, equation (6), the value of » being 0°5. This is practically the value before found, viz., 0°495, and is identical with what, as the results of subsequent experiments given in Part III., has been found lead to hold accurately up toa thickness of 20 cm. of lead. The hemisphere was 7°7 cm. diameter. Fig. 4 (Pl. XII.) shows the results obtained up to the first 2 cm. of lead, and fig. 5 the results for the complete hemispheres with both lead and zine. Dealing first with lead, the curves drawn are the theoretical curves calculated according to equation (6), with AX = 0°5. Fig. 4 may be first considered, as it is very remarkable. The observational points lie close on the curve, almost from the point at which all the §-rays are absorbed. Instead of the large irregularity in the initial part of the curve, as with ordinary dispositions, there is only a very slight departure from the theoretical curve, the slope of the curve initially being a very little greater than the calculated. In fitting the observations to the curves the following procedure was adopted :—By repeated trial with different values of A, that value was found which best fitted the results as a whole. In this way X was easily determined to about 2 per cent. Then in drawing fig. 4, one observation was put on the theoretical curve somewhere about the middle and the other points plotted from this point as the basis. Dealing now with fig. 5 for the whole lead hemisphere, only the logarithmic eurve is shown. Measurements were done with 0°47 and and 6°7 mg. of radium, and showed virtually no difference. To fit the results to the curve and to find the correction for secondary radiation, which at the end of the curve is of some consequence, two observational points, one at the end and the other near the middle, were arbitrarily placed on the theoretical curve (X= 0°5). A simple calculation then showed what the amount was that had to be subtracted from the observations to allow for secondary radiation, assuming the theoretical law of absorption held true, and this amount was subtracted from all the measurements. It has also been 3 B2 740 Mr. & Mrs. Soddy and Mr. A. 8. Russell on the subtracted in fig. 4, but there it is too small to be of any significance. With the 0°47 mg. radium this correction amounted to 1°78, and with the 6°7 mg. radium to 30°6. The corrected leak with the whole hemisphere in was 3°85 for the 0-47 mg., and 51°8 for the 6°7 mg. The natural leak was 2°8 in most of the experiments. This shows that the correction is very small and is nearly proportional to the quantity of radium. ‘The curve (fig. 5) shows that the observational points lie on the theoretical curve with very great accuracy. The gratifying result follows that according to these experiments the y-rays of radium are exponentially absorbed by lead as a single homogeneous radiation, with the value of the absorption coefficient 0°5, from the thickness at which the 8-radiation is completely absorbed up to 777 cm. This is so completely at variance with the result obtained with ordinary dispositions, and the conclusions of many inves- tigators who claim to have evidence of the existence of two and even three types of homogeneous y-rays from radium, that it may be regarded as a very remarkable result. It also follows that the y-rays are not scattered at all in passage through matter. The sharpness and delicacy of y-ray photo- graphs, when precautions are taken to remove #-rays by a magnet, also bear this out (Mme. Curie’s Thesis, ‘‘ Radio- active Substances,” p. 65). To indicate the difference between the case of the y-rays and that of the @-rays, which are well known to be scattered by thin films of absorbing materials, it is of interest to contrast the preceding result with some obtained with the @-rays of uranium X, also in a hemi- spherical ionization chamber. A copper hemisphere, 16 cm. in diameter, was mounted above an electroscope as shown in fig. 6, with the absorbing plates clamped on to the top. The uranium X preparation, usually on a very small thin piece of micro-cover glass, was laid on the centre of the top plate, and could be covered at will with thick disks of zine, lead, or aluminium, when it was desired to examine also the “reflected”? radiation. Sometimes the uranium X was placed directly on the centre of the uppermost plate. To compare with these results others were taken in a brass. electroscope of ordinary cylindrical pattern and size with the absorbing sheets clamped up to the base. The preparation was supported 9 cm. below the electroscope on a square of micro-cover glass 11 cm. sq. and 0°125 mm. thick by means. of a light frame attached to the table on which the electro- scope rested. The design was intended to minimize the reflected radiation as much as possible. The preparations. ee ge ee eee a ee _— a Question of the Homogeneity of y-Rays. 741 were in the form of films upon micro-cover glass. Some- times this was placed on the supporting square of micro- cover glass below the preparation. Sometimes the prepara- tion was fastened beneath the support, film downward, so that there was nothing below, but the rays had to penetrate 0:25 mm. of glass before entering the electroscope. Thick plates of lead, zinc, and aluminium could be clamped up immediately beneath the preparation at will. The curves for lead, for the hemispherical electroscope (fig. 6) only, are shown in fig. 7 (Pl. XII.). The middle curve refers to the bare preparation with nothing below. The upper curve was obtained by placing close above the preparation a disk of lead to reflect back the rays. The lower curve is the difference curve, referring to the reflected radiation only. The middle curve is practically a straight line, if anything slightly concave to the origin. There is thus nothing abnormal about this result due to the use of the hemi- spherical ionization vessel and the cone of rays of angle 180°. Unfortunately no actual comparisons were done for lead, but for zinc and aluminium the results in the hemispherical apparatus were compared with those obtained with the ordinary disposition with cylindrical electroscope described. The curves are shown in figs.8 & 9 (Pl. XII.). In fig. 8, for zine, the curves A, ©, E refer to the hemisphere, the curves B,D, F to the cylinder. As before, the middle curves of each set C and D refer to the bare preparation, the upper A and B to the preparation with a thick plate of lead covering 742 Mr. & Mrs. Soddy and Mr. A. 8. Russell on the it, and the lower E and F to the reflected radiation only. There is practically very little difference between the curves C and D, though one refers to a comparatively narrow cone — and the other to a cone of angle 180°. Fig. 9 shows similar results for aluminium, the lettering of the six curves referring to the same disposition as in the last figure. Here the hemisphere curves A and C are markedly more straight than the cylinder curves B and D, which, as is well known, are strongly concave to the axis (Levin, Phys. Zett. viii. 1907, . 985), the absorption coefficient, indeed, increasing 2°95 times before all the rays are absorbed. So the general result of these experiments is to show that, on the whole, the simple exponential law is more nearly followed with 8-rays when a hemispherical ionization chamber and cone of rays of angle 180° are used than in the common disposition. This suffi- ciently illustrates the great difference between the @- and y-rays. i view of the controversy that is in progress as to the nature of the law of absorption of @-rays, these results are of interest as showing how artificial the mathematical treatment of the question is which assumes a rectilinear propagation of the rays. The passage of @-rays through matter probably resembles more a diffusion than a radiation, and before much real advance can be expected to be made it would seem necessary to obtain some information of the free-path of the 8-particle in various metals. With the apparatus shown in fig. 3 several other measure- ments of the y-rays both of uranium X and radium, for zine hemispheres as well as lead, have been taken. Figs. 5 and 10, Pl. XII., show the result for radium with a zinc hemisphere 9 cm.in radius. Tor the latter two thirds of the hemisphere the curve agrees remarkably closely with the theoretical curve (7 = 0:28). This again is identical with the value (0°278) given in Table II. of the last paper. When dA was put equal to 0°30 the agreement was far less perfect. But for the first part of the curve the observed readings fall consistently below the theoretical. The lowest curve in fig. 10 is the difference curve between the observed and theoretical values of Ir, and it will be seen to be very approximately exponen- tial up to the point at which @-rays interfere. The value of the coefficient X’ of these supposed secondary rays is 1:25, or 4:5 times that of the primary (compare Madsen, Phil. Mag. 1909, xvil. p. 447). This result thus suggests that for zine a secondary radiation is generated which does not come into equilibrium with the primary until about 2 cm. of zinc have been penetrated. Again there is no evidence of a soft primary y-radiation. The remaining two curves (fig. 11, Question of the Homogeneity of y-Rays. | 743 Pl. XII.) refer to the uranium X y-rays for lead and zinc. The fact that the uranium X was in the form of a large surface 5 mm. below the base of the hemisphere somewhat complicates the interpretation of these results. This would have been fatal for the radium y-rays, but for the less penetrating rays of uranium X it is probable that the disturbance so produced is not very great. In addition it was not found practicable to determine the curves over more than the first 2 cm. of the hemisphere, so that the position of the curve with reference to the theoretical is more or less arbitrary. This makes their interpretation rather uncertain. As drawn, both curves agree well over the latter half with the theoretical curve (App = 0°66 and Az, = 0°35), while for the first half the observed values are consistently greater than the theoretical, the departure being greater for zinc than lead. The curves would not at present repay further discussion. All that can be said is that they bear out the view before expressed that if a soft y-radiation of uranium X exists it must be relatively feeble in comparison with the primary, while at least in the radiation of uranium X there is as yet no sufficient evidence either of its existence or of that of a secondary penetrating radiation. It has been thought advisable to append the observations, from which the y-ray curves in Pl. XII. figs. 4, 5, 10, and 11 have been obtained, in the following two tables. The observations represent divisions of the scale per minute corrected only for the natural leak of the instrument. TaseE I. Pb and Zn Complete Hemispheres. Ra y-Rays. See Pl. XII. fig. 5. ; Pb. 8/7/09. 9/7/09. | 22/6/09. | Zn. | 22/6/09. Zn. 22/6/09. T (cm.). ||6°7 mg. Ra./6-7 mg. Ra.|-47 mg. Ra.|||'T (cm.). “47 mg. Ra. T (cm.).|'47 mg. Ra. 0511 1147 1159 94 | 0-185 188 4-154 43-7 1-009 807 774 59:3 0:332 173 4-448 39:8 1:495 567 537 40-0 0°626 150 4°742 37:3 2005 397 392 28'8 0-920 1348 5036 349 2:491 298 295 21:0 1:214 1195 || + 5:330 33:1 2-980 231 227 16-2 1:508 107°2 5624 31:2 3:464 1847 181 12°6 1-802 93:4 5918 30°3 3:953 150°2 146°3 10:08 2096 | 835 || 6212 28°3 4446 128-7 125:3 8:09 2390 752 6506 27°75 4-946 108°5 107°5 7-63 2-684 68:2 || 6-800 27:05 5438 98-2 97°5 6-90 2978 61:5 7 094 26°25 5-940 90:7 90°2 6-25 3:272 | 556 7 386 25 80 7-44 82:4 82°4 565 3566 | 52:0 7680 a 3860 | 477 || 891 25°25 24°75 744 Mr. & Mrs. Soddy and Mr. A. S. Russell on the TABLE II. Pb and Zn Hemispheres (Initial Parts). Ra and UrX y-Rays. (See Pl. XII. figs. 4,10, and 11.) Fig. 4. Fig. 11. | Fig. 10. | Fig. 11. Pb. 18/6/09. Pb. 19/6/09. | 21/6/09.|| Zn. 30/6/09. | 29/6/09. T (em.). |'47 mg. Ra.|| T (em.).| UrX. | UrX. || T (cm.).|'47 mg. Ra.| UrX. 0316 253 295 245 || °O375 317 642 0632 209 81 70°83 || ‘O75 224 107°6 0947 191 65 HDD | 112 203 79'°3 1261 18] ae 5O3 47°3 sO | heehee veal "1573 172 Y 50°6 42-4 184 190°5 65°2 "1893 165 g 48°8 40-4 ||| °222 183°5 61°5 °3155 139 = 30°4 30°6 ‘260 181 57°6 4418 121 5 31°4 23°65 ||| ‘331 1755 49°6 “5674 106°7 a 24:0 19°8 i365 ee RN 47-9 6914 93°7 ne 20°3 16-7 406 165°5 46:2 °8156 84°4 < 17°6 146 ‘478 159°5 42°5 9419 14:5 153 12-4 553: 1569 39°3 1-066 66°4 13°6 11:02 625 Taig 36°7 1-190 61:4 11:8 9°35 ‘700 1469 345 1314 55°4 10-2 S35 i a I2 141°9 32°5 "994 69°5 543 24-8 21°7 || °847 137°4 30°7 1494 48°7 1041 13:0 LOO aii <9il9 133°4 29°6 1-987 35°4 1527 8:15 6°95 | 994 129°4 28-1 S| 502 2:037 55 4-62 || 1:066 125°4 26°6 1-558 ATED POETS RO RE CRN NES ees | 1:3860 113°4 22:0 1682 BOO Mile Bsc 3 /u) pI OARE SOR CARRS oe || 1-654 97°9 19:05 1-806 EDs te llnte deine saul ek qemeren ane ae eta | 2:097 83'4 15°50 aM hi ied Mami hee | 2977 63-5 10°50 Part IIl.— The Variations in the Value of the Absorption Coefficient of Radium y-Rays. In the former paper the absorption of the y-rays of radium by lead over a range of thickness of 1 cm. to 9 cm. was examined. The absorption was found to be strictly expo- nential, the value of X being 0°495 (cm.)—1. This investi- gation has now been pushed further by employing much larger quantities of radium than before, to see if the exponential absorption held over as large a range as could be investigated accurately. In the first of the new experiments the same y-ray lead electroscope was used and the same method of measurement employed as had been used previously. 6°7 mg. of radium bromide was placed immediately below the electroscope at a distance of 21:5 cm. from the upper surface of the base. The lead used for absorbing the rays was in the form of circular plates 12°5 cm. in diameter and about 1°25 em. thick. Question of the Homogeneity of y-Rays. 745 These plates were placed directly on the radium. The range of thickness of Jead placed over the radium was from 0 to 18°6 cm. Up to 11 cm. the exponential law of absorption held true. The value of X was 0°500 (cm.)-1. Beyond 11 cm., however, the curve continued no longer straight but became convex to the origin (fig. 12, Pl. XII.). The departure of the curve from the straight line occurred when the intensity leak (corrected for natural leak, which was 3°6 divisions per minute) was about one division per minute. This departure must be due, either to the presence of a very penetrating primary radiation from the radium, or to a constant secondary radiation entering the electroscope otherwise than through the base, and which, being very small, did not begin to affect the straightness of the curve until very great thicknesses of lead had been penetrated. The next experiment was carried out very similarly to the first except that an electroscope of brass was substituted for the lead one and no circular lead screens round the electro- scope were used. The electroscope was a cylindrical one of the ordinary type, 13 em. in height and 10°8 cm. in width. The thickness of the walls was 0°32 cm. and of the base 0-6 cm. The lead plates were placed on the radium as before (which was placed in a recess in a piece of wood) and a range of 0 to 14 em. of lead investigated. The curve obtained in the same way as the last is no longer straight over any part of the range but is convex to the origin, ’ varying from 0-62 (cm.)~! to 0:08 (em.)—1 over the range. The results are shown in fig. 13, curve A(P]. XI1.). The difference between the character of this curve and of the last it would be natural to ascribe to the nature of the material of which the electro- scope is made as the other factors of the disposition have been unchanged. Experiments as to the cause of this difference have shown, however, that it is due as much to the nature of the absorbing plates as to that of the material of which the electroscope is made. These will now be described shortly. A base 0°8 cm. was substituted for the 0°6 cm. base and the walls were increased in thickness to 1°8 em., but the general convexity of the absorption curve was not affected. It was found impossible with this disposition to obtain the straight line relation by thickening the brass electroscope with brass. A series of measurements was next done under exactly similar conditions as last except that the two circular screens of lead before used were placed around the electroscope. The effect of the lead screens on the character of the curve was very marked (fig. 13, curve B). The curve which had been convex before became now straight over a large part of the 746 Mr. & Mrs. Soddy and Mr. A. 8S. Russell on the range but finally became convex as before. The value of the leak at this point of departure from the straight line (T=5°5 cm.) is about 10 divisions per minute. ‘The departure in this case is not due to the same small effect as had caused the departure in the first experiments (fig. 12). Some measurements to illustrate the effect of the circular screens on the radiation are given in Table III. TaBie IIT. Thickness of Lead. Intensity, Intensity, (cm.) no lead screens. with lead screens. 7415 23°21 16°24 8-591 17°38 9°99 9°856 14°03 6°22 11°086 11°65 4°27 Thus at the end of the range the radiation entering the brass electroscope is reduced to 0°37 of its value by merely placing lead screens round the side of the electroscope. Similer screens of brass of equivalent thicknesses were sub- stituted for the lead but they bad practically no effect on the leak in the electroscope. It may be noticed that the difference in the values in columns 2 and 3 of Table I. is practically constant (7-4 divisions per minute), as though caused by a constant reflected or secondary radiation entering the instrument, which is cut down by lead but which is capable of penetrating brass. Such a type of radiation has already been indicated by Kleeman (Phil. Mag. 1907, xiv. p. 643) and Eve (Phil. Mag. Aug. 1909, p. 283). The origin of this effect was sought next. With disposition otherwise identical, the radium was now mounted not in a block of wood but was placed in a shallow groove in a lead plate of the same diameter as the absorbing plates and with 3°5 cm. thickness of similar plates below. When the absorbing plates are now put on the radium it is surrounded in every direction by at least 1 cm. of lead. The absorp- tion curve now obtained was exponential over the range 1 to 10°5 cm. (A=0°50). The sereens of lead round the brass electroscope have now practically no effect, a very small difference between the leaks with or without lead round the- electroscope being due to secondary y-radiation reflected from the wood of the stand, the existence of which has been established by Kleeman. This latter secondary y-radiation is quite distinct from the peculiar secondary radiation under discussion. Question of the Homogeneity of y-Rays. 747 Experiments were now conducted in the middle of the laboratory and the apparatus was set up so as to reduce secondary radiation effects to a minimum. Fig. 14 shows the disposition. The radium, mounted in a block of wood 2, Fig. 14. was placed on a wooden table 5. Over the radium were placed absorption plates of lead or of copper 1, and on the top of 1 was placed the electroscope. Two circular screens of metal could be placed either round the electroscope 4, or round the wooden stand and absorption screens 3, Through- out the experiments 4 was of lead and the electroscope of brass. 5 could be changed by lining the iable with some other material than wood, and 3 could be varied in nature and in thickness by placing plates and circular screens of various bodies in position. ‘The results are summarized in Table IV. (p. 748). For different thicknesses of 1 (lead) the value of the differences in leaks A and B, © and D, E and F varies, but in general it was about 5 to 10 per cent. when experiments were conducted in the centre of the laboratory and 10 per cent. and more when the electroscope was held in position by a stand of wood. Leaks H, G, B, and D are the smallest ‘ied correspond to the ionization due to the absor ption of the primary y-rays only. A and C are the greatest, and corre- spond to the maximum amount of secondary radiation obtainable by the disposition employed, plus that due to the primary y-rays. The values of E and F depend on whether the thickness of 3 (lead) is great enough to ensure perfect screening. H=F=G if this thickness be 2 em., but 748 =Mr. & Mrs. Soddy and Mr. A. 8. Russell on the H>E>G if thickness be only 0°3 ecm. From VIII. of Table IV. it may be seen that if brass is used as absorbent instead of lead the variation of the other factors of the disposition have no effect on the value of the leak. TaBLe IV. Disposition. Nature of the Effects. Heese coe 1 lead (7°5 cm. thick), | Leak greater without 4 (lead) than 2 wood (3 em. thick), with it.—(A and B.) 3 absent, 5 wood. fi as Cee 1 lead, 2 wood, 4 absent, | Leak greater without 3 (lead) than with 5 wood. it.—(C and D.) TH ee 1 Jead, 2 wood, 3 lead, | Leak greater without 4 (lead) than with 5 wood. it.—(E and F.) Vier he 1 lead, 2 wood, 5 lead. | Leak the same whether either or both 4 and 5 (lead) be present or absent.—(G.) IVs, oanisnes 1 lead, 2 lead. Leak the same whatever be the nature of 3, 4, or 5, and whether 3 or 4 be present or not.—(H.) Ale oat 1 lead, 2 wood, 3 brass, | Leak practically the same whether 5 brass. 4 (lead) be present or not.—(J.) VII. ...| 1 lead, 2 wood, 5 lead. | Leak the same without 4 (brass) as with it and greater than when 4 (lead) is present.—(K.) VIII. ...| 1 brass, 2 wood, 3 brass| Leak the same whether 4 (lead) is or lead, 5 wood, brass, present or not.—(L.) or lead. 1D. ees Thick lead electroscope| Leak the same whether 4 Mead is whatever bethe nature} present or not. Of), 2a: OL oO: From these observations the following results have been deduced. The process of absorption of the y-rays by lead causes a secondary radiation, probably an incidence radiation, to be produced by the primary in the metal. This radiation is absorbed by the lead also and so cannot emerge from the lead in appreciable amount. In the experiments described in Table LV. the radium is inserted between thicknesses of lead and of wood. Through the wood this peculiar radiation escapes, and when it has escaped it can penetrate with ease Pe ee ee a eT | - — i Question of the Homogeneity of y-Rays. 749 every body tried of density less than that of lead, the absorp- tion coefficient per cm. of equivalent thickness for brass, for instance, being at least 5 times less than that for lead. In the experiments in which the electroscope was mounted on a wooden stand this radiation is reflected from it through the sides of the electroscope. In those conducted without such a stand the radiation is reflected from the wood of the table, and after reflexion it still retains its capacity for pene- trating brass. The placing ef much lead, brass, or copper on the table cuts down the amount of this radiation, showing that reflexion from wood is an essential part of the phenomenon. It may be seen from Table IV. that this radiation may be prevented from masking absorption results by cutting it off, (1) at its point of production by surrounding the radium entirely by a sufficient thickness of lead, (2) at its point of reflexion by covering the table with lead, or (3) at the point it enters the electroscope by using either a thick electroscope of lead or by placing a 0°5 cm. circular screen of lead around one made of some other metal having a thick lead base. The thickness of lead required in (1) to prevent the escape of the radiation was found to be about 0°5 em. A thickness suff- cient to ensure the complete absorption of the B-rays of radium made no appreciable difference. The effect cannot therefore be due to a secondary y-radiation generated by the action of B-rays upon lead. It may be emphasized here that this peculiar radiation is not caused by the action of an untransformed primary radiation on a wood reflector, for if it were, the same or a similar effect would be given when brass absorbing plates were substituted for lead in I. (Table IV.). The important point about this peculiar radiation is that it appears to be produced only as an incidence radiation by lead, and that it is very penetrating to brass, retaining this power of penetration even after reflexion from wood. In addition to this radiation, secondary y-rays capable of penetrating the brass electroscope are generated by the action of primary emergent radiation from all bodies covering the radium when it falls upon wood, glass, and magnesia brick. Copper and brass produce such rays to a much less degree, while lead produces no appreciable amount. These points can be demonstrated by placing a thick plate of glass &c. vertically on the table and measuring the leak with it present and absent, taking care that none of the rays produced can get through the windows rather than the walls of the electro- scope. Like the peculiar radiation described, these rays also are incapable of penetrating 0°5 cm. of lead. Tuomikoski, working with a very strong source of radium emanation, has shown (Phys. Zeit. 1909, x. p. 372) that the 750 Mr. & Mrs. Soddy and Mr. A. S. Russell on the y-rays of radium are practically exponentially absorbed between a range of 2:2 to 12:0 cm. of lead (4=0°51), but that from 12:0 to 18:0 the value of X decreases continuously. The work described above suggests that the cause of this decreasing value of » is not due to the heterogeneous character of the y-rays, as has been supposed, but that the peculiar secondary radiation is giving an increasing effect relative to that due to the primary y-rays as the thickness of the absorbing material is increased. Tuomikoski worked with an electroscope of aluminium, which behaves towards this radiation very like brass. In the light of these results, experiments on the absorption of the y-rays, from 2 to 22 cm. in total thickness of lead, were undertaken. In order to prevent any secondary radiation from the stand, bench, or wails of the room from entering the one vital spot in the lead electroscope already described, namely the windows, the following experiments were conducted in the middle of the laboratory. Fig. 15 shows the disposition employed. An iron tripod of height 27 cm., its top being a narrow circular ring, Fig. 15. 18°3 cm. diameter, was placed on the work-table and all except the lower part of its legs covered with sheet lead (0°14 cm. thick). On the top of the tripod was placed a flat slab of lead 18 cm. square and 1°56 cm. thick. On the slab was placed the lead electroscope sur- rounded by the two circular screens of lead. The slab would absorb any secondary radiation due to the radium coming from the table, the reading microscope, or the iron stand, though such, if any, has been shown to be very small, while the circular screens protect the sides. 31 mg. of radium bromide were iin mounted in a lead disk and placed aril ‘ ina cylindrical standof lead 12°5cm. . in diameter and 4:1 cm. high. Over the radium were placed the lead absorbing screens, each of which was about 1:25 cm. thick and 12°5 cm. in. diameter. The range of thickness of lead laid on between 10 and 20 cm. was investigated. The absorption was at first exponential (A=0'50), but it departed latterly just as it had done in fig. 12. It was found that if the natural leak were increased by a constant amount equal to half itself and the sum Question of the Homogeneity of y-Rays. 751 subtracted from the gross leak obtained in the usual way the curve was exponential right up to 20cm. The amount of this constant radiation bore about the same ratio to the amount obtained in the first experiment as the ratio of the quantities used now and before, and it seemed quite possible that it was due to a primary radiation more penetrating than y-rays, which could only be detected when the latter were very much reduced in intensity. This, however, was disproved by the next experiments. The two small windows of the electroscope were blocked up by 0°3 cm. of sheet lead as thoroughly as could be done without interfering with the microscope or ‘completely shutting off the light from the lamp. This constant radiation was then greatly reduced. By removing everything from the table except the apparatus used, and by covering with sheet lead the microscope and also the rubber cork by which the leaf system and the charging rod were held in position, the constant radiation was entirely eliminated. Three sets of readings were obtained on different days for the absorption over the range of 10 to 20 cm., and in all the three cases the rays from the radium were absorbed exponentially (A= °050 as before), the leaks actually obtained being corrected only for the natural leak of the instrument, obtained by removing the radium out of the laboratory, and keeping everything else as it had been during the series of measurements. When it is considered that for the greatest thicknesses of lead employed the cor- rected leaks vary from about 0°1 to 1:0 division per minute, the natural leak being constant at 3°60 divisions per minute, the necessity of shutting off every secondary effect may be realised. An experiment was indeed made by exposing a corner of one of the windows, and the corrected leak was so doubled. As previous experiments indicate, brass or other metals would be useless for this work. Lead alone can be used with confidence in such measurements. Fig. 16 (Pl. XII.) represents the absorption of the y-rays by lead over the whole range explored. From 2 to 12 em. total thickness of lead traversed, 6°7 mg. of radium bromide was used as source. From 10 to 22 cm., 31 mg. were used in two series of ineasurements and 45 mg.ina third. By making use of the values of the rates of leaks at two points common to two curves the composite absorption curve shown in fig. 16 is obtained. The second half of the curve is the better of the two series of readings obtained with the 31 mg. source. It is plotted as obtained. The first half is that obtained with the 6°7 mg. source, each value of which has been multiplied by a constant factor. In curve A (PI. XII. fig. 16), obtained with 31 mg., it may be noticed that there is a decided irregularity 752. Mr. & Mrs. Soddy and Mr. A. 8S. Russell on the between 15°6 cm. and 17 em., but the end point at 22-1 cm. is on the straight line. This effect is, no doubt, due to in- effectual shutting out of a small amount of secondary pene- trating radiation which manifests itself at these points. As the air space between the top of the absorption plates and the electroscope base was filled in (thereby reducing the secondary penetrating radiation) the points came on to the straight line once more. In the main curve and curve B (fig. 16) greater precautions were taken to prevent any trace of such radiation from entering the electroscope. The value of » obtained from the principal curve of fig. 16 over a range from 2 to 22:1 em. of lead is 0°498 (cm.)~1, that is to say, 1°392 cm. of lead cuts down the y-rays of radium to half value. This value agrees (1) with that previously obtained (0-495, F. Soddy and A. S. Russell, loc. cit. p. 644); (2) with the value given by Tuomikoski, over a range of 2°2 to 12-0 em. (0°51) ; (3) with the value in the present paper, using, however, a brass electroscope (0°50); and (4) in Part II. of the present paper with truncated hemispheres (050). This value for X (cm.)~? 0°50, making A/d=0°0438, may be used with confidence in calculations of the pene- trating rays from the earth’s crust. A new electroscope, made entirely of lead, was constructed in order to save trouble in blocking up the windows and the cork with lead. It is shown in section in fig. 17. It consists of a cylinder of lead of internal height 12°9 em., internal diameter 9°0 em., thickness of walls 1°30 cm. and of top 1°25 cm. Two cylinders of lead, 6°5 em. long and 3°5 cm. in diameter, 0-4 cm. thick in the wall, were soldered into the sides of the electroscope to protect the windows, which were circular and of the same diameter as the lead cylinders. The latter were Just large enough to allow the microscope to be inserted. The sulphur of the leaf system was surrounded by an earthed ring of brass. Over the cork and charging rod a third cylinder of lead, 0°4 cm. thick, was placed while measure- ments were being taken. To the instrument a permanent base could be soldered, or the thickness and nature of the base could be altered at will by clipping up different thickness to two pieces of brass attached to the sides of the electroscope at its base. This may be considered a standard form of electroscope for work on absorption of y-rays. — oe Question of the Homogeneity of y-Rays. 153: Initial Part of the Absorption Curves of the y-rays of Radium. Experiments involving four different dispositions were. carried out with screens of lead, tin, zinc, and aluminium. The thick lead electroscope with base removed (fig. 17) was: mounted on an iron tripod. Various thicknesses of metals. were clamped up to form the base. Jn disposition 1, 6°7 mg.. of radium bromide were placed at the apex of a cone of height 11°5 cm. and of base 3 cm. in diameter, cut out of a cylindrical lead block 12°7 cm. long and 10°5 cm. diameter. The cylinder was placed immediately underneath the electro-. scope, the top of the former being about 3:5 em. from the: base of the latter. Disposition 2 was exactly the same as 1, except that 1°24 cm. of lead was placed on the lead cylinder over the base of the cone. In dispositions 3 and 4 the rays: were not confined by acone at all. In 3, 0°47 mg. of radium: bromide, lying without cover in a recess in a lead block 1 cm. thick, was placed 14 cm. below the electroscope. 4 was the same as 3, except that 1:24 cm. of lead was placed directly over the radium. Table V. gives a summary of the results obtained with these dispositions over the ranges of thicknesses. of metals described. TABLE V. See r Shape of Disposition.| Metal.| 100 x rr Range in cm. Absorption Curve. 1 Al. | 2°78 to 4:22 0°8 to 2°4 Continuously concave. 2. Al, 0 0°6 to 2-4 Parallel to axis of T. 3. A, 776 0°9 to 26 Straight. 4 Al. 4°04 0°3 to 2°6 Straight. Le Zn. | 3:01 to 421 0°35 to 2°6 =| Continuously concave, 2. Zn. | 1:29 to 3:85 0°35 to 2.6 | More concave than 1. 3. Zn. 5°05 0°35 to 2°6 | Straight. 4. Zn. 3°39 0°35 to 26 | Straight. i. Sn. | 4:11 to 535 0°5 to 2°05 | Continuously concave. 2. Sn. | 2°21 to 4°80 0°5 to 2°05 | More concave than 1. 3. Sn, 5°55 0°5 to 2°05 | Straight. 4. Sn. 3°65 05 to 2:05 | Straight. 1. Pb. | 105 to 667 | 0-3 to 24 Continuously convex. 2. Pb. -——-- 3. Pb. | 7:37 to 5°53 03 to 2-4 =| Continuously convex. 4. Pb 5°32 0:3 to 2-4 | Straight. Phil. Mag.S. 6. Vol. 19. No. 113. May 1910. 