PHILOSOPHICAL

TRANSACTIONS

I OP THE

EOYAL SOCIETY

OF

* ,

LONDON.

(A.)

FOR THE YEAR MDCCCLXXXVIII.

VOL. 179.

LONDON:

PRINTED BY HARRISON AND SONS, ST. MARTINS LANE, W.C.,

printers in Oruhurn to jptr

MDCCCLXXMX.

810250 4V

ADVERTISEMENT.

THE Committee appointed by the Royal Society to direct the publication of the
Philosophical Transactions take this opportunity to acquaint the public that it fully
appears, as well from the Council-books and Journals of the Society as from repeated
declarations which have been made in several former Transactions, that the printing of
them was always, from tune to time, the single act of the respective Secretaries till
the Forty- seventh Volume ; the Society, as a Body, never interesting themselves any
further in their publication than by occasionally recommending the revival of them to
some of their Secretaries, when, from the particular circumstances of their affairs, the
Transactions had happened for any length of time to be intermitted. And this seems
principally to have been done with a view to satisfy the public that their usual
meetings were then continued, for the improvement of knowledge and benefit of
mankind : the great ends of their first institution by the Royal Charters, and which
they have ever since steadily pursued.

But the Society being of late years greatly enlarged, and their communications more
numerous, it was thought advisable that a Committee of their members should be
appointed to reconsider the papers read before them, and select out of them such as
they should judge most proper for publication in the future Transactions ; which was
accordingly done upon the 26th of March, 1752. And the grounds of their choice are,
and will continue to be, the importance and singularity of the subjects, or the
advantageous manner of treating them ; without pretending to answer for the
certainty of the facts, or propriety of the reasonings contained in the several papers
so published, which must still rest on the credit or judgment of their respective
authors.

It is likewise necessary on this occasion to remark, that it is an established rule of
thr ifock'ty, t.i \\hkli tiny will always adhere, never to give their opinion, as a Body,

a -j

L iv 1

upon any subject, either of Nature or Art, that conies before them. And therefore the
tlniiks, which are frequently proposed from the Chair, to be given to the authors of
such papers as are read at their accustomed meetings, or to the persons through whose
hands they received them, are to be considered in no other light than as a matter of
civility, in return for the respect shown to the Society by those communications. The
like also is to be said with regard to the several projects, inventions, and curiosities of
various kinds, which are often exhibited to the Society ; the authors whereof, or those
who exhibit them, frequently take the liberty to report, and even to certify in the
public newspapers, that they have met with the highest applause and approbation.
And therefore it is hoped that no regard will hereafter be paid to such reports and
public notices; which in some instances have been too lightly credited, to the
dishonour of the Society.

LIST OF INSTITUTIONS ENTITLED TO RECEIVE THE PHILOSOPHICAL TRANSACTIONS OR

PROCEEDINGS OF THE ROYAL SOCIETY.

Institution* marked A are entitled to receive Philosophical Transactions, Series A, and Proceeding*.
„ „ B „ „ „ „ Series B, and Proceedings.

„ ., AB „ „ „ „ Scried A and B, and Proceedings.

„ „ p „ „ Proceedings only.

America (Central).
Mexico.

p. Sociedad Cicntifica " Antonio Alzate."
America (North). (See UNITED STATES.)
America (South).
Buenos A v res.

AB. Museo Publico.
Caracas.

B. University Library.
Cordova.

AB. Aoadomia NaciontU de Cionoias.
Rio dc Janeiro.

p. Imperial Observatorio.
Australia.
Adelaide.

p. Royal Society of South Australia.
Brisbane.

p. Royal Society of Queensland.
Melbourne.

p. Observatory.

p. Royal Society of Victoria.

AH. University Library.
Sydney.

p. Linnean Society of New South Wales.

AB. Rojal Society of New South Wales.

AB. University Library.
Austria.
Agram.

p. Societas Historico-Natnralis Croatica.
Briinn.

AB. Naturforschender Verein.
GraU.

AB. Natnrwisseuschaftlicher Verein fur Steier-

mnrk.
Hermanns tad t.

p. .SiolxMilmririsclier Verein t'iir die N;i'm-
wuBeuschaften.

Austria (continued).
Innsbruck.

AB. Das Ferdinandcum.

p. Natnrwissenschaftlich - Mbdicinischer

Verein.
Klansenbarg.

AB. Az Erdelyi Muzeam. Das siebouburgibche

Museum.
Prague.

AB. Knnigliche Bohmische Gesellschaft der

Wissenschaften.
Schemnitz.

p. K. Ungarische Berg- nnd Fdret-Akadeniie.
Trieste.

B. Museo di Storia Natural!.

p. Societa Adriatica di Scienze Natorali.
Vienna.

p. Anthropologische Oesellschaft.

AH. Kaiserlicho Akadcmic der Wissensclmften.

p. K.K. Geographische Gesellschaft.

AB. K.K. Geologische Reichsanstalt.

B. K.K. Zoologisch-Botanische Gesellschaft.

B. Naturhifitorisches Ilof- Museum.

p. CEsterreichische Gesellschaft fur Metooro-

logie.
Belgium.
Brussels.

B. Academic Royale de Medecine.

AB. Academic Royale des Sciences.

B. Museo Royal d'Histoire Natarelle de
Belgiqne.

p. Observatoire Royale.

p. Societe Malacologiqne de Belgique.
Ghent.

AB. University.
Liege.

AB. Societe des

[ vi ]

Belgium (continued).
Liege (continued).

p. Societe Geologiqne de Belgiqne.
Loavain.

AB. L'Universite.
Canada.
Hamilton.

p. Scientific Association.
Montreal.

AB. McGill College.

p. Natural History Society.
Ottawa.

AB. Geological Survey of Canada.
Toronto.

p. Canadian Institute.

AH. Univei-sity.

Cape of Good Hope.

A. Observatory.

AR. South African Library.
Ceylon.
Colombo.

B. Museum.
China.

Shanghai.

p. China Branch of the Royal Asiatic Society.
Denmark.
Copenhagen.

AB. Kongelige Dauske Videiiskabernes Selskab.
England and Wales.
Birmingham.

AB. Free Central Library.

AB. Mason College.
Bristol.

p. Merchant Venturers' School.
Cambridge.

AB. Philosophical Society.

p. Union Society.
Cooper's Hill.

AB. Engineering College.
Dudley.

p. Dudley and Midland Geological and

Scientific Society.
Essex.

p. Essex Field Club.
Greenwich.

A. Observatory.
Kcw.

B. Royal Gardens.
Leeds.

p. Philosophical Society.
AB. York>hire College.
Liverpool.
AB. Free Public Library.

ll torio >"<-.• i y nf Laoouhin and Cheshire.

England and Wales (continued).
Liverpool (continued).

p. Literary and Philosophical Society.

A. Observatory.

AB. University College.
London.

AB. Admiralty.

p. Anthropological Institute.

B. British Museum (Nat. Hist.).

A. Chemical Society.

p. " Electrician," Editor of the.

B. Entomological Society.
AB. Geological Society.

AB. Geological Survey of Great Britain.

p. Geologists' Association.

AB. Guildhall Library.

A. Institution of Civil Engineers.

A. Institution of Mechanical Engineers.

A. Institution of Naval Architects.
p. Iron and Steel Institute.

B. Linnean Society.
AB. London Institution.
p. London Library.

A. Mathematical Society.

p. Meteorological Office.

p. Odontological Society.

p. Pharmaceutical Society.

p. Physical Society.

p. Quekett Microscopical Club.

p. Royal Asiatic Society.

A. Royal Astronomical Society.

B. Royal College of Physicians.
B. Royal College of Surgeons.

p. Royal Engineers (for Libraries abroad, six
copies).

AB. Royal Engineers. Head Quarters Library.

p. Royal Geographical Society.

p. Royal Horticultural Society.

p. Royal Institute of British Architects.

AB. Royal Institution of Great Britain.

B. Royal Medical and Chirnrgical Society.

p. Royal Meteorological Society.

p. Ro \.-il .Microscopical Society.

p. Royal Statistical Society.

AII. iloyal United Service Institution.

AB. Society of A rts.

)>. Society of Biblical Archaeology.

p. Standard Weights and Mea- ur.'.- Depart-
ment.

AH. The Queen's Library.

AB. The War Office.

All. University College.

p. Victoria Institute.

ii. Zoological Society.

I vii J

England and Wales (continued).

AH. Free Library.

AH. Literary and Philosophical Society.

p. Geologiml Soi-iety.

An. Owuus College.
Netley.

p. Royal Victoria Hospital.
Newcastle.

AB. Free Library.

p. North of England Institute of Mining and
Mechanical Engineers.

p. Society of Chemical Industry (Newcastle

Section).
Norwich.

p. Norfolk and Norwich Literary Institution.
Oxford.

p. Ashmnlean Society.

AB. Raclcliffe Library.

A. Radcliffe Observatory.
Ponzance.

p. Geological Society of Cornwall.
Plymouth.

B. Marine Biological Association.
p. Plymouth Institution.

Richmond.

A. " Kew " Observatory.
Salford.

p. Royal Museum and Library.
Stonyhnrst.

p. The College.
Swansea.

AB. Royal Institution.
Woolwich.

An. Royal Artillery Library.
Finland.
Hrlsingfors.

AB. Societe des Sciences.

;>. Societas pro Fiiuna et Flora Fenuica.
Prance.
Bordeaux.

p. Academic des Sciences.

p. Facnlt4 des Scion

p. Societ^ de M6decine et de Chirnrgie.

y. Soci^t^ des Sciences Physiques et

Natnrelles.
Clierlxnirpf.

p. SocicSte des Sciences Nattm-llr*.
Dijon.

j>. Academe des Sciences.
Lille

p. Kncnltc ilos Sciei

France (contiiiiicil).
Lyons.

AB. Academic des Sciences, Bolles-Lcttres et

Arts.
Marseilles.

p. Faculte des Sciences.
Montpellier.

AB. Academic des Sciences et Lettres.

B. Facnlte de Modi-cine.
I'.u-is.

AR. Academic den Sciences de 1'Institnt.

p. 'Association Franeaise pour 1'Avancenicnt
des Sciences.

p. Conservatoire des Arts et Metiers.

p. Cosmos (M. i.'Amif: VALKTTK).

AB. Depot de la Marine.

AB. Ecole des Mines.

AB. Ecole Normale Superieure.

AB. Ecole Polytechnique.

AB. Faculte des Sciences de la Sorbjnne.

AB. Jardin des Plantea.

A. L'Observatoire.

p. Rovue Internationale de 1' Elect ricite.

p. Revue Scientifique (Mons. RICIIKIJ.

p. Socii'-te de Biologie.

AH. Societe d'Encouragement pour 1'Indnstrie

Nationale.

p. Soci6te de Physique.
AB. Siiciete de Geographic.

B. Sin-ieti'- Entomologique.
AB. Societe G6ologique.

}>. S<)cieU' Mathomut ique.

p. Societ^ Metuorologique de France.
Toulouse.

AB. Academic des Sciences.

A. Faculte des Sciences.
Germany.
Berlin.

A. Deutsche Chemischc Gesellschaft.

A. Die Sternwarte.

AB. Kdnigliche Akademie der Wissenschaften.

A. Physikalische Gesellschaft.
Bonn.

AB. Universitat.
Bremen.

p. Naturwissenschaftlicher Verein.
Breslau.

p. Schlesische Gesellschaft fur VaterlUndiscbe

Kultnr.
Brunswick.

p. Verein fur Natnrwissenschaft.
Carlsrnhe. See Karlsruhe.

fiesi-llsi'liaft.

AB.

[• * * T
via

Germany (continued).
Dresden.

p. Verein for Erdknnde.
Emdcn.

p. Natnrforschende Gesellschaft.
Erlangen.

AB. Physikalisch-Medicinische Societat.

Frankfnrt-am-Main.

AB. Senckenbergische Naturforschende Gesell-
echaft.

p. Zoologische Gesellschaft.
Frankfnrt-am-Odcr.

p. Natnrwissenschaftlicher Verein.
Freibnrg-im-Breisgau.

AB. Universitat.
Giessen.

AB. Grosshcrzoglicbe Universitat.
G«rlitz.

p. Natnrforschende Gesellschaft.
Gottingen.

AB. Konigliche Gesellschaft der Wissen-

schaften.
Halle.

AB. Kaiser) iche Leopoldino - Carolinische
Dentftche Akademie der Naturforscher.

p. Naturwissenschaftlicher Verein fur Sach-

sen und Thltringen.
Hamburg.

AB. Natnrwissenscbaftlicher Verein.
Heidelberg.

p. Naturhistorisch-Medizinische Gesellschaft.

AB. Universitat.

Jena.

p. Medicinisch-Naturwissenschaftliche Gesell-

Bchaft.
Karlsruhe.

A. Grossherzogliobe Sternwarte.
Kiel.

A. Sternwarte.

AB. Universitat.
KUnigsberg.

AB. Konigliche Pliysikaliscb - Okonomiscbe

Gesellschaft.
Leipzig.

p. Annalen der Physik nnd Chemie.

A. AgtronomiBche Gesellsclmfl.

AB. Konigliche Sachsische Gesellscbaft der

Winsenschaften.
Muj'i. :. • .-

p. Naturwigscnschaftlicher Verein.
M:irburg.

AB. Univorxitut.

Germany (continued).
Munich.

AB. Konigliche Bayerisehe Akademie der
Wissenschaf ten .

p. Zeitschrift fiir Biologie.
MUnster.

AB. Koniglicbe Theologische und Philo-

sophische Akademie.
Rostock.

AB. Universitat.
Strassburg.

AB. Universitat.
Tubingen.

AB. Universitat.
Wllrzburg.

AB. Physikalisch-Medicinische Gesellscbaft.
Holland. (See NKTHEELANDS.)
Hungary.
Pcsth.

p. Konigl. Ungarische Geologische Anstalt.

AB. A. Magyar Tudos Tarsasag. Die Ungarische

Akademie der Wissenscbaften.
India.
Bombay.

AB. Elphinstone College.
Calcutta.

AB. Asiatic Society of Bengal.

AB. Indian Museum.

p. Great Trigonometrical Survey of India.

p. The Meteorological Reporter to the

Government of India.
Madras.

B. Central Museum.

A. Observatory.
Roorkee.

p. Roorkee College.
Ireland.
Armagh.

A. Observatory.
Belfast.

AB. Queen's College.
Cork.

p. Philosophical Society.

AB. Queen's College.
Dublin.

A. Observatory.

AB. National Library of Ireland.

B. Royal College of Surgeons in Ireland.
An. Royal Dublin Society.

AB. Royal Irish Academy.
Galway.

AB. Queen's College.

Italy.
Bologna.

AB. Accademia delle Scienze dell' Istituto.
Catania.

AB. Accademia Gioenia di Scienze Natarali.
Florence.

p. Biblioteca Naziouale Centrale.
AB. Reale Museo di Fisica.
Milan.

AB. Reale Istitnto Lombardo di f Soienze,

Lettere ed Arti.

AB. Societa Italiana di Science Natarali.
Naples.

AB. Societa Reale, Accademia delle ScienM.
B. Stazione Zoologica (Dr. UOHKN).
Padua.

p. University.
Palermo.

A. Circolo Matematico.

AB. Consiglio di Perfezionamento (Societa di

Scienze Natarali ed Economiche).
A. Reale Osservatorio.
Pisa.

p. Societa, Toscana di Scienze Naturali.
Rome.

p. Accademia Pontificia de' Nuovi Lincei.
A. Osservatorio del Collegio Romano.
AB. Reale Accademia dei Lincei.
p. R. Comitato Geologico d' Italia.
AB. Societd Italiana delle Scienze.
Turin.

p. Laboratorio di Fisiologia.
AB. Reale Accademia delle Scienze.
Venice.

p. Ateneo Veneto.
AB. Reale Istitato Veneto di Scienze, Lettere

ed Arti.
Japan.
Tokid.

AB. Imperial University.
Yokohama.

p. Asiatic Society of Japan.
Java.
Batavia.

AB. Bataviaasch Genootschap van Knnsten en

Wetenschappen.
Buitenzorg.

/'. Jardin Botaaiquc.
Malta.

/•. Public Libnu-\ .
Mauritius.

p. Royal Socu-ij of Arts and Sciences.
Mix < vi. XXXVIII. — A.

]

Netherlands.
Amsterdam.

AB. Koninklijke Akademie van Wetenschappen.
p. Zoologisch Geuootschap ' Naturn Art is

Magistral
Delft.

p. ficole Polytechniqae.
Haarlem.

AB. Hollandsche Maatschappij der Weten-
schappen.
p. Musee Teyler.
Leyden.

AB. University.
Luxembourg.

p. Sociike des Sciences Naturelles.
Rotterdam.
AB. Bataafsch Genootschap der Proefonder-

vindelijke Wijsbegeerte.
Utrecht.
AB. Provinciaal Genootschap van Kansten en

Weteusc happen.
New Zealand.
Wellington.

AB. New Zealand Institute.
Norway.
Bergen.

AB. Bergenske Museum.
Christiania.

AB. Kongelige Norske Frederiks Universtet.
Tromsoe.

/'. Museum.
Trondhjem.

AB. Kongelige Norske Videnskabers Srlskab.
Nova Scotia.
Windsor.

p. King's College Library.
Portugal.
Coimbra.

AB. Universidade.
Lisbon.

AB. Academia Real dan Sciencias.

p. Seccao dos Trabalhos Geologicos de

Portugal.
Russia.
Kazan.

AR. Imperatorsky Kazansky Universitet.
Kharkoff.

p. Section Medicale de la Societe" des Science*
Experimen tales, University de Kharkow.
Kieff.

p. Societ^ des Nataralistes.
Moscow.

AH. Le Musee Pnblique.

B. Socie'te Imperiale des Naturalist**.

Russia (continued).

Odes*.
p. Societ4 des^Naturalistes de la Nouvelle-

Rnssie.

Pnlkowa.
A. Nikolai Hanpt-Sternwarte.

St. Petersburg.

AB. Academic Imperiale des Sciences.
AB. Comit^ Geologique.
p. Compass Observatory.
4. L'Observatoire Physique Central.

Scotland.
Aberdeen.

AB. University.
Edinburgh.

p. Geological Society.
A. Observatory.
p. Royal Medical Society.
p. Royal Physical Society.
p. Royal Scottish Society of Arts.
AB. Royal Society.

Glasgow.

AB. Mitchell Free Library.
p. Philosophical Society.

Servia.
Belgrade.
p. Acade'mie Royale de Serbie.

Spain.
Cadiz.

A. Observatorio de San Fernando.

Madrid.

p. Comision del Mapa Geologico de Espana.
AB. Real Academia de Cicncias.
Sweden.
Gottenburg.

AB. Kongl. Vetenskaps och Vitterhets Sam-

halle.
Lund.

AB. Universitet.

Stockholm.

A. Acta Mathematica.

AB. Kongligu Svenska Vetenskaps-Akademie.

AB. Sveriges Geologiska Undersokning.
Upeala.

AB. Universitet.
Switzerland.

Basel.

p. Naturforschende Gesellschaft.
Bem.

AB. Allg. Schweizerische Gesellschaft.

p. Naturforschende Gcsellschaft.

Switzerland (continued).
Geneva.

AB. Societ6 de Physique ot d'Histoire Naturelle.
AB. Institnt National Genevois.
Lausanne.
p. Societe' Vaudoise des Sciences Naturelles.

Neuchatel.

p. Societe des Sciences Naturelles.

p. Astronomische Mittheilungen (Professor R.

WOLF).
Zurich.

AB. Das Schweizerische Poly tecbniknm .

p. Natnrforschende Gesellschaft.
Tasmania.
Hobart.

p. Royal Society of Tasmania.

United States.
Albany.

AB. New York State Library.
Annapolis.

AB. Naval Academy.
Baltimore.

AB. Johns Hopkins University.
Boston.

AB. American Academy of Sciences.
B. Boston Society of Natural History.
A. Technological Institute.
Brooklyn.

AB. Brooklyn Library (N.T.).
Cambridge.

AB. Harvard University.
Chapel Hill (N.C.).

p. Elisha Mitchell Scientific Society.
Charleston.
p. Elliott Society of Science and Art of South

Carolina.
Chicago.

AB. Academy of Sciences.
Davenport (Iowa).

p. Academy of Natural Sciences.
Madison.

p. Wisconsin Academy of Sciences.
Mount Hamilton (California)'.
A. Lick Observatory.
p. University of California.
Newhaven (Conn.).

AB. American Journal of Science.
AB. Connecticut Academy of Arts and Sciences.
New York.

p. American Geographical Society.
p. American Museum of Natural History.
p. New York Academy of Sciences.
p. New York Medical Journal.
p. School of Mines, Columbia Colleg*.

x

United States (continued).
Philadelphia.

AB. Academy of Natural Sciences.
AH. American Philosophical SociVi\
p. Franklin Institute.
p. Wagner Free Institute of Scicnn-
St. Louis.

p. Academy of Science.
Salem (Mass.).
/>. Essex Institute.
AB. Peabody Academy of Science.

Francisco.
AH. California Academy of Science*.

United States (continued).
Washington.

p. Department of Agricnli mi-
A. Office of the Chief Signal Officer.
AB. Patent Office.
AB. Smithsonian Institution.
p. United States Commission of Fish

Fisheries.

AB. United States Coast Survey.
AB. United States Geological Surrey.
A. United States Naval Observatory.
West Point (NT.)

AB. United States Military Academy.

iiri

PHILOSOPHICAL TRANSACTIONS, VOL. 179 (1888). A.

On page 114, line 5 from bottom, for "1883," retul "1833."

-i.;-.i:.., r. ,¥,i,;^v. ^OPo in^^Piv advanced p\ir knowledge or tne rnystm
ERRATUM.

' PHIL. TRANS.,' VOL. 179 (1888), A.
Page 63, for equation (07) as printed read

. - AVI

\'a S"

ADJUDICATION of the MEDALS of the ROYAL SOCIETY for the year 1888,

by the PRESIDENT and COUNCIL.

The COPLEY MEDAL to Professor THOMAS HENRY HUXLEY, F.R.S., for his Investiga-
tions on the Morphology and Histology of Vertebrate and Invertebrate Animals, and
for his services to Biological Science in general during many past years.

The RUMFORD MEDAL to Professor PIETRO TAOCHINI for important and long-
continued Investigations, which have largely advanced our knowledge of the Physics
of the Sun.

A ROYAL MEDAL to Baron FERDINAND VON MUELLER, K.C.M.G., F.R.S., for his
long services in Australian Exploration, and for his Investigations of the Flora of the
Australian Continent.

A ROYAL MEDAL to Professor OSBORNE REYNOLDS, F.R.S., for his Investigations in
Mathematical and Experimental Physics, and on the Application of Scientific Theory
to Engineering.

The DAVY MEDAL to WILLIAM CROOKES, F.R.S., for his Investigations on the
Behaviour of Substances under the Influences of the Electric Discharge in a High
Vacuum.

The Bakerian Lecture, " Suggestions on the Classification of the various Species of
Heavenly Bodies," was delivered by J. NORMAN LOCKYER, F.R.S.

The Croonian Lecture, " On the Origin and the Causation of Vital Movement (Ueber
die Entstohung der vitalen Bewegung)," was delivered by Professor W. KIIIM .

Mix ' . I. \\.\VIII. A.

CONTENTS.

VOL. 179.

I. The Iiifluenfe of Strexs «n<l Strain in, tin' /'////\ n-nl I'rnprrtii'x of Mutter. —
Part I. Elnxlifify (continued).— The Effect of Magnetisation on the Elasticity
and the Internal Friction of M,'tnlx. By HERBERT TOMLINSON, B.A. Com-
municated l>n I'riifi-fixor W. GRYLLS ADAMS, M.A., F.R.S. .... page 1

II. Chi the Spectrum of the Oxy-hydroycn Flame. By G. D. LIVEINO, M.A., F.R.S.,
Pi;>t>'xx<>r of Chemist i- >i, <ii«f J. DKWAI;, M.A., F.R.S., .l<i<-kx>nii<in J'n>f<:-
Unmenity of Cambridge ................. 27

III. On the Motion of a Sphere, in a Vixcons Li'jm'il. llij A. B. BASSET, M.A.
Communicated l»j Lord KAYLKUJH, D.C.L., Sec. ft.S. ....... 43

IV. OH l/'intiltuHs Numbers.— Part II. By J. J. SYLVESTER, D.C.L., F.R.S.,
Savilian Profeuor of Geometry in the University of Qjcford, and JAMES
HAMMOND, M.A., ('<iat«l> ................. 65

V. I\i />"i-t on Hygrorfietric Mi-tliodx; Firxt J'm-t, incliidimj the Saturation Method
'i, i<l //!,• ( 'ln'inical Method, and Den--],,,,',,/ luxii-nnn-nts. By W. N. SHAW,
M.A. Communicated />// \\. H. SCOTT, F.R.S., ,v. •/>/.>;•// of the Meteorological
Con a, -i/ ...................... 73

VI. ()„ tl«' Din,,,*!,;-* ,./'(t !'/«„<• C»l>i<: By J. J. WALKER, M.A., F.R.S. . . 151

o -J

VII. On the Changes produced by Magnetisation in the Dimensions of Rings and

Rods of Iron and of some other Metals. By SHELFORD BIDWELL, M.A.,
/••./,'.>' page 205

VIII. On the Ultra-Violet Spectra of the Eletnents. — Part III. Cobalt and Nickel.
By G. D. LIVEING, M.A., F.R.S., Professor of Chemistry, and J. DEWAR, M.A.,
F.R.S., Jacksonian Professor, University of Cambridge 231

IX. The Conditions of the Evolution of Gases from Homogeneous Liquids. By V. H.

VKLEY, M.A., University College, Oxford. Communicated by A. VERNON
HARCOURT, M.A., F.R.S., Lee's Reader in Chemistry, Christ Church,
Oxford 257

X. On tlte Induction of Electric Currents in Conducting Shells of Small Thickness.

By S. H. BURBURY, M.A., formerly Fellow of St. John's College, Cambridge.
Communicated by H. W. WATSON, D.Sc., F.R.S. 297

XI. Magnetic Qualities of Nickel. By J. A. EWING, F.R.S., Professor of Engineering

in University College, Dundee, and G. C. COWAN 325

XII. Magnetic Qualities of Nickel (Supplementary Paper). By J. A. EWING, F.R.S.,

Professor of Engineering in University College, Dundee 333

XIII. On certain Mechanical Properties of Metals considered in relation to the
Periodic Law. By W. CHANDLER ROBERTS- AUSTEV, F.R.S., Professor of
Metallurgy in the Normal School of Science and Royal School of Mines, South
Kensington, CJiemist and Assay er of the Royal Mint 339

XIV. On the Specific Resistance of Mercury. By R. T. GLAZEBROOK, M.A., F.R.S.,

Fettow of Trinity College, and T. C. FITZPATRICK, B.A., Fellow of Christ's
College, Demonstrators at the Cavendish Laboratory, Cambridge . . . 351

^'\. /1,1-firiants, Covariants, and Quotient-Derivatives associated ivith Linear
Differential Equations. By A. R. FORSYTH, M.A., F.R.S., Fellow of Trinity
College, Cambridge 377

XVI. The Small Free Vibrations and Deformation of a Thin. Kl'istic Shell. Ji>/
A. E. H. LOVE, B.A., Fellow of St. John's College, Cambridge. Communicated
by Prvfeuor Q. II. [>.\I:WIN, F.R.S. 491

[ v J

XVII. Colour Photometry. — Part II. Tl,<- M,-<ixn,-i'ment of Reflected Colours. />'//
Captain ABNEY, C.B., R.E., F.R.S., and Mojor-Genem! KKSTINO, R.I-:.,
F.It.S. page 547

XVIII. Combustion in Dried O./ //.</'" />// H. BBERETON BAKER, M.A., Dulvnch
Colleye, late Sdmtnr <>f lt<illi<,l ( ',,11, •,/,-, Oxford. Comma ni<-<i ted by Professor
H. B. DIXON, F.R.S. 571

, 593

LIST OF ILLfsTKATIONS.

Plates 1 to 4. — Professors G. D. Li \ 1:1 M; ami .1. I »i:\\ AK on the Spectrum of t IK- ( >\y-
hydrogen Flame.

Plate 5. — Mr. W. N. SHAW'S Report on Hygrometric Methods ; First Part, including
the Saturation Method and the Chemical Method, and Dew-point Instruments.

Plates f> to 8. — Mr. J. J. WALKER on the Diameters of a Plane Cubic.

Plates 'J to 14. — Professors G. D. LIVKIN<; and J. DKWAK on the Ultra- Violet Spectra
of the Elements. — Part III.

Plates 15 and 16. — Professor J. A. EWINO and Mr. G. C. COWAN on the Magnetic
Qualities of Nickel.

Plate 17. — Professor J. A. EWING on the Magnetic Qualities of Nickel (Supplementary
Paper).

Plate 18. — Professor W. C. ROBERTS- AUSTEN on certain Mechanical Prop-rtit s of
Metals considered in relation to the Periodic Law.

Plate 19.— Messrs. R. T. GLAZEBKOOK and T. C. FJTZPATRICK on the Specific
Resistance of Mercury.

Plates 20 to 23. — Captain ABNKY and Major-Genenil FKSTIKG on Colour Photometry.
—Part II.

PHILOSOPHICAL TRANSACTIONS.

I. Hie Influence of Stress and Strain on the Physical Properties of Matter. —
Part I. Elasticity (continued). — The Effect of Magnetisation on the Elasticity
and the Internal Fi-i<-ti,,,, of Metals.

By HERBERT TOMLINSON, B.A.
Communicated by Pro/;.-..-..-/ W. GRYLLS ADAMS, M.A., F.R.S.

Received May 18,— Read May 27, 1886.
(Recast, with additions, December 16, 1887.)

< h-iifin inut I'lir/ii'*'1 i >f tin' Ii«; xti, /<

ACCORDING to Professor G. WIEDEMANN,* the main part of the internal friction which
occurs in a torsionally vibrating wire is due to the rotation of the molecules about
their axes, first to this side and then to that, as the wire vibrates to and fro. With
this view the author's own experimentst on the internal friction of metals had been
so far in accordance that he wished still further to test the matter by investigating
the effect of magnetisation on the internal friction.

The author has already made some experiments on the effect of magnetisation on
the torsional elasticity of metals,! but the results of these experiments did not entirely
satisfy him, inasmuch as the means of eliminating the heating effect of the magnetising
solenoid were imperfect. It is true that the observed changes of temperature wrought
by the solenoid were comparatively small, but so also was the apparent alteration of
torsional elasticity due to magnetisation ; and it seemed, therefore, advisable to reopen
the inquiry, and to devise more perfect apparatus, whereby the heating effect above
mentioned might be entirely done away with.

Description of Apparattis.

The wire was clamped at its upper extremity, a, into a T-shaped block of brass
resting on the top of the air-chamber, A (see figure). The air-chamber consisted of two

* ' WIEDBMANN'S Annalen,' 1879, vol. 6, p. 485.
t ' Phil. Trans.,' 1886 (vol. 177, Part II.).

* ' Phil. Trans.,' 1883 (vol. 174, pp. 34, 35).
MDCCCLXXXVIII. — A. B 31.1.88

2 MB. H. TOMLINSON ON THE INFLUENCE OF STRESS AND

concentric brass tubes, 4 feet in length, and enclosing between them an annular space,
about a quarter of an inch thick, which could be filled with water. The wire hung
vertically in the axis of the' air-chamber, and at its lower extremity & was soldered
to a copper rod, b c, about '3 centim. in diameter, which was in turn connected to the
horizontal bar, V V. From V V were suspended two cylinders of equal mass and
dimensions, placed at equal distances from the axis of the wire. The nature of the
cylinders and their mode of suspension to the bar V V have been described in a
previous memoir on this subject.* The box B permits of free oscillation of V V and
its appendages, and is provided with air-tight fitting doors, whilst the glass window,
C, allows the vibrations of the wire to be measured by means of the usual mirror-

lamp-and- scale arrangement, which is sufficiently shown in the figure. The base of
the air-chamber, A, is let into the top of a stout wooden table pierced with a circular
aperture, o, through the centre of which passes the rod 6 c ; the top of the air-
chamber is secured by a ring, H, clamped to the upright, K. Wrapped round the
air-chamber, to within two or three inches of each end, was a considerable length of
cotton-covered copper wire, ^th of an inch in diameter, and well soaked with shellac
varnish. The copper wire was wound round the chamber in one layer, thus forming
a magnetising solenoid in which there were 8 '25 turns in a centimetre ; since the wire
to be tested was well within the solenoid, the magnetising stress may be regarded as

* ' Phil. Trans.,' 1886 (vol. 177, Part II.).

STRAIN ON THE PHYSICAL PROPERTIES OP MATTER.

3

nearly constant throughout its entire length. In order to maintain the temperature
constant, water from a large pail was made to flow into the annular space through
the tube D, and out through the tube D2 during the whole of the period of experi-
menting. The magnetising solenoid was actuated by ten GKOVE'S cells, the current
from which passed through a resistance-box, a tangent galvanometer, and a commu-
tator (not shown in the figure). The precautions adopted in the previous experi-
ments were here reproduced, care being taken that the amplitude of the vibrations
should be well within the limits of elasticity.

Experiment I.

An annealed iron wire, 100 centims. long and '1 centim. in diameter. The current
was always sent through the magnetising solenoid in the same direction,* and at
least a hundred vibrations were allowed to take place, both when the solenoid was
excited and when it was not, before the actual testing began. The experiment was
carried on for two days for about six hours on each day, and the numbers given below
for the logarithmic decrements and times of vibration are in each case the mean
values resulting from 400 vibrations, first without excitation of the magnetizing
solenoid, then with, then without, and so on.

First Day.
MAGNETISING solenoid not excited.

Number of trial.

Mean logarithmic decrement
due to internal friction t

Vibration-period

in -• •••mil-.

i

3
5

7

•0009728

•0009826

•0009598
•0009575

2-4745
2-4790

MAGNETISING solenoid excited.

2
4

6
8

•0010722
•0011794$
•0010249
•0010099

2-4750
2'4745

* Except in the fourth trial, when it was reversed for a few seconds by accident.

t The damping due to the resistance of the air has in all the experiments been calculated in the
manner described in ' Phil. Trans.,' 1886 (vol. 177, Part II.).

{ The current was in this trial put on, in the first instance, in the wrong direction, but was afterwards
reversed while the trial was still going on ; this, no doubt, accounts for the logarithmic decrement being
larger than in the other trials.

B 2

MB. H. TOMLINSON ON THE INFLUENCE OF STRESS AND

Second Day.
MAGNETISING solenoid not excited.

Number of trial.

Mean logarithmic decrement
due to internal friction.

Vibration-period
in seconds.

2

4

•0009408
•0009464

2-4765

MAGNET r SING solenoid excited.

1

3

•0010037
•0010034

2-4740

The current used in the trials on both days was throughout fairly constant, the
mean deflection of the needle of the tangent galvanometer being 47°. The constant
of the tangent galvanometer was '3167, therefore the magnetising stress in electro-
magnetic units was

477- X 8-25 X tan 47° X '3167 = 35'21.

This magnetising stress is a large one, and was, no doubt, sufficient to develop
the greater part of the whole magnetism which the wire was capable of receiving. It
may be seen that even under such a magnetising stress as the above the internal
friction is not much altered ; but, if the first four trials made on the first day be
neglected, it appears that the internal friction is on both days greater by about 5^ per
cent, when the magnetising solenoid is excited than when it is not.

Having ascertained thus much, the author next proceeded to determine the effect
of an intermittent magnetising stress, but still always in the same direction as before.

Experiment II.

One of the little clockwork arrangements used with Professor HUGHES'S induction
balance was now put into the battery circuit, so that the latter could be rapidly
opened and closed whilst the wire was vibrating, the same battery power being
employed as before. In this case the logarithmic decrement was at first fairly
constant, and equal to '001442, showing that the internal friction was greater than
when the circuit was not alternately opened and closed. After a time, however, as
the clockwork began to run down, and the makes and breaks of the current in conse-
quence to proceed more slowly, the value of the logarithmic decrement rapidly
increased,* and finally became '002719, or nearly double its first value.

• The cause of the rapid increase is shown in Experiment VI.

STRAIN ON THE PHYSICAL PROPERTIES OP MATTER.

Experiment III.

The current was reversed a number of times whilst the wire was vibrating, the
reversals being timed by a pendulum vibrating once in a second ; the same battery
power as before was used.

Number of rerenalt of the
current in 18 aecond*.

Logarithmic decrement due
to internal friction.

1

8
16

•001175
•004393
•003546

Sxpei'iment IV.

After the above reversals the magnetising stress was applied as in Experiment I.,
the battery power being still the same.

MAGNETISING solenoid not excited.

Number of trial.

Mean logarithmic decrement
due to internal friction.

Vibration-period
in second*.

1
2

4

•0010891
•0009730
•0009551

Magnetising solenoid excited.

3

•0009569

After a rest of 24 hours.
Magnetising solenoid not excited.

1
3
5

•0009480
•0009071
•0009363

2-4782
2-4790

Magnetising solenoid excited.

2

4
6

•0009249
•0009463
•0009294

2-4785
2-4788

6 MR. H. TOMLINSON ON THE INFLUENCE OP STRESS AND

Expefriment V.

The day after the last experiment had been made a fresh series of trials was
instituted with currents of various strengths.

Number of
trial.

Magnetising strew in
C.O.S. units.

Logarithmic decrement
dne to internal friction.

Remarks.

1

0

•0009440

2

4-615

•0094650

Current rendered intermittent by clock-
work, but not reversed

3

4-615

•0034865

Current reversed every two seconds; no
clockwork

4

0

•0010095

This trial made immediately after Trial 3,
with only a few preliminary vibrations

5

0-104

•0009567

Current reversed every two seconds

6

0-035

•0009875

Ditto

7

0

•0009364

Remarks on Experiments I.- V. inclusive.

It has been already noticed that when a large magnetising stress is used there is
but a slight increase in the internal friction, provided the magnetising current is not
interrupted or reversed during the trial, whilst Experiment IV. shows that if the
current be previously reversed a great number of times even this small increase
vanishes.* Accordingly, it may be said that under the conditions mentioned above
the internal friction is quite independent of any sustained magnetic stress which may
be acting on the vibrating wire. Similarly, there can be little doubt, the internal
friction of a torsionally vibrating wire would be entirely independent of the amount
of sustained statical torsional stress to which the metal might be at the same time
subjected, provided the wire had been previously vibrated torsionally a great number
of times.

The torsional elasticity is also equally independent of even large sustained mag-
netising stress, for it may be observed that the mean value of the vibration-period
deduced from Experiment I. is for the magnetised iron 2'4745 seconds and for the
uumagnetised iron 2 '4767 seconds, whilst in Experiment I\r. it is for the magnetised
iron 2'47865 seconds and for the unmagnetised iron 2'47860 seconds. Other experi-

* If we take the last six trials of this experiment, we get for the logarithmic decrements, when the
magnetising solenoid is not excited, and when it is, the values '0009305 and '0009332 respectively ; the
difference between these two values lies within the limits of errors of observation.

STRAIN ON THE PHYSICAL PROPERTIES OF MATTER.

incuts, conducted m..-,t r.-m-rnllv. l.-d t,, vimil.-ii- rwniti :nnl :il'Uh>!:nit Iv
the above-mentioned independence.

When the magnetising current is interrupted, or when it is reversed whilst the wire
is vibrating, there is for the larger magnetic stresses an increase of the internal
friction which may become very considerable.*

Experiments II. and III. seem to prove that when the number of interruptions or
reversals in a given time exceed a certain limit the effect produced by them on the
logarithmic decrement begins to decline, but it would appear from Experiment II.
that even when the interruptions occupy only a small fraction of a second t their
effect in increasing the molecular friction is very sensible.

Experiment V. was made partly with a view of ascertaining how far the interrupted
or reversed magnetic stress might be diminished before it ceased to exercise any
perceptible influence on the internal friction. For the rather rapid interruptions
produced by the clockwork it appears that when the magnetising stress is diminished
to 4'615 its effect on the friction is nearly nil. This, however, is by no means the
case when the current is reversed every two seconds, the value of the logarithmic
decrement being then more than three times as great as it is when the wire is not
magnetised; and even when the stress has been reduced to 0*104 it still exerts a
sensible influence.

The experiments also show clearly that, at any rate for short periods of time, the
longer the time of action of the magnetising stress, the greater is the effect on the
internal friction, for otherwise in Experiment III. there would be a greater logarithmic
decrement when the vibrations are 16 in 16 seconds than when the vibrations are 8 in
16 seconds, whereas the contrary is the case. Nevertheless, this increased effect on
the internal friction which accompanies increased time of action does not extend
beyond a period of a few seconds, for the increase of logarithmic decrement is
considerably greater when the vibrations are 8 in 16 seconds than eight times the
increase of logarithmic decrement when the vibrations are 1 in 16 seconds. }

[Added Sept. 29<A, 1 887. — Experiment III. has shown that there is a definite
frequency of reversal of magnetising stress for which the damping effect is a
maximum, and it appeared to be of interest to ascertain more exactly what is the
frequency producing the greatest effect with the particular wire examined, and also

* See Experiments II. and III.

f The clockwork arrangement must have interrupted the current at least ton times in one second.
[Later experiments showed that the diminution of the effect of an interrupted magnetic stress as the
interruption-frequency increased arose from the fact that the difference between the vibration-frequency
of the wire and that of the interruptions increased.]

J The increases of the logarithmic decrement are '00346 and '00024 in the two cases respectively.
[September 25, 1887. — It was afterwards found that the results of Experiment III. are to be attributed
almost entirely to differences between the vibration-frequency t>f the wire and the interruption-
frequency.]

8

MR. H. TOMLINSON ON THE INFLUENCE OF STRESS AND

whether this special value represents a property of the material or is dependent on
the accidents of dimensions of the wire and moment of inertia of the vibrator. An
examination of the results recorded in Experiment III. seemed to indicate that the
damping was a maximum when the vibration-frequency of the wire was the same as
the reversal frequency. The following experiment was then made :—

Experiment VI.

A fresh piece of the same wire was tested with two different vibration-periods, the
moment of inertia being altered by shifting the cylinders along the bar ; the magnet-
ising stress throughout was 46'025 C.G.S. units.

Interval in seconds between
consecutive reversals
of the magnetising stress.

Logarithmic decrement due
to internal friction.

Vibration-period of the wire
in seconds.

Before exciting the solenoid

•000799

1

•002841

2

•004587

2-370

3

•003988

4

•001769

1

•001052

1

•001785

2

•003020

3

•003871

4725

4

•004319

5

•004470

6

•002984

The reversals of the magnetic stress were next made to synchronise with the
vibrations of the wire, the vibration-period being 4 '725 seconds.

Magnetic stress in C.O.S. units.

Logarithmic decrement due
to internal friction.

46-88
77-35
107-40

•00896
•01118
•01137

This experiment speaks for itself, and shows most conclusively that the damping
effect is a maximum when the reversal-period and the vibration-period synchronise.

The experiment also shows that, with these high values of the magnetic stress, the
ratio of the logarithmic decrement to the magnetic stress diminishes very rapidly as
the latter increases.

STRAIN ON THi: PHYSICAL PHOPKHT1KS OF MATTKi:

Hitherto the magnetic stress had been maintained during the whole period
between one reversal and the next, but it was presently ascertained that if the
magnetic stress be removed immediately after each reversal the damping is much
increased.

Experiment VII.

A reversing key was used, such that the magnetising circuit could be opened the
instant after each reversal ; * the magnetising stress was 4278 C.G.S. units ; the
vibration-period of the wire- was 4'725 seconds, and with this the reversals were
made to synchronise in one of the trials which were made.

Interval in I
between connccutive reversal*
of the magnetising sire*.

4-000
4725
5-000

Logarithmic decrement due
to internal friction.

•007986
•017416
•008759

On comparing these last results with those of Experiments V. and VI., it will be
seen that when the battery circuit is opened instantly after each reversal the effect
on the internal friction is twice as great as when the magnetic stress is maintained
between each reversal and the next.

Nothing has been said as yet respecting the phase of the torsional vibrations of the
wire at which the reversals of the magnetic stress were made. When the reversals
synchronised with the torsional vibrations the former were made when the wire was
nearly^ at the end of its swing on one side or the other, but it was soon discovered
that the effect of reversing the magnetic stress was greatest when the reversals took
place exactly at the end of each swing, and least when they occurred at each instant
the wire was passing its position of equilibrium.

* The time during which the circuit remained closed after each reversal must have been only a small
fraction of a second, as on reversing a sharp tap was given to the key, the spring of which was rather
strong, and the finger removed as quickly as possible.

f That is, no particular pains had been taken to ensure that the reversals should take place exactly
at the end of each swing.

MIX-a'KXXXYlII. — A.

10

MR. H. TOMLINSON ON THE INFLUENCE OF STRESS AND
Experiment VIII.

The vibration-period of the wire was made 4 seconds ; the reversals synchronised
with the vibrations of the wire, and the magnetising circuit was opened immediately
after each reversal.

Magnetic stress in
C.O.S. unit-.

Logarithmic decrement
due to internal
friction.

Reversals made at the end of each swing.

1-135
42-775

•001080
•017416

Beversals made when the wire passed its
position of equilibrium.

1-135
42775

•000796
•001432

The value of the logarithmic decrement when there was no magnetic stress acting
on the wire was '000782, or little less than when the reversals were made as the wire
passed its position of equilibrium and the magnetic stress was 1*135. Even with the
higher stress of 42775 the effect of reversing when the wire was at the end of each
swing is very considerably greater than when the reversals are made in the position of
equilibrium.

In the next experiment an attempt was made to ascertain to what extent the
magnetic stress could be reduced before it ceased to have any sensible effect on the
internal friction.

Experiment IX.

The reversals were made to synchronise with the vibrations of the wire, and as
exactly as possible when the wire had reached its extreme position on either side. The
vibration-period of the wire was 4 seconds.

Magnetic stress in
C.O.S. units.

Logarithmic decrement
due to internal
friction.

28-540

•018332

12-935

•016440

7-280

•008350

1-800

•001454

1-135

•001080

0-495

•000918

o-oo

•000782

STRAIN ON THE PHYSICAL PROPERTIES OF MATTER.

11

It is evident that, under the above circumstances, even when the magnetic stress is
reduced to an amount which begins to be comparable with that of the earth's
magnetic stress, there is a very appreciable damping' due to the reversals.

By subtracting '000782 from the numbers in the second column of the last Table, we
get the increase of the logarithmic decrement due to the reversals in each case, and it
is interesting to compare this increase with the stress producing it ; this comparison
is made below.

Inerewe of the

Magnetic streM.
A.

logarithmic decrement
due to the reremli

B:JL

of the magnetic ilreo.

B.

28-540

•017550

•000616

12-935

•015658

•001210

7-280

•007568

•001040

1-800

•000672

•000374

1-135

•000298

•000262

0-495

•000136

•000274

The increase of the logarithmic decrement due to the reversals of the magnetic stress
is, for small stresses, proportional to the latter. When the magnetic stress reaches
1'800 the friction increases in greater proportion than the stress, until the latter ex-
ceeds 12'935, when the ratio B : A begins to decline.* This is what might be expected
from the behaviour of the magnetic permeability as the stress is gradually increased :
in fact, with this particular specimen of iron, the permeability was found to increase
very rapidly as the magnetic stress rose from 3 to 5 C.G.S. units, and again fell very
rapidly when the stress rose from 12 upwards.

The damping effect of magnetic stress, when applied as above, is noteworthy for its
magnitude ; for the logarithmic decrement when the wire is subjected to reversals of
a magnetic stress of 28 '5 40 is actually between twenty and thirty times as great as
when the wire is free from stress or under the influence of a sustained magnetising
stress.t

Experiment X.

In this experiment the effects of reversals and of applying and removing the mag-
netic stress always in the same direction were compared together, the magnetic stress
being in each case 42775 C.G.S. units, and the vibration-period of the wire 4725
seconds. In both cases the circuit was broken instantly after it was closed, and the
reversals or closings were made to synchronise with the vibrations of the wire.

* See also Experiment VI.
f Experiment II.

c 2

12 MR. H. TOML1NSON ON THE INFLUENCE OF STRESS AND

i

Logarithmic decrement due Ditto when the magnelifl-
to internal friction when ing stress Tag alwayn
the magnetising stress applied in the game

wag reversed, direction.

•017416 -0064C4

The effect in the former of the two cases is very much greater than in the latter,
and this might have been anticipated, for the change of magnetisation with each
reversal is much greater than that produced by each repetition of the stress. No
doubt the difference in the effects in the two cases would be proportionately still
greater for a lower value of magnetic stress, for, with this particular specimen of iron,
when the magnetising stress was about 4 C.G.S. units, the change of magnetisation
produced by each reversal was from eighteen to twenty times as great as the change
produced by each repetition of the stress.]

It has been remarked that when the amplitudes of the vibrating wire are such as only
to produce very small molecular displacements neither the internal friction nor the
torsional elasticity is sensibly affected by sustained magnetisation. Since, however,
the author's investigations, mentioned in the outset of this memoir, and in which the
wire had been vibrated through comparatively large arcs, seemed to show that
magnetism did slightly affect the torsional elasticity, he proceeded to re-try some of
his old experiments relating to the effect of magnetisation on torsional elasticity, and
at the same time also to study the alteration of internal friction which might ensue
with large amplitudes of vibration.

Experiment XI.

The mirror was twisted round through an angle of 18°, then refixed ; afterwards
the vibrator was twisted so as to bring the reflected spot of light just on to one end
of the scale ; the vibrator was then let go, and as soon as the subsidence of the
amplitude had caused the spot of light to reach the division on the scale marked 100,*
or thereabouts, readings were taken. For the battery power 10 GROVE'S cells were
used, and the magnetising stress was very nearly 35 in electromagnetic units.

> The zero of the scale was, in this instance, at the end of the scale from which the vibrator was
started, so that as the amplitudes of the vibrations subsided the readings increased.

STRAIN ON THK PHYSICAL PROPERTIES OF MATTKR.

MAGNETISING solenoid not excited.

Number of trial.

Reading when the
vibrator had reached its
extreme position.

Number of ribrafioni
executed after the
readinga had commenced.

1

102
608

0
52

2

108
608

0
52

8

100
600

0
52

4

98
389

0
20

Magnetising solenoid excited.

5

101
709

0
52

6

101
500

0
20

Thus it appears that, whether the subsidence in 52 or in 20 vibrations be considered,
the internal friction is very perceptibly greater when the solenoid is excited than
when it is not,* and that the difference is greater, the greater the mean amplitude of
vibration.

Experiment XII.

With the same arrangement as in the last experiment, and with the same initial
arc of vibration, the following was found to be the effect of sustained longitudinal
magnetisation on the vibration-period, the battery power used being the same as
before.

* The subsidence for 52 vibrations when the solenoid is excited is 608, and when the solenoid is not
excited 500, the former being 21 per cent, greater than the other. Again, the subsidence in 20
vibrations is 3C per cent, greater when the solenoid is excited than when it is not excited.

14

MB. H. TOMLINSON ON THE INFLUENCE OP STRESS AND
MAGNETISING solenoid not excited.

Number of trial.

Vibration-period in seconds.

1
2

2-481
2-479

Mean

2-480

Magnetising solenoid excited.

3

2-483

From this last experiment it would seem that for these comparatively large
amplitudes the vibration-period is slightly greater for the magnetised than for the
unmagnetised wire, and that consequently the results previously obtained hy the
author with less perfect apparatus are so far confirmed.

The Limit of Magnetic Elasticity.

From Experiment V. it is evident that when the value of the magnetising stress
becomes as small as 0'104 the effect of continued reversals of it on the internal
friction is still perceptible. Now, as any such effect was attributed to permanent *
twist imparted to the molecules, first to this side and then to that, according to the
direction of the magnetising current, an attempt was made to ascertain how far the
magnetic stress might be increased from zero upwards before the molecules would be
permanently twisted by it.

Experiment XIII.

In the axis of a magnetising solenoid 11 '43 centuns. in length was placed a
bundle of well-annealed iron wires, consisting of twenty pieces, each 8 centims. in
length and '1 centim. in diameter. The axis of the magnetising solenoid was
coincident with that of a second solenoid, made up of 814 turns of cotton-covered
copper wire ; the second solenoid surrounded the first, and was connected up with a
very delicate THOMSON'S reflecting galvanometer. A single GROVE'S cell, with a box
of resistance coils in circuit, actuated the magnetising solenoid, and the magnetism
imparted was measured by the throw of the needle of the galvanometer produced by
the induced current which resulted on closing the battery circuit. The permanent

' The word " permanent," whenever used in this memoir in connection with strain, must be taken as
denoting merely that the strain does not immediately disappear on the removal of the stress. The author
believes that such strains as have occurred in his experiments are for the most part sub-permanent.

STRAIN ON THE PHYSICAL PROPERTIES OF MATTER.

15

magnetism for a given magnetic stress was measured by the difference between the
deflections which ensued when the battery circuit was closed for the first and for the
tenth times ; thus, if Dl and D10 be these two deflections respectively, the permanent
magnetism is assumed to be measured by D! — D10, and the temporary magnetism

b D

10.

Resistance in the
battery circuit
in B.A. uniU
B.

Permanent mag-
netism
D.-D,,.

Temporary mag-
netism
D,»

(D,-D,.)B.

Dwx B.

1002*

0

4

0

40 x 10*

502

0

8

0

40

302

0

15

0

45

202

1-1

23

2-2 x 10»

46

152

2-0

31

3-0 „

47

102

3-5

49

36 „

50

82

5-2

62

43 „

51

62

8-3

88

5-1 „

55

42

19-0

187

80 „

58

32

81-8

188

10-2 „

60

From this experiment it appears that until the resistance in the battery circuit has
become about 302 B.A. units there is no perceptible permanent magnetism. The
electromotive force of the GROVE'S cell was 1 '9 volts ; the number of turns per centi-
metre in the magnetising solenoid was ,-^j, and the total resistance in circuit was
302 X '9889 legal ohms. The magnetising stress in electromagnetic units would

consequently be

4ir x 1-9 x 10" x 70

or

302 x -9889 x 10» x 11'43 '

•04896.

The above magnetic stress is little more than one-fourth of that due to the
horizontal component of the earth's magnetic force. The magnetising solenoid was in
this experiment placed horizontally and at right angles to the magnetic meridian, so
that the iron was uninfluenced by the magnetism of the earth ; but other similar
experiments seemed to show that, in whatever position the iron was placed, the mole-
cules required a very appreciable magnetic stress to produce permanent molecular set.t

[October 6th, 1887. — It may be objected to the above experiment that the length
of the pieces of iron is not sufficiently great in comparison with their diameter ; J but

* The resistance of the battery and solenoid amounted to 2 B.A. units, which together with 1000
B.A. units added from the box makes the number here given.

f Later experiments, however, proved that even with the small magnetic stress quoted above a
permanent deflection is produced. (See the passage following.)

J Cf. EWINO, ' Phil. Trans.,' 1885, pp. 533-536.

16

M]R. H. TOMLINSON ON THE INFLUENCE OF STRESS AND

the following experiment shows that even when the length is so great as to render
such an objection quite out of the question there is an apparent limit of magnetic
elasticity.

Experiment XIV.

A magnetising solenoid, no less than 354'3 centimetres in length, was prepared,
having coiled on it, in a single layer, silk-covered copper wire ; the number of turns
per centimetre was 8'5. The secondary coil consisted of 5909 turns of silk-covered
copper wire, coated with shellac varnish, laid on in six layers ; the resistance of
this coil was nearly equal to that of the ballistic galvanometer. The bundle of iron
wires was increased in number to 40, and the length of each was 325 centimetres, so
that, regarding the bundle as one solid rod, the length of it was 512'5 tunes the
diameter. The mode of experimenting was the same as before.

Magnetising stress
in C.G.S. unite
&

Permanent mag-
netism in scale
divisions

Temporary ditto
D10.

D10:S

0100153

o-o

8-0

5229

0-00306

o-o

15-5

5074

0-00760

o-o

39-0

5134

0-01506

o-o

78-0

5180

0-01874

o-o

96-0

5124

0-02480

o-o

128-8

5193

0-02960

o-o

153-7

5194

0-03667

0-5

190-4

5193

0-04821

2-5

250-8

5202

0-05720

4-0

297-5

5201

In reading the deflections there was certainly no error equal to '5 of a division, and
the first trace of the molecules taking a permanent set is when the magnetic stress
has reached 0'03667, or about one-fifth of the value of the horizontal component of the
earth's magnetic force. The first decided permanent set does not apparently occur till
the stress has reached 0'04821, a value very close to that recorded in the last experi-
ment as the limit of magnetic elasticity.

A similar experiment to the above was tried with a wrought-iron rod in an unan-
nealed condition, and in this case there was not the slightest trace of permanent set
until a stress of 0'0482l was reached, whilst the deflection due to temporary magneti-
sation with this stress was 538 scale-divisions.

These last experiments seem to show that between the values 0 and ^H of magnetic
stress there is no permanent set whatever ; but in the next experiment it will appear
that, if we operate by the method of reversals, permanent set may be detected with
much smaller stresses.

STRAIN ON THE PHYSICAL PROPERTIES OF MATTER. 17

Experiment XV.

The bundle of soft-iron wires used in Experiment XIV. was subjected to gradually
increased amounts of magnetic stress according to the following plan : — First the
magnetising circuit was closed and the deflection D! noted, and after the spot of light
had come to rest the circuit was opened and the deflection D2 observed ; next the
battery was reversed, and the deflections Ds, D4, produced by closing and opening the
circuit respectively, were read off as before. When the magnetic stress produces no
permanent set the four deflections are all of equal value ; but, if any such set exists,
D3 — D4 will represent the change of magnetisation produced by the wrenching of the
molecules from their positions of permanent set on one side to their positions of perma-
nent set on the other. If the reversals be made a great many times, D3 — D+ becomes
less and less, but never vanishes, and finally becomes a constant. Experimenting in the
manner mentioned above, the permanent set could be just detected when the magnetic
stress had reached the value of 0'01690, a value only a little less than half of that at
which it had been detected by the other method.* The method of reversals possesses
two great advantages over the previous method in attacking the question of a magnetic
elastic limit. In the first place the amount of permanent set detectable is about
doubled ; and, in the second place, we may go on repeating the reversals so that any
failure to detect a small amount of permanent set at the first reversal may be rectified
in the second or subsequent reversals.

A very protracted examination of the values of D3 — D4 for different values of
magnetic stress extending beyond £H was now made,t and it was found that, provided
the reversals were for each stress continued long enough to make D3 — Dt constant, the
permanent set as thus measured was exactly proportional to the square of the magnetic
stress. We can now plainly see how mutters stand with respect to a magnetic elastic
limit, namely, that no such limit is mathematically existent ; but from the rapid falling-
off of the permanent set, with decrease of magnetic stress, as indicated by the above-
mentioned law, and from the observations with hard and soft iron which have been
already made, we may, in all probability, safely assume that even with the very softest
iron* no permanent set amounting to one per cent, can be detected by the ballistic
method, provided the magnetic stress does not exceed |H.§]

* This might be expected from what has been said above.

t It is not necessary to enter into the details of this examination, as they will be laid before tho Royal
Society in a subsequent paper.

* The soft-iron wire used in these experiments was specially prepared for the author by Messrs. JOHNSON
and NKPHKW ; it can bo permanently elongated 25 per cent, before breaking.

§ Before this note was written Lord RATLEIGH had by another method shown ('Phil. Mag.,' vol. -.'•.
1887, pp. 225-241) that up to JH there is no sensible permanent set. Lord RAYI.KILII has also shown
in his paper that when permanent set is produced by higher magnetising stresses the amount of set is
proportional to the square of the stress.

MDCXXILXXXVIII. — A. I)

I- Mil H. TOML1NSON ON THE INFLUENCE OF STRESS AND

The Effect on the Interned Friction and on the Torsional Elasticity produced by

an Electric Current.

Experiment XVI.

Some experiments made in the year 1880* had seemed to show that, unless an
electric current was very powerful, its influence on the torsional elasticity of iron
and copper was very nearly, if not quite, nil ; and even in those cases where, apparently,
slight changes were wrought by powerful currents the results were such as to leave
the question undecided as to whether these changes might be due, not directly to the
current, but rather to the consequent heating. Accordingly the author determined to
dispense with powerful currents, and to endeavour to compensate for the lessening of
the current by increasing the accuracy and number of the observations. The
following were the arrangements : —

The clamp which secured the upper extremity of the wire was provided with a
terminal screw, by means of which connection was made with one pole of a battery of
five cells of the LECLANCHE type.t To the bar of the vibrator attached to the lower
extremity of the wire was soldered a rather fine sewing needle, which, hanging
vertically downwards in a line which was a continuation of the axis of the wire,
dipped with its point into a cup filled with mercury, whilst a caoutchouc-covered
connecting wire, passing from the cup through a key and a tangent-galvanometer to
the otjier pole of the battery, served to complete the circuit. It having been
ascertained that the rotation of the fine point of the needle in the mercury did not
appreciably increase the logarithmic decrement, the effect of the current .on the
internal friction and the torsional elasticity of wires of nickel, iron, and tin was
investigated. As the lengths of the wires used were in all cases more than
600 centimetres^, and the diameters comparatively small, the strength of the current
did not exceed 0'30 ampbre. It will be hardly necessary to enter further into
details, since these experiments resemble very closely those already described, when
the effect of longitudinal magnetisation was tested. Suffice it that the examination
of the wires occupied several days, and the final conclusion arrived at was that when
the current was maintained constant there was no effect on the internal friction or on
the vibration-period produced by it apart from what might be expected from the very
slight rise of temperature resulting from the passage of the current,§ which rise could
not in any case have equalled 1° Centigrade.

* ' Phil. Trans.,' 1883 (vol. 174, Part I). Experiments 24 and 26. *

\ The current yielded by this battery did not vary so much as 5 per cent, during the whole of the
period of experimenting.

J The apparatus here used was described in the author's first memoir on Internal Friction.

§ This could be calculated within a sufficient degree of approximation from data furnished br Mr. J. T.
BOTTOMLET ('Nature,' Sept. 25 and Oct. 2, 1884).

STRAIN ON THE PHYSICAL PROPERTIES OK MATTEH.

19

It may be objected to the above experiments that too weak a current was employed
to bring out with sufficient distinctness any sign of what was looked for; but, as has
been already mentioned, the reason for employing such a current was to avoid as far
as possible heating effects. Moreover, it was evident that the nickel and iron wires
were in a state of circular magnetisation under the influence of the current, inasmuch
as a very sensible twist was produced by the combined action of the circular
magnetisation and the longitudinal magnetisation, resulting from the vertical
component of the earth's magnetic force. The current passed up the wires, and by
the earth's vertical magnetic force they were longitudinally magnetised with the
north-seeking magnetic pole downwards, the combined action of the two magnetising
stresses resulting in twisting the nickel wire in the direction of a right-handed screw ;*
by properly timing the impulses of the key used for closing the circuit a deflection of
eight scale-divisions could be easily produced. With the iron wire the twist was much
more perceptible than with nickel, and in the opposite direction, as might be expected
from the researches of JOULE and GOBE. By properly timing the impulses of the
key an amplitude of no less than 140 scale-divisions could be got up.

Experiment XVII.

[October, 1887. — It has been remarked that a sustained electric current passing
through a wire does not, except by heating, appreciably alter the internal friction.
The currents used in bringing out this result, though strong enough to produce very
sensible circular magnetisation, were comparatively feeble. As soon, however, as it
appeared, from Experiment VII., that the magnetising stress need only be applied
for an instant in order to produce an effect, it was resolved to ascertain to what
extent the internal friction could be influenced by circular magnetisation produced by
sending an intermittent current through the wire, the current being reversed in
synchronism with the torsional vibrations of the wire at the end of each swing of
the latter, and instantly removed again after each reversal. The vibration-period
of the wire was 4725 seconds.

Current in C.O.8. nniU lined to
circularly magnetite.

0-0199
0-0345
0-0616
0-1060
0-2056

Logarithmic decrement due to
internal friction.

•000845
•001102
•001210
•001564
•002624
•004591

* This shows a shortening in the direction of magnetisation, and therefore the result is in accordance
with that arrived at by Professor BARRETT, who has proved that nickel contracts in the direction of
magnetisation.

D 2

•20

MU. H. TOUfLlNSON ON THE INFLUENCE OP STRESS AND

By subtracting '000845 from the other numbers in the same column we obtain the
increase of logarithmic decrement due to the reversals of the current in each case.

Current in C.G.S. unite.

Increase of logarithmic
decrement due to reversals.

13: A.

A.

15.

0-0199

•000257

•0129

0-0345

•000365

•0106

0-0616

•000719

•0117

0-1060

•001779

•0168

0-2056

•003746

•0237

The increase of the logarithmic decrement increases proportionately to the current
until the latter attains a value of 0'0616 ; from this point upwards the ratio B: A
increases. Since the circuit was broken immediately after each reversal the current
might have been earned much higher without sensibly heating the wire,* and, in all
probability, with circular magnetisation as with longitudinal magnetisation, the ratio
B : A would, beyond a certain value of the magnetising stress, begin to decrease as
the stress increased. There was, however, the following difficulty in experimenting
with higher currents : — In consequence of the wire hanging vertically it was all the
time under the influence of the earth's vertical magnetic stress, and was thereby
longitudinally magnetised, so that when the current was sent up or down the wire
the combination of the two magnetisations caused the wire to twist to this side or
that ; and, since the reversals of the current were synchronous with the vibrations of
the wire, the amplitude of these last vibrations was, owing to the above-mentioned
cause, increased or diminished according as the phases of the two sets of vibrations
were the same or opposite. With the smaller currents the difficulty was sufficiently
overcome by first making a number of reversals with the two sets of vibrations in the
same phase, and then an equal number with the phases opposed ; but with the
current 0'205G this mode of compensation began to fail. It is evident, however,
that the internal friction of iron can be very largely increased by reversals of a
current passing through the wire.]

The Effect of Magnetisation on tJie Longitudinal Elasticity.

According to GUILLAUME WunTHEiMt long continued magnetisation diminishes both
temporarily and permanently the longitudinal elasticity of iron and steel, but
magnetisation continued only for a short time has no sensible effect. Since the
author's own experiments led him to view with considerable caution these results of
M.Wj£RTHEiM,and to believe that long continued magnetisation diminished the elasticity

• Rise of temperature causes decrease of the internal friction of annealed iron,
t ' Annales de Chimie,' vol. 12, 1844, p. 610.

STRAIN ON THE PHYSICAL PROPERTIES OF MATTER. ^

simply on account of the heat generated by the magnetising solenoid, he proceeded as
follows : —

Efl" 'in,.',d XVIII.