3 C 754. Mr. & Mrs. Soddy and Mr. A. 8. Russell on the It is obvious from the results set forth in the Table that for the initial part of the range the absorption curves may be straight, or may depart in either direction from the exponential form according to the conditions of experiment and the absorbing metal used. Thus, ifin the original investi- gations on y-rays, zinc or aluminium instead of lead had been used as the absorbing metal, the curve obtained over the first part of the range would have been found to be exponential. For disposition 1, the value of ) rises in the case of aluminium to about the normal value (A/d= 0-040), but for zine, and still more for tin, it rises beyond the normal value, while for lead it diminishes, but does not reach the normal value. In dis- position 3, which gives exponential curves for aluminium, zinc, and tin, the value of X is from 1°5 to twice the normal. A further result may be mentioned which has been obtained by a new disposition, in which a narrow cone of rays and a shorter cylindrical ionization chamber connected to a separate electroscope have been employed. The curves for aluminium and lead are concave and convex respectively, while those for zinc and tin are straight over the whole range. X for zine is 0°268, nearly the normal value, but for tin it is 0-355, about 26 per cent. too great. Contrasting this result for zine with those given in Part IJ. with truncated hemispheres, we see that lead is normal in the hemispherical and abnormal in the cylindrical ionization chamber, while with zinc the converse is true. Zinc, in the disposition last described, is the only case so far found of a metal obeying the simple exponential law with the normal value for A, from the thickness sufficient to absorb B-rays up to the greatest thickness tried (6 cm.). Finally, a number of experiments may be referred to on the variation of the absorption coefficient X with variation of the different components of the disposition employed, over ranges of thickness greater than the equivalent of 1 cm. of lead. The chief disposition used (denoted by A) was to place the absorbing screens directly over the source of y-radiation at a distance of about 14 cm. below the electroscope, though some experiments have been carried out by clamping up the absorbing screens to act as base and leaving the ionizing source bare (B). It was found in general that » for any substance varies within certain limits with practically every ‘important change in the disposition. The substances are divided into classes according to density. Class I. denotes lead and mercury, Class II. comprises substances in density from copper to magnesia brick, and Class III. from sulphur to pine-wood. For an electroscope made entirely of one material, if disposition A be employed with a radium source, the absorption coefficients vary slightly with the thickness of ° (+) 7/6 p(s.) =MXc (5) 1/2.,,,7/6 : a = BEe, Oo ort E denotes the potential energy per unit area and 2 the surface-tension of the surface of a liquid at the temperature T, and L denotes the internal heat of evaporation. x’, x’, B, H,, H., M, denote constants, each of which is the same for all unpolymerized liquids at corresponding states. T., Pec, Pc, denote respectively the critical temperature, density, and pressure of a liquid, }c, is the sum of a number of con- stants, each of which refers to an atom of a molecule, and is independent of all conditions except the nature of the atom. The values of c, for the ditterent atoms involved were determined in one case by means of equation (6) from data referring to the liquids ether, methyl formate, carbon tetra- * Communicated by the Author. T Phil. Mag. Oct. 1909, p. 491; Dec. 1909, p. 901. — = = ee a —— 784 Dr. R. D. Kleeman on the Nuture of the chloride, benzene, fluor-benzene, bromo-benzene, iodo-benzene. This gave, putting H=1, the values H=1, C=5:30, O=5°94, F—5°76, Cl=8 40, Br=10-65, Sn=14-66,. [=15-49) wis were then used for calculating the value of Xc, for the rest of the compounds. The above equations were applied to a Jarge number of liquids, and a fair agreement with the facts obtained. The constant ¢, on account of its independence of chemical combination, temperature, &c., one would expect to be of | fundamental importance. An endeavour was therefore made to connect it with some known fundamental quantity of the atom, as its atomic weight. The result obtained was that it is proportional to the square root of the atomic weight of the atom or to its chemical valency. This shows that the valency of an atom is proportional to the square root of its atomic weight, a result already obtained by Traube*. We may therefore write = /m, or Sv for Sea, where S,/m, denotes the sum of the square roots of the atoms of a molecule, and >v denotes the sum of the maximum valencies of the atoms. The molecular weizht of a molecule, it should be noted, is denoted by m. This statement will now be tested in different ways. 1/2, According to equation (2) the expression is propor- tional to Sc,, and should therefore be proportional to & iy A2m and =v. The ratio ; | ES given for a ae liquids in the third column of Table I., and the ratio al is given in the fourth column. Both of these ratios, it will be seen, are fairly constant. Itshould be mentioned that the quantities p and X relate to 3 Tc, and are taken from the first paper by the writer mentioned, being obtained from the surface- tension and density determinations of Ramsay and Shields. When any of the data used in this paper are not given they can be obtained from the above paper. The values of ¥ /m,and >v are given for a large number of liquids in Table IJ. The values of 2v were obtained by putting H=1, O=4, N=3, O=4) B=4, Cl=7, Br=10,,Sn=10,)2=13, the sueuee accepted maximum valencies of the atoms (Traube, loc. cit.). The sixth and seventh columns ot Table I. contain the values of H?m El? m ea Tass prr/ my pre respectively, corresponding to 3T7T,.. Both ratios are = Whys. Let. p. O61, Oct 1908: iy ACY ‘ie a" Forces of Attraction between Atoms and Molecules. 785 approximately constant, as should be the case according to ( equation (1), if Yc, is proportional to } /m, and w. ' TABLE I. : i ; eeeietenid, | 7 Pi | Pm Bem | Em |B m i j p p=Vm, | pzv p punvm, | p2v ] : POI Gah, sendy es'ode eee .0s 404°2 14:5 135 7526 27:0 25:1 i ( Methyl formate ...... 287°7 15:2 14-4 530°1 28°0 26°5 ! 4 Carbon tetrachloride ..| 487°5 16:0 13°7 813 29°8 25-4 - | BENZeNC ................5. 421°6 15°7 14-1 778 29:1 25-9" IK | Chloro-benzene......... 493:7 16°4 13-7 905 30°0 25°1 Wi Ethyl acetate............ 435°] 146 13°6 if Propyl formate......... 428 14:3 13°4 i Methyl propionate ...| 428°8 14-4 13°4 Propyl acetate ......... 502°5 14:2 13°2 | Ethyl propionate ...... 500°8 14-2 132 : Methyl butyrate ...... 500 14:2 13-2 ! | Methyl isobutyrate ...) 497°6 141 13:1 i g ie From equation (6) we have that the ratios lat p2®>,/m, a pu should each be constant for corresponding states. Table II. i gives the values of these ratios for a number of liquids. It lh will be seen that the values in the upper part of the table . are approximately constant. A comparison of these results with those obtained with Sc, instead of Y4/m, or Sw, | Table XIII. Phil. Mag. p. 507, Oct. 1909, shows that i equation (6) is better satisfied in the latter case. The reason for this will appear later. The lower part of the table contains the liquids which are known to be polymerized. They do not fit in with the other iT liquids, as observed before. When a liquid is polymerized | we should, in accordance with the assumptions on which the above equations are founded, use n&c,, n> Vm, nv, nm, instead of Se,, > /m,, Xv, m, where n denotes the number of i normal molecules combined into a single molecule, which it would then make the liquid fit in with the normal liquids. ri But n is usually not known. We may, however, assuming the theory true, do the opposite thing, and determine n for a polymerized liquid by means of one of these equations. In i Phil. Mag. 8. 6. Vol. 19. No. 113. May 1910. 3E in| | L1/2m7/6 T/2mi7/6 786 Dr. R. D. Kleeman on the Nature of the this way it can be shown that a water molecule at 2 in the liquid state consists on the average of two molecules of the normal form H,O. The ratio for di-isopropyl in Table IL, it will be seen, does also not fit in very well with the remaining liquids. This suggests that it is slightly polymerized. if TaBLeE II. i! LN mi “| Son s Loge Le ae () aie 2 it : a Dye ye, | i Name of liquid. 5 /3 U a By Waa eae Sy q Bo Fie! ce ee a Pate atl nol oe se | b DTH NV) One Cela ee) 1686 27°84 30 60°6 56°2 4 Methyl formate......... 1202 18°92 |, 20 63:4 60°0 Carbon tetrachloride .. 1766 27°30 By 64:7 55°2 | Benzene nace Paes 1716 26:76 30 64:1 57°2 | Fluor-benzene ......... 1932 30:12 | 33 64:1. 58°6 ' Bromo-benzene ......... 2154 3470 | 39 621 55°2 | Todo-benzene............ 2374 37°03 42 64:1 56°5 | Stanniec chloride ...... 2192 34°70 38 63:2 SV GT( i Dizisobubyle tee esses oe 2762 45°68 50 60°5 55°2 | Chloro-benzene......... 2060 31-72 | 36 64:9 572 | Bentane tin .acb.sce see 1784 29°30 .| 32 60°9 95°7 : le ptamte: sien ec tesnee 2494 36°76 40 67:8 62°3 i Octane ea oeasae ars 2770 45°68 50 63:4 55°4 Hexané’ 2.40 cake 1941 32°76 | 38 59:2 51-1 | Isopentane..............- 1734 29°30 32 592 53°0 | Hexamethylene......... 2128 34:76 36 61-2 59°1 | | Di-isopropyl* ......... 1782 24°76 | 28 72:0 63°6 Propyl formate......... 1860 29°84 32 62:3 581 Ethyl acetate ......... 1862 29°84 32 62°4 58°2 Methyl propionate ... 1861 29°84 32 62:4 58°2 Propyl acetate ......... 2197 35°30 | 38 62°2 578 Ethyl propionate ...... 2193 30°30 | 38 62:1 oT 7 Methyl butyrate ...... 2130 35°30 38 60-3 561 Methyl isobutyrate ...,. 2117 35°30 38 60-0 DDT Methy] acetate ......... 1565 24°38 | 26 64:2 60-2 Ethyl formate ......... 1556 24°38 26 63°8 59°8 UA CRbEC ACIG sf 0-(.c.oe ea! 1155 18:92 20 61-0 O78 NMA ORAGeS cick tie bvehaace aes 660 6 6 1100 110 Methyl alcohol......... 1082 11°46 12 94-4. 90-2 Propyl alcohol ......... 1821 22°38 | 24 81:4 759 Table III. contains the values of T2/22/3 1/24 2/8 Swim and Sa, corresponding to 27T,, and these should be constant ac- cording to equation (3). This, it will be seen, is approximately the case. Forces of Attraction between Atoms and Molecules. 787 Tae ITI. | : mpl? aoe ql? oe qe? ae Name of liquid. TRV yas ee aes i p" ps3 Vim | ow COLE VE a ee 598°3 14°3 13:3 Methyl formate ............ 290°2 15°3 14-5 Carbon tetrachloride ...... 434-2 15:9 13°5 PEUIAEMD uy clone secs enie auto nas 415°9 15°5 13°9 Hinjor-penzene ..........-..-. 427°3 14-2 13:0 Bromo-benzene ............04+ 522°6 151 13°4 | HOHO-OGAZENE, 2.50 ..0.000 00. 570°9 15°4 13°6 Staumic cliloride ............ 5176 14-9 13°6 Dian 0th) (ie 607°3 133 1271 Chloro-benzene............... 491°5 155 13-7 | BAAS so eee acco .cesaeenvecee 428°4 146 13:4 | EUSA ios sce canon cenicvnes 559°7 15°2 14:0 2a) 2S 623°6 14°3 125 RM cee ses | 495-1 151 1-0-4 LS 425°5 145 13°3 Hexamethylene ............ 465°4 13°4 12:9 -seppropyl* .........0c000. 448°8 181 160 Erapyl tormate,.........+.... | 422°4 14-2 13-2 Hithyl acetate.............s000. 417-9 14:0 13-1 Methyl propionate ......... 4166 14:0 13°0 Prapy! ACEUaLe ..........0000 482°8 13°7 12-7 Ethyi propionate ............ 480°1 13°6 126 Methyl! butyrate ............ | 479°6 13°6 12°6 Methyl isobutyrate ......... ee Piety 135 12-5 | Methyl acetate ........ ...... ! 3041 14°5 156 | Diy. (OPM .........00....| 3953 14°6 13°7 From equation (5) we have that the ratios L2 -LyN 1/6 1/2 7/6 m 2 em \i a - _(* and Pe_ (=) S/m Pe Sv Pp, should each be constant for corresponding states. These ratios are evaluated for a number of liquids in Table IV. (p. 788). They are approximately constant as we should expect from the foregoing results. But the deviations from constancy are greater than when Xe, is used instead of > Wm, or Xv, as will be seen by comparing the table with that given in the Phil. Mag. p. 904, Dec. 1909. We thus conclude from the foregoing discussion that the constant Sc, may be replaced by either } /m, or Sv, and on the whole a fair agreement with the facts obtained. The small deviations obtained with normal liquids are very probably in part due to slight polymerization, which one would expect to exist in the case of every liquid in a degree depending on its nature. 3 HK 2 & pc} * 788 Dr. R. D. Kleeman on the Nature of the Tasie LY. 7/6 1/2 Name of liquid. dw] Vm pe(=\"" : fei (2) Ha ea : pe Sm P| Sy Wicisobuty]...--asi---0-0 C,H, | 50] 45°68 | 304x10° 607 665 Di-isopropyl ............ C,H,, | 32] 29:3 a 658 719 Isopentane............... O,H,, | 382 | 29:3 208), 650 710 Hiexane cies tte senses C.Aop 38 |) a2767" 243 0 640 742 EVIE AMO |... 5 sate hanaese O,H,,. | 40 | 36°76 | 276 ,, 690 751 Oelanel ee... ..see Foaeese O,H,, | 50] 4368 | 311 ,, 621 711 Fluor-benzene ......... C,H,F | 338 | 30:12 209 _—s,, 635 695 Bromo-benzene ......... C,H;Br| 39 | 34-70 259 ,, 659 740 Todo-benzene............ C,Hsl | 42} 37:03 | 282. ,, 672 763 Chloro-benzene ......... C,H;Cl | 36 | 31-72 243 «Cs, 674 764 Hexamethylene ........ C,H,, | 86 | 38476 | 229 ,, 6385 658 Stannic chloride ...... SnCl, | 88) 3470 | 258 ,, 678 743 Benzene 12 cee eisess Cog) 30) 265764) - 205%, 676 758 Oh bt) gileienrhel co Saree C,H,,O} 30 | 27°84 | 199 ,, 664 716 Propyl formate......... C,H,0, | 32 | 29°84 | 220 ,, 688 37 Ethyl acetate............ O,H,0,) 32) 29:84 | 217 .,, 677 726 Methyl propionate ...C,H,O, | 82 | 29°84 | 213 ,, 666 714 Propyl acetate ......... C;H,,0,| 88 | 3530 | 239 ,, 628 676 Ethyl propionate ...... C,H,,0,| 88 | 35°30 | 254 ,, 668 720 Methyl butyrate ...... C;H,,0,| 88 | 35°30 | 253 ,, 667 718 Methyl isobutyrate ... C;H,,O,| 388 | 35°30 | 236 ,, 622 669 Isobutyl formate ...... C;H,,0,| 38 | 35:30 | 265. ,, 697 750 Methyl acetate ......... 0,H,O, | 26 | 2438 | 179 ,, 690 735 Ethyl formate ......... C,H,O,| 26 | 24:38 | 185_,, 711 758 Carbon tetrachloride .. CCl, SZ iP 2eeoOu! Zila Se 670 785 Ethyl] butyrate ......... O,H,,0,| 44 | 40°76 | 287 ,, 652 704 Ethyl isobutyrate ...... C,H,,0,| 44 | 40°76 2387 651 703 Isobutyl acetate......... C,H,,0,| 44 | 40°76 | 286 ,, 651 702 Methyl valerate ...... C,H,.0,| 44 | 40°76 | 289 ,, 658 709 Amyl formate ......... C,H,,0,| 44 | 40°76 IO 676 730 Monochlorinated 5) ’ eae aha } C,H,Cl,| 26 | 2984 | 391 ,, 684 778 Chloraldehyde ......... C,H,Cl,| 26 | 22°84 | 415 ,, 725 825 Carbon dioxide......... Co, 12} 11:46 fees 654 685 Methyl formate......... CHO. 20") 1892") Vs0an F. 694 733 Hydrochloric acid...... HCl 8 6796) 10", 622 715 The investigation which led to the equations given at the beginning of this paper was based on the assumption that the attraction between two molecules of the same kind is given by 6(z)(X¢a)?, where #(z) is a function of the distance. between the molecules. The present investigation shows. that the law of attraction may also be expressed in the form Jae Vm)? or $(2)(20). So far we have not rs obtained any information as to the form of the function ¢(z). In order to investigate this point it will be necessary to develop a different method of investi- gating the nature of surface-tension than the one adopted in Forces of Attraction between Atoms and Molecules. 789 previous papers. This method leads to the same equations as those given at the beginning of this paper in perhaps a more straightforward way, and besides brings out further relations and points of importance. The surface-tension of a liquid is defined as the work required to produce an increase of unit area of surface. This increase in area may be produced in many ways. The way which we will use in this investigation is to suppose a mass of liquid cut into two portions by a plane, and these then separated from one another by an infinite distance. If W denote the work done in separating the portions, and A the amount of new surface produced in each portion, A the surface-tension is given by A= — Let AB in fig. 1 be a plane which cuts a thick slab of Fig. 1. fo) liquid into two portions, and suppose the thickness of each portion greater than the sphere of attraction of a molecule. The work of separation per unit area will first be calculated on the assumptions that, (1) the matter is evenly distributed in space, (2) the attraction of one element of matter on another is not affected by intervening matter. The coordinates of an element of volume in the portion © will be denoted by (2, y, z), AB lying in the yz plane; and the coordinates of an element of volume in D will be denoted by (21, ¥1, 21), the axes y;, 2; being common to the axes yz. The attraction of a volume of matter whose mass is equal to that of a molecule on another equal volume will be taken as (Xca)’(z), where $(z) is a function of the distance of separa- tion of the volumes. According to condition (2), 2c, must 790 Dr. R. D. Kleeman on the Nature of the be the sum of a number of independent constants each of which refers to one of the atoms in a molecule. The com- ponent of attraction parallel to « or 2, of the element of volume dx,.dy,.dz,in D on the volume da.dy.dz in OC whose coordinates are (a, 0, 0) is N?$(z) (Se.)(A=*) dw .dy.dz.day,.dy,.dz, where c=V/{(atm)ryrter}, and N denotes the number of molecules per c.c. The attrac- tion of the whole slab D on the element dx.dy.dz is therefore +o +0 +0 N*(Sen)*| { { (2) 2420) de. dy dz. dey.dyy.dy=N%(So,)F 0 —0o — say. The work done in moving this element of volume to infinity is | N?(2e,)2F . de. The work done in moving to infinity a cylinder of unit cross- section and infinite length, standing with its base on the plane AB is, therefore, | +o +0 +0 +0 +0 N°(Sc,)? | \ ( ( {oe dz .dx.dx,.dy,. dz, Oe 10 + So me and this is equal to 2x. If the density of the liquid is denoted by p and the molecular weight by m, N= , and denoting twice the value of the above integral by «” the result may be written | Xr — Kl! - (Sc) z. which is a relation similar in form to that obtained previously. We have seen that an application of this equation to the facts showed that «is the same for all liquids at corre- sponding states. It follows from this investigation that the function ¢(z) must be of the form ¢(<, 4, 8), where p=ep- and T=8T,, which will make «’’ assume the same value for all liquids at corresponding states. Forces of Attraction between Atoms and Molecules. 791 But the assumption that the matter is evenly distributed in space is not true, and zero cannot therefore be a limit in the above integrals; moreover, the use of integrals is then not generally admissible. We will, therefore, develop a formula which takes into account the fact that the matter of a substance is not evenly distributed in space. Suppose the liquid cut up into squares by three systems of parallel equidistant planes one of which is parallel to the plane AB, fig. 2, and suppose a molecule is situated at each point of intersection of three planes. Let x, denote the distance between two molecules situated at the two corners of the edge of a square. Consider two molecules a and 6, a being situated in the slab C and 6 in the slab D. Let the coordinates of the molecule a be (naa, 0,0), using the same system of coordi- nates as before, and those of 6 be (weg, uta, vea). The com- ponent of attraction of the molecule a by 6 along a line at right angles to the plane AB is (Se, 8p (=) et ete where c= V{ (ntat+ waa)? + wee + van} =r,V{(ntw)+uw+e'}. The attraction produced by the whole slab of liquid D on @ is therefore v=0 uo W== 0 2 ecg OR a Gy == Ww) = (Be, Pl say. v=—®O U=—wBD W=—0 w The work done in moving the molecule a to infinity is (Se, | P dna (Sty) | ee Gay, Gd) Na U7 «@ 792 Dr. R. D. Kleeman on the Nature of the The work done in moving the molecules lying on a line at right angles to AB to infinity is (Sea = | F dn. n=l n Therefore the work done in removing a cylinder one cm.? in cross-section standing on the plane AB is n=l n A430)? S| F.dn La BL (x2 i ESNet V{(n-tu0)?-+ 02-40%} } Va n=1 n V=—-o U=—H w=0 (n+w) .dn f V { (n+ w)* +? 4-07} p \18 Since ad m this gives n=5 (2) (ee )? 2®Q o 2\m o7 eee where Q isa function of 2. We have seen that x°Q or x’ is the same for all liquids at corresponding states (equation 2). This is realized if ¢(z) is taken to be of the form st (2,8), where 2, is the distance between the molecules at the critical temperature. The surface-tension is then given by = by (Sea)? M= (2) (e,) M, La where M is a function of ae and this is the same for all liquids at corresponding states since —? and~ has the c c Pe ' same value at corresponding states. The preceding investigation gave that $(z) must be of the form ¢(<,a, 8), when the matter is evenly distributed in space. The result just obtained shows that ¢(z) must be of the form 1 at(= 8) _ which, it will be seen, is a special case of (z, a, 8). ee ee ee eS eee 4 — Forces of Attraction between Atoms and Molecules. 793 _ _ In this and the foregoing investigation we have neglected the influence of the surrounding vapour on the surface-tension of a liquid. If condition (2) is true—namely, that matter has no screening action on its attractive forees—we may suppose in moving the slabs apart that an amount of matter equal to the density of the vapour remains stationary. We should then be dealing with matter of density (p;— 2), where pi denotes the density of the liquid, and p, that of the vapour, and to this the foregoing investigation would apply. The formula for the surface-tension then becomes where «'’ as before denotes a constant which is the same at corresponding states. But since pp=ap,, where a is a con- stant which is the same for corresponding states, the equation may be reduced to the previous form involving p, only. A formula for the internal latent heat will now be developed on the same lines. We will assume (1) that the kinetic and internal energy of a molecule is the same in the liquid and gaseous states at the same temperature, (2) that the internal heat of evaporation of a molecule is equal to the work done in moving the molecule from the interior of the liquid to an infinite distance against the attraction of the molecules of the liquid, which is supposed to be identical with that producing surface-tension. We have seen that the work done in moving a molecule at a distance nx from a liquid surface to an infinite distance is Bete, | F . dm (equation 7). 2 When the molecule lies initially in the surface of the liquid this becomes Seq) | FE. dn= (2c,)*#a s > ¢ (z) a (n+w)dn 0 2/0 ; V=-0O U=—-wD w=] “ + (2i¢a)*2, (" {4 > x | $(2) = (n+) | + 13 E (2)(n + w) Jaze ban, a=) w—0 where Z=2, /{(n+w)?+u? +7}, and this is one-half of the molecular internal heat of evapo- ration. The other half of the heat of evaporation is expended in bringing the molecule from the interior of the liquid to - 194 Dr. R. D. Kleeman on the Nature of the the surface. This will be at once evident when it is pointed out that in bringing the molecule to the surface no work is done against the attraction of the surface layer of the liquid equal in thickness to the radius of the sphere of action of a molecule. Now the investigation of the surface-tension gave that @(z) must be of the form (258). It follows, therefore, that the internal heat of evaporation per gram of substance, which we will denote by L, is given by an equation of the form Ore! nal ay 1 P\O? ees L= a (Ze, ait? (7, p\= fe (Sce)*(F) ds ( B), the value of 3 (=. ) being the same for all liquids at cor- oD: responding states. The formula is of the same form as equation (6) given at the beginning of the paper. It was deduced from the equations 2 / mils oe rae"(Z) (ea Gag BLT ye (Phil. Mag. Oct. 1909, p. 491), and was found to agree well with the facts. It is necessary next to consider the effect of the surrounding vapour on the magnitude of the heat of evapo- ration of a liquid. The work done in removing a molecule from the interior of a slab of vapour of density p, to an infinite distance is, according to the above investigation, equal to A a (Se_)*(2)" (2, 8) where # is the distance of separation of the vapour molecules. The equation for the internal heat of evaporation, therefore, strictly is = (x,)?f (2) ga(%, 8)-(@) 6 (2 )f L==(Se,)?4 (2) $3(22 8)—-(@) #3 (2 8) p- But since pp=ap, the equation may be reduced to the previous form 4/3 L=B?(2)" (%e,) m where B? is a constant which is the same for all liquids at corresponding states. 5 Dbt xX 107 Forces of Attraction between Atoms and Molecules. 795 Since the above equation can also be obtained in another way, as already mentioned, and it has been found to agree well with the facts, we conclude that the assumptions made in deducing this relation with respect to the kinetic energy and attractive force of amolecule are true. Further evidence of the truth of this result will appear in a subsequent paper. In order to get a foothold in the determination of the nature of the function let us assume that ‘om La a 4/3 4/3 ek (*) io bs) Va, m m mM We will next compare the values of A with the correspond- ing values of {p,4?—p,‘/?}. Table V. contains the values of these quantities at different temperatures for four liquids, and the values of L, p;, ps, used in their determination. The values of L, p;, p2, were taken from a paper by Mills*, who has calculated the values of I. at different temperatures for a number of liquids by the help of thermodynamical formule, using the density and pressure data of Ramsay and Young. It will be seen that the variation of A with tempera- ture is small in comparison with that of {p,4°—p,**}. The . B)=$: (=, B)=A a constant, so that : Zz 1 former quantity corresponds to ¢, i 8) and the latter to =. “~ We may therefore assume that oo —, 8) is approximately a ee constant, which we will denote by K. The attraction between - (S/ m)?s two molecules is then approximately given by —. i. e., it is proportional to the square of the sum of the square roots of the atoms of a molecule and inversely proportional to the fifth power of their distance of separation. The variation of K with temperature—it varies propor- tionally with A—may be due to two possible causes: (1) the force of attraction may diminish with rise of temperature; (2) the inverse fifth power law may not be exactly true. Very likely the variations are due to both causes, the exact * Journal of Phys. Chem. vol. viii. p. 405 (1904), 796 Dr. R. D. Kleeman on the Nature of the TABLE V. Hither. Py. psi ip eee L ees 3 i ( Pi Sah )z Vim, ‘7362 | -0,8270 | -6648 86:16 3845 } ‘7135 |-001870 | —-6375 80°40 3742 | 6894 |-003731 | “6091 75°36 3670 6658 |-006771 | 5813 70°79 3613 | 6402 | -01155 5491 65°85 3558 | ‘6105 | -01867 5131 60:33 3488 5764 | 02934 ‘4706 54-91 3461 : 5885 | -04488 4299 48:31 3392 | ‘4947 | -06911 3528 39°74 3342 | ‘4268 | +1135 ‘2665 27-09 3016 | 3663 | 1620 1737 18-11 3094 | ‘3300 | 2012 ‘1101 12-03 3242 Isopentane. Pang 1 tl ee fle) ac ine lo ee a | 6392 |-001090| -5521 81:35 3713 } 6196 |-002358 | +5282 7521 3588 5988 |-004480| 5048 70°72 3530 ‘5769 |-007819 | -4799 66°38 3484 5540 | -01284 -4520 61-76 3443 | 5278 | 02022 ‘4211 56°67 3390 | -4991 | 03106 3862 50:89 3321 | -4642 | 04728 3424 44-04 3241 4206 |-07289 | 2845 | 35:39 3133 | 3694 |-1101 2199 24-99 2825 | ‘3311 |-1418 1563 17°36 2799 ‘3028 | -1676 "1108 11:95. 2718 | 2761 |-1951 0666 688 2602 = aS Forces of Attraction between Atoms and Molecules. 797 13215 1°2734 1:2192 11566 1:0857 “9980 ‘8666 9598 9294 8968 "8634 8264 "7860 "7403 6844 ‘6148 ‘D241 "4549 4157 1°0032 Table V. (continued). Carbon tetrachloride. 0,2984 007974 01304 02024 03021 04386 06250 ‘08787 1232 ‘1754 2710 0,6821 002225 004396 007968 01352 02153 03344 05063 07634 ‘1178 1862 ‘2451 2865 1°923 1649 1584 1519 1-442 1364 1:277 1175 1-055 8991 6504 bL 113°2 107°49 99°51 92°16 85°10 79°21 71°95 64:03 54°41 41:93 25°76 15°01 8:80 ta (0,5 =e.) JOM a, 4286 4200 4161 4104 4070 4009 3942 3893 3835 3765 3972 7/3 Li ’ (0,°°—p,) 3 m, 4440 4470 4423 4206 4091 4061 3968 3872 3752 3545 3206 3007 2865 ee ea eee ee ON aE SS SES ee ——— = a a 798 Dr. R. D. Kleeman on the Nature of the law being probably a complicated function of z and T. The nature of the law will be further discussed as we proceed. The foregoing investigation shows that the molecular attraction which gives rise to surface-tension is also respon- sible for the internal heat of evaporation. The fact that we may replace %/m, by Sv, the sum of the valencies of the atoms in a molecule, suggests that it is also the force operating in chemical combination. We may suppose, as is done by chemists, that each atom contains a number of cells each of unit valency. A cell will thus attract another with a force equal to 3. The study of chemical combination has shown that the avidity of an atom decreases with the number of atoms with which it is already combined. Thus the attraction of one valency cell on another depends on the presence of other cells. The nature of chemical attraction thus differs from that of gravitation, which is independent of all conditions. The chemical attraction of a molecule would not, therefore, be an exact additive property of the attractions of its atoms. The slight deviations of the facts from the relations expressed by the equations given at the beginning of the paper may partly be ascribed to this effect. Chemists suppose that an atom is linked to another in a molecule by the attraction of two valency cells, one belonging to each atom. Therefore at least one cell of each atom in a molecule is weakened as far as its attraction outside the molecule is concerned. Since a hydrogen atom contains one valency cell only its attraction would be much more weakened by combination than that of any other atom. One would therefore expect that if the value of the constant ca for an atom is given exactly by 4/m, or v when not in combination with other atoms, the value of c, for hydrogen in chemical combination relative to the values of ¢, for the other atoms ‘in combination should be less than that given by Wm, or v. This is strikingly shown by the results obtained by the writer. Using the equation LY2 7/6 epee = Cn the writer has in a previous paper calculated the values of ¢, for a number of atoms. These values are quoted at the beginning of this paper. Later in the paper the writer pointed out that these values give a somewhat better agree- ment with the facts than the values given by Vm, or v Forces of Attraction between Atoms and Molecules. 799 Dividing each value of ¢, by the corresponding value of Vm memobiain the: ratios H=1,)C= los; .O=1-49,. F=1°32, Cll, Br=119, Sn=1°35, -[=1°38.,, It will be seen that the value of c, for hydrogen is in comparison with the other atoms about 30 per cent. less than it should be accord- ing to the square root law, which fits in, according to the above, with what one would expect from chemical con- siderations. Let us now investigate the conditions when the interaction of the molecules or screening effect in different systems of molecules produces the same proportional change in the attraction, &. Let A, B, C, be three molecules not neces- sarily of the same kind. Remove the molecule A to an infinite distance and let the work done be denoted by Fa. The work is being done against the attraction of the mole- cule A by B and C modified by the interaction of the mole- cules on one another. Next remove the molecule B and let the work done be denoted by F. Now the removal of A and B may be carried out in a different way. Remove the molecule B first and let the work done be Fa’. Then remove the molecule A and let the work done be F’. Then we have Fa+F=Fa'+F’. Now suppose that the mass of each molecule is m times as great, the position of each molecule remaining the same. Let F become nF, then F’ will become nH’. According to the above equation (Fa—Fa’') will therefore become n(fa—Fa‘). This shows that the per- centage diminution of the attraction between the molecules in a system of molecules is the same as that in another system if the distances between the molecules and their arrangement is the same in both cases and the mass of each molecule in one system is the same fraction or multiple of that of the corresponding molecule in the other system, This is realized for different liquids at different temperatures when their values of 7? are equal to one another. M0 It does not seem improbable that the effect of a molecule A in diminishing the attraction of a molecule C on B is proportional to the attraction A exerts on B. In that case, the magnitude of the diminution of the attraction of the molecules on one another can be calculated if we are given the law of attraction that would exist if there were no inter- action between the molecules. Let us, for example, see what the attraction between two atoms of mass m ,, m», becomes if they influence one another in that way. The force of attrac- tion of the atom m, at the point occupied by the other atom 800 Dr. R. D. Kleeman on the Nature of the we will suppose is Kay my, 1£ not influenced by the atom mg, where Kj, is a function of the distance of separation of the atoms, being given by the above law. But if the attraction of the atom m, is influenced by that of m, in the way described, its attraction becomes {Ky/m, — Ki Ke, /my4/ mo} or Ky/ mf 1 —Kay/ mg}, where K, is a constant which is independent of the distance of separation of the atoms and their atomic weight. Similarly the attraction of the atom mp at the point occupied by the atom m, would be Ky Vm{1—Ky my}. The attraction between the atoms is then the product of these two expressions, that is K/ vm V/ m2 41—Ke Vimo} {1—K, Vm}. The effect of the interaction of the atoms is thus to decrease the attraction on one another in the ratio 1: {1—Ky/m}{1—Ke/m}. When the molecules have not the same mass, the decrease of the attraction of the one having the larger mass, it will be seen, is less than that of the other. This fits in with some results deduced previously in the paper from experi- mental results. We may obtain the above result in another way. The chemical attraction of an atom very probably represents a quantity of energy proportional to the attraction or Vm. Therefore, when two atoms separated by an infinite distance are approached to one another till they are separated by a given finite distance, a quantity of potential energy equal to K, Kv m/m, disappears from each atom or from the sur- rounding space, where Ky, is a constant and K, a function of the distance of separation of the atoms. The attraction of the atoms upon one another in this position is therefore (Ky /m, — Ky Ky, / n/m) (Ky Vm —K Ke Vm, m2), which is the same expression as obtained before. If the chemical attraction of an atom represents a corre- sponding quantity of energy, it follows that when an atom is broken up into two parts which are then separated by an infinite distance, a certain amount of energy must be trans- formed into energy of attraction. Thus if the mass of an atom is (m,-++m,) its chemical energy would be proportional to A/(27,+m,). When the atom is broken into two of mass m, Forces of Attraction between Atoms and Molecules. 801 and m, the chemical energy is proportional to ( Yu+ Vig). Now the latter expression is greater than the former, as is at once evident when both expressions are squared, and an increase of chemical energy of attraction has therefore taken place under these conditions. Part of this energy must be derived from the work done in separating the atoms. If the gravitational attraction of an atom on another cor- responds toa quantity of energy proportional to the attraction exerted it must be of infinite magnitude, and the factor with which it must be multiplied to give the attraction must be infinitely small. The decrease of energy when two atoms are brought near to one another is then infinitely small in comparison with the energies contained by the atoms, and the attractive force of each atom would therefore not be appreciably diminished, 7. e. the attraction of one atom would not be influenced by that of another, as we know from experiment is the case. The heat of formation of a molecule in the gaseous state decreases slightly with rise of temperature. This may be due to a decrease of the force of attraction caused by the rise of temperature. The force of attraction between two molecules is thus probably affected in two ways, viz. (1) by the temperature*, (2) by the vicinity of other molecules. K(x 4/4)? The absolute value of K in the expression giving the attraction of two molecules on one another when separated by a distance z, is an important quantity and will now be determined. We have BS ye © y=2 u=n t=0 = oe (2 vm) 2. 3 = - (2) = W) ; dn 0 V0 t= —08 U=—D w=] «“ aml i el v=) v=) i a (> vine | iz’ Eo math + w) | 44 > o(2)(u 4 ) Jace ; dn, where cata V{(ntw)?+wvt+v%?, | hg If on is substituted for }(<) the equation becomes eS eee s pm ge QO t=0 4=0 W=0 n+w dn L_ (> vm)’ a > > 2 i ( male pele 2 m Ui Jo v=-@ u=—w wl L(n + w) Ur U's =e K (" if v=o U=—o n t=n n ti d S a/, —, Seiad id SE ap ; nN +(& Vt) mua? \5 r . Rad a sn? +u? +0715 Bs i Ante} yea * But since the temperature of a gas is a measure of the kinetic energy of the molecules it is difficult to see how it can affect their forces of attraction. Phil. Mag. S. 6. Vol. 19, No. 113. May 1910. 3F ee ee = - 2 7 802 Dr. R. D. Kleeman on the Nature of the Now the value of the right-hand side of the equation will not be altered if we place the integral sign before the summation signs, the equation then reducing to Sy (> /m,) 2K a eh v= = u= = uso it ae mn m 1.2 v=—o U=—-Ho wW=1 fw? +u? ee O=D N==O 1 Sie ree eS raenar art ae ty, ie From this equation K can be determined. The value of the summation quantity obtained by expanding it in series and adding together the most important terms is within a few per cent. equal to 8°93. The value of L must be expressed in ergs, and for m the absolute mass of the molecule must be substituted, the value of m, in } /m, is, however, the atomic weight in terms of that of the hydrogen sien taken as unity. The absolute mass of a hydrogen atom will be taken as 71x 10-# grm. The calculation was carried out for ether and carbon tetrachloride at 2T,. For ether the equation becomes 754% 42x 10" KK (27°8)°(-6907)49_ 2 = 4 (T1x 10-% x 74) which makes K equal to 1°55 x 10-*°. In the case of carbon tetrachloride we have 39°87 X42 x10! KK (27:3) 4385)%* 2 Tas (7:1 x 10-® x 154) or KePi (x10, The value of K may thus be taken as 2 x 10-# (gram)(em.)(see.)~* inroundnumbers. It will be observed that K is the chemical attraction between two atoms of hydrogen separated by one centimetre. It will be interesting to compare this force of attraction with that due to gravitation. ‘T'wo masses each equal to one gram and separated by a distance of one centimetre attract one another with a force equal to 6°6579 x 10-§ dyne. The gravitational attraction of two hydrogen atoms on one another unit distance apart is therefore 3°356 x 10—°6 dyne. A comparison of this value with that of K shows that the chemical attraction is greater than the gravitational for dis- tances less than 100 em. from the attracting body. But this cannot be true since the gravitational force has been x Soa. x Oca. x , : 4 | | { Forcer of Attraction Vetween Atoms and Molecules. 803 determined for distanees less than 100 cm. between the attracting bodies and its value we see is less than that given by the expression for the chemical attraction. The chemical force must therefore decrease much more rapidly than that given by the inverse fifth power law for distatices greater than one centimetre. The result suggests, however, that the chemical force at a distance of one or two cm. from a mass of matter may een be comparable with that of gravitation. If that is so, it sould not be impossible to detect the fact experimentally. To obtain a legitimate comparison between gravitational and chentical attraction it is necessary to compare the at- traction between two molecules separated by a distance equal to their distarice of separation in the liquid state, say two molecules of ether at 31T., corresponding to which K has been determined. The distance between two molecules 1/3 of liquid ether at 2T, is i. , which gives 5-1 x 10-* cm. The gravitational attraction between two hydrogen atoms separated by that distance is 1°3%10-* dyne, while the chemical attraction is 5°6 x 10-9 dyne. The chemical at- traction is thus much greater than that due to gravitation. The gravitational attraction thus plays practically no part in producing molecular aggregation, and the various phenomena connected with it. It will also be interesting to compare the chentical attrac- tion with that of two opposite electrical charges each equal toe. When separated by a distance 10-% cm. the electrical charges attract one another with a force equal to 1/16 x 10-3 dyne, while the chemical attraction between two hydrogeri molecules for that distance is °2x10-° dyne. The electrical attraction is thus greater than the chemical attraction between two hydrogen atoms. But the chemical attraction between two atoms of lead would be about 200 times greater than that between two hydrogen atoms, and thus of the same order of magnitude as the electrical attraction between two charges e. The inverse fifth power law of attraction between two molecules has already been put forward by Maxwell. He arrived at the law from the following considerations. Measurements of the coefficient of viscosity have shown that it is proportional to the temperature, and the coefficient of diffusion is proportional to the square of the temperature, but according to the formule based upon the kinetic theory of gases these quantities vary as the 1/2 and 3/2 power of the temperature respectively. The law of force must 3F2 804 Dr. R. D. Kleeman on the Nature of the therefore be such that in the equations if: t= vm OHA a “ap ITD Aeeseet ate 12pa7" o” the square of the radius of sphere of action of two mole- cules is inversely proportional to v the velocity of a nol for then the coefficient of viscosity 7 is proportional to 1 and 6 the coefficient of diffusion proportional to vt, which corresponds to the first and second power of the ‘tempe- rature. Suppose the potential energy of two molecules varies in- versely as the nth power of their distance of separation. When a collision takes place and the force is repulsive as Maxwell supposed, their nearest distance of approach is given by where & is a constant independent of the temperature. Now in order that o? may be inversely proportional to v we must have n=4, and the attractive force therefore varies inversely as the fifth power of the distance between the molecules. If the force is one of attraction instead of repulsion, the value of o given by the above equation i3 the least it can have to prevent the colliding molecules from rotating round their centre of gravity. The Passage of the « Particle through Matter. Maxwell has given an investigation in general terms of the motion of a molecule through a gas consisting of mole- cules of a different kind (Maxwell's Collected Papers, vol. 1i. p- 36). It has been applied by Sir J. J. Thomson to the case of an @ particle traversing a gas (‘ Conduction of EHleec- tricity through Gases,’ second edition, p. 370). According to an equation deduced by Sir J. J. Thomson the decrease of the velocity V of a molecule of mass m, wheh it traverses a gas of molecular weight mw, for a pile dc is given by i d\ — Ms _ (m, + m2) \r=i N dz ~ m+ Mg M1Mo where A is a constant independent of mj, m., and V, and the force of attraction between two molecules separated by a n—5 fay shes 5 distance r is given by —!. v 7” Now, according to the result obtained in this paper, KG WOK: 5 Me et ‘ s Jim Gin V/ Mo. yr" n°” a — Forces of Attraction between Atoms and Molecules. 805 But this law probably applies only to a region corresponding to distances from the molecule of the order of the distances of separation of the molecules of a liquid. The « particle, on account of its great velocity and small mass in comparison with that of the molecules of the matter through which it passes, would approach a molecule very closely, and the law of attraction relating to these small distances of approach of the bodies might therefore be somewhat different from that given above. Since the attraction undoubtedly increases very greatly with the closeness of approach of the bodies, the phenomenon would be regulated by the law holding at closest approach. Let us suppose that the law of attraction is K “ 2 P J m,. > A/ Mo, the only justification so far for making this assumption is that it reduces the above equation toa simple form. We have then ‘ ary et Se OS may. 2 Wm, ee and integrating we get Dividing the equation by ae E eat K bes s Ve—V,’= a yg / my z > Mo - (x;— 2). l This equation states that an a particle traverses the distance (a;—az,) In a gas during which its energy falls from 2, T 2 Vim | Vom, i Ea! die “AAS ye te ene a O gr" and that this distance 1s inversely propor- tional to the sum of the square roots of the atoms of a molecule of the gas. Now Prof. Bragg and the writer* have carried out a set of measurements on the range of the @ particle in different gases at the same pressure, and found it inversely propor- tional to S4/m. Therefore, if the part of the energy expended by the « particle due to its carrying a charge is less than that expended in the way explained, the law found by experiment is at once accounted for. There are various reasons for believing that the energy expended on ionization is less than that expended otherwise, as imparting kinetic energy to the molecules it encounters, at any rate the loss of energy by the @ particle cannot be easily explained by ex- penditure on ionization only. Thus the ionizations per unit * Phil, Mag. Sept. 1905, p. 318. 806 Dr. R. D, Kleeman on the Nature of the length of path in different gases are not proportional to the corresponding expenditures of energy, The writer* has calculated the total energy expended by the @ particle on an ion made from different atoms, and found that some of the energies differ from one another by as much as 40 per cent. It appears very probable, therefore, that the above explana- tion is the true explanation of the law found by experiment. The chemical attraction close to a molecule thus seems to be inversely as the third power of its distance from the centre of the molecule, which gradually becomes the inverse fifth power as the distance approaches that of separation of the molecules in a liquid, and for distances much greater than this we have seen that the power of the distance must be greater than the fifth, otherwise the attraction will be greater than the gravitational at distances for which we know this is not the case, Diffusion of Gases. Assuming that the attraction between two molecules is ie , that is inversely as the fifth power of their distance of v separation, Maxwell has obtained a formula for the coefficient of diffusion of a gas 1 into a gas 2 which gives pp OO Ae LE Cae Wee: A rm) oe Ps MyMy(M, + Mz) J where 7, Po, P1, P2, are the partial pressures and densities of the gases, and p=pi+pe. The value of B according to the investigation in this paper is K DV fy > 4A my. Let us consider the case when the number of molecules 1 is small in comparison with the number of molecules 2, so that P=Px “We have Z: a = a and p2p2= — and the equation may therefore be written goat’ RAE iG Ea ~ pemymA {[ KY /m,. > W/2 MyMg(M, + Mg) 248 { my + Me Ao mK MyMgrrA/- mM, .2r/ Mm» : * Proc. Roy. Soc. A. vol, Ixxix. p, 220 (1907), a SS = Forces of Attraction between Atoms and Molecules. 807 where M is a constant, which is the same for all gases at the same temperature and pressure. This equation may be used to calculate the relative coefficients of ditfusion of gases into one another. 7 Table VI. contains the coefficients of inter-diffusion of a number of gases taken from Winkelmann’s Handbuch der Physik, Wérme, p. 759. The third column in the table contains the coefficients calculated by means of the above equation. The calculated coefficient for N,O—CO, has been put equal to the experimental and the others reduced cor- respondingly. The agreement between calculation and ex- periment, although not very good, is sufficiently close to show that the law of attraction elite Mila between two molecules is approximately true. TABLE VI, Observed coefficient Caleulated coefficient. | ra Name of Gases. of diffusion. of diffusion. LS 60 ‘O89 ‘089 oy See 142 "125 | be Mi ie thie a J vccccccccccsece “Lb SLA pee 00 Ae ‘161 *160 7 ee a “180 ‘116 H — SO, Guiliumavisl'weeis as "480 "655 oo Dee 556 723 ye 642 O04 i sewapeaces oye: T22 ‘871 The quantity K we have seen is only approximately a constant; therefore, even if the inverse fiith power law were exactly true, a good agreement could not be expected since the values of K for different gases are equal to one another only at corresponding states, and have ditterent values there- fore at the same temperature. Remembering this, and considering the extremely complicated nature of the process of diffusion, the agreement between observation and calcu- lation by a formula of such simplicity is perhaps better than can be expected. Inferior Limit of the Distance of Separation of two Molecules during Collision. When a molecule collides with another the kinetic energy of either must be greater than the potential energy when they are in contact due to their attraction, for otherwise they would remain adhering to one another. ‘The diameter of a — EEEeEeEeEeEeEeeEeEeEeee—eeee 808 Forces of Attraction between Atoms and Molecules. molecule cannot therefore be smaller than the value of 2 given by the equation mm vy > =i, ats vm)’, which is obtained by equating the kinetic and poteunel energies. In the case of two hydrogen molecules at OPC. this equation becomes 5) ‘ —25 aad _ soe x (184 x 10°)? 2 x 10-%.,, Ag+ which gives w=1:34x10-8 em, This is of the same order as the usually accepted diameter of a hydrogen molecule. Thus when two atoms on collision do not separate but combine to form a molecule they are separated from one another by a distance less than that of x given by the above equation. The distance w with which two molecules of the same kind would be separated when revolving round their centre of gravity with their velocity of translation, is given by the equation 2v?m = : (Z./my)?. ez ‘ This equation is the same as the foregoing. When the molecules are different they cannot both possess their original velocity of translation. Therefore, when two atoms are combined into a molecule they may rotate round their centre of gravity with a velocity comparable with that of their translation without the equi- librium being disturbed. Distances of Separation of the Atoms in a Molecule. In the investigation given in this paper the molecules in a liquid were regarded as points, so that their diameter would he small in comparison with their distance of separation. The effect produced by a molecule is then the sum of the effects produced by its atoms, if these are approximately independent of one another in their external action. Now we have seen that the constant Sc, of the atoms of a molecule is additive within less than 10 per cent. The diameter of a molecule must therefore be less than one-tenth of the dis- tance of separation of the molecules in a liquid at ordinary temperatures. In the case ot ether at 3 T. the distance of separation of the molecules is 5°1x10-° cem., and the } ‘ Relative Motion of the Earth and the Acther. 809 diameter of an ether molecule must therefore be less than alex LO? om. It follows, therefore, that since there are fifteen atoms in an ether molecule, the average diameter of an atom in com- bination is less than 5°1x10-! cm. Since this is less than the accepted diameter of a molecule of hydrogen the result suggests that the atoms contract when combining to form a molecule. Further evidence that such a thing occurs will be given in a subsequent paper. Cambridge, Feb. 11, 1910. LXXXI. The Relative Motion of thejEarth and the A&ther. By Prof. Harotp A. Wiuson, D.Sce., L.R.S., MeGill Oni- versity, Montreal *. TEXHE following paper contains an attempt to reconcile the phenomena of the astronomical aberration of light and the negative results of the experiments made to detect relative motion between the earth and the ether, without supposing that bodies change in length when they are set in motion relative to the ether. It is shown that this can be done, and that the facts available are sufficient to determine the relative motion at any point outside the earth. A theory is then described which appears to offer an explanation of the motion deduced. According to Sir George Stokes’ well-known theory, aberration can be explained by supposing that the motion of the zther outside the earth is purely irrotational. It appears to be generally supposed T that it is necessary in Stokes’ theory for the ether to be carried along by the earth so that at the earth’s surface there is no relative motion. That this is not the case was pointed out by Stokes himself. He made this supposition merely because he regarded it as unlikely that the earth could move through the ether without setting it in motion. The statement frequently made that Stokes’ theory fails because no irrotational motion of the ether can be found which satisfies the condition that the eether is at rest relative to the earth at the earth’s surface is therefore not really justifiable. As Stokes has shown, the path of a ray of light is unaltered by any irrotational motion of the ether, so of course aberration is unaffected by any such motion. Ivrotational motion of the ether can only * Communicated by the Author. + See, for example, H. A. Lorentz, ‘Theory of Electrons,’ p. 173. { Mathematical and Physical Papers, vol. i. pp. 189 & 156. Also Phil. Mag. vol. xxix, p. 6 (July 1846), 810 Prof, H. A. Wilson on the Relative produce, at any rate, second order optical effects, so that such motion, if it exists, will be extremely difficult to detect. If we do not admit that bodies change in length when set in molion relatively to the ether, then the experiments such as Michelson and Morley’s referred to above *, in which the apparatus was turned about a vertical axis, show that at the surface of the earth the motion of the ether relative to the earth has no component in a horizontal plane. Let us regard the earth as at rest and suppose that the velocity of the ether at a great distance away is parallel to the axis of # and equal tov. Take the centre of the earth as the origin and denote its radius by a. We require then to find a distribution of velocity having a velocity potential (@) such that ¢,=a isa constant and that whenr=2 —dd/dx=t, dd/dy=dd¢/dz=0. Regarding the ether as an incompressible fluid, it will be seen at once that this problem is precisely analogous to the problem of finding the electric field in the space surrounding a conducting sphere placed in an electric field which is uniform at a great distance from the sphere. ‘The solution is od=v(a*/r?—r) cos 0, where @ is the angle between 7 and the axis of x. At the surface of the sphere 6=0, so that the velocity has no component parallel to the surface while — dd/drpaa=3v cos 8, so that the velocity perpendicular to the surface varies from 3v where 0=0 to0 where 0=7/2. It appears, therefore, that the stream-lines crowd into the earth just as the lines of electric force crowd into a sphere of very high specific inductive capacity or into a conductor. It will now be shown that a physical explanation of this crowding in of the stream-lines can be given which is consistent with some modern ideas as to the nature of matter. Let us suppose that the ether near an atom is in motion as though the atom contained a point source of ether and a point sink of equal strength, so that the ether entering the atom at the sink comes out again at the source. If the source and sink are very near together they will form a doublet. We may, if we like, imagine many such doublets in each atom. It is easy to see that two sources will move away from each other, and by considering the hydrodynamical analogue of the Cavendish experiment in electrostatics it 1s easy to prove that the force between them is inversely proportional * Other such experiments are those of Rayleigh and Brace: also Trouton’s experiments on a suspended charged condenser, and on the resistance of a wire. Motiou of the Earth and the dither. 811 to the square of their distance apart. It follows from this that a source put in a stream of ether will be acted on by a force proportional to the velocity of the stream. This analogy between electrostatics and hydrodynamics is well known, so that it is unnecessary to discuss it further. 6 8 10 h2 Y-RAYS OF RADIUM LEAD BRASS ELECTROSCOPE A ALONE B WITH LEAO SCREENS O-5 CM. THICK. ca SODDY AND RUSSELL, . Phil, Mag, Sov, 6, Vol. 19, Pl, XII, Fie. 1. Vie, 18, 28 0 2 4 6 ny (ite! | Tc. 9 Me. 1. T&M) CURVEA —> a4 = on ee wa ’ = ‘ a y-Ravs ito [Phe oF URANIUM x PrRays OF URANIUM X \ as| Cf . LEAD ALUMINUM y-Rays oF URANIVAL X YAAYS OF RADIUM URANIUM — —> pis eee LEE Y- RAYS OF RADIYN °? (LEAD REFLECTeR) ~ (keap ReFLecrog) aa LEAD MUSPHERES OF Sin ee 4 A 2 BRASS LEAD HEMISFHERE 5 PREM ERIE: c} HEMISPHERICAL (E09 EY (VERE ee ' Gee be FLecrroscore € fecresnore /\): ty/s/ A Sts cy si/b/oy oy 4 ete 40/s/o: S rhe fo on ylae + dr [Ev ary} Ar [Ec(-aa)] . ; 1/4 a 2h ea nee, > TAT Ar[ecean]= [etre 8 WITH LEAD SCREENS bp ‘ e rlecr9] ~ dx [etre = ats Novos > ~ . eS vel OSM THIEN. . = | [4 S oe, . TS . * deg, ~ > I ~ ~ x te 16) le we 1% \ ~ . . ¥ <2, wa \y Ne ~~ ~ Lo NS “3 le ‘ we yo I \ 3S \ Nay > SS \ a aaa we 4 mee SS . 7 4 ux) os of ‘ S xt Na “ SS =) ho ~ Ra > = ; t i SS ee Nir ~ = L K S - ” > cs te aut o8| 2 rs nn — * 10 I 5 3 : i Be | ‘ 12 TS TS ~ re SS : A | : Oh 06 Ye ui : * 8 og] 7 (1) —> we . ™ | Rene —~ ob Lj a Bre re 7a Fol 7 (eu) —> 07 } . ’ ise ors ve 7) 7r Te 7 -- 4 J THANE (00 & DEM irY TR, > oql__Z(CA) — —— || Q 2 a i a [ee 19 Ira, 16, Tie. 8. Fra, 12, 28 “ah oe, = - et = = a THEORETICAL cob CURVES. os mi eR INET maa = a £ raars gies x Fic. 10, 24 : ye ——r—ee a ONG OF AnenER Cpe E ED YRAVS_OF RADIUM. ten rugeraoreere MOSPLLERES cf Hervarncarcns lisdsps) Ye RATS GER ZU 20) LEAD 10 Ke dn 0149 8(er) s 1°, . mare! bT Mm, mt Leo é quero I LEAD ELECTROSCOPE N eae ge EAD wD Zine ZING HEMISPHER By Cninmaear (7/563) mm RADIUS) @ APD gy ABLE OIAE OO tmyy syblalune 8/yfog (ey~a) 7/290479) 4 Ge 16 6:7 MG. RADIUM. Ly @ Aro 96 re AOsissA€ (41m) te Zr NT Bee feb sr) |r ee Bem EB dr lO A-Rod A= O29 a ] Asio2 tf \ > — : os 08 D ~ SS ; : A Te\yit 04 iN SO ; il : 0:0) “a . Seon ee ca eke ee oe ta Talk —77-* & e Z T6|S Mes S oe ors Ory OP a ya os eee ore oe oe SS 4 Te TCM)— = i 0 6 i2 (8 RUBENS AND HOLLNAGEL, Phil. Mag, Ser. 6, Vol, 19, Pl. XII. Fe, 3. wy, core a ieee f altel i, i im Jo. D : tL JO IL Pn a a i ll Z| ; imi Ett IL q . H 15} a Fee “ | 13} ai rR 7 alle ” tlk ar yok lt |p =i Fie. 3a. & a [S} E _| nm S Sr ee © oT i al —— Tia, 11. Fig, 9. (ea em en [= T 10 10} 10 i iS 9 IF 8 8 es) 3| é 7 7}—+ _| | J 6 6 6 | 5 6 6 4 | 4 4 4 il [eae I g 2 é & a | 1 7 8On 504 1004 110 Cn 10u a ee ff ¢ AIT THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE, [SIXTH SERIES.] FUN Ee void. LXXXVI. Radium D and its Products nl Transformation, By G. N. ANTONOFF* HEN the agivity of a body eee has been exposed to the radium emanation for some time is measured, it is found that after the initial products, radium A, radium B, and radium ©, have disappeared, there still remains a residual activity. This was first noticed by Mme. Curie and examined in detail by Rutherfordt, who found that the active matter was complex and consisted of at least three successive pro- ducts which he called radium D, radium FE, and radium I’, Radium D was found to bea rayless product of long ale of transformation (the “ period”’ being the time taken for : product to be half transformed), radium E a B-ray Sodinet with a period of about six days, and radium F an @-ray product with a period of about 140 days. It is now de- finitely known that the polonium of Mme. Curie is identical with radium I’, and that the primary source of activity in radio-lead separated from radioactive winerals is radium D. Rutherford’s results were extended and completed by Meyer and v. Schweidlert, who examined in particular the products of radio-lead, and showed that they were identical: with radium D, radium E, and radium I’. By comparison of the B-ray activity of radium HE and radium C, Rutherford § * Communicated by Professor I. Rutherford, F.R.S + Phil. Mag. yiil. p, 636 (1904); x. p. 291 (1905). t Wren. Ber. exy. Abt. ii a, May 1906. § ‘Radioactivity,’ 2nd edition, p. 405 (1906). Phil. Mag. S. 6. Vol. 19. No. 114. June 1910. = 3. H 826 Mr. G. N. Antonoff on Radium D deduced the period of transformation of radium D, which he found to be about 40 years; while by comparison of the a-ray activity of radium F with that of radium C, Meyer and v. Schweidler* obtained the value 12 years. In the latter experiments a correction was made for the difference in ionizing power of the @ particles from radium C and radium I’. Data are not yet available for the corresponding correction due to the difference in speed of the @ particles from radium C and radium E. As it is now a matter of importance to know the period of transformation of these products with accuracy, the period of radium D has been determined by a new method, and the question as to whether there are two products between radium D and radium F has been closely examined. From experiments on the initial rise of the B-ray activity of radium D, Rutherfordt found the time period of radium E to be 6 days, while the period of decay of radium EH, from which most of the radium D had been driven by heating, was found to be 4:5 days. Meyer and v. Schweidler found that a preparation of radium H lost its §-ray activity initially according to a period of 4°8 days, but finally decayed more slowly according to a period of 6°2 days. They suggested as an explanation of their own and Rutkerford’s experiments that there are two products between radium D and radium F, and that the first is rayless, and has a period of 6:2 days, while the second is a #-ray product, and has a period of A*8 dayst. The following experiments were made with the view of settling definitely whether or not there are two products. Haperiments with Radium E. Suppose that radium E is complex and consists of two successive products, H, which is rayless and H, which emits B rays. Then, if radium H, is completely separated from the mixture of radium D and radium H, and Hs, the B-ray activity of radium EH, will decrease exponentially with its characteristic period. Similarly radium D will recover its B-ray activity according to the same period. If, however, radium Hj, and radium KH, are separated together, the 6-ray * Wien. Ber. cxyi. Abt. iia, April 1907. T Doc. Ge. t H. W. Schmidt, Phys. Zeit. viii. p. 365 (1907). Also Meyer and v. Schweidler, Phys. Zeit. viii. p. 457 (1907). Also H. Hess, Ween. Ber. exvi, II. p. 1289 (1907). and its Products of Transformation. 827 activity of the two will obviously not decrease according to an exponential law at first, since radium H, is changing to radium Hy. Finally, however, the decay will be exponential, corresponding to the slower of the two periods. In the experiments described later, the §-ray product has been separated from radium D, and has been found to decay always according to an exponential law, with a period of 5 days, and only 5 days. At the same time the radium D, inactive immediately after the separation, recovers its @-ray activity according to the same period ; consequently if radium E, exist it mustalways have remained with radium D in equilibrium amount, while radium E, must have been completely separated. It is therefore impossible to disprove the existence of radium EH, by such an experiment. The question of its existence can, however, be definitely settled by another method, depending on the observation of the rise of the §-ray activity in a preparation of initially pure radium D. To this end the following experiment was made:— A piece of platinum foil was exposed to the emanation cor- responding to the equilibrium amount of 150 milligrams of radium bromide. ‘lhe quantity of emanation being very large it was sufficient, in order to obtain a strong encugh preparation of radium D, to expose for 24 hours only. The emanation occluded by the platinum was got rid of by dis- solving the active deposit in acid and evaporating on a watch- glass. The active deposit of short period decays in a few hours, and we have practically pure radium D left, with only a very small #-ray activity. The curve showing the recovery of the §-ray activity was very carefully determined until a maximum value was reached. This curve was found to be identical with the other recovery curves, being expo- nential, and showing a period of 5 days. The results are given in the following Table I., where the maximum activity (that after 40 days) is taken as 100. If the rise of the activity is due to the production of a single @-ray product the recovery curve of the activity is given by the equation e = fart. Ip where I, is the maximum activity and I is the activity at any time ¢. The theoretical numbers calculated for a period of 5 days are given in column B of Table 1. They show a very good agreement with the experimental values given in 3H 2 828 Mr. G. N. Antonoff on Radium D column A. This is shown in fig. 1, where curve 1 represents the theoretical values, and the points represent the experi- mental values. If the recovery is due to the formation of successive products radium H, and H,, which have constants A>, and Az, and of which only the latter emits @-rays, it can easily be shown that the activity I at any time ¢ is given by I (1 _ Wye Ay —) p) I, bi Ay— Ao where I, is the maximum value. Taking Meyer and v. Schweidler’s suggested periods for radium E, and radium He, viz. 4°8 and 6°2 days respectively, the activity at any time is given in column CO, and is shown in curve 2 of fig. 1. The difference between this curve and the previous one is so striking that it is impossible to confuse them. It may be remarked that in the first case the activity after 5 days is 50 per cent. of the maximum, while in the latter it is only 14 per cent. This experiment shows conclusively that only one product exists between radium D and radium F, and that this product emits @ rays. TABLE I. Activities in arbitrary units, final value taken=100. 23 B. : ay Ne Theoretical for two ue ae Das: Experimental, | Theoretical for one} products radium product radium H| £, and EH, with with period 5days.|. periods 48 and 6'2 days resp. 0 0 0 0 1:04 13°3 13°4 22 19 22°4 23°] 4°8 2-2 26°4 26'3 5°6 2°8 31-9 32:2 70 3°8 41:2 41°3 10-1 4°8 48 3 48°6 14:2 49 49-2 49°3 149 5:1 50°7 50°3 150 58 546 55'2 18°5 68 60°4 61:0 22°5 78 65°7 66:1 26:0 88 68°6 70°5 30°2 9°9 74:4 74°6 35°0 19 92°'5 92°8 69:0 55 100 100 99 115 100 100 100 en ae, and tts Products of Transformation. 829 Some other experiments on the decay of radium E will now be described. Some radium E was separated from radium D by precipitation of the latter with barium sulphate, Activities. Time in days. and its activity was found to decay, according to an ex- ponential law, with a period of 5 days. The results obtained are shown in Table II. See also fig. 2, curve 2. TABLE II, Activities in arbitrary units, initial value taken =100. Time in Days. Theoretical for 5 days period. —_———— eee Kxperimental. MAMI HOSOSCS MOOsdsmeOOSo= ~J oo bo So =f (0) &30 Mr. G. N. Antonoff on Radium D Another portion of the same preparation was heated in a blowpipe flame until its activity was reduced from 60 divi- sions per minute to 5 divisions per minute. This was again found to fall to half value in 4 days (see fig. 2, curve 4). Fig. 2. : Co : Seas Sean aoe PSSA Ko =p ne yeas CCERRERERES Soe HNN inna UT epee Sie eee a Time in days. Log. of Activities. Finally, radium D was separated incompletely from another preparation, and was divided into two parts. The @-ray activity of the one was measured until it bad reached a final constant value 6°5. Subtracting this value from the readings and plotting on logarithmic paper we get a straight line showing a period of 5 days. The results are given in Table III. (see also fig. 2, curve 1). The other portion was heated on a platinum crucible lid in a blowpipe flame, in order to get rid of the radium D, The time period of the residue was found to be 5°1-5:2 days ; the excess of -1-2 day over previously found values being probably due to a very small quantity of radium D lett behind after heating (see fig. 2, curve 3). Radium H separated from old radium solution gave again the period of 5 days (see fig. 2, curve 5). The variation with time of the B-ray activity of ‘the various preparations is thus consistent with the view that only one product is present between radium D and radium F. The absorption of the rays from and its Products of Transformation. 831 TABLE ITT. Activities in Arbitrary Units. | | Time in Days. Activities minus Activities Activities minus observed. final constant 6°5. annie ee 0 176 Lig 100 0:2 17-4 10°9 98°2 05 Lijek 106 95°2 ibe 161 96 86°5 14 156 9-1 819 16 15-4 89 8071 2°2 14°5 8:0 72:0 32 13°5 7:0 63:0 4°5 12°3 58 52°4 52 11:9 5-4 48°6 6-2 11°2 4:7 42:2 T-2 106 41 36°8 8:2 10°1 3°6 32'4 94 97 3°2 28°8 10°6 9°13 26 23°4 14:2 78 or 17 32 6:7 0-2 18 62 6°5 0 ) 122 6:5 0 0 | radium E was also investigated. According to Hahn and Meitner* the rays from a single radioactive product are 2 absorbed according to an exponential law. The rays from radium E in radio-lead were examined by Hahn, who found that the absorption curve departed from the exponential long before the y-ray effect could influence it, since this latter has | been found by Meyer and vy. Schweidler to be ‘03 per cent., : and by Schmidt 016 per cent. of the total activity. Hahn showed that this deviation was due to some radium emanation remaining in radio-lead, which could be got rid of by heating. Meyer and v. Schweidler found that the 8 rays from radium HK, are absorbed approximately according to an exponential law, the coefficient > for aluminium being 44 cm., and that some soft 8 rays also come from the long period ‘products, which are half absorbed by 1°5x10-* cm. of aluminium. The absorption of the 8 rays from radium EH was also measured in the present case and found to be exponential, and the absorption coefficient for aluminium found to be 43 cm.