The magnetising solenoid described in the first portion of this memoir was placed
horizontally, and an iron wire of rather more than twice the length of the solenoid
was stretched along the axis of the latter and fixed at both ends. Water, as in
the earlier experiments, was kept constantly flowing through the annular space of
the air-chamber, so as to prevent the heat generated in the solenoid from affecting
the wire. The wire, when rubbed along its length with a resined glove, gave a clear
note, which was taken on a monochord. The magnetising solenoid was now excited
by ten GROVE'S cells, and the wire, having been loosened and stretched by the same
load as before,* was again secured at the ends and rubbed. The note yielded
was precisely the same as before, and, though the current 'was allowed to flow
through the solenoid for some time, not the slightest change could be detected.
Several trials were made of the same kind, and all concurred in yielding the same
results. Of course, the value to be attached to an experiment of this kind depends
mainly upon the ability of the observer to distinguish small differences of pitch.
Now the author's assistantt has so frequently proved his skill in this respect^
that it can be fairly said that if longitudinal magnetisation does influence longitudinal
elasticity it does so to an extent which cannot be appreciated, even by a well-trained
ear, when tested in the above manner. It might, perhaps, be possible to detect some
effect of magnetisation on the temporary elongation produced by large loads when
the method of static extension is employed, but not with the arrangement used by
WERTHEIM. M. WERTUEIM seems to have taken no precautions to avoid the heating
effects of the magnetising solenoid, and the very fact that he was only able to detect
any change after the solenoid had been excited for sonic time points suspiciously to
heating as being the origin of what he observed. It is true that according to
WERTHEIM the longitudinal elasticity of iron is increased by a small rise of tempera-
ture, but in this the author has ventured to differ from him. §

The Influence of an Electric Current on Longitudinal Elasticity.

According to WEHTHEIM|| the longitudinal elasticity of wires is diminished by the
passage of an electric current, independently of the alteration which would result

* The full arrangement for effecting this will be described in a future paper,
t Mr. FUHSE, Uic Curator of the Museum of King George III., King's College, Strand.
J The author has already given (' Phil. Trans.,' 1883 (vol. 174, Part I.), p. 53 and elsewhere)
specimens of this.

§ ' Phil. Trans.,' 1883 (vol. 174, Part I.), pp. 128-131.
|| 'Annales de Chimie,' vol. 12, 1844.

22 MB. H. TOMLINSON ON THE INFLUENCE OF STRESS AND

from the elevation of temperature produced by the current. When, however, the
experiments were concluded which have just been recorded, and, moreover, no appreci-
able effect had been found to be produced on the torsional elasticity by the passage of
a current, the author was led to endeavour to ascertain how far WERTHEIM might be
justified in the above-mentioned conclusion as regards the effect of a current on the
longitudinal elasticity.

First, in order to avoid heating as much as possible, currents from '2 to '3 ampere
were employed — currents which, though comparatively weak, are, as has been shown,
capable of producing very sensible circular magnetisation. The same length of wire
and the same apparatus as that described in the author's former papers on Elasticity*
were used, with this difference, however, that now the wire to be examined and the
comparison-wire were secured at their upper extremities to separate clamps which
were insulated from each other and from the bracket on which they rested. Near
their lower extremities the wire and the comparison-wire were united by a short piece
of copper wire, so that the current from the battery might, after passing through a
tangent-galvanometer and a commutator, continue its course down one wire and up
the other to the other pole of the battery. By this arrangement it may be seen that the
heating effect of the current will not cause any error except that due to slight differ-
ence in the thermal expansibility of the wire and the comparison- wire, due to the differ-
ence in the load on the two wires.t Any change of elasticity wrought by the current,
amounting to *1 per cent., could have readily been detected, but after some five or six
hours had been spent, and loads of very different amounts, almost up to the breaking-
load of the wire, used, the attempt was abandoned, as it seemed certain that for these
comparatively small currents there was no appreciable change in the elasticity resulting
from the passage of the current when the latter was maintained constant.

Next the effect of much more powerful currents was tried, the same arrangements
being employed as in Experiment XVIII., with the exception that the magnetising
solenoid was removed, and in its place two terminal screws substituted, one near each
end of the wire, but, of course, beyond the points where the wires are clamped.
These terminal screws served to connect the wire with a battery of 10 GROVE'S cells
and a box of resistance-coils, together with a tangent-galvanometer and commutator.
The wire was rubbed along its length with a resined glove, and the pitch of the note
determined by means of the syren — first when the current was not passing through
the wire, next when it was, and finally, a second time, with no current. The following
were the results obtained with annealed iron wire and unannealed piano-steel wire : —

* ' Phil. Trans.,' 1883 (vol. 174, Part I.), pp. 2-4.

t Even this source of error can be eliminated by testing the wire, in the first place, with the permanent
load on the comparison-wire greater, and in the second place less, by an eqnal amount than that on the
wire to be examined.

STRAIN ON THK PHYSICAL PROI'KUTIKS OF MATTER.

23

Experiment XIX.
IRON wire 365 centimetres in length and '07035 centimetre in diameter.

Current in C.G.S.
unit*.

Calculated rise of
temperature in degrees
• igrade produced
by the current.

Xiim!>cr of longitudinal vibra-
tion* in two minute*.

Decrease of longitudinal
elasticity per unit per
degree rlie of tem-
perature.

0
0-323

45-31

4115-5 X 20
4085-0 X 20

•0003268

Experiment XX.
PIANO-STEEL 365 centimetres in length and '0824 centimetre in diameter.

Current in C.O.8.
unite.

Calculated rise of
temperature in degree*
Centigrade produced
by the current

.Numhor of longitudinal vibra-
tion* in two minute*.

DccreaMe of longitudinal
elasticity per unit per
degree rinc of tem-
perature.

0
0-2335

44-67

4255-3 x 20
4229-5 x 20

•0003020

The principal difficulty connected with the accurate reduction of these experiments
lies in the calculation of the rise of temperature produced by the current. According
to J. T. BOTTOMLEY * the rise of temperature may be obtained from the following
easily proved formula : —

where t is the temperature of the wire, 6 the temperature of the room, c the current,
<rt the specific resistance of the wire at the temperature t, J JOULE'S equivalent, d
the diameter of the wire, and e the emissivity of the wire. It would follow from
Mr. BOTTOMLEY'S experiments that with wires whose diameters vary from 0'085 centi-
metre to 0'040 centimetre the value of e ranges from ToW to 4^5, or approximately as
the inverse of the square of the diameter. On this last assumption the values of e
for the iron and steel wires were calculated to be TjW an<i 16*79 respectively. The
values of ov were calculated from a knowledge of the specific resistance of the two
metals at the temperature of the room, which was 18° C., and of the coefficient of
increase of resistance per degree rise of temperature. The specific resistances of the
iron and steel wires at the temperature of 18° C. were 9707 and 20,742 respectively,

• ' Nature,' September 25 and October 2, 1884.

24 MR. H. TOMLINSON ON THE INFLUENCE OF STRESS AND

and the coefficients of increase of resistance were approximately '005 and '004
respectively. The value of J was assumed to be 42,000,000.

The liability to error in the estimation of the number of vibrations executed by the
wire in a given time is comparatively small, and is certainly much smaller than that
arising from uncertainty as regards the emissivity. On the whole, then, it is only safe
to assume that, if the observed decrease of the longitudinal elasticity of the iron and
steel be entirely due to the rise of temperature caused by the current, this decrease
would be about '0003 per unit per degree Centigrade rise of temperature. Now, as
far as the author is aware, there have been no investigations of the effect of rise of
temperature on the longitudinal elasticity of iron and steel as determined by the
method of vibrations. The author has, however, by the method of static extension,
obtained for unannealed piano steel, and for soft iron, a decrease of longitudinal
elasticity per unit, resulting from a rise of 1° Centigrade, of between '0002 and "0003.*
Further, he has recently* obtained from experiments on torsionally vibrating iron wire
at various temperatures a decrease of torsional elasticity amounting to '0003 per unit
per degree Centigrade rise of temperature. Taking all these facts into consideration,
it seems probable that such effects as are produced by even powerful currents on the
longitudinal elasticity are solely to be ascribed to the heat generated by these currents.

Discussion of WIEDEMANN'S Theory respecting the Internal Friction of Metals.

It will be now advantageous to review very briefly some of the results recorded in
this and the author's two previous memoirs on the internal friction of metals, with the
object of ascertaining how far these results support WIEDEMANN'S ingenious theory
respecting the cause of the friction. According to WIEDEMANN, as the wire vibrates
torsionally in one direction or the opposite, so do the molecules rotate about their
axes in one direction or the opposite, and at each rotation are permanently deflected,
so that, if at the extremity of any one of the swings of the vibrator the latter could
be checked and afterwards quietly restored to its position of equilibrium without
allowing it to pass over to the other side, the molecules would be permanently
deflected. If, however, as in free torsional vibrations, the vibrator is allowed to pass
over the position of equilibrium, the molecules will be permanently twisted in the
opposite direction, and so on ; thus the loss of energy due to internal friction repre-
sents the work which has to be performed in order to twist the molecules from their
permanent positions on the one side to their permanent positions on the other.
WIEDKMANN founds his theory mainly on the results of his experiments on the torsion
experienced by wires as tested by the statical method. In these experiments he
finds that even the slightest torsional stress produces permanent torsion of the
wire, and also, as he assumes, permanent twist of the molecules. It will be as well

* 'Phil. Trans.,' 1883 (vol. 174, Part I.), pp. 132-133.

STRAIN ON THE PIIYSICAL PROPERTIES OF MATTKI!

25

to quote one of WIEDEMANN'S experiments, so that some idea may be formed of the
extent of the permanent deformation produced by a given torsional stress.

Experiment XXI.

An annealed brass wire 48 centims. in length and 2 centims. in diameter, ln:nli>d
with 10'46 kilos., and subjected to torsional stress, increased by small amounts at a
time. Scale and mirror were used, and a displacement of the image through
34'9 scale-divisions corresponded to 1° of rotation of the mirror.

Load in grammes oied
in producing the
tonioa W.

Total ton- ion imme-
diately produced, T.»

Total tor-ion
produced after tome
time, T,.»

Total permanent ton-ion
produced
after tome time, Pj.

1st 30

104 x 3

104 x 3

06

Next 10

104

104

07

Ditto

107

107

1-0

Ditto

105

105

1-5

Ditto

105

106

2-0

Ditto

111

111

4-0

Ditto

112-5

114

6-2

An examination of the fourth column shows that the consecutive values of
permanent deformation produced by consecutive equal loads form roughly a
geometrical progression, of which the common ratio is 1'42. It may then be roughly
calculated that the permanent twist produced by the first 10 grammes cannot be
greater than '14, whilst the total torsion is 104. Thus in this experiment the
permanent torsion produced by the smallest load is less than yooth of the total ; but
in some of the author's own experiments, in consequence of a much greater length of
wire being used, and that, too, of much smaller diameter, the temporary deformations
vary, on the average, between -fatii and g^th of the smallest in WIEDEMANN'S experi-
ments, so that the permanent torsion of the wire itself would escape observation should
the attempt be made to detect it by the method of statical torsion.

[October, 1887. — Nevertheless, the experiments described in the author's paper
seem to place beyond doubt the soundness of WIEDEMANN'S hypothesis that even
with the most minute torsional deformations the molecules of a torsionally oscillating
wire are permanently twisted first to this side and then to that, and that this is the
main cause of the internal friction.t For how else can the enormous increase of the
internal friction which can be effected by interrupted magnetisation be accounted for ?
Had a magnetic elastic limit been proved to exist, there would have been some little
difficulty in accepting WIKIUM \\s's views, for in the author's experiments on internal
friction, previously alluded to, the logarithmic decrement was found to be independent

• WIEDKMANN designates as temporary torsion what is here put down a* total torsiou.
t Provided the amplitudes of the oscillations do not exceed a certain limit.
MDCCCLX \\V11 1 . — A. E

26 MR. H. TOMLINSON ON THE INFLUENCE OF STRESS AND STRAIN, ETC.

of the amplitude even for the very minutest torsional deformations ; but, since it ha8
been proved that any magnetic stress, however small, must produce some molecular
permanent set, the difficulty vanishes.]

Summary.

1. When the deformations produced by the oscillations are small the internal
friction of a torsionally vibrating wire is not affected by sustained longitudinal mag-
netisation of moderate amount. The internal friction is also not affected by sustained
magnetisation, even when carried nearly to the point of saturation, provided the
magnetising current be, previously to experimenting, reversed a great number of times.

2. When the deformations are large the internal friction is increased by sustained
magnetisation of large amount.

3. The torsional elasticity is entirely independent of any sustained longitudinally
magnetising stress which may be acting upon the wire, provided the deformations
produced by the torsional oscillations be small. When the deformations are large the
torsional elasticity is very slightly decreased by sustained longitudinal magnetisation
of large amount.

4. When the magnetising current is interrupted, and, to a greater extent, when it
is reversed repeatedly whilst the wire is oscillating, the internal friction is increased
provided the magnetising stress be of moderate amount. The increase of internal
friction may become very considerable when the magnetising stress is great.

5. When the deformations produced by the oscillations are small the torsional
elasticity is not affected by either repeatedly interrupted or reversed magnetisation,
even when the magnetising current is very large.

6. There exists a limit of magnetic stress within which the magnetic elasticity is
sensibly perfect, but a mathematically true magnetic elastic limit does not exist.

7. The passage of an electric current through a torsionally vibrating wire does not
affect, except by heating, either the internal friction or the torsional elasticity, provided
the deformations produced by the oscillations be small.

8. The effect of longitudinal magnetisation, even when carried to the point of
saturation, on the longitudinal oscillations of an iron or steel wire is nil.

9. The passage of an electric current through a longitudinally oscillating iron or
steel wire does not, except by heating, affect the oscillation-frequency.

10. When the deformations produced in a torsionally oscillating wire do not exceed
a certain limit the internal friction mainly depends upon the sub-permanent rotation
to and fro of the molecules about their axes.

[ 27 ]

II. On the Spectrum of the Oxy-hydrogen Flame.

By G. D. LIVEING, M.A., F.R.S., Professor of Chemistry, and J. DEWAK, M.A.,
F.R.S., Jacksonian Professor, University of Cambridge.

Received January 18,— Bead February 2, 1888.

[PLATES 1-4.]

IN 1880 we described, simultaneously with Dr. HUGGINS, a series of linen furining the
strongest part of the spectrum of water, and again, in 1 882, a second, less strong and more
refrangible series, forming a second section of the same spectrum. Subsequently,
M. DESLANDRES discovered a third still weaker and more refrangible series, beginning
at a wave-length 2610*3. We find, however, that the spectrum does not end there,
but extends both on the more refrangible and on the less refrangible sides to a con-
siderable distance. By employing a large spectroscope with a single calcite prism and
a long exposure we have obtained photographs of the spectrum of the oxy-hydrogen
flame, showing closely set lines from wave-length 2268 to 4100, with traces of lines
beyond those limits. The whole spectrum appears to consist of a succession of
rhythmical series of lines, the lines of each rhythmical series being stronger and more
closely set at the more refrangible end of the series and becoming weaker and wider
apart towards the less refrangible end. The strongest of these series are those first
described, those on either side of them becoming fainter as they are more remote,
until the highest series gave us a measurable photograph only after an exposure of
five hours. In most cases two series begin near together and overlap one another,
producing a complication which cannot easily be unravelled, and the overlapping
appears in some cases to extend to more than two series.

M. DESLANDRES states (' Comptes Rendus,' vol. 100, p. 854,) that the first band
of the water spectrum (i.e., the group beginning at a wave-length about 3063)
includes a series of rays which reproduce, line for line, at the same distances and with
the same relative intensities, the band A of the solar spectrum ; and that the second
band (i.e., the group beginning at a wave-length about 2811) includes a series corres-
ponding to B, and that in the third a may be found to be reproduced. He does
not state at what wave-lengths in these bands we are to look for the more refrangible

E 2 i.r,.«8

28 PROFESSORS G. D. LIVEING AND J. DEWAR

edges of A, B, and a respectively ; and we have not been able to make out such an
exact correspondence between the lines of the water spectrum and those of A, B, and
a as M. DESLANDRES' words seem to imply. Nevertheless, the similarity of the
grouping is very remarkable, as may be seen from the accompanying map, on which
are given the lines of A, on a scale slightly reduced from Professor PIAZZI SMYTH'S
solar spectrum, side by side with the lines of the water spectrum. The corres-
pondence of B and of a to certain lines of the higher groups is less striking. We
have no doubt that the peculiar arrangement of the lines, commencing at the more
refrangible end with some closely set lines and continued in a series of doublets, is in
all these cases the result of a general law.

The tables of wave-lengths which follow were obtained by measuring the distances
of the lines from those of iron photographed on the same plate through a part of the
slit. The whole of the lines previously measured have been re-measured, and the
numbers assigned as their wave-lengths corrected by reference to more recent measure-
ments of the iron lines. The scale is, however, that of ANGSTROM. On account of
the faintness of the rays at the two extremities of the spectrum a wide slit had to be
employed, as well as a lengthened exposure in photographing them, consequently
the photographed lines are broad and somewhat ill-defined, and groups of closely set
lines could not be resolved. The wave-lengths in the first and last tables are
therefore liable to greater errors than those in the intervening tables. Moreover,
in these parts, the spectrum is so weak that the strongest line is less intense than
the weakest lines of the region about X 3] 00, and it has been found necessary to
adopt a different scale of intensities for different sections.

It is, perhaps, worth remarking that there is a broad diffuse line, the strongest
of the lines in that region, coincident with K of the solar spectrum. There is no
line coincident with H.

ON Till! SI'KiTRDM OF THK OXY-HYDKOGKN FLA.MI

29

TABLES of wave-lengths, and inverse wave-lengths, of the lines in the spectrum of
the oxy-hydrogen flame. The numbers prefixed represent the intensities 1 to 6,
No. 1 being the most intense. Diffuse lines are noted with a d.

I.

Intensity.

X

1
X

Int.-li-ilv

X

1
\

IiilcMily.

X

1

X

6

2268-0

44,092

6

2374-9

42,107

6

2412-1

41,459

6

2272-2

44,010

6

2:575-5

42,096

Gd

2412-6

41,450

6

2283-6

43,791

6

2:176-6

42,077

6

24143

41,420

6

2297-0

43,535

6

•j::78-6

42,042

6

2414-8

41,411

6

2300-8

43,463

6

2379-6

42,024

5

2416-2

41,387

6

2307-5

43,337

6

2381-9

41,983

6

2418-0

41,356

6

2310-1

43,288

6.

2383-0

41,964

6

2419-8

41,326

6

8816-2

43,174

5

23843

41,941

6

2421-6

41,295

5

•j::-J3-8

43,033

4

2385-7

41,916

6

2422-4

41,281

M

2331 1

42,898

5

2387-0

41,894

4

2425-7

41,225

6d

23322

42,878

6

2390-6

41,831

6

2427-0

41,203

5

2337-5

42,781

6

2391-6

41,813

5

2428-1

41,184

6

2342-1

42,697

6

23935

41,780

5

2429-7

41,157

6

2345-6

42,633

6

2394-8

41,757

6d

2431-2

41,132

6

2347-5

42,599

6

2396-3

41,731

6d

2431-8

41,122

6

2351-6

42,524

6

2398-0

41,702

6

2433-3

41,096

6

2354-1

42,479

6

2398-6

41,691

6d

24339

41,086

5

2355-5

42,454

5

2399-4

41,677

6

24359

41,053

6

2356-6

42,434

5

2402-4

41,625

6

2437-2

41,031

6

2357-7

42,414

6

2403-2

41,611

5

2438-7

41,005

6

2360-6

42,362

6

2404-2

41,595

5

2440-3

40,979

6

2365-1

42.2S2

6

2405-4

41,574

6

2441-6

40,957

6

2366-1

42,264

6

2406-6

41,553

6

2443-2

40,930

6

2368-6

42,219

6 2407-6

41,536

6

2445-4

40,893

6

23712 42,173

5 2409-0

41,511

6

2446-5

40,875

6

2372-8 42,144

6

2410-1

41,492

6

2448-4

40,843

6

2373-6

42,130

30

PROFESSORS G. D. L1VEING AND J. DEWAR

II.

The average intensities of the lines in this section are greater than those of the
lines in the preceding section, so that a line marked No. 6 in the following table is
generally quite as intense as one marked No. 4 in the preceding table.

Intensity.

X

1

X

Intensity.

X

1

X

Intensity.

X

1
X

4

2449-3

40,828

5d

2496-3

40,059

6

2542-7

39,328

5

2450-9

40,801

5

2498.0

40,032

6

2545-6

39,283

6

2452-2

40,780

6

2499-8

40,003

4

2547-7

39,251

6

2453-3

40,761

5

2501-4

39,978

4d

2550-3

39,211

5

2454-7

40,738

5

2503-1

39,950

3

2553-4

39,163

4

2456-0

40,717

6

2503-7

39,941

4

2556-4

39,118

5

2457-7

40,688

6

2504-0

39,936

3

2559-6

39,069

6

24592

40,664

6

2504-4

39,930

5

2562-6

39,023

6

2460-0

40,650

6

2505-2

39,917

6

2565-6

38,977 •

6

2461-7

40,622

6

2505-6

39,911

5

2567-0

38,956

4d

2462-8

40,604

6

2506-8

39,892

5

2569-1

38,924

6

2464-5

40,576

6

2508-1

39,871

6

2570-4

38,904

6

2465-9

40,553

6

2509-1

39,855

6

2572-9

38,867

6

2467-1

40,533

6

2509-8

39,844

6

2573-4

38,859

5

2469-6

40,492

6

2510-5

39,833

6

2574-5

38,842

4

2471-9

40,455

6

2511-1

39,823

6

2576-7

38,809

6

2474-5

40,412

6

2513-1

39,791

6

2578-3

38,785

6

2477-6

40,362

6

2515-1

39,760

6

2580-9

38,746

6

2479-3

40,334

5

2517-5

39,722

6

2582-1

38,728

6

2480-7

40,311

5

2519-8

39,686

6

2582-8

38,718

5

2482-6

40,280

5

2521-7

39,656

5

2584-4

38,694

6

2483-7

40,263

5

2524-2

39,617

6

2587-1

38,653

6

2484-9

40,243

6

2529-2

39,538

5

2589-1

38,623

6

2485-8

40,229

4

2530-2

' 30,523

6

2591-3

38,591

5

2487-2

40,206

6

2531-4

39,504

6

2592-8

38,568

5

2489-3

40,172

5

2534-1

39,462

6

2594-6

38,542

4

24911

40,143

3

2536-6

39,423

6

2596-4

38,515

6

2492-3

40.124

5

2537-7

39,406

6

2598-6

38,482

6

2493-8

40,099

6

2538-9

39,387

6

2600-9

38,448

6

2495-6

40,071

6

2540-2

39,367

6

26032

38,414

6

2605-2

38,385

ON THE SPECTRUM OF THE OXY-HYUROGEN FLAME.

31

III.

The average intensities of the lines in this section is a good deal greater than that
of the lines in the preceding section, so that intensity No. 6 in this table would
include lines of intensity No. 4 in the preceding section.

Intensity.

X

1
X

Illt.-ll-irv.

X

1

X

Inteiuiiy.

X

1

X

3

2608-4

sgjtt

4

26732

37,409

6

2734-3

36,572

6

2608-9

38,331

6

2675-8

37,373

5

•2735-5

86,557

6

2609-7

38,319

3

26773

87,868

4

2737-8

.Mi.526

5

2611-0

38,300

6

2678-2

37,339

4

2740-2

36,494

6

2612-6

38,277

6

2679-0

37,327

6

274-27

36,400

4d

2613-5

38,264

4

2680-9

37.3M-2

4

27i:. '.'

36,418

4

2614-6

38,247

6

2681-8

37,288

4

2748-3

:«;,387

6

2G15-6

38,2: :-2

Tianrl

/ 2683-0

37,272

5

2751-0

36,351

5

2616-5

38,219

BVBB

\ 2683-7

37,263

5

8758*1

36,323

6

2617-0

38,212

6

2684-8

37,247

6

27547

36,302

5

26177

38,201

5

2685-5

37,238

4

2757-0

36,271

6

2618-1

38,196

6

2686-5

37,'22i

6

2759-0

36,246

4

2618-9

38,184

6

2687-2

37,213

4d

•27r,9-8

36,235

5

•2 • -.20-6

38,159

5

2687-7

37,207

4

2761-4

86,814

5

•j<;-2l-4

38,148

4

2688-9

37,191

6 •

2762-6

30,198

6

2622-2

38,136

5

:y,!)0-6

37,167

4

8764-1

36,178

6

2622-8

38,128

5

2691-7

37,152

5

_>7i;0-3

36,150

0

26-23-3

38,120

6

2692-5

37,140

6

27673

36,137

&j

262 1-3

38,105

6

2693-2

37,131

6

2768-2

:i>;,125

5

20-25-9

38,083

5

26H3-8

37,123

6

270'.tl

36,113

G

2i 127-2

:K063

5

2695-4

37,101

6

2770-0

36,101

5

20-277

38,057

6

2696-1

37,091

6

2770-9

36,090

5

2628-3

38,047

6d

2697-8

37,067

6

2772-3

36,071

2d

2631-3

38,004

4

2698-8

37,054

6

2773-8

36,052

5

2632-4

37,988

6

2699-7

37,041

5

2774-9

36,037

6

2633-4

37,974

3

2701-6

37,016

5

2776-1

36,022

5

2634-8

37,954

6

•2704-3

36,979

6

•2777-4

36,005

4

2635-7

37,941

5

2705-2

36,966

6

2778-6

35,989

6

:r,-9

37,924

5

2706-2

36,952

5

27792

35,982

4

2638-5

37,900

6

2707-8

36,939

6

2780-7

35,962

3

2640-5

37,872

4

2709-6

36,907

4

27&J2

::.-.,930

6

2642-2

37,847

6

2710-6

36,893

6

2784-8

35,909

5

2648-1

37,834

4

•2711-6

36,879

4

2786-5

35,887

o

•2644-2

:!7.819

6

27136

36,851

6

2787-7

35,872

I

•264.V7

37,797

4

2714-5

36,839

6

2788-3

35.K--4

3

2648-2

37,762

5

2715-8

36,822

6

2789-1

35,854

6

2650-7

37,726

4

2717-2

36,803

6

2789-8

35,845

3

2651-3

37,717

6

•2718-2

36,789

5

2790-5

85,886

6

2652-6

37,699

4

2719-8

36,767

6

27<U 7

35,820

5

2653-7

37,684

5

27-21-6

36,743

5

•27H3-8

::.-.. 794

6

2654-3

37,675

4

272::-:.

36,717

6

•27;>.V7

::.\769

2

2657-4

37,631

5

2724-8

36,701

r

2796-9

::.V7-r.4

4

2659-7

37,598

5 2726-1

36,682

6

•7-6

85,745

5

•2-;r,o-9

37,581

5 L'728-2

36,655

3

•J7'.t9-8

35,717

3

iMr.3-9

37,539

6 2729-9

36,631

O

2802-9

35,677

6

2666-0

37,501

. 4 -2730-6

36,623

6

2804-2

35,661

5

•jr.'> 1

37,480

4 -2732-1

36,602

6

2805-4

35,646

:

2671-1

37,438

5 2733-0

36,590

5

2806-8

:i\6-28

PROFESSORS G. D. LIVEING AND J. DEVVAR

IV.

The lines of this section are generally much more intense than those of the
preceding section, so that intensity No. 6 of this section may include lines which
would be marked as of intensity 4, according to the scale employed in the preceding
section.

Intensity.

X

1
A

Intensity.

A

1
X

Intensity.

A

l

X

3

2811-2

35,572

3

2857-6

34,994

5

2918-2

34,268

6

2811-3

35,571

3

2859-4

34,972

2d

2918-5

34,264

6

28117

35,566

3

2860-3

34,961

4

2919-8

34,249

0

2812-1

35,561

3

2861-7

34,956

4

2921-5

34,229

6

2812-4

35,557

8

2863-3

34,925

5

2923-8

34,202

id

2813-5

35,543

6

2865-5

34,898

4

2924-4

34,195

5

2814-9

35,525

3

2866-0

34,892

4

2924-8

34,190

4

2815-6

35,516

3

2868-3

34,864

4

2926-3

34,173

4

2816-1

35,510

3

2869-5

34,849

5

2927-1

34,164

3

2817-1

35,498

3

2871-5

34,825

5

2927-6

34,158

6

2818-2

35,484

3

2871-9

34,820

4

2929-9

34,131

4

2818-7

35,477

' 2rl

2875-0

34,783

4

2931-0

34,118

6

2819-3

35,470

3d

2875-8

34,773

3d

29335

34,089

3

2820-1

35,460

2d

2878-3

34,743

5

2935-2

34,069

6

28-JO-7

35,452

5d

2880-3

34,719

6

2936-5

34,054

6

2821-2

35,446

6

288M

34,709

4

2937-2

34,046

3

2821-8

35,438

6

2881-8

34,701

3

29378

34,039

4

2*22-3

35,432

4

2882-5

34,692

6

2938-5

34,031

6

2824-0

35,411

6

2884-2

34,672

4

2940-3

34,010

3

2824-8

35,401

3

2885-3

34,658

4

2940-6

34,007

41

2825-2

35,396

6

2886-1

34,649

5

2944-2

33,965

3

2826-3

35,382

6

2886-3

34,646

3

2945-2

33,954

U

2828-3

35,357

4

2887-5

34,632

6

2D46-5

33,939

2

2828-7

35,352

6

2888-5

34,620

4d

2947-5

33,927

6

2829-2

35,346

6

2889-2

34,612

5

2948-5

33,916

44

2829-8

35,338

3

2889-8

34,604

6

2950-1

33,897

4

2831-1

35,322

3

2890-2

34,600

6

2950-7

33,890

3

2831-7

35,314

5

2890-8

34,593

6

2951-2

33,885

4

2833-3

35,295

1

2892-9

34,567

6

2951-7

33,879

4

2834-0

35,286

5

2893-5

34,500

6

2252-5

33,870

3

2835-0

35,273

6

2894-2

34,552

6

2953-2

33,862

8

2835-8

35,263

4

2896-1

34,529

6d

2954-5

33,847

5

2836-7

35,252

6?

2897-1

34,517

6

2955-5

33,835

1

2838-8

35,226

4

2897-6

34,511

6

2956-3

33,826

4

2840-1

35,210

4

2898-1

34,505

5

2957-1

33,817

6

2841-0

35,199

6

2898-8

34,497

6

2958-9

33,796

3

2842-2

35,184

4

2899-5

34,489

3

2960-0

33,784

3

2842-7

35,177

3

2900-2

34,480

6

2y62-l

33,760

6P

2843-1

35,173

4

2900-9

34,472

5

2962-9

33,751

2

2844-4

35,157

6

2902-5

34,453

4

2965-5

33,721

6?

2845-4

35,144

1

2903-7

34,439

6

2966-5

33,710

3

2846-3

35,133

6

2906-0

34,412

6

2967-1

33,703

3

2847-4

35,120

5

2906-6

34,404

6

2968-0

33,693

3

2848-8

35,103

3

2907-3

34,396

6

2968-5

33,(i87

3

2849-5

35,094

3

2908-3

34,384

6

2970-0

33,670

3

2850-3

35,059

3

2909-4

34,371

5

2970-7

33,662

6

2850-7

35,079

3

2911-4

34,848

6

2971-1

33,658

2

2852-2

35,085

3

2912-9

34,330

6

2972-2

33,<il-5

8

2853-9

35,040

5

2913-5

34,323

6

2973-9

33,626

3

2854-9

35,028

4

2915-7

34,297

3d

2975-1

33,612

3

2855-4 35.021

3

2916-3

34,290

3d

2977-8

33,582

iind I)r\\ in:

HtiJ.Tran*. 1«8H. A. flat, I

ON THE SI'KCTHr.M OF T1IK OXT-HTDBOOBN FLAMK.

33

Intensity.

X

1

X

Intensity.

X

,
I Int^t,.

1

*

1
X

td

29794

33,564

4

29978

33,358

6

3022-5

33,085

Id

2980-2 :.55

6

2998-7

33,348

6 ii'123-4

33,075

6

2982-2

33,532

4

3001-9

£3,312

6 3025-2

33,056

6

2982-9

33,524

6

3005-0

33,278

6 3027-6

33,029

6d

2983-8

33,f,14

6

3005-G

33,271

6 3030-3 3S.OOO

6

2985-7

33,493

6

3008-2

83,242

5 3033-1

32,970

6

2987-2

176

6

3008-8

33,236

5 3036-4

32,934

5

-S-5

33,462

6

3012-9

33,191

4 30399

32,896

6

2990-5

33,439

6

3016-6

33,150

4

3043-9

32,853

5

2991-7

33,426

6

3020-9

33,103

3

30484

32,805

6

2992-9

33,412

6

3021-4

33,097

3

3052-7

32.758

G

2994-8

33,391

6

30220

33,091

4

80674

32,708

4

2996-6

33.371

II ' i

V.

This section is the strongest part of the spectrum of the oxy-hydrogen flame.
Lines marked as of intensity 4, on the scale of Table IV., would for the most part
be marked as of intensity 6 on the scale of this table.

Intensity.

X

1

X

Intensity.

X

1^

A

Intensity.

X

1

X

2

3063-3

32,645

u

3105-3

32,203

6

3148-0

31,766

4

3063-9

32,638

5

3106-0

32,196

3

3149-5

31,751

3

3064-6

32,631

3

3107-0

32,185

3

3150-6

31,740

4

3065-5

32,621

a

31088

32,167

4

3151-7

31,729

2d

3067-2

32,603

3

3109-7

32,157

6

3152-7

31,719

&d

3068-2

32,592

1

3111-5

32,139

2

31540

31,706

3

3070-0

32,573

3

31 12-8

32,125

6

3156-4

31,682

2

3071-5

32,557

3

3114-3

32,110

6

3157-3

31,673

6

3072-6

32,546

4

3UC-6

32,086 4

3158-0

31,666

5

3073-8

32,533

2

3117-4

32,078 5

3160-3

31, 643

5

3074-4

32,527

4

3119-2

32,060

4

31615

31,631

5

3076-6

32,503

6

31213

32,038 6

3162-8

31,618

3

3077-9

32,490

2

312-2-2

32,029 Sd

31639

31,607

2

3079-3

3-2,475

3

3123-5

32,015 5

3166-0

31,586

«>

3081-0

32,457

4

31245

32,005

U

31691

31,555

3

3082-6

32,440

6

3121',, i

31,990 6

31728

31,518

3

3084-6

32,419

6

3127-3

31,976

3

3174-0

31,506

5

3085-8

32,407

3

31278

31,971

6

31746

31,500

3

3086-7

32,397

2

3129-9

31,950

5

3177-2

31,474

1

3089-3

32,:J7()

6

3130-8

31,941

6

3179-6

31,451

3

3089-8

32,365

5

31326

31,1.22

6

3181-0

31,437

2

30906

32,356

2

3133-7

31,911

4

3182-6

31,421

2d

3092-0

32,342

2

:: 136-3

31,885

3

3185-6

31,391

8

3094-2

32,319

6

3137-4

31,874

6

3187-6

31,372

3

30948

32,312

5

81887

31,860

5

3191-3

31,335

3

3095-8

32,302

6

3139-4

31,853

6

3191-9

31.329

3

3096-3

32,297

3

3140-3

31,844

4

3194-5

31,304

:<

3098-3

32,27n

6

3141 •:.

81388

5

3198-7

31,263

2

3099-0

32,268

6

3142-5

:51,822

5

3200-4

31.246

5

3100-6

18,258

3

3U:: 5

81,818

5

3201-9

31,231

2

3101-6

32,241

6

3145-1

31,795

5-

32035

31,216

3

Slo-2 7

:'.L'.-230

2<2

3146-9

31,777

M IM VCLX XXVI II. — A.

34

PROFESSORS G. D. LIVEING AND J. DEWAR

VI.

The lines of this section are weaker than those of the preceding section, so that in
the scale of intensity of this table, No. 4 is not stronger than No. 6 of the preceding
table.

Intensity.

\

X

Intensity.

A

1

A

Intensity.

A

i

A

4d

3205-7

31,194

2

3295-5

30,344

4

3435-4

29,109

6

32081

31,171

6

3298-7

30,315

4

3437-4

29,092

6

3209-7

31,156

6

3304-2

30,265

4

3439-7

29,072

6

3212-0

31,133

M

3304-9

30,258

4

3442-3

29,050

3

3213-1

31,123

4

3307-5

30,234

4

3444-7

29,030

6

3214-4

31,110

6

3308-9

30,2-22

5

3447-6

29,006

6

3215-9

31,095

6

2310-5

30,207

6

3450-1

28,985

4

3217-5

31,080

5

3311-4

30,199

5

3451-0

28,977

4

3220-0

31.056

5

3314-8

30.168

6

3455-4

28,940

4

3221-0

31,046

4d

3318-0

30,139

5

3458-2

28,917

4

3222-8

31,029

6

3319-6

30,124

5

3459-5

28,906

4

3224-6

31,012

5

3321-8

30,104

5

3461-0

28,893

4

3225-9

30,999

6

3323-0

30,093

5

3462-6

28,880

5

3229-3

30,966

4

3325-4

30,072

5

3464-2

28,867

6

3230-3

30,957

5

3329-1

30,038

4

3465-9

28,853

6

3230-9

30,951

3d

3332-5

30,008

4

3467-7

28,838

4

3233-2

30,929

5

3335-9

29,977

5

3469-6

28,822

4

3234-6

30,916

6

3337-2

29,965

2

3471-9

28,803

4

3237-9

30,884

6

3344-2

29,903

5

3474-1

28,784

4

3240-6

30,858

6

3346-2

29,885

6

3475-5

28,773

6

3242-3

30,842

6

3347-3

29,875

4

3476-9

28,761

2

3243-7

30,829

6

3349-1

29,859

5

3479-1

28,743

6

3-248-4

30,784

6

3352-3

29,830

6

3481-4

28,724

6

3249-8

30,771

6

3353-7

29,818

6

348-2-9

28,712

6

3250-5

30,765

5

3355-9

29,798

4

3483-9

28,703

4

3252-0

30,750

5

3359-7

29,765

5

3486-2

28,685

4

3253-2

30,739

6

3362-2

29,742

6

3488-0

28,670

4

3254-8

30,724

6

3368-8

29,684

4

3489-5

28,657

4

3256-4

30,709

6

33701

29,673

6

3192-3

28,634

6

3258-4

30,690

6

3371-7

29,659

6

3493-3

28,626

6

3260-7

30,668

6

3378-2

29,645

6

3495-8

28,606

6

3262-5

30,651

6

3375-4

29,626

4

3497-2

28,594

2

3263-6

30,641

6

3377-6

29,607

4

3502-1

28,554

2

3266-4

30,615

6

3380:2

29,584

5

3507-6

28,510

2

3268-5

30,595

5

3383-3

29,557

6

3509-9

28,483

6

3270-0

30,581

6

3384-7

29,545

6

3511-6

28,477

G

3271-4

30,568

5

3386-2

29,532

5

3513-3

28,463

id

3273-7

30,546

5

3390-3

29,496

6

3515-2

28,448

4

3275-5

30,530

£

3394-5

29,459

5

3517-2

•28,432

4

3276-3

30,522

5

3399-1

29,420

6

3519-0

28,417

4d

3278-3

30,504

5

3404-2

29,375

6

35231

28,384

6

3279-2

30,495

5

3409-7

29,328

5

3524-1

28,376

6

3281-2

30,477

6

3412-8

29,301

6

3525-6

28,364

0

3282-9

30,461

5

3415-4

29,279

5

3527-3

28,350

6

3284-3

30,448

6

3418-3

29,254

5

3530-0

28,329

6

3285-7

30,435

6

3421-5

29,227

G

3531-8

28,314

5

3286-9

30,424

3d

3427-7

29,174

6

3533-6

28,300

5

3289-9

30,396

6

3428-5

29,167

6

3534-7

28,291

5

3290-7

30,389

3

3431-8

29,139

6

3536-7

28,275

6

3291-8

30,379

5

3433-5

29,125

5

3538-2

28,263

0

3293-0

30',367

ON THE SPECTRUM OK THK OXY-HYDROGKN FLAME.

85

In'.'M-it).

A

1^

A

Intensity.

*

1

A

luleiuily.

A

l

I

5

35407

28,243

5

3577-3

27,954

4

3627-6

27,566

6

35424

L'-i.-J:.".'

6 3578-7

27,943

6

30316

27,536

6

3544-9

•28,210

5 3584-3

27,899

4

3637-4

27.492

5

35477

--.187

5 3592-0

27,840

6

3045-3

27,433

4d

3552-3

28,151

5 3G004

27,775

4

3647-7

27,415

5

3557-8

28,107

6 3605-0

27,739

6

3651-1

27,38*

6

3559-3

28,095

5

3609-0

27,709

5

36531

27,374

5

3564-2

28,057

4

361S-1

27,639

4

3659-1

27,329

5

3570-4

28,008

6

3625-3

27,584

6

3(568-2

27,261

4

86698

27,249

VII.

The lines in this section are less intense than in the preceding section, and a line of
intensity 4 on the scale of this table would be included in intensity 6 of the scale of
the preceding table (VI.).

Intensity.

A

1

A

Intensity.

A

1

A

Intensity.

A

1

A

6

S673-3

27,223

4d

3779-0

26,462

6

3942-2

25,367

6

3674-6

27,214

6

3786-8

26,408

5

3947-1

25,335

6

3676-4

27,201

6

3789-6

26,388

5

39526

25,300

6 :;r,7s-j

27,187

6

3792-0

26,371

6

3960-1

25,252

6 3678-8

27,183

6

3794-0

20,:'57

6

3903-5

25,230

6 3681-5

27,163

4d

3799-7

26,318

6

39664

25,212

6 3681-9

27,160

6

8806-9

26,268

5

3973-8

25,165

6

3683-8

27,146

5

381M

'20,239

5

3981 1

25,119

fi

3684-7

27,139

5

38152

'26,211

6

3990-3

25,061

6

3686-9

27,123

5

3821-7

26.166

5

3996-9

25,019

6

3688-0

27,115 5

3823-4

26 155

6

4006-7

'24,958

fi :;r,!»l-3

27,091 5

8828-4

26,121

6

4008-5

24,947

4J 3694-6 27,067 5

3832-5

20,0'.»3

5

4019-9

24,876

6 3700-1 27,026 6

38344

26,080

5

4026-0

24,839

6 370 VI 26.993 6

384M

26,034

5

4036-8

24,772

6 3707-0 26,976

3846-3

25,999 5 4049-7

24,693

5 3710-3 26,952

6

3850-8

25,'.»69 6 4001-7

24,620

6 3716-5 26,907

5?

38.52-0

25,961

6

4065-3

24.598

6 3719-7 2C..SM4

s*

3S55-5

25,937

6

4073-1

24,551

5 3722-9

26,861

5

3859-1

25,913

6

4075-6

24,536

5

3729-3

26,815 6

3864-8

25,875

6

40942

24,425

5

3734-1

26,780

6

3868-2

25,852

5

!• '.'- :,

24,399

5

37367

26,762

5

3873-2

25,818

5

4099-1

24,396

5

3741-5

26,727

5

3879-0

25,780

6

41044

24.364

6

3743-7

26,712

5

88839

25,747

6

4128-2

24,224

6

3745-1

26,702

6

3885-7

25,735

6

4143-0

24,137

5

3747-9

26,082

5

3886-6

25,729

6

4150-2

24,095

5

3751-0

26.660

bd

3893-0

25,687

6

4157-4

24,053

5

3753-0

26,645

bd

3900-6

25,637

6

41755

23,!>4!t

6

3756-5

26,621

5

3907-5

25,592

6 4182-0

23.912

5

87603

26,594

5

3915-5

25,540

5

4193-7

23,845

6

3761-7

26,584

5

3923-5

25,487

6

4228-4

23,650

•U

3769-5

26,529

U

39384

25,425 6

4235-9

23,006

5

3774-8

26,491

6

3940-7

25,376 5d

4265-0

23,447

'

F 2

36

PROFESSORS G. D. LIVEING AND J. DEWAR

An inspection of the map shows in several places an arrangement of the lines which
suggests an harmonic relation of some sort between them : groups of lines where the
successive lines are set at gradually increasing distances and have a gradually
diminishing intensity. Such a group may be seen between the wave-lengths 3431 '8
and 3450*7, another beginning at X 3368 '8.

In many cases we find that the distances between the lines of the group are in
arithmetic progression, so that the wave-lengths of the group may be represented by
a general formula anz + bn + c, the different lines of the group being deduced from
this expression by giving n successive integral values.

Taking the series of strong lines of the spectrum between X 3669'8 and X 3547'7,
we find the second differences of wave-length approximately constant, and equal to
0'443. Comparing then the observed wave-lengths with a series calculated to have
their differences in arithmetic progression with a common difference, 0'443, we find —

Observed.

Calculated.

Difference.

3669-8

3669-7

+ -1

3659-1

3658-5

+ •6

3647-7

3647-7

0

3637-4

3637-3

+ -1

3627-6

3627-4

+ •2

3618-1

3617-9

+ 2

3609-0

3608-9

+ •1

3600-4

3600-3

+ -1

3592-0

3592-2

>2

3584-3

3584-5

-•2

3577-3

3577-2

+ •1

3570-4

3570-4

0

3564-2

3564-0

+ •2

3557-8

3558-1

-•3

3552-3

3552-6

-•3

3547-7

3547-6

+ -1

In many other groups a similar relation between the wave-lengths may be observed.

GROUP with second differences of wave-length equal to '3.

Observed.

Calculated.

Difference.

3544-9

3544-8

+ 'I

3540-7

3540-7

0

3536-7

3536-9

- -2

3533-6

3533-4

+ -2

3530-0

3530-2

- -2

3527-3

3527-3

0

ON THE SPECTRUM OF T1IK OXY-HVDROGEX FLAME.

37

GROUP with second differences of wave-length equal to '133.

Oloervcd.

Calculated.

Difference.

352 H

3524-1

0

• •

35217

3519-0

3519-4

- -4

3517-2

3517-2

0

3515-2

3515-2

0

35133

3513-3

0

3511-6

3511-5

+ -1

35099

3509-8

+ 'I

GROUP with second differences equal to 76.

Oterred.

Calcu'ated.

Difference.

3507-6

3507-5

+ 'I

35021

3502-0

+ "I

3497-2

3497-3

- 1

34933

34!>3-3

0

3489-5

34901

- -6

3488-0

3487-7

+ -3

3486-2

3486-0

+ -2

GROUP with second differences equal to '923. The first line may not belong to the

group.

Observed.

Calculated.

Difference.

3497-2

3497-9

- -7

34!»4-9

• •

3492-3

3492-0

•f -3

3489-5

3489-2

+ -3

3486'2

3486-5

— *S

3483-9

3483-9

0

3481-4

3481-4

0

3479-1

3478-9

+ -2

3476-8

3476-5

+ -3

3474-0

3474-2

- -2

3171-9

3472-0

- -1

3469-6

3469-9

- -3

3467-7

3467-9

.2

3465-8 .

346C-0

- -2

3464-1

84644

- -1

34625

34625

0

8460-9

3460-9

0

3459-4

34594

0

84682

3458-0

+ -2

, ,

84M<7

3455-4

3455-5

+ -1

38

PROFESSORS G. D. LIVEING AND J. UEWAR

GROUP with second differences equal to '3.

Observed.

Calculated.

Difference. •

3497-2

3497-7

- -5

3488-0

3488-8

- -8

3480-2

3471-9

3471-9

0

3464-1

3463-9

+ -2

3455-4

3455-2 + -2

3447-6

3447-8

- -2

3440-7

^

3433-5

3433-9

-"•4

3427-7

3427-4

+ -3

3421-5

3421-2

+ "3

3415-4

3415-3

+ 'I

3409-7

3409-7

0

3404-2

3404-4

- -2

3399-1

3399-4

- -3

3394-5

3394-7

- -2

3390-3

3390-3

0

338(5-2

3386-2

0

GROUP with second differences of wave-length equal to '2143.

Observed.

Calculated.

Difference.

3451-0

3450-8

+ -2

3447-6

3447-7

- -1

3444-7

3444-8

- -1

3442-3

3442-1

+ -2

3439-7

3439-6

+ 'I

3437-4

3437-3 + -1

3435-4

3435-3

+ "I

3433-5

3433-5

0

3431-8

3431-9

- -1

GROUP with second differences of wave-length equal to '3.

Observed.

Calculat«d.

Difference.

3428-5

3428-8

- -3

. .

3425-0

t t

3421-5

3421-5

0

3418-3

3418-3

0

3415-4

3415-4

0

3412-8

3412-8

0

ON THE SPECTRUM OF THE OXY-HYDROGEN FLAME.
GROUP with second differences of wave-length equal to '3.

3J>

OfeMTcd.

Calculated.

Difference.

3394-5

3394-2

+ "3

3390-3

3:?90-3

0

3386-2

3386-7

-. -5

3383-3

33834

- -1

3380-2

3380-4

- -2

:W7-6

3377-7

- -1

:W75-4

3375-3

+ 'I

3373-2

3373-2

0

33717

3371-4

+ "3

3370-1

3369-9

+ -2

3368-8

33687

+ -1

GROUP with second differences of wave-length equal to 1 '467.

OWrved.

Calculated.

Difference.

33491

3349-0 + -1

33325

3332-8 - -3

3318-0

3318-1 - -1

3304-9

33049 o

3293-0

3293-1 - 1

3282-9

3282-8 + -1

3273-7

3274-0

- -3

3266-4

3266-6

- -2

3260-7

3260-7

0

32564

3256-2

+ "2

3253-2

32531 + -1

3252-0

3251-5 + -T,

GROUP with second differences of wave-length equal to '322.

OWrved

Calculated.

Difference.

3086-7

3086-5

+ -2

307D-3

3080-0

- -7

3073-8

3073-9

- -1

3068-2

3068-1

+ "I

30033

3062-6

+ -7

3057-4

3057-4

0

3052-7

3052-6

+ "I

3048-3

3048-1

+ -2

3043-9

30439

0

3039-9

3040-0

- 1

3036-4

3036-4

0

3033-1

30.33-2

- -1

3030-3

3030-3

0

3027-6

3027-7

- -1

3025-2

3025-4

- 2

8023-4

3023-4

0

30220

3021-8

+ "2

3020-9

8020-5

+ -4

40

PROFESSORS G. D. LIVKING AND J. UEWAR

GROUP with second differences of wave-length equal to '783.

Observed.

Calculated.

Difference.

2893-5

2893-6

- -1

2882-5

2882-7

- -2

2871-9

2*72-6

- -7

2863-3

2863-3

0

2854-9

2854-8

+ 'I

2847-4

2847-1

+ -3

2840-1

2840-2

- -i

2834-0

2834-0

0

2828-7

2828-6

+ -i

2824-0

2824-0

0

2820-1

2820-2

- -I

2817-1

2817-2

- -1

2814-9

2815-0

- -1

2813-5

2813-5

0

GROUP with second differences of wave-length equal to '457.

Observed.

Calculated.

Difference.

3020-9

3020-6

+ -3

2907-1

m f

2992-9

2992-9

0

2980-2

2980-2

0

2968-0

2967-9

+ 'I

2956-3

2956-1

+ -2

2945-2

2944-8

+ '4

2933-5

2933-9

- -4

2923-8

2923-5

+ -3

2913-5

2913-5

0

2903-7

2904-0

- -3

2894-2

2894-9

- -7

2886-3

2886-3

0

2878-3

2878-2

+ 'I

2871-5 1
2869-5 /

2870-5

2863-3

2863-3

0

2857 6 1
2855-4 /

2856-5

2850-2

2850-2

0

2844-4

2844-3

+ "I

2838-8

2838-9

- -1

2834-0

2834-0

0

2829-8

2829-5

+ "3

2825-2

2825-5

- -3

2821-8

2821-9

- -1

2818-7

2818-8

- -1

2816-1

2816-1

0

2813-5

2813-9

- -4

2812-1

2812-2

- -1

ON THE SPECTRUM OF THE OXY-HYDROGEN FLAME.

41

GROUP with second differences of wave-length equal to 2 -4.

OUerred.

Calculated.

Difference.

3160-3

3160-5

- -2

8180*4

3139-7

.3

3121-3

3121-3

0

3105-3

3105-3

0

3092-0

3091-7

-1- -3

3081-0

3080-5

+ -5

3071-5

3071-7

- -2

3065-5

30G.V3

+ '2

GROUP with second differences of wave-length equal to '522.

Observed.

CalcnUted.

L)ini*rcnc€.

3005-6

3005-4

+ -2

2991-7

2991-4

+ 3

.".'77 -

2977!'

.j

2965-5

2965-0

+ -5

2952-5

2952-6

- 1

2940-6

2940-7

- -1

2929-9

2929-4

+ -5

2918-5

2918-6

- -1

2908-3

2908-3

0

2898-81
2898-1 /

2898-5

2889-2

2889-2

0

2880-3

2880-4

- -1

2871-9

2872-1

- -2

2865-5 1
2863-3 /

2864-4

• * *

2857-6

2857-2

+ -4

2850-7

2850-5

+ '2

2844-4

2844-3

-1- 'I

2838-8

2838-6

-1- -2

28333

28334

- -1

2828-3

2828-8

- -5

2824-8

28247

+ -1

2821-2

2821-1

+ -1

2818-2

2818-2

0

2815-6

2815-7

- -1

2813-5

2813-7

- -2

2812-1

2812-2

- -1

2811-2

2811-2

0

Now, considering that the dispersion employed was that of a single prism, and
that many of the lines are weak and many diffuse, the difference between the
observed and calculated values falls in most of the cases above given within the limits
of probable error. Indeed, one cause of the difficulty of obtaining very exact
measurements of some of the lines arises from the overlapping of different groups,

Mix V.I. \\.\\ 111. A. ft

42 ON THE SPECTRUM OP THE OXY-flYDROGEN FLAME.

by which two or three lines fall nearly in the same place, and produce a hazy band in
the photograph which cannot be resolved into its constituent lines. We thiuk, there-
fore, that the groups do actually follow the law above enunciated. M. DESLANDRES
has observed the same law to hold in regard to inverse wave-lengths in the lines
forming the bands of nitrogen, in those of A, B, and a, and in the groups of several
spectra of compound gases ; and has remarked that it is the law of sequence of the
harmonics of solid rods. A. HERSCHEL and PIAZZI SMYTH had previously noticed this
law in the sequence of the rays in one of the bands of the spectrum of carbonic
oxide.

While the work of photographing and measuring this spectrum was in progress
we received from Dr. GRUNWALD, Professor of Mathematics in the I. R. German
Technical High School at Prague, a long list of lines which he had, on theoretical
grounds, predicted would be found in the spectrum of water. The interest attaching
to this prediction induced us to make a more extended and complete investigation of
the spectrum than we had originally intended. Many of these predicted lines,
though not all of them, agree closely with lines which we have recorded in the
spectrum of the oxy-hydrogen flame ; and in the weaker parts of the spectrum at
either end, Dr. GRUNWALD'S list includes many of the strongest lines. His results
have been (in part) published in the ' Astronomische Nachrichten.' We are not at
present in a position to discuss his theory, which is a far-reaching one, and will need
to be tested at many points ; but the coincidences between his predictions and the
lines observed are very remarkable, and will, no doubt, attract the attention of many
besides ourselves.

A map of the spectrum on a scale of inverse wave-lengths accompanies this Paper,
and also an enlarged photograph which gives the general characters of the spectrum,
though in the strongest part it has been over-exposed. The photograph includes
the regions of the lines enumerated in Tables III. to VI., and greater part of VII.

[ 43 ]

III. On the Motion of a Sphere in a Viscous Liquid.

By A. B. BASSET, M.A.
Communicated by Lord RAYLEIGH, D.C.L., Sec. U.S.

Received November 10,— Read November 24, 1887.

1. THE first problem relating to the motion of a solid body in a viscous liquid which
was successfully attacked was that of a sphere, the solution of which was given by
Professor STOKES in 1850, in his memoir "On the Effect of the Internal Friction of
Fluids on Pendulums," ' Cambridge Phil. Soc. Trans.,' vol. 9, in the following cases:
(i.) when the sphere is performing small oscillations along a straight line ; (ii.) when
the sphere is constrained to move with uniform velocity in a straight line ; (iii.)
when the sphere is surrounded by an infinite liquid and constrained to rotate with
uniform angular velocity about a fixed diameter : it being supposed, in the last two
cases, that sufficient time has elapsed for the motion to have become steady. In the
same memoir he also discusses the motion of a cylinder and a disc. The same class
of problems has also been considered by MEYER* and OBERBECK,t the latter of whom
has obtained the solution in the case of the steady motion of an ellipsoid, which
moves parallel to any one of its principal axes with uniform velocity. The torsional
oscillations about a fixed diameter, of a sphere which is either filled with liquid or is
surrounded by an infinite liquid when slipping takes place at the surface of the sphere,
forms the subject of a joint memoir by HELMHOLTZ and PiOTROWSM.J

Very little appears to have been effected with regard to the solution of problems
in which a viscous liquid is set in motion in any given manner and then left to itself.
The solution, when the liquid is bounded by a plane which moves parallel to itself, is
given by Professor STOKES at the end of his memoir referred to above ; and the solu-
tions of certain problems of two-dimensional motion have been given by STEA.RN.§
In the present paper I propose to obtain the solution for a sphere moving in a viscous
liquid in the following cases : — (i.) when the sphere is moving in a straight line under
the action of a constant force, such as gravity ; (ii.) when the sphere is surrounded by
viscous liquid and is set in rotation about a fixed diameter and then left to itself.

" ' Crelle, Jonrn. Math.,' vol. 73, p. 31.
t ' Crelle, Journ. Math.,' vol. 81, p. 62.
+ ' Wissenschaftl. Abhandl.,' vol. 1, p. 17J.
§ ' Quart. Journ. Math.,1 vol. 17, p. 90.

«. -J 28.5.88

44 MR. A. B. BASSET ON THE MOTION OP

Throughout the present investigation terms involving the squares and products of
the velocity will be neglected. This is of course not strictly justifiable, unless the
velocity of the sphere is slow throughout the motion. If, therefore, the velocity is
not slow the results obtained can only be regarded as a first approximation ; and a
second approximation might be obtained by substituting the values of the component
velocities hereafter obtained in the terms of the second order, and endeavouring to
integrate the resulting equations. I do not, however, propose to consider this point
in detail.

2. In the first place it will be convenient to show that the equations of impulsive
motion of a viscous liquid are the same as those of a perfect liquid.

The general equations of motion of a viscous liquid are

du du du du —. 1 dp „

-jl + u- +V—+W- -- X + -J1 — vv2u = 0,

dt da dy dz p dx

with two similar equations, where v is the kinematic coefficient of viscosity.

If we regard an impulsive force as the limit of a very large finite force which acts
for a very short time T, and if we integrate the above equation between the limits
T and 0, all the integrals will vanish except those in which the quantity to be inte-
grated becomes infinite when T vanishes ; we thus obtain

1 d

1 d (T

U — UQ+ - — I pdr = 0.
dx}Qr

Putting f pdr = us where CT is the impulsive pressure at any point of the liquid,
we obtain

p (u — u0) + ^ = 0, &c., &c.,

which are the same equations as those which determine the impulsive pressure at any
point of a perfect liquid.

3. Let us now suppose that a sphere of radius a, is surrounded by a viscous liquid
which is initially at rest, and let the sphere be constrained to move with uniform
velocity V, in a straight line. If the squares and products of the velocity of the
liquid are neglected, Professor STOKES has shown that the current function t/» must
satisfy the differential equation

where

<P sin0 d

and (/•, 9) are polar coordinates of a point referred to the centre of the sphere as
origin.

A SPHERE IN A VISCOUS LIQUID. 45

H, 8 be the component velocities of the liquid along and perpendicular to the
radius vector ; then, if we assume that no slipping takes place at the surface of the
sphere, the surface conditions are .\

Also, at infinity II and 8 must both vanish.
These equations can be satisfied by putting

......... (4)

where i/», and i/«2 are functions of r and t, which respectively satisfy the equations

?3h _

r* ' ' °' ' '

2fc_lrffe

dr» r» ' ~ v dt '

The proper solution of (5) is 1/1, =f (()/>', which it will be convenient to write in the
form

where x (a) IS an arbitrary function, which will hereafter be determined.

In order to obtain the solution of (6), let us put i/»2 = re"*1'"' dw/dr, where 10 is a
function of r alone ; substituting in (6), and integrating, we obtain

rw = A cos X (r — a + a).

where a is the radius of the sphere and A and a are the constants of integration.
Whence a particular solution of (6) is

d e"

e"*
»., = Ar — - oes X (r — a + a).

Integrating this with respect to X between the limits oo and 0, and then changing A
into F (a) and integrating the result with respect to a between the same limits, we
obtain

rj<* d r F(«) f (r-a + «

- --

46 MR. A. B. BASSET ON THE MOTION OP

Performing the differentiation and then integrating by parts, we obtain

We shall presently show that it is possible to determine F(a), so that F(0) = 0,
and F(«) «""' = 0 when a = oo ; hence the term in square brackets will vanish at both
limits, and we obtain

sin'fl

. . (8)

We must now determine the functions x and F so as to satisfy the surface conditions
(2) and (3).

Equation (2) will be satisfied if

(9)
Equation (8) requires that

exp- -

Integrating the last term by parts, the preceding equation becomes

f V X (a) + F («) + aF' (a) + «*F" (a) } exp. ( - £) «/«, . ( 1 U)

provided, {F (a) + aF' (a) } exp. ( — «3/4i>«) vanishes at both limits. This requires
that F(0) = F'(0)=0, and that F(a)e-a' and F(a)e-"' should each vanish when
a = oo . When this is the case (10) will be satisfied if

.,..,,;. (U)
Whence by (9)

7T

A SPHERE IN A VISCOUS LIQUID. 47

and, therefore,

The conditions that F (0) = F (0) = 0 require that C = D = 0 ; whence

,\

'

Va/3«J ,

= ~+

Also the preceding value of F(«) satisfies the conditions that F(«)e~*', and F'(«)e~*'
should each vanish when a = oo ; whence all the conditions are satisfied, and we
finally obtain

r/3«8 a«\ / «»\ .

\.(T + 3rta + 2 ) exp- (- «) d"

3Va8in»gf'/a« \ f (r - a + «)'

"

The first integral can be evaluated ; in the second put r — a + a = 2u^/(vt) and
we obtain

3Va sin' g r f

V77"

_ r + a| «-^dM. (13)

4. When t = 0 the second integral vanishes, whence the initial value of «/> is

Vo8 sin3 0
*-•—-'

which is the known value of t/» in the case of a frictionless liquid, as ought to be the
case.

When t is very large, we may put t = oo in the lower limit of the second integral,
which then

whence

48 MR, A. B. BASSET ON THE MOTION OP

This equation gives the value of i/» after a sufficient time has elapsed for the motion
to have become steady, and agrees with Professor STOKES'S result.
5. Let vt be any solution of the partial differential equation

Then, if v0 = 0, F(< — r)vr dr, where F(T) is any arbitrary function which is inde-

Jo
pendent of r and t, and does not become infinite between the limits, will also be a

solution of (14) ; for, substituting in (14), the right-hand side becomes

F(0)v, + f F(< - r)vr dr = F(«K + f F(< - r) f* dr

Jo Jo

if f»0 = 0.

6. The second expression on the right-hand side of (13) is the value of \jj.z sin2 6 ; and
it is easily seen that this expression vanishes when t = 0. Hence it follows that the
expression which is obtained from (13) by changing t into r and V into Y'(t — T) dr,
and integrating the result from t to 0, is also a solution of (1). Now, if F(0) = 0, it
will be found in substituting the above-mentioned expressions in (2) and (3) that F(t)
is the velocity of the sphere, supposing it to have started from rest ; hence this expres-
sion gives the current function due to the motion of a sphere which has started from
rest, and which is moving with variable velocity F(t).

In order to obtain the equation of motion of the sphere, we must calculate the
resistance due to the liquid ; but in doing this we may begin by supposing the velocity
to be uniform, and perform the above-mentioned operation at a later stage of the
process.

If the impressed force is a constant force, such as gravity, which acts in the direction
of motion of the sphere, and Z is the resistance due to the liquid, it can be shown, as
in Professor STOKES'S paper, that

and that

Z = braUpa cos 0 - p ^ sin2 0} sin 0 d6,
*oV *** /•

where p is the density of the liquid ; also, since

[p cos 0 sin edO=-% f" sin2 0 ^ d0,

J o J Q uv

A SPHERE IN A VISCOUS LIQUID. 49

we obtain

irpa a + 2

—£K»$ +*)+«•*

where M' is the mass of the liquid displaced. Now, if V were constant, we should
obtain from (13)

and
whence

We must now change t into r, V into F (t -T)dT, and integrate the result with
respect to T from t to 0, and we obtain

and the equation of motion of the sphere is

(M + JM> + ^ ^ £ F(l - r) (i^r + a ^ )r/T = (M - M'V/. (15)

Integrating the definite integral by parts, and remembering that F(0) = 0, the
result is

and, differentiating with respect to t, (15) becomes

Let <r be the density of the sphere, and let

MDCCCLXXXVIII. — A. H

50 MR. A. P. BASSET ON THE MOTION OF

then (16) becomes

(18)

This is the equation of motion of the sphere, from which F (t) or v must be
determined.

7. Up to the present time we have supposed the motion to have commenced from
rest, so that F (0) = 0. Let us now suppose that the sphere was initially projected
with velocity V. In order to obtain the equation of motion in this case we may
divide the time, t, into two intervals, h and t — h, where h is a very small quantity,
which ultimately vanishes. During the first interval* let the sphere move from rest
under the action of gravity and a very large constant force, which is equal to
(M + ^M')X, and then let the large force cease to act. This force must be such as to
produce a velocity, V, at the end of the interval, h, whence we must have V = XA,
v = Xt ; and, therefore, v = "Vt/h. Changing f into f-\- X in (18), multiplying by
cw, and integrating between the limits t and 0, we obtain

= - ha \/ - f du PV" F'(tt - T) ~ + fxe^M + / f f^du. (19)

V TJ-JO JQ vT JQ Jy

Now F'(t) is composed of two parts : a large part which depends upon X, and which
is equal to V/A ; and another part which depends uponf, and which we shall continue
to denote by F'(l). Hence (19) may be written

= * (e" - 1) -H {(e» - 1) - ka A/ - f du [>(« - T)

X X V V 7TJ0 JQ

u}du, . . (20)

where

Vdr

Now x (u) depends on X, and therefore vanishes when u > h. When u < h,

x(u) = 2Vtt»,Yi ;

therefore

ft f* 2V

^"X (") du — ~T u^'du = 0, when h = 0.
Jo Jo'1

Hence, in the limit when h vanishes, (20) becomes

v = Vc-" + {(1 - «-") - ka A/- f du re-*('—>F (u - T) -J- , . (21)

"• V 7T JQ Jo V T

* The following procedure, suggested in a Report upon this paper, has been substituted for the
remainder of this uectiou as originally written.

A SPHERE IN A VISCOUS LIQUID. 51

and the value of the acceleration is

8. It seems almost hopeless to attempt to determine the com pit te value of F from
the preceding equations, but, in the case of many liquids, v is a small quantity, and
(22) and (23) may then be solved by the method of successive approximation. For
a first approximation

r=F «)=/«-",
whence

fF'«-T)rfr f« , — rfr . }

-'--

The integral on the right hand side of (23) cannot be evaluated in finite terms, and
we shall denote it by <£ (t). Putting T = ty, we obtain

where

TI 1.3... (2«- 1)
U"- 2"n!

Now

fi i _ , - «

;.-*^=L^±

Therefore

and therefore

When t is very large we may replace (1 — e~x')/X< by (M)~l, and we shall obtain

which shows that <^ (t) = 0 where t = CD .

Another expression for <j> (t) may be obtained in the form of a series, for

n •_'

52 J1R. A. B. BASSET ON THE MOTION OF

by successive integration by parts. The above series is convergent for all values of t,
and is zero when t = oo.