—! and for copper 164 cm.~! * Phys. Zeit. ix. p. 821 (1908). 852 Mr: G. N. Antonoff on Radium LD The Constant of Radium D. The determination of the constant of transformation of radium D presents some difficulties, as its period is too long for direct measurements. Meyer and y. Schweidler found its period to be about 12 years, by comparing the initial activity of radium C with that due to radium F after a known interval bad elapsed. In the present experiments a similar method bas been adopted, with the difference that the a particles emitted from radium F have been counted by the scintillation method. This avoids the uncertainty due to the correction for the different ranges of the « particles from radium C and radium F, A known quantity of emanation was sealed up in a glass tube. The emanation was transformed into radium D, and this in turn gave rise to radium F (polonium). After a definite interval the tube was opened and the number of scintillations per second from the deposit of polonium in the tube was directly counted. An approximate estimate of the period of radium D can at once be made from such an observation. The initial quantity of emanation in the tube was determined in terms of radium by the y-ray method. The number nof a particles emitted per second from the emanatien is known trom the counting experiments of Rutherford and Geiger*. If dA, is the constant of the emanation the total number of emanation atoms present is — =N. Since the time periods of. the if emanation and its short-lived products are very small com- pared with that of radium D and radium I’, it may be supposed as a first approximation that the emanation changes at once into radium D, and that this changes slowly but directly into radium F. On this assumption the number of atoms of ° ae . radium D producedis —. The number of atoms of radium F LO d produced per second is —= , where A, is the constant of i rsdium D. Since radium D has a long period of transfor- mation, radium F is initially produced at a nearly constant rate, and the amount of it, S say, present after a time ¢, Is given by Aon ——— if = At S Ra ( € ); where A, is the constant of radium F. ‘The number of atoms * Proc; Roy. Soc..A. vol. lxxxi. (1908). and its Products of Transformation. 833 of radium F breaking up per second is 48, which is equal to Agi — past . (L—e-™*), The value of 4,8 is deduced experimentally from the counting experiments. Since the values of A, and Ay are known, the value of 2,, the constant of radium D, can at once be deduced. In the more complete theory it 1s necessary to take into account the period of the emanation and of radium E. The emanation may be supposed to be transformed directly into radium D, for the intervening products have short periods of transformation. Let 2 ,, A», As, Ay, be the constants of transformation of the emanation, radium D, radium EH, and radium F respectively. Let N be the number of atoms of emanation present initially, and S the number of atoms of radium FE present at time ¢ after the introduction of the emanation. ‘Then S=N (A ec? 4+ Be 4+ Ce 4+ De"). The values of the coefficients A, B, C, D were deduced for me by Mr. H. Bateman, to whom I am very much indebted. a at sig RNG WEA (Ag —Ayz) (Az —AL)(Ag—Ay) B= MiAsds Be ~ (Aya) (Ag — Az)(Ag— Ag) re AideA3 a (Ay —As3) (Ay —As) (Ay —As) [ie NEE (Ay Ay) (Ao—Ay) (Ag— Aa)” Tt can easily be shown that for the time interval in the experiment, Ae? and Ce-4 are very small and can be neglected. In order to obtain the values of Band D it 1s necessary to know the constant of radium D approximately. This can be done by the approximate theory already mentioned. Writing the expression for B in the form No J J I B= — Sse oe pou As us)? ; At A3 A4 : $34 Mr. G. N. Antonoff on Radium D and taking Ai = 2°08 X 10-° (sec.-) as the constant of emanation, Ne —about dsl 0G? : Hs radium D, Ns= 60«K 108s ~ is radium E, \s=9'°81 x 107° 5 fe radium F, we get B=1:02 Ae 2 V4 Similaily we find D=1-09%2. 4 Consequently ue =, (1-02 et 1-09 o- 9). Since the time period of radium D (about 16 years) is long compared with the time of observation (Jess than one year) the value e~# is very nearly 1. Its value is found with sufficient accuracy by taking the value of A, found by the approximate calculation. Since ),8 is equal to the total number of a particles expelled from radium F per second, and the values of N and ), are known, the value of }» can at once be deduced. Haperimental Method. The emanation tubes in which the growth of polonium was observed were of the shapes shown in figs. 3 and 4. Fig. 3. These had been used in experiments upon the decay of the radium emanation. The initial amount of emanation had been accurately measured by the y-ray method. The emana- tion decayed zm situ in the tubes, and had practically dis- appeared before the tubes were examined for their content ot polonium. fF or this purpose the tubes of the shape shown in fig. 3 were cut in two, and the number of @ particles PE ag ne re and its Products of Transformation. 835 emitted from each part was counted by the scintillation method. In the tube of shape shown in fig. 4 the mica plate was removed. The open tube was fixed centrally ina glass tube (as in fig. 5), at one end of which was placed a Fig. 5. ie SETUP CEeP aD wea Sooo OE - ree ee B. Zine-sulphide screen. A. Microscope. piece of zinc sulphide screen. The tube was so arranged that the « particles from each part of it could fall on the screen. The apparatus was exhausted to a low pressure. The number of scintillations which fell per minute on a detinite small area of the screen was counted by a low-power microscope in the usual way. The diameter of the circular area of the screen in the field of the microscope was 1°09 mm. and its area ‘00866 sq. cm. ‘The part of the screen used for the counting experiment had been previously calibrated by Dr. H. Geiger, who found that the number of scintillations observed was on the average 10 per cent. less than the number of « particles falling upon it. To correct for this, the number of scintillations observed is in all cases increased by 10 per cent. As the emanation tubes were somewhat irregular in shape the following method was adopted in deducing the total number of « particles emitted per second by the polonium in the tube. Suppose the tube to be divided into a number of rings of equal length. As a first approxi- mation, it may be supposed that the amount of polonium on the walls of the ring is proportional to the volume included in the ring. This should evidently be the case, for the deposit from the emanation should be proportional to the amount of emanation initially contained in that part. Since the diameters of the rings are in all cases small compared with the distance from the screen, the polonium may be supposed concentrated at the centre of each ring. Let 0}, Uo, ... be the volumes of the rings, d,, dy... the distances of the middle parts from the screen, then ()] v1} C9 d? ave d,* + as ee where v is the total volume of the tube, d the mean distance, to be determined on the assumption that all ihe polonium is concentrated at a point. If v=1, then t;, t,... are the ee 218: 836 Mr. G. N. Antonoff on Radium D respective fractions of the total volume for the rings. If m be the number of @ particles per sq. cm. per second falling on the screen, the total number of a particles emitted is Amd’m. A typical experiment is described below. A conical tube of the shape shown in fig. 4 was filled on December 7th, 1908, with an amount of emanation corresponding to the equilibrium quantity from 10°5 mgrm. of radium. This was opened and the scintillations counted on October 23rd, 1909, an interval of 320 days. The distance of the zine sulphide screen from the end of the tube was 12-4cm. The number of scintillations observed, with the 10 per cent. cor- rection added, was 31:2 per minute, or 0°54 per second, for an area of screen of ‘00866 sq. cm. The corresponding number m per second for 1 sq. cm. of the screen is 60:0. The tube used was 10°3 cm. long, and for the purpose of calculation was divided into five parts of equal length. If Uj, Vo) U3, Vy Us Yepresent the volumes of the parts, we have nya "ATA d= 13" dem, oo Ws do= lo aren: v,== "1560 d3=1 1 3:em, v,='066 v G4, == 19" cm: v5= "009 v ds. =32 1 }.em: where v is the total volume, and where dj, d., dz, ds, d5, are the mean distances of the rings. 1 =i). The total number of « particles emitted per second, S=47d?m=1:63x 10°. This is the number of «& particles from the polonium, at a time 320 days after the tube was filled. By means of the approximate theory already considered the constant of change \; of radium D is given by On NS M= Fee)’ Now S=1'63x10°, the total number of « particles emitted from the polonium per second ; A, for the emanation ‘= 2°08 x 10-6 (sec.-!) ; (le-“)=°7983, taking the jai and its Products of Transformation. 837 period of polonium as 138 days ; 10°5 ~ 1000 since Rutherford* has shown that 3°4 x 101° atoms of radium and its products break up per second. Substituting these values, A2=1:19x10-° (sec.-!). This gives for the half period of radium D 185 years. Using this approximate value of A, the correct value can at once be deduced from the more accurate formula, viz. =Ag( 102. eT 42? — 1:09 eM). Kol x Oe sora ho Substituting the approximate value of A, in e~** we find 1:02 e~*#=-9863. Remembering that N= : Le ee 1 we get A,=1:24x10-*% This corresponds to a period of transformation of 17:7 years. A number of experiments have been made with different tubes containing different quantities of emanation which had been allowed to stand for different lengths of time. The results are collected in the following table :— Mien Amount of Time | Total numer | | Half-trans- | F before of particles | formation of radium testi Nat eet | yi ‘od | et emanation, testing | expelled per | period in in days. second. | years. Ga 105 mgrm.| 320 | 1:63 x10° | 124x10° | 177 2 ....../262 119 | 236 x10° | 133x10°°] 164 -9 ERTS 25°1 118 2°25 <105 | 1:36x10 16:1 -9 © eves 9-4 126 0-867 x 10° 1°34 10 163 9 raert as 7-0 129 0°677 X 105 1°37 X10 159 Taking the mean of these determinations we find A2= 1°33 x 10-° (sec.-") and the half period =16°5 years, with a probable accuracy of +°5 years. * Loc. cit. 838 Mr. G. N. Antonoff on Radium D Method of Separation of the Long-period Products from Radium. Rutherford * effected a partial separation of radium E from radium D by heating a platinum piate containing a ae of radium D, radium H, and radium F. Meyer and 7. Schweidler f separated the different constituents of radio- lead by means of electrolysis or by depositing on different metals. Various reagents were used with a view of separating the radioactive products from each other and from radium. It was found that by precipitation with barium sulphate, radium and radium D could be separated from the radium EH and polonium in an old solution of radium. The following method was adopted. A certain amount of an old solution of radium was boiled down on a watch-glass to drive off the emanation. The activity of the watch-glass was measured when it had reached a minimum value and found to be 19-5 divs. per minute. This is the measure of the combined activity of radium and polonium left on the glass. Equal portions of the radium solution were taken. Barium salt in varying quantities was added and precipitated as sulphate. The activity remaining in the various portions of the solution was found to be less and less with increasing quantities of barium until a minimum value of 1°35 or 14 was reached, after which the larger addition of barium in the remaining portions was found to have no further effect. The exper iment was repeated on a larger scale. Where enough barium was added the a-ray activity of the solution measured over some months was found to decay according to an exponential law with a period of about 134 days, which is about the period of polonium. If insufficient barium was added to entrain all the radium the activity of the remaining solution was found to increase in the first few days. The fact that the quantity of polonium does not diminish with increase of barium salt shows that there is no noticeable entrainment of the polonium by barium sulphate. To test this point more definitely two portions of an acid solution of the active deposit of radium were taken. One portion was evaporated to dryness on a watch-glass, the activity on which was then found to be 21 divs. per minute. In the other portion barium sulphate was precipitated, and, the precipitate having been filtered off, the polonium was likewise evaporated * Phil. Mag, vill. p. 290 (1905). tf doe. Cte. ae and its Products of Transformation. 839 to dryness. The activity of this was found to be 19 divs. per minute. In the circumstances of such an experiment this must be regarded as a very satisfactory agreement, especially considering that in the second case the film is always thicker than in the first, thus cutting off a little e-ray activity. In another similar experiment the activities were respectively 4°5 and 4:2 divs. per minute, a loss of about 7d per cent. Similar experiments showed that the @-ray activity was almost completely separated with polonium. These experi- ments were repeated on a larger scale with different quantities of barium salt. In some cases, when small quantities of barium salt were used, the separation was incomplete (see curve No.1, fig.2). When the quantity of barium used was sufficient the @-ray product separated in this way was found to decay exponentially with a period of 5 days. The barium sulphate precipitated was at first inactive, but a 8-ray product was found to grow in it having the period of radium E, as shown previously. The barium sulphate was always preci- pitated in warm dilute solution in presence of hydrochloric acid, and the precipitate was filtered immediately afterwards through a special filter and washed with dilute hydrochloric acid. Thus radium D can be separated completely from radium E and polonium, and in the case of radium solution, radium and radium D are precipitated and radium HE and polonium remain in solution. Summary. (1) Radium E hasa period of 5:0 days and follows directly after radium D without any intermediate product. (2) The absorption curve for the 8 rays from radium E is exponential with a coefficient of 43 em.~} (3) The time period of radium D is about 16°5 years. (4) Polonium and radium E can be completely separated from radium and radium D. In conclusion I wish to thank Professor Rutherford for suggesting this research. Boe ee LXXXVII. On the Radius of the Sphere of Action of a Molecule. By R. D. Kiseman, D.Sc., B.A., Mackinnon Student of the Loyal Society; Emmanuel College, Cam- bridge *. FYNHE writer ft has shown that two molecules a given distance apart attract one another, apart from their gravitational attraction, with a force proportional to the product of the two sums of the square roots of the atomic weights of the atoms of the molecules. This attraction gives rise to the surtace tension of liquids, and the work done against it in evaporating a liquid is the internal heat of evaporation ; there is also evidence that it is the attraction which produces chemical combination. It was also shown that for distances of separation of the order of those of the molecules in a liquid the force of attraction between two molecules varies approximately inversely as the fifth power of their distances of separation. Physicists usually use the term “radius of the sphere of action” in connexion with the field of force surrounding a molecule. The attractive force, however, in reality does probably not end abruptly at a certain distance from the molecule ; but there is a distance corresponding to which the energy necessary to move another atom to infinity is small in comparison with the energy expended when the atoms are initially in contact; and in this seuse we will use the term in this paper. If we take the attraction between two molecules as in- versely proportional to the fifth power of their distance of separation, the energy expended in separating them by an in- finite distance when they are initially separated by 10—° cm., GG Jf 10-40" And when they are initially separated by a distance of 5x 10-8 em., which is of the same order as the diameter the average diameter of a molecule, is proportional] to 1 La of a molecule, the energy expended is 695 x 10-8 which is a small fraction of the foregoing value. This shows that the radius of the sphere of action of a molecule is of the same order of magnitude as its diameter. The object of this paper is to give a definite expression for the radius of sphere of action in terms of other quantities. It will be seen that further properties of this quantity are brought to light by means of this expression. * Communicated by the Author. t Phil. Mag, May 1910, p, 783, Radius of the Sphere of Action of a Molecule. 841 Tt will be necessary first to obtain an expression for the surface tension of a liquid by means of a method due to Laplace. The surface tension will be expressed in terms of the work done in separating beyond the range of molecular forces the two parts A and B of a slab of liquid into which it is cut by a given plane. Suppose the part B divided by planes parallel to the interface into thin layers whose thickness is dz. Let the attraction on unit area of a layer by the slab A be denoted by $(#).dz, where 2 is the perpen- dicular distance of the layer trom the surface of A. The work done in moving unit area of the layer outside the attraction of A is therefore dz ( Cy EASES if w= ( ‘b(x)de, where ¢, is the diameter of the sphere of action of a molecule. Hence the work required to move the whole of the liquid standing on unit area of the surface of the slab A outside the attraction of the slab is Cy w.d2. e 0 Integrating by parts we have g & My | Cy dw cw baad Wher dh ts oil a e/ The term within the brackets vanishes at both limits, and dw dz ash — f(z) ’ hence the work done in producing two new units of area of surface is Sy ( 2h(z) . dz. <- (0 Therefore te} AA ae | zo(:)dz, 9 where 2 is-the surface tension. Differentiating this equation with respect to ¢; we obtain > UN uae ee Gia hee Uist sai ee Phil. Mag. 8. 6. Vol. 19. No. 114. June 1910. 3 I 842 Dr. R. D. Kleeman on the Pa is of Stefan * has shown that the internal molecular latent heat of evaporation is equal to twice the work done in moving a molecule from the interior of the liquid to the surface. Let us obtain, reasoning on the same lines, an expression for the internal latent heat per unit volume, supposing the internal energy is expended only in overcoming the attraction between the molecules which produces surface tension. The energy expended in moving a layer of liquid of unit area and thick- ness dz from the surface of the liquid out of the range of the molecular forces is a) “A (Qe. 0 We will suppose the layer taken so thin that the energy necessary to further separate the molecules till they are out of each other’s sphere of action is small in comparison with the energy expended in removing the layer. Next let us calculate the work done in moving a layer of liquid of unit area and thickness dz from the depth c, to the surface of the liquid. A moment’s consideration will show that the alge- braical sum of the work done against the attraction of the surface layer of thickness c, 1s zero. The work done in bringing the layer to the surface is obviously therefore ae “.2) Paw. 0 which is the same expression as the above. The sum of these two expressions is the latent heat of a volume of liquid equal to dz, or ies 24" h(x). dex, where L is the latent heat of unit volume. Differentiating this equation with respect to c; we obtain dl dey == 2(¢;). . e ° . e . (2) Eliminating (c,) from equations (1) and (2) we obtain CN C= +71: {f L, denote the latent heat of unit mass Ljp=L, and dL=dl,.p+L,.dp, and the equation may be written a), dl, iagiahr Rae a cy * Wied. Ann, xxix. p. 665 (1886). the Sphere of Action of a Molecule. 843 The above investigation applies strictly only to the case when the matter does not consist of molecules, but is evenly distributed in space. But still the result obtained should very approximately represent the facts. The above equation may also be written a= a ee) where + denotes the temperature, T T in which form it can be used to determine c,; from experi- mental data. The values of c¢ have been calculated for different temperatures for a number of liquids and are contained in Table I. The values of L, used are taken from Tapia of. Ether. Methyl] formate. | | a AN ai ea Te ax | ULyp) | 4 a BAR eh ued d(L,o) ig) dr \42xX107 9dr | Oy alae de \SBSE00) Tde |) dey | ae a) ee S| on ae Ae 2 pee 313 |-112 276 | 3:88x10 em. || 803-154, 436 "3:02 1078 em. | 323 |-110 265 [sao ',, | 313 |-151 480 S00: | 333 |-110 255 | 412 ,, | 823 147 471 ry ae | 343 108 261 | 3:94, 333 |-144 | -480 | ang | mee ere I | | | Carbon tetrachloride. Benzene. | oT | PTS CN Ay hy bane ea 363 |-112 | 220 148 x10 em. spas -237 | 452x107 em. | 373 |-107 | 299 460 ,, 363 | 113 | 239 4:52, | 383 |:109_ 224 | 464 ,, | 373-114 | 243 ao ae 393 |'105 "225 444 ~~, || 383 |-114 | “249 436 ,, 403 |-103 2.25 | 436, | 393/114 | 255 | 424 |, | | Ni pe Chlorobenzene. | Kthyl acetate. 423 |-103 299 | 440x107 ° om. || 358-118 250 4°51X10 em. 433 |-101 221 ee | 363 |-116 260 425 | 443 |-099 223 Ne ee | 373 |-114 OT =e ee 453 |-097 224 oO ee 383 111 ‘287 S60 14 463 |:097 227 | AAS. Oh 393 |-109 -300 346, Pe it 844 Dr. R. D. Kleeman on the Radius of a paper by Mills * who has calculated the internal heat of evaporation at different temperatures for a number of liquids by means of Clapeyron’s equation. The values of Lip were formed for temperature intervals of 10 degrees, and the differences plotted, and a smooth ‘curve drawn through the points obtained. From these curves the values of — for different temperatures were obtained, being given by the ordinates corresponding to the given temperature abscisse. In a similar way the values of were obtained, the values of X used being taken from the tables given by Ramsay and Shields Tt. In the deduction of the above equation the effect of the vapour surrounding the liquid is neglected, and it therefore applies only to a liquid whose density is large in comparison with that of the surrounding vapour. The calculations have therefore not been carried out up to the critical point. The table gives the diameters of the spheres of action of a number of molecules, and the radii are there- fore obtained by halving these values. It will be seen that the diameter of the sphere of action is of the same order of magnitude as the usually accepted diameter of a molecule. It is very nearly of the same magnitude as the distance of separation of the molecules in a liquid: this for example in the case of ether at 311°7 C. is equal to 4-24 10-8 cm., if the mass of a hydrogen atom is taken ~ equal to 7°1x10-% gram. The effects produced by these intermolecular forces can therefore be of considerable mag- nitude only when the molecules are at a distance from one another of the same order as their diameters. There is a slight but unmistakable decrease of the radius of the sphere of action with the temperature. It is impossible to say whether this is due to a decrease of the force of attrac- tion of the atoms with rise of temperature, or to an alteration in the configuration of the atoms in the molecules. The observed decrease of the heat of formation of a molecule with increase of temperature, however, makes it appear probable that the force of attraction decreases slightly with increase of temperature. ; The radius of the sphere of action does not vary very much with the nature of the molecule. It would be the same for all kinds of molecules if the function which expresses the force of attraction between two molecules in terms of their distance of separation is the same for all molecules multiplied * Jour. of Phys. Chem. vol. viii. p. 405 (1904). * Phil. Trans, of the Roy. Society, A. vol. clxxxiv. p. 647 (1893). a the Sphere of Action of a Molecule. 845 by a constant which depends on the nature of the molecule. This, however, is probably true for atoms only, and the slight . . C . deviations of the values of - from constancy in the case of molecules are due to the fact that the centres of the atoms of a molecule do not lie ona point. The change in the law of force introduced in this way should, however, be small. _ The formula for the diameter of the sphere of action can be transformed into a form which brings out some further properties of this quantity. It can be shown that the internal latent heats of liquids at corresponding states are each the sume fraction of the corresponding latent heat at the absolute zero. The writer * has shown that the latent heat L is con- nected with other quantities by the equation pt i. L=B 27 (s vin) ; mi where p is the density and B a constant which is the same for all liquids at corresponding states, and S/m is the sum of the square roots of the atomic weights of the atoms in a molecule. At the absolute zero we have p) 3 2 a’! % 3 2 Ee * 0 PRR OEE 2G Pe La aoe b= EB, mis (Sv = Bo mi (vin) . since we may write py=ap- where a is a constant. We may also write p=«p, where «is the same at corresponding states, and we have 43,88 2 9 a ie oa L,=B “fe (s vin) mils which by the help of the above equation becomes 2 4/3 Lig = Lo pase 129 By? atls or L,= Ly where 7 is the same for all liquids at corresponding states. It is also shown in the paper cited that the surface tension at corresponding temperatures is the same fraction of the surface tension at the absolute zero, or N=pXo, where pu is the same for all liquids at corresponding states. We have then ee eb Be ak die! ze dL Lope a(n) . . . ly Thus if c; is independent of the temperature, Tina) must be a * Phil. Mag. p. 507, Oct. 1909, and Joc. cit. 846 Radius of the Sphere of Action of a Molecule. a constant ; and if c; varies with the temperature the value of <= is the same for all liquids at corresponding states, or the diameter of the sphere of action of a molecule is at corre- sponding states the same fraction of the diameter at the abso- lute zero. We have shown * that the law of attraction between two molecules is given by a function of the general form Plz, 4, BEV my, SV me, where a=p/p., 8=7/T,, and z is the distance of separation of the molecules. It foilows from this that the factor which expresses the variation of the diameter of the sphere of action with temperature must have the same value for all liquids at corresponding states, which agrees with the above result. We have seen that if the function which expresses the variation of the force of attraction between two molecules with their distance of separation is the same for all molecules, the radius of the sphere of action must be of the same magnitude, apart from the effect of temperature, for all kinds nN of molecules. In that case —2 should have the same value for all liquids, and ~ should nave the same value at corre- sponding states since the temperature factor is then the same. Now this is approximately realized. The values of ~ for a number of liquids at corresponding temperatures are given in Table Il. It will be seen that they are approximately TasBLE IT. | Name of liquid Tey are | ; j= \— oe | ie cs tice eed oe | 311:7| 14:19] -6907| 75°44] -272 | Methyl formate .....s:ceceseeeee | 324-7 | 19-80] 9283) 92°85] -230 | bapenvenn?, Pe 0. YS aN 3743 | 17-87 | 7826) 81-73) 279 | | Carbon tetrachloride ..... ........... 371°0| 16°69 |1°4385 | 39°87 | -291 | Gitcedhenzene oc cecil 4220 1778! 9611) 65:88 | 281 aphy tere ater ah) ei | 348°3] 17:01] 8305 7861 260 ) 1 constant. The slight deviations from constancy are probably due, as has already been remarked, to the fact that the centres of the atoms of a molecule are not concentrated at a point. Cambridge, March 10, 1910. * Locs-eit. By Sai LXXXVIII. The Constant of Uranium X. By FREDERICK Soppy, .A., and ALEXANDER S. RussE, .A., B.Sce., 1851 Exhibition Scholar, late Carnegie Research Scholar *. HE value of the coefficient of decay (A) of the @-rays of uranium X was originally found to be 0:031 (day)-! f. This result gives a period of half-change of about 22 days. Subsequent investigators of this body have not made special measurements of its decay, but have simply stated that such measurements as have been made point to a similar result. Moore and Schlundt t, however, examined over a period of about 50 days, a number of preparations of uranium X separated in various ways from the parent body, and found that the rays decayed to half-value in 21 or 22 days. With the intensely active preparations of uranium X, separated by us from 45 kilograms of uranyl nitrate, we have been able to examine the decay of both @- and y-rays, over periods of time up to 230 days in the case cf the latter, and for con- siderably longer in the case of the former. In 230 days the activity decays to 0°15 per cent. of the initial value. The investigation proves that the @- and y-rays decay at exactly the same rate, the value of the constant » being 0°0282 (day)~?. The period of average life of the body is therefore 35°5 days and its period of half-change 24°6 days. This value of the constant A is about 10 per cent. less than the value obtained originally. Ten preparations of uranium X, prepared in different ways, at different times, and of intensity varying over a wide range, have been examined. The @-ray measurements were carried out in a brass electroscope having a zine base 0°010 em. in thickness. The preparations were placed generally either at a distance of 115 cm. or of 13°5 em., covered with thicknesses of sheet zinc or not, according to their intensity. By altering the distance of the preparation from the electroscope, and the amount of zinc covering the active body, measurements could be made with one preparation over considerable periods of time, and yet with the rate of leak of the electroscope always within accurately measurable limits. The y-ray measurements were carried out in a thick lead electroscope with base 0°975 cm. thick. The distances from the pre- paration to the base of the electroscope varied from 10 to 2cm. ‘Two standards were employed in each of the two sets of measurements, so that, in the event of an accident to one of them, the measurements of decay might still be continued. * Communicated by the Authors. + Rutherford and Soddy, Phil. Mag. [6) y. p. 444 (1908), t Phil, Mag. |6j xii. p. 393 (1906). 848 Messrs. F. Soddy and A. S. Russell on The results obtained with the various preparations are summarized in Tables I. and II. TasLe I.—Decay of B-Rays. . Days under oval No. Preparation. Cover. AAA ricky A(day) . 1 ...| B (1st Separation, Dec. 1908)} _...... 0 to 286 | 0:0283 2 ...| A (4th Separation, Sept. 1909) | 0028 em. Zn |’ Oto 129} 0:0289 ...| B (4th Separation, y ) bare 0 to 58 0°0281 .| C (4th Separation, bare Oto 140} 0°0281 99 | D (4th Separation, _,, 0:028 em. Zn| Oto 140] 00283 3 es ) — ) 6 iy is Pia aaa eee 0-010 em. Zn | 79 to 220| 0:0282 ih Ditto., distance 13:5 cm. 0:010 em. Zn | 220 to 287 | 0:0282 8 9 A (4th Separation) distance 115 cm. Ditto., distance 13°5 em. 0023 cm. Pb| 132 to 193 | 00281 BCD (4th Separation) } Composition results f | oo" bare 0 to 1382 | 0:0287 0to 192} 00281 Fig. 1. DAYS 7 25 20 As 100 125 150 175 200 Nos. 6 and 7; 8 and 9; and 10 are the principal results in the above Table. Curves I., II. and III. of fig. 1 corre- spond to them respectively. In No. 10, B, OC, D were care- fully prepared with activity in the ratio of about 1:10: 100, by evaporating down aliquot portions of the same solution. 0 the Constant of Uranium X. 849 All three preparations (D covered with zinc) were measured from the start, so that the ratio of one to any other was accurately known. When B had decayed too much C was continued in its place, then D covered with zinc, and finally D bare. From these results Curve III. (fig. 1) was obtained. The separate results are shown in Nos. 3, 4, 5 of Table I. Of the preparation B of first separation (No. 1) only two measurements were taken, one just after preparation, and the other after the lapse of 236 days, so that this value of X is not of so much importance as the other values. The mean value may be taken as 0°0282, and the maximum departure in any single set of observations is less than 3 per cent. TaBLE I].—Decay of y-Rays. . Days under -1 | No. Preparation. rie Aces A (day) . | Dells she A (3rd Separation) | O to 69:5 00281 Brae vee.’ B (ditto) 0 to 69°5 0:0280 0 SS C (ditto) 0 to 69°5 0:0280 BEE CaN Ns ABC (ditto) together | 69°5 to 226 0:0281 Bs auc wa A (4th Separation) 0 to 143 00284 Fig. 2. 22 i 20 IS, 10 U 0S t OU Sa TS T3 Days — ~ 0 30 ee) 90 120 150 80 210 240 Nos. 1 to 4, and 5 are plotted in Curves I. and II. of fig. 2, The mean value of \ (day)~! obtained for the y-rays is 0°0282, §50 On the Constant of Uranium X. from 0 to 226 days, a value identical with that found for the -rays. The rate of decay of the @-rays of preparations A, B, C, and D of the fourth separation was higher than the normal value over the first ten days after preparation, the value of X (day)? varying from 0:033 to 0:029. No such initial abnormal decay was shown in the 4-ray measurements conducted at the same time on these preparations. From about the tenth day onwards the decay was quite normal. Such a result had been sought for, as it has been shown by one of us * that the apparent value of the radioactive constant of uranium X (0°031 day)—! might be composed, assuming a double disintegration of uranium X into two bodies in the proportion of 7 to 1, of two separate values, one of 0°027, and another of 0°004. The abnormalities were not confirmed, however, by a special experiment in which two different, newly prepared uranium X preparations were compared with another, known to be decaying normally, care being taken to get ‘all. the preparations into the same physical state. Measurements at intervals of 2 days for 16 days showed that all three preparations decayed at the normal rate (A=0°028 (day)—!). The abnormalities in the decay may be due to the fact that the preparations had not come immediately into equilibrium with the surrounding atmosphere with regard to their physical state. Thus a preparation which had been heated strongly in the course of mounting and which therefore was very dry on the first day of measure- ment, would tend to give diminishing values of 2, until it had assumed its state of physical equilibrium. This point will be further examined. The electroscope used for @-rays was tested to see whether two radioactive preparations gave together the sum of the leaks produced by each separately.. If A and B denote respectively the intensities of two radioactive preparations, the activity of which could be measured either separately or together, it was found that for the electroscope used, A and B separately gave a greater effect than A+B together, if A+B was greater than 215 divisions per minute, a smaller effect if ALB lay between 215 and 115, and exactly the same effect if A-+B varied from 115 down to the smallest leak tried (1 division of the scale =0:023 mm.). To avoid disturbances due to secondary rays from the radioactive preparations themselves, w hen both were used together, one of the sources was a y-ray preparation placed in a definite position at the side of the instrument, the other a @-ray preparation under the base. Between 115 and 215 the * Soddy, Phil. Mag. [6] xviii. p. 742 (1909). Pendulum swinging through an Arc of Finite Magnitude. 851 difference between A+B together and A and B separately varied only from 2 to 1 ‘per cent. Such a small difference could not of itself have produced the initial abnormal decay. Besides, two of the preparations whose intensities lay wholly within the strictly additive portion of the range showed the abnormal effect. Both the @- and the y-rays decay to zero. There is now no detectable residual B- or y-radiation in the first active set of preparations separated in November, 1908. The @-rays of the main preparation of Feb. 1909 ieee now (April 1909) just passed beyond the range of measurement, the decay, so far as could be seen, having been quite normal right up to the end. In this period of 14 months, the radiation decays i in the ratio of 100,000 tol. All samples of crystals of uranyl nitrate of third, fourth, and fifth separation tested against an equal weight of standard untouched crystals have regained their equilibrium content of uranium X quite normally. There is not the slightest evidence that in the numerous chemical operations to which the crystals have been at various times subjected, any concentration or impoverishment of a body, intermediate between uranium and uranium X, and acting as the direct parent of the latter, has taken place * Physical Chemistry Laboratory, University of Glasgow. \ —SS seks 7 LXXXIX. On the Motion of a Pendulimn swinging through an Arc of Finite Magnitude. By J. RosE-Innss, 1/.A., B. ‘Se.t ie the following paper it is proposed to investigate by comparativ ely simple methods the motion of a pendulum when the swing is not treated as infinitesimal. Fu The Elliptic Integral of the First Kind. The Jacobian theory of elliptic functions is developed primarily from an examination of the properties of the integral i da AL cee In the present section a new geometrical interpretation will ~* M. J. Danne has described the separation of such a body (Le Radium, 1909, vi. p. 42), but it is desirable to await further confirmation of his results betore basing any conclusion upon them. tT Communicated by Prof. F. T. Trouton, F.R.S, 892. Mr. J. Rose-Innes on the Motion of a Pendulum be offered of this integral, which, it is believed, is rather sumpler than any hitherto proposed. We may recall the well-known connexion between the independent variable wv of the above integral and the radical occurring in the denominator of the subject of integration, viz. :—that /1—x? sin? 2 represents the cosine of a side of a spherical triangle opposite to an angle of value a, provided that the constant ratio of the sine of a side to the sine of the opposite angle is «. We may conveniently introduce a simplification by assuming that one of the angles of the triangle is a right angle. Consider then a spherical triangle ABC of which the angle at C is a right angle ; and let sin AB be denoted by x. CG A a Then sin BC my, Spey. Dat and cos BC = 1 —k? sin? A. The fundamental elliptic integral may be therefore written ( O aaN “}, eos UB: But since C is a right angle we have cos AC x cos BC = cos AB = V1—«’, so that the integral may be also written jezee dA e¢ V1—£# ; Let AC’ be a neighbouring position of the arm AC, and produce AC and AC’ to meet the circle whose pole is A in the points D and D’ respectively. Let the radius of the sphere on which the triangles are drawn be denoted by R. “y swinging through an Are of Finite Magnitude. 853 Then by projecting the figure onto the circumscribing cylinder whose circle of contact is the polar of the point A, lod dD A after the manner of Archimedes, we find that the area of the strip CDD’C’ is equal to R* cos AC dA. If we take Aus Pt eee the area of the strip can be written eos ACdA Vi—e? R? and thus it appears that the integral will be equal to the area outside the locus of C and inside the polar of the point A, when the angle A proceeds from an initial value zero to a final value ¢. We next proceed to find the locus of C. 4 Ney Produce the side AC to meet the polar circle of A, and let the point of intersection be called D. In like manner produce 854 Mr. J. Rose-Innes on the Motion of a Pendulum the side BC to meet the polar circle of B, and let the point of intersection be called E. Then since cos AC X cos BC=cos AB, a constant, we have also sin CD x sin CH =a constant. This is the well-known property of a sphero-conic with regard to its two cyclic arcs. (See Salmon’s ‘Solid Geometry.) Hence we see that the required locus is a sphero-conic which passes through the poles of its two cyclic ares. Relations between the Elliptic Functions of any argument u and those of K—u. Let ABC as before be a right-angled triangle with the right angle at C. Produce the sides AB and AC until they meet the polar circle of A in the points D and E respectively; produce the sides BA and BC until they meet the polar circle oa A 2 of B in the points Ff and G respectively. Let the oval curve denote the locus of C, and let the intersection of the two polar circles which lies on the same side of AB as © be denoted by H. We see that the area of the quadrilateral CHHG remains constant as C moves, since each of the four angles contained in it remains separately constant. Subtract this quadrilateral and half the sphero-conic from the triangle HED; the remainder consists of two curvilinear quadrilaterals, whose sum is therefore proved independent of the position of C. By taking the particular case when A is a right angle and B is zero, we see that the value of this constant sum is equal to swinging through an Are of Finite Magnitude. 895 the quantity usually denoted by K. Let the elliptic integral corresponding to BAU be denoted by u, then the elliptic integral corresponding to ABC may be denoted by K—u. Thus angle BAC = am wu, angle ABC = am(K—u). The formuls concerning the angles and sides of the right- angled triangle ABC can now be translated into the language of elliptic functions. For instance ' h tan BAC x tan ABC = ae But BAC = amu, ABC = am(K—vwz) eos AB= V1—#, 1 so that tan utan (K—u)= Fr 9 ae: —K Other formule may be obtained in a similar manner. Theorem in Spherical Trigonometry. We now proceed to prove a theorem in Spherical Trigono- metry which will be found useful hereafter in studying the Addition-theorem. | If on the arc AB of a great circle there be erected two é c 0 F A triangles ACB, ADB, having the angles at C and D each equal to a right angle, and if perpendiculars AE and BF be Jet fall from A and B respectively onto the great circle passing through the vertices C and D, then (.) CF =DE, Gi.) CAB = DAR, Gii.) fee nov) — ete BY) sin AB ¢ 856 Mr. J. Rose-Innes on the Motion of a Pendulum Part I.—We have cos AB = cos CA cos CB = cos CE cos EA cos CE cos FB. Similarly cos AB = cos DE cos EA cos DF cos FB. Equating these expressions for cos AB, and cancelling out common factors, we obtain cos CE cos CF = cos DE cos DF. Hence cos (DE —CD) cos (CE CD) cos DE cos CF cos CD +tan DE sin CD = cosCD + tan CF sin CD, so that tan DE=tan CF which gives the desired result. Part I1.—We have cosDAB _tanADtanAB _ tanAD eos CAB’ tan AB tan WC * “fax: Also cosCAE _tanAHtanAD _ tan AD cosDAK tanACtan AH tan AC’ Therefore cos DAB __ ©08 CAE eos CAB cos DAE’ cos (CAB—CAD) _ cos(DAH—CAD) cos CAB cos DAE A cos CAD + tan CAB sin CAD =cos CAD + tan DAE sin CAD, so that tan CAB=tan DAH, which shows that the angles CAB and DAE are equal. In the same manner we may show that the angle DBA is equal to the angle CBF. Part III. We have : sin AH | ae sin sin AD aaa wat ig _ sin BF cos ADE = sin DAE cos AE = sin CAB cos AH, by Part IL., cos BUF = sin CBF cos BF = sin DBA cos BF, by Part I. q swinging through an Are of Finite Magnitude. 857 Hence sin(ADE+BCF) = sin ADE cos BCE + sin BCE cos ADE = ee sin AK cos BE + ee 7 sin BF cos Ah __ sin AE cos BF +sin BF cos AE fi sin AB en (AH + BP) fad aris alee The Addition-theorem. In order to prove the Addition-theorem, we may employ Legendre’s investigation on spherical triangles, adapting it to our present system of interpretation. Suppose that AB is the minor axis of a sphero-conic of the nature already discussed, Let C and D be two points on the sphero-conic lying on the same side of AB ; join CD, CA, CB, DA, DB. Drop the perpendiculars AK and BF on CD produced, and bisect CD in the point G. Draw the polar circles of A and B; let them intersect AB produced in H and K respectively, and let L denote that point of intersection of the two circles which lies on the same side of AB as Cand Ddo. Produce AD, BC to meet LH, LK in M and N respectively. L Suppose next that the great circle joining C to D suffers a slight displacement, in such a way that the area of the segment cut off from the sphero-conic remains unaltered. Pal. Mag. S. 6. Vol. 19. No. 114. June 1910. 3K $58 Mr. J. Rose-Innes on the Motion of a Pendulum Then the intersection of CD with its neighbouring position will be ultimately the point G, halfway between C and D. By means of the theorem in Spherical Trigonometry, Part L., we deduce that CF=DE, and since GC=GD we must have GF=GH. It follows that the small change in the value of the perpendicular BF owing to the displacement of the are CD tust be equal and opposite to the small change in the value of the perpendicular AE; so that the sum AE+BF will remain constant during the displacement. By means of the theorem in Spherical Trigonometry, Part III. we deduce that the sum ADE + BCF will also remain constant during the displacement. We can easily show that the spherical excess of the pentagon LMDCN is equal to KLH—(ADC+ BCD) ; hence, when the are CD moves as described above, the area of the pentagon will remain unaltered. From the triangle LKH subtract half the sphero-conic, add the segment of the sphero-conic cut off by CD, and subtract the pentagon LMDCN; the remainder consists of two elliptic areas AKNC and BHM, whose sum is therefore proved constant when CD moves under the prescribed conditions, Let wand v denote the elliptic integrals determined by the angles ABO and BAD respectively, so that am u = ABC = FBD, am = BAD Cn) Suppose that C and D move clockwise along the sphero- conic, subject to the condition already mentioned. Then, by letting D approach indefinitely close to B, we see that the elliptic area BHMD and the angle BCD can he made simultaneously as small as we please; we obtain in the limit am (u+v) = ADC+BCD. Take the two triangles ACE and BDF, and place them alongside each other so that the side DF coincides with the side CH, D coinciding with OC, and F coinciding with E. This is possible, since the side DEF can be proved equal to the side CE by the theorem in Spherical Trigonometry, Part I. Also, since the angles AEC and BED are both right angles, the sides AE and BF in the new positions would be portions of the same great circle. Hence the two triangles so placed would make up one triangle, whose angles would -be respectively equal to amw, amv, and swinging through an Are of Finite Magnitude. 809 a—am(u+v). We may call the triangle so formed Legendre’s triangle, because it is really the triangle in- vestigated by Legendre, though obtained in a manner different from his. By applying the ordinary formule o£ Spherical Trigonometry in Legendre’s fashion to this triangle we easily reach the addition-theorem ; the algebraic details are well known, and need not be repeated here. The two Eccentric Circles of Jacobi. We shall now bring the above investigation into con- nexion with the geometry invented by Jacobi to illustrate the addition-theorem. Let AB be the minor axis of a sphero-conic of the nature already described ; take the tangent-plane to the sphere at the point A, and project the spherical figure from the centre of the sphere onto this tangent plane. The point A is evidently unaltered by the projection ; the projection of B may be denoted by B’, and the are AB will then project into the straight line AB’. The sphero-conie will project into a circle on AB’ as diameter, since the tangent-plane at A is parallel to one of the cyclic ares of the sphero-conic. Tbe other cyclic are of the sphero-cunie will project into a straight line lying outside the circle; let us denote this straight line by HK. By symmetry, HK will cut B’A produced at right angles; and if G denote the point of intersection, we shall have G closer to A than to B’, The straight line B’G subtends a right angle-at the centre of the sphere. aK 2 860 Mr. J. Rose-Innes on the Motion of a Pendulum In our previous investigation we considered an arc of a great circle which moved about so as to cut off a segment of constant area from the sphero-conic. It is well known that such an arc will always touch an inner sphero-conic concyclic with the original one. If we project the smaller sphero-conic as above, we shall obtain a ‘circle which lies within the circle having AB! as diameter ; by symmetry we shall have the centre of this inner circle somewhere on AB!. Let the points in which the inner circle cuts the straight line AB’ be denoted by 0 B’ TG nile as satiate G L and M, and consider the plane figure obtained as the result of cutting the solid figure by the plane through the line GB’ and the centre of the sphere. Let us denote the centre of the sphere by O; and let OC be drawn bisecting the angle AOB’ and meeting the straight line AB’ in C. Then we have the following relations among the angles of the figure :— COB’ = COA, CB’/O = AOG. But OCG = COB’+CB'0, COG = COA+AO0G ; therefore OCG = COG. Because OC is the bisector of the angle AOB’, it must also bisect the angle LOM ; hence MOC = LOC. Again, OLG = OCG+ LOC, GOM = GOC+ MOC ; therefore OLG = GOM. swinging through an Are of Finite Magnitude. 861 It follows that the triangles LOG, MOG are equiangular, since they have the angle LGO common, and the angles OLG, GOM are equal. Hence, by Euc. vi. 4, LG. GO = GO-GM ; so that LG.GM = GO? The Jength of the tangent from G onto the inner circle is thus seen to be equal to GO. It follows that the system of concyclic sphero-conics will project into a system of eccentric circles, all having the same length of tangent from the point G; hence the system of circles has a common radical axis, viz. the straight line HK. The introduction of the rest of the Jacobian geometry is fairly obvious. Application to Pendulum-motion. We may now apply the foregoing results to investigate the motion of a pendulum. As in the last Section, let us project the sphero-conic onto a plane touching the sphere at A, one of the extremities of the minor axis. Let the tangent-plane be vertical, and let the straight line HK of the last Section be horizontal, lying above the circle formed by the projection of the sphero-conic. Let us next suppose that the circle is materialized in the form of a fine wire, and let us consider the motion of a frictionless bead moving on the wire under the action of gravity. Suppose the bead sent off from the lowest point of the circle with a velocity equal to that required to reach the level of the straight line HK ; the bead will arrive at A, the top of the circle, without losing all its velocity, and will therefore perform a series of complete revolutions. From elementary Dynamics we know that the kinetic energy of the bead at any point of its path varies as its perpendicular distance from the straight line HK. Let us eall this distance 7; and let @ denote the angle between B'A and the line joining the point A to the bead. Then f = BIG—B'A sin’ ¢. = "G (1 ~ oe sin” 6) 862 Pendulum swinging through an Arc of Finite Magmtude. Turning to the second figure of last Section, we see that BOs A ees ue BIG = BO (Eue. vi. 8) therefore so that f = B’G(1—«’ sin’ ¢). From the above we may deduce that the time occupied by the bead in passing from the bottom of the circle to any other position varies as We see from this result that the time is proportional to the area outside the origina! sphero-conic. The motion of the bead may therefore be described as follows :—Let the circle and the straight line joining the bead to the point A be pro- jected centrally onto the sphere. Let the are which arises from the projection of the radius vector be produced so as to meet the polar cirele of A. Then the portion of the great circle lying outside the sphero-conic will describe equal areas im equal times. | The theorem at the end of the second Section may be interpreted as follows :—Suppose two beads are making complete revolutions round the circle in such a way that when one is at the top the other is at the bottom, and tice versa ; then, if the straight lines joining the beads to the point A make angles ¢ and @¢’ respectively with B’A, we shall have i I tan d@ tan } re Prof. Greenhill has shown that the addition-theorem can be thrown into the following form :—If two beads are sent off one after the other from the lowest point of the circle with the same velocity, so as to make complete revolutions, then the straight line joining the two beads will always touch a circle lying within the original circle. The connexion between the motion of a bead making complete revolutions, as considered above, and the motion of a pendulum swinging through a finite are is well known, and need not be given here. Pr Se ae) ] * XC. An Automatic Toepler Pump, designed to collect the gas from the apparatus being exhausted. By Bertram D. STEELE, J).Sc.* URING the investigation of certain dissociating com- F pounds containing ‘sulphur dioxide, it became necessary to exhaust constantly certain apparatus fora prolonged period. After several days had been spent in manipulating a Toepler pump of the usual pattern, an automatic Toepier pump was designed and constructed. The pump that was required was one which not only should be capable of being worked automatically for long periods, bat also one by means of which it should be possible to deliver samples of the extracted gas when desired, and if necessary to collect the whole of such gas for examination. These requirements are satisfactorily fulfilled by the pump which is described in the following paper. It will be best described in three sections, ail of which are shown diagram- matically in fig. 1 (p. 864). The pump proper is an ordinary Toepler pump, and is represented in the figure by the parts A and B and various adjuncts. The collecting apparatus is represented by the parts © and D and conneetin ¢ tubes. The automatic controlling apparatus is represented by s, F, HE, and y. The pump proper consists of the stroke-cylinder A and the reservoir B, these being connected by the U-tube d. This U-tube, ehich is about 8 mm. in internal diameter, is pro- vided at its lowest point with a short side-tube, by means of which the mercury can be removed from the pump if neces- sary, and it is constricted at one point to a diameter of about 2mm, This constriction regulates the flow of mercury and renders it impossible to bre ak the apparatus by a sudden blow of the mercury, even at the highest exhaustion. The U-tube in its shorter arm is 800 mm. long, 1 in order that either A or B may be put into communication with the atmosphere whilst the other vessel is exhausted. The stroke-cylinder A is provided, in the usual manner, with a ground-glass valve, a, and a manometer, }, ana the apparatus to be exhausted is attached on the other side of the phosphoric-anhydride tube and the stopcock, c. * Communicated by the Physical Society : read March 11, 1910. 864 Dr. B. D. Steele on an Fig. 1. The reservoir B is connected with the water-pump by the tube e, and to prevent the possibility of water entering B, this tube is made 85 cm. long and is provided at its lower end with a mercury trap, z, of special design. Automatic Toepler Pump. S65 It has been found that very rapid currents of air can be sucked through this trap without a single globule of mercury being carried into the water-pump. Two other tubes are attached to the reservoir B, one of which, m, is connected with the regulating apparatus H, and the other, n, with the mercury trap, k. The collecting apparatus was designed to enable either the whole or a portion of the extracted gas to be withdrawn and collected. In this apparatus the receiving bulb C is in direct com- munication with the stroke-cylinder A by means of the millimetre capillary tube, f, and with the atmosphere by rneans of a similar tube, g, which is attached to the top of the bulb. The U-tube h connects C with the subsidiary reservoir D, and from the slight enlargement at 7 a tube, 2, is extended downwards and dips into the vessel F. The top of the bulb D is connected through the ground-glass valve and mereury trap & with the top of the reservoir B, and by the tube / with the middle of its under surface. The essential part of the automatic controlling apparatus consists of two portions, the bulb E with the test-tube p, joined together as shown by the 8-millimetre bore-tube 0, and the branched tube g, which is of the shape shown in the figure, and is constricted to a fine capillary at the point ». The longer arm of this tube reaches to the bottom of the vessel Hi, the shorter to the bottom of the tube p. The vessels s and F are arranged so that any overflow of mercury from either of them finds its way into p. The tube q is attached to m by rubber pressure tubing, which can be closed by a screw pincheock at w, and each of the tubes, / and w, are cut and the parts joined by pressure tubing, which in the same manner can be closed by screw pinchcocks. The volumes of the various vessels are approximately as follows :— The stroke-cylinder A ...... 800 cubic centimetres. The reservoir Baar, 270) 0 eer se The collector ods seh ERT ng 22 ‘: The reservoir Dyiaun a) ons ‘ The bulb Tt) geet BDU juz ‘s It is essential that the U-tube d, and the tubes e, f, and g sshouid be over 80 centimetres in length. | The relative levels of the various bulbs are as follows :— _ The bottom of the reservoir B is placed 30 centimetres below the top of the tube f, and 10 centimetres below the 866 Dr. B. D. Steele on an connexion between A and the valve a, atv. The bottom of the reservoir D is situated 16 centimetres above the highest point of the tube 4. The shorter limb of the U-tube A is 25 centimetres long, and the enlargement 2 is 75 cm. above the level of the side- delivery tube on IF. Twenty pounds of mercury are required to work a pump, of the foregoing dimensions. Filling the Pump. Before starting to fill the pump, the stopcock c and the pinchcock on / are closed, and mercury is poured into the vessels # and F, and into the traps z< and k&. The trap & is provided with a side-tube at the point y, to enable this to be done, and when sufficient mercury has been inserted this tube is sealed off. The water-pump is now connected at the trap z, and the mercury required to fill the pump is poured into EK. Under the action of the water-pump, which must be sufficiently powerful to reduce the pressure in the apparatus to 10 or 15 cm. (whilst the side-tube is open to the atmo- sphere), the mercury rises in the tube m. Air is, however, slowly leaking in through the constricted branch of g, and this air breaks the mercury in m into columns which are carried over into B. In this way the whole of the mercury required is pumped rapidly from E into B. Working the Pump. It has been already stated that the pump was designed (1) to discharge the extracted gas into the water-pump, (2) to collect samples of the gas for examination, and (3) to collect the whole of the gas from any piece of appzeratus. The manner of working of the pump will be best shown by describing in detail the course of one complete stroke when it is working as required for case (1). ) Starting trom the time when a small quantity of the mercury required for filling the pump remains in E, and when the pressure in A, B, UC, and D has been reduced to 10 or 15 centimetres, it will be seen that with the removal of the Jast globule of mercury from E into B, air can freely enter the latter vessel through the tube m. The water-pump is not capable of removing the air as quickly as it enters, and the consequent rise in pressure causes the mercury to rise in Automatic Toepler Pump. 867 A, and, driving the gas before it into the receiver c, it over- flows through the tubes h and ¢ into the vessel F, from this into p, and finally through o into the large bulb B. This overflowing mercury closes first the constricted and then the open end of the branched tube g, and thus cuts off the supply of air to the reservoir B. The water-pump now reduces the pressure in the reservoirs B and D, and as a consequence the mercury immediately begins to flow back through the U-tube d, from A into B, and the overflow into F and p ceases. About forty seconds after the closing of the tube qg the mercury will have fallen to the point v, and in the meantime ihe mereary in p has been slowly siphoning through the branched tube g into the bulb E. With the constriction used, the time required to empty p is eighty seconds, and there is therefore an interval of forty seconds between the opening of the communication at v and the emptying of the tube p. During this time the apparatus to be exhausted remains in communication with the stroke- cylinder A. Tollowing the last globule of mercury which siphons from p into E, air enters through the shorter branch of g, and the mercury in E is pumped back into the reservoir B, and finally, when all the mercary has been removed from H, air again enters freely through the tube m, and another stroke begins. The gas that accumulates in C finds its way through the tube h, the bulb D, and the trap 4, to the water- pump. Case (2). If it is desired to collect samples of the gas, this ean be done as follows :— A test-tube of the size required is filled with mercury, and inverted over the turned-up end of the tube g, the pinchcock t is closed, and at the following stroke of the pump mercury flows, by way of f, C, and h, into the bulb D. Whena quantity more than sufficient to fill the bulb C has accumulated here, the pressure in D is caused to rise by opening the pincheock on the tube /, when the gas in C is forced into the tube placed to receive it. The excess of mercury flowing into s finds its way as before into p and E. Finally, the pinchcock / is closed, that at ¢ opened, and the cycle proceeds as previously described. Case (3). Should it be desired to collect the whole of the gas from the apparatus being exhausted, tae pinchcock ¢ must be kept closed, and a vessel to receive the gas must he placed as before over the end of g. If the pressure in the apparatus is low, that is, if the quantity of gas to be collected 868 On an Automatic Toepler Pump. is very small, the gas is-displaced from C at every stroke of the pump, and the overflowi ing mercury finds its way from s, through F, into p and E. If the pr essure is more than 1 or 2 centimetres, it is neces- sary to manipulate the pinchcock on J. In the great majority of instances it is onlv during the first five or six strokes of the pump that this will be necessary, and in this case it is most convenient to open / at the end of each stroke, exactly as described in case (2). Should the vessel to be exhausted be very large, the pinch- cock ¢ must be regulated so that the mercury contained in the bulb uw flows into D at such a rate that the last drop, followed by air, enters D at the moment when the latter has become about half full of mercury. If / is properly regulated, it is found that mercury flows through it only when the tube m is open to the atmosphere, and not when the reservoir B is exhausted by the water-pump. In obtaining high exhaustion it is advisable to leave the stroke-cy ‘linder A in open communication with the apparatus to be exhausted for a longer time than 40 seconds. This object is obtained by attaching a side-tube to p, the capacity of which increases the volume of mercury which must siphon through g at every stroke. The quantity of mercury, and hence the time of flow, can be regulated by sliding a well-fitting glass piston in this subsidiary tube. The time occupied by a complete stroke of the pump that is described here is 160 seconds, during 40 of which the communication at v is open. By the use of the subsidiary tube on p (not shown in the diagram) the total time is increased to four minutes, that duriny which v is open is increased to two minutes. In order to stop the pump, all that is necessary is to close the pinchcock w when the tube m contains mercury, and then to disconnect the water-pump at < when the vessel A is empty of mercury. To start the pump afresh, the pam is attached to z and the pinchcock w is openec. Chemical Laboratories, The University of Melbourne. Pessean in| XCI. On Coherers. By W. H. Ecouns, D.Se., A.R.C.S.* T is well known that an imperfect contact between almost any pair of good electrical conductors suffers a change of resistance when electrical oscillations of sufficient intensity occur in the circuit containing the contact. Prof. Branly, in 1896, concluded after much investigation that the effects observed were due to a modification of the electrical con- ductivity of the film of dielectric that happened to separate the conductors. Tor example,a mass of metal filings melted down in resin showed appreciable alteration of resistance when a spark discharge occurred near it ; and a pair of pieces of metal in very light contact, yet separated by their surface films of condensed gas, showed the phenomenon excellently. Sir Oliver Lodge at that date held the view that the alteration of resistance, usually a diminution, was due to the piercing of the dielectric and a consequent cohesion of the metals. He gave the name “coherer” to instruments exhibiting or utilising the phenomenon, and this name has been uni- versally adopted and the suggested explanation widely accepted. It has often been pointed out that Lodge’s explanation is inapplicable to known cases where the resist- ance of the coherer is increased by the passage of electrical oscillations. In the present paper it will be shown that there is no need to adopt the “ cohesion” hypothesis, even in the case of typical coherers where the resistance is lowered by the action of electrical oscillations. Ten years ago the known ways of investigating a coherer required the device to submit itself to powerful electric discharges. We find investigators of filings coherers watching . « . , . © the sparks between the filings and boasting of fusing the co) filings into chains of quite considerable tenacity. But the engineers who were at that date nursing the infant wireless telegraphy were, on the whole, more gentle with the coherer than were the men of science. At most, the engineers required that the current through the coherer when it *“cohered” should work a delicate relay. A few years later they merely asked that the coherence current should move the diaphragm of a telephone receiver. The oscillatory voltages that need be applied to a coherer in order to produce a resistance change perceptible by an ear at the telephone are very small indeed ; and in these cases a good coherer spontaneously and very perfectly recovers its original re- sistance when the oscillations cease. The “cohesion” explanation appears improbable here, though in the cases * Communicated by the Physical Society: read March 11, 1910, 870 Dr. W. H. Eccles on Coherers. familiar ten years ago, where the alteration of resistance is permanent, that explanation seemed plausible. Within the past six years the problem of the coherer has often been attacked, and each observer has in turn found all the theories of the action of coherers insufficient to explain his experiments, and has added some suggestion of his own. For example, A. Blanc* in 1904 concluded that neither oxide nor condensed gases played a part in certain coherers, and believed that particles of the two. surfaces diffuse at a rate depending on the current, temperature, and pressure. On the other hand, Robinson ft decided that in single point coherers the resistance is that of an elastic film of oxide, and that permanent coherence implied injury or thickening of the oxide ; while Huth ¢ attributes coherence to the ionization of the film rather than to the thickening of the oxide. From the point of view of the results of some of the experiments to be described in this paper, the obser- vations of A. H. Taylor § are of special interest. He found that the current through a single point steel coherer shows three stages as the gap is diminished. In the first stage there is a feeble leakage current due, he thinks, to vapour conductivity ; in the second, conduction due to metallic ions (in this stage a small rise of H.M.F. produces a great increase of current); in the third stage ordinary conduction occurs. In a paper entitled ‘*‘ Recherches sur les Contacts Imparfaits”’? (Journ. de Physique, ser. 4, ui. 1904, p. 350), A. Fisch describes his experiments on the voltage-current curves for a coherer consisting of two spherical surfaces of steel whose separation was capable of fine adjustment. He found, from a study of numerous curves, that the average curve exhibits three stages :—(1) As the current is raised the potential difference across the contact increases more and more slowly to a critical value constituting stage (2), where, though the current increases, the potential difference remains constant. Then (3) mcreasing current is accompanied by continually rising potential difference. This curve of Fisch’s is precisely the curve II. of fig. 8, below. Besides these investigators there are others who have advocated that the electrostatic attraction between the conducting surfaces may account for the phenomena. Very few, if any, experiments with electrical oscilla- tions of very minute measured voltages have so far been * Science Abstracts, vol. viii. 1923, 2248. + Science Abs. vol. vii. No. 1185; Ann. d. Phys. xi. 4, p. 754 (19038). { Science Abs. vol. vii. No. 1186; Phys. Zect. iv. p. 594 (1903). § Science Abs. vol. vii. No. 157; Phys. Review, xvi. p. 199 (1903). Dr. W. H. Eecles on Coherers. 