For a second approximation, (22) gives

Let

r=¥'(t)=fe-»-fka

and

,. = V€-M + {(1 - «-»') —fka A/- [' e-*» <fr(t — v) du. . . (28)

A. » 7T JQ

Let

X(«) = | JV** {«-«)**• ....... (29)

and (27) becomes

Whence to a third approximation

and the last equation becomes

-u)dn, .... (31)

and

-") + V€-A( -fka *"^ ~ ") d«+1«-*^(* -«)<£«. (32)

We must now express all the above integrals in terms of <f>(t). From (29) we
obtain

o Jo

A SPHERE IN A VISCOUS LIQUID. 53

by (24). Changing the order of integration, the last integral

whence

(33)

Substituting this value of x(0 in (30), we obtain

*«-£«- M#W, + fo A/if

Now

f' <f>(r}dr _ f ' j fT e~*" rfu

J0v/(*-T)-J0 TJOV/{(<-T)(T-«)}

f' , f' e-A» dr

~' U

E =X(I ~e )' (34)

also

rfT _ ft p - *

T) ~ J0 Jo

,
= *

o y - T)(T
' -*»

. . . . . (35)
and

(36)

whence

^e-M ........... (37)

Again,

f e-^(f - «)</« = we-" [' (< - u) du

Jo Jo

= i^8€~A', ........ (38)

whence (31) and (32) finally become

54 MR. A. B. BASSET ON THE MOTION OF

v =/«-»' - VAe-*' - fka (£ - \t)<t>(t) + v/« +yW^-A' (1 - |Xi), (39)

MrM. . (40)

These equations determine to a third approximation the values of the acceleration
and velocity of the sphere, when it is projected vertically downwards with velocity,
V, and allowed to descend under the action of gravity. If the sphere is ascending
the sign of g must be reversed.

If no forces are in action we must put/= 0, and the preceding equations give the
values of v and v to a first approximation only; but, on referring to (21) and (22), it
will be seen that the values of these quantities to a third approximation may be
obtained in this case from (39) and (40) by changing / into — V\ and expunging the
terms/«~xt and/X"1 (1 — c~At). We thus obtain, since X = lev,

(4 !

9. It appears from the preceding equations that the successive terms are multiplied
by some power of A; as well as of v. If k is not a very large quantity, and the velocity
of the sphere is not very great, the foregoing equations may be expected to give fairly
correct results ; but if k is a very large quant:ty, it may happen that, notwithstanding
the smallness of v, kv may be so large that some of the terms neglected may be of
equal or greater importance than those retained. Now, from (17), k= 9/a(2o-+ p)~la~*;
if, therefore, the sphere is considerably denser than the liquid, k will be small provided
a be not very small ; but if the sphere be considerably less dense than the liquid, k
will approximate towards the limit 9a~z, and this will be very large if a be small, and kv
may therefore be large. On the other hand, it should be noticed that when kv or X is
large the quantities c~xl and <f>(t) diminish witli great rapidity, and it is therefore by
no means impossible that the formulae may give a fairly accurate representation of the
motion even in this case.

All that we can therefore safely infer is this, that in the case of a sphere ascending
or descending in a liquid whose kinematic coefficient of viscosity is small compared
with the radius of the sphere (all quantities being of course referred to the same units),
the formulae would give approximately correct results, provided the velocity of the
sphere were not too great. But, in the case of small bodies descending in a highly
viscous liquid, it is possible that the motion represented by the formulae may be very

A SPHERE IN A VISCOUS LIQUID. 55

different from the actual motion ; and if this should turn out to be the fact, the
solution of (18) applicable to this case must be obtained by some different method.

Equation (39) shows that after a very long time has elapsed the acceleration
vanishes, and the motion becomes ultimately steady; in other words, the acceleration
due to gravity is counterbalanced by the retardation due to the viscosity of the liquid.
When this state of things has been reached, the terminal velocity of the sphere is

This agrees with Professor STOKES'S result, who applies it to show that the viscosity
of the air is sufficient to account for the suspension of the clouds.

10. We shall now consider the motion of a sphere which is surrounded by an
infinite liquid, and which is rotating about a fixed diameter.

We shall begin by supposing that the angular velocity of the sphere is uniform
and equal to o>, and shall endeavour to obtain an expression for the component
velocity of the liquid in a plane perpendicular to the axis of rotation, on the supposi-
tion that no slipping takes place at the surface of the sphere.

Assuming that the motion of the liquid is stable, it is easily seen that none of the
quantities can be functions of ^, where r, 6, and <f> are polar coordinates referred to the
centre of the sphere as origin. If, therefore, we neglect squares and products of the
velocities, the component velocity, v, of the liquid, perpendicular to any plane con-
taining the axis of rotation, is determined by the equation

dt ~

and if in this equation we put v' = r sin 0, where r is a function of r and t only, the
equation for v is

«Pp 2 dr 2o 1 dr

jj + . — , = - -r (*•*)

The value of the tangential stress per unit of area which opposes the motion of the
sphere is

_, / 1 rfR <fi/ J\

T = — vp( . -,. + •: I,

rf$ dr T i

where R is the radial velocity ; but, since R is not a function of <ft, the value of this
stress depends solely on that of f*. Now Professor STOKES has pointed out that
unless the motion of the sphere is exceedingly slow, the motion of the liquid will not
take place in planes perpendicular to the axis of rotation, but the velocity of every
partu-le will have a component in the plane containing the particle and this axis. But

56 MR A. B. BASSET ON THE MOTION OF

since this component does not produce any effect on the motion of the sphere, which it
is our object to determine, we may confine our attention solely to the calculation of v.
In addition to (43), v must satisfy the conditions :

(i.) At the surface of the sphere v = aa> for all values of t.

(ii.) When t = 0, v = 0 for all values of r greater than a, the radius of the sphere.

Let v = Rc~AV< where R is a function of E, alone ; substituting in (43), we obtain

dr* r dr
the solution of which is

whence

R = A — j - cos X (r — a + a) I .

d f e~xly( 1

v = A Tr \~r cos x<r ~~ a + a)r

Integrating this with respect to X between the limits oo and 0, and then changing
A into F(a) and integrating the result with respect to a between the same limits, we
obtain

/*• d I f" _,, x j (r-a + «)

• v = * v rtSr -

Performing the differentiation and then integrating by parts, we shall obtain
1 /TT f JF(a) I J (r - a +

- ~ ~

provided F(0) = 0 and F(a)c-0' = 0 when a = oo .
The surface condition (i.) will be satisfied if

F(«) + aF'(«) = - - -,

7T

whence

F(«)=-^(i-«-n

the constant of integration being determined so that F(0) = 0 ; this value of F(a)
also satisfies the condition that F(a)e~a> = 0 when a = oo . We therefore obtain

a?<osme [° \a I a\ , 1 f (r - a + *)3]

v = —.-.- l^-f-ll -- 1« ***> exp. \ — - -\ da.. . (44)

ri/(int) Jo b' r/ 4V< J

A SPHERE IN A VISCOUS LIQUID. 57

Putting ?• — a -f a = 2uv/(»'f) this becomes

7-'=' J = +(i_=)exp. (-'"Vl "}\t-*du. (45)

«• / *ff I ** Iff I * A> * '

If r > a it follows that vf = 0 wheu t = 0. When r = a and t = 0 the lower
limit of the definite integral (45) becomes indeterminate ; but since, in this case, we
are to have v' = aw sin 6, it follows that if we put k = r — a the quantities k and t
must vanish in such a manner that when k = 0 and t = 0, k/2^/(vt) = 0.

When t = oo we obtain

, a*u> sin 0
v = • (40)

This equation gives the value of v' after a sufficient time has elapsed for the motion
to have become steady, and agrees with Professor STOKES'S result.

11. Since the tangential stress per unit of area which opposes the motion of the
sphere is

T=-^

dr
the opposing couple is

= — Zirvpa* j ( - 1 •
* dr \r/.

If, therefore, the sphere be acted upon by a couple, N', it« equation of motion will
be

A**5* + G = N',
or

<r«w d fv\ XT , ...

-„- — v . (- = N, ........ (4/)

•>p dr \r/«

where

N = 3pN'/8a4.

When the motion of the sphere commences from rest the value of v or v' cosec 6 will
be obtained from (45) by changing / into T, <a into F' (t — r)dr, and integrating the
result with respect to T from t to 0, where F (t) is the variable angular velocity of the
sphere.

MPCCCLXXXVin. — A. I

58 ME. A. B. BASSET ON THE MOTION OF

Now,

d /v\ 1 dv v
dr \r/a a dr a*

Hence, if <u were uniform we should have

(<lr\ 2a> [" „> T cat

, ) = — 2u + —T- exp. (— 2w </vt/a — tr) du
(If/a \/7r J o

Putting u + Y/ (vt)a = )8, the definite integral

.^/a.fvA[ ^A"0 , W 1

1 2 a 3a» ' J

y

= 2 a

if i>< be small ; whence

/dv\ 2co f .

[ -7-) = — o» --- 7- v^

\drja av^7T\v

ao>

2a
Changing t into T, and w into F' (« — T) dt, (47) becomes

Putting

(48) becomes

10p . 7

- A1, fe = X,

<ra-

" + x" + ; vb i >' <' - ' - " *•

((-r)i = 4^N. (49)

Now we have supposed the motion to have commenced from rest under the action
of the couple N' ; but if the sphere had initially been set in rotation with angular
velocity fl, and then left to itself, it can be shown in the same manner as in § 7 that
the equation of motion would be

A SPHERE IN A VISCOUS LIQUID. 59

where F (0) = fi. Putting 0 (t) for the last two terms, and integrating, we obtain

)0(«)dtt, ........ (51)

-x<'-">0(u)<ft< ....... (52)

For a first approximation we have

a. = fie-", « = - \flf~" = F' («).

Whence, if $, x, and «/» have the same meanings as in §8, a second approximation
gives

(53)

x<'-><£(u)du ........ (54)

™ V' * * 0

And a third approximation gives

. N ^i/» rf [i f«

(0 + ., <*«* f**~**/rdr

aay/TT t/<J0 J0

-X(i-1»*(«)<fc. (56)

Now we have shown in §8 that

Also

f. •-'—#(-) *- *(«)(« + i)-f

= J' rfr I* «-*»-*

And the value of the last integral in (56) is given by (38) ; whence

I 2

60 MK. A. B. BASSET ON THE MOTION OF

which determines the value of the angular velocity as far as v*.

1 2. HELMHOLTZ and PIOTROWSKI discovered from their experiments that in the case
of many liquids slipping takes place at the surface of the solid ; when this happens,
the surface condition is

........ (58)

where ft is the coefficient of sliding friction. Putting k = vpft~l, we obtain from

(43A)

d M A f f3F(«) . 2F(«)1 /

*L"* V *].* + *} exp- (~

1 /7T f fF(«) . _,,, J rf / «3

2* V 5 Jo {— + F (a)i s exp- r a

3F 3F

provided F(0) = F'(0) =0, and F(«)e-a' and F(«)e— ' are zero when a = oo
Equation (58) will be satisfied if

/3

the solution of which is

F = -

where p and 5 are the roots of the equation

The roots of (59) will be real if a > k, that is, if a > vp/fi. Now, if there is no
slipping, /3 will be infinite, and therefore, when there is comparatively little slipping,
ft will be large, and this relation will be satisfied unless a is small or v is large ; on
the other hand, if there were no friction between the surface of the sphere and the
liquid, ft would be zero, but it seems improbable that any liquid exists which possesses
the property of viscosity with regard to the internal motion of its particles, and which
at the same time is incapable of exerting any action in the nature of friction against
any surfaces with which it is in contact. If therefore ft were zero, v would probably

A SPHERE IN A VISCOUS LIQUID. 61

also be zero, and the liquid would be frictionless. We shall therefore assume that the
roots of (59) are real.

The constants A and B must be determined from the condition F (0) = F7 (0) = 0,
whence

^-fri

p-y J

also this value of F satisfies the conditions that F (a) e~*§, and F" (a) c~*' should
vanish when a = oo : whence the value of v is

«««8inf f-r_0«_ /

~ V(7rvO J0 [r (3k + a) \

p -

13. We shall lastly consider the motion of liquid contained within a sphere, which
is rotating about a fixed diameter, when there is no slipping, and when the angular
velocity is uniform.

In this case v must satisfy the differential equation (43), and also the condition (i.)
of § 10 ; but (ii.) becomes v = 0 when t = 0 for all values of r < a : also we have a
third condition, viz., that the velocity must be finite at the centre of the sphere.

A particular solution of (43), subject to the condition of finiteness at the origin, is

/T d If \ (

V= *A V * dr re*V' { ~

(r

whence if p and <j are any quantities which are independent of r and t, a solution of
(43) is

AT d 1 p.,. r \ (r-«)3l (r + «)91 1 7

V = * V ,7 dr r\q F W LeXP' { - « } - eXP' { - M \ \ <*"

= " l (°d\ f F (a) e- ™ (cos X (r - a) - cos X (r + a)} da.

tiT T 1 Q J q

If we put p = a, q = 0, F (a) = a, the double integral when t = 0 is equal to r by
FOURIER'S theorem, for all values of r between a and 0. If we put p = oo, q = a,
the integral when t = 0 is zero for all values of r which do not lie between o and oo .
The solution of the problem is therefore contained in the formula

«;•_' MR. A. B. BASSET ON THE MOTION OF

where A is a constant, which, together with the function F (a), must be determined
so as to satisfy the conditions of the problem.

14. Though I am convinced that a solution of the problem exists in the form of a
definite integral, I have not succeeded in obtaining it ; and therefore subjoin a
solution of a different character.

Let S(r) denote the spherical function d(r~l amr)/dr ; then a solution of (43),
subject to the condition of finiteness at the origin, is

v = 2Axe-xl"S (\r) + wa, ........ (62)

when r =a, v =o>a for all values of t, whence

S(Xa) = 0, . ,. ,,, . ...... (63)

and the different values of X are the roots of (63).
Initially v = 0, whence

<oa = - 2AXS (Xr).

Let X and p be different roots of (63), and let T = S(^tr), then, since S(Xr) satisfies
the equation

ePS , 2dS 2S

M + -J --- T -i- x2s = o,

ar* r dr r-
we obtain,

(X» - /i«) f STV dr + |>T f - r*S ^1 ' = 0, 64)

Jo \_ dr drJ0

•

and since by (63), S and T both vanish where r = a, we obtain

£ST»-*dr = 0, .......... (65)

provided X and /* are different. To find the value of the integral where X = p., let
p = K -\- d\; then from (64)

2 o ^

r + cH S -r-— — — — d\ = U,
o dr d\ dr d\j 0

or,

£SV> dr = ^a8S'2 (Xa), . ,,, ^ ..,,,, . . (66)

where the accents denote differentiation with respect to Xa ; whence

cA> /\ \ w'f f" d sin Xr ,

'2 (Xa) = , - dr,

\ 30dr r

w .

= - sm X«.

A,

A SPHERE IN A VISCOUS LIQUID 63

Therefore

. 2o> (sin Xa — A/»)

x = "a'XS'1 (\a) *

nnd

2w e - »V« (8in \*i -Xa) S (Xr)

-—

whence the velocity of the liquid, which is equal to rsin 6, can be found.

When the angular velocity is variable, the value of the retarding couple, and the
equation of motion of the sphere, can be obtained by a process analogous to that
employed in § 11.

[March 10th, 1888. — Since this paper was read, a paper has been published in the
' Quarterly Journal of Mathematics,'* by Mr. WHITEHEAD, in which he attempts to
develope a method of obtaining approximate solutions of problems relating to the
motion of a viscous liquid, when the terms involving the squares and products of the
velocities are retained ; and he applies his method (see p. 90) to obtain expressions
for the components in the plane passing through the axis of rotation, of the velocity
of a viscous liquid, which surrounds a sphere which is rotating about a fixed diameter,
when the motion has become steady. It will be observed, however, that the expressions
for these components contain the coefficient of viscosity as a factor in the denominator,
and therefore become infinite when the liquid is frictionless. It would therefore
appear that tho method of approximation adopted is inapplicable to the problem
considered.]

Vol. 23. p. 78.

IV. On Hamilton's Numbers. — Part II.

By J. J. SYLVESTER, D.C.L., F.R.S.,

Savilian Professor of Geometry -in the University of Oxford,

and JAMES HAMMOND, M.A., Cantab.

Received March 9, -Read April 19, 1888.

§ 4. Continuation, to an infinite number of terms, of the Asymptotic Development

for Hypotlienusal Numbers.

"This was sometime a paradox, bat now the time gives it proof."

(Hamlet, Act III., scene 1.)

IN the third section of this paper ('Phil. Trans.,' A., vol. 178, p. 311) it was stated,
on what is now seen to be insufficient evidence, that the asymptotic development of
p — q, the half of any Hypothenusal Number, could be expressed as a series of
powers of q — r, the half of its antecedent, in which the indices followed the
sequence

2, f, 1, },!,*,...

It was there shown that, when quantities of an order of magnitude inferior to that
of (q — r)1 are neglected,

P ~ 1 = (q - r)* + I (q - r)« + ft (q - r) + tf (q - r)' ;

but, on attempting to carry this development further, it was found that, though the
next terra came out i§f5 (q — r)*, there was an infinite series of terms interposed
between this one and (q — 7-)', viz., as proved in the present section, between (q — r)'
and (q — r)1 there lies an infinite series of terms whose indices are

B> T«» Sa» 64> TS8» • • • i

and whose coefficients form a geometrical series of which the first term is -riiy and
the common ratio f .

We shall assume the law of the indices (which, it may be remarked, is identical
with that given in the introduction to this paper as originally printed in the
' Proceedings,' but subsequently altered in the ' Transactions ') and write

MDCCCLXXXVm. — A. K 1.6.88

PROFESSOR .1. J. SYLVESTER AND MR. J. HAMMOND

D (q - r)« + $ E (q - r)* + &c., ad inf.
+ «* ................. (1)

The law of the coefficients will then be established by proving that

If there were any terms, of an order superior to that of (q — r)*, whose indices
did not obey the assumed law, any such term would make its presence felt in the
course of the work ; for, in the process we shall employ, the coefficient of each term
has to be determined before that of any subsequent term can be found. It was in
this way that the existence of terms between (q — r)* and (q — r)* was made manifest
in the unsuccessful attempt to calculate the coefficient of (q — r)*. It thus appears
that the assumed law of the indices is the true one.

It will be remembered that p, q, r, . . . , are the halves of the sharpened
Hamiltonian Numbers E» + 1, E«, 'EiH_v . . . , and that consequently the relation

.. .^.., -._!- 2)

1.2 1.2.3

may be written in the form

- 1 _L 1) _ r(2r-l)(2r-2) s(2s - 1)(2» - 2) (2s - 3)

1 2 2.3 2.3.4

t (2t - 1) (2* - 2) (2< - 3) (2< - 4) « (2» - 1) (2» - 2) (2« - 3) (2» - 4) (2tt - 5)
2.3.4.5 2.3.4.5.6

- .................. ....... (2)

The comparison of this value of p with that given by (l) furnishes an equation
which, after several reductions have been made, in which special attention must be
paid to the order of the quantities under consideration, ultimately leads to the deter-
mination of the values of A, B, C, . . . , in succession.

Taking unity to represent the order of q, the orders of

p, q, r, s, t, u, v, w, . . .
will be

2» li i» i» i» t^, 3*2, 3*¥» • • •

Hence, after expanding each of the binomials on the right-hand side of (!) and
arranging the terms in descending order, retaining only terms for which the order is
superior to £, we shall find

* In the text above Q represents some unknown function, the asymptotic value of whose ratio to
(? ~ r)* >8 n°t infinite.

ON HAMILTON'S NUMBERS. 67

< >rder 2 p = qz

t

„ V.,\ +UEJM + ...... (3)

Again, retaining only those terms of (2) whose order is superior to ^, we have

1> = 3»; -fH; -ig + ^+ia*; -a3; -A*8 .... (4)
Order 2 ; f ; 1 ; | ; f .

From (3) and (4) we obtain by subtraction

Order $ 0 = $ r3 - 257- + f 9'

1 - i ** - 2?'r + V 2

I + «" + ««•

I +A«3 + |iA»/

A + 1: B «A

H +f:c2»

» tt + ^ D 311

M +$E<r'. + ...... (5)

Changing 7^, q, »•,... into 5, J-, s, . . . respectively, equation (4) becomes

so that, if we assume 7 = r* (1 — a), the order of a will be the same as that of r"8*3,
viz., - | + | = - i;

Hence, if we substitute )A(1 — a) for q in (5), neglecting in the result quantities of
the order ^, we shall find

|y» _ 2qr + |gi - |«* - 23'/- + V?
= f r» - 2/-3(l - a) + |»-s(l _ |« + f «2 + ^tf «»)
- i S* - 2^ (1 - $ a) + Vs (1 ~ «)

+ A'V - *** + i'"8
K 2

68 PROFESSOR J. J. SYLVESTER A.ND MR. J. HAMMOND

while at the same time, since the order of r'ot does not exceed £, we have

and in like manner

<

Thus equation (5) becomes
Order 1 0

= rl, and so on.

1 s4 +

where

order
Let

then

H

a =

-i

' = -3r - « -

a =

+|:Dr«

+ HE »*

A . _ JLI

8 > 16

where terms as far as, but not beyond, — ^ (which is the order of s~3w5) have been
retained.
Now

p consists of terms whose orders are 2, f , 1, f, f , \, . . .
q 1; l; |( _fttf> £( _ ;\

a » )> » 4> 2' 8"> T6' 4> • • •

Thus the order of a' is — £, and in the above expression all terms of a superior to

— ^ have been retained, and consequently (rejecting the square of a' whose order is

— ^) in the first line of (6) we may write

- r-V -

+

+ ^ r-

ON HAMILTON'S NUMBERS. 69

In the second line of (6) we may reject the whole of a', since its order is — ^, and
write

After substituting then- values for the terms in (6) which contain a, and at the
same time dividing throughout by f , we shall obtain

Order 1 0 = |r-V - ±s* + £ r8

| +

f + r

„ A

We now write

r = «»l-8 and = *-

where, observing that the values of /8 and /8' can be immediately deduced from those
of a and a' by changing r, s, t, . . . into «, t, u, . . . , it is evident that £ and ft are
both of the order — £ ; for a and of are both of the order — £. Thus (neglecting
quantities whose order is equal to, or less than, £) we have

Order |; f ; A ; tf

Order

70 PROFESSOR J. J. SYLVESTER AND MR. J. HAMMOND

Ar» = st*

and so on.

Hence (7) becomes

Order | 0 = f«6 -

Dividing this throughout by -| s, and then writing

6- = <a (1 - y) and y = | t~*u* (1 + y),

we obtain in exactly the same manner as before, merely altering the letters in the
previous work,

9 it I 1 .1 1 Q ^ Q

— . .1 ,*_ fl/W • •*• /i/'J — ^ J/3

^^~ J) J ^^^ ^T S ft

Order | ; ^

where quantities of the order \, or less, are now neglected.
Similarly

| Bs* = tit*

and so on.

Thus (8) becomes

ON HAMILTON'S NUMBERS. 71

Order f 0 = $«• - * tu* + (\ A - ^)l*

A + A «-v - A «8 - f f«v + ( AV - § A) >«3 - it-

A

Now the terms of the highest order in this equation must vanish when we write
t = u9, and therefore f — i + fA — -^5 = 0, which gives A = H- Substituting
this value for A, we find

Order f 0 = $«« - \tu* +

» A + A<-V-

„ A +

which is a mere repetition of equation (8), with all the letters moved forward one
place. Hence it is evident that, if we treat this equation as we treated (8), we shall
find B = ^J-, arriving, at the same time, at another equation which will be merely a
repetition of (8), with all its letters moved forward two places ; and this process can
be continued as long as we please.
Thus we arrive at the result —

A = B = C = D = E=... =H,

and the asymptotic development for Hypothenusal Numbers

is established.

Comparing this with the corresponding formula for Hamiltonian Numbers,

given at the beginning of the third section (at the top of p. 302, where the last term
is incorrectly printed H), it will be noticed that each of the two developments begins
with an irregular portion consisting respectively of four and one terms, followed by a
regular series. In the one case the regular portion is ^ (q — r)*, multiplied by a
series whose general term is f I (q — r)(4>" ; in the other it consists of a series of terms
of the form <?{1>" multiplied by —

/'//// 7'nin .s 1HMHA.

5.

v'

V. Report on Hygrometric Methods; First Part, including the Saturation Method
and the Cfiemical Method, and Dew-point Instruments.

By W. N. SHAW, M.A.
Communicated by R. H. SCOTT, F.R.S., Secretary of the Meteorological Council.

Received January 17,— Bead January 26, 1888.

[PLATE 5.]

IN August, 1879, at the request of the Meteorological Council, I undertook an
experimental comparison of the various methods of determining the hygrometric
state of the air, with the following instruction, " The chemical method to be
employed, and with it to be compared the dry -and- wet-bulb hygrometer, REGNAULT'S,
DINES'S, ALLUARD'S, and the hair hygrometer." Since that time I have devoted to
the subject the time that was at my disposal, and I now beg leave to lay before the
Council a statement of the experiments I have made and the results I have arrived
at with respect to the chemical method and the dew-point instruments.

The arrangement of the experiments was left to my discretion : I had, therefore,
first to practise myself in the use of the different methods in order to arrange in
some sort of order the means by which the various inquiries should be undertaken ;
and, further, it was necessary for me to know and to carefully consider the very
numerous contributions made by other observers to the discussion of hygrometric
methods, so that I might be able to distinguish between those points which had been
satisfactorily and permanently settled and those upon which further experimental
investigation might throw additional light. The distinction proved to be not a very
easy one to draw, and I shall therefore append to this report a summary of the work
done iu the subject since the time of DANIELL. I made a summary of this kind for
my own use at the outset, but since then many important memoirs have been
published, chiefly on the Continent, which bear particularly upon the question of the
trustworthiness of observations with the wet and dry bulb. I have taken account of
those memoirs in the summary which I now offer. It will, I believe, be found to
justify the following general conclusions : —

1. There is no hygrometric method of which it has been proved that an observer
following out definite written instructions with due care and skill can obtain measure-
ments of vapour pressure which are accurate to within 1 per cent. The accuracy
claimed by RENAULT for the chemical method is only " about a fiftieth," and to

MDOCCLXXXV1II. — A. L 2.7.88

74 MR. W. N. SHAW ON HYGROMETRIC METHODS.

this method authors nearly always refer, as being the ultimate standard of reference
for the other methods. There is no evidence to show that, speaking generally, the
various absolute hygrometers, such as those of SCHWACKHOFER, EDELMANN, NEESEN,
DINES, and others, can be relied upon to that degree of accuracy, whatever may be
done by one particular observer after laborious trial and testing with one particular
instrument.

2. It is possible by the use of suitable desiccating tubes to absorb the moisture
from air passed through them, and thereby to determine the weight of water
contained in the air ivhich actually passes over the desiccating substances. The
degree of accuracy attainable in this measurement is limited only by the un-
certainty of the weight of the drying tubes. As the drying tubes are somewhat
bulky and their weights are liable to alterations in consequence of variations in the
state of their external surface, this uncertainty is quite appreciable. No data are
given with regard to it by REGNAULT or other observers who have employed the
method. From a number of experiments of my own I may, however, conclude that
it is not safe to assign an accuracy to the weighings greater than that of
1 milligramme, but that with due precautions this limit need not be exceeded.
This may, of course, be the same, whatever the total amount of moisture absorbed
may be, and the fractional error will therefore depend upon that total amount of
moisture. If 1 gramme be taken up, the limit of error will be one part in one
thousand, so that an accuracy of 1 per cent, is well within the reach of an observer.

3. Air may be saturated by vapour arising from water in a vessel with glass sides
so that the vapour pressure reaches a value agreeing to within about 2 per cent, with
the vapour pressure in vacuo at the same temperature. There are no observations to
show whether a still closer approximation to the vacuum saturation-pressure would
be obtained by using air drawn from an enclosure surrounded by wet muslin.
REGNAULT'S experiments bearing upon this question do not give a decisive answer, in
consequence of complications arising from the uncertainty as to the density of
saturated vapour, and the effect of the glass.

4. The dew-point instruments, in the hands of skilled observers, give readings of
the so-called dew-point which are sensitive to within 0°'l C., but the reading may
depend on the skill of the observer, and there is no evidence to show within what
limits of accuracy the temperature so observed may be regarded as the true satura-
tion temperature of the air. Suggestions have been made as to causes of error, but
no measurements of the effects of those causes have been made which would enable
an observer to specify the degree of accuracy of the inferences from his observations.

5. With reference to the wet-and-dry-bulb method, two points are clear : —
(i.) Different observers use different tables* for the reduction of their observations,
and in certain cases these different reductions lead to very serious differences of
results; (ii.) The ordinary method of exposure when the two thermometers are

* See note B., p. 146.

MH. W. N. SHAW ON HYf .KOMKTRIC METHODS. 75

freely exposed without any provision for a definite circulation of the air cannot be
expected to give results which are accurate to within 2 per cent. ; in fact, no
satisfactory formula of reduction can be found.

The errors are especially serious when the air is nearly or quite saturated and the
wet bulb coated with ice.

The most accurate results with this method are obtained when an artificial
circulation of air is maintained, and in that case the highest accuracy claimed for
the method allows an error of ± 2 per cent, with temperatures above zero, and twice
that amount with lower temperatures, the comparisons being made with ALLUARD'S
form of REGNAULT'S dew-point instrument.

6. The case of the hair hygrometer is very perplexing ; opinions are very conflicting,
and it seems to be a question upon which meteorologists take sides. An immense
amount of experimental work has been done, and it is possible that an observer
might make very useful observations with the instrument, but whether the inferences
from his observations would be regarded with any degree of confidence by others
seems to be still an open question.

With the exception of REGNAULT'S researches on the chemical method (p. 121), the
experimental work that has hitherto been published has consisted in the simultaneous
observation of the vapour pressure in free air by two different hygrometric methods.
This plan is open to uncertainties arising from the two following causes : —

(i.) The air is taken from two different positions for the two instruments ; in
other words, the observations are made upon two different specimens of air, whose
hygrometric states are assumed to be identical, not upon the same air. Perhaps the
assumption is sufficiently justified when the air has immediate access to the reading
parts of each instrument, but I have pointed out in the summary of results certain
reasons for considering this uncertainty to be serious when the access is not direct, as
in the case of the comparison of a dew-point instrument with a ventilator-psychro-
meter, in which case the air may suffer alteration in passing over the vanes of the
ventilator. And, indeed, in any case, in an experiment designed to compare the
efficiency of different methods, the uncertainty arising from this cause is a dis-
advantage, for the liability to error in the instruments themselves is quite sufficient to
make the comparisons difficult ; and, moreover, if we could rely upon the identity of
the results of two instruments, a similar arrangement might be employed to determine
whether there is any local variation in the b} grometric state of the air, an independent
question not without interest.

(ii.) The observations obtained with different instruments differ in character. The
chemical method gives the mean value of the vapour pressure during the period of
the experiment, and does not indicate any small variations from time to time. The
wet bulb takes a certain finite, though it may be short, time to reach its equilibrium
state, and therefore does not give the vapour pressure at any particular instant. The
dew-point instrument, on the other hand, may be taken as giving the vapour pressure

L 2

76 MB. W. N. SHAW ON HYGROMETRIC METHODS.

at the instant of the formation of the dew deposit. These differences may, perhaps,
best be illustrated by considering what would be their effects upon the ideal curve
which would represent the continuous variation of the hygrometric state of the air, if
we suppose the air to be subject to a series of rapid alternations of moisture and
dryness ; the dew-point instrument would give a series of points on the true curve ;
the observations with the wet and dry bulb would give a series of points, not on the
true curve, but a somewhat modified curve, the slopes being more gradual ; and, pro-
vided the alternations were sufficiently rapid, the curve of the wet bulb would
generally be smoother than the true curve. A series of observations with the chemical
method would give a series of points on a third curve quite different from the other
two, and one from which all effect of rapid alternations would have disappeared, and
only the more permanent changes would be shown. Hence, two instruments of
different kinds can only be compared by taking the mean of a number of consecutive
observations, and assuming that the effect of all rapid alternations disappears ; and
this is probably the case under all ordinary circumstances. This must, however, be
borne in mind in considering the comparisons, because the rapid alternation is
possibly of not infrequent occurrence. When using a dew-point instrument it is
sometimes observed that at the same temperature the dew forms and then disappears
again. This phenomenon has been differently interpreted by different observers,
some considering that it shows the extreme sensitiveness of the instrument to small
atmospheric changes, while others have attributed it to the variation of the observed
dew-point with variation of wind velocity (see p. 142).

It is, therefore, evident that it would be of some advantage to eliminate if possible
these two causes of uncertainty. I have endeavoured to do so in the experiments I
am now communicating, in the following manner : The instrument to be tested was
enclosed in a glass vessel, B, which was connected on the one side with an apparatus,
which I will call a saturator, A, designed to saturate air at a given known tempera-
ture, and on the other side with drying tubes, C. An aspirator drew air through the
whole arrangement. The hygrometric state of the air is given (1) by the temperature
of the saturator, A ; (2) by the instrument ; (3) by the drying tubes, C. I first ascer-
tained (§ 1 to § 10) that when B was cut out and A and C put in immediate connexion,
the two methods — namely, the saturation method and the chemical method — gave
concordant results. Then B was introduced, but the dew-point instrument was not
worked, and the results of A and C were again compared and found to be equally con-
cordant. This showed that the state of the air was not altered by the mere presence
of B ; and, finally, observations were taken with the dew-point instrument while the
air was being drawn through, and the results of A and C were again compared. It
was found that they were still concordant, and since the state of the air was known
before it passed over the instrument, and was proved to be the same after passing the
instrument, we are quite safe in assuming the hygrometric state of the air while it
was passing the instrument to be that given either by the saturator A or by the

MR. W. N. SHAW ON HYGROMETRIC METHODS. 77

drying tubes C. Hence observations made with the instrument may quite fairly be
taken to be observations upon air whose hygrometric state is really accurately known ;
the degree of accuracy will be clear from the particulars of the experiments. The
experiments accordingly group themselves in the following manner : —

I. Experiments to ascertain the limits of accuracy of agreement between the
method of saturation and the chemical method.

These experiments are practically a repetition of REGNAULT'S work with my
arrangement of the apparatus and absorbent substances. This repetition is necessary
(1) as a preliminary, in order to make sure that the apparatus is in satisfactory
working order, and as a test of the drying substances used (Table II.) ; (2) because
UEGNAULT'B observations were undertaken with the view of determining the density
of vapour in saturated air, and nearly always the air was practically saturated when
it reached the drying tubes. In order to complete our knowledge, we have to extend
the observations to cases in which the temperature of the air when it reaches the
tubes is considerably above the temperature of the saturator. In this case one would
expect a priori that the agreement would be very nearly the same, but I know of no
published experiments to directly test the point.

II. Experiments to determine whether the interposition of the vessel C interferes
with the concordance of the results obtained. The only apparent reasons for such
interference are condensation upon the connecting tube or the vessel C, or leakage
from the dew-point instrument. The second part of I. may, therefore, be taken with
these. The results of I. and II. are shown in Tables II. to V.

III. Comparisons of the results of the dew-point instrument with those of the
saturator and chemical method when all three are taken together. (Table VIII.)

IV. Observations with the dew-point instrument compared with the results of the
saturator only. A few such observations were made for the purpose of testing special
points under circumstances that made the double testing of the air inconvenient or
undesirable. (Exp. 73 to 83.)

I may now proceed to give the details of the experiments.

THE METHOD OF SATURATION AND THE CHEMICAL METHOD. TESTS OF THE
EFFICIENCY OF THE APPARATUS AND OF THE DESICCATING SUBSTANCES.

§ 1. The method of saturation which is here referred to simply means passing air
through some form of apparatus by which it is saturated, the temperature of satura-
tion being read by a thermometer placed in the saturator. This is, of course, not a
method of measuring the pressure of moisture in a given specimen of air, but simply a
means of obtaining air the pressure of vapour in which is known from its temperature.
The saturators used will he described later.

§ 2. The chemical method consists, as is well known, in causing a known volume, v,
of air to pass through weighed tubes capable of abstracting the whole of the moisture

78 MB. W. N. SHAW ON HYGROMETRTC METHODS.

from the air and determining the weight of aqueous vapour thus absorbed by the
tubes. Then, if f be the amount of moisture per cubic metre in the air which enters
the tubes ; A, the density of dry air at 0° C. and 760 mm. pressure ; d, the specific
gravity of aquecus vapour referred to dry air at the same temperature and pressure ;
t, the temperature of the entering air ; a, its coefficient of expansion per degree
Centigrade ; e, the pressure of the aqueous vapour, then

In order to determine the volume, v, of air which enters the tubes, a known volume
of water is allowed to flow out of an aspirator, its place being taken by the air which
passes through the drying tubes. Let V be the volume of water which is allowed to
run out of the aspirator, this volume will then be occupied by saturated air, the dry
part of which has passed through the drying tubes ; if T be its temperature when in
the aspirator, E the corresponding pressure of aqueous vapour, the pressure due to
the dry air will be B — E, where B is the height of the barometer at the time. Its
pressure before entering the tubes was B — e, and its temperature was t ; its volume,
therefore, was —

T, B - E 1 + a t

V = V • - - • - — •
B-e 1 +«T

Hence, if w be the number of grammes of moisture absorbed,

w B - e 1 + «T

= V'B -E' i + «< '

and we get for determining e the following equation : —

_ 760 (1 + a. t) w B - e. 1 + aT

Arf V B -E 1 + »t

or

e 760 1 + a T w

(2)

B -e " B -E V Ad

I + « T) w

A d (B - E) V

= x

therefore

or

e =

approximately, neglecting squares not likely to be larger than

( E - X wfV\

\ ~* ~B / aPProximate'y- .iff,-. •"« • ",: :-•• (4)

Mil. \V. N. SHAW ON HrGHOMKTKIC METHODS.

79

The application of this formula involves the assumption of a value of d, the specific
gravity of steam. The difficulties connected with this are discussed in Note A
(p. 121). The value '622 has been assumed throughout my experiments. For Xw/V
inside the bracket the value of e derived from the temperature of the savurator may
be substituted without appreciable error.

To facilitate computation I have tabulated the values of X, i.e., 760(1 -4- aT)/Ae£,
for each degree Centigrade as below : —

TABLE for Reduction of Aspirator Observations.

Table of the value of 7GO (1 + aT)/dA for every degree of temperature between

- 10° C. and + 30° C.

d = 0-G22.

A = 1293 grammes per cubic metre.

a = -003GG.

Temperature
T.

7«0(1 + oT)

u*

Temperature
T.

7«0(1 + aT)

log.

*A

<a

o

-10

•9104

1-9592313

o
11

•9830

1-9925666

- 9

•9139

•9608781

12

•9865

•9949919

- 8

•gin

•9025187

13

•9899

•9956119

- 7

•9208

•9641531

14

•9934

•9971266

- 6

•9242

•9657*13

15

•9969

•9980360

— 5

•9277

•9074035

16

1-0003

0-0001402

- 4

•9311

•9690196

17

1-0038

•0016398

- 3

•9346

•9706298

18

1-0073

•0081380

- 2

•9381

•9722340

19

1-0107

•0046258

- I

•9415

•9738525

20

1-0142

•0061054

0

•9449

•9754-247

21

1-0176

•0075839

+ 1

•9484

•9770114

22

10211

•0090575

+ 2

•9519

•9785921

23

1-0245

•0105260

+ 3

•9554

•9801C73

24

1-0280

•0119898

+ 4

•9588

•9817458

25

1-0314

•0134484

+ 5

•9623

•9683004

26

1-0349

•014H023

+ 6

•9657

•9848586

27

1-0383

•0163117

+ 7

•9692

•9864112

28

1-0418

•0177550

+ 8

•9726

•9879583

29

1-0542

•0191955

+ 9

•9761

•9894998

30

1-0487

•0200303

+ 10

•9796

•9910359

The fraction (E — t')/B is always small, and B has therefore always been taken equal
to 760. The practical error thus introduced is only of the order '00002.

§ 3. Before describing the apparatus first employed, it will be well to call attention
to the temperature measurement. For this purpose I have been able to avail myself
of eleven thermometers ; of these, two, by NEGRETTI, are the property of the Meteoro-
logical Office, graduated to half degrees FAHRENHEIT ; two of iny own, by HICKS, are
graduated to fifths Centigrade ; two, by HICKS, graduated to fifths FAHRENHEIT, one,
by HICKS, graduated to fifths Centigrade, one, by CETTI, graduated to fifths Centi-
grade, and three, by GEI&SLER, graduated to tenths Centigrade, are the property of

80 MR. W. N. SHAW ON IIYGROMETRIC METHODS.

the Cavendish Laboratory. All these thermometers were, through the kindness of
Mr. WHIPPLE, specially compared for me at Kew for every degree Centigrade in
January, 1882. Their freezing points have been determined from time to time, and
the tables of corrections revised. The temperature readings may be regarded as
accurate to within 0°'l C.

§ 4. The apparatus used for the first series of experiments (Table II.), of which fig. 1
(Plate 5) shows the general arrangement, consisted of three distinct parts : —

(i.) The aspirator (C) for causing the passage of a known volume of air over the
desiccating substances in the drying tubes.

(ii.) The weighed drying tubes (B, B) for determining by their increase of weight
the quantity of moisture in a known volume.

(iii.) A saturator (A) for supplying saturated air at the temperature of the room.

Between the drying tubes and the aspirator was placed an additional drying tube,
or a bottle filled with chloride of calcium, to prevent moisture reaching the weighed
tubes from the aspirator.

(i.) The aspirator was of the ordinary form, a copper cylinder with conical ends ; a
tap was fixed at the bottom, and through a cork at the top passed three glass tubes :
the first, for the delivery of the air from the drying tubes, passed nearly to the bottom
of the aspirator to ensure a uniform flow of air ; the second, connected with a similar
one entering at the bottom, served as a gauge ; the third passed to the lower surface of
the cork, and was used to fill the aspirator by being connected with an aspirating
pump. The aspirator was also provided with a thermometer, passing through the side
of the upper cone.

The volume of each aspirator was determined by completely filling it with water as
for an experiment, and running the water out gradually into a litre flask of known
weight when empty, and weighing the flask each time when full, the flask being care-
fully dried by means of a hot cloth between successive fillings.

Two aspirators were employed marked A and B, and the quantity of water in each
was determined twice with the following results : —

Aspirator A, 1st observation 163f>5'9 grammes, temperature 18° C.

2nd „ 16371-2

Aspirator B, 1st „ 16384-0 „ „

2nd „ 16383-8

We have therefore, allowing for the density of the water used and the temperature,

Volume of A at 0° 16384 cc.
B at 0° 16400 cc.

The volumes at any temperature t will then be

Aspirator A, 16384 (1 + '000052 t) cubic centimetres.
B, 16400 (1 + -000052 /)

MR. W. N. SHAW ON HYGROMKTRIC MKTIK >I»S.

81

One of these two gives the volume of water run out in any experiment. The water
would be replaced partly by air which had come through the drying tubes and partly
by the vapour of the water formed in the aspirator. We may assume that the air in
the aspirator at the end of the experiment in saturated with moisture.

(ii.) The drying tubes, — A specially constructed form was used. Instead of being
closed with corks perforated with glass tubes, glass connexions of wider bore were
used, which were thickened and ground into the U-tubes. These latter were of the
ordinary size, about 6 inches long and i^-inch internal diameter. The long tubular
stoppers were bent over, in the case of sulphuric acid and pumice tubes through two
right angles, and in the case 'of phosphoric acid tubes (shown in fig. 2, Plate 5),
through one right angle. The wide ends of these tubes could then pass over narrower
tubes coming vertically through the bottoms of small mercury cups, and thus forming
the connexions between the drying tubes and the other parts of the apparatus. The
connexions were thus made by means of mercury joints. These joints were tested,
and found to be quite tight for differences of external and internal pressure many
times greater than those occurring in the experiments. The arrangement is very
convenient, as the tubes can be simply lifted from their places and as easily replaced ;
they require careful brushing to remove the adhering mercury, and the ends are closed
for weighing with small india-rubber stoppers. The liability to error in consequence
of moisture on the surface of the mercury is probably not so great as that to which
the tubes would lie exposed by using india-rubber connexions. The drying tubes
were filled either with phosphoric anhydride, or with rather coarse fragments of
pumice saturated with the strongest sulphuric acid (sp. gr. T84). Experiments will
be detailed below to show that either of these substances is perfectly efficient for the
purpose of withdrawing all the moisture from the air passed over it. A number of
experiments have shown me that drying tubes filled with recently fused chloride of
calcium, although in many ways convenient, are not capable of extracting alt the
moisture from air.*

Correction of the iceight of the drying tubes for weighing in air. — The main part of

* The experiments were of two kinds : — '

(1.) Two chloride of calcium tubes were arranged in front of two sulphuric acid tubes and nn
aspirator- ful of saturated air passed through «11 four. The gain of weight in each of the four for three
observations is given below : —

lat tube.

2nd tube.

3rd tube.

4th tulw.

<C»C1,) (C»CW

(H,S04.)

•

(HJBO,.)

gin.

gin.

em.

gin.

•2025

•0010

•0208

•22'20

•oooo

•0245

•0013

•2226

•0000

•0190

•0000

(2.) Tho two aspirators were used to determine by two independent .simultaneous observations the
MDCCCLXXXVTII. — A. M

82

MR. W. N. SHAW ON HYGROMETRIC METHODS.

the weight of the tubes is the weight of the glass, and the specific gravity of the
pumice and of the strong sulphuric acid does not differ much from that of glass ; we
may, therefore, calculate the correction for weighing in air on the assumption that the
specific gravity of the whole tube is that of flint glass, which we may take to be 3 '5.
This would make the correction to weight in vacuo for a tube of 150 grammes equal
to 30 milligrammes. The effect of a barometric variation of 1 cm. upon such a tube
would therefore be to alter its apparent weight by '39 mgm., and a variation of 1° C.
of temperature would produce an alteration of '11 mgm. in the apparent weight.
The changes in barometric pressure and temperature between two successive weighings
may therefore be such as to cause the apparent weight of the tube to alter by a
considerable fraction of a milligramme. Now the amount of moisture is determined
by the difference of weight of the tube at the two weighings, and accordingly any
error in the weighing, due to neglecting to correct for weighing in air, will be of the
same absolute magnitude, and of very much greater relative importance, in the weight
of moisture absorbed by the tube. The variations of pressure and temperature,
however, between the successive weighings of the tubes during the observations were
not sufficient to produce any appreciable effect upon the results.

(iii.) The saturator. — This part of the apparatus was similar in principle to that
used by REGNAULT. Two long bell-jars stood in a shallow dish of distilled water ;
the one jar was filled with well-washed sponge, and the second contained a wire
frame covered with muslin, which, with it, stood in the water at the bottom of the
dish. The air supplied to the aspirator was drawn by means of a glass tube passing
through the cork in the top of the jar from the middle of the muslin cage, and close
to the opening of the tube was the bulb of the thermometer, which also passed
through the cork ; the place of the air thus removed was supplied by air passing
from the outside through the sponge vessel, and delivered into the second vessel
outside the muslin cage. During an observation, which lasted generally about two
hours, the thermometer was read every quarter of an hour by means of a telescope
placed at some distance, in order to avoid any error of parallax, and the mean of the
readings taken as the temperature of the saturated air. The muslin cage served to

amount of moisture in the air of the room, first by pumipe tubes, and secondly by chloride of calcium
tubes. The amounts are given below : —

By pumice tubes.

By chloride of calcium
tubes.

Difference.

gm. •

gm.

gm.

•1550

•1435

•0115

•1498

•1352

•0146

Since the above was written I have found that previous observers have expressed the same opinion
about calcium chloride. (See ANDREWS, ' Phil. Mag.,' vol. 4, 1852, p. 330, and MULLEK-EK/DACU, ' Deutsch.
Chem. Gesell. Ber.,' vol. 14, p. 1093.)

Ml! W. N. SHAW ON HTGROMETRIC METHODS.

83

protect the thermometer from external radiation, as well as to ensure the complete
saturation of the air.

§ 5. A series of observations was first made, with the view of testing the drying
tubes and other parts of the apparatus. There were two points to be determined :
first, whether the desiccating substances used could be regarded as completely drying
the air passed over them ;* and secondly, whether one drying tube was sufficient for
the purpose, or two or more were necessary. For this purpose four drying tubes
were mounted, two being filled with phosphoric anhydride in the ordinary form of
white powder ; the other two were filled with pumice moistened with commercial
sulphuric acid (sp. gr. T84). The pumice was broken into course fragments, and
before being used was saturated with sulphuric acid, and heated to redness.

The saturated air was then sent through all four tubes, and usually the gain in
weight of each tube determined. From the gain in weiyht of flic first tube alone,
the pressure of vapour in the saturator was calculated by the formula—

e =

760 (1 + at)

tc B [ — e
V" B-E

L*L"?

1 + at

and this was compared with the pressure as given by REGNAULT'S table for the
temperature indicated by the thermometer in the saturator, corrected by the table of
Kew corrections, and a determination of the freezing-point during the course of the
series of observations. The subjoined tables give the results obtained. Table I. is a
specimen of the observations as they were taken, and Table II. gives the collected
results.

TABLE I. Experiment 10. Aspirator A.

Time.

Tempera-
ture of
raturator.

Tempera-
ture of
aipiiator.

Barometer.

Weight of
pumice tube,
I., II.

Weight of
pumice tube,
III., IV.

Weight of
1' In Miihi trie lube,
III., IV.

355

19-5

o

176-4613

177-5045

146-9780

4-10

19-5

4-30

19-5

4-44

19-45

5-00

19-45

5-30

19-45

5-50

19-40

19-4

29750

176-7315

177-5061

146-9798

Means and
differences.

19-46

••

••

•2702

•0016

•0018

Corrections.

- 12

-•3

Tension of vapour calculated froui Tube I., II., 16'66 mm.

• See note at end (p. 148).
M 2

84

MR. W. N. SHAW ON HYGROMETRIC METHODS.

TABLE II.

Number
of expe-
riment.

Dale.

A*pi-

i-.u jr.

Gain in
weight,
tube 1, in
grammes.

Gain In
weight,
tube 2.

Gain io
weight,
tube3.

Gain in

weight,
tube 4

Mean
corrected
tempera-
ture of

saturator

Cor-
rected
tempera-
ture of
aspirator

Tension
in mm.
calculated
from
tube 1.

in mm.

ealculateJ
from tem-
perature
t of satu-

Differ-
ence.

Dura-
tion of
experi-
ment.

(0-

(T).

rator.

Phosphoric acid.

Sulphuric acid.

0

O

h. m.

1

July 4

B

•2915

-•0050

+ '0048

+ -0028

21-05

22-7

18-28

18-55

-•32

1 20

2

4

A -2928

+ -0008

+ •0117

+ •0018

20-98

22-2

18-29

18-41

-•12

2 15

8

6

It

•2916

+ -0023

+ •0080

+ -0040

21-05

21-8

18-17

18-55

-•38

2 23

4

7

ft

•2944

+ -0005

+ •0011 --0008

21-06 21-7

18-34

18-56

-•22

2 3

5

7

•3074

+ -0023

+ -0033

+ •0017

2159 ; 21-5

19 12

19-18

-•06

2 7

6

9

tt

•2931

+ -0007

+ -0022

+ -0006

20-91

21-45

1825

18-39

-•14

2 45

7

" 9

t*

•8020

-•0001

-•0002

-•0009

21-31

21-0

1877

18-85

-•08

1 15

Sulphuric acid.

Phosphoric acid.

8

„ 10

ft

•2870

+ -0027

+ -0019

,_

20-64

21-2

17-85

18-09

-•24

3 2

9

» 11

B

•2785

-•0005

+ •0018

••

20-00

21-4

17-35

17-39

-•04

2 0

10

„ 13

•2702

+ -0016

+ -0018

19-34

19-1

16-66

1670

-•04

1 55

11

»> •*•*

n

•2557

+ •0013

+ •0010

18-55

187

15-74

15-80

— •08

2 0

12

„ 12

M

•2682

+ -0023

+ -0013

••

19-08

?

16-53

16-53

-•00

2 0

In the first seven experiments the two phosphoric acid tubes were placed first, and
in the last five the sulphuric acid tubes were in that position. The columns giving
the increase of weight in the different tubes show that these nearly always gained a
small amount, but that amount is very irregular, and is about the same whether the
phosphoric acid tubes or the sulphuric acid are placed first, and the calculated tension
is in nearly every case within 1 per cent, of the tabulated tension. This being about
the same error as that which occurs in RKGNAULT'S observations, we may take it that
the first tube was sufficient to completely dry the air passed through it, and that the
increase of weight hi the other tubes was due to some other cause.

§ 6. The different connexions were made partly by glass and partly by india-rubber
tubing, and this suggested itself as a possible source of the observed differences. A
number of observations were therefore taken with a view of determining how far this
might be the case.

I. A glass tube about five feet long was mounted as a connexion between two
mercury cups ; and air, first dried by passing through a phosphoric acid tube, was
passed through a long glass tube and then through a sulphuric acid tube, and the
weight of the latter determined before and after the passage of the air. There was
accordingly nothing but the glass tube between the two drying tubes. The results
were as follows for four experiments :—

MR. \V. N. SHAW ON HYGROMETRIC METHODS.

85

Date.

Increase ia weight of
the lulpharic avid tube,
in gramme*.

.) ul v 18
„ 19 ....
„ 20 ....
„ 21 ....

•f-0029
+ •0002
-•0004
-•0002

It appears, therefore, that after the tube had been once dried no further increase of
weight of the sulphuric acid tube occurred, and hence that, with due precaution, glass
tubes may be used as connexions without any fear of error.

II. The glass tube was replaced by an india-rubber tube six feet long, and similar
observations taken, fifteen experiments being made. The sulphuric acid tube always
gained in weight, although every precaution was taken to keep the india-rubber tube
dry between the experiments. The least amount of moisture was obtained when a
second observation was taken, immediately after the completion of a first. The
increase of weight of the sulphuric acid tube generally amounted to about 15 milli-
grammes. An india-rubber tube cannot, therefore, be used with any security for
connecting two drying tubes.

§ 7. We may accordingly conclude that the increase in weight of the drying tubes
after the first, in the table of results given, was due for the most part to moisture
derived from the india-rubber connexions. For the last three observations in the
table, p. 84, these connexions were made as short as possible, so that the amount of
india-rubber surface exposed to the dry air might be small. With the apparatus in
that form the drying tubes gained very much less in weight than before, and we may
give the results obtained from these three observations as instances of the accuracy
which may be expected by this method.

THE METHOD OF SATURATION AND THE CHEMICAL METHOD. — EXPERIMENTS WITH

UNSATURATED AIR.

§ 8. With the experience gained by the experiments just described, I proceeded to
arrange the apparatus by which the result of the chemical method could be compared
with that of the saturation method on unsaturated air, and by which moreover dew-
point observations could also be taken during the progress of an experiment such as
those described. At first a REGNAULT dew-point instrument alone was introduced.

It was fixed by means of its glass tube into the cork of a three-necked globe in a
manner which will be sufficiently indicated by a glance at the figure (fig. 3) ; on each
side of the globe, between its long horizontal neck and the rest of the apparatus, was
attached a coil of copper tube to ensure that the temperature of the internal air was
the same as that of the external, except in the vessel containing the hygrometer

86 MR. W. N. SHAW ON HYGROMETRIC METHODS.

itself; attached to the second copper coil was a second globe with three necks, which
contained a thermometer inserted in exactly the same manner as the REGNAULT
hygrometer, and likewise shown in the figure. This sequence, consisting of (l) a
copper coil k (fig. 3), (2) a globe containing the hygrometer, h, (3) a copper coil, k',
(4) a globe containing a thermometer, t, was simply interposed between the saturator
and the aspirator, so that the air passed through the whole series.

Fig. 3.

70 ASPIRA TOR

In order to fit the REGNAULT thimble tightly into the globe it was cemented by
means of gelatine to a glass tube, over which it would just slip. This was found not
to be quite tight when there was a considerable difference of pressure between the
inside and outside, and the leakage might have interfered with some of the observa-
tions. The joint was therefore strengthened by lapping the line of junction of the
silver and glass with pure india-rubber strip pulled very thin and having each
successive layer painted over with benzene. After a sufficient thickness was lapped
on, the whole was painted with a solution of india-rubber in benzene and varnished,
and no further trouble was caused by any leakage at the joint except in the five
experiments noted in Table VI.

Fig. 4.

Subsequently a DINES hygrometer enclosed in a cylindrical glass tube, fig. 4, was
added to the apparatus immediately following the globe containing the REGNAULT.
The metal box, connecting tube, and thermometer of the DINES were detached from
the wooden frame in which they are usually mounted. The delivery end of the tube

W. N. SHAW ON HYORO1IETRIC METHODS. 87

was easily slipped through a piece of thick india-rubber sheet forming a stopper to
the 3-inch glass cylinder, and the other end, having the tube and thermometer very
close together, was enclosed by half slitting an india-rubber bung previously carefully
drilled, the slit passing along the common diameter of the two drilled holes; the
half slit bung was then slipped on the tube and thermometer, carefully pressed
together with shellac varnish, and the bung coated with varnish and pressed home.
It is perhaps surprising that there was no leak, but repeated tests showed that the
DINES was in this way fixed air-tight in the cylinder. At first a leakage took place
where the blackened glass of the instrument was fixed into the metal box. This was
closed by painting round the edge with a solution of india-rubber in benzene, and
after a good thick layer had been formed it was varnished with shellac varnish.

These dew-point instruments were sometimes excluded, the REONAULT by with-
drawing the hygrometer and filling up the neck of its globe with a bung, the DINES
by removing the cylinder altogether.

All the connexions between the different parts of the apparatus were made by glass
and very short pieces of india-rubber tubing carefully wired and varnished.

The rest of the apparatus was similar to that already described, but was modified
in the following details : —

i. The saturator. — This was required to saturate air at a temperature below that of
the room. For this purpose a large galvanized iron water tank jacketed with 3 inches
of sawdust in a second galvanized iron tank was introduced to form a bath in which
the saturator could be completely immersed. The saturator proper will be understood
by reference to fig. 5. The glass tube a was open to the external air, and led to a
spherical glass vessel, b, filled with moist sponge ; from this vessel a bent glass tube, c,
led through a cork to the interior of the copper cylindrical vessel, V, through a metal
lid bolted to it, and made air tight by an india-rubber washer round the rim. The
vessel V was loaded with shot so that the whole apparatus just sank in water, and
above the shot was a thin glass cylinder containing a layer of water at the bottom, in
which stood a wire frame covered with well- washed muslin. The tube C opened into
the vessel V outside the muslin frame, but inside the glass. The air was drawn out
of the interior of the vessel through the tube d, close to which was the bulb of a
thermometer ; the tube d was the extension of a glass three-way tap T, which was
fixed just outside the copper vessel ; by it I could send air through the rest of the
apparatus without its passing through the saturator, and thus dry the connecting
tubes, Ac., when necessary. This part of the apparatus was immersed in water, in
the final experiments so that the tap was covered (the projecting nozzle being
extended by an india-rubber tube), in the tank already spoken of, and the temperature
of the water therein was reduced by stirring ice in it to any required extent.

ii. The drying tubes. — Two sulphuric acid or two phosphoric acid ones were used
for the observations. These remained as before, except that the connexions (see
fig. 1) between them were made by glass tubes and shortened as much as possible.

88

MR. W. N. SHAW ON HYGROMETRIC METHODS.

The ends of the connecting tubes coming through the mercury cups were moreover
covered with little glass caps between the observations to protect them from damp

during that time.

Fig. 5.

One-tenth natural size.

With these precautions it was assumed that any deposit received in the second
tube ought to be added to that in the first tube, and this departure from the practice
adopted in the first series of experiments is further justified by the fact that the new
aspirator, which will be described presently, more than doubled each of the old ones
in volume, so that the tubes had to take up twice the quantity of moisture (for the
same temperature) and in the same time as that required for the smaller amount in
the previous series. The variable amounts of moisture received by the second tube
showed that this part could not be neglected (see § 15).

iii. The aspirator. — The pattern of this was similar to those already described,
but its volume was more than twice as great, and provision was made for increased
accuracy in reading ; it is accordingly figured in fig. 6. Between it and the rest of
the apparatus there was, moreover, a water-pressure gauge attached, which was found
very useful, first, as a test of the tightness of the joints of the apparatus ; and,
secondly, to see that communication was quite free between all the parts. The larger
aspirator was used, as it was intended to supply in some cases comparatively dry air,
and a larger volume was required to avoid the effect of errors of weighing.

MR. W. N. SHAW ON HYGROMETRIC METHODS.

Two determinations of the capacity of the new aspirator were made on the same
plan as for the old ones (see p. 80), with the following results : —

1st observation, 35987'44 grammes of water at 12°7 C.
2nd 35989-28 120>2 ,

Fig. 6.

TO ASPIRATING

'-=^

,'

irtS(j

Ca

C/,

)'/"

"

1

One-tenth natural size.

These, corrected for temperature expansion of water and copper, give volumes : —

(1.) 36049-33 c.c. at 15° C.
(2.) 3604973 „

Mean . . 36049 '53 c.c. at 15° C.
The volume of the aspirator is therefore at t°

= 36049-5 { 1 + -0000525 (t — 15)}.
MDCCCLXXXVIII. — A. N

90

MR. W. N. SHAW ON HY/GROMETRIC METHODS.

§ 9. We have first to discuss the comparison of the results given by the saturator
with those of the chemical method for air which is not saturated.

The apparatus was mounted on November 7 with the REGNAULT dew-point instru-
ment in position, and the first four observations were wasted by the washer between
the copper cylinder and its lid being cut through and letting the water in. It was
put in order again on November 1 7, and gave no trouble afterwards. An observation
was taken with the temperature of the water as it stood in the tank, with the
following result : —

No. 5. November 17. Vapour pressure, by saturator . . .

„ ,, chemical method

Difference
Temperature of air, 12°'98 C.

. 10-63 mm.

' I0'65 »
. +'02 mm.

Ice was then added, and when it was melted a second observation was taken, as
follows : —

No. 6. November 19. Vapour pressure, by saturator 8'54 mm.

,, ,, chemical method . . 8'59 „

Difference . . +'05 mm.
Temperature of air, 14°'74 C.

These results seemed to give quite satisfactory agreement between the two indica-
tions of the states of the air. Passing over No. 7 for the present, the next five
gave results which were increasingly divergent, viz. : —

TABLE III.

No.

Date.

Temperature
of saturator.

Temperature
of air.

Pressure by

saturator.

Pressure by
chemical method.

Difference.

8

Feb. 9

o

1176

14-91

10-27

10-42

+ -15

9

., 9

11-92

18-08

10-38

10-75

+ -37

10

,, 11

13-14

15-83

11-24

11-84

+ -60

11

10

)» x —

13-07

14-37

11-18

12-20

+ 1-02

12

„ 14

13-43

M

1145

13-30

+ 1-85

§10. I was unable at first to assign any cause for this increasing difference, and
thought that in No. 9 it might be due to the air being supersaturated in passing into
the saturator, so in the next experiment, No. 10, I passed the air through a tube of
ice before it entered the vertical tube a of the saturator (fig. 5). I tried, moreover,
whether it was due to the action of the carbonic acid of the air, and interposed a

MR W. N. SHAW ON HTGBOMETRIO MKTH<>|>> 91

potash tube in No. 11, but neither of these trials showed that the cause was
discovered. The next observation, however, furnished an explanation. The tempera-
ture of the saturator was above that of the air, and the chemical method gave a result
•53 mm. below that of the saturator. On inspecting the connecting tubes t found a
very fine deposit of dew upon them ; and, as the apparatus was left in connexion with
the saturator during the night, I have no doubt that during the cold part of the night
the vapour was condensed in the connecting tubes, and that these were not dry when
the observations marked 8 to 12 were made, although I took the precaution of
running dry air through them, generally for a quarter of an hour, before each experi-
ment. Thenceforward the connecting tube was left in communication, by means of
the three-way tap, with a calcium chloride bottle over night, and the tubes were dried
more carefully before the observations. The next set of four observations, however,
showed that the amount of moisture in the drying tubes was still too great, the
pressure differences being '11, '20, '20, *40 mm. respectively. I then thought that the
dew-point instrument might be producing some effect by leakage or otherwise, and
accordingly removed it. The temperature of the bath was reduced by ice to 9°'96 C.,
and the observation showed a difference of '07 mm. only. More ice was added, and
the temperature reduced to 7°'39 C. : the difference at that observation was 0'09 mm.;
with further reductions of temperature, however, to 4°'74 C., 4°'82 C., and 3°'06 C.,
the drying tubes got too little moisture, the differences being '47 mm., '68 mm., and
'81 mm. respectively.

Then the temperature was allowed to rise gradually, and the difference changed
sign, having the values '19 mm. and '18 mm. at 6°'55 C. and 10°'42 C. respectively.
It appeared that the amount of moisture obtained by the chemical method was
practically identical with that given by the saturation temperature, when the
saturator was not cooled below 8° C. ; below that, when the temperature was falling,
the chemical method gave too small a result ; but, as the temperature rose again, too
large a result was obtained. The next series of observations, Nos. 31 to 38
(Table IV.), shows this order most clearly. Starting with the temperature of the
saturator at 16°78 C., and gradually cooling it down to 2°'94 C., then allowing it to
rise to 6°'47 C., I got the following results : —

N 2

92

MR. W. N. SHAW ON HYGROMETRIC METHODS.

TABLE IV.

No.

Temperature of
saturator.

Temperature
of air.

". ' .

Vapour pressure by
gaturator.

Vapour pressure by
chemical method.

Difference.

mm.

31

16-78

20

14-20

14-29

+ -09

32

16-33

18

13-80

13-88

+ -08

33

16-33

18

13-80

13-82

+ -02

34

11-96

17

10-41

10-47

+ -06

35

7-65

17-5

7-78

7-81

+ -03

36

4-81

16

6-42

5-64

- -78

37

2-94

18

5-63

4-93

- -70

38

6-47

18-2

7-20

7-50

+ -30

It then occurred to me that the effect was probably due to differences of temperature
in different parts of the bath.

I had not paid much attention to the uniformity of temperature, as the air was
drawn from a point in the interior of the copper cylinder quite close to the bulb of
the thermometer, and I considered that the air so withdrawn would be saturated at
that temperature, but I had neglected to take sufficient notice of the fact that it was
saturated air I was dealing with, and that the effect of the tube which led away
the air ought to be considered. It seemed likely that if the water in the bath was
not kept thoroughly well stirred, the water, cooled by the melting of the ice, would
sink rapidly until the temperature got near 4°, and that then a layer of cold water
would be formed on the top of the bath, and cool the top of the cylinder and the
tube through which the air came ; if this was colder than the saturated air, a part of
the moisture would be deposited in the tube ; the layer of moisture thus formed
would, on the other hand, be at a higher temperature than the rest when the bath
gradually got warmer by contact with the air at the top, and the air for the experi-
ment would be slightly warmed and receive an addition to its moisture in passing over
this band of moisture. At the end of the series of observations there was no moisture
visible on the parts of the tube above the cylinder, but on drawing the tube out of its
cork a little way a collar of moisture was distinctly visible where the tube had been
cooled by conduction through the cork, and thus my suspicion was confirmed. In
consequence of this discovery, considerable attention was devoted to the stirring of
the bath, to keep it at a uniform temperature. During an observation I could not
hope that the tube would be kept free from deposit, and the only thing to do was to
prevent its having any effect on the temperature of saturation. The next six
observations show the result of this precaution ; the temperature began at 6°'44 C., was
reduced to 3°'lO C., and then allowed to rise to 7°'93 C., and therefore included that
part of the scale in which the differences were previously most conspicuous.