871 published, yet only minute voltages nowadays arise in practice. The purpose of the present investigations is to examine the action of a certain kind of coherer under “ weak signals,” especially in respect of their energy relations. Method of Faperiment. The self-restoring coherer to be examined was placed in a circuit comprising itis necessary source of steady electro- motive force and a telephone receiver. At the same time it was also connected in series with a condenser, to form, as in practical wireless telegraphy, a shunt across the condenser of an oscillatory circuit (fig. 1). This oscillatory circuit was made a secondary circuit in relation to a tuned primary oscillatory circuit. The coupling was always extremely loose. Trains of damped oscillations, of maximum amplitude of the order of a few hundredths of an ampere could be excited in this primary ; and hence feeble electrical oscilla- tions of calculable magnitude could be excited in the secondary. ‘The amplitude of the oscillatory electromotive furce at the coherer contact, and the energy delivered to the eoherer, can be computed from the amplitude of the primar y Eaidlevons . ; but, besides other considerations, the lack of facilities for measuring the small mutual inductance between primary and secondary made another method (described below) of determining the energy seem preferable. The trains of damped oscillations in the primary were produced by making and breaking a small current running through the primary inductance. In cons equence, the coherer yielded a seund in the telephone connected in series. Asa general rule, the telephone sound is greater the greater the amplitude of the oscillations in the primary, because the response of the coherer is greater. To measure ‘the response of the coherer, the intensity of the sound was balanced against Etother sound which was produced in the same telephone by the interruption of an independent source of direct current, which could be switched into the circuit of telephone and interruptor in the place of the coherer. In this way the behaviour of the detector could be quan- titatively recorded when the steady current through it, or the oscillatory energy delivered to it, underwent desired variations. The method has enabled me to attain some degree of eae a in measuring the effects of oscillatory voltages as low as j4, volt, and energy transformations of the order of a few thousandths of an erg. _ Only one type of coherer has been examined closely. This ‘is a mercury and oxidized iron coherer—a kind I have used 872 Dr. W. H. Eccles on Coherers. for many years—and chosen for these experiments after careful comparisons with a number of similar coherers of more modern vogue. As is well known, almost any con- ductor that can be coated with a closely adherent, tenacious, and very thin film of solid or liquid of not too high re- sistivity makes a good coherer: for example, oxidized or sulphided copper against any solid conductor ; oxidized iron, steel, nickel, tantalum, or any similar metal, against mercury or other conductor are all good*. I have tried also iron- mercury coherers made by the action of organic acids and other reagents on iron. For example, even the graphite (?) films left on a steel knife that has been unwisely used as a fruit-knife make coherers. Amongst all these and others I have found that the best for my present purposes is the coherer made by dipping the end of a carefully oxidized fine iron wire or steel needle into a pool of clean mercury. It 1s important that the layer of oxide on the wire should be of uniform very small thickness. Seven or eight years ago I found that this can be accomplished very well by heating the wire in a glass tube carrying steam from a boiling flask. The wire should be watched till the film of oxide yields one of those interference colours familiar in the tempering of steel. For instance, a blue film makes a splendid coherer, which works well for a month or two in the open air. Most of the experiments discussed below were carried out on coherers made in this way. Details of the Apparatus. ih Fig. 1 is a diagrammatic plan of the apparatus. The primary circuit consists of the coil L’ and condenser C’; the Fie. 1. a D won ss came I ger imaiies CHEE CG way Was FF eae c = T o RL aD zr P3 : secondary of Land ©, The detector D and condenser K are arranged shuntwise tothe condenser C. Cand C’ are variable * See Lranly, Comptes Rendus, exxxiy. pp. 347, 1197 (1902). Dr. W. H. Eecles on Coherers. 873 condensers of the Marconi pattern, K is an air condenser, Land L' are coils wound on waxed cardboard tubes. The six-contact key 8 is designed to throw the telephone T quickly from the detector circuit to the direct current circuit. In the position of S shown in the diagram the telephone is in series with the detector and part of the potential-divider P,, and at the same time the primary inductance L! is connected through portion of the potential-divider P, and the interrupter I. When the key S is depressed the telephone is connected directly through the inter: uptor [and the potential-divider Ps, and the secondary circuits are disconnected. The mutual inductance between primary and secondary can be varied by running L’ along a straight slipway to or from the coil L; but usually the coils were held so that their mutual inductance was a few hundreds of centimetres. The resistance r is an important detail. Its purpose is to prevent the formation of oscillations in L/ at the make of the primary current; calcu- lation (as well as experiment) shows that if 7 lies within certain limits oscillations occur in L’ at the break only. The amplitude of the oscillations is governed by aid of the potential- divider P;. The frequency of the interruptor I was chosen of the low value 22 per second ; thus resonance with the telephone diaphragm was avoided, and the balancing of the sound pro- duced by the detector with that produced by P3 facilitated. The electrical dimensions of the apparatus were as follows :— feo oo, (00 em., Co = 9020 en, L = 42,700 em., oe syem., K=-1210 cm., r=20 ohms, T=311 ohms. The mutual inductance between Land I/ ranged from 200 to 2500 cm. in obtaining the results described below. These sizes were settled upon after many changes, partly because they were practical sizes (wave-length about 820 metres), but chiefly because certain later calculations were made easier. Details of Method. In the experiments the primary and secondary were used slightly out of tune. This is very necessary when detectors whose resistance varies are under study ; for the variations of a coherer’s resistance are quite wide enough to carry the secondary circuit erratically from one side to the other of perfect resonance if the circuits are initially very near that condition, with the result that inconsistent measurements would be inevitable. It is, of course, as easy to deduce formule for the mistuned as for the tuned circuits. The Phil. Mag. 8. 6. Vol. 19. No. 114. June 1910. ie 874 Dr. W. H. Eccles on Coherers. expression for 2, the current through the detector at break of | primary current, is of the form xv = le~-“+he-™ cos (pt+q) +ke- cos (ct +d), where m and p belong to the primary circuit and 6 and ¢ to the secondary circuit, the coupling being very feeble. The coefficients /, h, k are complicated functions of the electrical dimensions of the circuits. Hence the energy delivered to the detector at a break appears as six terms, of which, with the dimensions adopted, only two are of numerical importance, namely, h?/4m, 2lh/m. The other terms are less than 1 per cent. of the larger of these. Both f and / involve m, the damping factor of the primary, but not prominently. This fact enabled m to be determined easily and with sufficient accuracy. A calibrated thermo- galvanometer with a heater and series resistance totalling 1000 ohms was put in the usual place of the detector ; the mutual inductance between the circuits was increased (but the coupling was still very small), the speed of the interruptor was raised, and heavier primary currents were used than was customary when a detector was in position. Thus a con- siderable deflexion of the thermogalvanometer was obtained and recorded. Then a short loop of very thin copper wire of known high-frequency resistance was put in series with L/, and the observation repeated. Since the energy received by the thermogalvanometer is proportional inversely to the resistance of the primary on each occasion, the high-frequency resistance of the coil L’ can be calculated from the obser- vations. The mean of several determinations gave 0°78 ohm. The steady-current resistance was 0°22 ohm. The power delivered to the detector in the standard con- ditions of the circuits and primary currents can be computed as just indicated. Tor accurate results it is necessary to know M, the mutual inductance between primary and secondary, and m, the primary damping factor, very accurately. As neither of these could be determined with the precision desirable, the thermogalvanometer was invoked to determine the power passing from primary to secondary. In fact, the experiment to determine m, just described, may be taken as determining the factor of transference of power. Similar observations to the above were therefore made with primary currents of different values and with rather high speeds of interruption (nearly 200 per second), and the fact was established, obvious theoretically, that the power passed to the thermogalvanometer was proportional to the square of the primary current and to the frequency of interruption. Dr. W. H. Eccles on Coherers. 875 Finally, by observation of the thermogalvanometer de- flexion when a primary current of 51, ampere was interrupted 168 times a second, the result was established that with the primary and secondary in their standard relation a quantity of energy amounting to very approximately 4°82 x L0—" joule, or about 0°0005 erg, is passed to a detector of 1000-ohms resistance at each break of a primary current of 0°01 ampere. This corresponds to a power-delivery of 0:01 erg per second when the interrupter is running at the standard frequency. Corresponding figures were obtained for the cases where the detector has different resistances between 100 ohms and 2000 ohms. In the actual experiments on detectors two things were chosen for variation, namely, the electromotive force applied to the detector and the oscillatory energy passed from the primary to the detector. The effect observed was the loudness of the sound in the telephone. ‘The curves of fig. 2 (p. 876) have as abscissz the varying external E.M.I’. applied to the coberer, and as ordinates the consequent effect in the telephone. Hach curve in the diagram is obtained by supplying to the coherer the constant power marked onthe curve. The curves in fig. 3 (p. 877) have as abscissee the power supplied to the coherer when its applied E.M.F’. is unvarying ; the ordinates are the effect in the telephone. ‘The telephone effect is plotted as estimated power given by the coherer to the telephone circuit. The numerical connexion between the power passed into the telephone circuit and the measured intensity of the sound in the telephone was determined as follows. The thermo- galvanometer was put in the telephone circuit in series with the telephone, and a large primary current was interrupted at the normal rate. The intensity of the loud sound in the telephone was measured by the potential divider, and at the sane time a reading of the galvanometer deflexion was taken. A resistance was then put in series with the telephone and the galvanometer, and the measurements repeated. These readings give sufficient data to determine the amount of energy being dissipated in the telephone when the intensity of a sound is equal to that due to any assigned length of the potential divider P;. The measurements thus made showed, besides, that the power delivered by the detector to the thermo- galvanometer and telephone was practically proportional to the square of the length of P; required to produce equal sounds in the telephone. It is rather a big extrapolation to extend these results obtained with currents large enough to affect the thermogalvanometer, to currents that give only faint sounds in the telephone ; but if we do this, we find that when the intensity of sound produced by a detector is equal 3.1 2 876 Dr. W. H. Eccles on Coherers. to that obtained by interrupting a telephone current of one microampere 22 times per second, then the power passed from 12 1:0 Iron Positive 0°6 03 Or4 0°2 5| 107? Watt irig. 2; Ord 0 Tron-Mercury Detector. 0°6 0:8 Iron Negative 1:0 1:2 ==] ° > the detector into the telephone circuit is 4:38 x 10—” watt, 2.e. 4°38 x 10-° erg per second. By the aid of this factor, the curves of figs. 2 and 3 were drawn. | Dr. W. H, Eccles on Coherers. 877 Discussion of Results. The most striking outcome of the measurements is embodied in fig. 3. The curve connecting the power delivered to a coherer as electrical oscillations and the power passed by the Fig. 3. 10-8 Watt WwW A 8 Iron-Mercury Detector. W =power supplied as oscillations. Resistance about 1000 ohms. Oscillation amplitude at coherer=442 x WW volt. coherer to the telephone turns out to bea straight line passing nearly through the origin. The line is different for different values of the steady electromotive force applied to the detector. The upper curve of fig. 3 was obtained from a rather insensitive coherer, which happened to have a resist- ance of about 1000 ohms at its most sensitive voltage, the iron being positive. The lower curve is one selected under similar considerations with the iron negative. The approxi- mate equation of the former is w = 0:024( W —0°9 x 10-5), of the latter is w = 0:019CW —2°2 x 10-8), the unit being a watt. All the other curves that have been obtained with coherers of various construction, resistance, applied E.M.F., are very like these. The more sensitive the coherer the steeper the line, is the only rule that has yet come to light (and in some iron-mercury coherers the gradient of the line is 0°06) ; there is norule yet proved for the constant 878 Dr. W. H. Eecles on Coherers. term subtracted from W. It may be mentioned again that the absolute value of the multiplier of W in the above equations is rather doubtful, because of the big extrapolation necessary in calibrating the telephone. But doubt on this point does not affect the character of the curves ; it affects only their gradient. As written above, the approximate equations for the curves suggest that for a particular de- tector under invariable conditions there is a fixed wastage of oscillation energy amounting constantly to about one-tenth of an erg per second, however large or small the oscillation energy given to the detector may be. If, however, the equations be written w +0°022 x 10-§=0:024W and w+0:042 x 10-§=0:019W, the inference is that a small quantity of energy, which is invariable while the detector is undisturbed, is delivered by the detector to the telephone circuit in a form that never makes any proportion of itself manifest as sound. Whatever the cause of this small energy wastage may be, the curves of fig. 3 show that the coherer must be put amongst those detectors that have been called “integrating” detectors. Thus the coherer is not, as has usually been supposed, a ‘“‘ voltage-operated ”’ detector. As detectors go, it has a very good efficiency (7. e. ratio of energy conversion), but evidently its efficiency falls rapidly as signals get weak. The curves of fig. 2 show how the sound heard in the telephone varies with alteration of the potential difference applied to the coherer and telephone in series. Hach curve is obtained by submitting the coherer to the fixed excitation whose value in watts is marked on that curve. As the excitation diminishes the maxima accompanying variation of applied voltage become less marked. The resistance of the coherer was kept as constant as might be during these experi- ments, but wide variations of resistance are inevitable. The asymmetry of the coberer under positive and negative voltages shown in these curves obtained by the action of electrical oscillations on the instrument, is no doubt connected with the similar phenomenon mapped in fig. 4, where typical steady current curves, plus and minus, are drawn. In these curves the abscissee are the potential differences applied at the terminals of the coherer. In both cases the lack of symmetry is due to the use in the coherer of two metals so different as iron and mereury. In order to examine whether this asymmetry was responsible for the failure of the curves of fig. 3 to pass through the origin, a new series of experiments on a different form of iron-oxide coherer was undertaken. ; Dr. W. H. Eccles on Coherers. 879 Fig. 4. 10-* Ampere } + Point Positive © Point Negative ‘ 10 0 = 0-2 Or4 Iron-Mercury Detector.—Steady-Current Curves. Additional Hxperiments. A coherer was made by heating a polished iron plate in air till rings of colour round the small heated area indicated that the plate was covered with a thin film of oxide. The end of a clean piece of iron wire was so held as to press on the plate af any desired place with a constant force. The plate was made one terminal of the coherer and the wire the other terminal, and this coherer was examined in precisely the As i Fig. 5. + Point Positive © Point Negative 107? Watt 0 2 Ww ; 8 Tron Point on Oxidized Iron Plate (light blue film). Applied voltage, 0°32. same way as the others. A typical series of measurements is set forth in figs. 5 and 6. The curve connecting the energy 880 Dr. W. H. Eccles on Coherers. given to the coherer and the energy passed to the telephone is again a straight line passing some distance from the origin. Fig. 6. 10-9 Watt 15 - + Point Positive ° Point Negative 1°) + W=5-53%10% 0°5 0 0°2 074 oo 0-8 Tron Point on Oxidized Iron Plate. The equation of the curve is w = 0°028(W —1'4 x 10-8). The curve is the same whether the current passes from point to plate or in the opposite direction, which clearly indicates that the asymmetry of the other curves was due solely to the use of two metals. The curves connecting applied voltage and steady current for these iron point iron plate coherers are also the same whatever the direction of the current. Fig. 7 shows such a curve, which has been selected from a large number as typical. A EHypothesis of Coherer Action. The experimental results described above lead me to offer a simple hypothesis of the mode of operation of the class of coherers in question. When an oxidized wire is dipped into mercury or touched against a conductor, the oxide dielectrie between the conductors is so thin and so extended that a considerable current, relatively speaking, can be passed through it. The films of oxide used in the experiments had a thickness of the order of a wave-'ength of light and an electrical resistance usually between 50 and 2000 ohms. When a current is passed through the film, heat is generated in the film and raises itstemperature. The rise of temperature may be considerable if the mass heated is small, but is limited by the escape of heat to the surrounding metal]. If the rise of temperature does not affect the resistance of the heated material appreciably, the relation between the applied electro- motive force and the consequent current will be a linear one: Dr. W. H. Eccles on Coherers. 881 but if the resistance of the material varies with its temperature the relation between electromotive force and current will not be linear. A direct experiment on a film of oxide of iron showed that its resistance coefficient was negative and nearly I per cent. per degree centigrade. Calculation shows that a resistance-temperature coefficient of this magnitude can cause the voltage-current curve to deviate considerably from a straight line. The equation of the curve is deduced below. If a train of electrical oscillations be passed through the minute mass of oxide traversed by the steady current, its energy appears as heat in the oxide. The resistance falls in consequence, and the equilibrium of the direct current in its circuit is disturbed. Since the rise of current enhances the heating of the oxide, the subsidence of the current to equi- librium is somewhat prolonged when, as in the case of iron- oxide, the resistance-temperature coefficient is negative. Hence the effect on a telephone diaphragm, whose natural period of vibration must be regarded as great in comparison with the duration of a train of oscillations, may be very large. It remains to be seen whether the disturbance of the direct current, caused by a small variation of resistance, can account for the phenomena disciosed in the course of the above- described experiments. Let W, be the fraction of the energy of a single train of oscil- lations given to the variable resistance p of the detector; let p=po(l—aé@) where py is the value of p at the temperature of the surroundings. Then kW,=(6@)=—(6p)/pox, where k is a constant involving Joule’s equivalent and the specific heat and the mass of the oxide. This initial disturbance (ép) of resistance is accomplished in less than 1/10000 of a second and gives rise to the perturbations of current to be traced immediately. Let the direct current circuit of the detector comprise a total invariable resistance r (which includes the resistance of the telephone), the variable resistance p of the detector, and the inductance L of the telephone. Let c¢ be the current at any moment and e the applied electromotive force. Assume that the rate of loss of heat from the warmed mass of oxide to its surroundings is m@, where, as implied above, @ is the temperature of the oxide above the temperature of the sur- roundings, taken as zero. Then at any time ¢ dé : a — kpe See aly, > el” ons Cileng ¢b) and ne OR ep GG eek ey pe dt Seat Dr. W. H. Eccles on Coherers. Since p=p,(1—«@), (1) becomes ] at + kpyx . po? + mp=mpo, | The simultaneous equations (1’) and (2) contain the complete history of the fluctuations of ¢ and p from any assigned equilibrium condition. We need only investi- gate here small fluctuations. First, however, let us obtain the equilibrium value of c¢, say c,, corresponding to a given value of the applied electromotive force «. Put dce/dt=0 and dp/dt=0 in (1’) and (2) and eliminate p. We get BY NLUDG ‘ ATE € + hapoc? re ae This equation agrees perfectly with the results of numerous experiments on the iron-oxide-iron coherer. For example, the experimental curve of fig. 7 has the parameters r=390 ohms, p,=970 ohms, af or locity Fig. 7. fold | 10°? Ampere 20 + Point Positive © Point Negative 10 0 0:2 04 0°6 0:8 Iron point on oxidized iron plate. Steady current curve. In general it would appear that the curves drawn from experiments on detectors should fall into the two classes Dr. W. H. Eccles on Coherers. 883 indicated in fig. 8 by curves I., III., separated by the case IT. In class I. the curve has two vertical tangents; the condition Electromotive-Force Ideal Curves suggested by theory. for this is p)>87. If when the coherer is in use the values of e and c pass those corresponding to the lower bend of the curve, the original values of ¢ and e can only be regained by interrupting the current by external agency: the coherer is not “self-decohering.” In class III. p)<8r: there is no vertical tangent, and the coherer is self-restoring. In ease II. p»=8r. All these cases can be realized, with patience. To return to the problem of the small fluctuations of the current from its equilibrium value when the arbitrary small variation (6p;) is imposed on the system by the arrival of a train of oscillations at the time t=0. Equations (1’) and (2) become 16 La FE arch apley = ,00g. a4), (4) ddp 0 “ i - nop = —2(n—m) a Gea +. Cay Here Steer rm er} 4 ts (6) and p;, ¢; are corresponding equilibrium values of p and c. Equations (4) and (5) give d°8p, dt? + (Ln+ rtp) + {n(r+,) —2(n—m)p,} 6p; =9, (7) L which can be written in the form Pee + G1) +5097 = 0. so, 2) asd Pl Fe ee A se ee He ~ 2 at =~ _- a ae I lk ae ~ | on 884 Dr. W. H. Eccles on Coherers. Here a is essentially positive. The quantity 6 has the same sign as de/dc, as may be seen by differentiating (3) and comparing (7) and (8). Since my experiments have been for the most part confined to self-restoring coherers, } has been in practice always positive. Thus (8) yields do, Ae-™4+Be-, . . 1 aa where ae #(144/1-%). It appears later that a? is always much greater than 46, so that, approximately, m,9=b/a or a—bd/a. The only datum available at the moment for determining the arbitrary constants A and B in (9) is that the initial value of dp; is (601). Therefore (dp;)=A+ le : 5 5 2 A (10) Using (9) in (4), LP + (r+ M)B4= —e (A em + Bem) whence ; EPig t ay TTP), Cy e~™ 1’ —e L e—™ti—e L qn a [ aot 2 po Lu PANO, A Tt On L 1 L (11) which vanishes at t=0. Let the work done in the telephone by the fluctuation of the current through it be 2, and let the effective resistance of the telephone be P. Thus the rate at which the telephone current is working at any moment is Pe,’, and, therefore, iy ole i. cfat= 2RG; ( 6c} dt. e 0 ev Equation (11) now gives w= : a, a pal {+2 Ms Re 2P ve; Meg )} en aE +Ae ig a5 by using the condition (10). Dr. W. H. Eccles on Coherers. 885 Now, as stated before, (69,;)=—koyx Wy, where W, is that portion of the energy of a single train of oscillations usefully converted to heat in the detector. Therefore (12) becomes w= ee fw (yh. ca) . ar ia i a e r+p, “mM, kpyx \m, | The experiments have shown that the term involving the arbitrary constant A is always positive. Hence, if A be positive, m, must be greater than m,; moreover, (10) shows that B must be negative when, as was always the case, (6p;) is negative. These considerations applied to equation (9) imply that 6p, runs through a series of diminishing negative values, passes through the value zera, and becomes positive. This is not possible. On the other hand, if A be negative, mz must be less than m,, and B may be either positive or negative. If B be positive, (10) shows it must be numerically less than A, and then the condition m, ov mer 9 Sa This equation regarded as an equation between the energy W, spent in the detector and the energy w, delivered by the detector to the telephone, is precisely of the form required by the experimental curves of figs.4 and 5. ‘Those curves may be regarded, in fact, as determining the constant of integration /?. The methods of measurement are not yet sufficiently refined, however, to make it worth while to examine this intercept term in detail. The coefficient of W,, on the other hand, may be profitably enquired into. In this coefficient P, £, py, 2, and » (which includes P, of course) are to be taken as constants, while p,, a, b have values dependent on the square of c¢,, the direct steady current passing round the detector circuit. By aid of equations (6), (7), and (8) the part of the coefficient which varies with ¢, may be expressed in terms of the quantity » defined by (6). This quantity n, it should be noticed, is itself a function of the square of ¢,, and therefore the following discussion applies equally to positive and negative steady currents through the S86 Dr. W. H. Eecles on Coherers. detector. It will be found that Lan= Lin? + mp) +7n and Lb=n?r—mnpy+2mo, ~ » ~ (15) when, as implied by (3), we remember that MPy= NP; - Now if (3) be differentiated (15) will be seen to become ee de Hence the coefficient of W, in equation (14) ‘ Ln L\de ay eee Vee Se It is convenient to think of ¢, the impressed current through the detector, as the independent variable. It is, of course, readily connected with e, the impressed voltage by equation (3) or by fig. 4. Then as ¢,; is varied the factor »—m has as graph a para- bola with its vertex at the origin and with the tangent there horizontal. This parabola rises only slightly within those values of the current ever passed through detectors. ‘The second factor has as graph, when values of c, are again taken as abscissee, a slowly falling curve. As a rule the graph of the product of these two factors appears to be a slowly rising curve. Now the third factor dc/de is the gradient of the €, c, curves shown in figs. 7,8. Hence, for a self-restoring coherer we expect that if we draw curves connecting w and e—that is the watts in the telephone, and the voltage applied steadily to the detector—we expect to find w rising to a maximum ata value of ¢ rather beyond the steepest part of the steady-current e«, c, curve. This deduction is amply borne out by the experimental curves of figs 2 and 6. Jam not able at the present moment to enter into an exact numerical examination of the whole matter ; but such rough values as have as yet been obtained leave little room for doubt that the quantitative adequacy of the above reasoning is as perfect as its qualitative sufficiency. It may be re- marked, in closing, that the term subtracted from W, in the bracket of equation (14) has its maximum at about the same value of c, as gives the maximum gradient in the e, ¢, curve of the detector. This deduction also has to some extent been experimentally confirmed, but the lability of coherers is so Dr. W. H. Eccles on Coherers. 887 remarkable and the difficulty of carrying a coherer through a variety of experiments without altering its internal state is so yreat, that full confirmation will only be attained by selection of the best from a very large number of observations. I have to thank the Royal Society’s Government Grant Committee for a grant of money to defray the cost of con- struction of the apparatus used in the above experiments. Summary. A method of investigating detectors is developed with special reference to the relations between the energy given to the detector in the form of electrical vibrations and the energy delivered by the detector, as direct current, to the circuit of the indicating instrument. The stream of energy supplied to the detector was always of the same order as that usual in telegraphy. The detector under examination was placed in a circuit containing suitable inductance and capacity, which was secoudary to a primary circuit. The primary could be set into electrical vibration by breaking a known current init. The coupling was very small, so that when a current of a few milliamperes was broken in the primary, the energy delivered to the detector was of the order a thousandth of an erg, and the electromotive force at the coherer terminals was of the order a tenth of a volt. The response of the detector was measured by comparing the sound in its telephone with the sound produced in the same telephone by interrupting a measurable direct current. A special switch key enabled the comparison to be made quickly. The power delivered to the detector and to the telephone was determined by extrapolation from measurements on stronger currents with the thermogalvanometer. The results of experiments on coherers made of oxidized iron wire dipping into mercury, and on coherers made of a clean iron point touching an oxidized iron plate, are exhibited as curves connecting: (1) the steadily applied E.M.F. and consequent current through the coherer; (2) the steadily applied E.M.F’. and the power given to the telephone, for various rates of delivery of vibration energy to the detector ; (3) the power delivered to the detector and the power passed to the telephone, the E.M.F. applied to the coherer being constant. Curves (1) show that in a self-restoring coherer the current increases more and more rapidly as the E.M.F. is raised, till, in general, a point of inflexion is reached, and then the current increases more slowly. Curves (2) show 858 Prof. A. W. Porter on the Inversion-Points jor the rise and fall of sensitiveness to oscillations as the applied H.M.F. is increased. Curves (3) show that if W represents the power in watts delivered to the coherer, and w the power passed to the telephone circuit, then w=m(W-—a) where m and a have values settled by the magnitude of the current through the detector. The quantity m for a good low resistance iron-mercury coherer has been found to be as high as 0-06 ; while a is usually near 1:0 x 10-8 watt. These curves show that these coherers are not ‘“ voltage-operated ” detectors but “‘integrating ” detectors. The author puts forward the hypothesis that the properties of an oxide coherer may arise solely from the temperature variations caused in the minute mass of oxide at the contact by the electrical oscillations and by the applied E.M.F. He examines the hypothesis mathematically, and shows that most of the phenomena recorded in the curves (1), (2), '3) above, can in this way be accounted for as perfectly as the present state of the measurements permits. XCIL. On the Inversion-Points for a Fluid passing through a Porous Plug and their use in Testing proposed Equations of State. PartIl. An Evamination of Experimental Data. By ALFRED W. Porter, 6.Sc., Fellow of, and Assistant Professor of Physics in, University of London, University College *. | a former paper (which will be referred to as Part I.) published in this Journal (April 1906, p. 554) I showed that the usual equations by which the behaviour of real gases is represented indicate that the inversion temperature for the Joule-Kelvin effect must depend upon the pressure, as had previously been found by Witkowski; and that moreover (as had not been before observed) there must be two inversion-temperatures for each value of the pressure. Of course only such equations were dealt with as indicate the existence of critical phenomena; it would be a very retrograde step at the present day to take others into consideration. In the present paper I gather together as much evidence as I can as to how experimental data support these conclusions. This I have found can be exhibited easily in a form which appeals at once to the eye, and enables one ai sight to diseri- minate between proposed equations of state. * Communicated by the Author. a Fluid passing through a Porous Plug. 889 The criterion of an inversion-point is the satisfaction of the equation CoN me ih ST v=), where T' is the absolute temperature and v the specific volume. The implication of this equation is that if isopiestic values of v be plotted against T, any point of contact of the tangent through the origin (T=0) corresponds to an inversion temperature. Pigs’. a This statement and diagram should replace those on the upper half of Part I. p. 560. By sufficiently accurate draughtmanship therefore the real inversion-points could be determined from the experimental values of Amagat, Sidney Young, and others. But such curves would be unsuitable for accurate work. Sufficient accuracy is obtainable, how- ever, if a difference curve be drawn as follows:— Let A=v— "X, where v, is the specific volume at any 0 standard temperature Ty. We shall call A the discrepancy because it represents the difference between the actual volume and the volume which the gas would have had if it had changed from that at the standard temperature according to the law for perfect gases. It follows easily that pov y= TA — A; Wier aT Phil. Mag. 8S. 6. Volo Wo. Liddy June 19:10. 