MU. \V N SHAW ON HTGROMKTRIC METHODS.
TABLE V.

No.

Temperature of
•Mrator.

Temperature
of air.

Vapour prearare
from Mtnratar.

Vapour preMore by
chemical method.

Difference.

39

6-44

1- -

7-18

7-25

+ '07

40

6-00

19-3

6-97

7-01

+ -04

41

310

19-9

5-70

:....:

- -07

42

476

19-fl

6-40

6-43

+ -03

43

7-44

21-4

7-69

7-71

+ -02

44

7-93

211

7-95

-'".•

+ -14

These six observations gave results agreeing very closely, the mean difference
between the tabular saturation pressure and the pressure as deduced from the chemical
method being '06 mm only. It will be observed that the same degree of accuracy of
agreement is given by the observations in Table IV., which are free from the difficulty
of the deposit on the leading tube, the mean of the differences for the first five
observations in that table being "056 mm.

As the temperature readings could not be relied upon to an accuracy much beyond
0°'l C., it seemed unnecessary to endeavour to press the comparison of the saturator
and the chemical method further.

§ 11. I concluded that I then knew enough about the methods to be justified in
expecting the results of the chemical method to agree with the tabular result obtained
from the temperature of the saturator within O'l mm., and in regarding the air under
those circumstances as being in a known state, and in using it in the investigation of
the action of the dew-point instruments. It was, however, necessary to determine
whether or not the presence of these instruments affected the air by leakage or other-
wise : so that for the succeeding observations I always determined the pressure by the
chemical method. First, the REQNAULT was replaced : the result appeared to be
disastrous, as the first two observations gave differences between the saturator and
chemical method results of 1'OG and — '37 respectively, but an investigation disclosed
a deposit of moisture in the tube above the tap. The rest of the observations taken
are all given in the following table, and notes are appended in cases where some

disturbing cause was acting.

TABLE VI.

No.

Date.

Hygrometer
included.

Tempera-
ture,
saturator.

Tempera-
ture,
air.

Vapour
prewiurc,
gatnra tor.

Vapour
preiwure,
chemical
method.

Difference.

Remarks.

47

July 19

RKOXAULT

16°85

19-14

14-26

14-28

mm.

+ -02

48

„ 19

i

16-89

19-3

14-30

14-27

- -03

r.'

,, 21

2-94

19-6

5-63

5-90

+ -27

50

„ 22

2-98

20

5-65

5-75

+ -10

51

,, 22

368

20

5-93

6-01

+ -08

52

„ 23

i

7-"'.'

20-5

7-51

7-53

+ -02

94

MR. W. N. SHAW ON HYGROMETRIC METHODS.
TABI.E VI. — (continued).

No.

Date.

Hygrometer
included.

Tempera-
ture,
xatnrator.

Tempera-
ture,
air.

Vapour
pressure,
saturator.

Vapour
pressure,
chemical
method.

Difference.

Remarks.

53

July 24

REGNADLT

3°22

20°

5-75

5-73

mm.
- -02

54

„ 24

M

3-89

20

6-02

6-00

- -02

55

» 25

7-08

19

7-47

7-57

+ -10

56

„ 26

8-73

17

8-39

8-52

+ -13

Satnrator not com-

pletely covered by

water.

57

„ 28

11-74

18-18

10-26 10-29

-f -03

58

„ 29

,,

13-09 19

11-21 11-48 + '27 Water not stirred.

59

„ 31

15-25

21-5

12-88 13-35

+ -47

The dry ingtnbes wei-e

left in connexion

.

with the appa-

ratus for sometime,

and moisture was

gained by diffusion.

60
61
62
63
64
65

Aug. 1

", 2
„ 2

» •*

,. 5

»

16-49
16-85
17-62
18-12
18-82
18-45

24
24
24
24
22
21

13-94
14-26
15-00
15-45
16-14
15-86

14-00
14-72
15-45
15-84
16-50
16-08

-f -06
-1- -46^
+ -45
+ -39 L
+ -36

'Hygrometer showed
increasing signs of
I leaking at the joint;
| was accordingly
taken out and re-
placed.

66

„ 10

REONAULT

21-60

27

19-16

19-42

+ -26 The globe containing

and

the REGNAULT had

DINKS.

been washed and

i

not thoroughly

dried.

67

„ 11

22-02

25

19-65

19-72

-1- -07

DINES read during

the observation.

'

Its tube became

68

„ 12

»»

22-06

23

19-70

17-04

-1-96^

covered with mois-

69

„ 12

22-09

24

19-74

19-16

- -58

ture that was not

70

„ 16

B

20-27

23-5

17-66

16-90

- -76 I

<( taken up subse-

71

„ 19

ii

20-63

23

18-05

16-56

-1-49 1

quently, and there-

72

„ 21

20-09

23

17-46

17-73

+ -27J

fore lost to the dry-

ing tubes. The

method of reading

was therefore

_ altered (see § 22).

73

„ 22

()

20-03

24-5

17-39

17-34

- -05

74

.. 27

n

15-75

18

13-30

13-47

+ -17

75

„ 28

n

15-72

18

13-27

13-23

- -04

76

„ 29

n

6-38

18-5 7-15

6-87

— -"28

84 Ibs. of ice were

melted to cool the

bath, and it is

probable that the

leading tube re-

mained too cold

during the observa-

tion.

77

„ 30

n

7-41

17

7-68

7-76

+ -08

78

.. 30

7-96

18

7-97

7-95

- -02

79

Sept. 1 | „

11-71

18

10-24

10-33

+ -09

80

„ 1

12-90

18-5

11-06

11-14

+ -08

81

,, 3

14-50

18-5

12-27

12-31

+ -04

82

„ 5

"

1-09

17

4-94

4-91

- -03

MR. W. N. SHAW ON HYGROMETRIC METHODS.

95

It will be seen that there are definite reasons for rejecting the experiments
numbered 56, 59, 61-65, 68-72, and of the rest there are five only for which the
differences reach O'l mm., and the mean difference for the whole number (without
regard to sign) of those not rejected amounts to "077, or again less than '1 mm.

§ 12. Rejecting from Table VI. then the numbers indicated, and arranging the
observations in the order of humidity of the air, we get the following re-arrangement
of the Tables V. and VI. :-

TABLE VII.

Number of
experiment.

Temperature of
air.

Percentage
humidity.

Difference Iwtwecn result of chemical
method and utnration prewure.

Poiitire.

Negative.

50

o

20

32

mm.
+ •10

mm.

49

19-6

33

•27

41 19-9

33

t t

-•07

53

20

33

f t

•02

51

20

34

•08

, .

82

17

34

B f

•03

54

20

35

« •

•02

42

19-9

37

•03

43

214

41

•02

40

19-3

!•_•

•04

52

20-5

42

•02

44

214

43

•14

39

18-8 45

•07

76

18-5 45

. .

•28

55

19

46

•10

78

18

52

. .

•02

77.

17

53

•08

60

24

63

•06

57

18-5

64

•03

79

18

67

•09

80

18-5

75

•08

73

24-5

76

. f

•05

81

18-5

78

•04

67

25

82

•07

48

19-5

86

. ,

•03

74

18

87

•17

75

• 18

87 n

. .

•04

47

19-14 87 -02

•_N

15

88

•01

27

14-8

91

. ,

•05

5

13

96

t f

•07

13

127

100

••

•18

The last four observations are not among those quoted in Tables V. and VI. ; they
were taken before the behaviour of the saturator and its difficulties were fully under-
stood. They are, however, observations for a very high degree of humidity, and in
such cases the peculiar difficulties of the apparatus do not come in ; they may there-
fore, I think, be fairly included in the table of final results.

96 MB. W. N. SHAW ON HYGROMETBIC METHODS.

§ 13. I have now accounted for the whole number of 82 experiments made with the
modified apparatus (including those in which dew-point readings were taken) for
comparing the saturator and the chemical method at different degrees of humidity,
with the exception of three, Nos. 7, 25, and 26. No. 25 gave a difference of — '07
when the humidity was 70, and simply confirms the general result ; No. 7 gave a
difference of + 1'73 when the humidity was 87 ; and No. 26 a difference of — '57
when the humidity was 86 ; these show a wide divergence, for which I am quite
unable to account. There is nothing in my note-book which places those observations
on a different footing from the rest ; it is true that they were made at a time when I
had not fully appreciated the necessity of keeping the bath at a uniform temperature,
but the difficulties did not occur with other observations at corresponding humidities.
I can only attribute the large differences to clerical errors in entering the weights.

§ 14. The reductions of the observations have been made by formula 3 of p. 78,
with the aid of the table of p. 79. The value there adopted for the specific gravity
of steam referred to air at the same temperature and pressure is '622. The table of
vapour pressures which I used at first was REGNAULT'S ; those given in the results are
derived from the re-calculation in LANDOLT and BOUNSTEIN'S table. The effect of
this change is to diminish the saturation pressure by '02 mm. or '03 mm. in each case,
and has consequently given to the Table VII. a preponderance of + differences which
did not show itself when REGNAULT'S numbers were taken. As the table now
stands, 20 observations give a positive difference and 12 a negative one. If the
•03 mm. be restored 14 observations will have a positive difference, 15 a negative
one, while 3 will give zero difference. Almost precisely the same result would
be attained by using the value '624 for the specific gravity of steam instead of
•622. The one observation on saturated air with the negative difference '18 mm. in
the second series of experiments bears out the results of the previous experiments
with the original apparatus (§5).

The net result of the whole investigation upon the chemical method would seem to
show : —

(1.) That if the specific gravity of saturated aqueous vapour referred to air at the
same temperature and pressure be assumed to be '624, the calculation of the pressure
from the weight of moisture absorbed gives a value agreeing to within O'l mm. (the
mean difference being '07 mm.) with that derived from the temperature of saturation,
provided that the air is superheated after passing through the saturator.

(2.) That there is no reason for assigning the differences in the observations to
other than experimental errors.

(3.) That for saturated air, the result of the chemical method is slightly less than
the tabulated saturation pressure. This is shown in the first series of experiments,
and agrees with REGNAULT'S result. But I think that the later experiments and the
observations with the dew-point instrument which follow clearly show that when
saturated air is passed along glass tubes at the same or even a slightly higher

\ll{ \V \ <H\\V ON HYGKOMKTUir .MHTHui-s 97

temperature a deposit of moisture is formed on the gloss. The observations of
l!i:t;NAULT and HKI:\\ n, are in favour of this view; and, if that be so, it is clearly
impossible to conduct saturated air along glass tubes to the drying tubes ; some of
the moisture will be deposited on the way. How much moisture will be required to
saturate the glass is, I believe, unknown ; but if the conducting tubes are maintained
at a temperature a degree or more above that of the air when it is being saturated
the deposit will not take place.* Thus, in the observations upon air saturated at a
temperature below that of the glass, the air could only travel a very short distance
before it came in contact with glass at a temperature higher than its own, and the
amount of moisture necessary to saturate this short length of glass would l>e supplied
by the saturated air passed through the apparatus as a preliminary part of the
experiment. So that this suggestion seems to me to explain satisfactorily what is
otherwise a great difficulty, namely, that the saturation pressure agrees with the
tabulated pressure only if the air is heated after being saturated.

To assume that the result of the chemical method should agree exactly with that
given by the temperature of the saturator is to assume not only DAL/TON'S law to be
true, but also that the expansion of vapour with rise of temperature from the
saturation point takes place, according to the law of GAV-LusSAC, with a coefficient
of expansion the same as that of air. The specific gravity of steam which must be
substituted, therefore, in order to obtain identical results by the two methods is not
the actual specific gravity of steam in the unsaturated air experimented on, but what
would be the specific gravity if the vapour obeyed GAY-LUSSAC'S law in the manner
indicated, or (since the specific gravity is referred to air at the same temperature and
pressure) the specific gravity of the saturated vapour. Thus the observations may be
employed (as RKGNAULT employed similar observations) to determine the specific
gravity of saturated steam at the temperature of the saturator. The observations
do not admit of sufficient accuracy to trace the variation of the specific gravity with
temperature, but the mean specific gravity for the 32 experiments of Table VII. for
temperatures of steam between 1° C. and 21° C., if referred to air at the same tem-
perature and pressure, is '6245.

CLAUSIUS has calculated the specific gravity of saturated steam by thermodynamic
reasoning from other known constants of steam ('Mechanical Theory of Heat,' p. 153).
Assuming the value at 0° C. to be '622, he gives the value at 50° C., '631. From
his results, therefore, we obtain '6240 as the mean value between 1° C. and 21° C.,
a result agreeing very closely with that obtained from my observations.

If, however, the specific gravity of saturated steam be greater than that of unsatur-
ated steam, the vapour cannot obey GAY-LlJSSAc's law in the manner stated, and in
consequence the pressure of the vapour of unsaturated steam, calculated by the
chemical method, with the true specific gravity of unsaturated steam ('622) ought to be
somewhat greater than that given in the table of pressures of saturated vapour ; and
the results of Table VII. may accordingly be held to prove that the true pressure of

* Sec note at end of paper (p. 149).
MIXtVLXXXVIII. — A. O

98

MR. W. N. SHAW ON HYGROMETRIC METHODS.

vapour in unsaturated air is greater than the tabulated pressure for the temperature
of the saturator by 2/622 of the saturation pressure, that is, by an amount varying
from '02 mm. to '06 mm. for vapour pressures between 4'94 mm. and 19'65 mm. In
other words, to obtain the true pressure the proper value to substitute for d in the
formula of reduction of the chemical method is '622, at any rate when the fraction of
saturation of the air under experiment is less than '8, but if the air is saturated the
value '624 must be substituted.

NOTES. (November 11, 1887.) — (l) In order further to verify the condensation of
moisture upon a surface when the air surrounding it is not fully saturated, I have
tried to detect the actual difference of weight produced by the deposit. For this
purpose 1 have suspended a glass globe from one arm of a balance, and have counter-
poised it, surrounding the globe by a cylinder with a lid formed of two closely-fitting
pieces of glass, with a small hole left for the suspension wire. Inside the cylinder
were placed alternately a vessel of water and a vessel of sulphuric acid. After making
all allowances for the density of the air displaced, the experiments show an excess of
weight in the moist air corresponding to the condensation of about 1 X 10~5 gramme
per square centimetre. More accurate experiments are, however, in progress.

(2) In this discussion DALTON'S law has been assumed to be strictly accurate. If
the differences observed by REGNATJLT (p. 120) should prove on further investigation
to be real differences, the specific gravity of the saturated vapour must be increased
to about '645 for saturated air, and '643 for non-saturated air.

§ 15. In conclusion, it may be well to give a specimen of the observations from
which the tables are compiled, and some notes of the practical conduct of the work.
I take one experiment at random from my note-book : No. 68, August 12.

Initial weight of tubes V., VI.
Final

Gain

grammes.

142-8090

143-3856

•5766

III., IV.

grammes.
147-9447
147-9638

•0191

Barometer.
30°-00
30°'00

Temp. Balance Case.
24°' 70
25°-00

Total gain '5957 gramme.

Time.

Saturator
temp.

UEONAULT

DINES.

Air
temp.

12.00

22°89

o

O O

— 23-80

12.15 22-88

—

— 2380

12.30 22-88

—

2378

12.45

22-37

—

23-78

Final temp, of aspirator 24°-00

1.00
1.15
1.30

22-87
22-38
22-37

21-15

2116 23-80
23-80
23-82

1-0280 x -5957 x 1-00823

e (tubes) - 86066-6

1.45

2287

—

— —

= 19-78

1.53

22-37

—

2400

e (sat.) =1970

Duration Ihr. 63min.

Means

22-37

21-15

21-15

_

Diff. . . -08

Temp, correction . .

••

31

•27

—

•11

22-06

20-88

_

1

1

MB. W. N. SHAW ON HYOROMETRIC MKTIIODS. 99

The balance was a short-beam OERTLING, and the weights a box of OERTLINO'S,
compared with the Laboratory Standard. The weighings were carried to O'l mgm.
The drying tubes after an experiment were transferred to a desiccator to cool, and
then weighed. The barometer and temperature of the balance case were read at
each weighing, and recorded, so that a correction for weighing in air might be
introduced if necessary. (See § 4.) For Experiments 1-27 the tubes were filled with
sulphuric acid and pumice, and those for 28-82 with phosphoric acid. The alterations
in the state of surface of the tube are sufficient to prevent the accuracy of weighing
to O'l ingm. being entirely trustworthy, but the error is less than 1 mgm. (See
p. 74.) This limit gives to the accuracy of the weighing operations in the chemical
method a limit of '6 per cent, for the smallest amount of moisture, viz , '1737 gramme,
and of '16 per cent, for the largest amount, viz., '6880 gramme.

The duration of the experiments was generally about two hours, varying, in fact,
from 1 hr. 10 min. to 2 hr. 55 min. During that time 36 litres of air passed through
the tubes. The amount received by the second tube was very variable. In 29 experi-
ments out of the 82 it was less than 1 mgm. ; in one experiment, however, it reached
99 mgm. The amount was always added to that received in the first tube, for reasons
previously stated (§ 6-8).

Before an experiment was commenced dried air was drawn through the whole of
the apparatus, except the saturator and aspirator, by the use of an aspirating water-
pump for from a quarter to half an hour, to clear the tubes of deposited moisture ;
for this purpose the third aperture of the three-way tap was employed. After this
the plug of the tap was turned, and the air passed through the saturator for a
quarter of an hour or more to fill the vessels with air in the state required for the
observation. The pump was then cut off, the aspirator connected, and the drying
tubes put in their places ; the entry tube to the saturator was then closed, and the
tightness of the apparatus tested by reading the pressure gauge and observing the
cessation of drip from the aspirator. At the close of an observation the moist air was
swept out of the globe, &c., by dried air (using the water-pump), unless it was
intended to take a second observation immediately ; when left over night, the globes,
<&c., were connected with a calcium chloride bottle.

The temperature readings were by the following thermometers, compared at Kew,
us described in § 3 :—

Saturator ;. .- . . Exp. 1-14 GEISSLER, C.L.O. 21 graduated to 0°'l C.
•.*. . :,.,». „ 15-82 HICKS 79,916 „ 0°'2 C.

Air ... ,<.,.,„ o|, [.{,„„ . . . . HICKS 87,400 „ Od'2 C.

REOHAULT..,,!,, v., ^..l,.ir.^ .... . HICKS 79,915 „ 0°'2 C.

DINES. — Its own thermometer, by HICKS (not compared).

Aspirator. — An ordioary chemical thermometer compared in the Laboratory, .

graduated to 0°'5 C.

Other small points of detail, employed to secure as close an accuracy as possible,
may be passed over.

o 2

100 Ml!. W. N. SHAW ON HYGROMKTRIC METHODS.

DEW-POINT HYGROMETERS.
1. RKGNAULT'S HYGROMETER.

§ 16. I pass on now to consider the observations rmicle with REGNAULT'S hygrometer
upon air in a known state of humidity. The method by which the thimble was
exposed to the air to be investigated has already been described (§ 8).

I need now only give particulars of the apparatus adopted for cooling the ether and
for reading. The thermometer had a very short cylindrical bulb, and was well
covered by the ether in the thimble. The stem of the thermometer was passed
through a cork fitting into the glass tube which held the thimble, and through the
cork passed also two very narrow copper tubes, one to the bottom of the ether, the
other just through the cork. The end of the latter was connected to a MAGNUS
aspirator, which was provided with a tap, so that the rate of passage of the air was
under very easy regulation. As the thimble was enclosed in a glass globe, it could be
viewed quite closely without any fear of the presence of the observer altering the
dew-point.

§ 17. So far as I can gather from published accounts, REGNAULT'S instrument has
always been regarded as a standard (see Note A, p. 130); that is to say, a dew-
point determination has been held to give a final verdict as to the pressure of vapour
in the atmosphere. REGNAULT'S directions for the use of the instrument are, first of
all, to cool the ether so as to obtain a deposit, and note the temperature ; then let the
temperature rise until the deposit is gone ; then cool slowly by tenths of a degree
until two-tenths are determined, for the lower of which there is a deposit and for the
higher no deposit ; the mean gives the dew-point, from which the pressure of vapour
in the air is at once obtained by reference to the table of vapour pressures. No calcu-
lation is necessary on account of the temperature of the air, because a change of
temperature of the air would not cause any appreciable change of pressure, either of
dry air or vapour, but only a change of volume. With regard to this method of
observation, I may remark that it is not always possible to confine "deposit" and
" no deposit" temperatures within the limits of a tenth of a degree. It is compara-
tively easy to do so when the circulation of air in the neighbourhood of the polished
surface is brisk, and I have found from some experiments on dew-point instruments
in the strong current produced by a rotary fan that the air current simply improved
the facility of reading the instrument, and did not seem to alter the final reading ; but
this opinion is, I believe, not shared by all experimenters, and I do not pi-ess it now,
as my observations assumed to some extent the permanence of the condition of the
air operated on.* In the experiments I am now dealing with the circulation was
slow. The thimble of the hygrometer was contained in a globe of 5 cm. radius,
holding therefore about half a litre ; through this 36 litres of air were passed on an

* On this point, see below, pp. Ill and 142.

MR. W. N. SHAW ON HYGROMETRIC METHODS.

101

average in two hours, giving a minimum motion cf 3 cm. per minute — assuming, that
is, that the motion was uniform over the whole central section of the globe. The
motion at the thimble in the middle of the globe was probably considerably greater
than this, but at any rate could not be called rapid. Under such circumstances the
temperature of the thimble and ether does not rise uniformly, so that one had to keep
the ether mixed by an occasional bubble of air, and the practical temperature limits of
deposit and no deposit, though sometimes two-tenths, were sometimes four-tenths of
a degree centigrade.

§ 18. Let us call the temperature of saturation of the air the theoretical dew-point.
Then, in order that the observed dew-point may coincide with the theoretical, the
following assumptions seem to be required : — (1.) That as soon as the temperature of
the thimble is below the theoretical dew-point a deposit of moisture is formed.
(2.) That the observer can see it. (3.) That there is no deposit when the temperature
is above the theoretical dew-point. These three assumptions may be regarded as
independent, though their consideration must to some extent overlap. I will take
No. 2 first. The possibility of seeing a deposit depends upon circumstances. In the
first place the thimble must be highly polished. In one observation, No. 15, when the
thimble was tarnished, the saturator being at 16°'96 C., and the chemical method corre-
sponding to a temperature of 17° '2 C., two readings of temperature of earliest visible
deposit were 15° C. and 16° '3 C. respectively, either of them much lower than the
theoreticid dew-point, and indicating uncertainty in the readings, which did not occur
when the thimble was well polished. Secondly, the illumination is of importance. On
this point no general direction seems useful ; my observations were made indoors, in a
room with two windows. I found that I could see the deposit most easily sitting
with my back to the windows, and having a reflector or a lamp placed almost behind
the thimble, so that the light grazed the side of it, a uniform dark background being
immediately behind. Small specks on the polished surface seem to facilitate the
reading, the first indication of a deposit being a slight fringe round the specks.
Thirdly, different observers do not take the same readings, although practice may do
away with the differences.

I was assisted in the observations by one of the students at the Cavendish Labora-
tory, who had had no previous experience in dew-point readings ; at first our readings
dirtered considerably, mine being the higher. The following cases show this :—

8»tur»tor.

Dew-point.

ObMTTW.

o

174
13-96

16°92
17-17
13-53
13-68

S. H.
W. N. S.
S. H.
W. N. S.

102

MR W. N. SHAW ON HYGROMETRTC METHODS.

The differences were still larger in the earlier observations, beginning indeed with
a whole degree, but with continued practice they diminished, and after a time became
no longer perceptible. As a rule, however, the observation of the dew-point fell to
my share. It was necessary, in order to prevent unconscious bias towards a known
result, that the observer who read the dew-point temperature should not know what
the saturator thermometer was indicating, and in the division of duty I took the
dew-point readings ; but, as I have said, in the later observations our readings agreed,
and if we exchanged the duties there was no recognisable difference.

§ 1 9. With the utmost anxiety to be quite fair and to observe only what was really
to be seen, it was sometimes difficult to form a satisfactory opinion as to whether a
deposit was on the thimble or not. Here, for instance, are some notes made on dew-
point readings during Experiment 6 (air temperature 15° C.) : —

Time.

Saturator.

Thermometer
readings.

Remarks.

0

o

12.40

9-55

8-7

Very faint deposit.

1.0

9-50

8-6

i! »

. .

, ,

9-7

Deposit gone.

1.15

9-35

8-9

Deposit round specks.

1.25
1.30

9-50

9-0
10-9

Fine deposit specks.
Deposit gone.

10-0

? Deposit.

2.0

9-50

9-45

Dew 1 , .

2.15

9-50

9-70

Dew j ver^

3.0

9-50

7-3

Heavy dew.

Thermometer 1
correction . J

•50

•27--24

In this case the observation was exceptionally difficult in consequence of the slower
rate of aspiration. According to the assumption, I ought to have seen the deposit
when the temperature reading was below 9°'24, and not otherwise. At other times
observations seemed to be easier ; the following is extracted from my note-book for
the next experiment : —

Saturator temperature 12°-6-12°75. Mean 12°'68. Correction 0°'42.

Mean theoretical dew-point 12° '2 6.

Dew-point observations.

Dew on.

Dew off.

12-2

12-4

12-2

12-5

12-1

12-5

12-4

12-6

12.4

12-8

12-5

12-5

MR. W. N. SHAW ON HYP.ROMETR1C METHODS.

103

Mean dew-point 12'42

Correction ..... ... '22

Observed dew-point 12*20

Difference from theoretical . . i '06

The impression that is left on the mind by observations with the instrument is a
sense of insecurity and want of confidence in the observations, that is sometimes
followed by surprise at the concordance of the reading with the theoretical dew-point,
and the closeness with which the indication of the dew-point follows any change in
the temperature of the saturator, as, for instance, in the example just quoted, where
the saturator readings rose from 120>18 C. to 120-33 C., the dew-point readings
followed from 12°'08 C. to 12°'28 C.

§ 20. The other two assumptions referred to in § 18 may be tested by the accuracy
of agreement of the final dew-point readings deduced from the observations with the
theoretical dew-point given by the saturator, and the chemical method. The results
of the comparison for a number of observations are given in the following table ; the
numbers of the experiments are the same as those of Table VI.

TABLE VIII.

Number of
experiment.

Temperature
of air.

Percentage
humidity.

Theoretical
dew-point.

Observed dew-point
by RBQHAULT.

Difference.

60

0

20

32

2-98

•
2-90

o

-•08

49

19'6

33

2-94

2-91

-•03

53

20

33

3-22

2-94

-•28

51

20

34

3-68

363

-•05

82

17

34

1-17

1-08

-•09

54

20

35

3-89

3-76

-•13

76

18-5

45

6-14

6-10

-•04

55

19

46

7-08

6-85

-•23

78

18

52

7-96

7-88

-•08

77

17

53

7-41

7-33

-•08

60

24

63 16-49

16-35

-•14

57

18-5 64

11-74

11-71

-•03

79

18 67

11-73

11-71

-•02

80

18-5 75

12-90

12-91

+ •01

73

24-5

76

20-03

20-19

+ •16

81

18-5

78

14-50

14-62

+ •12

67

25

82

22-02

22-19

+ •17

48.

19-5

86

16-89

16-92

+ •03

74

18

87

15-75

15-72

-•03

75

18 87

15-72

15-72

•00

47

19-14 87

16-85

16-80

-•05

5

13 96

12-38

? 12-38

r"00

104

MR. W. N. SHAW OX HYGROMETEIC MKTHODS.

The theoretical dew-point is taken from the temperature of the saturator, but the
observations included in the table are only those for which the result of the chemical
method agreed very closely with that of the saturator.

It will be seen that the differences are most frequently negative. They are
generally very small, so that it would appear that the assumptions specified on p. 101
are generally justified by the experiments. There are two cases in which the dew-
point is considerably below the saturator temperature, namely, Nos. 53 and 55. I
quote the observations for these two experiments :—

No. 53.

Time.

Salt! ra tor.

Dew-point.

12.34

3-57

3-10

12.49

3-40

3-05

1.10

3-38

3-05

1.40

3-39

3-10

1.55

3-40

3-05

2.10

3-43

3-30

2.25

3-47

3-35

2.40

3-50

3-30

If in this case we take the last three readings only, we get-
Mean temperature of saturator . . 3°'47
Correction '23

Mean dew-point 3°'32

Correction '22

3°-24

3°-10

Difference : '14,

so that a nearer approximation would be obtained. It appears that the last three
observations correspond to no physical change in the air, but simply refer to some
alteration in what was taken as indicating visible dew. This example well illustrates
the uncertainty that rests with dew-point determinations.

MR. W. N. SHAW ON HYGROMETRIC METHODS.

105

No. 55.

Time.

Satunttor.

Dew-point.

3.5

O

7-56

7-15

3.20

7-59

7-05

:;:;:,

7-21

6-9 7-4

7-15

BJO

7-22

6-8 7-2

7-0

4.10

7-27

6-8 7-2

7-0

4.25

731

695 7-3

7-12

4.40

736

71 72

7-15

4,35

7-40

7-30

Mean ....

7-40

7-11

Correction . .

32

•26

7-08

6-85

In this case no reason can be readily assigned for the difference. It must be
attributed to mere errors of observation, but the rapid and unusual variation of the
temperature of the saturator may account for part of the difficulty of obtaining
concordant dew-points.

The other noteworthy cases are those in which the dew-point was higher than the
saturator temperature, Nos. 80, 73, 81, 67, and 48. In the first four of these experi-
ments I was paying special attention to the very faintest indications of the presence
of dew. I had come to the conclusion, from observations with DINES'S hygrometer
conducted in the manner detailed below, that it was possible to get a dew deposit
upon a glass surface at a temperature above the theoretical dew-point, and I was
endeavouring to ascertain if the same occurred with the REGNAULT instrument. It
would appear that to a certain extent that is the case. I give my observations
and notes for Experiment 81 : —

MDCocuutrvm. — A.

106

Ml!. W. N. SFIAW ON HYGROMETBIC METHODS.

No. 81.

Time.

Saturator.

Dew point

Remarks.

5.30

o

1479

14-80 P4'6
0 \15-8

Deposit
Deposit gone

5.45

14-80

,, 7rJ14-6
1470 \14-8

Deposit
Deposit gone

6.0

14-80

14-72

6.15

H-80

14-80

6.27

» •

A dew-point reading taken, cooling very slowly.

Dew seen at 15°"0. Dew-point 15°'02. Obser-

vation repeated : dew at 14°"9

6.30

14-80

14-9

Dew at 14°'9, very definite indication

6.45

14-81

14-85

Dew at 14°-85, very definite indication

Thermometer "1
correction J

0-30

0-18

The dew-point is put down as the temperature of appearance of dew in the last two
readings, as the deposit was extremely transient. It would appear from this that
dew may, with very great care, be seen at a temperature from 0°'16 C. to 0°'32 C. above
the dew-point. The other observations are merely ordinary dew-point determinations,
and may be affected by too high a reading being set down for the temperature of dis-
appearance ; but I had no doubt in the case quoted above. I find also the following
note in ray note-book for Experiment 82, when the saturator read 1°'18 C. (corrected) :
" There is a change in appearance of the silver at 2°'l (unconnected), and the dew
deposit is very slight even when the temperature is considerably below the dew-point.
The dew-point as ' naturally ' determined is given in the table. The plan of reading
deposit both on and off is quite useless, as the dew often disappears when the thermo-
meter shows a temperature below that at which dew was previously deposited." If
the change in appearance at 2°'l indicated a deposit, allowing for thermometer correc-
tions, the deposit would have been seen at 1°'88, when the theoretical dew-point was
1°'18, in other words at a temperature of 0°7 above the saturation temperature.

§ 21. It would appear, therefore, that the third assumption of p. 101, viz., that
there is no deposit unless the temperature of the thimble is below the theoretical
dew-point, is not strictly true if the very first indications of a dew deposit are taken
into account.

With the exceptions thus mentioned, the theoretical and observed dew-points are
in very close agreement. It may be said that at first an observer would probably
obtain too low a reading with REGNAULT'S instrument, and that with a fair amount
of practice his observed dew-points would probably agree on the average with the
temperature of saturation, although there might be exceptional observations with
somewhat widely diverging readings, depending possibly upon the illumination of the

MH. W. N. SHAW ON HYGROMETRIC METHODS. 107

thimble. Finally, if very closely observed, the dew-point would be some tenths of a
degree too high, in consequence of a slight deposit formed above the true dew-point.

DINES'S HYGROMETER.

§ 22. The method of introducing the instrument into the air to be investigated has
been already described, § 8. The presence of the instrument caused at first consider-
able interference with the comparison of the saturator and chemical method. In
order to cool the glass surface upon which the deposit is formed, water, cooled by
dissolving ice in it, is made to pass along a metal tube (fig. 4), which runs close to
the stem of the thermometer t, and ends in the metallic box B, into which the ther-
mometer dips. The small cistern which holds the cooled water is in the ordinary
course screwed on to the leading tube at A. I found, however, that the neighbour-
hood of this cold reservoir caused a deposit of moisture in the tubes of the apparatus
near it, and in consequence spoiled Experiment 71, and the previous Experiments 68,
69, and 70 had been rendered useless by the deposit formed in the long metal tube,
which must, of course, be cooled below the dew-point and cause condensation before
the glass surface can be'cooled. So that the observations of dew-point with DINES'S
hygrometer reduced from these experiments are not of much value. In order to
avoid these defects the flow of water in the DIN EH was reversed, so that it had only
to pass through a very short length of tube before reaching the box, and the cistern
was separated from the rest of the apparatus by a considerable length of india-rubber
tubing. Further, the observations were only taken with DINES'S instrument at the
close of an experiment, after the drying tubes had been detached from the aspirator.
The apparatus was then connected with an aspirating air pump, and air drawn
through the saturator and the vessel containing the hygrometer.

§ 23. The observation of a dew deposit, upon the surface of the blackened glass is
a very easy matter, and much more satisfactory than with the REGNAULT thimble.
The arrangement that I have found to give the easiest observation is to attach a
piece of black paper to the middle of the window of the room, and look at the glass
surface from such a direction that it reflects only the black paper when there is no
dew on the surface. Under these circumstances a deposit shows itself with surprising
facility. The observations taken with DINES'S instrument were as follows :—

After Experiment 73. Saturator reading 200-05

DINES, 68° F. = 20°'00

0°'05 diff.

„ 77. Saturator reading, 7°'49.

DINES first showed dew at 10°'6, but no more at 8°'3 ; on a
second trial dew at 7°'8.
P 2

108 MR. W. N. SHAW ON HrGEOMETRIC METHODS.

After Experiment 78. Saturator reading, 8°-00.

DINES : — Water run through slowly, dew appeared at the
first edge (nearest to the entering water) at 9°'6 ;
water stopped : — temperature gradually sank to 9°'00,
and dew appeared at the other end, the first deposit
vanishing. The second deposit was visible only in the
best light and vanished without any measurable altera-
tion of temperature. Observation repeated : — Dew
found at first edge at 9°'72, but no dew formed on
other parts, although the temperature was reduced
to 9°'06. The temperature was run down to 7°'78
before anything like a wide deposit of dew was formed.
An estimate of 8°'3 was made as the dew-point, and
confirmed by repetition of the observation.
„ 79. Saturator reading, 11°'61.

DINES : — Dew at further end at 15°'61. Thermometer went
down without any more cold water to 13°"6, and the
dew extended partly over the glass. Dew at both
ends at 13°'06; vanished from the one end without
alteration of temperature ; disappeared altogether,
although the thermometer fell to 12°'67.
„ 80. Saturator reading, 13°'83.

DINES : — Observation taken without air being drawn
through, i.e., without circulation ; a deposit was
formed at the near end only at 9°'28.
„ 81. Saturator reading, 14°'51.

DINES : — Temperature reduced to 1 40-00 without drawing
any air through ; on starting the air, dew was deposited
at first on the edge nearest to the air inlet, and it
gradually extended over part of the glass surface
•without alteration of temperature.
„ 82. Saturator reading, 0°'80.

DINES dew-point, 10>22.

With ice only, the temperature of DINES could not, be
reduced below 1°'22, at which temperature the dew
was just visible. Calcium chloride and sodium chloride
were then added to the ice, and the following observa-
tions made. The faintest visible deposit was at 1°'2.
Cooling down to 0°'56 increased the deposit, but not
very greatly. This was repeated many times, and in
one experiment the thermometer was taken down to 0°,

MR. W. N. SHAW OX HYGROMETRIC METHODS.

10U

but the deposit was not then very large, though quite
distinct.

On September 4, the day after Experiment 81, observations were made with DINER'S
instrument, but no observation with the chemical method was made.

The object of the experiment was to watch for the deposits of the dew when the
temperature of DINES was maintained as nearly as possible at constant temperatures.

The following are the observations :—

Experiment 83.

Time.

b. in.

;{ -28

3 30

3 45

3 52

3 55

4 0
4 6
4 20

4 35

4 46

5 7

hfanfar.

15-09
15-01

15-00
14-98
15-00
15-00
15-00
15-01

15-02

Air
temperature.

16-Ot

Thermometer ran slowly down : dew 6rst showed at the near end at

l.Y'-U'i. and gradually extended all over the further side while the

thermometer fell — ultimately to 12°'2; the deposit continued to

increase, but only on the same portion of the surface.

Dry air passed through and deposit removed.
Dry air stopped and damp air sent through. DINES reading 12°'89.

Deposit formed on a portion only of the face.

Deposit removed by dry air.

DINKS, 13°'89 ; damp air turned on and gave a deposit after some time.
DINKS, 14°'44; damp air put on; dew visible iu 1m. 45s. Dry air put

on ; dew gone in 11s.
DINES, 15°'00. First sign of dew visible at 1m. lls. after starting air.

DINKS wab then standing at 15°'22; it rose to 15° 44, the dew still

remaining.
DINKS, 150'56; damp air sent through, a very faint deposit of dew

appeared, which did not disappear until the thermometer had reached

15*72.
DINIS cooled again slowly. First indication of dew at 15°7, very

faint; water stopped, and thermometer fell to 15°'4, and dew slightly

increased. Still on at 15r'6, rather fainter at 16*'l.
The deposit did not completely disappear until the thermometer

reached 16°-94.
The water was then run through still more slowly. A deposit of dew

was visible when the thermometer reading was 16°"2, and the water

was then kept just flowing. The thermometer reading never went

below 16°'l, but the dew deposit was quite distinct.
Damp air turned on, and last observation repeated. Started at 16°7 ;

dew began to form almost immediately, and the flow was stopped,

the thermometer fell to 15°'8, showing increase of dew without

further supply.

To verify that there was a rail deposit, and not an incidental

reflexion, the tap was turned and dry air sent through, the spot on

the glass being carefully watched. The cloud distinctly cleared off.

§ 24. It will be seen from these experiments (Nog. 78 and 79) that the temperature
of the water in the interior of the metal box, of which the glass surface forms the lid,
is not necessarily uniform. If the flow is very slow a narrow current is established
through the box and a deposit forms along it, the other part being left bare (Experi-
ment 81). The facility of reading is much improved by a circulation of the air

110 MR. W. N. SHAW ON HYGROMETRIC -METHODS.

(Experiment 80). Certain of the experiments show that a dew deposit is formed at
a temperature above the theoretical dew-point. Thus in Experiment 82 the deposit
is obtained at 1°'22, the theoretical dew-point being 0°'80 ; and in Experiment 83 a
deposit was obtained at 16°'22, the theoretical dew-point being 15°'02. These read-
ings were obtained many times successively, so that they cannot be assigned to want
of uniformity in the temperature. They show that a visible deposit is obtainable
upon a glass surface when the air is not saturated, and the difference is larger than
in the case of the REGNAULT hygrometer, amounting to a whole degree in Experi-
ment 83.

It, therefore, seems that the result obtained from observations with DINES'S hygro-
meter is likely to give a very easy determination of the dew-point, that is within
small limits of error ; but, that if it is observed with the closest attention, the result
will be considerably too high, in consequence of the premature formation of a dew
deposit, and it may be erroneous in consequence of the variations in temperature of
the different parts of the box containing the thermometer. This latter effect may be
almost entirely avoided by using for forming the deposit water which is itself of a
uniform temperature and very slightly below the dew-point, so that a large quantity
may be required to flow for a small change of temperature, or by the method sug-
gested by Professor CHRYSTAL, and in use by some of the Scottish observers, as by
Mr. H. N. DICKSON (see ' Edinburgh, Roy. Soc. Proc.,' vol. 13, pp. 199 and 951).
But the instrument can hardly be regarded as a standard one.

The deposit upon the glass surface above the dew-point is, I think, well established ;
it was suspected by REGNATJLT (see p. 121), and appears to be less obvious in the case
of silver than of glass. The difference may, however, be due to the greater difficulty
of observing it.

3. — ALLUARD'S INSTRUMENT.

§ 25. A description of this modification of REGNAULT'S instrument is given in
Note A (p. 141). The difference between the two lies entirely in the facility of
reading. ALLUARD encloses his thermometer in a brass box with gilt and polished
sides, and surrounds one side with a brass plate so cut that the side of the box is
surrounded by a surface of brass in the same plane with it. It is easier and less
uncertain in reading than REGNAULT'S instrument. The best way of illuminating it
is similar to that suggested for DINES'S hygrometer, namely, to set up a small patch of
black paper in the window and arrange the reflecting surface of the hygrometer so
that the eye sees the reflection of the black paper. A very faint deposit of dew is at
once visible. I have not been able to arrange the instrument so that I could observe
its behaviour in air of a known state. The instrument is too bulky for me to do that
without apparatus on a much larger scale than mine. I cannot, however, think that
such a course is necessary. The observations cannot differ from those with REGNAULT'S
instrument, except in regard to the facility of seeing deposits. For that reason it is

AfK. W. N. SHAW ON HYGROMETRIC METHODS.

Ill

more suitable for use as a standard instrument, as the feeling of uncertainty, that is
very strong while working with REQNAULT'S, is much less with ALLUAED'S.*

4. — BOGEN'S INSTRUMENT.

§ 26. I need spend but few words over this instrument. It is a dew-point instru-
ment in which the cooling is produced by solution of ammonium nitrate contained in
a silvered glass vessel. The water is squeezed into the glass vessel by means of a
flexible india-rubber ball, and the temperature of deposition is read in the usual way.
It affords a fairly good lecture experiment, but continuous observations are quite
impracticable, as the apparatus has to be washed out as soon as one observation is
taken, and the delay and trouble are too great. There is, besides, no possibility of
graduating the fall of temperature. I have made a few observations with it, but it is
not worth while to recall them.

* On turning back to some results that I obtained in 1883, I find considerable differences in the
readings of different dew-point instruments. I was at that time comparing all the different instruments
that I had, viz., ALLUARD, RKONAITLT, DINES, hair-hygrometer, and a large number of wet and dry bnlbs,
whose indications were reduced by two sets of tables, viz., JELINEK'S and GLAISHEB'R. They were honest
attempts to obtain a comparison of the indications of different instruments, but the results were such
that they showed that a very systematic investigation was required before one could hope to obtain
concordant observations. I quote a table of results here. 1 made the assumption that by driving the
air of the room over all the instruments by means of a rotary fan I should obtain all ttie instruments
exposed to air in the same state. I fear that could not have been correct. I used the chemical method
*• a check, but I have since that time seen means of improvement, and I only offer the table as a
specimen of the difficulties of a hygrometrio observer.

TABLE of Vapour Pressure in Room, as deduced from Observations with various Instruments, 1883.

Date.

Chemical
method.

Au.t AHD.

RWXAI'LT.

D15E8.

Hair.

Wet and dry by

• il.MMII H.

Wet and dry by

JELIXKK

April 10

mm.
4-81

mm.
4-91

mm.
7-34

mm.
5-81

mm.

mm.
6-24

mm.

:. t-

„ 11 ...

4-87

5-49

5-85

, .

5-25

7-39 (P)

7-04 (?)

„ 12 ...

5-4

5-69

5-53

6-9

6-35

6-98

6-49

„ 13 . . ".

574

5-6*5

7-54

5-44

b-93

6-38

May 2 . . . .

5-1

5-10

5-40

5-82

5-14

,,:,7

5-5)6

„ 3 . . . .

6'5

5-53

6-01

6-24

,5-68

645

5-81

„ 5 ....

6-1

5-89

6-23

>•,!.-,

5-31

6-40

5-85

„ 8 ....

7-65

7-57

7-88

8-35

727

8-23

8-14

I had previously attempted to compare all the dew-point instruments together in the manner described
in my preliminary report of May 10, 1881, but after a few observations I -came to the conclusion that any
such attempt was worthless unless a circulation of air was provided, observations of different hygrometers
in a closed vessel not giving comparable results.

112 MR. W. N. SHAW ON HT/GROMETRIC METHODS.

NOTE A.

SUMMARY OF THE RESULTS OF WORK ON HYGROMETRIC INSTRUMENTS AND METHODS
SINCE THE TIME OF PROFESSOR J. F. DANIELL.

The subject of Hygrometry was considered in an essay by DANIELL, a third edition
of which was published in 1845, after the author's death ; it discusses the method of
hygrometric observation founded on the deposition of dew upon an artificially cooled
surface, and describes the method of using and the advantages of the dew-point
hygrometer invented by DANIELL* in 1820. The following paragraphs may be quoted
as bearing on the subject which I have now in hand : —

" It was also an important object to ascertain whether any hygrometric property of
the glass, or difference between it and the metal in attraction of moisture, would have
any appreciable effect upon the condensing power.

" Long experience has, however, convinced me that the metallic hygrometer
possesses no real superiority over the glass one. The visibility of the deposition in
the latter is rendered perfect by making the condensing ball of black glass, and
viewing it by reflected light in the manner of a mirror ; and I never could perceive
any difference in the sensibility of the two instruments."

The essay also discusses tables of pressure of aqueous vapour, &c., which have been
entirely modified by subsequent and more accurate experimental data. Hygrometers
founded upon the variation of organic substances are considered, and evidence to show
the superiority of the dew-point methods is adduced. The chief reasons against
organic hygrometers, as DE SAUSSURE'S and DE Luc's, are quoted from DE SAUSSURE'S
essay. Setting aside the difficulty of preparation of the hairs, it would appear that
DE SAUSSURE did not expect two hair hygrometers to give identical readings, even
when similarly prepared and mounted, and the following instance of a case in which
identical readings would not be obtained was taken by DANIELL from DE SAUSSURE.
If two otherwise identical hygrometers are both exposed for a long time to very dry
air, say at 40° of the hair hygrometer scale, and then one of them is exposed to a
humidity of 30° while the other is in an atmosphere of 50°, when the two are replaced
in the same atmosphere of 40° humidity, neither will return to the 40° indication ;
one will read about 37° or 38°, and the other 42° or 43°.

DANIELL had used DE Luc's hygrometer, and considered it as unsatisfactory as
DE SAUSSURE'S.

He discusses also the wet-and- dry-bulb method (which was introduced by
Dr. HUTTON and modified by Sir JOHN LESLIE) in the form given to it by Dr. MASON.
We need not trouble about previous modifications, as MASON'S form is now practically
adopted.

* ' Quarterly Journal of Science,' 1820.

MB. W. N. SHAW ON HYOROMETBIC METHODS. 113

At this time APJOHN'S formula /" =/' — c//88 X p/30 for calculating the pressure
of vapour from the wet-and-dry-bulb indications was known. UANIELL calls atten-
tion to the following disad vantages attending their use : —

(1.) The smallness of the scale of the instrument, so that the probable errors of
observation bear a very high proportion to the required result, particularly about
the freezing point.

(2.) The liability of the indication to be affected by causes other than the humidity
of the air and its temperature. The suggestion of PELTIER, that it may be affected
by the electrical state of the atmosphere, is referred to as a possible explanation of
the cases of the temperature of the wet bulb being found above that of the dry
" which occur in all long series of observations." (See pp. 129 and 138.)

(3.) The uncertainty in the results computed from the observations ; instances are
given to show that the results deduced by APJOHN'S formula are different from those
obtained from the Greenwich table of factors, derived from experimental comparison
of the wet-and-dry-bulb readings with direct observations of the dew-point.

There is another discussion, in English, of the various questions relating to hygro-
metric measurements, occupying 22 pages of the seventh edition of the ' Encyclopaedia
Britannica,' and published in 1842, from which we extract the following information
about Balance Hygrometers:—

" Hygrometers have frequently been formed by suspending from one arm of a
balance some substance which strongly attracts moisture from the atmosphere, and
nicely counterpoising it by a weight on the other arm. The changes in the humidity
of the air are then meant to be indicated by the changes in the position of the beam,
arising from the gain or loss of weight in the suspended body. A great variety of
substances have been used for this purpose, such as sponge, caustic potash, the
deliquescent salts, sulphuric acid, &c. These, like the former instruments, are all too
late in their indications, though some of them might scarcely be liable to lose their
sensibility were it not that they soon become useless from the accumulation of dust,
soot, &c., especially if in or near a large city."

DE SAUSSURE'S hair and DE Luc's whalebone hygrometers are carefully described,
and a comparison of their scales is given, showing very considerable divergence, and
indicating very peculiar behaviour of the hair hygrometer, as though it had a
maximum near the saturation point. I give half the table of comparison to make
the divergence clear. This is DE Luc's comparison, and was repudiated by
DE SAUSSURE, because DE Luc saturated the hygrometer by actual contact with
water.

MDi '« « I.\\\\ III. — A.

114

MR. W. N. SHAW ON HTGROMETRIC METHODS.

Di Luc.

D« SAUSSUBE.

o

55

88-8

60

91-6

65

93-8

70

95-6

75

97-2

80

98

85

100

90

100

95

99-3

100

98-3

We may perhaps disregard WILSON'S hygrometer, which is a mercury thermometer,
with a rat's bladder bulb. A good deal of the article is taken up with a discussion
of some of the therraodynamic properties of air, not much to the point.

We have next a discussion of dew-point methods, and a description of DANIELL'S
hygrometer, and an exposition of its defects, as well as those of the various sugges-
tions made as improvements on DANIELL'S, of which there are many.

A table of results of comparisons of DANIELL'S and various instruments of this
kind is given, and the mean errors (28 experiments) from the results obtained by
LEROY'S method (cooled water in an open vessel) are given as follows :—

A DIE'S . . ,
DANIELL'S . .
Spherical bulb
Long bulb .

- O'l
+ 2-9

- 478

- 6-6

The author of the article then turns his attention to the wet-and-dry-bulb method,
which he says has been sadly neglected for 30 years. The following extract will
show the application of a formula of reduction to very wide variations in the condi-
tions of observation ; the result is very fairly satisfactory :—

" The results of experiments determining the dew-point for a considerable number
of indications of the wet and dry thermometers, and under various pressures, though
principally at pretty high temperatures, are given in a Calcutta journal, ' Gleanings
in Science,' Nos. 2 and 3, 1829, and in the 'Edinburgh Phil. Journ.' for October,
1883, from which we have obtained the following table. The sixth column is derived

from the formula—

(/, + -66372) (t- f) _ f
1 75-438 ft

where t is the Fahrenheit temperature of the air, t' that of the moist bulb, and t" the
dew-point ; andyj, ft,, ft,, are the forces of aqueous vapour in a state of saturation at
these temperatures respectively.

MR. W. N. SHAW ON HYGROMETRIC METHODS.

115

l';ir< .ii1.' ' »T

Ttmpera-
ture of
the air.

Tempera-
ture of
moift bulb.

Difference
or

(lepreniion

ObMnred
dew-
point

Computed
dew
point

Difference.

Remark*.

2975

30-025
29-35

67°2
56-4
65-0

52*0
49-5
51-5

15°2
6-9
13-5

35°7

:;:•.-,
35-45

357

40-7

:;.;.;

0

0
+ 1-2
+ 1-1

3

^Dr. AKDKBSOK'S experi-
ments

29787

82-0

76-8

5-2

74-0

75-0

+ 1-0

"] Observations made in India,

29-83

81-0

72-1

8-9

68-0

,;-,;

+0-6

I at the level of the sea, by

29-78

81-5

70-9

10-6

66-5

665

0

[ means of LESLIE'S and

29-8

7475

67-3

7-45

63-0

63-C

+ 06

J DANIKLL'S hygrometers

28-739

91-5

69-2 21-3

60-5

60-1

-0-4

]

28-739

91-5

70-7 20-8

62-5

62-7

+ 0-2

28-807

87-5

71-48

16-02

64-0

65-4

+ 1-4

•

24-342

70-25

60-0

10-25

54-0

53-6

-0-4

[ Ditto on hills in the south

22-945

61-75

54-37

7-38

48-0

47-98

-0-02

of India

22-921

63-0

53-46

9-54

46-0

45-3

-0-7

22-917

61-75

46-09 15-66

265

25-0

-1-5

22-909 57-75

47-75 10-0

36-0

35-7 -0-3 J

1 1

" The dew-points in the sixth column do not differ very materially from observa-
tion ; but the temperatures from which they were computed had first to be corrected
for the barometric pressure being different from 30 inches. The precise rule for
estimating such a correction is as yet unknown ; but it appears that, for the same
temperature of the moist bulb, the difference between it and the dry thermometer,
when the pressure amounts to 30 inches, is to their difference under any other pres-
sure, B, nearly in the inverse ratio of 57 to 27 + B. On this supposition, t — t', the
observed depression in the fourth column, before being used in the formula, lias been
multiplied by (27 + B)/57, and the difference between the product and t — t' has like-
wise been applied, with its sign changed, as a correction to t, the temperature of the

air.

Passing by a discussion of the theory of the instrument, which may require modifi-
cation in the light of subsequent progress in the theory of heat, and a graphic method
of reducing observations proposed by Mr. MEIKLE, the other points of interest in the
article are a brief account of DE LA RIVE'S suggestion (since developed by Mr. WILD-
MAN WHITEHOUSE) to apply the rise of temperature produced in a thermometer
moistened with sulphuric acid, to determine the hygrometric state of the air.

"While the sulphuric-acid hygrometer displays considerable ingenuity, the other
instrument, the wet-bulb, is on several accounts so decidedly preferable that the
invention of M. DE LA RIVE is not likely ever to come into general use. Water can
be more readily obtained everywhere, and is much more safe and portable than
sulphuric acid. Besides, owing to sulphuric acid freezing at an uncertain or variable
temperature, depending on its strength, such an instrument would be apt to give
doubtful results at low temperatures. For, whatever be the strength of the acid
at first, it will continue to decrease in nn uncertain manner on the bulb by gradually
absorbing moisture. However, the heat derived from the condensation of the vapour

Q2

116 MR. W. N. SHAW ON HYGEOMETRIC METHODS.

will sometimes be sufficient to keep the acid in a liquid state at a temperature which
would freeze it in a close vessel ; and whenever it happens that sulphuric acid remains
liquid on the bulb of one thermometer, while water is frozen on that of another, a
comparison of the two instruments might throw some light on the influence of frost
on the temperature of the latter. We presume, therefore, that the most important
use likely to be derived from this hygrometer of M. DE LA RIVE would be to assist
in perfecting the theory of the moist-bulb hygrometer ; and possibly some other
absorbent substances might answer even better for this purpose than sulphuric acid
does."

The British Association Report of 1832 includes a report on meteorology by
Professor J. D. FORBES, the chief points of which are recapitulated in the Encyclopaedia
article already cited ; it is, however, mentioned that DE SAUSSURE'S " views of hygro-
metry were, in some respects, so very imperfect that he was not aware of the fact that
the coolness produced by the evaporation of water from porous bodies was independent
of the rate at which the moisture was carried off by currents of air — a want of know-
ledge which gave him much trouble."4 There is in addition a passing reference to the
labours of GAY-LUSSAC and MELLONI upon the scale of DE SAUSSURE'S hygrometer.
These, I think, it is needless to discuss, as the subject has since been taken up by
REGNAULT.

A supplementary report of the same authority appears in the British Association
Report of 1840, in which the advantages of the wet-and-dry-bulb method, as compared
with the dew-point method, are pointed out. A good deal of attention had been paid
to the former method about that time, and from the results of the work FORBES
considered that " we may now consider the moist bulb problem as practically solved."
The solution appears to be as follows : —

Theoretical considerations lead to the assumption of a formula :

e"= e'-m(t-t')b~e'.

Where e" represents the actual pressure of vapour in the air.

e' the saturation pressure at the temperature t'.

t and t' the readings of the dry and wet bulbs respectively.

b the height of the barometer.

B a standard barometric height.

m is a constant to be determined, either by calculation or by auxiliary hygrometric
determinations.

" This formula, employed by AUGUST and BOHNENBERGER, coincides essentially with
that of IVORY ('Phil. Mag.', vol. 60, 1822, p. 81,), who first gave a proper theory of

* 'Brit. Assoc. Report,' 1832, p. 239.

MR. W. N. SHAW ON HYGROMKTRIC METHODS. 117

the moistened-bulh hygrometer. His value for m is not far from the truth, being -j^
for Cent, degrees or -fa for FAIIKKMIKIT."

Ai-.ioiiN ti -t i -i I i lii- v.-iln.- ..r m iii .-i rariei | of ttnuuAuaam '•;> bu wi n MgpimMDti
and by those of others. Assuming DE LA ROCHE and B£RARD'S vulue of the specific
heat of air, he finds m for English inches and for FAHRENHEIT'S degree to be
-£r = '01149 ; a posteriori he has determined it—

(1.) From experiments on the dew-point . . , . . . ..»*.!;.• .i ;£'• j '01151
(2.) „ ,, refrigeration in dry air . . . . V > . . . '01150

(3.) „ „ „ air once saturated, then wanned '01140

BOHNENBERGER'S value of the constant in reduced to inches and FAHRENHEIT
degrees ia m = '0114, which is practically identical with APJOHN'S values.

KAMTzt employs a formula which differs only slightly from that given above, and
from observations on the Fuulhorn obtains a value '0118 for m when reduced to the
same units.

KUPFFER I gives the value in = '01135.

PIUNSEP § has furnished us with a large number of valuable test experiments in a
warm climate. They are already referred to in the table on p. 115. The report
continues ; " When we find that Mr. PRINSEP once more coincides with Dr. APJOHN'S
numbers, only hesitating whether to prefer ^ to gV for the value of m, we are
prepared to admit that this problem is, practically speaking, completely resolved, and
this being the case, it is scarcely worth while to disentangle the various imperfect
steps by which so happy a consummation has been attained, and the hygrometer
rendered as commodious and as accurate as the common thermometer."

These citations and references will perhaps be sufficient to show what were accepted
in England as the well-established facts in hygrometry before REONAULT introduced a
degree of accuracy into thermal experimental measurement that necessitated a revision
of the experimental work done in all branches of science connected therewith. It
would, however, appear that there was some valuable work done on the Continent,
which, if known in England, had not found its way into the summaries of the subject
that I have been able to consult.

There is a very good account of the state of hygrometritfd science in the second
volume of the ' Corso Elementare di Fisica Sperimentale,' by GIUSEPPE BELLI,
published in 1831. The various rough methods (including DE LA RIVE'S sulphuric
acid arrangement) are described, and their weak points are thoughtfully laid bare.
DANIELL'S hygrometer is examined, and the reasons which may make its indications

* ' Brit. Assoc. Report,' 1840, p. 98 (note).

t ' Poggendorff's Annalen,' vol. 30, 1836, p. 43.

J ' Bulletin de 1' Academic des Sciences de St. Petersbourg,' vol. 6, No. 22.

§ ' Journal of the Asiatic Society of Bengal,' 1836.

118

MR. W. N. SHAW ON HYGBOMETBIC METHODS.

untrustworthy are set forth. DOBEREJNER'S modification* is cited as one of many
given in GEHLER'S dictionary. The author then goes on to describe a method of using
a dew-depositing arrangement which consists in reading the temperature of the
boundary of a dew deposit upon a surface whose temperature varies continuously,
instead of reading the temperature of a surface assumed to be at a uniform tempera-
ture at the instant at which a deposit occurs.

It is unnecessary to give a lengthy description ; it will be sufficient to say that the
lower end of a vertical column of mercury in an iron cylindrical tube is maintained by
ice or by a freezing mixture at a very low temperature ; after some time the tempera-
ture of the column becomes steady, gradually increasing in the vertical direction.
Dew is deposited up to a certain height, and the temperature at that height is read by
a thermometer with a small bulb sunk in the mercury to the required level. The
height of the thermometer bulb can be adjusted, and its position defined by two index
points which are on the same level as the bulb. The precautions necessary for
accuracy are described in due course. This instrument was used by BELLI as a
standard hygrometer.

After this discussion he passes on to his evaporation hygrometer, and the wet-and-
dry-bulb. His experiments showed (1) a depression of the wet-bulb reading of about
0° '7 R. as corresponding to a relative motion of the air of three metres per second ;
(2) in still air wet-bulb thermometers with large bulbs take a longer time to arrive at
their final temperature than those with small bulbs, and the final temperature is
higher in the former than in the latter case. I was interested to notice that he
included in his comparison observations a brass ball 80 mm. in diameter, containing
the bulb of a thermometer, as I had made a similar experiment with the same object
before I knew of BELLI'S work. I quote his results. He concludes from them that,
provided there is a sufficiently good air current, the reading is independent of the size
of the bulb.

TABLE I. — Air still.

A.

B.

C.

D.

Temperature of
Boom.

8 mm. (Ham.

Cylindrical.
44 mm. and
7 mm.

16 mm. (limn.

Brass ball.

h. m.
3 45

10-9 B.

775

0

8-0

820

9°'2

3 55

10-9

775

8-0

8-15

8-9

4 5

1075

775 8-0

8-1

8-0

4 15

107

775

8-0

8-05

8-6

4 25

10-67

775

8-0

8-05

8-6

* 'Gilbert's Annalen,' vol. 70, 1822, p. 136.

MR. W. N. SHAW ON' HYOROMKTRir METHODS.

119

TABLE II. — Air agitated by a Cardboard Fun.

Time.

Temperature of
Room.

A. B

C.

D.

Air still ....

10-67 R.

7-75

0

8-0

8-05

o

8-6

After 4 m. agitation

10-85

7-5

7-5

7-7

7-9

. . 6m. more.

10-9

7-48

7-5

7-5

7-65

„ some more .

11-0

7-48

7-6

7-6

7-65

Freshly bathed, and the air then agitated for 6 m.

h. m.

4 55

11-05

7-6

7-7

7-7

8-2

5 0

11-1

7'5

7-7

• •

7-9

5 5

, .

7-55

7-.::,

7-7

7-7

5 10

11-1

7-6

77

7-7

7-8

Left standing without agitation of air for 10 m.

• •

10-9

8-0 8-1

1

8-1

8-5

Some experiments to determine the effect of radiation upon a wet bulb are then
detailed. A wet-bulb thermometer was observed alternately in the sun's rays and in
shadow, in two positions in the same room a short distance apart. The air surrounding
the bulb was kept in motion for each of the positions ; the bulb in the sun gave, a
mean temperature of 20° '66 C., and in the shade 20° '14 C., the temperature of the air
being 25° '25 C. From this observation a calculation is made that the effect of radia-
tion upon a wet bulb under ordinary circumstances would not exceed 0° '2 C.

BELLI therefore considered that a satisfactory observation with a wet bulb that
could be used as a basis of calculation could be obtained, provided that the bulb is
small, not greater than 7 mm. in diameter, and that the air in its neighbourhood is
kept in a state of brisk agitation.

These conditions being satisfied, sets of tables might be formed, each table corre-
sponding to a particular pressure, and showing, for that pressure, the pressure of
uqueous vapour for a given temperature of the air, and a given difference between the
wet and dry bulb.

BELLI then proceeds to calculate a formula for the psychrometer, which comes out
for the pressure 760 mm. :—

e = E' - -592 (T - T').

This agrees remarkably well with AUGUST'S formula, e = E' — 0'0007832(T — T') P,
for the same value of the air pressure.*

Some comparisons of results obtained by this formula are given. They are not very
consistent as they stand., but the tables used in their reduction have been considerably

* See below, p. 124.

120 Ml{. \V. N*. SHAW <)N' HVr.KOMKTHir METHODS.

modified since that time, so that the observations require to be re-calculated before
any opinion can be formed about them.

An apparently complete enumeration of hygrometrical methods is given in the article
"Hygrometer" in GEHLER'S ' Worterbuch' (1829), already alluded to. Upwards of
50 different instruments and modifications are described. A section is devoted to the
theory of the different methods, which it is needless to reproduce. The author prefers
the psychrometer to DANIELL'S hygrometer, and is of opinion that it is impossible to
construct a truly scientific instrument founded on the alteration of dimensions of animal
or vegetable substances.

I pass on now to the consideration of the work bearing upon hygrometric measure-
ments by REGNAULT. That which was directly intended for the elucidation of the
subject is contained in two papers published in the ' Annales de Chimie,' and entitled
" Etudes sur 1'hygrome'trie."*

The first part of the first memoir is devoted to the discussion of the fundamental
data of all hygrometric calculation, viz. :—

(l.) The saturation pressure of aqueous vapour in the air at different atmospheric
temperatures.

(2 ) The specific gravity of aqueous vapour in saturated air referred to dry air, at the
same temperature and pressure.

(3.) The specific gravity of aqueous vapour referred to air, at the same temperature
and pressure, when the fraction of saturation is less than unity.

The discussion of the first question amounts to an investigation of the truth of
DALTON'S hypothesis with regard to saturated air, namely, that the saturation pressure
of water vapour in air is the same as it would be at the same temperature in vacuo ;
so that in any hygrometric calculation the saturation pressure of vapour may be taken
from the table of pressures of water vapour in vacuo for different temperatures.

For the experiments we are now referring to, REGNAULT used a slightly modified
arrangement of his apparatus for determining the vacuum pressures, and obtained the
saturation pressure in air and nitrogen gas. The pressure in air was observed for 34
temperatures lying between the limits 0° C. and 38° C. The results obtained show a
pressure in air less than that given by the table for vacuum, by amounts varying
between '10 mm. and 74 mm., the mean of the differences between the two tables of
observations being '44 mm.

The results obtained with nitrogen gas are very similar, the mean differences for
the first of two sets of observations being '56 mm. The differences, though very
irregular in amount, are considerable, and are always in the same direction, and might,
therefore, be held to show that DALTON'S law is only approximately true. REGNAULT,
however, suggests that they may be due to some constant error which he could not
discover, and later, in the same memoir, and in a subsequent paper on the pressures of
ether and other vapours (' Memoires de 1'Institut,' vol. 26), he adduces reasons in
* 'Annales de Chimie,' vol. 15, 1845, p. 129, and vol. 37, 1853, p. 257.

MK. \V. N SHAW OX HYGIM.MKTKIC MKT1K •!>> 121

favour of that suggestion ; attributing, in fact, the diminished pressure of vapour to
the molecular action of the glass side of the vessel upon the saturated vapour contained
in the air, producing a condensation upon the glass, the slowness of diffusion preventing
the pressure reaching its maximum value by consequent evaporation.*

The second and third questions may likewise be disposed of theoretically, or,
perhaps I should say, hypothetically, provided aqueous vapour may be regarded as
behaving like a perfect gas when its pressure and temperature are made to vary.

In that case the specific gravity of steam referred to hydrogen may be determined
from its molecular weight, and the specific gravity referred to air may then be
calculated from the known specific gravity of hydrogen ; this theoretical value would
of course be constant for all temjwratures and pressures, and equal to 0'622. Several
series of experiments to ascertain if such were the case are detailed in the paper we
are now considering. The first two series of experiments were made on water vapour
in vacuo, and showed that the number quoted, '622, was, within the limits of error of
experiment, the true value of the specific gravity, provided the fraction of saturation
of the vapour experimented on did not exceed 0'8, but the specific gravity is sensibly
greater when the state of saturation is more nearly approached. The third series of
experiments was made by applying the chemical hygrometric method to air artificially
saturated with moisture at a known temperature. A volume of this air was made to
pass through two drying tubes of sulphuric acid and pumice.t by means of an aspirator
whose capacity was accurately determined ; the gain in weight of the drying tubes
gave the quantity of water vapour contained in unit volume, and this could also be
calculated from the known pressure of saturation at the temperature of the saturating
vessel, assuming the theoretical value '622 for the specific gravity of the vapour. A
comparison of the results obtained serves to show whether this assumption of '622 as
the value of the specific gravity is justifiable or not.

The series comprises 68 experiments on air saturated at temperatures varying
between 0° C. and 27° C. The following table shows the mean percentage difference
between the observed and calculated values of the mass of moisture per unit of volume
at the different temperatures : —

' This method of accounting for the discrepancies between hypothesis and experiment in reference to
DALTON'S law has been since confirmed by an experiment of HKKWIO (PoouKN DO BIT'S ' Annalen,' vol. 137,
1869, p. 592) upon compression of vapours. He found that the pressure of the vapour could be increased
beyond the point at which a deposit was first formed on the sides of the vessel, and that the vacuum
saturation pressure was the increased pressure, and not the pressure at which the deposit is first formed.
(See also p. 96.)

t A number of preliminary experiments were made to verify that the sulphuric acid was efficient as an
absorbent.

MDCCVI \\XVIII. — A. R

122

MR. W. N. SHAW ON HYGROMETRIC METHODS.

Temp'Tature of
saturated air.

Number of
experiments.

Percentage
difference.

o°-o

9

•44

14-0

9

•77

7-0

7

•86

20-5

21

•45

24-5

22

•90

The differences thus tabulated lie all in the same direction, the observed weight of
moisture being in each case too small. They may be accounted for in four different
ways : — (1) the pressure of aqueous vapour in the saturated air may be slightly less
than that given by the table of pressures in vacuo ; (2) the density of vapour in
saturated air may be less by about 1 per cent, than the theoretical density ; (3) the
temperature readings of the saturated space may be slightly inaccurate, errors in the
temperature readings of from 0°'06 to 0°'15 C. being sufficient to account for the
differences ; (4) the differences may be due to the incidental errors of the experiments.
The irregularity in the numbers for different temperatures tends rather to show that
either of the first two suggestions is insufficient completely to account for the
observations, and there is evidence that suggestion (2) is directly contradicted by
experiment.* The amount of thermometric error required in each case to account for
the differences is very small, and REGNAULT gives no details as to the method of
correcting the thermometers he employed in the experiments, but his known familiarity
with thermometers of every kind and their errors makes it highly improbable that
he could have overlooked these in the case before us. The precautions adopted in
order to secure the observations against the possible sources of error that are suggested
by the arrangement of the apparatus are not definitely stated.

A table of results .of some experiments to determine by the same method the
specific gravity of saturated steam at temperatures above that of the air of the room
is also given, but their author does not regard the result as satisfactory. t

The second part of the memoir considers the various methods employed to determine
the fraction of saturation of the air. The chemical method is passed over, as being
fully discussed in the previous part of the memoir.

With regard to the behaviour of DE SAUSSURE'S hair hygrometer, REGNAULT set

* For another suggestion as to the cause of the difference based upon my own observations, see § 14
(ante).

f REGNAULT sums up this part of his work in the following words : — " Quoi qu'il en soit, on voit qn'en
prenant pour base ma table des forces 61astiques de la vapeur aqueuse dans le vide, et admettant que la
densite de la vapeur est constamment dgale a 0'622, celle de 1'air dans les memes circonstances etant 1,
le poids de la vapeur d'eau caleule ne pent differer de la quantite reelle que d'une fraction tres petite, un
centieme environ."

MR. W. N. SHAW ON HYGROMETRIC METHODS. 123

himself to determine whether the indications of hair hygrometers are strictly com-
parable under the following different conditions :—

1. When they are constructed of the same kind of hair, prepared at the same

operation.

2. When they are constructed of different kinds of hair, but prepared at the same

operation.

3. When they are constructed of different kinds of hair, and prepared at different

operations by different processes.

A considerable number of hygrometers were compared with each other, their " fixed
points " being determined by means of saturated air, and ajr dried by sulphuric acid.
The results are summed up by REONAULT as follows : —

" Two hygrometers, mounted with similar hair, may be non-comparable solely on
account of their not being stretched by equal weights."

" Hygrometers constructed with hair of the same kind, prepared at the same opera-
tion, do not give identical results, but they do not differ to such an extent that, for
the majority of observations, they may not be regarded as comparable," The maximum
difference between different instruments was about three hygrometric degrees.

" Hygrometers constructed with hairs of different kinds arid prepared in different
ways may show very great differences in their indications, even when they agree at
the fixed points."

It is consequently impracticable to construct a single table for the calculation of
results which is applicable to all instruments, and it is desirable that observers should
have at their disposal a simple process which enables them to make for themselves a
table for their own instrument, and by which they can verify the graduation of the
instrument as often as they wish.

With this object REGNAULT proposes to enclose the hygrometer in a cylindrical
vessel at the bottom of which can be placed successive layers of mixtures of sulphuric
acid and water of different known strengths ; he gives formula and tables for the
pressure of aqueous vapour due to such solutions at different temperatures. The
process of forming a scale for the hygrometer is described. REONAULT proposes to
abandon the attempt to determine the point of the scale corresponding to absolute
dryness, and indeed to use instead the fractional humidity of about '20, as given by a
suitable solution of sulphuric acid. The effect of changes of temperature upon hair
hygrometers was not determined, in consequence of the breaking of the apparatus
intended for that purpose, (See p. 139.)

REGNAULT gives as his final opinion about hair hygrometers, in his second memoir :
" II est & de"sirer que les observateurs renoncent ddfinitivement a un appareil sur le
bon e'tat duquel ils ne peuvent jamais compter."*

* ' Annales de Chimic," vol. 37, 1853, p. 258. (For later information about the hair hygrometer,
see p. 139.)

R 2

124 MR. W. N. SHAW ON HYGROMETRIC METHODS.

After laying down the following objections to DANIELL'S hygrometer, viz. : —

1. The temperature indicated by the thermometer may be different from that of

the surface upon which the deposit is formed.

2. The observer is too close to the instrument.

3. The evaporation of the ether on the bulb may alter the temperature and hygro-

metric state of the air.

4. The ether generally contains some water, which evaporates and changes the

hygrometric state of the neighbouring air still more.

5. In dry weather it is generally difficult, and sometimes impossible, to get a deposit

of dew at all.

REGNAULT describes his own instrument, proposing to produce the evaporation of the
ether in the capsule by causing air to bubble through it by means of an aspirator,
which enables one to adjust the temperature with very great nicety. No comparisons
of results with those of the chemical method are given.

The memoir then describes a large number of experiments upon the pyschrometer,
the object of which was to determine to what extent AUGUST'S formula —

, _ 0-4290 - Q .
J~J 610 -f

is applicable to psychrometer observations.

It is first made clear by direct experiment that when perfectly dry air is made to
pass over a dry bulb and wet bulb successively the temperature of the latter depends
upon the rate at which the air passes ; from this it is inferred that a similar result
will be found with air more or less moist, and this is verified by direct experiment.
Several sets of experiments, however, including a series upon two wet bulbs exposed
in an open court 2 metres from the wall and 7 metres above the ground, one of them
being supplied with a current of air of known velocity by means of a ventilating fan,
and the other left to itself, lead to the following conclusions : — Psychrometer observa-
tions in moving air are practically independent of the size of the bulb. The agitation
of the air certainly influences the formula for the psychrometer, but when the instru-
ment is exposed to free air the same formula can be adopted, so long as the velocity
of the wind does not exceed 5 or 6 metres per second.

REGNAULT suggests the advisability of determining only the form of a reduction
formula theoretically, and then assuming constants, determining their values by actual
comparison with some other method.

His psychrometer observations were all reduced to absolute measure by comparison
with the results of the chemical method, and in consequence of the comparison the
constant of AUGUST'S formula was altered, so that it became —

MR. W. N. SHAW ON HYGROMKTR1C MKTIKMiS 125

0-480 (t-f)
J—J 610 -I7

This represents very satisfactorily the results obtained for fractions of saturation
exceeding '40.

For values of t' below 0° C. the number 610 must be replaced by 689.

Observations by M. IZARN in the Pyrenees are quoted to show the applicability of
the formula at high-level stations.

The second memoir (' Annales de Chimie,' vol. 37, 1853) is mainly devoted to the
discussion of the formula for reduction of psychrometer observations.

AUGUST'S theoretical formula, recast and calculated with the new values of the
constants which REGNAULT was then able to supply (regarding the latent heat of
vaporisation of water at temperatures likely to occur in meteorological observations as
being constant and equal to 600), becomes —

x =/' — 0-000635 (t — t') H ;

but, instead of using this formula for calculating his results, REGNAULT assumes the
form

and determines the value of the constant A for psychrometers exposed in such situa-
tions as are likely to give rise to various conditions, by obtaining simultaneous readings
of the humidity by means of the chemical method.

The mean results of his various series of experiments are given in the following
table :—

126

MR. W. N. SHAW ON HYGROMETB1C METHODS.

Number
of the

-. ii. -

Number
of expe-
riment*
in (he
series.

Situation of the
psychrometer.

Barometric
range
during the
series,
in mm.

Range of dry-
bulb tempera-
ture during
the series.

Range of
fractional
humidity.

Most
suitable
value
of A.

Extreme percent-
age error.

Mean
per-
centage
error.

Positive.

Negative.

1

10

Closed room, 100

755 to 760

C.
21-4 to 23-4

•480 to -688

0-00128

+ 28

-3-3

17

cubic metres.

2

8

Closed room, 1,000
cubic metres.

735 „ 765

8-06 „ 15-25

•589 „ -758

o-ooioo

4-3

1-6

1-5

3

7

Same room as No. 2,
but with two

751 „ 756

13-01 „ 17-49

•452 „ -672

0-00077

1-1

0-9

•4

opposite windows

open

4

41

The large open
court of the Col-

748 „ 764

7-18 „ 29-78

•197 „ -745

0-00074

6-1

6-6

2-7

lege de France,
1,000 square
metres in area

[Protected from
the sun ; 7 metres

above ground, and
2 metres from the

the wall.]

5

15

The long court of
the College de
France, planted
with trees.

748 „ 772

0-85 „ 9 84

•624 „ '984

0-00100

10-5

20

2-8

6

9

Same locality, but
exposed to full
sunshine.

759 „ 762

29-06 „ 30-46

•228 „ -328

0-00090

107

100

7

16

Same locality . .

739 „ 774

-7-63 ,,-0-13

•611 „ -879

0-00075*

11-8

28

8t

16

The large court of
the Taverne inn,

694 „ 703

10-43 „ 26-15

•350 „ -780

0-00090

23

2-6

Eaux Bonnes,

Pyrenees.

18

Pyrenees. . . .
[.Sometimes ex-
posed to sun.]

644 „ 649

9-52 „ 23-67

•417 „ -980

000090

9-5

8-6

The conclusion which REGNAULT draws from these series of observations is that
the psychrometer may be satisfactorily employed, and give the fractions of saturation
to within one-fortieth, provided that it be regarded as an empirical instrument. The
instrument should be protected from the direct action of the wind, and its indications
should be interpreted by means of a formula similar in form to the one given, the
constant A being determined for each locality where observations are to be taken,
and, if necessary, a different value should be taken for different parts of the scale.

A series of experiments to compare the indications of the psychrometer in a closed
room with the results of the chemical or BRUNNER'S hygrometric method was made by
VOGEL (' Abhandlungen der MUnchener Akadernie,' vol. 8, 1860, p. 295). The wet

* For degrees of saturation above '7 the number quoted is apparently too small for ice-covered bulbs.
The number A = G'0013 would have been more suitable for such observations,
t Experiments of M. IZARN.

MR. W. N. SHAW ON HYGROMETRIC METHODS.

127

bulb was read as it stood at rest, and also after it had been swinging pendulum
fashion at the end of a string. The mean of differences observed between the
stationary and vibrating readings was 0°'59. It is difficult to follow the results of the
75 experiments included in the table, in consequence of the unusual units employed.
The results are, however, more widely divergent than REGNAULT'S. REGNAULT'S
constant A is calculated for each observation, and its value varies between '00217 and
•00094.

The errors from the true percentage humidity, as given by the chemical method, lie
between + 2 and -f- 13 for the swinging instrument, their mean value being 7 '4.

The subjoined table for computing the dew-point from psychrometer observations is
extracted from a letter from Lieutenant NOBLE, of Toronto, published in the ' Pro-
ceedings of the Royal Society,' vol. 7, 1855, p. 528. It was obtained by means of
simultaneous observations with REGNAULT'S hygrometer. The factor,/, of the second
column, corresponding to the Greenwich factor,* is that by which the difference of
tempemtures of the wet and dry bulb must be multiplied in order to give the difference
between the air temperature and the dew-point, the formula being T = t — f(t — t').

TABLE of Factors by NOBLE and CAMPBELL.

Temperature of air
(0.

Factor
(/)•

Number of
obwnraiiou*.

Probable error e f
a tingle datum.

F.
48 to 51

231

21

•30

46 ,. 47

2-38

13

•26

42 „ 46

253

41

•40

40 „ 41

2-63

17

•41

88 „ 39

2-83

25

•48

34 „ 37

3-02

64

•43

32 „ 33

333

25

•63

30 „ 31

3-81

22

•61

28 „ 29

4-40

27

•66

24 „ 27

5-46

43

•82

22 „ 23

6-06

15

1-20

20 „ 21

6-93

6

1-40

18 „ 19

7-13

21

1-44

16 „ 17

7-60

20

1-76

14 „ 15

8-97

17

172

12 „ 13

10-30

20

•_'.-,::

10 „ 11

1150

11

2-19

8 , 9

13-06

8

4-63

6 , 7

15-20

7

3-66

0 , 5

1623

14

1-87

- 1 , - 4

19-37

10

411

- 5 , -10

21-64

6

465

- 11 , - 16

37-83

6

10-96

This method of reducing results is quite different from those already mentioned,
and is therefore only comparable with them by taking some actual observations and

• See Note B., p. 147.

128 MB. W. N. SHAW OX HYGBOMETBIC METHODS.

applying each method. Nothing is said in the letter as to the manner of exposure of
the wet bulb, except, parenthetically, that the thermometers were protected from the
full force of the wind. It would appear, therefore, that the table should correspond
to REGNAULT'S value of A = '00074.

NOBLE points out that the factors for temperatures below the freezing point do not
coincide with those deduced from the Greenwich observations. Without assigning
any cause for the difference, he mentions two circumstances which may bear upon the
question : —

1. If the air be a little above and has been below 32° F. there will frequently be
found a small button of ice at the foot of the wet-bulb thermometer which is not
easily perceived, and which will keep it at 32° F. when the temperature of evaporation
is really above that point.

2. The water may be cooled below 32° F. without freezing.

The hygrometric methods have also been discussed by KAMTZ, in a paper published
in 'Kamtz, Repertorium/ vol. 2, pp. 341 to 361. I have (December 20, 1887) only
lately seen the original paper ; the following notes are taken from an abstract given
in the ' Fortschritte der Physik' for 1861.

Comparing REGNAULT'S hygrometer in the original form with one consisting of
glass test-tubes silvered inside, and with DANIELL'S, and testing the results by the
chemical method, KAMTZ prefers the test-tube form of REGNAULT, and considers that
the difference of radiation and the difference in the surface may account for different
temperatures at which dew is deposited.

Concerning the psychrometer, we have a discussion of a large number of observa-
tions by this method, including the Greenwich observations and some of REGNAULT'S,
as well as his own.

Taking the general theoretical psychrometric formula —

=

(where e is the required pressure of vapour, e^ the pressure at the temperature of the
wet-bulb, T the psychrometric difference, h the barometric height, and a and ft are
constants), instead of adopting REGNAULT'S approximate formula—

KAMTZ expands the general form in powers of T : thus —

c = e1+otTP1 + ^T+yT2+8 — j

where d = 745 — h.

The various series of observations were then separately employed to determine the
constants of this formula, with the following results : —

Mil. \V. N. SHAW ON HYGBOMKTRIC METHODS. 129

A. RBONAULT'S observations (with a large psychrometer) —

= el- 0-6698 r + 0'015253 re, - 0'00522 r + 0'58299 r ' ~ - .

B. REGNAULT'S observations (small psychrometer) —

e = el — 1-24442 T + 0'05881 T e, + 0'009006 r + 0'58299 r ^^ .

C. Greenwich observations —

e = el — 0-27842 T — 0'02309 -r e, + 0'002437 r + 0'58299 r-~

745

D. KAMTZ'S observations. Psychrometer and DANIELL'S hygrometer comparison

e = el- 0-64569 T - 0'003861 re, + 0'012397 r + 073932 T 74' h .

745

E. KAMTZ. Chemical method and psychrometer —

e = el — 0-50051 T - 0'021641 T^ + 0-015548T2 + Q-58299T74,! 7 * .

74o

F. Condensation experiments—

e = el- 0-67894 T - 0'0045928 T el + 0'010690 T* + 0'58299 T

G. All the observations together—

e — el — 0-57515 (t — tj — 0'005989 (t — «,) Cl
+ 0-002664 (t - trf -f 0-58899 (t —tj

Apart from the concordance of the bai-ometric correction, there is nothiug in these
formulae that seems to indicate any advantage in introducing the additional constants,
nor does the reduction of observations, specimens of which are given, by means of
them, make them any more reassuring ; the differences in the pressure calculated by
the different formulae amount frequently to 20 per cent, of the whole, and KAMTZ
assigns one-sixth as the extreme error probable in using the general formulae, G. He
points out that the size of the bulb has some influence on the reading, and considers
that comparable results are only to be obtained by making comparisons between the
separate instruments and the condensation or the chemical method.

To another paper by KAMTZ on the psychrometer below the freezing point, in
KAMTZ'S ' Repertoriura,' vol. 3, I have not been able to obtain access,* nor yet to

* Since the above was written I have seen the paper. The chief points of it are referred to in the
extracts quoted from PKRNTER'S memoir (p. 137). KAMTZ proposes to add 0°'5 C. to the obeerved
psvehrouiutric difference for ice-covered bulbs.

MDOCCLXXXVIII. — A.

130

MR. W. N. SHAW ON HYGROMETRIC METHODS.

" Observations Anormales des Psychrom^tres" (' Bulletin de la Socidte Vaudoise,'
vol. 9, p. 234) nor the " Rapport de la Commission Hygrome"trique" (' Actes de la Soc.
Helvetique des Sciences Naturelles,' 1866).

CHISTONI, at the request of the Italian Meteorological Office, undertook a compa-
rison of the ordinary psychrometer and a ventilation psychrometer with a standard
hygrometer. The ventilation psychrometer is an instrument provided with two
centrifugal fans on the same axle driven by clockwork. Each fan leads to a separate
vertical tube, and the bulbs of the two thermometers are placed in the mouths of
these tubes, so that air passes over them at a rate which is sensibly constant, and the
same at each observation. The experiments were carried out at Ostiano. REGNAULT'S
hygrometer was chosen as the standard instrument with which to compare the psychro-
raeters, for the following reasons : — (l) The chemical method was too cumbersome for
use in the country ; (2) BELLI'S standard hygrometer might be supposed to affect the
air in its neighbourhood by its large cold mass. The air was drawn through
REGNAULT'S hygrometer by means of a very ingenious aspirator formed by allowing
sand to fall from a cloth funnel down a long vertical tube, provided simply with
a side opening connected with the hygrometer.

The observations were made in a school-room with three windows, with two psychro-
meters, each used in turn as a ventilated and non-ventilated instrument. They were
reduced by the ' Tavole ad uso della Meteorologia.'*

A summary of the results is given in the following table : —

Exlrcme

Instrument.

Size of
cylindrical bulb.

No. of
obser-
vations.

Range of
barometer.

Range of
dry bulb.

Range of
humidity.

error
from

REGNAULT,

Absolute
mean error
per cent.

per cent.

No. 1. Ventilated .

6 mm. x 25 mm.

46

750-766

12° to 30°

30 to 70

(Cir.) 10

(Cir.) 2-6

No. 2. Ventilated .

6 „ > 69 „.

51

?J

»*

»»

„ 7

„ 2-8

No. 1. Unventilated

6 „ x 25 „

51

)l

n

)»

„ 15

,, 6-6

No. 2. Unventilated

6 „ x69 „

40

)>

ji

))

,, 14

., 5'8

The constant A of REGNAULT'S formula was calculated for each of the ventilator
observations ; its value varies between 0'00072 and 0'00098 for No. 1, and between
G'00069 and O'OOllO for No. 2. These results are strong evidence in favour of the
ventilation method.

The wet bulbs were covered with muslin and moistened before each observation, so
that the results are not necessarily applicable to cases in which the wet bulb is kept
continually moist by means of a wick or other capillary arrangement.

CHISTONI extended his observations to temperatures of the dry bulb below 15° C. in
the winter months of 1877- 8 at Pavia. The summary is as follows :—

* See below, p. 148.

MK. \V. N. SHAW OX HYGHOMKTRIU MK I HMD-

1

Extreme

Instrument.

Sixe of bulb
(cylindrical).

No. of

ol»er-
ratioQi.

Range of
barometer.

Range of
dry bulb.

Ranjie of
humidity.

error
from

RlGMAL'LT

AbMlute
mean error
per cent.

(per cent.).

No. 1. Ventilated .

6 mm. x 25 mm.

69

739-765

-1-2 to 19-6

44 to 92

(Cir.) 15

(Cir.) 3-3

No. 2. Ventilated .

6 „ x 69 „

69

tt

- 1-0 „ 18-9

„ 13

„ 2-6

No. 1. Unventilatcd

6 „ X 25 „

70

M

- 37 „ 19 4

H

,, 22

„ 5-0

No. 2. Unventili.t.-d

6 „ X 69 „

70

«

- 3-6 „ 19 4

n

„ 18

„ 37

The values of the constant A of REONAULT'S formula lay between 0*00062 and
0-00233 for No. 1, and between O'OOOGO and 0'00199 for No. 2.

CHISTONI has continued his work by making a series of comparisons between the
ventilated psychrometer and REGNAULT'S hygrometer readings for temperatures below
15° C. The observations published in the ' Annali della Meteorologia' for 1879 comprise
191 comparisons for temperatures mostly below 15CC., when the wet bulb stood above
0°C. ; 32 comparisons when the psychrometer was in the so-called critical state, i.e.,
when the dry bulb stood above 0° C. and the wet bulb below that temperature, and
32 comparisons when the dry and wet bulbs were both below zero. The observations
were made at Collio, where the mean barometric pressure for the six months over
which the observations lasted was 680 mm.

The following table is taken from CHISTONI'S paper : —

Maximum Maximum

Temperature

actual iiotitlre actual negative

Mean

interval.

error from error from

absolute error.

HlONAl'LT

RCUIAOLT.

t< ° t' < o

19

29

9-8

t > o t' < o

20

51

131

t > 0 t' > °

21

29

6-4

The errors given are the actual errors in the computed humidity. I have not
calculated the percentage errors ; a rough idea of them may be got by taking the
humidity at 50, when the percentage errors would require each number given in the
columns of the table to be multiplied by 2.

These errors are very serious ; they are only briefly discussed in the paper ; the
generalisation is made, namely, that the errors are positive and small when the
humidity is great, and they are negative and larger when the humidity is small. The
psychrometer observations were reduced, as before, by means of HAEQHENS'S ' Tavole
ad uso della Meteorologia.1

CHISTONI has also published* a paper discussing the various psychrometrical

* ' Memorie e Notizie della Meteorologia Italiana,' 1878.

132 M|{. W. N. SHAW OX HTGROMBTBIC METHODS.

formulae and their application. He first explains that the assumptions made by
AUGUST and others in order to obtain a theoretical formula cannot practically be
justified, and he then deduces the formula originally given by BELLI* and supplies it
with the more accurate values of the constants obtained since BELLI'S time, and
reduces it by omitting negligible quantities to the form : —

. fl _ (0-237 B - l-79)(< - Q
377-2 - 0-490 t'

Following REGNAULT'S suggestion that the theoretical formulae should be regarded
only as general forms, he has the three following : —

REGNAULT'S form, /=/' — A (< — f)B . . .... (I.

BELLI'S form, /=/ - (~-+ '-*><•' ~~f) (2.)

AUGUST'S form, f = f - * ('~_f- • . • • .T (3.)

+ n(t-f)'V (4)

n_
to these he adds as a suggestion : —

where A, m, n, p are constants to be determined by experiment.

He then applies these four formulae to the computation of the pressure of aqueous
vapour from psychrometric observations, determining the constants by the method of
least squares, using the results given by REGNAULT'S hygrometer as standards. He
then tabulates the errors ; if we denote by A/*j A/2 . . . the errors from REGNAULT of
the formulae (1) (2) ... we get the following values for the mean absolute error, i.e.,
the mean of the errors without regard to sign :—

A/! = 0'33 mm.
A/2 = 0'32 „
A/3 = 0-29 „

He concludes that his own suggestion most nearly represents the actual case, but
at the same time he remarks that the psychrometric method is, so far as he can see,
not capable of giving the pressure of aqueous vapour to within O'l mm.

Certain practical points with reference to psychrometer observations are discussed
in a paper by CANTONI.* By a series of observations upon the temperature of the air
as recorded by thermometers exposed freely to the sun, or protected by various screens
or metallic tubes, he shows that the best results are obtained without screens, provided
that a current of air is made to pass over the bulbs, and a similar advantage results

* See PKRNTEB'S paper, quoted below.

MH. \V. N. SHAW OX HYOKo.M KTKK ' MKTIIODS.

in the case of a thermometer which has its bulb covered with a thin layer of any
material. From this he is led to test the effect of ventilation upon the psychrometer,
and the observations given show a considerable depression of the temperature of the
wet bulb in consequence of the current of air produced by the double-fan arrangement
by the Tecnomasio of Milan. BELLI originally suggested the reading of wet-bulb
thermometers contained in a metal tube through which air was driven, and the
suggestion was practically carried out in an instrument designed by BUZZETTI, but
the new arrangement of the Tecnomasio gave a much stronger current than the
original apparatus of BUZZETTI, and from CANTONI'S observations it would appear that
the temperature of the wet bulb might be depressed often more than a whole degree
by the stronger current of air.

A further contribution to the discussion of psychrometric observations lias been
made by ANGOT,* whose report is printed in full in the ' Annales du Bureau Central
Me'te'orologique,' 1880. Starting from REGNAULT'S general form—

and writing it

x=f — A.(t — t')h,

A-
-

he determines the value A for each of 3670 comparisons between psychrometer
and dew-point observations made at Paris (height 40 metres), at the observatory of
the Puy de D6me (height 390 metres), or at the station on the summit of the Puy de
D6me (height 1470 metres). Of these, 282 observations gave the temperature of the
wet bulb below 0° C. The range of the wet bulb temperatures was from — 20°'5 to
+ 23°'6 C., and the difference of the wet and dry bulb readings reached 16° C. The
dew-point instrument was ALLUARD'S modification of REGNAULT'S arrangement. t

Having thus obtained a series of values for A, ANGOT found the mean of the values
for temperatures of the wet bulb lying between 0° C. and 1° C., between 1° C. and
2° C., and so on for each station.

The means in every case showed that the value of A depended on the value of
/ — t'. The following is cited an an example :—

Puy de Doma (Plain).
t-f

A.

t - 6°-42.

0-35

0-001022

1-43

0-000948

241

0-000821

3-40

0-000818

4-40

0-000792

6-77

0-000705

• 'Journal de Physique,' vol. 1, 1882, p. 119.
t See below, p. 141.

134 MR \V. N. SHAW ON HYGROMETRIC MP]THOUS.

He therefore assumed that A was a function of t — t', and, writing—

A = a + b (t — t'),

determined the values of a and 6 for each series by CAUCHY'S method, giving to each
equation a weight equal to the number of observations from which the values A and
t — t' were obtained.

The 36 values of b thus obtained presented no regular variation, whether arranged
with reference to t — t', t', or h.

He therefore supposed b to be constant and found it equal to — 0'000028.

The values of a increased when t', and consequently f, increased, and when h
diminished and could be represented by the equation —

a= 0-000776 +

n

The psychrometric formula therefore became —

x = /' [1 - 0-0159 (t - t')~\ - 0-000776 h (t - t') [l - 0'0361 (t - t')]
for temperatures above 0° C., and

«=«'[! — 0-0159 (t — t')] — 0-000682 h (t — t') [1 — 0'0411 (t - t')]

for temperatures below 0° C.

From these formulae tables have been constructed ; these are not given in the
abstract in the ' Journal de Physique,' but a diagram which serves the purpose is
appended at the end of the paper.

The results with these tables seem to be very satisfactory, as compared with those
given by HAEGHENS'S tables, the mean percentage error of 90 observations at Paris
being -f- 1*1 by the latter and only — 0'5 by the former; while for 91 observations of
CHISTONI'S the mean error by the old tables is 2'5, and by ANGOT'S only + 0'5 ; and
further, applying the new tables to REGNAULT'S observations below the freezing point,
the mean error is only + 0'7, as compared with +3*5 obtained when REGNAULT'S
formula is employed. On the whole ANGOT states that the error of the computation
from a single observation above zero is about i 2 units in the relative humidity, and
the absolute mean error of a series of some 20 to 30 observations 0'5 ; for tempera-
tures below zero it would probably be necessary to double these figures, but at any
rate even for very hot and dry regions the tables never give negative values for the
humidity, which is a possibility with the old tables. No particulars are given as to
the mounting and moistening of the thermometers.

An account of an interesting series of observations with the psychrometer in very
dry atmosphere is given by H. F. BLANFORD in the ' Journal of the Asiatic Society
of Bengal,' vol. 45, 1876. The observations were taken in various stations in India,

MR. W. N. SHAW ON HYGROMETRIC METHODS.

LSI

and the dew-point was computed from them by APJOHN*S formula,* by GLAISHER'S
factors, and by AUGUST'S theoretical formulat supplied with RKGNAULT'S values of the
constants occurring therein and adapted to English units. The thermometers were
either a pair with small pea bulbs or a pair of CASSELLA'S Kew pattern, with small
spherical bulbs. Particulars are given with reference to the verification of their indi-
cations.

The wet bulbs were exposed either (I) in a thermometer shed, i.e., " in a frame with
one or two cross bars (generally protected by wire netting at back and front) under a
thatched shed open on all sides to the wind," or (2) by the " sling " method. For the
Upper India series the bulbs were covered with a single thickness of old thin calico.
A water bottle was placed two inches to the side of the bulb with the water level a
quarter-inch below it, and communication was made by a well-washed lamp-wick of
some dozen threads of coarse yarn.

The dew-point instrument was CASSELLA'S form of REGNAULT.

Assuming the direct dew-point determinations to be correct, the following are the
errors shown by the several computations :—

No. of

Mean

M.I.TI

Dew-

Errors.

Place.

Series.

obser-
vations.

air
temp.

wet
bulb.

point
below
air.

Baro-
meter.

Conditions.

Aco.

APJ.

GLAIBH.

Secunderebad .

1

8

n

92-4

67°-9

44:

1

+ 2-6

+ 6-1

+ 6-4

"I

w •

n •

9
8

1
8

93-4
94-8

70-1
708

41-0
440

S 28-07

+ 8-3

+ 4-9

+ 5*
+ 8-0

+ 84
+ 5-6

> In thermometer shed

I» *

4

12

92-7

89-6

419 J

+ 3-9

+ 7X)

+ 4-6

J

Bellary . . .

5

7

94-0

697

407

"

-0-4

+ 2-8

+ 0-9

1

1

2

l>47

68-9

187-0

Oft. I

1-7-0

1-83

1-47

I

*» • • •

7

8

94-7

684

487

• _ * t

-20

+ 1-9

+ 0-9

f "

If * * '

8

9

958

666

483

•-4*

+ 0*

+ 1-8

J

Coimbatoor . .

9

8

968

689

481

+ 0-5

+ 4-6

+ 8-8

1

ft ' *

10

1

96-4

68-5

496

28 39

+ 1*

+ 6-1

+ 65

\ .,

tf

11

10

95-8

699

IN

+ 0-3

+ 88

+ 1-9

1

Trichinopoly .

12

6

898

77-2

185

29-35

+ 0-7

+ 17

-1-6

In verandah. Sling

thermometer

Madras . . .

13

5

97-0

74-1

811

29-72

-0-3

+ 23

-05

In verandah. Sling

thermometer

Calcutta . . .

14

9

842

794

130

2960

-0-4

-0-1

-1*

}In sitting-room.

15

12

838

79-7

68

2955

+ 06

+ 08

-OS

(•line thermometer

Allahabad .' '.

18
17

1
5

77-4
873

612

• :.••

S9-3
41-4

•2954
29-45

+ 02
+ 1-8

+ 3i>
+ 6-4

+ 2-9
+ 5-5

V In verandah.

Agra". . . 18

7

881

60-8

45-1 29-85

-6-1

-0-2

+ 87

»i

Lahore . . \ 19

8

81-9

62-3

429 2908

+ 4-6

+ 8-8

+ 8-7

n ...
„ •

20
21

8

7

730
74-9

556
56-9

3J 4
87-6

2918
2918

-2*

+ 0-8

+ 1-4
+ 4-6

+ 3^
+ 67

V In thermometer shed

Mean . .

••

••

••

••

••

••

+ 0-46

+ 870

+ 3-00

83 ' 30'

t««/-

32)

»,/ -

'fc. 0'>:

136 MR. W. N. SHAW ON HYGROMETRIC METHODS.

The results show that AUGUST'S formula gives the dew-point from psychrometer
observations with considerable accuracy, even when the dew-point is very far below
the temperature of the air. The further conclusion to be drawn from the observa-
tions is already made clear, namely, the necessity for the exposure of the wet-bulb to
a sufficient current of air. The readings are too high by all the computations when
the exposure was in the verandah or room.

BLANFORD does not regard his dew-point observations as unexceptionable, and calls
attention, among other practical points in connexion with this observation, to the
necessity for a highly burnished surface for the thimble and freedom from microscopic
scratches, as these make it very difficult to seize the moment of definition when the
humidity is very small and amount of deposit consequently light.

The attempts at deducing theoretically a formula for the computation of psychro-
metric observations, to which allusion has hitherto been made, have been based upon
the assumption that there was a continuous replacement of a layer of air, no matter
how thin, surrounding the wet bulb. This layer of air was supposed to be reduced to
the temperature of the wet bulb, and completely saturated with moisture. The
effect of the varying rapidity of motion of the air was not taken into account, and
the effect of radiation was regarded as insensible. An equation is obtained by
AUGUST, APJOHN, and others, between the amount of heat supplied to the surrounding
air and the amount lost by the evaporating liquid. BELLI, on the other hand,
equates the mass of air reduced, per unit time, from the temperature, t, of the dry to
the temperature, t', of the wet bulb, to the mass of air saturated by evaporation per
unit time. This slight divergence between the methods of obtaining the equation
leads to no difference in the results, because the theory evidently represents ultimately
the same physical state of things. The matter is, however, treated from a different
point of view in MAXWELL'S article on " Diffusion " in the ' Encycl. Britann.' (9th
edit.). In the discussion of the physical problem, as there treated, he considers a
" steady " distribution of moistui-e and temperature round the wet bulb, supposed to
be maintained in a perfectly still atmosphere of indefinite extent, the distribution of
heat and moisture being brought about by conduction, radiation, and diffusion.
MAXWELL calls this the conduction and diffusion theory in calm air, in contradistinc-
tion to the convection theory referred to above. The mathematical solution is
identical in form with that of the corresponding electrical problem to find the distri-
bution of potential, due to a charged body in a field, and consequently, with the
correction for radiation, leads to the following equation : —

PS [K All

^^-^{D + I^S

where

PS is the vapour pressure in the air undisturbed by the presence of the

wet thermometer bulb.
pl is the vapour pressure at the surface of the bulb.

MB. W. N. SHAW ON HYGROMKTRIC METHODS. 137

P the whole pressure of the air.

S is the specific heat of the air.

L is the latent heat of the vajKnir at the temperature 6.

<r is the specific gravity of the aqueous vapour referred to air.

K is the coefficient of thermal conductivity of the air.

D the coefficient of the diffusion of the vapour.

R is the coefficient of radiation per unit of surface of the bulb.

A is the area of the bulb.

C is identical with the electrostatic capacity of the bulb, and is there-
fore equal to the radius for a spherical bulb.

p is the mass per unit of volume of air.

#0 the temperature of the air undisturbed by the wet bulb, i.e., the
dry- bulb temperature.

Ql is the wet-bulb temperature.

This equation, which has also been obtained by STEFAN, for a spherical bulb, differs
from the convection formula, which is p0 = pl — PS (00 — 0^/Lcr, only by the factor
in the last term. The factor K/D is said, by MAXWELL, to be less than unity, and
probably about 77. STEFAN, in the ' Zeitschrift der Oesterreich. Gesell. fiir Meteoro-
logie,' vol. 16, 1881, assigns the value unity to K/D. He, moreover, deduces the
special case of the formula for a rotational ellipsoidal bulb.

It appears from the formula that for spherical bulbs the effect of radiation is
proportional to the diameter of the bulb. Further, if we take account of convection
currents, their effect would practically be to increase R and D, keeping their ratio
constant, so that the relative importance of the radiation term would be diminished
by the effect of a current of air.

The application of this formula is further considered by PERNTER,* who points out
that the usual assumption that the effect of radiation is negligible leads to serious
error for bulbs of considerable magnitude. Substituting in the formula, he finds that
for still air the radiation effect is equal to the conduction and diffusion effect if the
radius of the bulb (supposed spherical) is equal to 0*57 cm. With a convective
current the effect of radiation would diminish, so that for a current of given velocity
the formula might be written —

where A is a constant and probably equal to O'OOOGSO, a is constant for a given
velocity of the air, and for a given pressure. Its value, as deduced from a considera-
tion of observations by BLANFORD, ANQOT, CHISTONI, and the author of the paper, is
3'0 P/7GO. Moreover, PERNTER considers a certain further correction in the following

* ' Sitzungsberichte der Wiener Akademie,' vol. 87, 1883.

Ml MVi-I.X XXVIII. —A. T

138 MR. W. N. SHAW ON HYGROMET1UC METHODS.

paragraphs, which I have translated directly from his memoir : — " A further influence,
of which no account was taken in the formula, lies in the muslin covering of the wet
thermometer. In establishing the formula no regard was had to any such effect.
KAMTZ has, however, called attention to the frequent case, which he had occasion to
observe, that the moist thermometer stood sometimes as high, sometimes higher, than
the dry, when a comparison with the hygrometer did not show a saturation of the air
with water vapour. He had investigated the cause of this phenomenon, and had
found it in the muslin covering of the wet thermometer. According to his experi-
ments at low temperatures, the wet thermometer stood 0°'46 C. too high.

" That this should be an effect of radiation, as is no doubt the case with dry
muslin, would require as a consequence that the radiation effect should increase with
increase of the difference (t — t'). The fact is, however, that this effect in propor-
tionate comparisons diminishes (mehr verschwindei) as t — t' becomes greater, at any
rate in moving air. This behaviour will require further searching investigation. I
will, however, at present mention that I am of opinion that this phenomenon is to be
ascribed to an ' inertia ' of the wet thermometer when near the saturation point of
air, so that in very moist air the evaporation does not take place rapidly enough in
relation to the in-rushing air. This is confirmed also by experiment. If account is
to be taken of this ' inertia' of the psychrometer in the formula, a correction can be
so applied to the psychrometer difference (t — t'), that it is a maximum when
t — t' = 0, and is chosen inversely proportional to the difference (t — t'). If v be the
maximum value, this correction would be = v/(t — t' + 1) ', if we, with KAMTZ, take
v = 0°'5 C., this correction will only be of appreciable value up to differences of 9°,
practically only up to those of 6°."

I have translated this directly, as the point is an important one, and this is the first
explanation of it that I have seen attempted, but I cannot say that I fully understand
the method of introducing PERNTEK'S correction. The quantity v is said to be the
maximum value of the correction, which I should take to be the temperature excess
of the wet bulb over the dry when the air is actually saturated, but this would
evidently not be the case when t — t' = 0. The complete formula, as thus corrected,
stands thus —

= -Px 0-000630

There is evidently something wrong with this, as the condition for saturation, viz ,
Po — Pn leads to the imaginary difference —

t-t'=--5{l ±v/-l}.

No comparisons of dew-points computed by the formula with actual observations
are given, although the observations of BLANFORD, CHISTONI, and others are discussed
in the course of the paper, and the values of the constants deduced from them are given.

MR. W. N. SHAW ON HTGROMKTRIC METHODS. 130

PERNTER has also made some comparisons between the results of observations with
WILD'S ventilated-psychrometer and REONAULT'S hygrometer on the Obir (2048 metres
high) ; these observations are 30 in number, the dry-bulb temperature lying between
60<2 C. and 14°'4 C., the temperature difference between 0°'6 and 3°'6, and the
barometer between 591 mm. and 600 mm., so that they all refer to states of great
humidity. The values of REGNAULT'B constant A are computed for each observation ;
they vary between '0008432 and '0014946.

In spite of the recommendation of REGNAULT and others, the hair hygrometer has
not been entirely abandoned. In 1852 KREIL used a hair hygrometer as a hygro-
graph and found a difference of ± 4 per cent, between its indications and those of a
psychrometer, but he laid some part of the errors at the door of the psychrometer. In
1858 PICHOT suggested the graduation of a hair hygrometer by using always the same
air, and producing changes in the humidity by changing the pressure or temperature.
An improved form of hair hygrometer was introduced by HERMANN and PFISTER*
in 1870, and during the same year R. WOLF compared the indications of two such
instruments with those of a psychrometer ; the results are in favour of the hair hygro-
meter, especially below the freezing point, inasmuch as the two hair hygrometers are
nearer together than either and the psychrometer, but no comparison with a standard
method was made, so that the evidence of superiority is not quite complete. In 1878
Dr. KABL KOPPE of Zurich published a small bookt in which the whole question of
hygrometric measurement is briefly discussed and the advantages of a hair hygro-
meter with provision for adjustment are more elaborately described. The adjustment
is carried out by means of a screw turned by a watch key, and in order to get a fixed
point the hygrometer is enclosed in a metal case provided with a glass front and back.
The front can be removed for observations, but for adjusting it ia put in its place and
the interior of the case is saturated by a wet cloth. KOPPE regards hair hygrometers
as quite sufficiently comparable, and gives the following particulars as to the
expansion of hair by heat alone. DE SAUBSURE found an expansion for 1° R. of
19 millionths of the length of the hair, and KOPPE, by observing the readings in
completely saturated air at various temperatures, found no change of length indicated
for changes of temperature' up to 20° C.

REGNAULT has taken up the subject of hygroraetric methods again in CARL'S
' Repertorium,' vol. 8, 1872, p. l.J He, however, only recapitulates his former con-
clusions, adding some practical directions for applying the chemical method con-
tinuously in meteorological observatories so as to obtain the mean quantity of
moisture for every three hours. VON BAUMHAUEB,§ referring to REGNAULT'S paper,

• ' Carl, Repertorium,' vol. 6, p. 117.

t ' Die Messnng des Feuchtigkeitsgehaltes der Luft.' Zurich, 1878.

I Translation of part of a paper entitled " Instructions pouvant servir a 1'etablisscment des observa-
toires mc'te'orologiques " in the Geneva 'Archives des Sciences,' vol. 40, 1871.

§ ' Poggendorff's Annalen,' vol. 148, 1873, p. 448. (Translation of a paper in the ' Archives Necr-
Inndnises,' vol. 6, 1881.)

T 2

140 MR. W. N. SHAW ON HYGROMETRIC METHODS

agrees with that author in preferring the chemical method, and recalls the attention
of meteorologists to an apparatus, described by him 17 years previously, intended to
give continuous observations by the chemical method. It is a sort of hydrometer
containing pumice and sulphuric acid, and is floated in an oil bath, and there is an
arrangement for passing the air through without interfering with the free flotation.

A similar apparatus is described by SNELLEN.* I am inclined to look upon all
apparatus requiring anything like a long leading tube with very grave suspicion. A
sudden change in the temperature such as sometimes covers all walls and furniture
with a deposit of moisture would supply a layer of water to the leading tube that
would nearly saturate the air for a considerable time and render the instrument worse
than useless at a time when its results are wanted to be most accurate. Vox BAUM-
HAUER suggests leading air by means of a lead tube from a captive balloon to his
instrument in an observing room. It is almost painful to try and think what an
observation with such an arrangement would really mean.

A short paper by LEFEBVREt gives the results of experiments to compare
REGNAULT'S hygrometer with the chemical method. One series of experiments com-
pares the hygrometric state as given by the chemical hygrometer with that deduced
from the dew-point reading of REGNAULT'S instrument, taking the dew-point to be : —

(1.) The temperature of the first appearance of dew.
(2.) The temperature of disappearance.
(3.) The mean of the two preceding.

The collective results may be tabulated as follows, calling differences from the
results of the chemical hygrometer errors : —
Number of observations, 1 5.
Range of temperature, 17°'3 C. to 24° "0 C.

„ f — 2-00 in percentage humidity.

Extreme error of first appearance observations •<

1 + '60

— 1-83

•18

— T92
+ '75

Mean error of " first appearance " observations . '47

„ "disappearance" „ . '74

"mean" '55

" disappearance " ,, •<

" mean ,, •<

A second set of experiments gives a comparison of two chemical hygrometers with
each other and with REGNAULT'S. I give a table of the differences in the fractional

* 'Archives Ni'ei-landaisop,' vol. 9, p. 477.
t ' Annalcs de Chimie,' vol. 25, 1819, p- HO.

Mil. W. N. SHAW ON UYGKOMETRIC METHODS.

141

humidity between the two chemical hygrometers, and between their mean and
RENAULT'S, the fractional humidity in each case being not far from '50 :—

Difference between (he

Difference between

No. ..f

fractional humidity

BKUNAULT and the mean

Experiment.

a* g Ten I>T two chrm cal
hygrometer*.

of chemical
hygrometer*.

1

-f "0007

-f -00l>0

2

-•0058

+ -0061

3

+ -0031

+ -0157

4

+ -0016

+ -0002

5

+ -0055

+ •0047

6

t m

• •

7

+ -0016

+ •0018

8

4- -0004

+ -0062

9

. .

. ,

10

+ -0004

+ •0062

11

+ -0000

+ -0049

12

-•0031

-•0087

These results are very satisfactorily concordant. The aspirator for the chemical
method contained 4 '278 litrtes, and the tubes were filled with sulphuric acid and
pumice, the air was taken from out of doors, and the REGNAULT, placed at the
window, was observed at the beginning, middle, and end of each experiment, and the
mean of the three dew-point readings taken.

Observations were also taken with DANIELL'S instrument, but they were so dis-
cordant that they are not recorded.

For many years REGNAULT'S form of condensation hygrometer was regarded as a
satisfactory standard instrument. In 1877 ALLUARD introduced a somewhat modified
form, which was used by ANGOT in the researches already discussed (p. 133). The
modification consisted in replacing the silver thirnble by a brass tube of square section
provided with various metal tubes to allow of the passage of air through the ether
contained in the tube. Instead of the glass upon which REGNAULT'S thimble was
mounted, there are two windows in opposite sides of the square tube, near to the top.
These enable the bubbling of the air through the ether to be watched. The sides of
the tube are gilt and highly polished, and one of them is framed by a broad band
of brass, gilt and polished in like manner. This surrounding band is very near to,
but does not touch, the brass tube. The dew is therefore deposited on a flat gilt
surface, and the identification of a deposit is rendered easier by the proximity of an
unaltered surface with which to compare the cooled one.

A dew-point instrument was also introduced in 1871, by Mr. DINES, in which the
dew deposit was caused by cooled water, and took place on blackened glass ; this was
modified by its inventor in 1879,* so that it could be used with either water or ether.

• Sec Symons's " History of Hygrometer*," 'Meteorol. Soc. Quart. Jonrn.,' veil. 7, p. 161.

142

MR. W. N. SHAW ON HYGROMETRIC METHODS.

These instruments, however, may be regarded as being intended to facilitate the
reading of the dew-point rather than as questioning the results given by REGNAULT'S
instrument. The case is different with a new condensing hygrometer, introduced by
CROVA. * This consists of a tube of thin brass, the interior of which is nickel-plated
and carefully polished ; one end is closed by a plate of ground glass, and the other by
a lens. The air to be experimented on is drawn through this tube by means of two
tubulures entering it at right angles, and the tube is cooled by surrounding it with a
brass box, partly filled by carbon disulphide, through which air is made to pass. The
temperature is read by means of a thermometer dipping into the liquid. The dew
deposit is seen by means of the lens, and the arrangement is such that the mean
temperature of appearance and disappearance can be read to less than 0°'l C. This
arrangement was adopted by CROVA, because he was dissatisfied with the behaviour
of REGNAULT'S instrument when the dew-point was very low, and when there was a
considerable wind.

Having arranged the instrument, he made a series of comparisons of the results of
REGNAULT'S instrument, observed in the open air or window, with those of his own,
the air being in that case conducted through a fine lead tube to the instrument inside.
The thermometer employed had been previously corrected.

The following are some of the results : —

SERIES I. —Wind north-east, fairly strong. Barometer 759.

Dew-point.

Hour.

Temperature of air.

BEONAULT.

CBOVA.

h. in

o

o

o

9 40

21-0

4-6

47

9 50

21-0

5-3

5-9

10 0

21-0

5-9

7-4

10 30

21-5

6'5

7-5

11 0

22-0

6-5

7-0

1 30

22-5

67

8-3

SERIES II. — Wind north-west, gradually increasirg. Barometer 758.

humidity 0'54 to 0'58.

Psychrometer

Dew-point

Hour.

Temperature of air.

BlCNAl'LT.

CBOVA.

li in

0

o

0

8 15

19-4

9-5

9-5

8 30

19-9

9*7

10-5

8 £5

J97

8-3

9-8

9 0

196

8-1

9-5

* 'Journal do Physique,' vol. 2, 1883, p. 166.

Ml!. W. N. SHAW ON IirGKOMKTRIC METHODS.

143

SERIES V. — North-west to north, variable. Barometer 751. Psychrometer

humidity 0'405.

Dew-point

Temperature of air.

RCGKAULT.

CKOVA.

h. m.

0

o

o

•2 30

1675

f2-4:,

Succes- 1 3-05

4-05
Constant

sivelv. "] 1

U •-'••'

3 0

• •

/2-05

565

\ 1-35

These differences in the dew-jx>int readings by the two methods are very serious,
as, if this be the real state of affairs, the work done with REGNAULT'S hygrometer as
a standard instrument is so far invalidated.

It will be noticed that the methods of finding the pressure of aqueous vapour in
air hitherto described are all indirect : that is to say, when the instrument has done its
work, a calculation has to be gone through in order to obtain the value of the vapour
pressure. The dew-point method approaches, perhaps, the nearest to a direct method,
but it must be remembered that what is actually observed, or rather what is supposed
to be observed, is the temperature at which a deposit takes place on a certain cooled
surface. It has always been assumed that this temperature corresponds to the
temperature of saturation of the air, and that in consequence we may take the corre-
sponding pressure of vapour from REONAULT'S table. The experiments of REGNAULT
referred to above, page 120, do not completely justify this assumption, although the
error is but small, but those of CROVA just described throw doubt, for different
reasons, upon this fundamental point.

The calculation of the pressure of vapour from the weight of water contained in a
given volume of air — the chemical method — depends in like manner upon assumptions
for which there is at present no absolute experimental verification. Such experiments
as there are show discrepancies between observation and calculation which are beyond
the limit of experimental errors. And yet a direct experimentsU determination seems
in principle extremely easy. Assuming only that the pressure of dry air in moist air
is independent of the pressure of the vapour, which must be assumed if we are to
assign any meaning at all to the " pressure of aqueous vapour in the atmosphere,"
it follows that, if we abstract the moisture without altering the volume, and measure
the resulting pressure of the dry air, the pressure of the vapour is simply the difference
between the initial and final pressure of the air. Moreover, the dry air is known to
obey the law of BOYLE with extreme exactitude for small variations of pressure and
volume, and hence the measure of the diminution of volume of air produced by the

144 MB. W. N. SHAW ON HYGIIOMETRIC METHODS.

*

removal of its moisture without altering its pressure furnishes an equally accurate
means of finding the original pressure of the vapour. Only one stage less direct is
the method of finding the increase of pressure produced by saturating a specimen of
air whose volume is kept constant and assuming the saturation pressure from
REGNAULT'S tables. Calling such determinations " absolute," an instrument for
making them is called an absolute hygrometer, and of these very many have been
suggested by various experimenters in recent years. Before describing them, I will
just mention that the simplicity of the method is entirely illusory. The differences of
pressure to be measured are small, never practically exceeding 30 mm. in our climate,
and being generally about 10 mm. Taking the latter amount, and supposing that an
accuracy of 1 per cent, is required, the pressure difference must be measured to O'l mm.,
which is less than J-^QQ of the ordinary barometric pressure. Consequently the volume
or the pressure must be kept constant during the measurement to the same degree of
accuracy ; and, since a change of temperature of 1° C. alters the volume or pressure by
about '00366 of its amount, the temperature of the apparatus must be kept constant,
or its change compensated for, to within 0°'02, which is a very difficult matter.
Further, any taps or connexions must hold sufficiently tight to guarantee that the
vessel will not lose or gain air to the extent of -shroff °f its whole volume during the
course of an experiment. These are a few of the difficulties.

The best known of these new instruments is that of SCHWACK.HOFER, described and
figured in JELINEK'S ' Zeitschrift' for 1879. In its modified form it is an apparatus
for finding the diminution of volume of air in consequence of the absorption of the
vapour by sulphuric acid. The air is first enclosed in a burette, with a graduated stem
communicating with a mercury cistern ; the burette is then, by a suitable arrangement
of taps, brought into communication with a second vessel, open at the bottom and
shaped like an elongated bell, containing a number of vertical glass tubes, and filled
up with sulphuric acid. The mercury is then forced by means of a plunger dipping
into its cistern into the burette, and the air passes over into the second vessel, dis-
placing the sulphuric acid, leaving, of course, a large surface of acid exposed, which
dries the air after four or five transfers backwards and forwards. The sulphuric acid
is then driven back to its original level, which is carefully marked ; the pressure of the
dried air is adjusted to be the same as before by means of an oil gauge, and the
diminution of volume is read off ort the graduated stem of the burette. The tempe-
rature is read by a thermometer sealed into the burette, and kept uniform by a jacket
of glycerine round the burette.

I have omitted a large number of small precautions which are quite necessary in
taking an observation. The apparatus is very delicate, and is therefore available only
for laboratory use.

PEENTER worked with an instrument of this kind. He says : " Bei fiinfzig Versuche
waren resultatlos, bis es mir gelang die Hiihne zu dichten, das Quecksilber luttfrei
und rein zu halten und die ganze Rohre vor aller Verunreinigung zu schiitzen.

WB. W. N. SHAW ON HYGROMETRIC METHODS. 145

Endlich, nachdem ich drei Wochen daran mich abgemllht, schienen die Versuchs-
ergebnisse einigermassen entsprechend. Die . . flinfzig . . Versuche lieferten
alle eine zu kleine absolute Feuchtigkeit und zwar im Mittel um 2*5 mm. circa zu
klein."

A similar apparatus was used by SWORYKIN, who seems to have had more success
with it

An apparatus somewhat similar to SCHWACKHOFER'S was designed and made by
Mr. DINES. It is described in the 'Meteorological Magazine,' September, 1883, and
is said to work satisfactorily.

The next instrument of the kind is EDELMANN'S, described in WIEDEMANN'S
' Annalen,' vol. 6, 1870, p. 455. It is of very much simpler construction. A glass
cylinder is closed at each end with india-rubber corks perforated so as to take a tube
containing a thermometer and connected with a mercury pressure-gauge at one end,
and at the other a second tube communicating with the top and bottom respectively
of a small vessel of sulphuric acid by means of glass cone joints. The mercury gauge
is also attached to a cone joint ; there are stop cocks between the cone joints and the
cylinder. The cylinder is provided with a metal jacket. By the double connexion the
sulphuric acid is allowed to run into the cylinder without altering the volume of the
air. Between each operation the cylinder has to be cleaned and dried, and washing
out with alcohol and ether is recommended. It is said to give good results, though
none are quoted in the paper, and if, as suggested, the vessel is finally dried by passing
through it air from a " Wasser-trommel-geblase," good results ought certainly to be
received with unusual gratitude. The description of the instrument, and its use,
reads rather like an instrument maker's solution of the problem " to make an absolute
hygrometer." VAN HASSELT* has described an instrument somewhat similar to
EDELMANN'S, but he uses phosphoric anhydride instead of sulphuric acid. The
anhydride is enclosed in a thin glass bulb, which is broken by shaking when the
moisture is to be absorbed ; the difference of pressure is observed by means of an oil
manometer, which consists of two glass tubes connected by a flexible india-rubber one.
This arrangement allows the volume to be reduced to its original value before reading
off the pressure.

RuDORFFt uses a WOULFF'S bottle, with three necks, in an ingenious manner. Into
the middle neck is fixed a burette, which supplies the sulphuric acid ; this is allowed
to pass into the vessel until the pressure reaches its original value, as indicated by a
manometer of sulphuric acid, sp. gr. 1'300. Thus the volume of sulphuric acid run
in gives at once the diminution of volume of the air due to the absorption of the
moisture.

This plan was somewhat improved by NEESEN.J who has two WOULFF bottles, one

* ' Beiblattor,' 1879, p. 697
t ' Beiblatter,' 1880, p. 349
J ' WIKDEMANN, Annalen,' vol. 11 1880, p. 526.

MDCCCLXXXVIII. — A. U

146 MR. W. N. SHAW ON HTGROMETRIC METHODS.

connected with each limb of the manometer. By this means the temperature correc-
tion is very ingeniously compensated, and, moreover, a second determination can be
made with the same apparatus by using the other bottle.

MATERN* also uses the compensation by a second vessel, but the vessels are glass
cylinders, provided with lids ground flat, so that they can simply be laid on. The
necessary connexions, &c., are sealed into the lids. The method he suggests is
that of finding the increase of pressure, due to the saturation of the air, by allowing
water to drop into one of the vessels on to blotting paper. He also givest a simplified
form of the instrument for more rapid use.

I have not been able to find any series of observations showing the comparative
results obtained with these different instruments : they have been tried by their
inventors, but means of comparison are not at hand, except for SCHWACKHOFER'S
instrument, as already mentioned.

NOTE B.
HYGROMETRICAL TABLES.

It may be useful to add a statement with reference to the tables at present in use
by meteorological observers in different countries for the reduction of wet-and-dry-bulb
observations. Mr. R. H. SCOTT has very kindly supplied me with the following list
of original hygrometrical tables : —

HAEGHENS ('Annuaire de la Socie'te' Me'te'orologique,' 1849), followed by RENOU,
France, and DENZA, Italy.

WILD (' Repertorium filr Meteorologie,' vol. 1, St. Petersburg, 1870), followed by
JELINEK, Vienna, 1876.

BLANFORD. — ' Tables for the Reduction of Meteorological Observations in India.'
Calcutta, 1876.

GLAISHER. — 'Hygrometrical Tables,' 7th edition. London, 1885.

GUYOT. — " Psychrometrical Tables," ' Smithsonian Miscellaneous Collections.'
Washington, 1887.

Of these HAEGHENS'S and WILD'S tables are millimetre-centigrade tables founded upon
REGNAULT'S table of pressures and the psychronaetric formulae of the same authority,

viz. : —

_ 0-480 (*-Q
610 - if '

0-480 (t - Q
689 - t' '

* ' WIEDEMANN, Annalen,' vol. 9, p. 147.
t ' WIEDEMANN, Annalen,' vol. 10, p. 149.

MB. W. N. SHAW ON HYQROMETRIC METHODS. 147

BLANFORD'S tables are bused upon the same observations and formulae, but they are
adapted for English inches and FAHRENHEIT degrees (see p. 135). The following
information respecting them is extracted from the introduction to the tables (p. 4) : —

""TABLE III. — This table gives the tension of saturated aqueous vapour, in decimals
of an inch of mercury at the temperature 32°, in latitude 22°, at the level of the sea*.
It has been reduced from the original table for the latitude of Dublin, computed (from
REONAULT'S observations) by the Rev. ROBERT DIXON, by correcting his values for
the difference of gravity, viz., multiplying them by the constant factor 1 '00286184.

"The psychrometric tables which follow are all based on this table, and the
computation has been chiefly made by the aid of the arithmometer.

" AUGUST'S formula (modified by REGNAULT), which has been used in computing
the Tables IV. to XL, is as follows :-r-

" For temperatures of the wet bulb below 32° :

-430 (<-Q
~J " 1240-2 -f

and for temperatures of wet bulb above 32° :

-480 (t-f)

X=S - 1130 _ ,' h-

wherein t' and t are the temperatures of the dry and wet bulb thermometers respec-
tively, in FAHRENHEIT degrees, f the tension of vapour at temperature t', h the reading
of the barometer in inches, and x the tension of the vapour present in the air at the
time of the observation."

GLAISHER'S tables are based upon REGNAULT'S table of pressures, and the table of
Greenwich factors, which the following extract from the introduction to GLAISHER'S
tables (p. 4) sufficiently describes. These are adapted for English inches and FAHREN-
HEIT'S scale of temperature :—

"Determination of the Dew-point from, obsemations of tJie Dry and Wet Bulb

Thermometers.

" TABLE I. — Factors by which it is necessary to multiply the excess of the reading
of the dry thermometer over that of the wet, to give the excess of the temperature
of the air above that of the dew-point, for every degree of air temperature, from 10°
to 100°.

" The numbers in this table have been found from the combination of many
thousand simultaneous observations of the dry and wet bulb thermometers and of
DANIELL'S hygrometer, taken at the Royal Observatory, Greenwich, from the year
1841 to 1854, and from observations taken at high temperatures in India, and others
at low and medium temperatures at Toronto. The results at the same temperatures

u 2

148

MR. W. N. SHAW ON HYOROMETRIC METHODS.

were found to be alike at these different places ; and therefore the factors may be
considered as of general application.

" By means of the numbers in this table the temperatures of the dew-point in the
general tables have been calculated ; and these were for many years checked by direct
observations with DANIELL'S hygrometer made by me at the Royal Observatory,
Greenwich."

GUTOT'S tables are a very complete set, comprising :—

i. Extended tables based upon REGNAULT'S formulae already given, adapted

for millimetres and Centigrade degrees,
ii. The same adapted for inches and FAHRENHEIT degrees. (These are given

" for ordinary use.")
iii. GLAISHEB'S table of Greenwich factors and an old set of GLAISHER'S

psychrometrical tables, based upon them and a Greenwich table of

pressures. (These are given for the purpose of " comparing results.")

Besides these there are ANGOT'S tables referred to on p. 134, but I am not aware
that they are in general use in any country.

The information may be summarised in the following table : —

Country.

Tables.

Formula of reduction.

Units.

E norland

QLAISHEB

Greenwich factors

Inches FAHREVHEIT

India

BLANFOBD

REGNAULT'S formulce

America

GUYOT

Both

France

HAEGHENS .

"

Italy

"

Germany

WILD

"

Austria

JELINEK . . *

"

" "

Russia

WILD . .

. " "

NOTE, APEIL 23, 1888 (see p. 83).

I have assumed REONAULT'S investigation of the behaviour of sulphuric acid tubes (' Annales do
Chimie,' vol. 15, 1845) to be conclusive evidence that his acid was efficient for its purpose. His experi-
ments wore of two kinds. First, four sulphuric acid tubes were mounted " in series." The first two
were at the ordinary temperature, the third at 0° C., and the fourth at — 30° C. The first tube gained
1'235 gramme in weight when moist air was sent through the four ; the weights of the other three tubes
were unaffected. Secondly, air was dried by passing it through three sulphuric acid tubes, each one
metre long, the third being in a freezing mixture. The dried air was then led through a weighed vessel
containing moist sponge, and thence through two weighed sulphuric acid tubes. The first of these two

MB. W. N. SHAW ON HYGROMETRIC METHODS. 149

took up 0767 gramme of moisture, being precisely the same amount as that lost by the sponge vessel.
My attention has lately been called to the papers by E. W. MOBLKT (' Amer. Journ. Sci.,' vol. 30, 1884,
p. 140 ; vol. 34, 1887, p. 199), in which the question of residual moisture in air passed over sulphuric
acid or phosphoric anhydride is more rigorously treated. It appears that the residual moisture would
not appreciably affect the weighings in the observations here recorded, for sulphuric acid only leaves
nnabsorbed " not far from a fourth of a milligramme of moisture in 100 litres of a gas," and " the
moisture left unabsorbed by phosphorus pentoxido, if capable of determination, may be very roughly
stated as possibly a fourth of a milligramme in 10,000 litres."

NOTE, APBIL 23, 1888 (see p. 97).

The researches of WABBURO and IHJJORI (' Wiedemann, Annalen,' vol. 27, 1886, p. 481) afford some
information about the formation of a deposit. On a specimen of fresh lead glass at about 16" G.
exposed to the vapour of water in vacua a deposit of 18 X 10 * grammes per square centimetre was
observed when the vapour pressure inside the vessel corresponded to the temperature 4°'71 C. ; the
deposit increased to 194 X 10~8 grammes per square centimetre when the vapour pressure was increased so
as to correspond to a temperature 0°'87 C. below that of the glass surface. No results are given for lead
glass which enable one to form an opinion as to what happens when the difference of temperature is
made very small, but for Thuringian glass a sudden increase of 50 per cent, in the deposit is shown for
a diminution of the temperature-difference from 0°'27 C. to 0°'17 C. With lead glass, of which even a
fresh surface showed comparatively very feeble absorption of vapour, no absorption at all waa detected
after treatment with boiling water. The apparatus referred to in the text was, excepting the three-way
tap, constructed of lead glass that had been frequently washed with cold water.

VI. On the Diameters of a Plane Cubic.
By J. J. WALKEE, M.A., F.R.S.

Received June 16,— Read June 16, 1887.
Revised February 9, 1888.

[PLATES 6-8.]

I. ABSTRACT.

1. THE object of this Memoir is to develop relations which subsist between a cubic («)
and the complex of lines, in its plane, which are the polars with respect to it of the
points on any transversal (L). This complex becomes the system of NEWTONIAN
DIAMETERS of the cubic, when the points on the plane are projected on a second plane
parallel to that containing the vertex of projection and the line L (Ix -f- my + nz = 0).
This development involves frequent reference to the envelope of the complex in
question, the conic

which, in analogy with the " pole " of a line in the theory of conies, I propose to call
the " POLOID " of the cubic u and the line L ; and, in particular, when the line L is at
infinity, the " CENTROID " of u.