3M ———— —— Temp. | 150 Atmos. 0 10085 0 1:0390 0 1-0825 0 || 1:1360 99:45 || 1:4500 742 1:4890 890 Prof. A. W. Porter on the Inversion-Points for and that if A is plotted against T the inversion-points are given by the same rule as before. The curves thus obtained enable fairly accurate estimates of the inversion temperatures to be made. At the same time the present writer has for some years found them to be of great use in exhibiting the properties of real gases in other respects. It is convenient to take zero centigrade as the standard temperature ; the value of A is then necessarily zero at that temperature. If the origin be shifted to this temperature the tangents for finding the inversion-points must be drawn from the centigrade tem perature corresponding to the absolute zero ; this has been taken as —273°1. Since the pressure p is a constant for each curve the product pA may be dealt with instead of A. The following table, based on Amagat’s experiments (Ann. de Chem. et de Phys. 1893), exhibits the data for nitrogen in full :— TABLE I.—NITROGEN. 200 Atmos. 250 Atmos. a oe ——— 16:03 || 1:0815 138 11145 145 11575 115 12105 199°5 | 1-8620 1186 19065 1086 19585 852 | 20145 488 300 Atmos. pu. \pAxXlOl| py. |pAx107|| 9) po: pA SCs es | pA x 10% 717 15376 609 1-5905 409 Temp. 350 Atmos. 400 Atmos. 450 Atmos. 950 Atmos. pe. |pAxX101 || pe |pAx 104 post pax 104, pv. |pAx10'. : 0 [11950 | 0 || 12570|/ 0 ||13230 | 0 | 20015 | o | 16:03 | 1:2675 24 1:3290 —18 || 13940 —66 || 2:0690 — 500 99°45 | 16465 | 163 1-7060 —88 | 1:7665 | —382 || 24230 | —3064 199°5 | 20730 50 2:1325 | —418 || 2-1940 | —954 |] 28380 | —6256 The corresponding curves are shown in fig. 2; and from these the inversion-points are found by drawing tangents through the absolute zero. The values so found are given in Table II. In the columns headed “reduced ” the pressures and temperatures are given as fractions of their critical values, viz. :—p,=27 atmos., T,=127° abs. C. | a Fluid passing through a Porous Plug. 891 Fig. 2. NITROGEN b A ©1590 Atm ©200 ‘) - / 4 / 2250 Z aw fu / “05 y 2300 TABLE I].—Nrrrocen. Inversion-points. Pressure. Temperature. Dae Weve, Peetinbed Centigrade. | Reduced. 200 740 | about 190 about 3°60 250 92 | 17 3°50 300 11:10 142 3°34 300 12 95 111 3°02 400 14:80 12 2°24 cabal 16°65 None None upwards. 892 Prof. A. W. Porter on the Inversion-Points for These numbers must be regarded as approximate only; partly on account of the uncertainty in the curves themselves for each of which there are five points only; and partly owing to the uncertainty of the critical data. The uncertainty in regard to the critical data might be removed by employing actual instead of “reduced” values of p, v, and T, but at present it is general relations only that are sought for, and the results are accurate enough to give valuable information. Plot the reduced pressures and temperatures of the in- version-points against one another (fig. 4). Notice first that there is an indication of a maximum pressure for which inversion can occur. In fact the 400-atmosphere curve (fig. 2) is not far from this maximum pressure. ‘This is the expla- nation of the wriggle in this particular curve. Although it was perfectly easy to draw curves with curvature of the X,, NITROGEN. © CARBON-DIOXIDE. ww KS ® = i Q u S 3 ak & ReouceO PRESSURE same sign for the other pressures represented on fig. 2, it was impossible to do so for the 400-atmosphere curve. The point for 16° seemed to wander froma smooth curve drawn through the remaining points as though there was greater error attached to it. The recognition of the possibility of two inversion-points for the same pressure cleared away the diffi- culty that was felt at first, and revealed the faithfulness with which Amagat’s work had been carried out. The apparent error of this point was precisely such as might easily have been removed by the smoothing-out processes to which so many observational results are subjected. On fig. 3 is also ploited the curve corresponding to van der Waals’ equation of state. It will be seen that there is no quantitative agree- ment between it and the curve derived from experiment. On a Fluid passing through a Porous Plug. 893 the same diagram are shown the inversion-points corre- sponding to the equation of state which Dieterici has studied in connexion with isopentane, viz.:— A pw—b)=RTe Rte, or, in the reduced form, a(28 = ity = yetlt-Viby2))), The inversion-points corresponding to this equation are given by the equation any? =(10— y°) e2(12—2/72)_ It is obvious that there is remarkably close correspondence between this and the experimental curve as regards their general trend. A much closer fit can be obtained by modi- fying the index of I’ in the exponential term, from 3/2 to about 1°65; but in view of a modification to be proposed in a later paper no stress will be laid on this fact. Carbon Dioxide. Carbon dioxide has been studied more fully by Amagat. In the case of this gas we can obtain more definite curves because of the greater number of experimental points. The discrepancies calculated from Amagat’s are given in Table III. and are plotted in fig. 4 (pp. 894 & 895). The inversion-points obtained graphically from the curves in fig. 4 are tabulated in Table 1V. TasBLe [V.—Inversion-points. CQs. Pressure. Inversion-Points. Atoms. Reduced. Cent. Reduced. 200 2°74 0 ‘90 250 3°42 5 92 400 5°48 30 1 nearly 500 / 6°85 | 45 1:06 600 ie ae | 81 1-17 700 | 9°59 | 135 | 1°35 800 | 10°96 | 168 1-44 850 | 11-64 198 1:55 Prof. A. W. Porter on the Inversion-Points for 894 1186— | OISLS|| 9FES— | O8OI-% P8es— | OG16-1] Le0s— | GGe8-T 116I— | 0892-1] ZO9T— | 0689-1 ISLI— | 82-1) OUGT— | 099-1 9I9I— | 4269-1] SISI— | OFT9-1 IPFI— |G¢9-T}) OSZI— | O8L9-T EQSI— | %Z9-T!) SZII— | SPST 0 |OFSrT) O |osge- orxvd| ced |yorxvd] cad ‘soulyY 048 ‘SOV 008 QLLI— | 0880-2 GLOL— | S9CL I LOFL — | 00T9+1 e68I— | 9049-1 PEGI — | Se69-1 PITI— | S866-1 066— | OF9F-T 9F8— | SISF-T z69— | OOOF-1 Esg— | OOLE-T gce— | 9688-1 6LI— | S0T8-1 Q0 | ST8%.1 01xva| vad ‘SOWTY OGL | GEsI— |09¢6-1]] LE + | OBIT GPEI— | 0919-1] 6L4— oste-t F8lI— | ¢8z¢-1|| c89— ean 8ZII— | 006F-T]| 869— | 29Z8-T IGOI— | ee¢r-1]] 699— | 0068-1 C1g— | OALP-1]| GEO— | OFSZ-T g4g— | ¢zse-1|] OI9— | O6TZ-T ZoL—- |00Ge-1] Tg¢— | S98T-T 1g9— |o61¢-1|| Gor— | OLGT-T 68r— |068%-1]| G4E— | SL2T-T she— | 0692-1|| 89c— | €660-T 98I— |0B8%-T|| GFI— | 020-1 @ (econ. “0s ~ \cero1 oLxyd a “OLX Vad Sai ‘sowyy 0OL || “sourY 009 ‘ACIXOIQ] NOCUVO—'T I] AIIV], 861+ | GLL9-T Shot | OZ9E-T Lol | O0cH 0so—_ | OLD FLG— | OFGI-T LOS— | 0880-1 I@E— | OPGO-T GoE— | O1Z0-T 60E— | 0066- &96-- | O&96- LLI— | O8&6- IOI— | OSI6 0 GO68: 01x va] ad ‘BOMTY OOS 190e | 0€9¢-1 E621 | GLIB-T Geb | €880-T isc+ | 0966: stit+ | o9se. re + | oste. 6g — | OF8S- 86 — | gI¢e. QII— | 0gze: ogt— | 0962. G0I— | OTLL- GL — | LPL 0 |08zd ‘501 a 2 ad ‘sOw}Y OOF aie cae a sear oe aoe eae era Baye) ‘dway, a Fluid passing through a Porous Plug. 895 In calculating the reduced values the critical temperature is taken as 30°'3 C. and the critical pressure as 73 atmospheres. Fig, 4, pA > A00 900 tm. 1 y wae 200 Atmos Z 2B 6750 2 2 i 600 150° Temperativre Cent DA ere ees 700 7/50 me x Q) S 800 \ % ~3'350 —*|5 CARBON DIOXIDE These inversion-points are also shown by the small circles on fig. 3; so that comparison may be made with Dieterici’s equation. Again, the general alignment of the curve is the same as that of Dieterici’s theoretic curve. Moreover, the agreement can again be made much closer by choosing a somewhat different value for the index n. It is most interesting to 896 Lnversion-Points for Fluid passing through Porous Plug. note, however, that the experimental points for CO, corre- spond to a range of temperatures lying on the lower portion of the theoretic curve ; while those for N lie on the upper part. These facts tend to substantiate the claim put forward in my previous paper that there are probably two inversion- points corresponding to each pressure. This point is of im- portance; for whereas other investigators have readily admitted that the inversion-point varies with pressure, yet I believe there is a tendency to regard the two-fold value for each pressure as merely a result of pressing to more than a legitimate extent the validity of the theoretic equations which had been considered. The inversion-points have also been calculated for other gases. Those for isopentane, ethylene, and ether lie on the lower part of the inversion-curve, and the curve for CQO, passes nearly through them all. General properties of the Equation of State JR gues cee E€ RTv. —l The fact that by a proper choice of na very much closer agreement can be obtained, warrants one in further examining the general properties of the family of curves for which n is the parameter. i Whatever the value of n may be, this equation indicates the existence of critical phenomena. | If n is the same for all gases, there are three constants which may be different for different gases; hence, with the same restriction the law of corresponding states will be satisfied. The reduced equation is (00 1) yeep {2(1- =a) f qv e s ° e ° ° The equation connecting @ and y at an imversion-point 1s BL(y'—4(n 4+ 1] =—2(n41)5 and the equation between a and ¥ is 9 2 \ ay’ t=[4(nt1)—9"] exp. Cie Ls a ie yy” —— —— For each value of n there is a maximum pressure for which inversion can occur; this is given by the value of « corre- sponding to y"=4; hence the maximum value of « for any On Secondary Homogeneous X Radiation. 897 one value of n is 1 - ete he ome = A" wexp. 12(F 9) f nti 2n _ The least value that this maximum pressure can have is when 7 is about 1°8; the value of this particular maximum pressure is about 15:1; andthe corresponding critical volume is about ‘777. The values of y corresponding to «=0 are respectively zero and that given by y*=4(n+1). It will be seen that increasing n lowers both the upper and lower parts of the curve obtained by plotting the reduced temperature against the reduced pressure as in fig. 3; these portions are remarkably straight except near the turning-point and for small values of « on the lower portion. and the corresponding value of @ is Conclusion. The foregoing results show unmistakably that Dieterici’s equation gives a suitable first approximation to the positions of the inversion-points of many gases. In a further part we shall discuss a second approximation which at the same time suggests a law for those gases for which Dieterici’s equation is not OS gene XCIII. On ee Homogeneon X fFadiation. By J. \ ©. Cuarmay, B.Sc. and 8. H. Prrer, King’s College, (.) London * T is well on that all bodies while exposed to Réntgen radiation emit secondary X rays. Barklat has own that these secondary rays are of two types—a scattered radiation having the same penetrating power as the primary beam and resembling it in that it is heterogeneous, and an X radiation characteristic only of the element emitting it and independent of the penetrating power of the primary beam. Barkla and Sadler ¢ have shown that this radiation is homogeneous and is only excited by a primary beam more penetrating than itself. The former type of radiation has been found to be such as would be emitted by electrons when accelerated simply by the electric forces in the primary pulses §. Though much is known of the homogeneous Communicated by Prof. C, G. Barkla. Phil. Maz. June, 1906, Phil. Mag. Oct. 1908, ‘Conduction of Electricity through Gases’ (Second Edition), tat Se : i i i i iq if al >» 898 Messrs. J. C. Chapman and 8. H. Piper on radiation the experimental evidence has been insufficient to lead to an exact theory of the processes resulting in its pro- duction. Several possibilities suggest themselves, which we wish to examine and upon which a few simple experimental results throw some light. It has been shown by Townsend, Perry, Dorn, and others, that another type of radiation is emitted in addition to the secondary X radiations. This consists of negatively charged particles identical with those found in a vacuum-tube. Sadler * has recently shown that in most cases they only appear with the homogeneous X radiation, and it is possible there are no exceptions to this rule. This connexion between the two types of radiation suggests the possibility that the homogeneous X radiation may be produced as the result of the sudden alteration in velocity of an ejected electron, or again, it may be due to the subsequent bombardment of other atoms by the ejected electron, the radiation taking place either from the colliding electron or from the atom struck. Several objections can, however, be urged against the hypothesis that the radiation arises from the ejected electron. Firstly, the velocities with which the electrons leave the - substance, that is, the velocities of ejection, have been found to vary over wide limits by using various primary beams. Yet, whatever these velocities, the secondary X radiation given off is cf constant penetrating power and perfectly homogeneous. Secondly, the same velocity of ejection from the atoms of different substances is accompanied by homo- geneous beams varying greatly in penetrating power. These facts seem to show that the radiation cannot proceed from the electron at its ejection. Another explanation is that the characteristic secondary radiation results from the bombardment of atoms by these ejected corpuscles. In this case the radiation may proceed from the colliding electron or the atom so struck. The same arguments that have been used above go to disprove that it is the expelled electron which is the source of the radiation, for although the velocities of these electrons differ greatly before collision when a heterogeneous primary beam is used, yet the secondary radiation is of constant penetrating ower. The following experiment in addition points to the cen- clusion that it is not the atom which is bombarded that emits the homogeneous radiation. * Phil. Mag. March, 1910. Secondary Homogeneous X Radiation. 899 Secondary Radiation from an Alloy. This experiment was made to determine whether an appre- ciable part of the homogeneous secondary radiation from an element is due to the subsequent bombardment of atoms by the electrons liberated by the primary pulses. A primary beam of X rays, of a degree of hardness sufficient to excite homogeneous copper radiation without directly stimulating the homogeneous silver rays, was allowed to fall on an alloy plate consisting of two parts of silver to one of copper, and the resultant secondary beam was received by two electroscopes, one of which was used to test the penetrating power of the beam while the other served to standardize the intensity. Since the copper radiation was excited, corpuscular radiation accompanied it and the object of the experiment was to see if these electrons ejected from the copper on bombarding the silver atoms made them give off their characteristic secondary radiation. If this had happened the silver radiation would have been superimposed on the copper, and the difference between the absorption coefficients by aluminium of the copper and silver radiations in this composite beam would have enabled each to be dis- tinguished. If on the other hand this effect was small or possibly non-existent, the radiation from the alloy would have corresponded to that from a pure copper plate. When the radiation from the alloy was allowed to pass into the electroscope without previously passing through any absorbing plates, the percentage absorption of the beam by a thin sheet of aluminium placed i in the path of the rays was found not to differ from that of a beam from a pure copper plate ; examined in this way the radiation from the alloy was indistinguishable from a homogeneous copper radiation, as shown by the following mean readings : — Coprrr. ALLOY. Percentage absorption by aluminium sheet (‘00163 cm. thick) after 33 per 33-0 39-7 cent. absorption of homogeneous r MORMON HOT TACIALION) 6... idee. vacsnsegeces In order, however, to magnify the effect which would have been given by any silver radiation in the rays from the alloy, the beam was allowed to pass, before being analysed, thr ough a thickness of aluminium which absorbed 70 per cent. of ite incident copper radiation, while this thickness only cut down the much more penetrating homogeneous radiation from a a * SS a er —————— eS Se ee ee ee 900 Messrs. J. C. Chapman and 8. H. Piper on silver plate by 7 per cent. On testing the radiation coming through this thickness its penetrating power was found to differ only slightly from that of homogeneous copper radiation. Thus :— Corrprr. | ALLoy. sheet (‘00163 cm. thick) after 70 per 2:5 29-9 cent. absorption of homogeneous i Percentage absorption by aluminium | 3 COpPeRMadIallON yy. ..ceewee es see eceee ate The slight decrease in penetrating power was fully accounted for by the presence of radiation scattered from the silver in the alloy, this scattered radiation having the same penetrating power as the primary beam, which was of course harder than the radiation characteristic of copper. This experiment showed that the electrons leaving the copper and bombarding the silver atoms did not themselves cause the latter to give out an appreciable homogeneous secondary X radiation. This result indicates that it is im- probable that the homogeneous radiation from a metal is the result of bombardment of atoms, for it has been shown by Campbell Swinton * and by Kaye f that the intensities of X radiation produced at anticathodes of various metals when subject to the same cathode stream are at any rate com- parable. Usually the X rays from the heavier metals were more intense (as measured by the ionization produced) than those from light metals. If then an appreciable X radiation were produced in copper by the bombardment of the cor- puscular secondary radiation, we should expect a radiation of comparable intensity to be set up in the silver by the bombardment of electrons from copper atoms in the alloy. The alloy experiment also shows that the silver atoms when separated only by molecular distances from atoms which are emitting homogeneous radiation are not stimulated so as to emit their own characteristic radiation. It is still just possible that the corpuscles from the copper excite neigh- bouring copper atoms, but have not sutticient velocity to stimulate the silver atoms. That is, a critical velocity higher in silver than in copper may be necessary to the production of the characteristic X radiation. This point will need further investigation. * Proc. Roy. Soc. 1897, pp. 222-226. + Camb. Phil. Soc. Proc, xiv. Oct. 15th, 1907. Secondary Homogeneous X Radiation. 901 Since performing this experiment a paper has been published by Sadler * showing that with silver there is a corpuscular radiation in the absence of the characteristic secondary X radiation which has been observed. ‘This also shows that electrons ejected with speeds approaching the velocity of emissions of the electrons which accompany the characteristic homogeneous silver radiation when just excited, cannot produce homogeneous secondary radiation by collision with an atom. The above considerations point to the conclusion that the atom from which the electron is ejected is the seat of pro- duction of the radiation. As the result of the loss of an electron the atomic system will have to adjust itself, and in so doing the electron or an electron may vibrate with an emission of the homogeneous secondary radiation. It seems probable, therefore, that, as has been pointed out, an electron is made to vibrate by the passage of a Rontgen ray pulse over it, with the consequent emission of secondary X rays in somewhat the same way that certain fluorescent bodies give out radiations when suitable light is incident upon them. The agreement between the law found by Barkla and Sadler in the case of secondary radiation and Stokes’ law of fluor- escence, and the fact, discovered by Stark +, that fluorescence is accompanied by a corpuscular radiation, both strongly support this theory. The possibility of an appreciable phosphorescent effect, that is, a persistence of secondary radiation after the removal of the radiating atoms from the exciting beam, as observed in the case of ordinary light by Becquerel, led to the following experiment. An Attempt to Detect a Possible Persistence of Secondary X Radiation. A brass cylinder (5°5 cm. in diameter and 6 cm. high) mounted on ball bearings was rotated rapidly about a vertical] axis (O in figure). An X ray bulb inside a lead box was placed as shown in the diagram so that the rays fell directly on the cylinder. Lead screens, §;, 82,83, were adjusted so that with the X ray bulb working and the cylinder at rest no leak of the Bp osPOpe, as shown by a gold leaf, was observed. Between §, and 83 a particular part of the cylinder was subject to the X rays and gave off the homogeneous X radia- tion. After passing S,and 8; this part was no longer subject to the rays but came into a position in which it could affect * Phil. Mae. March 1910, t Phys. Zeit. ix. pp. 481-495, 1908, + 902 On Secondary Homogeneous X Radiation. the electroscope A, and had the disturbance in the atom con- tinued for a long enough period ionization should have been KN S2 produced in the electroscope with consequent deflexion of the gold leaf. Varying speeds were used but no leak effect was observed. : The number of revolutions per second made by the cylinder was measured by an estimation of the note obtained by a prong fixed to the cylinder striking a card once in each revo- lution. The highest number of revolutions of the cylinder which, by gearing, the motor was capable of giving, was approximately 270 per second, and since the angle subtended by S, and S83 was 30°, the time which elapses between a certain fixed part of the cylinder leaving the X rays and coming into a position in which it was able, if still giving off this secondary radiation, to affect the electroscope, works out at approximately 1/3000 second. With the cvlinder rotating at this speed no appreciable leak was observed in the electroscone over a period of five minutes, though a de- flexion of 0-2 of a division could have been easily read. On the other hand, had the homogeneous secondary X radia- tion excited in the cylinder not decayed appreciably in the De) Radioactive Minerals in Common Rocks. 903 small time taken for half a revolution, a deflexion in the gold leaf electroscope of about ten divisions per minute would have been observed. This experiment shows that of whatever nature the effect inside the atom may be, it certainly is in the case of copper and zinc ey damped and cannot continue in sutiicient strength to give 1/250th of the original secondary radiation aiter a period ‘of 1/3000 second. In conclusion, our best thanks are due to Professor Barkla for his valuable advice and his interest in the experiments. Wheatstone Laboratory, King’s College, . April, 1910. . —— == - 2 Se ———— = XCILYV. Radioactive Minerals in Common Rocks. iby). W. Waters, B.Sc., A.R.C.S.* N the ‘ Philosophical Magazine’ for October, 1909, the author described a physical separation of the minerals in some common rocks, according to their radioactivity. It was found that the radioactivity was in each case con- centrated in certain accessory minerals. The rocks were crushed, and the minerals were separated by taking ad- vantage of their different densities and magnetic properties. The same methods have now been applied ‘to a specimen ot granite from Dalbeattie, and also to a tertiary granite from the Mourne Mountains. In each case the radioactivity is found to be concentrated in the heavier minerals. In the Dalbeattie granite, the most radioactive constituent was a feebly magnetic mineral, which was shown by micro- scopic and spectroscopic examination to be allanite. When the mineral was spread over the tray of a gold-leaf electro- scope, the rate of leak of the electroscope was 10() scale- divisions per hour. The other heavy minerals present were magnetite, sphene, and apatite, which were only feebly radioactive. In the Mourne granite the most active constituent was very heavy and feebly magnetic. Jt was found by micro- scopic and spectroscopic examination to consist of zircons mixed with a small amount of a titanium mineral. The rate of leak of the electroscope in this case was 600 scale-divisions per hour. The mineral was radioactive enough to produce frequent scintillations on a zinc sulphide screen. The object of these experiments was to accumulate material for finding the helium ratio and hence the age of ordinary * Communicated by the Hon. R. J. Strutt, F.R.S. GOL Radioactive Minerals in Common Rocks. rocks. . There is, however, an unforeseen difficulty owing to the small size of the crystals in which the minerals occur. A large number of these were measured under the micro- scope, and their mean dimensions obtained. They are given in the following table :— Average greatest Average least a dimension. °“ dimension. RET ASRINT, ROR Mia Vn Tec | Allanite from 2 O55 Dalbeattie granite POPS ae 01035 amas | Rutile from 0-130 ,, 0-082, | Cornwall granite | | ae from — } i 0-140 , 0:052.73 | | Lewisian gneiss | | | | Zircon from . 02 | | Mourne granite } TONE | One | The range of @ particles in these minerals is about 0-04 mm. It will be seen that the dimensions of the crystals are of this order of magnitude. If the helium in radioactive minerals consists of a particles shot out by the radioactive substance, the greater part of it will therefore be found, not in the erystal, but in the ground-mass surrounding it. (This is independent of the gaseous diffusion that takes place in the course of long periods of time.) The helium ratio for minerals occurring in small crystals isolated by this method will therefore be smaller than we should be led to expect from their age. Prof. Strutt has kindly devemine the quantity of helium present in some of the minerals, and finds the following values :— Helium in c.c. per gm. X 10~4. Sphene from Dalbeattie granite ......... 3-4 Rutile from Cornish granite ............ D3 Zircon from Lewisian gneiss ............ 36°0 As might be expected, these values are much lower than is usual for such minerals. As any deductions of the age of rocks based upon radioactive measurements for minerals isolated in this way are unreliable, the experiments have been discontinued. Ror Sen XCV. The Rate of Decay of the Radioactivity of Polonium. By J. W. Waters, B.Sc., A.R.C.S* OLONIUM is generally regarded as identical with radium F’, one of the products of the disintegration of radium. ‘The nature of the product of disintegration of polonium is a matter of considerable interest. If the product is a solid substance it will accumulate in the course ot time, and if it is radioactive its presence will be shown by the deviation of the curve of decay of radioactivity of the polonium from an exponential curve. The rate of decay of radioactivity of a specimen of polonium about five years old has been found during the past few months. Taking the age as five years the activity of the specimen was about 700 of its initial activity. The rate of decay was found to be nearly the same as that found by Mme. Curie for the initial rate. The polonium was in the form of a thin deposit on a rod of bismuth. This was mounted inside the case of a gold-leaf electroscope, and the time taken by the image of the gold- leaf to move over a given range of the field of view of the observing microscope was determined at intervals of a few weeks, Let the time of transit of the gold-leaf be T seconds. The time taken by the gold-leaf to fall over the same range before the polonium was present was about 12 hours = 43,000 seconds. Then, the radioactivity at any time is 1 43,000° following table. Hach is the mean of several observations :— ‘ 1 : : proportional to ;,- The values are given in the No. of days. A ig Radioactivity. o sec, 0 799 100 00 20 876 91-07 21 879 90°72 a 1005 79:11 81 1171 67°61 86 1185 66°77 96 1224 64°58 114 1331 59°27 123 1388 56°74 * Communicated by the Hon. R. J. Strutt, F.RS. Phil. Mag.8. 6. Vol. 19. No. 114. June 1910. oul. a 906 Mr. W. J. Harrison on the A curve has been drawn showing the relation between logo and the number of days since the beginning of the 0000 PN ee ae a CCE ae eT LPN EET La oat CLE EEN a se eee 10 20 30 40 50 60 70 80 90 100 is 13 No. OF DAYS. 9500 “9000 L0G, O 8500 8000 7500 0 experiment. From the curve it is found that the activity falls to half value in 148 days. This differs slightly from the initial value which is given by Mme. Curie as 140 days, but the accuracy of the experiment is not sufficient to justify any deduction therefrom. T have to thank Prof. Strutt for suggesting this experiment and for the loan of the polonium. XCVI. The Decay of Wavesina Canal. By W.J. Harrison, M.A., Fellow of Clare College, Cambridge*. } iG Dr. Houstoun’s reply ¢ to my paper f it is clearly evident that his main contention must be wrong. It is necessary for the validity of his argument that waves of the type sin 27” sin fee a b should have different periods and rates of decay (moreover, that waves of the type sin — cos —” should have the same b * Communicated by the Author. t+ Phil. Mag. [6] vol. xix. pp. 205-206 (1910). t Phil. Mag. {6} vol. xviii. pp. 483-491 (1909). elt aw , and those of the type sin a egal Decay of Waves in a Canal. 907 period as these of the type sin — These two modes have the same periods and rates of decay that they would have in an unlimited (laterally) mass of liquid of the same depth. The lateral boundary conditions simply determine whether they shall be present or absent. It will be noticed that equations (1) to (6) on page 488 of my paper represent the conditions at the free surface and bottom only, the remainder are satisfied identically. Now in an unlimited mass of liquid these periods and rates of decay are the same, it is simply a question of choice of origin. The combination of my analysis with Dr. Houstoun’s expe- rimental results shows that the mode p=1, g=0 (for notation liere see my previous paper) undoubtedly exists and is pre- dominant, since this is the only mode whose periods agree with those observed. My explanation is that this mode represents the motion in the body of the water, but that it does not do so near the sides. It cannot do so at the sides since there is no slipping (if this be the assumption made). The experimental fact that the logarithmic decrement is increased by only 10 per cent. when the breadth of the box is decreased by one-half does not render invalid my con- tention that the effect of the sides is confined to their neigh- bourhood, but appears to support it. In any case, whatever explanation be given of the increased rate of decay of the observed over the calculated motion, I still maintain that the method of approximation by which Dr. Houstoun arrived at his equation is wrong. He says in his former paper, “‘ Let the motion be in one horizontal direction?’ *. If v be assumed to be zero, then everything follows as he states, except that there is an un- resisted traction tending to force a velocity v into existence, which he has neglected; this traction depends on a : ae There is no reason why 0 should be neglected and Y 07u ' : 70 / e i 37 be retained, as in equation (3), on page 155, unless the length of the box be small compared with the breadth. It is impossible for u to have the form given in equation (11), page 156, and v to have a zero value. If this is so, then Dr. Houstoun’s approximation breaks down, and we have here the reason for the discrepancy between the two results. Clare College, Cambridge. * Phil. Mag. [6] vol. xvii. p. 154 (1909). 5 N 2 Nt 908 (tq XCVII. Further Tests of Brittle Materials under Combi Stress. By, Waurer A. Scopue,.A.R.C.Se¢., sae Whitworth Scholar *. Introduction. NHERE are three theories of elastic strength which are recognized by enyvineers,—that a material fails under a definite (a) Maximum principal stress, (6) Maximum shear stress, (c) Maximum strain. A large number of tests on ductile metals have been made, and these have shown that the maximum shear stress approximately determines the failure of such materials, and consequently that theory 1s justified. Although tests in pure tension and torsion indi- eate that ductile materials fail ey shearing, whereas brittle materials fracture across the plane of maximum principal stress, it has not been usual to make any distinction between the metals when the results of tests under combined stress have been recorded, or, al most, following the lead of Guest, the metals have been termed ductile. Brittle materials have been almost entirely neglected. Yet there are two important theories which have been disproved for ductile materials, which lead to very similar results, and which are supported by the forms of the fractures obtained with brittle materials. Hiven without any new experimental evidence, it appears to be very probable that brittle materials obey the maximum principal stress law. Previous Tests. The author used cast iron bars for his original tests of a brittle material under combined stress+. Fracture is the most satisfactory criterion of strengh for a brittle material. But it is doubtful whether cast iron follows Hooke’s law with sufficient closeness to allow the ordinary elastic theories to be applied to it; certainly it is not perfectly eiastic to fracture. Consequentiy it is not a satisfactory material from the standpoint of an elastician, but it was adopted because it is the brittle material which is most commonly used in engineering practice. The results of the tests on cast iron showed that both the maximum principal and shear stresses at fracture varied considerably if they were calculated on the assumption that the metal was elastic to rupture. But the planes of failure approximated very closely to those of the maximum principal stress. The metal certainly yielded somewhat, and it was * Communicated by the Physical Society: read April 22, 1910, + Proc. Phys. Soc. London, vol, xx. Tests of Brittle Materials under Combined Stress. 