2. HESSE* first appears to have used the equation to s in the theory of the ternary
cubic form — but without any recognition of its geometrical significance — to obtain
the equation to the cubic in " line-coordinates:" viz., in the form of the resultant of

the system

3s 3« 8s .

a* : ay : az = ' : m '• "-

with

Ix -f- my -f nz = 0,

and this resultant will plainly be, to a factor, the equation to s itself in line-
coordinates £, 17, £, with /, m, n substituted for f, rj, £ respectively.

* ' CRKLI.K, Jonrn. Math.,' vol. 41, p. 285 . . . , under date January, 1850.

18.688

152 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

3. In the first edition of the 'Higher Plane Curves' (1852) SALMON, under the
subject " Poles and Polars " (of cubics), showed that

*= 0

is the envelope of the polar lines of points on L, touching each tangent to u at the
points L = 0, u = 0 in the point harmonic conjugate to its contact with u relatively
to its intersections with the other two, and proposed for the locus of s = 0 the
designation " Polar conic of the line " L, for which I have ventured to suggest, and
use, the shorter name " Poloid " of (u and) L.

In CAYLEY'S Memoir -" On Curves of the Third Order," 'Phil. Trans.,' 1857, the
additional property is given of the polar lines of points on L all passing through a
point (viz , the intersection of the two right lines into which s, " the lineo-polar
envelope of the line," then breaks up) on the Hessian of u, when L joins corresponding
points on that curve. Beside these I have not been able to find any notice of the
conic s.

4. If through any point P in its plane chords are drawn meeting the cubic in O1;
Og, O8, and a point 0 is taken on the chord determined by the relation

3 . ^_ J_ J
pn ' PA ~T~ w

PO " PO! ir PC, ' POS'

the locus of 0 is a straight line, according to a theorem of COTES'S, communicated by
his friend, Dr. ROBERT SMITH, Master of Trinity College, Cambridge, to MACLAURIN,
after COTES'S lamented death, and proved by MACLAURIN* as a case of a more
general theorem which presented itself to his mind when " meditating on this com-
munication." For shortness I have called the point O the " CoTES-point " on the
chords through P, the locus of which is now well known to be the polar line of the
point P relative to the cubic u.

5. If P describes a line L, the CoTES-points of the polar lines of the points on L —
regarded as chords of u — relatively to their intersections with L, may be considered.
SS 28-32.

•Jv

The locus of the CoTES-points of this complex of lines is shown in the sequel to be

a nodal cubic (37) and (40),

v = 0,

which covariant of u and the line L, I propose to call their " Cotesian."

6. Considering (§§ 33-37) more generally the locus of CoTES-points on chords of u
subject to the condition of touching s, the result comes out as a concomitant breaking
up into two factors, one the cubic v just referred to, the other being the equation to

* ' De Lin. Geom. Prop. Gen. Theor. IV.,' p. 24, ed. 1748. The Theorem is not given in the ' Harmonia
Mensurarnm,' as sometimes erroneously stated, with the subjects of which treatise it has no connexion.

./ J. \\ntt,,

Phil .

. 1888. A.PltUf 6 .

s

J.J. Walker.

Phil. Trnn*. 1888.A./Y«* 7.

J.J.Walker.

Phil. 7m»*.lH88.AJ%ofe8.

MB. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 153

the three tangents to u at the points in which it is met by L, now first obtained in
a general form. In the Memoir referred to ('Phil. Trans.,' 1857, p. 439) CAYLEY
obtained, with his peculiar skill, the equation of the three tangents in question
for HESSE'S canonical form of u ; viz.,

ax3 + &/ 4 v? + Qexyz (2)

finding for the " satellite line " of L, or line in which the tangents again meet u, the
equation that, further on (§ 36) will be shown to verify the general form at which I
have arrived ; and in terms of which (if it be taken as a fundamental covariant of u
and L) and of 6- the equation of the nodal cubic r may be expressed.

7. Whereas through any point in the plane of the cubic u and the line L there can
be drawn in general (§§ 40-43) two chords having it as their COTES- point relatively to
those in which they meet L, viz., those determined by the polar-conic of the first
point ; if, however, that point be taken on s the two chords coincide, and thus a
complex of double chords is obtained, which are the polars with regard to s (§ 4'2) of
the CoTES-points of the polar lines of the points of L, and are shown (§§ 58-60) to
have as their envelope a tricuspidal quartic (81)

w= 0,

the equation of which canuot be found explicitly except for specifically assigned forms
of u.

The point in which a double chord meets s being its CoTES-point, that in which it
touches its envelope is shown to be harmonic conjugate to the former with respect to
the intersections of the chord with the line L and with the conic s again ; and the
two intersections of the double chord with s are thereby discriminated (§ 59).

8. Again, considering (§ 61) the points of a line having its pole on L, the chords of
which those are CoTES-points constitute two groups : viz., one a pencil through the
pole of the polar line, the other having as its envelope a conic (§ 88) touching the line
L as well as the polar line in question (in virtue of its being the line of the latter
complex through its own CoTES-point) and the double chord through its point of
contact with s, which is both a ray of the pencil and a line of the complex.

This system of conies (§ 62) has as its envelope the sides of the quadrilateral formed
by the transversal L and the tangents to the cubic at the points in which L meets it.

9. A question of some interest is considered (§§ 38-39) : what, if any, of the complex
of polar lines of points on the transversal L are conjugate, in the sense of their inter-
section being the CoTEs-point on either ? Discarding the tangents to u at the points
in which it is met by the transversal L, each of which is conjugate to itself, the o.dy
distinct conjugate polar lines of points on L are the two tangents to the poloid (s) of L
from the pole of that line with respect to the poloid.

10. As the chord of contact with the poloid of the two tangents through its pole

MDCCCLXXXVIII. — A. X

154 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

L forms a coincident, pair of "double chords" (of the cubic) ; and these three lines,
viz., the two tangents and their chord of contact, form a triad of lines of reference by
means of which the properties of the complexes of lines here considered may be
deduced with far greater facility than through the use of the canonical form (2).

The two tangents to s from the pole of L with respect to s are the nodal tangents
of the Cotesian v (§ 50), the line L being its inflexional axis ; a circumstance which
explains the unique character of this triad of lines, and marks them out as the best
system of lines of reference for the discussion of properties connected with this con-
comitant of the primitive cubic u (§§ 48-62).

11. But, whereas these tangents to s, the poloid of L, are only real when L cuts s
in real points (which is shown to occur only when it meets the cubic u in but a single
real point), three other triads of lines exist, of which one at least is always real, con-
venient as lines of reference in many of the questions which arise : viz., the sides of
one of the three triangles whose corners are the pole of L, with respect to s; one of
the three points in which L meets u ; and the pole of the connector of these two
points. Each of these being self-conjugate triangles in respect of s, the equation of
that conic is reduced to a trinomial form ; and since the cuspidal tangents of the
envelope w (§ 76) all pass through the pole of L, the equation of that curve is of a
comparatively simple character for one of these triangles of reference. In Plate 6,
ABC is one of these triangles (§§ 63-76).

12. If the line L touches u it touches its poloid s also ; and consequently other lines
of reference have to be looked for. These are found in the line L, the tangent to u
at the point where L cuts it (itself a tangent to s also, it will be remembered), and the
chord of contact of these lines with s.

In this case the cubic v degenerates into the line L, and a conic having double
contact with s; while the envelope w also degenerates into a conic having double
contact with s and the Cotesian conic at the same points (§§ 77-81).

13. When L becomes the line at infinity the pencil of chords through any point on
it is to be replaced by a system parallel to a given line, and the polar line by the
diameter which is the locus of the mean point on any chord of the system relatively
to its intersections with the cubic.

The envelope of these diameters I have called the "Centroid" of the cubic, from
its evident analogy with the centre of conies, apprehending no confusion in such a
connexion with the sense of a " mass-centre," which it sometimes bears.

The consideration of the Centroid and associated curves occupies the concluding
part of this Memoir.

14. The method of treatment of the discussions, an abstract of which has just been
given, is uniformly analytical, trilinear coordinates being employed. The results will be
found to be arrived at without much difficulty, or tedious calculation, considering the
great generality of most of them. With a view to simplifying three important discus-

MR. J. J. WALKER ON THK DIAMETERS OP A PLANE CUBIC. 155

sions as much as possible, a preliminary investigation of a form into which the result
of substituting for the variables in a ternary quadratic form the expressions

ty ty ,0d S^> 3A ,ty

n ~T — m v» I -f — n ^-, m ~r — I ft

oy as vz ox ox < i

<£ being any ternary form and /, m, n any three constants, is introduced (§§ 24-27) ;
and the special cases of its application in the sequel are considered.
15. The notation employed throughout is as follows :—

(i.) The right line, considered generally, is written

£* + W + fc = 0 :

(ii.) The particular transversal considered in connexion with a cubic u is always
written

L =E lx + my + nz :

(Hi.) The cubic u being, in point coordinates,

« = ax* + fry3 + ez3 + Sajarty + 3a^z + . . . + Gexyz,
is written in line coordinates — or its reciprocal is —

u =

(iv.) The poloid of L with respect to the cubic in point coordinates is represented
by s as in § 1, or (§ 19)

and in line coordinates — or its reciprocal — by or, where
4<r = 4u23M - «-£* + . . . + 2 uu -

(v.) Any special point on L regarded as the " pole " of a pencil of chords of the
cubic u is marked (x'y'z), and any special point on the polar line of a/v/V is marked
x"y"z" ; a " chord " being a line, or transversal, considered particularly in connexion
with its intersections with the cubic u.

II. PRELIMINARY.

16. The intersections of a line £r + r)y -f £z = 0, with a curve u = 0 of order p,
may be studied through the equation

UV-'M + P'f'~1 . p"-' . D"-2u + +/Y>Du + u = 0, . . (3)

x2

156 MB. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

where, x'y'z being a given point on the line, and D defined by

p(p- 1) . . . (p - r+ 1) D'= {(w - tf)^, + (£a - tft± + (fi8- ,») £,

if, for shortness, a, ^, y represent the sines of the angles of the triangle of reference,
p is equal to the length of the segment of the line between the point x'y'z' and any
one of the p points in which it meets the curve u (' London Math. Soc. Proc.,' vol. 9,
p. 227) ; provided f2 + tf + £- - 2??£ cos A - 2£f cos B — 2ft cos C = 1.
1 7. Thus the sum of the reciprocals of thgse segments is equal to

— p Du/u ;

while, if (xyz) is any other point on the same line, & + • • • , and p similarly equal to
the segment between x'y'z1 and it,

-r,a). . . (5)

But if xyz is the CoxES-point (§4) on the line in respect to x'y'z' and the curve u,
p/p' = — p Du/u

whence (5)

(x — x') 5-7 + (y — y') ~— , -j- (z — z') ~-;
or,

viz., the locus of xyz is the polar line of x'y'z.

18. If now, instead of considering £ 17, £ as variable and (x'y'z') as fixed, £ 77, £ are
regarded as constant — say equal to I, in, n — and (x'y'z') as any point on L, or
Ix + my -f- nz = 0, the envelope of the line (6) will obviously be the same as its
equation in line coordinates, regarded as a curve of order p — 1 in point coordinates
x'y'z'. Thus, if u were a quartic, (3) having been written in the form

« , , ,o , . / / /

x^ +....+ Zx*y ^ +...+ Gxyz ^^ = 0,

its envelope might at once be written down as

&v«/3v\2 i . r/av\s9v\* .

viz., the poloid of the line L and the quartic u is another quartic. I have made the
foregoing remark to draw attention to the perfect generality of the theory of the

Mil. J. J. WALKER OX THE DIAMETERS OF A PLANE CUBIC.

157

poloid, and, at the same tune, to show that it is of equal or higher degree than u for
values of p exceeding 3.

19. In the case, then, alone contemplated in this Memoir, of u being a cubic, and
(6) being, therefore, otherwise written

I3&u

a^"1

the envelope takes the form (1)

as given by SALMON ('Higher Plane Curves,' 3rd edition, p. 15G), /, m, n here
replacing a, ft, y.

If written in the normal form of a conic the coefficients wn . . . ?<23 . . are the contra-
variant conies of the triad of conies,

-o

w'=a,/3>

a«/0
^3'

(7)

taken singly and in pairs, viz., the cubic being written as in the ' Higher Plane
Curves,' only with the substitution of e for m,

cz8 +
and the poloid of u and L being

s = «„«• +
then

«n = (blcl — e2)? + (ac, — O

2 + njgyz + ?<312x -f

-f 6ezyz, (8)
(9)

= (063 — c22)P + (ca3 — c

— ae)mn
+ 2(a2e — a^b^nl + 2(a3e —

— 612)/r + 2(6^ — a263)m»
+ 2(/>,63 — be)nl + 2(63e — 6,c2)Znj

— e2)n2 + 2(0^ — c2as)wn

-j- 2(c2e — tgC^nZ + 2(c1c2 — ce)lm

ca2 + C2a3 — 2c,e)m2 -f (ba3 -j- a263 — 261e)»t2
+ t2(blcl + e8 — a2c2 — asb3)mn-\- ^(b^ — bc^nl-^- 2(63C!— cb^l

= (cfc, -r- tgC! — 2c2e)/2 + (ac — a^m- + (a&3 + r/36, — 2a^e)nz

+ 2(a2rj — ac2)7nn + 2(a2c2-f c2— ftjCj — a3bs)jil -{- 2(c2«8 — ca2)/

= (^G! + ijCj — 263e)/2 -f (acs + a2Cj — 2aye)m2 + (at — a2fe,)/r

+ 2(a361 — abjmn + 2(/>3a8 — ba3)nl + 2(0,63 -f e2 — c^ — ^iC

(10)

158 MR. J. J. WALKER ON THE DIAMETERS OP A PLANE CUBIC.

20. The invariants of the equation (3), which forp = 3, is

p3 D3w + 3/>2 D2u + 3/3 DM + u = 0, .- ..... (11)
and, for one of the conies, s, u{ . . ., or 9xw/3 . . .,

p- D3»/., + 2p D», + M! = 0, ....... (12)

give the fundamental invariants of L and u, Uj . . ., in a very succinct form, involving
x'y'z' regarded as parameters connected by the equation lx' -\- my' -f nz' = 0 ; thus, if

A' = ax' + fty' + yz,
then (11) — £, i), £ being replaced in the operator D (4), § 16, by /, m, n—

ttt DX - (Du^2 = A'Xj . . . ; , . i ... (13)

«8 D2tt3 + «3 D2M3 - 2 DM, D?f3 = A'-«.,3 . . . ; . . . . (14)
(' London Math. Soc. Proc.,' vol. 9, p. 232.) Also (1)

u D2u - (Du)2 = A'3*, . '. ' ...... (15)

(w D3w - Dw D2M)2 - 4{(Dw)2 - ttD'tt} {(D%)2 - DM D3w} = A'V . (16)
if

v = b2cH6 + . . . = 0

is the standard form of the condition that L shall touch u.

21. The above forms are very convenient for the comparison of related concomi-
tants essential to the objects of this Memoir.

Thus, the condition that L should touch s (9)

(wjaWjs - zt223/4) l* + • • • -I" ( Via/2 - «ii««2s) win +...( = 0), (17)

multiplied by A'2 is (if lx + my + «z' = 0) equal to

* D2s - (D.s)2.
Now, since D* (A'*7) = 0, whatever integers k, kf may be,

A'*{* D2* - (Ds)2} = A'2s6 D2 (A'2*)* - 4 (D (A'%)}2 ;

* It is to be observed that if u is of order p, then Dr« is of order p — r ; and that if 0 = ^X> X being
of order q, -^ of order r, then

y D0 = qty Dx 4- rx DV',

p.|) - 1 D30 = q.,f - If D2X -I- 2qr DX D^ + r.r - 1X
and so on : thus

MR. J. J. WALKKR ON THE DIAMETERS OF A PLANE CUBIC. 159

or (15)

4 A'*{«D-« - (Ds)3} = 4{u D2u - (Du2)}6 D2{« D2u - (Du)2}

-[4D{uD2«-(Dw)2}]s (18)

But

4 D{tt D2w - (D«)2} = 3 Du D2u + u D3u - 4 Da D«u

= uD*« — DuD*u; (19)

and

12 D2{u D2n - (Du)2} = 3 DM D3u - 2 (D2w)2 - D« D3?*,

6 D8{« D'-w - (D»)2} = D« D3u - (D2u)2 (20)

By substitution from (19) (20) in (18)

4 A'<{« D2* - (D*)2} = 4{u D2w - (D«)2} {Du D3u - (D2w)2}

- (u D3* - Du D3u)2,
whence (16) (17)

...), ... (21)

i.e., the condition that L shall touch s, is equal to one-fourth of that for L touching the
culnc u, with changed sign . (i)

22. Considering next the discriminant of s; viz. (9),

if, for shortness (7),

vl = Dt*i, v3 = Dug, i»g = Dw3, T *

WL = D2!^, u>2 = D2Wj, w3 = D2t/3, J

V, = -U73«3 — tt'3W;j, V2 = WSU1 — W\U3, V3 =: M>jt*2 — W'2«1( ^ . (24)

then (13, 14),

A'*(4t*isw33 — Ujg3) = 4(u3u>i - ra8) (w3u>3 - r32) - («2ws -f- u.,u>3 — 2ty>8)3

= 4U1W1-V12; . . (25)

* By means of these expressions the equation of the poloid («) may be thrown into a form which
exhibits it explicitly as the envelope of the polar line of points in L: viz., by (9), (13), (14), (23),

Of the right lines in this form, (i) xu\ + ... is the polar line of the point x'y'z' in L; (ii) xwl -f ... or
(.11 — f£) *i +• • • • is the Newtonian diameter of chords of u parallel to L (shown in fig. 1 touching the
poloid at D'); (iii) jn\ + ... or x'vl + ... is the chord of contact with («) of (i), (U).— (April, 1888.)

160 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

= 211^.3 + 2U2Wt - VjV, ........ (26)

Hence (22),

8A'6D.'(s) = (tt,u>, + UjW! - 21;^) (2U1W1 + 2U1W1 - V, Va)

- 2t;1v!

= 2A'6P3, . . ,. ... ...... ...... ...... (27)

where,

P = (Ultfl + . -)/A3 = (V,^ + . .)/A3 =

viz., P is the condition that L should be cut in involution by

wx = 0, % = 0, u3 = 0,

which condition, multiplied by A3, has been shown (' London Math. Soc. Proc.,' vol. 0,
p. 233) to be

?<1(D«2D2?<3 — D«3D%2) + w2(D«3DX — Di^D-Mg) + M8(DM1D8t/3 — DM8DX) = 0.

Thus it is proved (27) that four times the discriminant of s is equal to the square of
the Caleyan of u ...................... (ii)

23. If the line Ix + my + nz = 0 joins corresponding points on the Hessian of u its
coefficients satisfy the Cayleyan

P=0;

hence its poloid s breaks up into two right lines through the intersection of which the
tangents of s all pass ; and the locus of this Intel-section, as the line L varies in
position, is the Hessian itself. This is the property mentioned in the Introduction
(§ 3) as proved by CAYLEY, ' Phil. Trans.,' 1857, p. 432.

24. A theorem on the result of certain substitutions, which will be of use in
subsequent investigations, may be conveniently considered before entering upon
them : —

If <j> is any ternary form of order p, and «/> a quadric — say

^ = ax2 + by2 + cz2 + Zfyz + 2g2a; + 2hoy ...... (28)

MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 161

and if in \fi for xyz are substituted respectively

30 30 ,30 30 30 ,30

n =* — m «- » / 5^ — n - • • m ~ - — I , »

dy cte d* dc dc . /

the result may be thrown, if L = Ix -+- rny -j- &z, into the form

L r r /a^ a^_ / ^_\«\ / a»f a«^ ^ a^ \ / a»^ a-> a2^ a2^ \ i
ro> - 1)- Ll W <>* taw / W^^ ^3^ a^" S^y" h ^S aya2~a/ a*ar/ " j

The proof of this theorem depends on the identities

\fs

y2-

with similar values for

But (28)

MDCXXLXXXVin. — A.

162 MR. J. J. WALKER ON THE DIAMKTERS OF A PLANE CUBIC.

, / a<£ a# ,o4> a<j> a^> ,a</.\

\b (n ." — m ^ » t -r — TO ^ » mf—lJ
\ oy oz oz ox ox oy/

= (cm2 + bft2 - 2fmn) /W + (an2 + cZ2 - 2gnl) (^V+ (W2 + am2 -
4- 2 (- amn - ft2 + glm + hnZ) ^ ^ + 2 (~ b»^ + flm - gm2 + hmn)

2 ( — c?m + fw J + gw1'1 —

the substitution of the values (30), (31) of (d<f>/dx)z, . . . , c<f>/dy 3<f>/3z, . . . , in which,
with the addition and subtraction of the terms, wherein L = Ix + my -\- nz,

a/

Lh

gives an expression identically equal to the form (29).

25. The cases of the application of this theorem which occur are : — (1) When
(§ 28) <f> is a cubic u, and i/r is the polar line of a point whose coordinates are
ndu'/dy' — mdu'/dz . . . , (x'y'z') being also a point on Ix + my + nz = 0 > so that
a = 32<£/8a;2 . . . f = d"<f>/dy 9z . . . ; in which case

and the quadratic functions in Z, m, n which multiply <f> and \fi respectively become
(bc'+ b'c - 2f f ') P + . . . , +2 (gh' 4- g'h - at" - a'f) mn + . . . , and 36*'

rpesectively, where a' = dzu'/dx'* . . . f ' = d'2u'/dy' dz'. . . .

Now, otherwise, the substitutions may be made in the conies (da/ox) ... of the
other form of the polar line,

MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 163

and then the multiplier of 3u'/2 (the present value of p<j>l(p — 1) in (29)) becomes,
for i/» = Zdu/dx,

or, identically, that multiplier becomes

a*7 a^ ,.3*'

oc i<; - — . *H\

OO-\ w W •% / ' Ov t\f

ox oy o~

for

OU < '' , ' •'

3x 3y ~ 3c

respectively.

Thus the result of the substitutions becomes

''• **•' . . (32)

whereas, by making the substitutions in the first way, the result would be

Identifying these two results,
(bc'-f b'c - 2ff')P + . . . + 2 (gh' + g'h - af - a'f)«w» + . • •

so that the polar of x'j/z with respect to the poloid of L may be written

«» av ah* , yu' »«\ „ /y^ a a

• ay + ay" a^ " ^

26. In the second case which occurs of similar substitutions (§ 30)

= 6u

the doubles of the coefficients of s ; and

Ix + my -\- nz= L
Y 2

164 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

no longer vanishes. The result (29), § 24, thus becomes

92w 8%, &u

{
(4«22«33

-f 4L2 {(4

«22 W33 —

mn}. . *. ..... (34)

It is to be observed that the values of d\jj/dx . . . are to be obtained by substituting
92M/3x2 ... for a ... after differentiation ; thus, generally, in the present case

= 2ax + 2h«/ + 2gz

27. Lastly, the case which will occur at § 38 of the substitutions,

differs from the first case only in the differential coefficients of u being multiplied by
x', y', z instead of x, y, z, so that (29) becomes simply equal to — us' ; i.e., — us
with x'y'z' for xyz.

III. GENERAL FORMULAE AND EQUATIONS.

28. The polar line of a point (x'y'z'} on L, or Ix -f- my + nz = 0, meets it in a
second point, the coordinates of which are

x : y : z

oy' 9z/ 9z' 9a/ * 9«c' 9w' '

and if the coordinates of this latter point are substituted for xyz in the quadric
forms (9w/9a;), (9w/9y), (9w/9z) of

/9w\

the polar line ot this point will be expressed in terms of the coordinates of the
original point (x'y'z').

MR. J. J. WAI.KKK ON TI1K DIAMKTKHS OF A PLANE CUBIC. 165

Referring to the general form for the result of such substitutions, (29), § 24 in the
present case it is (32), § 25,

3u' 3u' . 3u

the form of which shows that the polar urith respect to the poloid (.s) of any point on L
cute the polar line with respect to u of that point in its CoTES-point.

29. The relation just proved gives at once the coordinates of that CoTES-point in
the form

And, plainly, the Corrvs-point of the polar line of a point on L is the pole (with respect
to (s) the poloid of L) of that chord of the pencil through the point on L, which passes
through the contact with s of the polar line of that point. (Fig. 1.)

30. From considerations founded on the relation just established the locus of the
CoTES-point (xyz) of the polar line of a point (x'y'z) on L may be at once obtained in a
general form, viz., by the elimination of x'y'z' among

3tt' 3u' Sit'

or

3%

or

, '"-

*

\yf + m,/ + nz' — o.
From the last two

3s 3* , 3* - - - - , ' ••<

= n^ w« :*5 n ., :m= t^-'

3y 3a Sz 3« &r 3y

and the substitution of these values in the first gives the locus of xyz in the form
(34) § 26, of six times v, if

166 MR. J. J. WALKER ON THE DIAMETERS OP A PLANE CUBIC.

v = - ±u (AZ2 + Bm2 -|- Cn2 + 2Fmn + 2Gnl + 2HZm)

S + 'H&} ; "
(AZ + Hm + Gn + (HZ + m + n) | + (Gl + Fm + Cn)|

2s

where, as above (7), MU . . . , r% . . . are the contravariants of

du , du ,dn

' W = ' W='

taken singly and in pairs, these being the coefficients of the poloid, viz.,

s = unx* + w22y2
and A, B, C, F, G, H are the coefficients of its reciprocal, viz.,

4A =

4F =

31. The coefficient of ,s in the above expression admits of an important transforma-
tion, viz., it may be written

and this (71 ), § 55, is identically equal to

— 4P (Ix + my + nz), ........ (38)

where P is the Pippian, or Cayleyan, of the cubic u.
Also (55)

Ifi j(AZ + Hm + Gn)~ + (Rl + Bm + Fn) || + (Gl + Fm + Cn)| I + 24Ps

s-{(A'+ 12A) ?;+...+ 2 (F+^F)^ +...}(/*+ my + »«), . (39)

MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 1G7

6A', . . . 6F', . . . being the second differential coefficients with respect to /, m, n of
the reciprocal (v) of u, 6W . . . ; while, otherwise, (21) § 21,

v = — 4 (A/2 + Bm8 + Cn* + 2Fron + 2Gid + 2HJw).

With these substitutions the equation to the locus of the CoTE8-points on the polar
lines of points in L, or Ix + my + »«, takes the form

2(F'+ 16F)- + . . .- 8PL* = 0, . (40)

a cubic, degenerating into a conic when the line L touches the cubic u, which will be
traced further on (§§ 50-53) by referring it to the line L and the two tangents to s at
the points L = 0, s = 0. But previously it will be of interest to show the signifi-
cance of the part

. . (41)

in the equation to v; and to add a few remarks on the relations (38) (39).

32. It is not easy to verify these relations by means of the invariants of (12) § 20,
because the variables which enter into them are perfectly general — except satisfying
ax + /3y + yz — constant. They are verifiable, the former with slight, the latter with
moderate labour by means of the canonical form of the equation to the cubic ; but
much more readily by means of the simultaneous forms of u and s, to which every
form of u and its concomitant are reducible, referred to in § 10. The verification is
therefore deferred until the reduction to those forms is explained in the sequel, and
will be found in the paragraphs cited.

33. The tangent to u at any one of the three points in which it is met by the
transversal L, being the polar line of that point, also touches the poloid s, the point
of contact with s being its CoTES-point — viz., it is that determined on it by the polar
with respect to * of the point in which it meets L ; or, otherwise, as the point in
which it is cut by the coincident tangent.

The general equation to the three tangents, at the points u = 0, L = 0, may be
obtained without the difficulties which would attend the direct investigation — hitherto
unattempted, at least successfully — through the property of their touching the poloid
also, as follows : the point of contact of one of the tangents being (x'y'z), its equation
is

or, u being a cubic,

168 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

But in virtue of its touching s also, if (xyz) is any other point on it,

yz — zy, zx — xz', xy' — yx'
must satisfy the reciprocal (§ 30) or tangential of s,

i.e.,

(Cy* + Bza - 2Fyz) z'2 + (Az2 + Cz2 - 2Gzx) ^ + (Ex2 + Ay» - 2H*7/)z'2

+ 2 ( — Ayz — Fz8 + Gxy -f Hzz) y'z' + 2 ( - Ezx + Fxy - Gy* + HJ/Z) zV
+ 2 ( - Cxy + Fzx + Gtyz - Hz2)«y = 0. • r: ». f&u* ..... (43)

The eliminant of these two quadrics (42, 43) with

lx -\- my -f- nz' = 0
in the known form

{(be' + b'c - 2f f)i3 + . . . + 2 (gb/ 4- g'h - af - a'f) mn + . ..}*
- 4 {(be - P)P + . . . + 2 (gh - af) mn + . . . } {(b'c' - f2)/2 + ...} = 0, .(44)

is, if

L = lx + my + ns,

(t> + 4PL*)2 - 16FLV.

34. For

(be - f2) Z2 + . . . -f 2 (gh - af ) mn + . . . = 36s,

4{(b'c'-f/2)Z2 + . . ,}^4{(BC-F2)x2 + . . . + 2(GH-

~ 4 (Discrt. of s) L2«

Hence

4 {(be -£*)?+. ..}{(b'c'- f'2)/2 +...} = 36P2LV. . . (45)

Observing now that

4A= — — -.(— V 4F-.

~" '

_

3; 3j; Bx By d

it appears at once from (30, 3 1 ) § 24 that — substituting s for <f>—

MB. J. J. WALKKK ON THE DIAMETERS OF A PLANE CUBIC. 169

4 ( _ Ayz - Fx> + Gxy + Hzx)
so that

4 (be' + b'c-2ff')/2 +...
a*w a*M 3

+ u» §? - w

9»M \

1

+ • • • f
\&u(d*\* ,^f/3«v 0 3^ a»a»\ K,

' 1 9^ h

o _ . • j

J » " *" + "

the negative term in which being the result of substituting n ds/dy — m dsjdz for x' . . .
in

is (§30)

-Ct>; .... (46)

consequently, (45) (46), sixteen times the eliminant (44) is

0 ..... (47)

As was mentioned, (38), §31, the terms multiplied by 4s within [ ]a in (47) are
equal to

-6PL,

so that the elimiuant is simply, after division by 36,

or

v (v + 8PLs) = 0.

MDCCCLXXXVIII. — A. Z

170 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

35. The presence of the factor v in this result is accounted for by the fact that the
solution has been really that of the more general question : "to find the locus of the
CoTES-points of all chorda of u (relatively to their intersections with L) which
touch s " ; discarding, therefore, this factor, the equation of the tangents to u at the
points

u = 0, L = Ix + my + nz = 0,

•

is

CT = v + 8PLs = 0 (48)

Reverting now to the form (40) of v, in which

the equation of the three tangents to u at the points

u = 0, Jx + my + nz = 0
is found in its standard form ; viz.,

wherein £ 77, £ are to be replaced in v and its second differential coefficients by I, m, n.

36. It will be satisfactory to verify the general expression for the satellite chord of
L by applying it to the canonical form of u, for which CAYLEY has obtained the form
referred to in § 6.

For

u EE OKI? -f- fty3 + C2;3 + 6ea;ys,

- 8(abc

8 ' = - 4&W + 32es(cm*l + Zm3/) + 48&ce%»»,

, 3%

i8-s = «*;

whence,

+ 48 ^) ^ = {abc(bd* - ZcamH - 2al>nH) - S(abc + 2e3)oem2n2}x. . . (50)
Again,

MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 171

4cam8/ + 4«&"S0 - 32(a6c + 2e»)^mn - 6(abc
+ cam8/ + a&n8/) + 32(a6c + 2e*)ePmn -f 8(abc +

whence,

Zn2 ) x • • • ( 5 1 )

From (oO) (51) there results, as the equation of the satellite chord of L,

(abc + %<?)S (bcl* — caw8/ — ZabiM — 6aem2n2)ar; .... (52)

which, ifa=6 = c=l, and e, I, m, n be replaced by m, a, ft, y respectively, agrees
exactly with the form given in the " Memoir on Curves of the Third Order" ('Phil.
Trans.,' 1857, p. 439).

37. The values of the second differential coefficients of tr and v used in the
preceding verification may be calculated from the formulae (10) given, § 19. Making
all the coefficients of u having suffixes vanish, «u . . . «ig ... for the present form of M
are found ; taid thence 2dio-/3£2 = ^u^u^ — Wgj8 . . . obtained. Next v, expressed
in terms of I, m, n as line coordinates, is determined as

viz., therefrom it is found that

v = 26W - 2 (abc + IGe3) 2am3n3 - 24e2/mn2 (bcP) - 24 (ale + 26s) eWn».

Finally, the second differential coefficients of v may be found. And the value of v so
found affords an independent verification of the relation (i.), §21,

v = — 4cr,

when in <r and v f, 77, £ have been replaced by /, m, n.

I may remark, in conclusion, that I first obtained the general form of v by considering
the question mentioned in § 35, determining in the resulting form (47) which of the
two factors represented v, which m, by examination of the special forms when the line
L and the tangents to * at the points L = 0, * = 0, were the lines of reference.
Subsequently, I observed the method of arriving at the general form of v given
§§ 30, 31, independently ; which completed the general proof in as simple a manner as
could be expected.

38. The triad of tangents m = 0 reappear as a part of the complete solution of a

2 2

172 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

question of considerable interest in connexion with the subject of this Memoir : what, if
any, pairs of polar lines of points on L are conjugate, in the sense of having a common
CoTES-point ?

The polar line of the point x'y'z' on L, viz.,

meets L in the point

, N , . . . 9?*' du' , 3tt' du' 3w' , du'

(^•(y)'(^ = n^-m^:l^--n^:m^-l^, . ,. . (52)

and the condition plainly is that the polar line of (x) (y) (z) should pass through the
original point x'y'z', i.e., that

when for (a:) (y) (z) in (du/dx) . . . the values just given in terms of x'y'z' (52) are
substituted. The result is at once obtained by means of the general theorem, § 24

(and § 27) ; viz., it is simply

u's = 0,
with

lx' + my' + nz = 0.

Now, at the three points u' = 0, lx' -f my' + nz' = 0, [(«) (y) (z)] plainly coincides
with (x'y'z') ; thus each tangent to u at those points is a double conjugate line, which
accounts for the factor u' in the result above.

39. The points

s' =0, Id + my1 + nz' = 0,

determine the unique pair of distinct or proper conjugate polar lines; viz., these are
the tangents to the poloid at the points in which it is met by the transversal L ; which
will be real only when L meets s in real points.

These two conj ugate tangents to s and their chord of contact L form, as before
remarked, an unique triad of lines, to which the cubic u and its system may be con-
veniently referred, in questions involving *, v, and certain other concomitants of L and
u particularly, as will appear in the sequel. Plainly the equations thus referred will
only be symmetric as regards two coordinates, however.

40. Through any point (x"y"z") in the plane of the cubic u and the transversal L two
chords of u can, in general, be drawn having it as their CoTES-point, relatively to their
intersections with L ; viz., the connectors of (x"y"z") with the points (x'y'z'} (...) in
which its polar conic meets L.

If (xyz) is any other point on one of these chords, then the coordinates of the
point in which it meets L, or

lx -{- my + nz = 0,

MB. J. J. WALKER ON THE DIAMKTERS OF A PLANE CUBIC. 173

will be proportional to

n (x"z - z"x) - m (y"x - x"y) . . .
or

Lx" - L"x, Ly" - L"y, Lz" - L"z,

and the substitution of these coordinates in

,

+. .. = 0

gives the equation of the two chords in question, in the form
. (Lx" - L"*)* * + . . . + 2(L/' - L"y) (Lz" - L"z) ,, + . . . = 0,

wherein L stands for Ix + my + nz, L" for Ix" + my" + nz".
The equation of that one of the pair which cuts L in the point (x'y'z) being

(y'z" - z'y") (Lx" - L"x) + (z'x" - x'z") (Ly" - L"y) + (x'y" - y'x")(Lz" - L"z) = 0,
that of the other will be either

+ {(x'z" - z'x") ^ + 2 (y'z" - z'y") ^} (Ly" -L"y)

(Lz''-L"z) = Q, . . (53)

or one of the two analogous forms obtained by interchanging x", y" or *", z".
41. The reciprocal of the equation (52) beiug, to a factor,

ra^av' /avMfi J^L.^L ** w \

I a/'» a*"* ' \$y" w)\l * \$# &<' &> sy" ~ w a/' a^y

" • u>

if x"y"z" should be a point on the poloid *, the two chords having it as their COTES-
point will coincide — as would otherwise be inferred from the consideration that any'
point on s might be regarded as the point of intersection of two coincident tan en ts
become a " double chord," and its equation will be either

"-L-,)=0, . (54)

or one of two analogous forms resulting from the interchange of x", y' or x", z".
Otherwise, the equation of the double chord through (x"y"z"), a point on * is

174 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

42. But considering a point (x'y'z) on L, the double chord through it is, plainly,
that connecting it with the point in which its polar line touches the envelope s, viz.,
this is the polar with respect to s of the CoTES-point on that polar-line (§ 29). Its
equation, thus considered, is therefore

/3s \ /a»\ /a»\

with the coordinates of the CoTES-point referred to substituted for xyz in (ds/dx)
(3s/3?/) (3s/3z); those coordinates being (36). § 29,

a/ ;k' _ a^a_^ a«' ay 3«' a?*' a/ aj</ jv ay
a7 37 ~ a/ a/' a^ ^ 7 a** 3a/' s/ 3^' ~ fa W

The resulting equation to the double chord is, therefore,

_
a* a*' ay "~ a/ 3 M ~ a* a a?

, >

=

43. This being cubic in x'2/V, and of five dimensions in the coefficients of u, its
envelope, obtained as the condition that Ix + my' + nz' = 0 shall touch the ternary
cubic in x'y'z', will be a curve of order 4 in xyz, of order 6 in Imn, and of order* 8 in
the coefficients of u.

This envelope, as well as that of the complex of chords of u, connected with the
polar line of a point on L by having their CoTES-points on it, but not passing through
its pole, will be more conveniently considered by means of special forms of s and u,
to which every cubic may be reduced.

* This would appear to be 20, but another form of the equation of the double-chord, into which the
coefficients of u enter only in the second degree, may be obtained from that of s (8 bis) given in the
Note to § 22. Combining (i) and (iii) of that Note, the coordinates of the point of contact of the
polar line of (x'y'z) with the poloid (D, fig. 1) are at once given as proportional to

respectively ; hence, the line joining this point with (x'y'z') is

w' (aDw'j + y Dw'2 + z DM'S) — DM' (xu\ + yu'3 + zu's) = 0.

But, if 6a . . . 6/ . . . stand for rftuldx- . . . &u/dy 3« . . . ; \, /«, v for ^m — ftn, »n — 7?, fil — am,
respectively,

DM'! = \a' + ph' + vg', D«'s = \h' + fti + •/', Du's = \g' + /.

while, since Ix' -f my' + nz' = 0

t'ft — y'v = z' (an — <yl) — y' (fil — urn) =— I (a: + fly' + 72') =— ZA'; x'v — z'\= — mA'; y'\ —x'fi= — »iA'.

Substituting in (i) the equation of the double-chord (DE, fig. 1) becomes divisible by A', and is

a»'\ , a^u ,a»« , a»«\

(5b bls)

— (Note added April, 1888.)

MK. J. J. WALKER ON THE DIAMETERS OF A PLANK CUBIC. 175

44. The question of the angle (6) between the polar line of a point (x'yV) on L and
the " double " chord of the pencil through that point is of some interest, as a generali-
sation of the question of the angle which a Newtonian " diameter " makes with its
" ordinates."

The polar line being

8u' du' 8u'

and the double chord (56)

what may be called the " direction coordinates" of the two lines are (the angles of the
triangle of reference being A, B, C)

and, using A' as in § 20 ; A ... H as in § 30,

The angle between two lines whose direction coordinates are X/tj/, X'^V is given,
generally, by

tan 6 = 2 (nv — p.'v) sin B sin C/3SXX' sin A cos <4, ... (57)

and in the particukr case now under consideration

. . . + 2F^'^+ . . .11

. . ,)~j I Sx'sin-.-l cos/41 sinC y-, — sin^ 5-7)

'

• (58)

45. If x"y"z" is the point of contact with its envelope, the poloid *, of the polar

line of a point x'y'z on L or

Ix + my + nz = 0,

from the forms of the equation of that polar line it is evident that

176 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

whence at once x", y", z" are given in terms of x'y'z' by

x" : y" : z'

A ... 2F ... being the coefficients of the reciprocal of s.

Conversely, if x, y', z are to be expressed in terms of x", y", z", they are the
coordinates of the point common to the two conies

3s 9w 9s 9w _ 3s 9?* 9s 9w _ 9s 9w 9s 9?*
3z 9y 9y dz 8x 9z dz dx 9y tte dz 9y
and the line

Ix + my + nz = 0.

46. But the point x'y'z' is more readily determined by the double line of the pencil
of chords through that point, having their CoTES-points on the tangent to the poloid
at x, y, z, which has been shown to be (54)

- L"») + 0*" - L-«) = o.

At the intersection of this and L

3V' fflu" 9V

'"H 2//"= U)

gving

x' : y • *

9V 9V , 9V 9V 9V . 9V

or either of two other analogous sets of values.
47. Thus, when L is taken as z = 0, and

u =

of the coefficients A ... 2H of the tangential equations of the poloid s, all vanish
except

40 = - «262, 4H = 2a6e2 ;

MH. J. J. VVALKKR ON THK DIAMKTKRS OP A PLANE CUBIC. 177

giving for the point of contact of the polar line of (x'rfo)

x" : y" : z" = betf* : neat* : - ob-sij : ..... (61)
and conversely

x : y' = - ez" :ox" = by": - ez". = '•; • WM? 3 (62)

IV. SPECIAL FORMS OF CUBIC AND POLOID.

48. T now treat some of the questions, hitherto considered without reference to any
particular form of «, in more detail, by means of special forms in which the line L is
used as one of the lines of reference — say

L = «z = 0; ........... (63)

and for the two other lines of reference I take 1st, the tangents to the poloid * from
the pole of L, as

x = 0, y = 0.

The conic s is now (10) § 19 — »s being dropped — reduced to

(a363 — e'-)z2 + (ab — a^xy = 0,
with the conditions

a^-a^O, (i.)

&a2 — &J2 = 0, . ........ (ii.)

ba3 + a263 — 261e = 0, . . ; •; j ...... (iii.)

a63 + ctg&j — 2aje = 0- • ... - . ; .-., (»v.)

From the first and second of these,

a2(ab — Uj&j) = 0,
whence

a3=0,

since ab — asbl = 0 is excluded by the form of s.
Hence, from the second

*>i=0,
and from the third and fourth

fls=0, 63=0,

MDCCCLXXXVIII. — A. 2 A

178 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

since 6 = 0 or a = 0 is excluded by the form of s, now reduced to

s/n*= -e*z*+abxy, ... ...... (64)

and u is reduced to the terms

cz

3

Gexyz ...... (65)

49. The line z = 0 meets u in the points

x : y = 6* : - a*, ' x : y = 3&* : - S'a*. x : y = $'&» : -
(£, A' being the imaginary cube roots of — 1, so that

a + 3'=l, A»'=l, S3 =-.»', ys=-A, ^ + ^
and the tangents to u at these points are

o*6*y — 2ez,

(66)

the common equation to the three being, by actual multiplication,

a*ba? + a&V — SeV + 6a6ea«/2 = 0, ...... (67)

or, restoring lapsed factors,

w = o?Wn*u - { SaWcjn*x + SaWcjfy + (a262cw* + Babe V)z] nV =0. . (68)

Further, the same equation may be thrown into the form

ra = a262n6 (oua? + &/ - 2«ryz) + 8a&en3 ( — e2n2z2 + a6n2a;?/) 712 = 0. . (69)

50. The tangential equation of s is now simply

= 0,

where

4C = - a262n4,

4H = 2abe*n*.

MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 179

The general value (37) 5 30, of v, the Cotesian of L, now reduces to

v = _ (03* + ty 4. „? + 3Clz»x + 3cjZ8y + Gexyz) ( — «8ft8u«)
- n8z8 { - a8ft*n* (c^ + c*y + cz) + 2afte8n* (ez)\
+ 2iiz { — aWn* (cz2 + 2^zx + 2c$z)}

+ 2 (e*n*zs + aft/i%) 2 ( - aftn8) em*,
i.e.,

v = aW (no* + ft?/3 — Zexyz) ...... . . (70)

a cubic having a node at the pole of L (now 2 = 0) with respect to *, the tangents to
s being also the nodal tangents of the Cotesian v.

51. The comparison of the equations (69) (70) of m and v verifies the relation,

(48) §37,

m = v + 8PL».

since, in the present form (65) of u, the Cayleyan P, for the values of the line
coordinates / = 0, m = 0, is simply

P = often3 ........... (71)

52. Eliminating y and x successively between

•

v = oar5 -|- by3 — 2exyz = 0,

.s = — c222 + abxy = 0,
there result

(a*ba* - e3?8)8 = 0,

showing that v has triple contact with * ; viz., when the loop of v is real it touches s

at the point

x:y:z = (««&)-» : (aft8)-* : e'1 ;

or at the point determined by the real lines, or any pair of them,

(azft)»a; = (aft*)'y = ez.

Now, making 2 = 0 in u and v, the points in which L meets these cubics are
determined by

ax* + fey3 = 0,
the real one being

o»x + ft'y = 0,

Z=0,
2 A 2

180 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

so that the connectors of the pole of L (with respect to s) with this point and the point
of contact of v ivith s are harmonic conjugates relatively to the tangents to s from that
pole.

53. The polar line of P (xfy'o), a point on L, meets the Cotesian v in three points
(say, D0 its CoTES-point) and two other points (say, Dlf D2), while it touches the
poloid s in a point D. Now D0 is also the CoTES-point on this polar line relatively to
the point D and the cubic v : to prove this —

The coordinates of D are, (59), § 45,

x : y : z = bey1'2 : aex'2 : — abx'y', . . . ..... (72)

and

. t ..{'.. (73)
becomes, for those values of the coordinates of D,

3,y 82

' : Sax'*

viz., the u-polar line of D is

3 (ax'3 + fy'3) (y'x + x'y) - x'y' (ax'*x + by'2y + Zex'y'z) = 0 ; . . (75)

but

00;'% + by'2y + Zcx'y'z = 0 ........ (70)

is the w-polar line of (x'y'o), and

y'x + x'y =0, . . . .tl.^j aaiJ", /B.iJ x*' (77)

the s-polar of the same point, cuts it in its CoTES-point, D0 (35), § 28. Thus the
three lines (75) (76) (77) meet in one point (D0), and this is the «-CoTES-point
relatively to P, and the v-CoTES-point to D of (76) ; whence

3 ' + '

I

DD0 • DD0 n DD,
or

2 1 1

DD0 ~ DD, "

as verified very exactly in figs. 2, 3, i.e., the CoTEB-point on the polar line of a point
in L is harmonic-conjugate to its point of contact with the poloid of L and u relatively

Ml! J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 181

to the two other points in which it meets the Cotesian v ; and is thus discriminated
from them, when all three are real If one only is real, that of course is the COTES-
point.

54. The general form given above (49), § 35, for the satellite chord of the line L,
may be verified from the form it takes for the present one of u (65), the reciprocal of
which — as far as the terms giving those which do not vanish in its second differential
coefficients for £ = 0, 17 = 0, £ = n — is

v = a a>,C- + 6as&fi,£V - 24«te«£*£, ; ... (78)

viz., for those values of £ 77, £,

while the reciprocal of s ( = — e22z + abxy) being now

4<r = -

Also

3*« 3*i» , 3*jt .

^j = 6ar, ^=66y, ^ = 6 (Cl« + c# + cz),

9*U

With these values of the second differential coefficients, which do not vanish, for the
present forms (64) (65) the satellite chord of L, or nz = 0, is n4 multiplied by

(2alri\ + *) ax + (2crbc2 + *) by + (Sazb- - 4a«68)

+ 2( - 4afee8 + Safte3) « + (* + *) (ey + cp) + (* + *) (ex + c,?),

or, identically,

Cja: 4- 3a262Cjt/ + (a262c + Sate8) z,
which agrees with the form in cr (68).

182 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

55. What has preceded is, substantially, a verification of the relation (39) § 31 ;
but, to make it more clear, observing that the only one of the coefficients A ...
F ... of the reciprocal of s which does not disappear from the sinister of (39) is C, in

the term

*\

96C« P = - 24a262n5 (3

and adding to this

144Ps =

that sinister becomes
— 72 nz

Again, the dexter is (L being now nz)

— nz {(12o68c1n*) 6ax + (12a2&c2n4) 6% + (30a262/i4 - I8a^2«*) 6 (op + c.2y '+ cz)

+ 2 (— 24a6e2n* +
or, identically, also

- nz {UtaWn^CiX + cjy) + (72a262n4c + 144a&e2n*e)z}.

Lastly,

o w

the only term in the sinister of (38),

= — 4aben*z = (71) § 51, — 4PL.

56. It has been shown that v and u meet the transversal L in the same three

points (§ 52),

aa? + by3 = 0, z = 0,

of which one only is real when the line L meets its poloid s in real points, the condi-
tion of the lines of reference x = 0, y = 0, at present used, being real. That this
would be the case appeared from the fact that the discriminants of the cubic (12) and
quadratic (13) § 20, whose roots are proportional to the intercepts of L between a
certain point on it and its intersections with u and s respectively, are of opposite
signs.

57. The tangents to v at the points in which it is met by L are plainly obtained
from those to u at the same points, by changing e into — e/3 ; viz., they are (66)

I

MB. J. J. WALKER OX THE DIAMETERS OF A PLANE CUBIC. 183

eliminating z between this equation and v, the result is

= 0,

viz., the points in which v is met by L are points of inflexion on v, as was manifest
from the form of its equation. When the node of v is real, only one of these points of
inflexion is real ; but if v is acnodal, all three will be points of real inflexion. Thus
much is known of v generally. In Plates 7, 8, it is figured for two cases when L is
the line at infinity, as will be further described.

58. The coordinates of the CoTES-point on the polar line of (x'y'O)

by'*y + Zex'y'z = 0
may be found as those of its point of intersection with

y'z + z'y = 0,
the s-polar of (x'y'O) ; viz., they are

x : y : z = — 2ea/y : 2ex'y'* : «*'8 — by"A • ..... (79)
or, what is the same thing (36), p. 165,

i a*' a«' &' du

x :y :z =

a/ ay' ay

The double chord (56) §42, through (x'y'O) is determined as the polar with respect
to * of that CoTES-point (79) on the polar line of (x'y'O). Its equation is, therefore,

abx?y'(— y'x + x'y} + e(ax^ - by'3)z = 0.

Arranging this as a binary cubic in x'y1,

aezx* + abyx'*y' — abxx'y" — bezy* =0, ..... (80)

the discriminant will be the envelope of the double chord as the point (x'y'Q) varies on
the transversal L, viz.,

3w = 4ab (a.c- + 3eyz) (by* + 3exz) — (abjcy - 9e*z*)°~ = 0, . . . (81)
or, developed,

wSa'Mzy + 4d-bex*z + 4a62ey3z + I8abe-xyz2 — 27«V = 0.

59. If the first derived of the cubic (80), i.e.,

3ex'*z + Zbx'y'x- by'-x = 0,
-f 2ux'y'x-ax'-y = 0,

184 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

are combined, the coordinates of the point of contact of the double chord with its
envelope will be obtained in terms of those (x'y') of the point on L, through which it
is drawn, viz., eliminating z between the equations (82), the point of contact is deter-
mined by the line

(Vox'3 + by'3) y'x — (ax'3 + 2by'3)x'y = 0. . ... . . (83)

The intersection of the double chord with L lies on the line

y'x - x'y = 0, . . .., ;.,.., . . ;. . . (84)

and its two intersections with the poloid on the lines

bi/*x - ax'*y = 0, ... .'"" !,'' : . " . . (85)
ax'2x — by'*y = Q, (86)

the latter being its CoTES-point, viz., the point of contact with s of the polar line of
(*', y', 0)-(61), §47.

Now these four lines through x = 0, y — 0, form a harmonic pencil, since, writing
(84) (86)

y'x — x'y=p,

ax'zx — by'~y = q,
[giving

(ax'3 — by3) x= — by'-p + x'q,

(ax'3 — by'3) y = — ax^p + y'<l\
then (85)

bi/*x — ax'*y = (a2x6 — b2y*) p + (%'3 — a%'s) x'y'q,
and (83)

('2ax'3 + by*) y'x - (2by'3 + ax'3) x'y = {(- 2ax* - by'*) bij'3 + (2by'3 + ax*) ax'3}p

+ {(2ax3 + by'3) x'y - ( 2by'3 + ax'3) x'y'} q
= (a-x'& — lry'5)p + (ax3 — by'3) x'y'q ;

viz., the later pair are harmonic-conjugate with respect to the former pair_p, q. Thus
it appears that the CoTES-point on a double chord, and its intersection with the line L,
are harmonic-conjtigate points with respect to its second (or " empty ") intersection with
the poloid s and its contact with its envelope w, as stated, § 7. This is shown in
Plates 7, 8, in the case in which L is the line at infinity, by D being the mid-point
of CCi.

60. Returning to the equation (81) of the quartic w, if y and x be alternately
eliminated from either

MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 1- •"'

ox9 + 3eyz = 0, or by* + 3ezx = 0,

and

abxy — 9e?z* = 0,

there result

a*bx* + 27e82s = 0,

ab-if + 27c823 = 0,

showing that w is a tricuspidal quartic, two of the cusps being imaginary when the
line L meets the cubic u in one real point only, the real cusp being at the point

x:y :z =
on the line

a'x — bky = 0,

which is the cuspidal tangent, and is harmoiiic-conjuyate to the connector of the s-pole
of L and its real intersection with u, relatively to the tangents to a from that pole.

The curve w is shown in the figure, Plate 7, in the case when L is at infinity and
the cubic has a single real asymptote.

61. Through any point x"y"z" on the polar line of a point x'y'z' in L, two chords of
u pass which have the first (x"y"z") as their Corns-point ; viz., the chord which forms
one of the pencil through x'y'z' and the other (53), § 40 the line,

(yV ~ a

"

x"y"z" being subject to the relation

and the envelope of this line, or " alien " chord, may be found for any assigned
form of «.

In the case when L is 2 = 0,

s = — e*z* + abxy,

u = ay? + by'A + cz3 4- Sc^x + 3c.,z-y + Gexyz,

i 8*u »i &" "

« ate"* ~ C ' * 3^%" = Z '

MDCCULXXXVUI. —A. 2 B

186 MR. .7. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC,

and

since

The equation of the " alien " chord is then simply

ax"z"y'(zx" - xz") + (ax"z"x' + 2ez"Y) (zy" - yz") = 0
with (76)

ax'V + fy'V + 2ezyz" = 0,

in virtue of which the eqtiation of the " alien " chord reduces to

ax'x"(xz" - zx") - by'y"(yz" - zy") = 0 ; . . . ..'(87)
the envelope of which is

ab (y'x + x'y}- + 8e (ax'-x -f by"2y + Zex'y'z) z = 0, . . . (88)

a conic touching the line L at the point which is harmonic -conjugate to (x'y'O) with
respect to the intersections of L with the poloid s ; and touching the polar line of
(x'y'O) at the point in which it is met by

y'x + x'y = 0,
viz., the point

x : y : z = — 2ex'2y' : 2ex'y'2 : ax'3 — by'3,

i.e., at the CoTES-point on that polar line (79), § 48.

62. The conic envelope just found may be called the "satellite-conic" to the polar
line of x'y'z'. Of course, this conic touches also the double chord of the pencil through
x'y'O, since it is at once an " alien " and a proper chord of the polar line of (x'y'Q).

Arranging the equation (88) to the satellite conic as a quadratic in x'y' it is

a (by2 + 8ezx) x*2 + 2 (abxy + 8e222) x'y' + b (ax2 + 8eyz) y® = 0,

giving at once, for the envelope of the system of conies satellite to the complex of
polar lines of the points on L, the quartic

ab (ax2 + 8eyz) (by2 + 8ezx) — (abxy + 8<rV)3 = 0,

which developed is

8ez (atba? + a&V - 8e*z* + Gabexyz) = 0 ; ..... (89)

viz., the envelope of the satellite conies is the system of four lines L and the tangents
to u at the points L = 0, u = 0 (67), § 49.

The explanation of this is 'that at the point of intersection of the polar line (76)

MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 187

with one of the tangents (66), that tangent is itself the " alien" chord for that point :

for (76) (66)

aa»x" + by'sy" + Zex'y'z" = 0,

a«6».T" + aWy" - Zez" = 0,

give

x" : y" : z" = -2&»ey' : 2a*cx :

and the substitution of these values in (87) gives for the " alien " chord, after dividing
out the extraneous factors 2ftbex'y'(a*xf — fcty), simply

aWx + cWy — (2ez = 0 ;

viz., the tangent to u at the point x : y : z = 6* : a* : 0.

Thus the system of satellite conies is inscribed in the quadrilateral formed by the
line L and the three tangents to u at the points in which L meets it. A satellite para-
bola is shown in Plate 8, when L is the line at infinity.

63. The tangents to * from the pole of the transversal L being real only when L
meets the cubic u in a single real point, it is desirable to use as lines of reference
another pair in connexion with L, which shall be real in all cases for purposes in which
their reality is essential.

Consider the connector of the pole (C, fig. 1) of L (z = 0) with a real point (A) in
which L meets u, taking it, say, as y = 0 ; and let the s-polar of the point z = 0,
y = 0, be taken as the third line (BC) of reference, x = 0. The equation of s is then
reduced to the three terms, remembering that still / = 0, m = 0,

(a&,-a8«)^+(&a2- V)2T + W>3-«2)«2=0 ..... (90)

since the triangle x = 0, y = 0, z — Ois self-conjugate with respect to that conic ;
and as conditions for the terms yz, zx, xy disappearing from its equation

&a3 + «A = 2 V> ]

«&3 + «s*>i = 2a^, I ......... (91)

ab — aA = 0.

But since the equation of the three tangents to u at the points

u = 0, z = 0
is now of the form

ku — Jfc'z2 = 0,

and one of them passes through the point

y = 0, 2=0,
2 B 2

188 MR. J. .T. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

the term y? must disappear from u, or

a=0,
which (91) necessitates either

a.2 or ?>i = 0 ;

but the supposition . a., = 0 would imply a second of the tangents to u at the points
u = 0, 2=0 passing through the point y = 0, 2 = 0, so that

6, = 0,
and therefore, by the second of the three conditions (91)

e = 0 ;
whence, by the first,

b:a2 = -&3:«3 ....... • • (92)

64. Thus the equation to the cubic is reduced to the terms

by* + C2s + sc^fy + SagO^z + 3b<y*z + Sc^x + 3c222y =0, . . (93)
with the relation (92) ; and (90) the poloid to

+ «3&322 = 0,

or ....... (94)

the reciprocal of which is
where

A = 6a2a363n4' = — a32632n*,
B = — a22a3&3n4 = ba2a^n*,

C = - bafn*,
or

A : B : C = asb3 : a32 : — a^.

(95)

65. For the above forms (93) (94) of u and s, the values of their first differential
coefficients, with the coordinates (x'y'O) of a point on L substituted for xyz, are

9s 9s 9a

S'Jy'S-

8tt OU OU , , 7 /n , I* '9 I 7 „ ,'2 .

a? : 97 : a? = 2asa;y : &y + a^ : a'** + & '

MB. J. J. WALKER ON THE DIAXIKTKKS OF A PLANK CUBIC. 189

giving (36) for the CoTES-point on the u-polar line of that point

x : y : z = bajfy + My3 : a^'8 + o^y'2*' : - a,V8 - S&a^V. . (97)
The values give

...... (98)

in virtue of the relation (92)

viz., the above is the equation of the Cotesian for the form (93) of u now used.

66. The Cotesian v for the forms of u and. s at present employed (93, 94) has been
found quite independently of any application of the general equation (37) given
above ; but it may be of interest to test the general formula by this result.

The only terms in the general expression which do not disappear in virtue of

/ = 0, m=0, F = 0, G = 0, H = 0,
are four times

\ -

n>

which, taking the values of u, s, A, B, C (93, 94, 95), for the present case, is identi-
cally equal to four times

8V (by* + cz8 + Sa^y + Sa^z + 3b3y*z + 3c,22.r + 3c2z2y)

+ a^) + as2a363n4 (by + bsz) -f bafn* fax + c# + cz)}
a,6,z2) n* [ba^(a^y + a3z) + (- a22n2) (by + 63z)}
= feaz3n6 (by3

Rejecting the factor 6a33Ha, and applying the relation (92),

a8&3/& = — 03, or 6o3/a2 = — 68,
this result is, as before (98) ; or, with the rejected factors restored,

v = (by3 -f Sogafy - a^z - b^z) X 4&a3sna, ... (98 bis)

a cubic having a node at the point

x = 0, y = 0,

190 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

viz., the s-pole of L or z = 0 ; the nodal tangents being

or (92) - ......... (99)

- by* = 0.

The node, therefore, will be real or unreal, as »2, 6 are of the same, or unlike signs.

67. Since, now (94)

as&322,

it appears that the nodal tangents (99) of v are the tangents to s at the points
* = 0, L, or 2 = 0, and will therefore only be real when L cuts its poloid in real
points, as shown before (§ 50).

The transversal L meets the Cotesian v, as well as the cubic u, at the points

2 = 0, y = 0, z = 0, byt+Sa^^O, . . . (100)

which last two will be real only when v is acnodal, or when L does not meet s in real
points.

68. The tangents to •

u = &y3 + ex3 + 3ckffy + 3a3o:22
at the points u = 0, L, or z = 0, are (48)

CT = v + 8PLs
if, as just above,

viz., in this case, then,
tsr/4 = ?>a2s

- 4a326323/a22), . . (101)

since, for the above form of u, with £ = 0, 77 = 0, £ = n,

P = — 2&a2a3w3. .. . . . , .. :. . (102)

Now, the terms within brackets in CT are equal to

(by - &32) (&V + 3&a2a;2 -f 46632/2 + 4&3222)/&2

or J. , (103)

(by - 632) (6y + v' - S&tt^ + 2&32) (6y - vx

J

Mil. .1. .1. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 191

which factors may be identified with the three tangents to u at the points u = 0,
2 =r 0 ; viz., the points

2=0, y=0; 2 = 0,i/=±v — Sa^/b x.

These will be real points only when ba3 are of unlike sign ; i.e., when L does not
meet s in real points, and v is acnodal.

69. The Cotesian (98) — discarding the factor 4&ai8»<—

by* + Sa^y — a^z — b^z == 0

and the primitive cubic u (93) are plainly intersected by L, or z = 0, in the same
three points, and the tangents to v at these points will be derivable from those to u
by changing therein

3«3 into — a,, 3&s into — 68 ;
viz., their equations are

3by + 6,2 = 0, 1

)> ...... (104)

± 3 V — 36ojO; + 3by — 26g2 = 0. J

70. The equation to s (94) being

- ba#? + 6y — V2* = 0,
shows by its form that the tangent to u at the point

(L or) z = 0, y = 0,
viz., the line (103)

ty — &32 = °.
or

<W + <hz = 0,
touches s at the point

x : y : z = 0 : 63 : 6.

71. Again, throwing the equation to s into the form

3s = - (&V + 36aj^ + 46%2 + 4 bjz*) + 462y2 - 4&&3yZ + 63228,
= { - 36 (a,a!« + 36y2) + 4 (26y + 6,2) (% - bsz) } + (26y

the terms within brackets being (103) the tangents to u at the other two points at
which L meets it, it appears that these lines touch s on the line

26y+ 632 = 0; ......... (105)

192 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

i.e., at the points determined by it, and

= 0, ....... '. . (106)

the polars of those two points of the three common to L and u.
For these points are determined (93) by (z = 0) and

67/2=0,
or _

x:y:z = b:± \/ — 3ba.2 : 0,

the polars of which, with respect to (94)

a<?x* + ajyf — a36322 = 0,

are _

a2bx ± b \/ — Sbag/ = 0,

or

2 = 0.

72. The equation of the double chord (§ 41) of the pencil through x'y'z on L being

generally (56)

^ Bs 1 3s ck 3s 3jA _

^" u>

is, in the present case, from the values of the differential coefficients given (98), § 65,
(a^-b^)(y'x-x'y) + (3l^2-a^)x'z = 0, . . . (107)
or, arranged as a binary cubic in x'y',

3M x'y"2 - ^ = o. .

The discriminant of this last form gives the envelope of the double chord as the
point x'y' describes the line L, or z = 0 ; viz., it is

3w =

- {afx* +3 (a2y + a3?) (by + 3bsz)} (Bba^ + (by + 3b3z)*} . (108)
or

3w = — 36a2 (agx3 + 36y2)2

+ 2a2 (a<pr + 3%2) (2by — 3&3z)2 - 3 (2a^ - a3z)(2% - 363z)3, ( 1 09)
in vktue of the relation (92)

MB. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 198

The former value of w shows that the polar of y — 0, 2 = 0, one of the three points
common to L (2 = 0) and n, viz.,

*=0,
is the tangent at a cusp at the point

x=0, by + 3bsz = 0, . . ''; ' ..... (110)
the tangent meeting w for the fourth time at the point

x = 0, a.,y + a32 = 0, or by — 63z = 0 .....

73. Now, since any real intersection of L and M might be taken as the point y = 0,
2 = 0, it follows that when these three intersections are real there will be three real
cusps to w, the cuspidal tangents being the polars, with respect to the poloid s, of the
points common to L and u.

This is shown, independently, for the two intersections of L and u other than
y = 0, 2 = 0, by the second form of w, which gives simultaneously (109)

\ ^F • 3 / \ »/ 3 /

of which the former has been shown to be the (square of the) s-polars of the two
intersections in question (106).

74. It is plain from the forms of w that the triad of coordinates (110), (112) satisfy
the first differential coefficients of w ; in fact these are

- (2by - 3bsz)2}

- 6632) -y(by + 363z)2},
(by - Gb^) - y (2by -

fk

~ = 4a3 (Sajftx8 + (by + 3&82)3} (2by -

- 40, {36(a^8 + 3&y«)(2fy - 3b3z) - (iby
Also

, i = 4a,8 { - 36 (a^* + by'-) + (2by - 86,2)*},

MDCCCLXXXVIII. — A.

194 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

giving for the cuspidal tangent

2

when in -., • • • are introduced the values x = 0, by = — 3bsz,

as in (110) ; when the values a/x* = — 3by~, 2by = 3bsz,

= 0

as in (112) above. The same values substituted in the second differential coefficients
will be found to make

_ „
"~-

75. Considering the intersections of w and s, the elimination of x2 between their
equations leads to

(Wy* - 3bbs*yz2 - b^z*) z = 0,
or

(by - b,z) (2by + 63z)2 z = 0. ^ ^,..(113)

Keferring to the equation of w (108), § 72, it appears at once that

«a2/ + «32 = 0>
or (92)

ty — &sz = 0,
gives

x2= 0,

viz., the tangent to u at the point y = 0, z = 0, (103), § 68, touches w at the point

x :y:z= 0 : bs : b,

in which w meets s ; and the same line has been shown (§ 70) to touch s at that point.
From this it would follow at once that the other two tangents to u, at the points
L or z = 0, u = 0, touch both w and s at the same points ; but this is shown by (113)
above, since w and s appear at once from it to have double contact at the points in
which s is met by the line

2by + b.z = 0,

MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 195

which has been proved (§ 71) to be the chord of contact with s of the two tangents

referred to

(by + 2&sz)2 = 0.

76. Generally then, when the transversal L meets the cubic u in three real points,
w the envelope of the double chords of pencils of lines through points in L, is a
tricuspidal quartic having triple contact with s, the poloid of L, at the points in which
it is touched by the three tangents to u at the points L = 0, « = 0 ; the cuspidal
tangents being the s-polars of those three points; and the fourth pair of points
common to w and s lie on the line L.

In Plate 8 the envelope w is figured for the case of L being the line at infinity
and the cubic u a modification of the Cissoid, as more particularly described
below (§ 91).

The property of the double chord touching w in a point which is harmonic to its
second intersection with the poloid * relatively to its CoTES-point and intersection
with L, has been proved in § 59 by the use of other lines of reference, with the reality
or imaginariness of which it is, plainly, unconnected.

77. When L touches w, it and the tangent at the point where it again meets the
cubic being taken as

2 = 0, y=0,

and their chord of contact with the poloid s as

x= 0,
the equation of the poloid (10) is reduced, since 1 = 0, m = 0, to

(06, - as8) a2 + (6a3 + as&3 - 2\e) yz = 0,

the conditions for which are (ib.)

60, -&,« = <>, ........ (i.)

«,863 — e2 =0, ........ (ii.)

a&s + «A — 2<V = 0, ....... (iii.)

ab — a2&! = 0 ......... (iv.)

But since y = 0, z = 0 is on u,

a=0;
therefore, by (iv.),

&i = 0,

since o2 = 0 is excluded by the form of s ;

2c2

196 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

therefore, by (i.),

6=0,
and by (iii.)

e=0;
finally, by (ii.)

as=0,

since 63 = 0 is excluded by the form of s.

Thus, it appears that the cubic u and poloid s are reduced to

, '". . . . (114)
'^V . . (115)

The form of u shows that x = 0 passes through the point in which 2=0 touches
it; or the poloid touches the cubic at the point of contact of the transversal L.
78. The coefficients of the reciprocal of s which do not vanish are

4A = - a22632< 4F = 2a2363n* ;

which values, with I = 0, m = 0, give (37), § 30 (dividing out nc),

v= — 22{— a.2

c222

b&z) ; .......... (116)

the locus degenerating in this case into the line L and a conic having double con-
tact with s at the points where the s-polar of the point of intersection of L and u
meets it and s.

79. The double chord through any point x'y' on L or z = 0 (56),

,

^ "-" =

in this case, wherein

3s 3s 3s
£ : By : £ =

18

^^m O ft nf» t jj, ^^^ fi i\ /y» *)/ I _L /j >y | O/~| /» 'V* <j/ •* JL O/Tf /l T* -a/ *I ' /l •J/ i ___ O^v */>« o j> \ _!.-.. /^

««'2 \ vto"Qt*' T / \ "Q^ \ ^jLt-o'-'Q'*'' */ r^ ^(tnt/o*l/ w I ^^ C/o u I — , •>(/ ' 1* I ^^— ^/,

or

= 0 (117)

MR. J. J. WALKER ON THK DIAMETERS OF A PLANE CUBIC. 197

the envelope of which as x'y describes the line L or 2 = 0, is

w = ajz8 + 8&37/z = 0, ........ (H8)

another conic having double contact with * and v at the same points.

80. Lastly, the chord through a point (x"y"z") on the polar line of x'y, any point
on L which has the former as its CoTES-point, but which is not one of the pencil of
chords through (x'y'z'}, is (53) — when 2' = 0, and L is 2 = 0 —

^' {,/z" (zx* - *-) + x'z" (z,f - yz") } + 2 y'z" (z,j" - yz") - 0.

But here

9V'

whence the chord in question is

-y'y"z"x-(x'y"+1y'x")z"y + (x'y" + 3y'x")y"z = 0. • . (119)
81. The polar line of x'y'Q being

the envelope of the chord (119) above, as x'y'z" describes the polar line just
mentioned, determined as the condition that

ya!" + ajy + b&'*z" = 0
shall touch

x'2y"2 - (y'x + x'y)y"z" - 2yyz"x" + 3y'zx"y" = 0,

is — dividing out the factor i/* —

(2a.fr'x -f Sfejy'z)8 + 8a.2bsxt3yz = 0, ...... (120)

or, as it may be otherwise written,

'x - btf'z)* + 86, (Za^y'x + a^ + b^z) 2 = 0, (121)

a conic inscribed in the triangle formed by the Hues y = 0, L or 2 = 0, and the polar
line of the point x'y on L.

198 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

V. NEWTONIAN DIAMETERS.

82. If L is the line at infinity, then

1 : m : n = sin A : sin B : sin C,

= a. : j8 : y\

and to the CoTES-point of a chord through a given point on L (x'y'z') corresponds
the " mean point " on a chord parallel to a given line, the " direction coordinates " of
which are, § 44,

m-ftl *£-y£ ft-™i\ ..... (122)

the line being written

& + yy + & - 0 ;

viz., the quantities (122) may be considered as the coordinates of the point in which
this line, and all parallel to it, meet the line at infinity.

To every equation in the preceding part of this Paper involving as parameters
(x'y'z') the coordinates of a point on L, a finite line, now corresponds one in which
x'y'z' are replaced by

X : /* : v = yr? - ft : at, - y£ - : ft - ay. ' ." . . (123)
Thus, for the polar line of x'y'z' (7) now appears D2?< (4), § 17, i.e.,

which is, plainly, the locus of a point 0 on a chord parallel to £r + . . . = 0, and
meeting the cubic u in the points O1( O2, O3, such that

OO, + O02 + 003 = 0 ; '. . !' " . . . (125)

viz., O is the " mean point " on the chord relatively to the triad 0^ 0.;, O3.

The line (124) is the Newtonian Diameter of the system of chords (V1') '> an<^ ^
envelope, since (123)

«X + /8/x + yv = 0 . .. . .. ^ ( . . .. . (1 20)

the " centroid," i.e., the " poloid " of the line at infinity, is

or

uny? 4- ... + Uy$)Z + . . . = 0,

where «,, = 0 is now the condition that d,u should be a parabola . . . ,

MB. J. J. WALKKll ON THK Dl \\IKTK1JS OF A PLANE CUBIC. 199

ttjg = 0, that the line at infinity should be cut harmonically by the conies 3,u,
3jtt . . . .

The condition that the cubic u should touch the line at infinity in the standard
form (b*c* 4-...)a8-f... = 0 is equal to four times that for the centroid being
parabolic and touching the finite asymptote of u ; and

83. When the cubic meets the line at infinity in three real points, or the three
asymptotes are real, the centroid is an ellipse inscribed in the triangle formed by
these lines, so as to touch them at their middle points (Plate 8).

But if the cubic u has only one real intersection with the line at infinity, then the
centroid is a hyperbola, having the single real asymptote as a tangent ; and

The asymptotes of the hyperbolic centroid are the only real pair of conjugate
diameters of the cubic u, viz., each cuts every chord parallel to the other in its mean
point, and in particular divides the tangential chord of the cubic parallel to the other,
or the parallel nodal chord if the cubic is nodal, in the ratio of 2 : 1. Thus in Plate 7
the chords BBj, B'B'j, B'2, parallel to one asymptote of the centroid, are divided in
such wise by the other asymptote in the points Bg, B'0.

84. The mean point on any diameter of u regarded as a chord of that cubic is the
point in which it is met by the diameter of the centroid conjugate in direction to its
chords, the "double" one (§ 7) of which is the s-polar § 42 of that mean point.

The locus of the mean points of diameters of the cubic is the nodal cubic v, the
Cotesian of the line at infinity, having as its nodal tangents the asymptote* of the
centroid, and being therefore acnodal when the three asymptotes of the cubic are all
real (Plate 8), the acuode, or conjugate point, being the centre of the elliptic centroid.

The asymptotes of the cubic v meeting, two and two, on the diameters of s through
its points of triple contact with v, are parallel to those of the primitive cubic u ; and
the line at infinity is its inflexional axis, these inflexions being at the points in which
the asymptotes of « meet that line.

In Plate 8 the Cotesian is represented with three hyperbolic branches, each touching
the centroid ; in Plate 7, with a real loop touching the centroid, and a single real
asymptote parallel to that of the cubic u.

85. The envelope of the " double ordinates " of the Newtonian Diameters is repre-
sented in Plate 8 as a quartic (w) with three real cusps, the three diameters of the
centroid conjugate to the asymptotes of u being the cuspidal tangents. This quartic wt
as well as the Cotesian v, has triple contact with the centroid at the points of contact
of the asymptotes of u. In Plate 7, the cubic u having only one real asymptote, the
quartic w has only one real cusp ; also it and v have only one real contact with * at
the point of contact of the real asymptote with *. Within the limits of the diagram
the only parts of w visible are (i.) that adjoining the contact referred to, terminating
on one side with the point of contact (C7) of the double ordinate CD of the diameter
DD0 — in this case (of L being at infinity) the point on CD conjugate to D (§ 59)
being at infinity, D is the centre of the segment CC' — and (ii.) the cusp of w with the

200 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

contact C'] of another double ordinate C'D', of the diameter D'D'0, D' being the centre
of C'C\. It will be observed that D0, D'0 are the mean points of the diameters DD0,
D'D'0, D0 being harmonic conjugate to D with respect to DjDo, the other twx> real
intersections with v (§ 58).

86. In Plate (8) is shown an "alien" ordinate EE0 of DD0, having its mean point
EQ on that diameter (§ 61), but being an ordinate of the other tangent to s which
might be drawn through E0. The parabola, which is the envelope of these " alien "
ordinates through the different points on DD(J, is also shown, touching DD0 at the
point D0, and having the connector of that point with the centre of the centroid as
diameter (88), this connector being now the representative of the

y'x + x'y = Q

of the equation just cited, the connector of the CoTES-point on the polar line of
(x'y'O) with the pole of the transversal L, now become the line at infinity.

87. The asymptotes of the centroid offer themselves as an unique pair of Cotesian
axes to which the cubic may be frequently referred with advantage, its equation being
then

u = ax3 + bf -f- 6exy + Sc^x + 3c2y + c = 0, . . . . • . (i.)

and the centroid, with changed sign — which is now immaterial —

e~=-0. . , ....... (ii.)

For many discussions also it is convenient to define the diameter by the coordinates
(x"y") of its point of contact with s.

The equation of its double ordinate will now be

ax"(x - x") + e(y- y"} = 0 ; . . ... . (iii.)

and that of its " alien " ordinate at the point x1yl

x — x^+ey](y — yJ=Q. . ,•;•.-.-,. . (iv.)

It will be sufficient to indicate the steps by which these may be verified indepen-
dently of the general formulae given in the earlier part of this Memoir.
The diameter is the tangent to (ii.) at x"y" or

y'x + x"y - 2x"y" = 0, . ,, ,. -,,-*. ,., . fa (v.)

the coordinates of the mean point of which may be found as one-third of the sums of
those of its intersections with u. The polar of this mean point is (iii.) ; and that
of the mean point of the second tangent to s through ,rlyl on (v.) is (iv.)

MB. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 201

88. The secondary chords through points lying on a given diameter, and having
them as their mean points, being inclined at varying angles with that diameter, to
determine that one which makes a given angle — say 6 — with the diameter

tan 6 = y"(ax"xl — eyj sin oj/fay"*^ + ex'yl — y"(ax"xl -}• ey^ cos a>} ,

01 being the angle between the asymptotes of the centroid : viz., it is the chord which
meets the diameter in the point (x^y^ such that

xl:y1 = e (y"(cos <a — sin o>) + x" tan 6} : ax" [x" (cos <a -f sin w) — y" tan 6}.

In particular, the point at which the secondary chord is perpendicular to the diameter
is determined by

xl:yl = e (x" + y" cos w) : — ay"(y" + x" cos «).

The angle between a diameter and its primary chords or ordinates is given by
tan 4> = (ax"3 - ey") sin <o/[x"(ay" + e) — (ax"2 + ey") cos w),

and those diameters which are perpendicular* to their chords are the tangents to the
centroid at the points determined by

x (ay + e) = (ax9 + ey) cos o>,
i.e., the intersections of the conic

a (x2 cos a> — xy) -f e ( — x + y cos a») = 0
with the centroid ; or, multiplying by 6, those of the parabolas

abx* cos at + be ( — x -f- y cos o>) — e2 = 0,
aby2 cos ta -\- ae(x cos <a — y) — e* =. 0.

The finite intersections in question are only three in number, their coordinates being
determined by the cubics

a*bx? cos w — abex* — ae*x + & cos o» = 0,
ab2y* cos o> — abeyz — bezy -\- e3 cos <a = 0.

89. When the asymptotes of the centroid are imaginary, two convenient Cotesian
axes of coordinates are found in the diameter parallel to one asymptote of u and its

* " Diameter antcm ad Ordinatas rectangnla si modo uliqua sit, etiani Axis dici potcst." NEWTON,
' Knnmeratio Lim arum Tortii Ordinis,' § 2.

MIXXJCLXXXVIII. — A. 2 D

202 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC.

conjugate ; to which lines the other two asymptotes will form a conjugate harmonic
pair. These axes are the special form which those of the self-conjugate triangle of
reference employed, §§ 63-76, take ; and all the equations there deduced will be
adapted to the above Cotesian axes simply by substituting unity for z.

90. When the cubic u is parabolic, i.e., touches the line at infinity, its "centroid" s
also touches that line, viz., it becomes a parabola, which may then be referred to the
tangent parallel to the finite asymptote of u, and the diameter through its point of
contact.

The Cotesian v will then be made up of the line at infinity and a parabola, and the
chords of u having their mean points on a diameter of which they are not ordinates
will envelope another parabola, while the double chord will envelope a third.

In this case all the equations and formulae of §§ 77-81 are applicable by simply
making 2=1.

It has not been thought necessary to add a figure in illustration of this case, the
curves being all of familiar character.

VI. — DESCRIPTION OF PLATES.

91. The two types of cubic which have been drawn to illustrate results arrived at
in this Memoir have been constructed geometrically with great accuracy as follows : —

(i.) Those in Plate 6 and Plate 8 from two conjugate diameters of a hyperbola, as
the well-known Cissoid of DIOCLES from two rectangular diameters of a circle ; viz.,
from a vertex of one diameter a pencil of lines was drawn, each to the extremity of an
ordinate parallel to the other diameter, and then its intersection with the equidistant
ordinate on the other side of that diameter determined a point the locus of which
gave the cubic u as represented, with one cuspidal branch and two hyperbolic. For
the figure in Plate 6 the line L was drawn arbitrarily, cutting u in three real points
A, A', A", and the tangents to u at these points traced, the accuracy of their directions
being vouched by their tangential points proving to range in a right line KKjK2.
The poloid s was next inscribed in the triangle formed by the triad of tangents,
touching them at points Alf A2, A3, harmonic-conjugate severally to A, A', A" with
respect to the corners of the triangle. Another arbitrary line OD having been drawn
touching the conic in D and meeting the cubic in three real points, its CoTES-point O
was found by actual calculation from the measured lengths of the segments between L
and the cubic u. The polar of O, being the double chord of the pencil through the
point on L of which OD was the polar line, of course determined that point (x'}.
[Otherwise x might have been taken arbitrarily and the CoTES-point on L relatively
to it have been found by measurement ; and so the polar-line of x' with the double
chord have been arrived at.] The accuracy of the figure so far was tested by
examining the agreement of the CoTES-points of L itself and another chord with
their intersections by the polar line at O', O". For the figure in Plate 8, corre-

MR. J. J. WA1.KKR ON THE DIAMETERS OF A PLANE CUBIC. 203

spondingly, the " centroid " was inscribed in the triaugle formed by the asymptotes of
u so as to bisect the sides. The Cotesian v and the quartic w were laid down from
their equations without much difficulty, their triple contacts with s, and the parallelism
of the asymptotes of v to those of u, combined with the simply defined positions of the
cusps, and directions of the cuspidal tangents of IP, enabling them to be traced with
great accuracy. The parabolic envelope shown in connexion with the diameter DD0,
touching that line at its mean point and having an axis parallel to the connector of
that point with the centre of the centroid, as well as touching the asymptotes, was
readily constructed geometrically from these data, and its accuracy tested by a
tangent drawn arbitrarily, meeting u in three real points E, Et, E2 and cutting DD0
in Ey, proving to have the last as its mean point, as determined by actual measure-
ment of segments, with great exactness.

(ii.) For the second illustration, Plate 7, the figure of a cubic u having only a single
real asymptote was obtained by constructing a Cissoid from two conjugate diameters of
an ellipse, precisely as that of DIOCLES from the circle, of which in fact this, as well as the
figure previously described, may be regarded as projections. The hyperbolic centroid
was then constructed from its equation referred to the two conjugate diameters of the
generating ellipse, which are also conjugate diameters of the centroid, and to one of
which the real asymptote of u is parallel. The Cotesian v with its real loop and
single real asymptote, parallel to that of u, having also the asymptotes of the centroid
as nodal tangents (§ 50), was constructed from its equation, the loop touching the
centroid at the contact with it of the real asymptote of u. It will be remarked in
this as in the case preceding where they are all three real, how soon the cubic v
approaches its asymptote, and its curvature becomes inappreciable. As regards the
little of w visible within the limits of the figure some remarks have already been
made, § 60. Two diameters, besides the conjugate pair which the asymptotes of *
form, § 83, have been introduced, DD0 touching one branch of s and having three real
intersections with v ; D'D'0 touching the other branch of s and meeting v in only one,
its mean point, D'0. The " double " chords of these diameters, polars of their mean
points, touching their envelope w at the points Cj, C'lf equidistant from D, D'
respectively as C, C', are typical of their kind. It will be remembered that they are
" double " chords of u, with which, however, in this figure, they have only one real
intersection. The mean point of the chord AA'A", drawn parallel to the double
chord CD, as determined by actual measurement of segments, coincides with Ay, its
intersection with the diameter DD0, to a nicety. Some remarks have already been
made, § 83, on the chords BBj, B'B',8'., parallel to one asymptote of s and cut by the
other in their mean points B0, B0' with great exactness. It has not been thought
desirable to introduce the parabolic envelope connected with either of the diameters
drawn, into this figure.

2 D 2

[ 205 ]

VII. On the. Changes produced by Magnetisation, in the Dimensions of Rings and Hods

of Iron and of some other Metals.

By SHELFORD BID WELL. MA., F.R.S.

Received February 9,— Bead March 1, 1888.

IN a paper communicated to the Royal Society in 1885* I described some effects
of magnetisation upon the dimensions of iron rods. It was there shown that the
elongation which such a rod at first undergoes when magnetised (a phenomenon which
had been carefully studied by JOULE and others) does not, as had been believed,
remain unchanged at a maximum when the magnetising force exceeds that which is
sufficient to produce so-called saturation. On the contrary, it is found that, when the
magnetising force is continually increased beyond this limit, the elongation becomes
gradually less until the rod is at last actually shorter than it was in the unmagnetised
condition.

Though there could be little doubt as to the general qualitative reality of the effects
described, the experimental evidence was nevertheless not free from certain defects
which I was not at the time able to overcome.

In order to obtain a magnetic field of the highest intensity possible with a limited
battery power, a solenoid was used which was not much longer than the rods them-
selves, and the field, therefore, was not quite uniform. The effect of the ends of the
rods was also uncertain, and might have played some material part in the pro-
duction of the phenomena in question. Another serious element of uncertainty
arose from the fact that the rods with which the experiments were made were
permanently magnetised. Since it was impossible to remove a rod from the instru-
ment and demagnetise it before every one of a series of observations, it was thought
desirable that the amount of residual magnetism which it contained should at least be
the same throughout the series. This object was attained by passing through the
coil at the beginning of an experiment a current equal to the strongest which it was
intended to use subsequently, the permanent magnetism thus imparted to the iron
having been found to be quite unaffected by the passage through the coil of currents
weaker or not stronger than the first. The effect of this residual magnetism upon the
elongations and retractions was not easily ascertainable. Lastly, the magnetisation
might with advantage have been carried considerably further if sufficient battery
power had been available.

• 'Roy. Soc. Proc.,' vol. 40, 1886, p. 110.

25.6.88

206 MR. S. IttDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

It is clear that if such experiments could be made with iron rings instead of rods
several of these objections would at once disappear. The field due to a current through
a wire wrapped evenly round the ring would be very approximately uniform and easily
calculable, while the ends, with their attendant disadvantages and uncertainties, would
no longer exist. I have from time to time during the last two years made attempts in
this direction, but the practical difficulties encountered turned out to be unexpectedly
great, and it is only recently that I have succeeded in surmounting them.

To bring the strong fields required within the compass of a battery of reasonable
size, it was essential that the rings should be small. On the other hand, if they were
too small, the exceedingly minute changes in their diameters due to magnetisation
would not be measurable. Ten centimetres had been fixed upon for the length of the
rods used in the old experiments, as being the shortest which was advantageously
possible ; but a ring 10 cm. in diameter would require more than three times the
number of turns of wire necessary for the rods to give an equal field, and the increased
resistance thus introduced would demand a correspondingly larger battery. It was
finally determined to give the rings a mean diameter of 6 cm., but with this small
diameter the number of turns of wire per centimetre which could be properly
wound on was less than in the case of the straight solenoid. The strongest field yet
obtained with such a ring, vising a battery of 30 GROVE'S cells, was 450 C.G.S. units.
When the same battery is employed with the straight solenoid the strength of the
field at the centre reaches 840 units.

Again, the troublesome and misleading effects of heat are much less easily avoided
in a ring than in a straight rod. The importance of securing a steady temperature
may be inferred from the fact that, taking the coefficient of expansion of iron to be
0 '0000 122, the elongation accompanying a rise of temperature of a single degree
is more than eighty times greater than that which it is desired to measure. Several
rings were prepared before the difficulty was surmounted of sufficiently protecting the
iron from the heat generated in the surrounding coil by strong currents. Unless an
interval of at least half a second intervenes between the closing of the circuit and
the commencement of the consequent heat expansion, no satisfactory observation
can be made.

In order to get rid of the permanent magnetism which remains after every obser-
vation, a modification was adopted of the device employed by Professor EWING, and
described by him in 'Phil. Trans./ vol. 176, p. 537. For the variable resistance it
was found advantageous to use a long German silver wire, wound in a close spiral
upon a wooden cylinder, instead of a tube containing sulphate of zinc solution.
Contact with the wire was made by a transverse spring which was capable of sliding
from end to end of the cylinder. The arrangement is indicated in fig. 1. It will be
seen that the German silver spiral AB acts to the coil wound upon the ring S as a
shunt, the resistance of which gradually diminishes from 26 ohms* to nothing as the

* The resistance of the whole sp'ral.

DIMENSIONS OF RINGS AND RODS OF IRON AND OTHKR METALS. 207

spring C slides from A to B. While the spring is so moving the current from the
battery E is being rapidly reversed by the commutator D : thus, the coil upou
the ring is traversed by a series of alternating currents of decreasing strength, the
effect of which is to completely demagnetise the ring. In practice the slide and the
commutator were operated together by simply turning a handle.*

Fig. l.

Passing over the earlier and altogether unsuccessful attempts in which the insulated
wire was wound directly upon the iron, I give below particulars of three rings which
yielded results of value, and which will be distinguished as No. 0, No. 1, and No. 2.
The first was of use only for comparatively weak currents : the second and third
could, with a little care, be employed in connection with 30 GROVE'S cells.

Ring No. 0. — This was made of soft iron rod of a superior quality, commercially
known as " bright rod." The joint was carefully welded and the ring was turned in
the lathe to a circular section. It was then covered with four layers of flannel, cut
into strips and wound with three coils of insulated wire.

External diameter of ring 6'50 cm.

Mean diameter 5*85 cm.

Diameter of section 0'515 cm.

Diameter of copper wire . 0*07 cm.

Total number of turns 450

Fieldt due to coil 30 X C nearly.

C being the current expressed in amperes.

Two brass rods, 6 cm. and 8 cm. in length, were attached by clamps to the exterior
of the coils, forming prolongations of a diameter of the ring. These rods served to
communicate the variations in the diameter of the ring to the measuring instrument,
which will be described further on.

* The iron was, of course, demagnetised without being removed from the instrument, the slide
apparatus being connected when required by means of a switch.

t Tho field at a distance r from the axis of the ring = 2nt'/r, where n is the number of turns of wire
and i the current in C.G.S. units.

208 MB. S. B1DWELL ON CHANGES PRODUCED BY MAGNETISATION IN

Ring No. I. — This was made from a piece of the same rod as the last. The joint
was welded and the whole ring was filed smooth, but it was not turned. The ring
was covered with a boxwood case of circular section (something like a WINTER'S
ring), which was formed of two rings of semicircular section, having grooves of semi-
circular section cut upon their flat sides. The grooves in the two portions of the
boxwood ring enclosed a space slightly larger than was necessary for the reception
of the iron ring, allowing a play of about 1 mm. Five layers of copper wire were
wound over the boxwood ring.

External diameter of iron ring 6'7 cm.

Mean diameter 6 cm.

Diameter of section 0'7 cm.

Diameter of section of boxwood ring . . . . 1 '8 cm.

Diameter of copper wire 0'07 cm.

Total number of turns 644

Field due to coil 42'9 X C.

Brass rods were screwed into small holes drilled at opposite ends of a diameter of
the iron ring and passed freely through the boxwood casing and the coils, pieces
of india-rubber tubing being inserted between the rods and the wires of the coils.

Ring No. 2 was a welded ring made from a different specimen of iron. It was, like
No. 0, turned to a circular section, and was remarkable for the extreme facility with
which it could be cut in the lathe. Rods of iron (instead of brass) were screwed into
two opposite holes, and care was taken to make the screws fit very tightly, with the
object of securing nearly perfect magnetic uniformity throughout the ring. This ring
was, like No. 1, covered with a wooden case of circular section, but the holes made in
the wood to admit the rods were furnished with short spouts of glass tubing, having
;m external diameter of 0'57 cm. and projecting about 2'5 cm. ; the rods were
turned down to such a size as to pass freely through the tubes. The wire with which
the wooden ring was covered was wound in six layers closely up to the tubes, and the
break in the uniformity of the coil was very slight. The tubes also served to shield
the rods from the heat radiated by the coil when a current was passing.

External diameter of iron ring 670 cm.

Mean diameter ' , . .. . . . G'03 cm.

Diameter of section 0-G7 cm.

Diameter of section of wood casing •, . . . 1 '3 cm.

Diameter of copper wire 0'07 cm.

Total number of turns 869

Field due to coil 57'G X C

Some experiments were also made with two iron rods.

DIMENSIONS OP RINGS AND BODS OP IRON AND OTHER METALS. 209

Rod No. 1. — Commercial iron win :

Length 10cm.

Diameter , 0'27 cm.

Rod No. 2. — Soft charcoal iron : —

Length ...'.. 10 cm.

Diameter 0'33 cm.

Each iron rod had rods of brass, 5'5 cm. in length soldered to the ends ; the
total length of the compound rod thus formed being 21 cm.
The magnetising solenoid used with the rods was as follows : —

Length 11 '6 cm.

Diameter of wire 0'12 cm.

Number of turns 876

Field at centre 92 X C. nearly.

The rods were fixed axially in the solenoid by means of corks at the two ends.

Fig. 2.

IX

The nature of the measuring instrument employed is indicated in the diagram, fig. 2.
A detailed description of it, with a drawing, is given in the former paper. The ring S,
round which is wound the wire, whose ends are shown at W, W, is placed in a vertical
position, the lower end of one of its rods R resting in a conical recess in a brass base
plate E. The upper end of the other rod R', which is chisel-shaped, acts at B upon a
brass lever, one end of which abuts upon a fixed knife edge A, and the other upon a
short arm fixed perpendicularly upon the back of a small circular mirror M, which
turns on knife edges D about its horizontal diameter. Shallow obtuse-angled notches
are cut in the lever at A and B, and in the mirror arm at C. By means of a lantern,

MDCCCLXXXVin. — A. 2 E

210 MR. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

illuminated by a lime light, the image of a horizontal wire is, after reflection from the
mirror, projected upon a distant vertical scale. A very slight deflection of the mirror
causes a considerable movement of the image. The actual dimensions are as follows :
-The distance AB =10 mm., AC =170 mm., MC = 7 mm. ; the distance from the
mirror to the scale was in all the experiments described in the present paper 7320 mm.
(24 feet). The multiplying power therefore (remembering that the angular deflection
of the beam of light is twice that of the mirror) was

7320 x 170 x 2

7 x 1Q = 35,554 times.

Each division of the scale = 4^ inch, = '04 mm. Therefore a movement through one
scale division indicated a difference in the diameter of the ring, or the length of rod,
under examination of

1/35554 of 0-64 mm. = O'OOOOIS mm.*

Since the mean diameter of the rings is 60 mm., one scale division corresponds to

0-000018/60 = 0-0000003 of diameter,

or three ten-millionths.

The rods being 100 mm. long, one scale division indicates 0*00000018, or l-8
ten-millionths of their length.

The optical arrangements were very perfect, and it was easy to read to half a scale
division, or less, with accuracy.

The currents were measured by one of AYRTON and PERRY'S ammeters, having a
commutator by means of which the coils could be arranged either in series or in
parallel, the deflections in the former case being ten times as great as in the latter.
The instrument had been recently calibrated and was checked by reference to a tangent
galvanometer. The ammeter being nearly dead-beat, the circuit needed to be closed
only for a very short time in taking a reading. But it was found necessary in the
experiments with the rings to switch the current through a coil of the same resistance
before measuring it. Th3 coils employed for this purpose were wound with the same
kind of wire as that used for the rings. The ammeter was read to a quarter of a scale
division.

The experiments were conducted as follows :—

(a) The ring or rod was demagnetised.

(b) A certain current was passed for a moment through the magnetising coil and

the deflection indicating elongation or retraction was noted.

(c) The same current was passed a second time and the deflection noted. This was

generally different from that given by (6).

(d) The iron was once more demagnetised.

* Seven ten-milliontbs of an inch.

DIMENSIONS OP RINGS AND RODS OF IRON AND OTHER METALS. 211

(e) The same current waa again sent through the coil, and the consequent deflection
was the same (or nearly so) as in (b).

(/) The current was passed through the coil for the fourth time, and the resulting
deflection was about the same as in (c).

The mean of the deflections found in (b) and (e) was taken as giving the true
elongation or retraction produced in a previously demagnetised ring or rod by a given
current The mean of (c) and (f) gave the changes which occurred when the iron had
been permanently magnetised by the same current. To these latter no great
importance is attached.

The above series of operations was repeated with every different current that was
passed through the magnetising coil.

Table I. gives the results of experiments made with the three iron rings and with
the rod No. 2. The magnetising forces are expressed in C.G.S. units and the
elongations as ten-rnillionths of the diameter of the rings und the length of the rod.
These results, except as regards ring No. 0 (which are omitted to avoid confusion),
are plotted as curves in fig. 3, the abscissae representing the magnetising forces and
the ordinates the corresponding elongations. The curve of ring No. 0 would, if plotted,
ascend very near to that of No. 2, and descend almost in a straight line with that of
No. 1, its apex being about midway between the two.

TABLE I. — IRON.

Magnetising forces are expressed iu C.G.S. units; elongations in teu-niilliontliK of the diameter or

length.

Bin* No. 0.

King No. 1.

Ring No. 2.

Rod No. J.

Monetising
force.

Elongation.

KnetiMDg
force.

KlongMion.

Magnetising
force.

Elongation.

Migncti.ing
force.

Elongation.

1

3

10

5

6

4

16

1-8

9

24

26

13

8

11 22 5-4

13

30

49

30

12

26 •-".»

6'3

23

42

66

35

20

44

47

15

39

45

152

29

31

50

65 23

08

42

198

n

56

53

90 23

97

36

-'1-

14

107

53 128 20

123

33

L".'J

6

172

41 187 12

321

0 234

29 237 45

:::." -:i 351

5 287

-1-8

3(54 - G 391

-3

337

-10

449

-12

374

-14

2E2

212 MR. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

Fig. 3

Upon examination of these curves it at once appears that the general character of
the phenomenon under discussion is just the same in rings as it is in rods. In both
cases a continually increasing magnetising force is accompanied by elongation, which
reaches a maximum, then falls to zero, and ultimately becomes negative. And both
sets of curves appear to follow a similar law, ascending in a form suggestive of a
parabola, descending in a straight line. Greater differences might easily be found to
occur in rings of different specimens of iron than those which exist between the rings
and the rod in the present instance. The maximum elongation (23 ten-millionths) of
the rod which happened to be chosen for these experiments is indeed less than that of
any of the rings, and its retraction begins at an earlier stage. This, no doubt, is
partly owing to the effect of the ends and to want of uniformity in the magnetisation,
some portions of the rod having attained the state of maximum extension while others
had either not yet reached it, or had passed it. But it may also be to a great extent
due merely to the quality of the iron, for I have myself observed in an iron rod a
maximum extension of as much as 45 ten-millionths* (equal to the maximum extension
of ring No. 0 and greater than that of ring No. 1), while an elongation of 56 ten-
millionths (greater than that of any of the rings) has been recorded by JouLE.t On
the other hand, the maximum extensions of the rings would undoubtedly have been
less if they had been made of harder iron, instead of the softest which was con-
veniently procurable.

* ' Roy. Soc. Proc.,' vol. 40, 1886, p. 117.

t See JOULE'S paper, ' Phil. Mag.,' vol. 30, 1847, p. 76, or the Physical Society's ' Reprint,' p. 235.
The greatest elongation recorded occurs in Exp. 2, p. 240. It amounted to 28 scale divisions, each of
which corresponded to an extension of 1/138528 inch in a rod 36 inches long.

28/36 x 138528 = 0-0000056
or 56 teu-millionths of the length of the rod.

DIMENSIONS OP RINGS AND RODS OP IRON AND OTHER METALS. 213

It may, perhaps, be well to state briefly here, though more will be said on this
point in an appendix to the paper, that each ring or rod has its own peculiar curve,
which is perfectly definite and is at any time obtainable by the method above
described without material variation.

Being thus satisfied that these curious effects of magnetism are, except as to mere
details, independent of the form of the iron, and having regard to the fact that it is
much easier to obtain intense fields with straight than with circular coils, I thought
it worth while to continue the experiments upon rods only. The metals used, in
addition to iron, were cobalt, nickel, manganese-steel, and bismuth.

The iron rod was that already described as No. 1.

The cobalt rod was obtained as a rough casting from Messrs. JOHNSON and
MATTHEY. It was turned in the lathe to a cylindrical form, and, when finished, its
diameter was 7'1 mm., and its length 67 mm. A single scale division, therefore,
corresponds to 27 ten-raillionths of its length.* Brass rods were soldered to its ends
as usual.

The nickel used was in the form of a strip 1 00 mm. long, 9 mm. wide, and
075 mm. thick.

The manganese-steel rod was given to me by Dr. FLEMING, who obtained it from
Messrs. HADFIELD. It was said to contain 12 per cent, of manganese. Its length
was 100 mm., and diameter 4'5 mm. A scale division corresponds to 1'8 ten-
millionths of both the nickel and the manganese-steel rods.

The bismuth was a cast rod procured from Messrs. JOHNSON and MATTUEV. Its
length was 132 mm., and diameter 7 mm. One scale division represents 1'4 ten-
inillionths of its length.

The results of a series of experiments with the iron, nickel, and cobalt are given in
Table II. and plotted in fig. 4. The mode of operating was exactly the same as that
before described, each rod being demagnetised before every observation recorded in
the Table.t

* For each scale division corresponds to O000018 mm., or, since the length of the cobalt is 67 mm.,
to 0-000018,67 = 27 ten-milliontbs of length.

t The experiment with nickel was made immediately after that with iron, and the number of battery
colls used was, after the first two currents, increased similarly in both cases. It was fonnd that, owing
to the constancy of the battery, the ammeter readings in the nickel experiment were, from the third to
the ninth cui-rents, practically identical with those obtained in the iron experiment. It was, therefore,
assumed that they would be the same with the last three currents, and these were not actually read.
Thus the metal was not unnecessarily heated by strong currents, which was a matter of great impor-
tance, the magnetic behaviour of nickel being far more sensitive to slight changes of temperature than
that of iron or cobalt.

214

MB. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

TABLE II. — Iron, Nickel, and Cobalt.

Iron.

Nickel.

Cobalt.

Magnetising
force.

Elongation.

Magnetising
force.

Retraction.

Magnetising
force.

Betraction.

16

1-8

13

0

31

0

22

2-7

18

1-8

47

1-4

31

6-3

29

7-2

65

27

49

20

50

14

125

16

69

23

69

23

175*

31

125

20

125

42

224

42

231

7-2

231

63

352

50

362

-11

362

83

393*

50

468

-24

468

96

462

50

592

-35

592

105

561*

46

686

-40

686

111

592

46

780

-42

780

113

686

38

842

-45

718*

36

749

34

811

31

Magnetising forces are expressed in C.G.S. units ; elongations and retractions in
ten-millionths of length.

Fig. 4.

* Observations made with descending currents.

DIMENSIONS OP RINGS AND RODS OP IRON AND OTHER METALS. 215

It appears that the retraction undergone by the iron continues to increase with
increasing magnetising forces until its amount is nearly twice that of the greatest
elongation. But towards the end of the experiment there are indications that a limit
is nearly reached.

The nickel curve is of the same character as that which I have already published,
but it goes further, and the retraction is finally as much as 113 ten-millionths of the
length. The beginning of the curve is slightly different from that given in my former
paper. This difference was found to be accounted for by the fact that the nickel
there referred to was permanently magnetised.

The curve showing the results of the cobalt experiment is of the most remarkable
nature. No evidence of any change of length appears until the magnetising force
exceeds 30 or 40 units. Then the length of the rod begins to diminish, and continues
to diminish until the force reaches about 400. But beyond this point the rod
gradually becomes longer, and the retraction with a force of 800 units is only three-
fifths of its maximum amount.

[In showing some of these experiments at the conversazione of the Boyal Society
on May 9th, 1888, I had the privilege of taking my current from the large secondary
battery used for lighting the building. By means of this it was found possible to
carry the field up to about 1350 units, a far higher degree of intensity, I believe, than
had ever been previously obtained for any experimental purpose without the use of
electro-magnets. In this field the cobalt rod was elongated, the extent of its elonga-
tion being about half that of its greatest retraction in weaker fields. The cobalt
curve in fig. 4 would therefore, if prolonged, cut the horizontal axis, probably at
about 1100 (at which point there would be neither retraction nor elongation), and
would continue on the upper side of the axis. Iron and cobalt, therefore, behave
oppositely. With continually increasing magnetising force iron is at first extended
and afterwards contracted, while cobalt is at first contracted and afterwards extended.
-May 14th, 1888.]

It is difficult to imagine what can be the physical meaning of this effect. The
possibility suggested itself that cobalt might acquire a maximum magnetisation with
a magnetising force of about 400, further increase of the force resulting in a diminu-
tion of the magnetisation. The importance of establishing such a property in any
substance* induced me to make some experiments upon the magnetisation of cobalt
in strong fields, t but there was no indication of any such maximum as was looked for,
the magnetisation being still on the increase when the experiment was stopped with
a force exceeding 700.

It would be interesting to ascertain whether the retractions of iron and nickel

* See MAXWKLL'S ' Electricity,' vol. 2, § 844.

t In ROWLAND'S experiments (' Phil. Mag.,' vol. 48, 1874) the magnetising force was not carried
beyond 147 C.G.S. nnits.

216 MB. S. BIUWELL ON CHANGES PRODUCED BY MAGNETISATION IN

would under higher forces exhibit the same peculiarity as that of cobalt. A com-
parison of the three curves seems to favour such a conjecture, but I cannot test it
with the apparatus at present at my disposal. w

No result was anticipated from the experiment with the bismuth ; but with high
forces it was found to exhibit elongation. The effect, though small, was quite
unmistakable, and even roughly measurable. It was certainly not due to heat, the
elongation occurring instantly when the current was turned on, and disappearing as
suddenly when the current was stopped. Nor do I think it could be accounted for
by the presence of iron as an impurity. The results may be stated as follows :—

Magnetic force. Elongation.

280 .:.:.•....,. .... Suspected.

470 .•,-..-,. Quite perceptible.

680 . f , 0'25 scale div. = 0'3 ten-millionths.

842 1'25 scale div. = T5 ten-millionths.

Thus it appears that when the experiment was discontinued the elongation was
increasing more rapidly than the field.

No alteration in the rod of manganese steel could be detected with forces up to
700 units. With 850 units there was a perceptible elongation, estimated at about
one-tenth of a scale division, or one fifty-millionth part of the length of the rod.

Before the experiment there was no indication of any attraction between the steel
and a delicately-pivoted compass needle, 22 mm. long, at a distance of two or three
mm. After the magnetisation the steel had acquired sufficient polarity to draw the
needle 20° from the magnetic meridian when held about 3 mm. from the opposite pole
of the needle.

It must be admitted that it is not easy to form a plausible conjecture as to the
physical causes of these magnetic extensions and retractions. It has been customary
to explain the extension of soft iron by supposing its component molecules to be
small magnets of elongated form, which, under the influence of a magnetising force,
tend to set themselves in one direction ; but to this view there are obvious objections.
In any case, it seems probable that there are at least two influences, or sets of
influences, at work, the one tending to cause extension, the other retraction. In the
early stages of the magnetisation of iron the former predominates, but as the
magnetisation advances it is overcome by the latter.

One of the influences tending to produce retraction must certainly be of a purely
mechanical nature.

Suppose a uniformly magnetised rod to be transversely divided through the middle.
The two halves, if placed end to end, will be held together by their mutual attraction,

* [I hope to have the opportunity of repeating all my experiments with more powerful currents than
those hitherto used.— May 14th, 1888.]

DIMENSIONS OP RINGS AND BODS OP IRON AND OTHER METALS. 217

pressing against each other with a certain force per unit of area, which can be
measured by the weight necessary to tear one half from the other. The same pressure
will exist between any two portions of the rod separated by any possible cross-section,
and a certain longitudinal contraction of the rod will be the consequence. If now the
rod, having been first demagnetised, be placed in a vertical position upon a fixed base,
and loaded at the upper end with a weight equal to the greatest it could support
when magnetised, it will undergo the same contraction as before, the compressional
stresses being equal in the two cases. The amount of contraction occurring in this
latter case is well known, and does not differ greatly in different kinds of iron. Does
then this mechanical contraction completely account for the magnetic retraction which
is observed in high fields ?

With the view of answering this question, I carried out a series of experiments
upon the lifting powers of electromagnets in the form both of rods and of divided
rings, the results of which have been communicated to the Royal Society.* It was
found that the lifting power did not, as was generally believed, reach a practical limit
with a comparatively small magnetising force, such as 135 or even 250 C.G.S. units.t
It is true that after this latter point was passed the lifting power increased much
more slowly than at first, but with a force of (500 units it was nevertheless upwards
of 10 per cent, greater than with a force of 250, the weight supported being in the
two cases 13,500 and 15,000 grammes per square centimetre, allowance having been
made for the electromagnetic action between the iron and the coil.J Between these
limits the increments of lifting power and magnetic force appear to be approximately
proportional, the curve expressing their relation being a sensibly straight line.

Here, then, we have the means of comparing the mechanical with the magnetic
effect. Taking 2040 X 106 as the value of YOUNG'S modulus for wrought iron, in
grammes weight per square centimetre, the contraction produced by loading the
upper end with a weight of P grammes per centimetre is expressed as a fraction of the
original length by P/2040 X 108.

If P = 13,500 grms. (the weight per centimetre supported in u field of 250 units),
the contraction will be

13500/2040 X 106 = 0'0000066,

or 66 ten-millionths of the length of the rod.

Again, if the rod be compressed by a load of 15,000 grms. (the weight supported
in a field of 600 units), the contraction will be

15000/2040 X 10" = 0-0000074

* ' Roy. Soc. Proc.,' vol. 40, 1886, p. 486.

t See JEHKIN'S ' Electricity,' 7th edition, 1883, p. 124.

J It is, perhaps, questionable whether any deduction should be made on this account. The figures
without such deduction would be respectively about 14,000 and 15,000 grms. But the difference is not
sufficiently great to affect the argument.

MDCCCLXXXV1II. — A. 2 F

218 MB. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

or 74 ten-millionths of the length ; i.e., the contraction of the rod when longitudinally
compressed by the weight which it could support in a field of 600 is only 8 ten-
millionths greater than when it is compressed by the weight which it carried in a
field of 250.

Now it appears from the tables and curves contained in the present paper that the
magnetic contraction of an iron rod (or ring) is greater in a field of 600 than in one of
250 by as much as 40 ten-millionths. It follows, therefore, that what I have called
the mechanical contraction only accounts for a fifth part of the total observed effect,
or even less if it should be the fact, as is not unlikely, that the elongating influence
upon which the contracting influence is superposed, does not altogether cease to
increase after the net elongation has reached its maximum.

The main cause of the retraction, therefore, still remains a mystery.

It is proposed to continue these experiments, which must certainly tend to throw
some light upon the molecular effects of magnetism.

APPENDIX.

The value to be attached to the results given in this paper depends greatly upon
evidence of the precision with which measurements can be made of lengths commonly
regarded as infinitesimal. In order that an estimate may be formed of the degree of
accuracy arrived at, I have thought it desirable to give in the form of an appendix
certain details of the experiments, which if contained in the paper itself would have
been tedious and cumbersome.

As elsewhere stated, the optical arrangements are so perfect, and the edge of the
focussed image of the wire is so sharp that, under favourable conditions, it is possible
to read to a quarter of a scale division (1 scale division = 0'64 mm. = 4JQ inch) even
when the reflected beam of light is 732 cm. (24 feet) long. But during an experi-
ment it unfortunately happens that, in consequence of the heating effect of the
currents, which no amount of care can completely obviate, the image is in constant
movement up or down the scale, a change of one degree in the temperature of the
magnetised ring or rod, together with its brass connections, causing a rise or fall of
more than 150 scale divisions, This not only renders it difficult to make an accurate
reading, but it is the source of another and more serious inconvenience, namely, that
there can be no fixed zero. Every observation of a deflection, therefore, involves two
scale readings, in each of which there is a chance of error. The following was the
method adopted. At the moment when the upper edge of the wire appeared in the
course of its wanderings to coincide exactly with a certain previously determined
division which it had been seen to be approaching, and the number of which was
recorded as the temporary zero, the contact key was depressed, and the division

DIMENSIONS OP RINGS AND RODS OP IRON AND OTHER METALS. 219

nearest to which the image was deflected was quickly noted.* The difference
between the two readings gave the deflection in scale divisions. The first of these
readings was believed to be correct within less than a quarter of a scale division ; the
second within half a scale division. It is clear that the results of two such observations
made under similar conditions and not unequally affected by any source of disturb-
ance, such as temperature changes, ought not to differ by more than one scale division,
and their mean should be correct to half a scale division. Now half a scale division
corresponds to O'OOOOOU mm., or about 1 three-millionth of an inch, which is equivalent
to l£ ten-mil lionths of the diameter of the iron rings and (approximately) of the
length of the cobalt rod, and to a little less than 1 ten-millionth of the length of
the rods of iron and nickel. This was the degree of precision aimed at. How nearly
it was actually attained may be gathered from the detailed notes of some of the
experiments contained in Tables III., IV., and V., which are given as specimens, and
from the diagram fig. 5.

The tables consist of a transcript from the note-book of the actual figures recorded
in the experiments to which they relate. Nothing has been altered or omitted, and
nothing added, except the two final columns showing the results when reduced to
magnetising forces and ten-millionths of length. The figures printed in heavy type
are those which appear in Table II. Such of the figures as have been rejected (or,
rather, amended) owing to evident mistakes are printed in italics, t This occurs in
only two instances.

The experiments selected are those which relate to the " Iron rod No. 1 " and the
cobalt rod. The first is chosen both because it goes further than any experiment
with the rings, and also because it affords an opportunity of comparing measurements
taken on two different occasions. The figures relating to the cobalt are given in full,
on account of the novelty and probable importance of the results obtained with that
metal.

* This could only be done by a momentary glance, the circuit being necessarily closed for no more
than about half a second to avoid heating : it was, therefore, not attempted to read to a fraction of a
division.

t The error which most frequently occurred in making a rapid reading was to mistake a long " 5-line "
for the long " 10-line " which came next above it on the scale. This mistake was generally detected and
rectified the moment it was made, but in three or four cases, all of which are specified in the text, the
wrong number was recorded. In these cases the observation was not actually rejected, but was corrected
by subtracting five.

2 F 2

220 MB. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

00 tO

• " '/-

(M <M <>J i-H IM CQ -

oo
oo

I

o

I

1

ffe

11

!!

E-S

PH t,

o>

Por 2nd
current.

•s-g

o

hj
5

1411

d*5!

•". •"

i-l 7Q <M <N (N !M i-H r-H

1 1 1 1 1 1 1 1 1

fc

CO-*«OQO»O^-l<N
lM(MlM<MS1(Ni-(

1 1 1 1 1 1 1 1 1

a

i— Ii-Hr— li— IIMNOi— ItNOOO^i— ICM(N<Ni-HrH

OO'JD
O^i— I

M

pa

o

t— l(MOi»OO(M-

rH rH O <N <M (M <M rH rH

1 1 1 1 1 1 1 1 1

rH I-1 rH rH (M <N <N <*> OJ <M rH rH

1 1 1 1 1 1 1 1 1

H3

OOOOO5OrHrH(MO5-*-*Oai(MOl«COO«OO
rHpHrH(N(M(M(M(MrHIN(M (M<M(NrHr-lr-l

000OOJO
rHrHrH(M

(N(NCOrH(N!MrHIM(N»5l-lrHp— 1

rH(NO>»OO(M-'Jl-*OoaJO
I-HrHrH I-H(M(NIM<MCirHi-H

1 1 1 1 I 1 1 1 1

MB

i-H rH rH (M <N (M OJ (M rH (N rH

t^iOWi-HO N-<tO

rH rH <M IM

OCO-*
CO 01 rH

DIMKNSIONS OF RINGS AND RODS OP IRON AND OTHER METALS. 221

oo
oo

00

1

:SJ

I

I

y

s

5

*

i!

SI

H

M

M

H

QOOCO

«.

CM <N <N rH <N CO ^« rf *• if
1 1 1 1 1 1 1

14 0 14 U9
i4—<'*«'5I:*CO
i-H Oi »J <M 'M G-l

1 1 1 1 1 1 1

•4 >4

14 14 14 14

I <N *l

1 1 1 1 1 1 1

1 1 1 1 1 1 1

r» o» o a> —i o» (N 1-1

o a> —i o» (N 1-1

co«-Hi>-«aOi-c

I i— 1 i— I >-l (M (JJ ol

1 1 1 1 1 1

CO «4 «4 •* d i-c <N i-t «O CC CO
rHi— (PHI— (i— (rHi— i r-c i-c <N CO

O OJ 14 d •* rH O I-» «4 1
^ •* COCJ (N •* W 00 "5 >

t-H 6l !M OJ !N
1 1 1 1 II 1

rt -N (M -N (M IM

1 1 1 1 1 1 1

I rH 1-1 •?! 11

rHr-li-Hi— I rl p- rH r-4 i-H ~H i— I <N CO Cl

MR, S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

S

I

1

gfc
II

I

I

§

R1

o

g

i)

o sr
?l

o

GO

I

W

i-H •* l£5 10 ^ « CO CO CO Tl< 10 CO

• CO ^ tO « CSI 10 CO 05 00 Tj< ^H I-H CD 05 t-

i-H rH <N CO CO i

«, . . .OrH5O«5O5asts.iOCOr-<

»O *O O '". i~ ' ~ > ". • :

O • ^ ^ O rH <O "5 00 CO I>- •* <N rH CO Jt^ 00 rH i-H <M

. . . . rH rH CO rH rH OO O !>• OS CO CO «O OS .O

• ...., . as <o <o oo I-H co t^. oo cc o » •* o . «o

rH OJ <M (N CO (M rH <M rH CO rH <N CO rH

'rH(M<NCOCOCOrHCOrHCOrHC<lCO rH

. rH 5* <D OS OS !>. •* CO <M CO !>• CO <M . (M

* I-HrHrHrHl— IrHi— IrHrHI—lrH .1—1

s 50 r^ o oo ••*! «o r^ co <M

.as

.

r-l (N (M <N CO W rH !M rH CO r-l <M M i-H

O rH «O «D O OS t^

OS t^ "5 CO <M CO Is. «5 rH r
r- IrHrHrHrHrHrHO^rH

"

i-H rH (N (N (N O) OJ rH <

r-l 0* (M CO rH

1-H i-H iH 1-H rH IN (M (N <M <N rH CN CM rH CM <N CO rH

I— li— li— lrHf-l<NC<l(MCO

<NCO<MrH<NrHG1rH<N<NCOi— <

00 tn<n rH O

l<N'}lrs.Oit>O>OOCO-*COCOrH<
rH I-H (M (M CO <*i rH (

I >i: MKNSIONS OF RINGS AND RODS OF IRON AND OTHER METALS. 223

The tables contain twenty columns, which, for convenience of reference, are lettered
from A to T.

A gives the number of GROVE'S cells used.

B the resistance in ohms inserted to vary the current when only one cell was in use.

C shows the scale divisions indicated immediately before the circuit was closed (i.e.,

the temporary zero), the metal having been recently demagnetised.
D gives the readings obtained on closing the circuit for the first time after

demagnetisation.
E contains the differences of C and D, giving the elongations or retractions in scale

divisions.
F gives the scale readings immediately before the circuit was closed the second

time (another temporary zero). Owing to small changes of temperature, the

position of the wire was nearly always different by several divisions from that

denoted in column C.
G gives the deflections when the circuit was closed for the second time after

demagnetisation.

H contains the differences of F and G.

I, J, K are repetitions of C, D, E, the rod having been again demagnetised.
L, M, N are repetitions of F, G, H.
O gives the means of E and K.
P the means of H and M.

Q contains the ammeter readings of the currents.
It the currents in amperes obtained by multiplying the ammeter readings by the

factor 0'1356 or I '356, according as the coils of the instrument were arranged

(by the commutator) in series or in parallel.
S gives the field at the middle of the coil, obtained by multiplying the amperes

by 92.
T shows the elongations and retractions in ten-millionths of length obtained by

multiplying the figures in column O by 1 "8 for the iron and by 27 for the

cobalt.

The first of the experiments with the iron rod was made on Dec. 19, 1887, the
second on Jan. 4, 1888, the whole apparatus having been in the meantime dismantled
and the rod removed from the coil, which had been used for another purpose.

In the first experiment currents of descending as well as of ascending strength
were used. In the second only one smaller current was applied after the strongest,
the experiment being then stopped.

The following points should be noticed : —

(I.) The very close agreement between the results obtained on different days,

224 MR. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

particularly with the smaller currents. Such variations as exceed the specified limits
of observational errors may be fully accounted for by small differences of temperature
affecting the susceptibility of the iron.

(2.) The close agreement between the deflections obtained with ascending and with
descending currents. As before, changes of temperature would account for the
greater part of such differences as exist.

(3.) In the thirty-three pairs of observations recorded in columns F and K, there
is only one instance in which the discrepancy exceeds a single scale division, and in
that case the full battery of 30 cells was in use.

This shows that the degree of precision aimed at was practically attained, at least in
the final results, which are the means of two observations.

The greatest reasonably possible combination of errors in the measurements of the
parts of the apparatus and of the distance of the mirror from the scale would not,
even assuming that they all conspired together in one direction, affect the results to a
greater extent than about 1 per cent. It may therefore be taken as exceedingly
probable that the figures recorded in the table correctly represent to one ten-millionth
part of the length of the rod the elongations and retractions which actually occurred.
The slight variations in the results obtained upon different days or when the order of
the currents was reversed are mainly due to differences in the physical condition of
the iron (thermal or magnetic), and not to instrumental or observational errors.

Pig. 5.

In order that an idea may be formed of the extent of such variations, the diagram
fig. 5 has been constnicted, in which the three series of observations have been plotted
together, the points belonging to each different series being distinguished by a different
kind of mark. Those of the first experiment with ascending currents are marked
with crosses ; those of the same experiment with descending currents are marked
with crosses surrounded by circles ; and those of the second experiment with ascending
currents are distinguished by dots. A smooth curve is drawn through all the marks.

DIMENSIONS OF EINGS AND BODS OF IRON AND OTHER METALS. 225

There is only one instance in which a point appears to deviate from the curve by a
distance equal to as much as half the height of one of the small square spaces, namely,
when the magnetic force is about 425 units. Now the height of one of these spaces
corresponds to five ten-millionths of the length of the rod ; or, since the rod is four
inches (10 cm.) in length, to an absolute length of two-millionths of an inch. Half
the height of a space, therefore, represents a length of one-millionth of an inch,* and
this is the widest deviation which occurs in the whole of the three series of obser-
vations. It is clear that in the great majority of cases the deviation from the mean
curve is very much less, not often being so great as one-fifth of a space height, or one
two-and-half-millionth part of an inch.

The close agreement of columns E and K also shows how perfectly the demag-
netising apparatus fulfilled its object. It appears from columns H and M that the
elongating effect of the second current after a demagnetisation was generally different
from that of the first, the difference being of course accounted for by permanent
magnet ism .t

The figures giving the remarkable cobalt curve are just as regular as those which
relate to the iron. In the results of the 14 pairs of observations in columns E and K
there is not a single discrepancy of more than 1 scale division. Observations were
made both with ascending and with descending currents, all of which are plotted in
fig. 4. It was not until after the experiment was nearly completed that my attention
was called to the fact of the retraction reaching a maximum with currents of medium
strength, and falling off when stronger currents were applied. Thinking that this
might possibly be an effect of heat upon susceptibility, I put on all the 30 cells at
once, the metal having been first allowed to become quite cool. The deflection was,
however, just the same as that which the 30 cells had produced before.

There can be no doubt that the curve in fig. 4 correctly represents the effect of
magnetisation upon the length of my rod of cobalt. Whether all specimens of the
metal would behave in exactly the same manner is not so certain. Professor BARRKTT
found { that cobalt was elongated when magnetised. From the analogy of iron it
seems not unlikely that a softer rod of cobalt than mine would in the earlier stages of
magnetisation undergo some elongation ; but it is more than probable that with
sufficiently high magnetic forces such elongation would be ultimately converted into
retraction of the kind indicated by the curve in fig. 4.

In the experiments with ring No. 0 the observations were only made once, so that
their individual accuracy cannot be tested by comparison.

* About one-twentieth part of the wave-length of green light.

t [It appears from a comparison of the figures in columns O and P of Tables III. and IV. that the
permanent elongation which remains when the magnetising force has been withdrawn increases until
the magnetising force which produced it reaches about 130 units, after which it is nearly constant. It
has been pointed out that this fact tallies well with the known constancy of residual magnetism when
the fields are strong.— May 14th, 1888.]

$ ' Nature,' vol. 26, 1882, p. 585.
MIX i i I. \\XVIII. —A. 2 G

226 MR. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

In the case of ring No. 1 (the experiment upon which, as upon all others except
ring No. 0, was conducted exactly in the manner described for the iron rod No. 1
and the cobalt rod) there was no discrepancy gi-eater than 1 scale division in the
two series of 13 observations (columns E and K), except in one instance when there
was evidently a mistake.

The number of pairs of observations with ring No. 2 was 19, in 14 of which the
discrepancy did not exceed 1 division. Of the five greater discrepancies all but one
occurred with descending currents, when the ring was suffering from the heating effect
of the full battery. There is some reason to believe that the demagnetisation was
not so perfect in this experiment as in the others.

Eleven pairs of observations were made with the iron rod No. 2. In two cases
there were discrepancies of 2 divisions ; no other discrepancy exceeded 1.

In consequence of the very great effect produced by small changes of temperature
upon the susceptibility of nickel, the pairs of observations with this metal were not
nearly so concordant as those with iron and cobalt. The number of double observa-
tions made was 11. With 5 of these (including a case in which there was an evident
misreading of 5 divisions) the discrepancy did not exceed 1 ; twice it was 2, three
times it was 3. One observation of a pair was rejected on account of the evident fact
that the demagnetisation had been accidentally omitted.*

But, though the discrepancies axe great in comparison with those which occurred
with the other metals, their absolute magnitude is very small, and it is not probable
that any point in the nickel curve, fig. 4, is in error by so much as a millionth of an
inch,t or two ten-millionths of the length of the strip.

ADDENDUM.

Received June 5, 1888.

THROUGH the great kindness of Mr. W. H. PBEECE, F.R.S., who allowed me the
unrestricted use of the large battery of secondary cells employed in lighting his
residence at Wimbledon, I have been able to repeat some of my experiments with
much stronger magnetising currents.

The instrument and the magnetising coil used and the mode of working were
exactly the same as before, but it was thought desirable to vary the samples of the
metals, in order that it might appear how far the peculiarities already noticed were
independent of the particular specimens which had been examined.

* The demagnetising process involved eight separate operations, including the closing and opening of
switches. The omission of any one of these might render the process of no effect.

f When the discrepancy was 3 scale divisions the probable error of the mean = 0'67 X 1'5 = 1 scale
division = 0-000018 mm. = 0'00000072 inch (or less than a millionth) = O'OOOOOOIG of the length (or
less than two ten-millionths).

DI.MK.NSIONS ()K KIMiS AND KoDS OK II{o.N AND OTIIK.K MKTAI.S 227

The iron was a piece of soft charcoal wire, 7'5 cm. in length, and 0'32 cm. in
diameter. The nickel was also a drawn wire, its length being 7 '5 cm., and diameter
0*3 cm. The cobalt was a short rod turned from a cylindrical casting. When finished
its length was 4'1 cm., and diameter 0'5 cm.

The nickel and cobalt were supplied by Messrs. JOHNSON and MATTHEY, who
prepared them expressly for these experiments. Both were probably purer than the
specimens previously used. The new cobalt rod in particular was much softer than
the old one, and no great difficulty was experienced in turning it in the lathe — an
operation which I performed myself.

The results of a series of experiments with these rods are given in Table VI., and
plotted as curves in fig. 6, the points at which observations were made being distin-
guished by crosses.

It was not attempted, and, indeed, under the circumstances, it would not have been
possible, to make the observations with the same degree of precision as in the first
described experiments. The values of the elongations and retractions are therefore
given in two-million ths of the lengths (expressed for convenience as multiples of five
ten-millionths) instead of in ten-millionths. To this degree of approximation identical
values of the elongations could be obtained for the same magnetising forces with
tolerable certainty, and the accuracy of the tabulated results may be accepted as
amply sufficient for the purpose in view.

TABLE VI.

Magnetic field in

Elongations in tcn-millionths of length.

C.O.8. unit*.

Iron.

CoUlt

Nickel

35

- 85

100

15

-10

-135

220

f

-25

ri

310

0

-45

-210

500

-35

-30

-225

»c>:.

t <

-20

-235

7 t-.~i

-50

0 -245

875

15

960

-60

30 -2*5

112(1

-65

45 -240

i-ji:.

t 4

55 -250

1806

-65

65 -245

1400

-66

75 -245

1500

-65

'••'

NOTE. — Where no observations are recorded none were made.

A second series of observations with magnetising forces up to about 400 units was
subsequently made in my own laboratory (a battery of seven GROVE'S cells being

2o2

228 MR. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION IN

employed) in order to arrive at a more definite determination of those portions of the
curves where greater detail seemed to be required. The corresponding values of
magnetising force and change of length thus obtained are given in Table VII., and
indicated by dots in the diagram fig. 6.

TABLE VII.

Elongations in tcn-millionths of length.

Magnetic field in

C.G.S. unite.

Iron.

Cobalt.

Nickel.

65

13

0

-104

90

17

, .

. .

112

• •

- 8

• •

125

19

t t

-167

181

13

-19

-199

237

7

-31

-218

293

0

-37

-233

343

- 6

-44

-240

393

-13

-43

-242

Fis*. 6.

Referring to this diagram, it will be seen that in the cases of iron and cobalt the
coincidence of the two sets of observations is as close as could be expected. The
earlier portions of these curves have, in fact, been drawn through the dots without
any regard to the position of the crosses. But for nickel the two series of results
diverge very considerably. The curve is here drawn as smoothly as possible through

DIMENSIONS OF RINGS AND RODS OP IRON AND OTHER METALS. 229

the crosses, and the dots are seen to lie evenly and regularly a little below it. A
discrepancy of this kind might, under the circumstances, have been foreseen. It is
beyond doubt due to the fact which I have frequently noticed and already remarked
upon in this paper, that the degree of retraction which nickel undergoes when
magnetised is materially affected by differences of temperature,* a fact which may
be explained by the well-known influence of heat in diminishing the magnetic suscep-
tibility of that metal. Now the temperature of Mr. PRBECE'S engine-room on the
afternoon of May 25, when the secondary battery experiments were carried out, was
most uncomfortably high, whereas the air pf my laboratory when the other experiments
were made, three days later, was bitterly cold. Unfortunately no thermometer
readings were taken, but the difference of the temperatures on the two occasions
could not have been less than 10° C. Such a difference is, I think, sufficient to
account for the discrepancy of the two series of results.

The results of the experiments referred to in this addendum may be stated as
follows.

Save as to mere details, which may be expected to vary more or less with different
specimens of the metals used, according to their purity and physical condition, the
results lastly obtained are in agreement with the former ones, so far as these go.

With regard to details, the new specimen of cobalt reaches its maximum retraction
at an earlier stage than the old one, namely, at about 300 units instead of 400. Both,
however, agree in not yielding the smallest indication of either retraction or elongation
in weak fields.

The retraction exhibited by the new nickel rod is enormous, being for equal mag-
netising forces more than twice as great as that of the old one, and ultimately
amounting to nearly l/40000th part of its length. I believe this to be accounted for
by its superior purity.

The new iron curve differs from that in fig. 4 in showing a somewhat smaller
maximum elongation and a greater amount of retraction, but the two are of just the
same general character.

The latest results are so far merely confirmatory of the earlier ones. But they go
further, and afford information concerning the behaviour of the metals in far stronger
tields than were before obtained.

They show clearly that the retractions of iron and of nickel reach a limit in fields
of a certain intensity, about 1000 for iron and 800 for nickel in the specimens
examined, the retraction in stronger fields being neither greater nor less. My conjec-
ture that a minimum length might possibly be passed is therefore not supported.

On the other hand, it appears that the length of the cobalt rod, after passing through
a minimum, regains its original value in a field the strength of which is for this
particular rod about 750 units, and then rapidly increases, the increments of length

• I find that Professor BARBKTT had observed this effect of heat. ' Nature,1 vol. 26, 1882, p. 586.
—June 13, 1888.

230 MB. S. BIDWELL ON CHANGES PRODUCED BY MAGNETISATION.

and of magnetising current being nearly proportional when the experiment was stopped
with a field of 1400 units. The actual elongation was then about four times as much
as the maximum elongation of the iron rod.

It is clear that so far as cobalt is concerned the subject is not yet exhausted, though
the further experimental investigation of it would not be easy. A limit to the inten-
sity of the magnetic field, which can be produced by a coil of given dimensions, is
imposed by the heating effect of the current,* and this limit was practically reached
in my experiments. With a larger coil, and a suitably adapted instrument, stronger
fields could undoubtedly be obtained without either serious heating or sacrifice of
uniformity ; but the battery power necessary for such a coil would be dispropor-
tionately great, so great, indeed, that for any considerable increase of the field it
would not be easily obtainable.