909 not possible to find the exact stress distribution, but it appeared to be reasonable to assume that the yield was sufficient to cause the stresses which are directly due to the bending and torsion to become uniform. For example, fig. 1 fl UniFORM Evastic Limit. STRESS represents the centre of a bar under a central bending load. AOA! is the section midway between the supports. The tensile or compressive stress (at a point in the section), due to the bending, is set out perpendicularly to AA’. The com- pressive stress at A is represented by AB. When the material has not been stressed at A or A’ beyond its elastic limit, BOB’ indicates the stress distribution over the section. If the material has yielded at the outside, the inner Jayers are more fully stressed, and a line like BCO replaces the straight line BO. A small yield causes BOO, which represents the true compressive stress at any point, to follow BDO more closely than OB. But BD represents a uniform stress over the section, and this condition was therefore assumed. The assumption that the shear stress due to torsion was uniform may be similarly justified. The Specimens for the Present Tests. The behaviour of cast iron, therefore, appeared to support the maximum stress theory, but it was necessary to assume that the known yield caused a redistribution of stress. This was not entirely satisfactory, and further to test this law it was decided to use a material which was elastic to fracture, or was more truly brittle, and which therefore obeyed the usual laws of stress distribution at rupture. Specimens of brittle alloys were obtained, but were condemned after a careful examination, without being tested under combined stress. The metal finally selected was Vickers, Maxim “Cast 7” steel, which contained 0°7 per cent. of carbon. The total length of a bar was 36 inches, and it was #‘inch diameter, with # inch squared ends. In order to obtain a 910 Mr. W. A. Scoble: Further Tests of material which was perfectly elastic to fracture it was neces- sary to harden the bars. Unfortunately a suitable furnace was not available for heating them. The assistance of an experienced toolsmith was obtained, and a spread fire was arranged on a smith’s hearth. A long iron tube was heated as uniformly as possible in the fire over a greater length than that of a bar. A bar was placed inside the tube, and was frequently moved along it, and rotated, further to ensure uniform heating. When the bar had attained a suitable temperature, about 1400° Fahr., it was removed from the tube and lowered into dilute vitriol. The bar was kept vertical in order to keep it straight ; it was washed later in strong soda water to neutralize the acid from the hardening bath. The temperature of a bar before it was quenched was not measured, it was only judged from its appearance. It was, therefore, unlikely that the bars were of equal strength after hardening, and it was decided to make more than one test on each bar, so that each specimen would afford inde- pendent evidence on the law of failure. The Apparatus and Method of Testing. The apparatus employed has been fully described else- where”. One end of a bar was supported so that it was free to take its natural slope under a bending load, but it was not allowed to twist. The bar rested on rollers at 30 inches from the other support, and therefore was free to twist at this end. A wooden pulley fitted on the squared end of the specimen and was twisted by means of ropes which carried dead weights. A bending load was applied at the centre of the bar. The maximum shear stress due to the torque was confined to the whole of the outer surface of a bar, and the bending load produced its maximum compressive stress only at the highest point, and its maximum tensile stress only at the lowest point, of the mid-section of the specimen. ‘The test of a bar under combined loading was made first. for a further torsion test one part of the original bar was held in two hardened, ser- rated grips, or clamps, bolted together. The clamps were bolted to a lathe bed with the bar just resting on the roller bearing. The squared end of the half bar then carried the torsion pulley as before. When it was necessary to make a further bending test, a part of the original bar was supported on knife edges and loaded by weights applied at its centre. * Proc. Phys. Soc. London, vol. xx.; also Phil. Mag. Dec. 1906. Brittle Materials under Combined Stress. 911 Calculation of the Stresses. The maximum stress due to bending was calculated from the formula oe: =7, ers i in which p is the maximum stress. y is the greatest distance from the neutral line of the section = —. g M is the maximum bending moment acting on the beam. I is the moment of the inertia of the section about its ad’ 64° d is the diameter of the bar. neutral line= The maximum shearing stress due to the torque was given by as Lér =o a TC S is the maximum shearing stress. T is the torque. , anne : OD tGr ) p “ The maximum principal stress P; was : +a/ = +7, and 2 is . : D / ‘ the minimum principal stress P; was £ = - +87. The oS stress difference was 24 // +S?, and the maximum shear a : stress i +S", If the maximum principal stress be represented by P,, and the least principal stress at the same point by Ps, then the maximum stress law states that P,;=constant; the maxi- mum strain theory is represented by P,—7P;=constant, since in these tests the second principal stress, Ps, is zero; and the shear stress being constant corresponds to P,—P;=constant. Data from the Tests. Stress-strain diagrams for a pure torsion test and a simple bending test have been plotted in fig. 2. The diagram shows Se 912 Mr. W. A. Scoble: Further Tests of that under either kind of loading, stress and strain were proportional to fracture, and therefore it was permissible to calculate the stresses from formule based on Hooke’s law. eee Smee By enpinc i | Va a Re fee abel akc Abs Rice my Leahey It was not intended to compare the results from different hars, but the several tests on each particular specimen only. It was convenient, however, to divide the bars in three groups depending on their strengths. The remarks were written against the data immediately after the tests were made, before the results were calculated and plotted. They are intended briefly to record observations made during the tests. or example, the bending tests were made in certain cases on twelve inch lengths, and therefore the bending moments recorded at fracture were probably high. In some of the pure torsion tests the bar fractured at two sections, and the test was marked correct. Twelve specimens were prepared, but four of them developed flaws in hardening. Brittle Materials under Combined Stress. 913 Bar. —a— Co G5 CO Bending Moment. lbs, in. 2,960 3,970 1,770 2,880 0 2,330 0 3,830 2,360 0 3,450 0 3,090 4,500 4,650 Hardened Cast Steel Bars. 2 Torque. Ibs. in. 6 ,200 ) 3,940 5,860 0 4,750 6,280 0 6,020 0 2,960 0 1,660 4,190 0 3,900 Tensile stress inch diameter, Shear stress due to Bending.| due to Torque. Ibs. /sq. in. 12,720 72,000 86,850 43,000 70,000 0 109,400 113,000 53,850 0 lbs./sq. in. Minimum Principal Stress. bos. /sq. im, | Maximuin Principal Stress. lbs./sq. in. | 79,400 82,060 | —69,340 i) 72,000 0 | ) 86,850 | 0 61,200 86,450 | 43,450 | 0 70,000 ) | 78,000 78,000 —78,000 | 47,900 84,350 — 27,150 71,250 71,250 == 71,95) Q 81,000 0 57,700 93,200 — 35,800 76,400 76,400 = 76-400 4 0 83,900 ) 73,200 73,200 | =—7o2ny | 0 75,400. | 0 | 36,000 120.200 | —10,800 0) 113,000 Q 20,200 59,600 — 6,800 50,S00 50,900 | —50.900 ) 53,850 0 47,500 47,500 — 47,500 Stress difference = twice Max. Shear Stress. 151,400 72,000 86,850 129,900 70,000 156,000 111,500 142,500 81,000 129,000 152,800 83,900 i a 146,400 75,109 131,009 113,000 66,400 101,800 CS 53,850 95,000 Kittective length as a beam 80 inches. Remarks. Low. Correct. High. Low. Correct, Low. Correct, Low. ; Sam | High. Correct. Low. High. Correct. | High. Correct, High. Correct. Correct. ical. Correct. Correct. 914 Mr. W. A. Scoble: Further Tests of The Results Considered. It is evident that the stress difference or shear stress law does not apply to brittle materials. The maximum principal stress is approximately constant, but as the results have been set down in the order in which the tests were made, it is not easy to determine whether the value of this stress at fracture depends on the least principal stress, P3. The bending and twisting moments at fracture have been plotted in fig. 3. The number of the bar tested is given ot a hh eee Sh a “F000 ""¢ uA fe 4000 Benoinc Moment O 1000 a LBS.INS. against a point, and the letter indicates whether the result was probably high, correct, or low, and corresponds to the remark in the table. If the maximum principal stress were constant at rupture, the points for a bar would lie on an Brittle Materials under Combined Stress. 915 ellipse. Three such mean curves have been drawn in, one for each group of specimens. Considering the difficulties encountered in making the tests, the points ‘lie fairly evenly about the curves. The deviation is most noticeable for the group which contains bars 6 and 11. The author has expressed the opinion that the behaviour of all isotropic, homogeneous materials may be expressed in one form, that P,;~—mP;=constant, at the failure of the material, in which equation “‘m” is a constant whose value depends on the degree of ductility of the material*. P, does not appear in the equation because it is zero for the system of loading adopted, but even in the more general ease, in which there are three principal stresses, the available evidence indicates that the intermediate stress P, does not affect the values of P, and P; at failure, with the loadings which are met with in engineering practice. The advantage of the above equation is that when m=0, it represents the maximum principal stress hypothesis, when m=~y, it indicates a constant maximum strain, and m=1 corresponds to the stress difference law. ‘The maximum and the least principal Tig. 4. Haroeneo Cast Steet X2H 3 Maximum &Mavitum Paincipar Stresses 3S & Maximum Principal STRESS /ta00. S ~3 -7 -6 -5 ~4 3 =p aT Minimum PrincipacStress /ioo00 = 485/S.IN. stresses at rupture have been plotted in fig. 4. This least principal stress is the P; at the point at which the maximum * Proc. Phys. Soc, London, vol. xxii. ; also Phil. Mag. Jan. 1910. eee —— 916 Tests of Brittle Materials under Combined Stress. principal stress operates, and although it has been called the minimum principal stress, it is not the algebraical minimum for the whole bar, but only at the point at which rupture commences. The notation in fig. 4 is similar to that of fig. 3. It is easy to distinguish the three groups of tests. An increase of the minimum, P;, does not regularly affect the maximum principal stress, Py}. The maximum strain is certainly not constant for a particular bar, and varies more than the maximum principal stress. Although the points are very irregular in this diagram, they are not more so than those usually obtained for a ductile material. The angles between the surfaces of the fractures and the axes of the bars were measured, and agreed with those of the planes of maximum stress even better than in the case of the earlier tests on cast iron. For example, ina pure torsion test the angle which the plane of maximum stress makes with the axis of the bar is 45°. The corre- sponding angles measured from the fractured surfaces in five pure torsion tests were 43, 46, 44, 44, and 44 degrees. It was possible to measure the angles very accurately because the fractures were wonderfully clean, and as true as if the edges had been cut with a tool. Conclusion. The tests show that the maximum principal stress is the best criterion of strength fora brittle material under combined stress. It is interesting to compare the strengths of the bars before and after hardening. None of these specimens were tested unhardened, but other steel bars, of exactly the same size, yielded at 2400 lbs. inches torque, or 2660 Ibs. inches bending moment*. The values for the present material should probably be rather higher. It therefore appears that when the bars are made very hard the bending moment which they are capable of withstanding without fracturing is slightly greater than that at yield in the unhardened con- dition. But sometimes the strength to resist bending is diminished by the quenching. Similarly, as an average result, it may be stated that a hardened bar can withstand about double the torque which would cause it to yield when in its ductile state. * Proc. Phys. Soc. London, vol. xx. ; also Phil. Mag. Dec. 1906. EA O17 XCVIIL. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 823.] January 12th, 1910.—Prof. W. J. Sollas, LL.D., Se.D., F.R.S., President, in the Chair. a following communications were read :— 1. * On the Igneous and Associated Sedimentary Rocks of the Glensaul District (County Galway).’ By Charles Irving Gardiner, M.A., F.G.S., and Prof. Sidney Hugh Reynolds, M.A., F.G.S.; with a Palzontologieal Appendix by Frederick Richard Cowper Reed, M.A., F.G.S. 2. ‘On the Gneisses and Altered Dacites of the Dandenong District (Victoria), and their Relations to the Dacites and to the Granodiorites of the Area.’ By Prof. Ernest Willington Skeats, Pesce, 4.4.0.8., F.G.S. The area described lies about 25 miles south-south-east of Melbourne. ‘The earlier literature is discussed, and it is shown that the early geological surveyors regarded the dacites as Paleozoic ‘traps’ passing gradually into the granodiorites. Prof. J. W. Gregory first described the rocks as dacites, probably of Lower Tertiary age, resting upon the denuded surface of the granodiorites and of the adjoining Lower Palxozvic sediments. ‘The author describes the field-relations of the rocks, and shows that gneiss occurs between the dacite and the granodiorite in places. Else- where at the contact the dacite appears slightly altered. The contact with the plutonic rock is everywhere abrupt. No foliation or banding occurs in the granodiorites, but acid veins pass from the junction into the altered dacite and also cut across the foliations of the gneiss. ‘The field-evidence, therefore, shows that the dacites are older than the granodiorites, and also that the gneiss was formed before the intrusion of the acid veins. Chemical analyses of the rocks and of the coloured minerals of the dacites are recorded. The chemical evidence indicates that slight differentiation of a magma took place: the dacite was first erupted, and, following shortly on that, the granodiorite (of slightly more acid composition) was intruded into the dacite. ‘he microscopic characters of the eranodiorite, the dacite, the altered dacites, and the gneiss are described. In the altered dacites a slight banding or schistosity occurs near the contact, ilmenite is changed to secondary biotite by reaction with the felspar in the microgranular ground-mass, biotite is corroded by the attack of the ground-mass, and hypersthene is altered at its margin to secondary biotite and secondary quartz. 918 Geological Society. Finally, minute granules of blue tourmaline occur in the contact- rocks, All the changes enumerated above are attributed to contact- metamorphism caused by the intrusion of the granodiorite. In the gneiss, hypersthene is not found, ilmenite is rare, and the rock is completely foliated. It shows a granular ground-mass similar to, but coarser in grain than, the ground-mass of the dacite. Besides occurring at the contact, it has also been found in parallel zones intercalated in dacite near the contact. The author believes tnat the gneiss is a peculiar modification of the dacite, but direct evidence as to its mode of origin is as yet incomplete. It may possibly be the result of extreme contact- metamorphism of a dacite of peculiar character, such as a tuff. It is possible that it was produced by differential movement in the dacite before complete consolidation, and certainly before the intrusion of the granodiorite. Since, however, dynamic effects are present in some sections, and are accompanied by changes found in ~the dacites altered by contact-metamorphism, the author is rather inclined to support the view that primarily the gneiss is due to differential movements in part of the dacite series, complicated by effects due to contact-metamorphism by the later intrusion of the granodiorite. 3. ‘Recent Improvements in Rock-Section Cutting Apparatus.’ By H. J. Grayson, Demonstrator of Petrology and Assistant in the Geological Department, University of Melbourne. The apparatus described has been designed and constructed by the author, for use in the University of Melbourne. It comprises a slitting disc of mild steel and two bronze grinding laps, mounted on a very substantial wooden table. ‘The discs and laps are each 10 inches in diameter, and revolve at about 900 revolutions a minute. The discs and laps are connected with endless belts, which in turn are connected with wheels driven by a 1-horsepower electric motor. Special clamps are used to attach the rock-specimen and to cut the slice. A goniometric crystal-holder, permitting of slicing in any desired direction, is described, and can be fitted to one of the clamps. Clamps swinging radially across the grinding laps permit the parallel grinding of the slice to any required thinness. A polishing lap can be placed in the position of one of the grinding laps. The finishing of the slice is done by hand on a slate disc. In the second part of the paper, the author describes in detail the method which he employs in making a rock-section, and refers to a number of improved methods or variations of the usual processes which he has in practice found advantageous. a a ee ze - Pear eee ee wl itl ti a eee in - “EB Mee > &. i a Ce-LN DEX#To VOL. XIX. ~~ so [3 & a *, “os oe sit ge a Dy apm Ro —_<_ ABSORPTION spectra of uranous and uranyl compounds, on the, 566. Ather, on spontaneous generation of electrons in an elastic solid, 129; on the, 181; on the sun’s motion with respect to the, 417; on the relative motion of the earth and, 809. Air, on the effect of dust and smoke on the ionization of, 657. Alpha rays, on the action of, on glass, 192. Alternating current spark potentials, on, 203. Andrew (A. R.) on the geology of Nyasaland, 653. Anomalous dispersion by potassium vapour, on, 200. Antonoff (G. N.) on radium D and its products of transformation, 825. Atmosphere, on the relation between periodic variations of pressure, temperature, and wind in the, 26, 448, Atmospheric electricity, on, 715. Atomic vibration, on the funda- mental constant of, 1. Atoms and molecules, on the nature of the forces of attraction between, 783. Bailey (E. B.) on the cauldron subsidenee cf Glen Coe, 334. Bailey (T. E. G.) on the geology of Nyasaland, 653, Balmer’s formula, analysis of, 21. Bateman (H.) on the velocity of propagation of an_ earthquake wave in the interior of the earth, 576; on the reflexion of light at a plane mirror, 824. Baynes (R. E.) on saturation specific heats with van der Waals’s and Clausius’s characteristics, 407. kinematical Ber and bei functions, tables of the, 49. Berry (G. H.) on the striking point of pianoforte strings, 648. Bessel functions, on the asymptotic expansions of, 228. Beta rays, on the absorption of, by solutions and liquids, 605. Bevan (P. V.) on the absorption spectrum of potassium vapour, 195. aa law of probability, on, 712. Books, new:—Bouasse’s Oours de Physique, pt. vi., 331; Results of Observations made at the Coast and Geodetic Survey Magnetic Observatory at Cheltenham, Mary- land, 1905 and 1906, 331; Raffety’s Introduction to the Science of Radioactivity, 331 ; The Scientitic Papers of Sir William Huggins, 441; Cavalier’s Lecons sur les Alliages métalliques, 444; Cramp and Smith’s Vector and Vector Diagrams applied to the altern- ating Current Circuit, 445; Cool- idge’s Klements of Non-Euclidean Geometry, 445; Kohlrausch’s Lehrbuch der Practischen Physik, 446; Clark’s Physical Science in the time of Nero, 651; Joly’s Radioactivity and Geology, 819. Borodowski (W. A.) on the absorp- tion of 8 rays from radium by sulutions and liquids, 605. Burbury (8. H.) on Boltzmann’s law of probability, 712. Burton (Dr. C. V.) on a gravita- tional problem, 91; on the sun’s motion with respect to the ether, 417. Cadmium amalgams and the Weston normal cell, on, 250 Callendar (Prof. H. L.) on electrical recording thermometers for clinical work, 538, aD OO EEE e_=-=-=---- 920 INDEX. Campbell (A.) on the use of mutual inductometers, 497. Campbell (Dr. G. A.) on telephonic intelligibility, 152. Campbell (N.' on the principles of dynamics, 168 ; on the ether, 181. Carus-Wilson (C.) on the pitting of flint-surfaces, 336. Chalmers (S. D.) on the sine con- dition in relation to the coma of optical systems, 356. Chapman (J. C.) on secondary homo- geneous X radiation, 897. Chattock (Prof. A. P.) on the pressure of the electric wind in hydrogen containing traces of oxygen, 449. Cloud method of determining the elementary electrical charge, on a new modification of the, 209. Clough (C. T.) on the cauldron sub- sidence of Glen Coe, 334. Coherers, on, 569. Coil, on the effective resistance and inductance of a helical, 77. Coma of optical systems, on the sine conditions in relaticn to the, 356. Comets, on the size of the tail- particles of, 626. Condenser, on the energy of a double- layer, of electronic origin, 573. Conductivity, on a new method of determining thermal, 587. - Copper under combined stress, on, 116. Corpuscular radiation, on homo- geneous, 337. Crystals, on the regularity of structure of actual, 96. Dielectric capacity, on the nature of, 1. Dielectrics, on the polarization of, in a steady field of force, 390. Disk resistance, on the variation of, with temperature, in water, 513. Disturbance, on the instantaneous propagation of, in a dispersive medium, 160. Double-layer condenser of electronic origin, on the energy of a, 573. Dow (J. S.) on the physiological principles underlying the tlicker photometer, 58. Ductile materials under combined stress, on, 116. Dust, on the effect of, on the ioniza- tion of air, 657. Dynamics, on the principles of, 168, Earth, on the bending of electric waves round the, 276, 435, 757; on the relative motion of the, and zether, 809. Earth-air electric currents, on, 715. Earthquake-wave, on the velocity of propagation of an, 576, Eccles (Dr. W. H.) on coherers, 869. Electric charge, on a new modifi- cation of the cloud method of measuring the elementary, and its most probable value, 209. currents, on earth-air, 715. effect of the ultra-violet spec- trum, on the, 290. field, on a theory of the struc- ture of the, and its application to Hontgen radiation and light, 301. recording thermometers for clinical work, on, 538. resistance, ou the change of, of metals in a magnetic field at dif- ferent temperatures, 685. spark, on the constitution of the, 285. waves, on the bending of, round the earth, 276, 435, 757 ; rourd a large sphere, on the, 516. wind, on the pressure of the, in hydrogen, 449. Electricity, on rays of positive, 424. Electrons, on spontaneous generation of, in an elastic solid zther, 129; on the charge of, 438. Electroscopic measurements, on yari- ations observed in, and their pre- vention, 278. Electrostatic effect of a changing magnetic field, on the, 461. Elsden (J. V.) on the geology of the neighbourhood of Seaford, 333. Eve (Dr. A. 8.) on the effect of dust and smoke on the ionization of air, 657. i Fearnsides (W. G.) on the Howsgill fells, 447. Finlayson (A. M.) on the nephrite and magnesian rocks of New Zealand, 207; on the metallogeny of the British Isles, 821. Fletcher (A. L.) on pleochroic halos, 630. Flicker photometer, on the physio- ea principles underlying the, 58. INDEX. 921 Flint-glass, on the polarization of, in a steady field of force, 392. Fluid, on the inversion-points for a, passiug through a pcrous plug, 888. Friction in gases at low pressures, on, 376. Gamma rays, on the question of the homogeneity of, 725. Gases, on the theory of the small ion in, 201, 817; on friction in, at low pressures, 376, Geological Society, proceedings of the, 206, 352, 447, 653, 820, 917. Gibson (Prof. A. H.) on the vari- tion of disk resistance with tempe- rature in water, 513. Gill (E. W. B.) on the electrical effect of the ultra-violet spectrum, 290. Glass, on the action of the a rays on, 192; on the polarization of flint, in a steady field of force, 396. Gold (E.) on the relation between periodic variations of pressure, temperature, and wind in the atmosphere, 26, 448. Gravitational problem, note on a, ol. Grayson (H. J.), recent improve- ments in rock-section cutting apparatus, 918. Halos, on pleochroic, 327, 630. Hargreaves (R.) on the ignoration problem, 486. Harrison (W. J.) on the decay of _ Waves in a canal, 906. Havelock (Dr. T. H.) on the instan- taneous propagation of disturbance in a dispersive medium, 160. Helical coil, on the effective resistance and inductance of a, 77. Heslop (Miss M. K.) on the dyke at Crookdene, 654. Hoge (J. L.) on friction in gases at low pressures, 376. Hollnagel (H.) on measurements in the extreme infra-red spectrum, 761. Houstoun (Dr. R. A.) on the damping of long waves in a rectangular trough, 205. Hume (Dr. W. F.) on the granite- ridges of Kharga oasis, 448. Hydrogen, on the pressure of the electric wind in, 449. Ignoration problem, cn the, 486. Phil. Mag. 8. 6. Vol. 19. No. Inductometers, on the use of mutual, 497, Infra-red spectrum, on measure- ments in the, 761. Inversion-points for a fluid passing through a porous plug, on the, 888. Ion, on the theory of the small, in gases, 201, 817. Ionization of air, on the effect of dust and smoke on the, 657. Tron wires, on effects of bending overstrain in soft, 619. Je1vis-Smith (F. J.) on an optical method of reading the torsional angle of a rotating shaft, 300. Joly (Prof. J.) on pleochroic halos, 327, 630, Jones (Prof. H. C.) on the absorp- tion spectra of uranous and uranyl compounds, 566. Jones (O. T.) on the Hartfell- Valentian succession around Plyn- limon, 332. Jupiter, on photographs of, taken at the Lowell Observatory, 488. Ker and kei functions, tables of the, Kleeman (Dr. R. D.) on the nature of the forces of attraction between atoms and molecules, 783; on the radius of the sphere of action of a molecule, 840. Kuehne (J. M.) on the electrostatic effect of a changing magnetic field, 461. Kuenen (Prof. J. P.) on Pirani’s method of measuring self-induct- ance, 439. Kundt's tube, on Ko6nig’s theory of the ripple formation in, 476. Laws (8. C.) on the change of resist- ance, of metals in a magnetic field at different temperatures, 685. Lees (Dr. C. H.) on the laws re- garding the direction of thermo- electric currents, 508. Lewis (Dr. W. C. MeC.) on the energy of a double-layer condenser of electronic origin, 573. Light, on a theory of the structure of the electric field and its appli- cation to, 391 ; on the reflexion of, at a moving mirror, 824. Linities of the mth order, on the theory of, 144. 30 114. June 1910. 922 LN DEX. Liquids, on the absorption of f rays by, 605. Lowell (Dr. P.) on photographs of owe taken at the Lowell Observatory, 488; on the limits of the oblateness of a rotating planet, 700. McAulay (Prof. A.) on spontaneous generation of electrons in. an elastic solid ether, 129. Magnetic field, on the electrostatic effect of a changing, 461: on the change of resistance of metals in a, 685. Makower (Dr. W.) on the recoil of radium OU from radium B, 100. Marr (Dr. J. E.) on the Howgill fells, 447. Materials, tests of brittle, under combined stress, 908. Mennell (F. P.) on the geological structure of Southern Rhodesia, 822. Metals, on the change of resistance of, in a magnetic field, 665. Millikan (Prof. R. A.) on a new modification of the cloud method of determining the elementary electrical charge and the most ' probable value of that charge, 209; Molecular diameters, on, 25. Molecules.and atoms, on the nature of the forces of attraction between, 783; on the radius of the sphere of action of, 840. Muff (H. B.) on the cauldron sub- sidence of Glen Coe, 384. Nettleton (H. R.) on a new method of determining thermal conduc- tivity, 587. Nicholson (Dr. J. W.) on the effec- tive resistance aud inductance of a helical coil, 77; on the asymp- totic expansions ‘of Bessel func- ‘tions, 228; on the bending of electric: waves round the earth, 276, 435, 757; on the bending of electric waves round a large sphere, 516; on the size of the © tail-particles of comets, and their scattering effect on sunlight, 626. Obiateness, on the limits of the, of a rotating planet, 700. Optical x method of reading the tor- sional angle of a rotating shaft, on an, 300. Optical systems, on the sine con- dition in relation to the coma of, ~ 356. Orstrand (C. E. Van) on reversion of power series, 366. Overstrain, on effects of bending, in soft iron wires, 619. Pendulum, on the motion of a, swinging through an arc of finite magnitude, 851. Perrin (Prof. J.) on the charge of the electron, 438. Perry (Prof. J.) on telephone cir- cuits, 673. Photometer, on the physiological principles underlying the flicker, 58. Pianoforte strings, on the striking point of, 648. Piper (S. H.) on secondary homo- geneous X radiation, 897. Pirani’s method of measuring self- inductanée, on, 439. Pitchblende, on the rate of evolution of heat by, 314. Planet, on the limits of the oblate- ness of a rotating, 700. Planetary distances, on a new binary progression of the, 597. Pleochroic halos, on, 327, 650. Polarization of dielectrics in a steady field of force, on the, 390. Polonium, on the rate of decay of the radioactivity of, 905. ~ Poole (H. H.) on "the rate of evolu- tion of heat by pitchblende, 314. Porous plug, on the inversion-points ~ for a fluid passing through a, 8&8. Porter (Prof. A. W.) on the inver- sion-points for a fluid passing through a porous plug, 888. Positive electricity, on rays of, 424. Potassium vapour, on the absorption spectrum of, 195. Power series, on reversion of, 366. Probability, on Boltzmann’s law of, 712. Pump, on an automatic Toepler, 863. Quartz, on the polarization of, in a steady field of force, 392. Radioactive minerals in common rocks, on, 908. Radioactivity of polonium, on the rate of decay of the, 905. Radium, on the absorption of 6 rays from, ’ by solutions and liquids, 605. INDEX: 923 Radium ©, on the recoil of, from radium B, 100. D and its products of trans- formation, on, 820. Rastall (H. R.) on the boulders of the Cambridge drift, 206; on the Skiddaw granite and its meta- morphism, 820. Rayleigh (Lord) on the regularity of structure of actual crystals, 96. Reversion of power series, on, 366. Ripple formation in Kundt’s tube, on K6nig’s theory of the, 476. Robinson (Dr. J.) on Konig’s theory of the ripple formation in Kundt’s tube, 476. Rocks, on radioactive minerals in common, 903. Romanes (J.) on the boulders ot the Cambridge drift, 206. Réntgen radiation, on a theory of the structure of the electric tield and its application to, 301: on,337. Rose-Innes (J.) on the motion of a pendulum swinging through an arc of finite magnitude, 851. Royds (1.) on the constitution of the electric spark, 285. tubens (Prof. H.) on measurements in the extreme infra-red spectrum, 761. Russ (Dr. 8.) on the recoil of radium C from radium B, 100. Russell (Dr. A.) on alternating current spark potentials, 203. Russell (A. 8.) on variations ob- served in electroscopic measure- ments and their prevention, 278: on the question of the homo- geneity of y rays, 725; on the constant of uranium X, $47. tutherford (Prof, Ie.) on the action of the a rays on glass, 192. Sadler (C. A.) on homogeneous cor- puscular radiation, 337. Savidge (H. G.), tables of the ber and bei and ker and kei functions, with further formule for their computation, 49. Scoble (W. A.) on ductile materials under combined stress, 116; on tests of brittle materials under combined stress, 908. Serivenor (J. B.) on the rocks of Pulau Ubin and Pulau Nanas, 654; on the turmaline-corundum rocks of Kinta, 655. Self-inductance, on Pirani’s method of measuring, 439. Shaft, on an optical method of reading the torsional angle of a rotating, 300. Simpson (Dr. G. C.) oa earth-air electric currents, 715. Sine condition in reiation to the coma of optical systems, on the, 356. Skeats (Prof. Ii. W.) on the gneisses and altered dacites of the Dan- denong district, 917. Smith (F. HE.) on cadmium amal- eams and the Weston normal cell, 250. Smoke, on the effect of, on the ionization of air, 657. Smythe (Dr. T, A.) on the dyke at Crookdene, 654. Soddy (F.) on the question of the homogeneity of y-rays, 725; on the constant of uranium X, 847. Soddy (W. M.) on the question of the homogeneity of y rays, 725. Solar system, on the mutability of the, 597. Solutions, on the absorption of 8 rays by, 605. Spark, on the constitution of the electric, 285. Spark potentials, on current, 203. Specific heats, on saturation, with van der Waals’s and Clausius’s characteristics, 407. Spectra of uranous and uranyl com- pounds, on the absorption, 566. Spectrum, on the absorption, of potassium vapour, 195; on the electrica! effect of the ultra-violet, 290; on measurements in the extreme infra-red, 761. Sphere, on the bending of electric waves round a large, 516. Steel under combined stress, on, 116. Steele (Dr. B. D.) on an automatic Toepler pump, 863. Steven (A. I.) on anomalous effects on first loading a wire, 619. Stress, on ductile materials under combined, 116; on tests of brittle materials under combined, 908. Strong (W. W.) on the absorption spectra of uranous and uranyl compounds, 566. ; alternating eee te et) et os oe 924 INDEX. Sun's motion with respect to the iether, on the, 417. Sutherland (W.) on the fundamental constant of atomic vibration and the nature of dielectric capacity, 1; on molecular diameters, 25; on the theory of the small ion in gases, 817. Telephone circuits, on, 673. Telephonie intelligibility, on, 152. Thermal conductivity, on a new method of determining, 587. Thermo -electric currents, on the direction of, 508. Thermometers, on electrical record- ing, for clinical work, 538. Thomson (Sir J. J.) on a theory of _ the structure of the electric field and, its application to Rontgen radiation and to light, 301; on rays of positive electricity, 424. ‘Thornton (Prof. W. M.) on the polarization of dielectrics in a steady field of force, 390. Toepler pump, on an automatic, 863. Torsional angle of a rotating shaft, on an optical method of reading the, 300. Tyndall (A. M.) on the pressure of the electric wind in hydrogen . containing traces of oxygen, 449. Ultra-violet spectrum, on the elec- trical effect of the, 290. Uranium X, on the constant of, 847. Uranous and uranyl compounds, on the absorption spectra of, 566. Water, on the variation of disk resistance with temperature in, 513. Waters (J. W.) on radioactive mine- rals in common rocks, 903; on the rate of decay of the radio- activity of polonium, 905. Waves, on the damping of long, in a rectangular trough, 205; on the bending of electrical, round the earth, 276, 4385, 757; round a large sphere, on the, 516; on the decay of, in a canal, 906. Wellisch (Ii. M.) on the theory of the small ion in gases, 201. Weston normal cell, on cadmium amalgams and the, 250. Wilde (Dr. H.) on a new binary progression of the planetary dis- tances, and on the mutability of the solar system, 597. Wilson (Prof. H. A.) on the relative motion of the earth and ether, 809. Wind, on the relation between peri- odic variations of pressure, tem- perature and, in the atmosphere, 26, 448; on the pressure of the electric, in hydrogen, 449. Wire, on anomalous effects on first loading a, 619. : X radiation, on secondary homo- geneous, 897. END OF THE NINETEENTH VOLUME, Printed by Tayior and Francis, Red Lion Court, Fleet Street. SMITHSONIAN INSTITUTION LIBRARIES MADINA 3 9088 01202 4857