It is unlikely that a field suitable for these experiments could be produced by means
of an electro-magnet. For the present, therefore, our knowledge as to the behaviour
of cobalt under magnetisation must remain incomplete.

In conclusion I have to express my hearty thanks to Mr. PREECE for having so
freely placed at my disposal the resources of his valuable electrical installation.

The quantity of heat generated in the coil per unit of time varies as the square of the magnetic field.

[ 231 ]

VIII. On the Ultra-Violet Spectra of the Elements.— Part III. Cobalt and Nickel.

By G. D. LIVEING, M.A., F.R.S., Professor of Chemistry, and J. DEWAB, M.A., F.R.S.,
Jacksonian Professor, University of Cambridge.

Received February 27,— Read March 15, 1888.

[PLATES 9-14.]

IN our first communication to the Society on this subject we gave a reference map of
the iron lines in the ultra-violet region, based on measurements of the wave-lengths
of nearly a hundred of the principal linea These measurements were made by means
of a diffraction grating by RUTHERFURD. Since that time we have obtained some of
ROWLAND'S fine gratings, having a much larger ruled surface, though a somewhat
smaller number of lines to the inch, than the RUTHERFURD grating. With larger
gratings it was advantageous to use larger telescopes, and it was a matter of some
interest to determine whether the improved appliances led to the same numerical
values of the wave-lengths as before. In each case we determined the constant of the
grating — that is, the distance between successive lines — by ANGSTROM'S scale. We
had, for instance, measured the deviations by the RUTHERFURD grating of several
lines — namely, C, D, b, F, and H — for as many orders of spectra as we could on both
sides of the collimator, and then calculated what must be the distance between the
lines to produce these deviations, on the assumption that ANGSTROM'S wave-lengths of
these lines were correct. We have since used the solar lines E, or the corresponding
lines of the spectrum of the spark taken between iron terminals, for the purpose of
gauging our gratings, because ANGSTROM had bestowed the greatest pains in the
measurement of the wave-lengths of these lines. PEIRCE and BELL have corrected

o

ANGSTROM'S measures, but, for convenience in comparisons, we have retained the old
scale. In order to make our numbers correspond to the new scale they must all be
multiplied (according to ROWLAND) by 1 '00016.

The first comparison of measurements with the different gratings was made with
respect to the cadmium lines, which have frequently been used as lines of reference.
Four pairs of photographs of the cadmium line No. 17 in MASCABT'S notation had
given us the following wave-lengths : —

3.8.88

232 PROFESSORS G. D. LIVEING AND .T. DEWAR

2748-58 . .at 4th order.

2748-27 . . „ „ „

2748-20 . . „ 5th „
2748-16 , 6th

Mean . 2748'30

Six pairs taken with a ROWLAND grating and the same goniometer gave us —

2748*43 . . at 3rd order.

2748-41 . . ,-, 4th „

2748-50 . . „ 5th „

2748'35 . . „ 6th „

2748-41 . . „ 7th „
2748-29 , 8th

Mean . 2748'40

BELL'S* value of the wave-length of the same line is 2748'45. The photographs
were all taken with a collimating eyepiece. Considering that the cadmium lines,
though strong, are very diffuse, so that there is room for considerable error in esti-
mating the centre of the photographic image, and that an error of 0'013 mm. in
measuring the distance between the two images of the line on the photograph would
make an error of 1" in the deviation, or very nearly "03 in the value of the wave-
length, the values above are fairly concordant. However, besides the difficulty of
exact measurement arising from the ill-defined character of the cadmium lines, there
is a source of error arising from the gratings themselves, and affecting both of them.
It is that the two images of any given line, on the two sides of the normal to the
grating, are not in focus at the same distance from the object-glass of the telescope.
Hence, if the photographic plate is adjusted to the focus of a particular line when the
grating is turned in one direction, it will not receive a sharp image of the same line
when the grating is turned in the other direction.

With a collimating eyepiece it is comparatively easy to place the grating so that
the normal to its plane may coincide with the axis of the collimator. Hence, fair
measures of the wave-length of any line may be obtained from the deviation for that
line, as measured on one side only of the normal. But, since every change of focus
makes it necessary to take a fresh reading of the position of the grating when it is
normal to the axis of the collimator, it is better, as a rule, because errors of reading
thereby become of less consequence, to measure the angle between the two images,
one on one side and the other on the other side of the normal. If this is done, the
difficulty arising from difference of focus on the two sides must be faced, as any
alterations of adjustment between the times of taking the pair of photographs would
be wholly inadmissible.

* ' Amer. Journ. Sci.,' vol. 31, 1886, p. 429.

ON THE ULTRA-VIOLET SPECTRA OP THE ELEMENTS.

Again, it seldom happens that a grating gives the two spectra of the same order
equally bright. With one of our gratings the spectra of the 3rd and 5th orders are
bright on one side only of the normal, while those of the 4th and 6th orders are
bright on the other side only. Where faint lines were in question, therefore, we have
used spectra of different orders on the two sides, and computed the wave-length by the
formula (m + n)\ = 4a sin £ (a -f ft) cos £ (a — ft) ; where ra and n are the orders of the
spectra employed, a and ft the deviations of the ray from the normal, (a -f- ft) is, of
course, the angle directly measured ; (a — ft) has to be calculated from the reading of
the instrument when the grating is normal to the axis of the collimator. As £ (a — ft)
is always a small angle, any error in determining it will affect the value found for X
but little.

Measures of other cadmium lines gave the following results :—

RUTHEBTUBO gr»t ing

ROWLAND grating.

No. of line.

No. of

No. of

BILL'S value of

A

pairs of
photograph*.

A

pain of
pbotographi.

A

18

1

2572-95

5

2572-75

2572-95

22

2

2329-67

5

2320-92

2321-14

23

2

2312-66

5

2312-75

2312-83

24

2

2264-91

4

2264-64

2264-88

. ,

2

2240-10

25

1

2194-06

?1

2194-28

2193-98

Previous measures of lines Nos. 24 and 25, by interpolation between the copper
lines of which the wave-lengths had been determined by RUTHERFORD'S grating, had
given the values of X 2265'0 and 2194'2.

Other cadmium lines, measured by ROWLAND'S grating, gave the following results :—

No. of the
line.

No. of
pairs of

photographs.

A

BILL'S value of

A

9

3

/ 3612-15
\ 360975

3611-75
3609-39

10

3

/ 3467-03
I 3465-61

3466-70
3465-22

11

5

3403-08

3402-68

1-2

5

{3260-46
3252-08
3249-83

3260-12
3251-77
3249-40

M DCCCLX X X VIII. — A.

2 H

234 PROFESSORS G. D. LIVEING AND J. DEWAR

The wave-lengths of this last group of lines are all too large by about 0'35 as com-
pared with BELL'S, and they seem to be affected by some common error, but whence it
arises we do not know. Our measures of lilies in the visible spectrum, where, of
course, there are fewer sources of error, agree very closely with BELL'S. Thus our
measures of the blue lines gave the figures for X 5085 '26, 4799*40, and 4677*59, and
these, when multiplied by the factor 1*00016, come very close indeed to BELL'S
numbers, which are 5086*09, 4800*15, and 4678*39. On the whole, we are inclined to
think that a really good goniometer with a plane grating and a telescope of moderate
dimensions, focal length 8 or 9 decimeters, will give extremely accurate wave-lengths,
while the greater angular aperture of such a telescope gives it a considerable advantage
in point of light over the concave grating used by BELL. The method of measuring
wave-lengths by the coincidences of lines in spectra of different orders, for which a
concave reflecting grating is admirably adapted, could not be easily applied to the
ultra-violet spectra of cobalt and nickel, because the lines are so crowded that the
overlapping of two or three spectra, all in focus together, would produce a complication
which it would be nearly impossible to unravel, except by dispersing the spectra in a
direction at right angles to the dispersion produced by the grating. The chromatic
aberration of our quartz lenses is a positive advantage in dissipating the light of the
spectra of those orders which are not under examimation.

For the determination of the cobalt and nickel lines specimens of those metals were
prepared so far spectroscopically pure that the spark between fragments of the cobalt
showed none of the characteristic strong lines of nickel, and the spark between pieces
of the nickel showed none of the characteristic strong lines of cobalt. As the metals
after reduction were fused with an oxyhydrogen blowpipe in lime crucibles, they were
not free from all traces of other metals. For the arc lines much labour in the purifi-
cation of the metals would have been wholly thrown away, because a variety of metals
are present in the carbon electrodes as well as in the limestone used for crucibles. In
the arc, therefore, we used samples sold as " pure," and identified the lines either by
their coincidence with spark lines photographed at the same time through a part of
the slit, or by their making their appearance, or being notably strengthened, on the
introduction of the metal into the arc. The list of arc lines is much less complete
than the list of spark lines, because weak lines in the arc are more easily overlooked
in a photograph crowded with lines, and when noticed their origin is with difficulty
identified. The wave-lengths of the spark and arc lines which were not measured
directly by means of a grating were determined by interpolation from photographs
of refraction spectra. In the highest region the copper lines were used as lines of
reference in this interpolation. For the direct determination of the wave-lengths of
the nickel lines about 170 photographs were taken, measured, and the results reduced ;
for the cobalt lines about 200. In many cases several lines could be measured on the
same plate, but we have rarely been satisfied without getting two or more independent
measures of the deviation for each line, and in many cases the measures have been

ON THE ULTRA-VIOLET SPECTRA OF THE ELEMENTS. 235

made in more than one order of the spectra. Owing to difference in the strength of
the spectra on the two sides of the normal to the grating, it sometimes happened that
faint lines appeared in one of a pair of photograplis, that is on one side of the normal,
but not on the other. In such cases, when the reading of the circle for the normal
position of the grating (which is liable to vary with every adjustment of focus) could
be accurately determined by the help of stronger lines, which could be measured in
both photographs, the wave-lengths of the faint lines have been calculated from the
deviation as measured on one side of the normal. Such a measurement, though it
gives a valuable result, is plainly not quite independent.

From the close chemical relationship between cobalt and nickel we should have
expected that their spectra would closely resemble one another. In regard to the
large number of lines which they exhibit, they certainly resemble each other, and
resemble iron ; and the resemblance goes a little further, inasmuch as the lines of all
three spectra are much more crowded in certain regions than in others, and the
crowded regions are approximately the same for all three. But, beyond such a general
resemblance, we have been unable to trace any definite correspondence in the spectra.
The number of lines of cobalt which according to our measurements have wave-lengths
identical with those of nickel lines is small, but in a record of 580 lines of cobalt and
400 of nickel it would be surprising if there were not many close coincidences, and, in
fact, we note forty-six cases where lines of cobalt do not appear to differ in wave-length
from lines of nickel by more than a tenth of a tenth-metre. Now, if the cobalt lines
were uniformly distributed over the whole region mapped and the nickel lines distri-
buted at random amongst them, it would be an even chance that twenty-six nickel
lines would not be more than a tenth of a tenth-metre distant from the nearest cobalt
lines. A glance at the map will, however, show that the cobalt lines are by no means
evenly distributed, and in the regions where they are most closely packed the nickel
lines are also for the most part closely packed. Hence, the chance of merely accidental
coincidences is very much greater than that above mentioned ; and we find that in
the region between the wave-lengths 2250 and 2550 the number of lines of the two
metals, which, as measured, are not more than a tenth of a tenth-metre distant from
one another is twenty-five, more than half the whole number of coincidences to that
degree of approximation. On the whole we are unable to conclude that the coin-
cidences are more than fortuitous. It should be observed that we have not yet had
the opportunity of comparing the spectra of the two metals photographed on the
same plate with high dispersion. On many of our plates the iron spark has been
photographed simultaneously with the spectrum of another metal, and we have noted
on the table of wave-lengths the cases in which these photographs show an unresolved
coincidence between an iron line and a cobalt or nickel line. These photographs,
however, have all been taken with a prismatic spectroscope, and it is probable that a
higher dispersion might resolve some of these coincidences.

2 H 2

236

PROFESSORS G. D. LIVEING AND J. DEWAR

TABLE of Cobalt lines.

In the first two columns are given the intensities, 1 to 6. of the lines as observed in
the spark and arc respectively, 1 being the most intense. In the third column are
recorded the wave-lengths. A figure in the fourth column indicates that the wave-
length against which it is placed was independently measured by means of a grating,
and at the same time records the number of separate determinations combined in the
given result.

Intensity.

X

Remarks.

Spark.

Arc.

G

2190-2

The wave-lengths of the lines in this region

6

2191-9

were obtained by interpolation between the

6

2193-1

copper lines photographed at the same

6

2198-2

time through a part of the slit

6

2205-7

6

2214-1

6

2215-9

6

2219-6

6

2229-5

6

2231-5

G

2234-4

3

2244-8

6

2253-2

2

2256-4

2

2259-7

2

2266-2

. .

A weak Ni line

6

2270-5

6

2272-0

5

2273-3

Also a Ni line

6

6

2274-2

. .

A stronger Ni line

G

2275-1

6

2275-9

6

2278-1

5

2280-1

4

2281-5

6

2281-9

5

2283-1

2d

2

2285-7

1

6

5

2287-8

6

6

2289-9

6

6

22UO-9

3

4

2291-5

1

3

2

2293-0

5

4

2295-5

5

5

2296-9

6

6

2298-3

4

6

2299-3

•

5

5

2300-3

5

4

2300-8

6

2303-8

6

t

2305-6

, .

Also a Ni line

6

2306-4

Id

2

2307-4

2

ON THE ULTRA-VIOLET SPECTRA OP THE ELEMENTS.

237

TABLE of Cobalt lines (continued).

Intensity.

A

Remark*.

Spark.

Ate.

6

6

2310-4

1

3

2311-1

2

Also a Ni line

5

5

2312-1

4

..

23131

2

3

2313-5

2

Also a Ni line

2

5

2314-5 2

6

4

2315-5

Also a Ni line

3

5

2316-8

Also a Ni line

6

2318-2

6

4'

2319-6

6

5

2321-0

Also a Ni line

3

4

2324-0

2

Also a Ni line

3

5

2325-9

2

8

4

9826-1

1

Also a Ni line

5

2327-3

6

6

2328-7

3

5

2330-0

1

Also a Ni lino

6

6

2333-7

3

4

2335-9

1

5

2336-6

Also a Ni line

2

4

2337-6

i

6

4

2338-4

5

4

2338-8

2

2340-8

1

Also a Ni line

a

2344-0

1

Also a Fe line

4

2344-3

5

•

2345-2

4

6

2346-2

Also a Ni line

6

5

2346-7

3

5

2347-0

I

5

2347-4

6

2348-1

5

5*

2350-6

Also a Ni line and a Fe line

6

5

2351-5

6

3

2352-1

1

1

3

2353-0

I

4

4

2357-7

1

1

6

2360-0

5

6

2360-2

2

4

88603

6

'(5

2360-8

6

6

2361-2

1

3

2363-3

2

6

5

2366-6

5

4

2370-1

3

4

2371-5

2

5

4

2371-3

5

5

2372-6

4

6

2374-8

2

Also a Fe line

1

2

2378-1

3

6

6

2380-3

2

4

2381-3

2

5

• •

2381-7

Also a Ni line

238

PROFESSORS G. D. LIVEING AND J. DEWAR

TABLE of Cobalt lines (continued).

Intensity.

X

Remarks.

Spark.

Arc.

2

4

2382-9

2

2

4

2385-9

2

3

6

2386-1

1

5

4

2388-3

2

A rather close double, very near a Fe line

1

6

2388-4

3

4

2389-1

6

4

2391-5

4

4

2392-1

> t

A weaker Ni line

4

6

2393-4

4

4

2395-1

1

5

2396-9

3

4

5

2397-8

2

6

}/

2401-3

1

6

I

2401-6

t t

Also a Ni line

6

6

2402-4

6

6

2403-3

3

4

2403-8

3

4

6

2404-0

4

6

2405-1

1

6

2406-9

3

Id

2407-1

3

3

t t

2407-8

2

3

4

2408-3

2

2

1

2411-2

2

6

3

2412-2

2

Also a Ni line and an air line

3

4

2413-7

3

5

2

2414-1

5

2

2414-8

1

3

,.

2415-5

4

1 t

2415-7

2

3

5

2416-5

2

3

3

2417-2

2

3

4

2418-1

2

1

6

2420-3

3

6

6

2421-6

6

4

2422-1

3

6

2423-2

2

4

Id

2424-5

2

5

6

2425-7

2

3

t t

2427-8

2

5

5

2429-6

2

6

f t

2430-0

1

1

2432-0

4

5

2

2434-6

Close to a Si line

6

5

2435-8

6

2436-2

3

2436-5

3

6d

B >

2437-9

5

3

2438-5

6d

, ,

2439-7

6

4

2440-6

5

4

2441-2

2

2442-0

3

ON THE ULTRA-VIOLET SPECTRA OF THE ELEMENTS.

289

TABLE of Cobalt lines (continued).

Intensity.

\

Kemarks.

Spirt

Are.

3

5

2443-3

2

3

6

2445-6

3

2

5

2447-3

2

Also a Fe line

4

t ,

2448-7

2

2

. .

2449-4

8

6

6

2452-0

6

. .

2452-7

6

5

2453-3

6

. ,

2453-6

6

2

2455-7

3

. ,

2459-0

1

6

3

2460-3

6

. .

2460-8

1

4

2463-7

2

2

. .

2466-5

2

6

6

2469-0

6

3

2469-7

6

2

2472-5

<U

6

2473-5

6d

, .

2474-9

6

. .

2476-0

6

3

2476-2

6

. .

2476-9

4

t t

2477-1

2

Also a Fe line

4

f m

2477-8

5

• •

2478-6

t u

Close to a carbon line

6

4

2483-2

6

. .

2484-1

6

, .

2484-4

3

^ w

2484-8

I

3

• •

2485-9

I

Also a Fe line

6

, B

2486-7

6

. .

2486-9

3

3

2489-8

1

6

3

2490-4

6

5

2494-4

6

4

2495-1

6

2

2496-3

3

3

2497-1

1

4

. ,

2498-2

1

5

3

2500-2

U

6

2501-7

6

3

2504-1

1

1

2505-8

3

.4

4

2507-5

1

6

, f

2509-4

1

1

2510-5

2

Also a Ni line

6

. .

2511-4

6

6

2511-7

4

. .

2516-9

5

1

2517-3

1

. .

25193

2

5

1

2520-7

1

240

PROFESSORS G. D. LIVEING AND J. DEWAR

TABLE of Cobalt lines (continued).

Intensity.

A

Remarks.

Spark.

Arc.

4

2522-5

1

Close to an air lino

6

i'

2524-2

1

Close to a Li line

2

* •

2524-5

1

2

2

2528-1

1

Also a Si line

a

3

2529-6

1

6

1

2531-7

2

m >

2533-4

1

Close to a Fe line

5

4

2535-5

1

6

4

2536-1

5

6

2537-0

3

\ 9 /

2540-2

1

2

) 2 I

2541-5

1

6

1 o /

2543-9

6

) 3 i

2544-2

5

4

2544-6

. .

6

3

2545-7

2

t ,

2546-3

1

6

4

2548-9

6d

4

2549-7

t t

Also a Fe line

4

..

25501

t t

Also a Fe line

4

2552-2

6

4

2552-7

t f

Also a Ni lino

6

4

25531

3

, .

2556-3

I

4

4

2556-9

1

2

6

2558-9

1 .

3

6

2559-6

1

6

1

25617

1

6

2563-6

1

6

• •

2565-0

6

3

2567-0

3

* •

2569-3

1

6

4

2571-9

6

4

2573-1

3

5

2574-4

1

Also an Al line

1

5

2579-8

1

Also a Ni line and close to an air line

3

6

2581-7

1

5

t f

2582-6

1

6

5

2584-8

2

6

2586-8

t %

Also a Ni line

6

6

2592-9

6

6

2598-8

t f

Also a Fe line, or close to it, and close to an

6

5

2600-3

air line

5

5

2603-9

4

4

2605-2

6

5

2605-3

4

, B

2613-0

3

6

2613-8

4

2

2618-5

4

6

2619-3

6d

4

2621-7

6d

6

2626-6

6

3

2627-3

.

ON THE ULTRA-VIOLKT SPECTRA OF THK ELEMENTS.

241

TABLE of Cobalt lines (continued).

Intenilty.

\

Remark*.

Spark.

Anx

6d

2628-4

A Ni line, doubtful Co line

3

6

2631-9

5

t ^

2«34-5

6d

6

2642-7

6

6

2644-4

5

3

9646-1

2

1

2648-4

3

2653-3

2

6

2662-7

I

5

6

2669-7

1

6

2670-1

. ,

Also a Ni line

3d

4

2675-4

5

t B

2677-4

6

4

2679-0

5

2679-8

6

( g

2681-5

3d

t t

2684-0

1

Also a Ni line

5

t B

2689-2

1

5

, ,

2692-5

1

2

6

2694-1

2

6

3

2695-3

1

6

t

2695-9

6

2696-0

6

^ t

2696-4

1

4

m m

2701-9

1

id

27062

1

Also a Fe line

6d

2706-9

1

Also a Fe line

6d

2707-4

1

6

6

2708-6

1

Also a Fe line

3

t f

2713-9

3

Very close to a Fe line

6

. ,

2714-5

5

3

2715-3

4

, .

2720-6

1

Very close to a Fe line

4i

m t

2727-5

i -

m t

2728-8

6

6

2730-7

6

, .

2732-6

H

, ,

2734-3

6

m t

2738-6

5

5

2744-7

6

6

2757-1

6

5

2761-0

1

5

4

2763-9

5

4

2766-0

5

t r

2766-5

1

4

6

2768-6

1

U

6

2774-8

Also a diffuse line in the Ni spark

2

m t

2775-7 1

6

6

2778-5

tt

, f

2785-2

U

. .

2785-7

U

r f

2786-9

1

6

••

2789-1

••

Also a Fe line

Ml" i i I. \\.\\ III.- \.

2 I

242

PROFESSORS G. D. LIVEING .AND J. DEWAR

TABLE of Cobalt lines (continued).

Intensity.

\

Remarks.

Spark.

Arc.

3

2793-4

6

• •

2795-8

b'

f f

2796-3

6

, t

2796-6

6

6

2798-4

U

2801-7

1

6

5

2803-3

f t

Also a Fe line

4

f t

2806-7

2

Also a Fe line

3

f t

2810-3

4

4

3

2815-2

1

6d

r t

2815-8

6

6

2818-3

6

6

2819-4

6d

5

2821-1

bd

, .

2822-7

6

, .

2823-2

1

2

2

2824-5

5

3

m t

2834-3

1

6

6

2836-7

6

f m

2845-2

bd

f r

2847-9

• •

Also a Fe line

6

5

2849-8

6

5

2862-2

6

. ,

2865-1

• •

Also a Ni arc line

2

. .

2870-4

1

Very close to a Fe line

(yd

. ,

2879-9

6

3

2881-3

1

6d

f t

2883-1

5

4

2886-0

2

ti

2890-0

bd

t t

2897-5

6

3

2899-3

6

5

2906-5

3d

. .

2918-1

6

6

2927-2

6

6

2929-0

2d

, ,

2930-0

2d

2

2942-5

1

Id

. .

2954-1

1

Gd

. .

2971-2

6d

6

2983-3

3

3

2986-5

3

3

2989-1

6

. .

2994-7

6

5

3000-1

6d

6

3008-5

6

5

3010-3

4

4

3013-2

6

6

3015-2

3

, .

3017-0

. .

Very close to a Fe line

2

; 5

I 5

3033-8
3034-0

6

5

3042-2

ON THE ULTRA-VIOLET SPECTRA OP THE ELEMENTS.

TABLE of Cobalt lines (continued).

IntoiuUy.

A

Remark*.

8p»rk.

Arc.

3

3

3043-6

3

3

3048-6

4

3

3050-6

t t

Very clone to a Ni lino

6

6

3059-6

2

3

3061-4

6

5

3063-0

6

4

3064-0

2

{ S }

3071-8
3072-0

6

G

3073-4

6

6

3078-9

3

3

3082-1

2

2

30863

4

4

3089-0

4

3

3097-6

6

. ,

3101-8

5

5

3103-3

6

6

3109-0

6

6

3109-5

5

5

31130

2

2

31211

I

6

6

3126-7

4

6

3130-4

1

3

2

3136-8

1

3

3

3139-5

1

3

3

3146-6

1

5

4

3148-9

6

6

3152-3

2

3

3154-2

1

3

3

3158-2

1

5

5

3159-2

6

6

31613

6

6

31643

5

5

3169-5

6

4

3174-8

4
5

5
5

3176-6
3181-7

1

Oat of focus in photograph with grating

5

5

3188-0

6

6

3210-1

5

4

3218-7

6

6

32265

5

4

3232-4

1

5

5

3235-2

5

4

3236-7

1

4

5

32434

1

Also a Fe line

3

• •

3246-7

1

Also a Fe line

6

5

32496

4

4

3253-7

1

4

5

3260-1

1

6

5

3261-7

6

• •

3262-7

u

5

3264-4

5

6

3271-3

2 I 2

•244

PROFESSORS G. D. LIVEING AND J. DEW AH

TABLE of Cobalt lines (continued).

Intensity.

X

Remarks.

Spark.

Arc.

6

6

3276-0

6

4

3277-2

5

6

3278-5

2

3

3282-9

1

6

3284-2

5

5

3286-6

6

3294-2

1

6

6

3303-2

6

6

3306-5

1

6

6

3308-2

5

3309-1

5

'5

3311-7

1

Also a Ni line

4

5

3313-6

4

3

4

3319-0

1

4

4

3321-7

3

Also a Ni line

5

5

3324-8

3

5

5

3326-4

3

6

6

3329-0

1

2

1

3333-6

4

5

5

3339-3

2

6

3340-2

1

6

3340-8

2

5

5

3342-2

2

5

5

3346-4

1

5

5

3347-7

1

6

5

3348-9

1

4

t g

3352-3

4

2

1

3353-9

4

6

3

3360-8

1

Also a Ni line

6

6

3362-3

3

1

3366-6

5

4

4

3370-4

4

6

5d

3376-6

1 t

5

6

3378-0

2

6

1

3380-0

2

Also B, Ni line

4

2

3384-7

3

6

3387-1 •

3

3

1

3387-6

5

6

1

3394-2

1

3

1

3394-8

5

2

1

3404-5

4

Also a Ni line and in OH flame with CoClj

6

3406-1

3

i

3408-6

3

In OH flame with CoClj

2

3411-7

3

In OH flame with CoCl2

3

3412-0

3

t t

2

3414-2

3

6

f t

3415-2

3

4

2

3416-5

3

In OH flame with CoCl,

42

3

3423-2

3

Also a Ni line

4

3430-9

3

In OH flame with CoCl2

5

3431-3

3

1 I

3432-4

4

In OH flame with CoCl3

3

[ ]

3432-9

1

Also a Ni line

ON THK ULTRA-VIOLET SPECTRA OF THK KI.KM KNTS.

•• i :•

TABLE of Cobalt lines (continued).

Intentlty.

A

Remark*.

Spark.

Arc.

u

4

3436-8

Also a Ni lino

. ,

6

3438-2

6

3

3442-3

3

3
4

3

:?

3443-0
3445-7

3
2

In OH flame with CoCL
Also a Ni line and in OH flame with CoCl,

3

3448-6

3

In OH flame with CoCl,

5

3448-9

2

2

2

3452-9

3

Also a Ni line and in OH flame with CoCl,

4

3

8454-6

2

5

4

3460-6

2

2

2

3462-2

3

In OH flame with CoCl,

2

1

3465-2

5

Abo a Ni line and in OH flame with CoCl,

and a Fe lino

1

1

3473-4

5

In OH flame with CoCl,

6

2

3476-0

1

6

5

3478-0

3

1

3482-7

2

In OH flame with CoCl,

1

4

3484-7

2

1

1

3488-8

5

In OH flame with CoCl,

5

4

3490-6

2

3

1

3495-1

4

In OH flame with CoCl,

5

8

3496-0

3

4

~] r

3501-0

2

1

\ 1 /

3501-6

3

In OH flame with CoCl,

5

J |^

3502-0

2

6

3503-4

1

2

3505-6

3

In OH flame with CoCl,

4

3509-3

5

In OH flame with CoCl,

6

3509-7

4

Also a Ni line

4

3

3512-0

3

In OH flame with CoCU

3

2

3517-7

5

5

3

3519-5

2

3

2

3520-9

4

Also a Fe lino

3

3

3522-9

4

In OH flame with CoCl,

4

3

3526-3

4

6

4

3528-4

3

In OH flame with CoCl,

3

1

3529-3

4

Also a Ni line

4

8

3532-8

4

In OH flame with CoCl , and a Fe line

4

5

3542-8

3

6

6

3544-7

1

6

fi

3548-0

4

3

3550-1

3

6

6

3552-4

1

6

f f

35623

2

3

3560-5

3

In OH flame with CoCl,

3

1

3564-5

2

In OH flame with CoCl, and close to a Fe

line

1

1

3568-9

3

In OH flame with CoCl,

1

1 {

3574-5
3574-9

2 1
2 /

In OH flame with CoCl,

6

5

3577-4

1

3

2

3584-7

3

Also a Fe line

1

2

3586-7

3

246

PROFESSORS G. D. L1VEING AND J. DEWAR

TABLE of Cobalt lines (continued).

Intensity.

X

Remarks.

Spu*.

Arc.

3

1

3594-4

3

Observed in OH flame with CoCIa

3

1

3601-6

3

Observed in OH flame with CoCI2 and close

to a Ni line

3

2

3605-0

4

Also a Fe line

6

6

3611-3

4

6

6

3614-8

3

1

3627-3

4

Observed in OH flame with CoCl,

6

6

3632-2

4

4

6

3634-2

2

6

6

3636-1

1

6

5

3638-9

3

.

6

6

3641-1

1

5

5

3642-7

3

6

m t

3648-8

3

6

* •

3654-0

1

6

5

3656-1

1

3

4d

3661-6

3

6

3680-8

3

3

2

3682-5

3

6

6

3690-2

1

3

3 {

3692-4
3692-8

1
3

3

6

3701-7

4

2

1

3703-5

6

Observed in OH flame with CoCl9

6

3711-6

5

5

3729-8

4

4

5

3731-8

4

6

5

3732-8

2

6

f t

3735-2

2

1

m f

3745-8

2

Observed in OH flame with CoCl3

6

. .

3753-9

2

Arc lines obscured in this region by bands of

6

. .

3769-7

cyanogen

6

5

3774-0

6

5

3777-0

6

5

3807-3

6
6

••

3815-1
3815-7

2 \
2 /

A Fe line here

6

3830-3

2

4

3841-4

7

Observed in OH flame with CoCl2

1

2

3844-8

9

Observed in OH flame with CoCl2

4

. .

3860-5

5

Arc obscnred by bands of cyanogen

2
3

} ' {

3872-4
3873-2

4 1
5 I

Observed in OH flame with CoCl2

6

5

3876-1

4

3

1

3881-0

5

6

t f

3884-0

1

1

3893-4

4

Observed in OH flame with CoCl2

5

t f

3894-3

2

6

4

3905-2

6

4

3909-0

Observed in OH flame with CoClj

6

6

3916-2

3

5

3935-5

1

Observed in OH flame with CoClj

6

3

3940-9

1

Observed in OH flame with CoCl2

ON THK ULTRA-VIOLET SPECTRA OF THE ELEMENTS.

247

TABLE of Cobalt lines (continued).

Infinity.

8p.rk.

Are.

6

6

3944-9

1

6

6

3952-4

1

6

6

3955-7

1

6

4

39577

1

6

. .

3968-8

1

t t

5

3974-1

2

2

3978-7

. .

Observed in OH flame with CoCl,

. .

6

3987-1

. .

5

3990-2

, ,

5

3991-4

1

2

3994-7

3

In OH flame with CoCU

6

1

3997-3

3

TABLE of Nickel lines.

In the first two columns are given the intensities, 1 to 6, of the lines as observed in
the spark and arc respectively, 1 being the most intense. In the third column are
recorded the wave-lengths. A figure in the fourth column indicates that the wave-
length against which it is placed was directly measured by means of a grating, and
the figure records the number of separate determinations combined in the given
result.

lnton-.it}-.

A

Remarks.

Sptrk.

Arc.

4

3

2173-8

4

3

2174-4

5

• *

2176-0

1

5

. ,

2176-7

6

, .

2179-4

4

. .

2179-9

6

8

2182-8

3

3

2184-2

3

6

2185-0

5

6

2188-2

1

6

4

2190-0

6

4

2190-6

6

. .

2193-2

U

3

2197-2

u

, .

2198-0

•

5

4

2198-4

2

4

2200-8

1

u

2203-0

248

PROFESSORS G. D. LIVEING AND J. DEWAR

TABLE of Nickel lines (continued).

Intensity.

A

Remarks.

Spark.

Arc.

3d

3

2205-2

2

2

2206-1

1

2d

3

2209-8

1

4

4

2210-5

t t

Also a Fe line-

6

5

2211-4

m f

Also a Fe line

4

5

2212-5

1

2

1

2215-8

2

A weak line in Co

3

5

2216-0

5

5

2217-4

2

6

2219-0

3

6

2219-8

2

5

f

2220-6

6

5

2221-3

6

f t

2221-7

3

2

2222-3

2

3

t t

2223-8

1

3

2

2224-3

2

6

2225-3

Also a Fe line

3

3

2225-8

2

6

2226-7

6

2

2227-2

Also a Fe line

U

3

2229-6

. .

A weak line in Co ; also a Fe line

6

t t

2231-2

5
6
6d

• '

2233-5
2235-5
2237-6

i;j";;

The mean of this pair was measured by the
grating

6d

2238-2

6d

t ,

2239-8

6

t t

2241-2

6

5

2242-2

i

A weak line in Fe

6d

2

2244-4

6d

2245-9

i

5

2246-6

i

6

2247-4

6

4

2248-8

Also a Fe line

6

m f

2249-2

6

m r

2250-2

6

6

2250-5

Also a Fe line

6

6

2251-1

Also a Fe line

6

4

2251-4

6

t t

2252-6

Id

2

2253-5

2

6

f r

2253-9

3

3

2254-7

1

3

5

2255-7

1

4

3

2257-6

5

. ,

2258-9

6

3

2259-4

6

t t

2260-3

Also a Fe line

6

4

2261-1

6

6

2262-6

. .

4

2263-1

f t

A weak Fe line ; ? if Ni

2

1

2264-1

2

Also a Fe line

ON THE ULJKA-VIOLET £PKCTHA OF T1IK KKKMKXTS.

249

TABLE of Nickel lines (continued).

Intensity.

A

^

Spark.

AM.

5

2264-8

Also a Fe line.

6

5*

2266')

, .

A stronger line in Co

6

3

2269-1

1

Id

1

2269-9

2

6

f t

2270-3

6

5

227M

6

• •

2272-8

6

6

2273-2

Also a Co lino

3

3

2274-1

t

A weak Co lino

4

5

2275-0

• •

A weak Co line and a Fe line

5

4

2275-7

1

Also a Fe line

. .

5

2276-3

3

6

2277-0

2

3

4

2277-8

M

3

2278-4

1

6

1

2279-2

u

. m

2280-6

0

6

22837

6

t m

2284-8

2

3

2286-8

1

2

6

2287-4

1

4

4

2289-6

6

t f

2290-7

*

6

6

2292-7

6

, t

2295-3

1

2

, t

2296-2

1

3
3

6

2296-71 2 /
2297-1 /

Tho mean of this pair was measured by the
grating

3(2

6

2298-0 4

3

t t

2299-2 4

3

t t

2299-8

4

6

. .

2301-5

1

2

. .

2802-0

4

2

t t

2302-5

5

3

4

2303-3

2

3

f t

2304-8

2

6

r f

2305-7

Also a Co line

3

6

2308-1

2

U

3

2310-6

1

6

. .

2311-2

Also a Co line

4

3

2311-8

3

3

• •

2312-5

3

6

3

2313-4

5

3

2313-6

1

Also a Co line and a Ft> line

Id

3

2315-6

3

Also a Co line

6

3

2316-8

.

Also a Co line

3

t f

2318-0

a

3

3

2319-3

3

4

2

2321-0

2

Also a Co line-

6

5 2321-6

6

3

2322-3

6d

t t

2323-3

6d

2324-0

Also a Co line

.M DCCOLXXXVUL — A.

•2 K

250

PROFESSORS J. D. LIVEING AND G. DEWAR

TABLE of Nickel lines (continued).

Intensity.

A.

Remarks.

Spark.

Arc.

4

2

2325-5

2

3

2326-0

2

Also a Co line

4

2

2329-6

2

6

• •

2330-1

. .

Also a Co line

2

6

2334-1

3

3d

6

2336-2

3

6

3

2336-6

, .

Also a Co line

6

5

2337-1

2

2340-7

3

Also a Co line

4

6

2343-0

3

6

2343-5

2

3

2344-7

3

Also a Fe line

6

• •

2345-0

6

4

2346-2

. .

Also a Co line

6

4

2347-6

1

6d

6

2349-8

t f

Also a weak Fe line

6

6

2350-5

f t

Also a Co line

3

3

2355-9

2

6

3

2358-5

1

4

6

2366-1

1

Also a Fe line

4

5

2367-0

1

5

6

2368-9

4

2369-5

i

6

6

2370-9

2

4

2375-0

2

6

3

2375-6

6

6

2378-6

2

5

2381-8

Also a Co line and a Fe line

6

3

2386-3

1

3

4

2387-5

1

4

6

2388-5

1

Also a Fe line

6

6

2388-7

6

6

2392-0

t t

Also a Co line

4

3

2392-6

2

2

2394-0

1

Also a Fe line

2

a

2394-3

1

6

3

2394-7

t t

Also a Fe line

6

..

2397-2

6

2400-1

1

Also a Fe line

6

3

2401-7

Also a Co line

6

6

2404-8

1

6

3

2412-1

1

Also a Co line and an air Hue

5

o

2412-8

Id

2

2416-0

2

6

3

2419-0

6

2

2420-8

6 3

2423-4

6

2426-8

6

2431-2

•

4

2433-2

. .

Also a Fe line

6

3

2433-9

. .

Also a Fe line

14

3

2437-5

2

6

1

2441-5

f t

Also a Fe line

ON THE ULTRA-VIOLET SPECTRA OP THE ELEMENTS.

251

TABLE of Nickel lines (continued).

Ii.U-n-ity.

X

Remark*.

Sprk.

Are.

5

2448-1

6

2'

2463-7

4

6

2455-4

6

2

2471-8

Also a Fe line

2

3

2472-8

1

Also a Fe line

6

6

2476-6

3d

3

2483-6

1

6

6

24969

3

• •

2505-9

3

Also a Co lino

6

2509-6

2

1

2'

2510-6

3

Also a Co line

6

6

2520-0

6

2524-1

Close to a Si line

6

3

2539-5

5

2543-2

1

Also a Fe line

8

3

2545-4

3

6

6

2549-1

VJ

6d

4

2552-6

1

Also a Co line

4

.,

2554-7

1

6

2557-5

4

5

2559-8

1

Id

2565-7

1

6

6

2568-2

1

6

6

2571-7

4

4

2575-7

6

6

2579-9

• •

Also a Co line

4

• •

2583-5

5

2584-4

1

6

6'

2586-7

Also a Co line

5

2593-1

Also a Fe line

6

2600-8

i'

6

2606-1

i

Also a Fe line

6

2606-7

• •

Also a Fe line

3

1 1

2609-6

i

3

f t

2614-9

i

Also a Fe line

6

• •

2626-3

i

6

2628-4

A doubtful Co line

6

2632-4

i'

6

2636-8

3

2639-5

i

Also a Fe line

6

t

2641-0

6

• •

2643-4

3

3

2646-8

i

6

6

2648-6

3

6

2655-6

i

5

5

2659-5

i

6

2664-9

5

2670-0

• •

Also a Co line and a Fe line

6

2672-1

6

2674-4

3

2678-8

2

2684-0

Also a Co line

6

2690-2

2 K 2

252

PKOKKSSOKS G. D. LIVKlXf! AND J. DKWAR

TABLE of Nickel lines (continued).

Intensity.

X

Remarks.

Spark.

Arc.

6

6

2700-4

5

6

2701-2

, .

Also a weak line of Fe

4

4

2708-3

4

2758-7

1

6

2760-4

4d

, .

2774-7

2

Also a diffuse line in the Co spark

2

3

2805-0

3

6d

t t

2806-0

6

m f

2807-8

3

1

2820-8

6

2823-9

2

2863-3

1

, .

3

2865-1

, .

Also a line in the Co spark

6d
6d

••

2880-9
2882-2

> These are perhaps air lines

6

2889-1

6

2898-8

6

2900-6

5

3

2906-9

2

2

2913-2

1 .

.

6d

2918-8

Bd

2928-4

1

Also a Fe line

6

2934-3

2d

2936-3

1

Also a Fe line

6

6

2938-7

f t

Also a Fe line

2

1

2943-5

1

4

2947-1

I

5d

2954-5

6

2957-8

5d

2968-7

3

3'

2981-2

3

Also a weak Fe line

4

3

2983-6

3

5d

, ,

2987-7

3

6

t

2988-0

3

2

2992-2

3

3

3

2994-1

3

Also a Fe line

2

2

3002-1

4

2

2

3003-2

4

1

1

3011-5

6

3

3018-8

4

4

4

3031-4

2

2

3037-5

5

4

4

3044-5

5

Also a Fe line

2

2

3050-4

4

2

3

3053-9

6

2

3057-2

6

Also a Fe line

3

3'

3064-2

4

3

3

3080-3

4

2

3086-6

4

4

3096-6

6

4

m t

3098-6

5

3

3

3101-1

5

2

2

3101-4 5

ON THE ULTRA-VIOLET si'l'CTRA OP THK KI.KMKXTS.

T.\i;i i of Nickel lines (continued).

Intwuity.

A

Remark*.

gptrk.

Are.

5

3

3105-0

5

Also a Fe line

5

4

3113-7

6

1

1

31336

6

Also a Fe line

6

6

3134-0

4

5

4

3145-5

2

U

5

3158-9

1

M

3

3179-2

2

6

I

3181-2

1

6

6

3182-6

1

Also a Fe line

6

3

3183-8

1

6

4

3194-9

4

3

3

3196-6

4

Also a Fe line

6

i •

3201-5

4

t t

6

3212-3

5

t f

3213-7

3

Also a Fe lino

6

t t

3216-0

6

6

3216-6

5

, .

3217-4

3

5

5

3221-1

3

4

4

3224-6

3

6

6

3226-3

3

2

1

3232-6

1

Also a Fe line

3

3

3234-2

3

3

3

3242-6

3

6

6

3247-8

4

5

3250-1

3

G

6

3270-6

1

Also a Fe line

6

t f

3274-4

5

4

3282-2

6

6

3290-1

6

5

6

3311-8

3

Also a Co line

6

r t

3312-4

3

3

3

3315-1

4

3

3

3319-7

4

3

4 3321-6

4

Also a Co line

5

3349-8

3

6

5

3358-1

2

3

2

3360-9

3

Also a Co line

5

3

3361-0

3

4

4

3365-1

3

4

4

3365-5

3

6
2

2
3

3367-2
3368-9

1
4

In explosions of OH with Ni
Also a Fe line

3

4

3371-3

4

..

3

3

3373-3

3

6

4

8373-6

2

4

4

3374-0

4

1

1

3380-0

4

Also a Co line

2

2

3390-4

3

2

2

3392-4

4

6

5

3400-5

2

6

• •

3402-8

2

5

6

3404-5

2

Also a Co line

254

PROFESSORS G. D. LIVEING AND J. DEWAR

TABLE of Nickel lines (continued).

Intensity.

A

Remark a.

Sptrk.

Arc.

3

3

3406-6

3

6

5

3409-0

2

2

2

3412-9

3

4

1

3413-4

2

In explosions of OH with Ni

1

2

3413-8

4

6

6

3420-6

1

1

2

3423-1

4

Also a diffuse Co line

1

1

3433-0

4

Also a Co line

2

2

3436-7

5

Also a Co line

6

, ^

3441-6

1

1

3445-7

4

Also a Co line and in explosions of OH

1

4

3452-3

3

with Ni

5

4

3452-9

2

In explosions of OH with Ni

4

4

3453-5

3

5d

2

3457-7

In explosions of OH with Ni

1

1

3457-9

6

1

1

3461-1

5

In explosions of OH with Ni

3

2

3465'!

4

Also a Co line

5

5

3466-8

2

5

4

3468-9

4

5d

6

3470-8

3

2

2

3471-9

4

2

2

3483-1

4

5

5

3485-2

2

1

1

3492-3

3

In explosions of OH with Ni

2

2

3500-0

3

5

4

3501-8

6

5

3505-9

6

6

3507-3

1

1

3509-7

4

Also a Co line and in explosions of OH

with Ni

2

, .

3513-3

4

Also a Fe line

1

1

3514-4

4

In explosions of OH with Ni

6

• •

3518-0

3

3

3519-1

1

1

5

1

3523-9
3526-0

2

In explosions of OH with Ni

5
6

6
5

3527-1
3529-2

1
1

Here is a series of fine closely-set lines
Also a Co line

6

6

3529-9

1

3

3

3547-5

2

6

, .

3550-8

2

6

, ,

3552-8

1

6

. ,

3561-1

2

1

3

3565-7

3

In explosions of OH with Ni

2

2

3571-2

3

Also a Fe line and in explosions of OH

2

. .

3576-1

3

withNi

5d

5

3587-2

2

1

1

3597-0

3

In explosions of OH with Ni

5

. .

3601-4

4

6

6

3608-6

4

2

2

3609-8

5

3

3

3612-1

5

In explosions of OH with Ni

(i.\ THK ri.THA-VIOLKT SPKCTUA OK THK KLKMKNTS.

255

TABLE of Nickel lines (continued).

InUniitj.

A

•

Rwmrkji.

fcgfc

Arc.

1

1

3618-8

5

In explosions of OH with Ni

6

5

3624-1

4

4

3634-9

3

3653-0

6

3655-2

6

3657-5

5

3659-3

V

* •

3663-4

1

• •

6

3666-9

6

2

3669-7

1

. ,

6

3671-5

5

4

3673-4

1

6

t i

3687-6

1

( ^

6

36946

. .

6

3697-2

. .

6

3710-9

3

t ^

3721-6

1

Also a Fe lino

( m

6

3724-2

2

2

3736-1

1

2

3768-9

5

3

3

3775-0

5

In explosions of OH with Ni

3

4

3783-0

5

In explosions of OH with Ni

2

3

3806-6

3

In explosions of OH with Ni

6

t f

3831-7

5

t t

3837-5

2

5

, ,

3848-9

2

2

2

3857-8

7

256 ON THE ULTRA-VIOLET Si'KUTRA OF THE ELEMENTS.

NOTE ON THE MAPS OF THE SPECTRA OF COBALT AND NICKEL.

These maps contain many lines which are not in the list in the text. These
additional lines were observed in the arc between carbon electrodes when metallic
cobalt and nickel respectively were introduced into it ; but they were not included in
the list in the text because the authors were unable, for the reasons given in the text,
to be quite sure that they were not due to other metals contained in the carbon
electrodes, or in the crucible in which the arc was taken. It is most probable that
the greater part are really due to the metals to which they are assigned in the maps.

The reader is referred to the list in the text for the measured wave-lengths, as,
owing to a parallactic error in ruling, many of the lines in the maps are placed a
fraction of a tenth-meter out of their true places.

.

<-5

1— I
(rf

o

fi

&4

o

+j

t— i
o

ri

i — i
^

/'////. 7m //.v. 1888 A. Pltttr 9

»-=

8 =

ft— =

s =

8—=

Liveiruj &•!>•

3

tparl-

— ^

? J

~^

"-|

""'

8—=

8 — =:

-=

ss-^

~^7*

*-=

—

$—=

—

%—=

3 —

—

«! — —

8 =

—

s —

Si — —

—

-=

"~i

E

$ — =

<», — •

—

-=

*p trlr

Zmtiff.l88&AJ>&zfe 10.

*jr«*»Jt a*c.

| EE

q

1

W».t Hnrmul * C?

*-=

l-i

„•»••*'•-•

/'// //. 7m //.v. 1888 A. / 'la All

NT

i. _ I ==

••or.

/%//. T/W/M. 1888 A. Plate 12.

o

(15

PH
en

0)

r— I

O
• t-l

fc

3

P-.
Bj

9-=

N C> .

o

-H

£

o

Q)

0)

1 — I

o

CM
O

PH •
06

!

8—=

(/*.! ft .

Phil.Trtins. l£8&AJ2afe.l3.

a a rfl..

tr

Phil. Tran*. 1888 \.P/nff> IK

8 *p*'k. em

[ 257 ]

IX. 77jf Conditions of the Evolution of Gases from Homogeneous Liquids.
By V. H. VELKY, M.A., University College, Oxford.

Communicated by A. VERNON HARCOURT, M.A., F.ff.S., Lee's Reader in Chemistry,

Christ Church, Oxford.

Received May 5,— Read May 31, 1888.

I. Introduction.

THE evolution of a gas, as a product of a chemical change, from a homogeneous liquid is
among the earliest and most common experiences presented to a student of chemistry.
Thus, oxygen and nitrogen, among the elementary gases, and nitrous oxide, nitric oxide,
carbonic oxide, and hydrogen sulphide, among the compound gases, can be readily
prepared either from certain salts in a state of fusion, or from aqueous or acid
solutions of certain compounds. The effects produced on the rate and magnitude of
these changes by varying the conditions of mass, temperature, pressure, and material
of the containing vessel have not, hitherto, attracted the attention of investigators.
In preference, similar dynamical problems have been studied of chemical changes
occurring either between two or more gases, forming a gaseous product, or between
homogeneous liquids, or between solutions of solids.

In all these instances the reagents and their products are in the same physical
condition, gaseous or liquid, as the case may be. Of such a character are the
investigations of BUNSEN, v. MEYER, HORSTMANN, and the more elaborate researches
of DIXON on the conditions of chemical change occurring between hydrogen and
certain combustible gases. While among changes between homogeneous liquids or
solutions of solids, should be mentioned the experiments of BERTHELOT and PEAN
DE SAINT GILLES, and of MENSCHUTKIN, on the formation of ethereal salts from
organic acids and alcohols, of HARCOURT and ESSON on the reactions between oxalic
and permanganic acids, and between hydrogen iodide and hydrogen peroxide, and
of GLADSTONE on the reaction between ferric salts and sulphocyanides.

Experience thus far has shown that a chemical change from which a gas results is
often complicated in itself, or is accompanied and modified by other changes yielding
products not susceptible of convenient estimation. Sometimes, also, some portion of
the compound whose rate of decomposition is being measured is vaporised unchanged.
In the present paper I have the honour of laying before the Royal Society an account

MDCCCLXXXVITI. — A. 2 L 23.8.88

258 MB. V. H. VELBY ON THE CONDITIONS OF THE

of experiments on the rate of formation of gases from liquids. For convenience, the
paper will be divided into three parts. In Part I. the effect produced by the presence
of finely divided particles is investigated ; in Part II. the result of variation of pressure
is studied ; in Part III. the particular case of the decomposition of formic acid into
carbonic oxide and water, under the influence of concentrated sulphuric acid, is
investigated, the only conditions varied being those of temperature and mass. The
mathematical representation of this change is also discussed, and the results observed
are compared with those calculated according to the hypothesis adopted.

Before passing on to details, I should wish at once to state that these investigations
were commenced at the instigation of Mr. HAKCOURT, who was kind enough to devise
for me the various apparatus herein described, and without whose constant co-operation
and advice this research could not have been completed. The experiments were con-
ducted in the laboratory of Christ Church, Oxford.

II. Previous Experiments on the Evolution of Gases from Liquids in ivhich they

are formed.

The principal chemical changes of this nature, at present investigated, are the
decomposition of ammonium nitrate into nitrous oxide and steam, of potassium
chlorate into potassium chloride and oxygen, and of hydrogen peroxide into water
and oxygen.

The first of these changes has been studied from the thermo-chemical side by
BERTHELOT,* and from a chemical point of view by myself, t BERTHELOT concludes
from his experiments that ammonium nitrate, when heated to 200° C., is decomposed
initially into ammonia and nitric acid, which in some degree distil in the gaseous
form, recombining in the cooler parts of the apparatus to reproduce the original salt,
but to a greater degree react together to form nitrous oxide and steam. These
changes are represented as follows : —

(i.) NH4N03 = NH3 + HN03.
(ii.) NH3 + HN03 = N2O +2H2O.

My investigations established that the rate of the second change is conditioned by
the extent of the first : that is to say, the rate of formation of nitrous oxide, taken
as the measure of the decomposition, depends upon the proportion either of ammonia
or of nitric acid present in excess over the quantities of base and acid required for
their complete neutralisation. Thus, an excess of ammonia, other conditions remaining
the same, retarded, or even completely stopped, the decomposition, and an excess of
nitric acid produced the same effect, though to a less marked degree. Unpublished

* ' Annales de Chimie,' [4], vol. 18, pp. 68-82, and [5], vol. 10, pp. 362-365.
t ' Chem. Soc. Journ.,' 1883, pp. 370-383.

EVOLUTION OP GASES FROM HOMOGENEOUS LIQUIDS. 259

•

investigations also point to the conclusion that the rate of decomposition of ammonium
nitrite in solution, into nitrogen and water, is also conditioned by the presence of free
nitrous acid. The decomposition of potassium chlorate into potassium chloride and
oxygen has been studied by TEED and P. F. FEANKLAND with DINGWALL,* whose
investigations were published while the present research was being carried on. The
results show that of the two changes commonly represented in the text-books, namely—

(i.) 2KC10S = KC10, + KC1 + 02 ;
(ii.) KC104 = KC1 + 2O2

— the former at the temperature of boiling sulphur is incomplete, not more than 6 '3 to
876, instead of 13'05, parts of oxygen being evolved for every 100 parts of potassium
chlorate originally taken. FRANKLAND and DINOWALL noticed that when the chlorate
is mixed with half its iveight of powdered glass the decomposition of the chlorate
according to equation (i.) is practically realised. The effect of certain metallic oxides
in favouring the evolution of oxygen from potassium chlorate has been studied by
W. A. MILLER and by MERCER. The former writes.t " Other oxides produce a similar
effect (to that of manganese peroxide) ; thus, I find when the chlorate is mixed with
ferric oxide it requires a temperature of about 260° C., with plumbic oxide a somewhat
higher temperature is needed, whilst magnesia and zinc oxide do not aid the decom-
position at all. I have found also that powdered glass and pure silica are equally
inert."

Mr. HARCOURT, in the course of his investigations on the rate of chemical change
made several years ago, studied the decomposition of hydrogen peroxide into water
and oxygen, of ammonium nitrite in aqueous solution into water and nitrogen,
of oxalic acid into carbonic oxide, carbonic anhydride, and water, and of the formation
of nitric oxide from a gently warmed mixture of potassium nitrate, ferrous sulphate,
and dilute sulphuric acid. The observations made in the course of these unpublished
investigations, and those especially on the effect of roughened surfaces on the decom-
position of hydrogen peroxide, led to several experiments detailed in this paper, and
will be alluded to in the sequel.

III. The Apparatus used.

The method adopted in most of the experiments for collecting a definite quantity of
gas, taken as the measure of the velocity of the chemical change, may briefly be
designated " the tivin \J-tube method." Though described in my paper "On the Rate of
Decomposition of Ammonium Nitrate," it will be convenient here to add a brief des-
cription. The apparatus consists of two inverted U-tubes (U fig. 1, page 288), made

* ' Chem. Soc. Jonrn.,' 1887, pp. 274-286.
t ' Elements of Chemistry,' fifth edition, vol. 2, p. 847.
2 L 2

260 ME. V. H. VELEY ON THE CONDITIONS OF THE

from the same piece of glass tubing ; the two limbs of each are in close contact, being
wired together above and below. Joined to the centre of the upper part of one of
these tubes is a stopcock, through which, by means of flexible tubing connecting it
with an aspirator, the collected gas can be drawn out till the tube is filled with water
when the open ends are standing in a pneumatic trough. Into one limb of this tube
the gas is delivered in minute bubbles from the open end of a capillary tube, joined
to the decomposition flask ; in the other limb the unbroken surface of the water falls
steadily. The twin tube contains a quantity of air, measured once for all ; the level
of water in one limb serves to mark the point at which an equal volume of gas has
been collected in the other tube, the level of water in the adjoining tubes being the
same. The actual volume varies with conditions of temperature and atmospheric
pressure, but, since the contents of both tubes are under precisely the same conditions,
this variation is immaterial.

The standard quantity of air taken throughout these experiments was 10 c.c. of
dry air under standard conditions of temperature and pressure. This was measured
out as follows : — The volume occupied by 10 c.c. under the conditions of temperature
and pressure on the day of measurement was at first calculated ; this was corrected
for diminution of pressure due to tension of aqueous vapour and the dip of the
delivery tube under the surface of the water. This volume was delivered into the
U-tube by means of a capillary tube, from a flask in which that volume of air was
displaced by water dropping from a burette. The temperature of the flask was kept
constant by immersing it in a vessel of water ; while the effect of draughts was
minimised by using the capillary tube. In reactions in which a gas insoluble
in water was formed, the small volume of gas evolved between each observation
of level and the refilling of the tube by means of the aspirator was collected in a
separate tube (C, fig. 1), and decanted up during the interval between this and the next
observation. Thus the whole process was continuous. In cases in which the gas was
partially soluble in water, suitable modifications were introduced or different methods
of collection adopted. The observations were made with the help of a clock, beating
seconds, stationed close to the apparatus ; and, by counting, the exact second could be
noted at which the slowly descending water was at the same level as the similar
column of water in the twin U-tube. The flask in which the reaction was carried on
was heated in an air or water bath, the temperature of which was kept constant by
means of an automatic mercury and oil gas-regulator ; in most cases the variation of
temperature did not exceed one- tenth of a degree. Long-range thermometers were
used, graduated to tenths of degrees, in which differences of a twentieth of a degree
could easily be noted ; some of the thermometers were calibrated according to the
method of BESSEL.

EVOLUTION OF GASES FROM HOMOGENEOUS LIQUIDS. 261

PART I.

EFFECT OF FINELY DIVIDED PAKTICLES ON THE RATE OF EVOLUTION OF GASES,
RESULTING FROM A CHEMICAL CHANGE OCCURRING IN A HOMOGENEOUS LlQUTD.

The effect of finely divided particles or of roughened surfaces in promoting the
evolution of gases from their solutions in liquids, and that of vapours from liquids in a
state of ebullition is well known. The former phenomenon has been investigated by
OERSTED, SCHONBEIN, LIEBIO, GERNEZ, SCHRODER, and TOMLINSON, and the latter by
WATT, GAY LUSSAC, MARCEL, MAGNUS, DONNY, and GROVE. GERNEZ* attributes the
effect of these particles in increasing the evolution of carbonic acid from its super-
saturated solutions to air bubbles contained within them, an explanation shown to be
inadequate. ToMLiNSON.t by similar experiments, distinguishes between surfaces
which are and are not " chemically clean." From clean surfaces, such as that of a
freshly broken flint, or of a metal carefully washed with alcohol, no gas bubbles are
given off; but when these surfaces are rubbed with a cloth and reintroduced into the
solution of carbonic acid, they are immediately studded with bubbles of gas. He
writes, " Make the solids chemically clean, and the solution adheres to them without
any disengagement of gas ; make them unclean, and then the adhesive force of the
solid becomes more energetic for the gas than for the liquid, and there is a consequent
separation of gas from the liquid." The experiments of SCHRODER}; are of a similar
kind. These phenomena, though analogous to those to be described below, are in this
sense different, that in the one case we are dealing with a ready-made solution of a gas,
but in the other with a gas in the course of manufacture from the material contained
in a solution. §

In an experiment an aqueous solution of sodium formate, containing 1 gram of
the salt, was added to such a previously-made mixture of sulphuric acid and water
that the composition of the whole, before the reaction set in, was in the ratio
1H2() : 3H3SO4. An evolution of carbonic oxide resulting from the chemical change —

HCOOH = CO + OHj

* ' Compt, Rend.,' vol. 63, 1866, p. 883.

t ' Phil. Mag.' vol. 34, pp. 136 and 229 : vol. 38, p. 204 ; vol. 43, p. 205 ; and vol. 45, p. 276.

J ' POOOENDOBFF, A imiili-ii,' vol. 137, p. 76 ; and Erganznngsband 5, p. 87.

§ Mr. HARCOURT tells me that his attention was called to this subject nearly thirty years ago, when, in
attempting to concentrate hydrogen dioxide under the receiver of an air-pump, he found that the decom-
position of the liquid was hastened by removing the atmospheric pressure. Subsequently, ho tried to
measure tho rate of decomposition of hydrogen dioxide under constant conditions, and had, finally, to
abandon the attempt, in consequence of the impossibility of fixing one condition, namely, the state of tho
surface of the glass vessel which held the liquid. If the vessel was scrupulously cleaned, the decompo-
sition took place more quickly than if any trace of grease adhered to the glass, and more quickly with
some kinds of glass than with others. — July 26th, 1888.

262

MR. V. H. VELEY ON THE CONDITIONS OF THE

occurred at a temperature of 350t8-350>9. But some finely divided particles, afterwards
shown to be silica, were suspended in the solution of sodium formate ; these were,
doubtless, due to some process adopted by the manufacturer for obtaining the salt.
It was observed that these particles were continually being borne up by the bubbles
of gas, and seemed to serve as nuclei for their formation.

The experiment was then repeated under conditions precisely the same, with the
exception that the solution of the sodium formate was Jittered into the sulphuric acid,
to free the solution from the coarser and visible particles. No appreciable quantity of
gas was given off at 35°'8, and it was found necessary to raise the temperature about
38°, viz., to 73°'8, to obtain a rate of evolution of gas similar to that in the above
experiment.

As these results appeared to be similar to those which Mr. HARCOURT had observed
in the decomposition of hydrogen peroxide, he conducted the two following experi-
ments. A mixture was made of 300 c.c. sulphuric acid (sp. gr = 17877) and 130 c.c.
of water ; to this, when cool, was added 10 c.c. of a filtered aqueous solution of sodium
formate (containing 1 gram of the salt). Of this mixture two portions of 200 c.c.
were taken, and introduced into two similar flasks, of which one contained a quantity
of well- washed, recently ignited pumice suspended in 10 c.c. of water, the other 10 c.c.
of pure water. The flasks, fitted with delivery tubes, were heated side by side in a
water bath, and the carbonic oxide gas collected in two graduated tubes, standing
in a pneumatic trough. The intervals of time between successive observations were
approximately equal ; the volumes collected in the two tubes were read simul-
taneously ; the temperature varied throughout the experiment from 75°'8 to 76°'2.

Mixture, with pumice,
c.c. of gas collected.

Mixture, without pumice,
c.c. of gas collected in same time.

Total

Difference.

Total.

Diflferencc.

10

5

15

5

5-8

•8

20

5

7-1

1-8

25

5

8-5

1-4

30

5

10

1-5

35

5

11-9

1-9

41

6

14-3

2-4

45

4

15-9

1-6

50

5

18-1

2-2

The first observation includes the volume, probably about 4 c.c., given off in conse-
quence of the expansion of the liquid and of the enclosed air.

The rate of evolution of gas from the mixture containing the pumice is to that
from the mixture without the pumice in approximately the ratio 5 : 2.

EVOLUTION OF GASES FROM HOMOGENEOUS LIQUIDS.

2G3

In order to determine the effect produced by variation of the mass of pumice, an
experiment precisely similar to the one described above was conducted ; in one flask
was placed '1 gram, in the other 5'0 grams of pumice. Temperature 76°.

Mixture with -1 gram.

Mixture with 6 gram*.

Interval of time

in minutei

Collected in
each interval

ToUl.

Collected in
each interval.

Total.

and decimal*.

C.C.

5-0

5-0

c.c.
4-5

4-5

4-b

5-0

10-0

4-5

9-0

385

5-0

15-0

6-0

15-0

4-02

5-0

20-0

5-4

20-4

4-12

5-0

25-0

47

251

4-28

5-0

30-0

4-9

30-0

4-47

5-0

35-0

5-0

35-0

4-85

5-0

40-0

5-0

40-0

5-4

5-0

46-0

5-1

46-0

6-05

The rate of evolution of gas from both mixtures is proceeding pari passu ; thus
the mass of pumice added, whether '1 gram or fifty times that amount, makes no
difference.

The effect of finely divided particles, not only upon the decomposition of formic
acid into carbonic oxide and water, but also upon other chemical reactions in which a
gas is evolved, was studied more fully by aid of the twin U-tube apparatus.

Decomposition of Formic Acid into Carbonic Oxide and Water.
HCOOH = CO +

Effect of Graphite.

Mixture taken

175 c.c. sulphuric acid.
75 „ water.

Into this was filtered slowly 50 c.c. of an aqueous solution of sodium formate, con-
taining 5 grams of the salt. From this quantity 165 observations of 10 c.c. each
are obtainable, so that in the earlier stages of the decomposition the effect due to loss
of mass in each successive observation is inconsiderable.

After a preliminary stage of initial acceleration (see Part II.), the rate of evolution
of gas became constant ; when this point was reached, the cork through which passed
the thermometer was taken out and about half a gram of finely-powdered graphite
was introduced into the mixture ; the containing flask was shaken, the cork restored
to its former position, and observations recommenced at the same temperature.

Temperature 80°'5-80°7.

264

MB. V. H. VELEY ON THE CONDITIONS OF THE

Intervals in minutes and decimals required for evolution of
10 c.c. carbonic oxide.

Before introduction
of graphite.

After introduction of graphite.

t

t

t

I

4-35

•83

3-03

3-25

4-20

2-37

3-12

3-25

4-40

2-80

3-13

3-22

4-30

277

3-22

3-37

4-17

2-83

3-15

3-40

4-17

2-87

3-0

335

4-22

3-03

3-15

3-42

4-22

3-15

3-25

From these observations it appears that the rate of evolution of carbonic oxide is
much accelerated by the presence of the graphite, for even after twenty-three
observations, corresponding to a loss in mass of nearly 14 per cent., the rate was
quicker than that observed before the addition of the graphite. At the conclusion
of the experiment the graphite was found to be uniformly distributed throughout
the acid liquid, and there was no perceptible odour of sulphurous oxide, thus
indicating that no reduction of the sulphuric acid had occurred.

Effect of Silica.

The experiment above was repeated with silica, obtained by decomposition of
silicon fluoride with water, which had been dried and recently ignited ; it was in
the usual form of a fine fluffy powder.

Variation of temperature 800>4-800<6.

Intervals in minutes.

Before introduction
of silica.

After introduction of silica.

5-28

i-9

3-06

5-0

2-62

3-17

5-08

2-85

3-06

5-1

2-66

3-37

5-08

2-7

3-33

2-56

3-33

3-06

3-33

On adding the silica and agitating the flask containing the decomposing mixture
an effervescence of gas took place, so tumultuous that some small quantity of liquid
frothed over; it was necessary to wait a few minutes until this had subsided
sufficiently to allow of observations being taken.

EVOLUTION OF GASES FROM HOMOGENEOUS LIQUIDS.

2G5

Effect of Glass Dust.

A quantity of glass dust was obtained by breaking RUPERT'S drops, and sifting the
finely divided glass through a sieve.
A mixture was made up as follows :—

300 c.c. sulphuric acid,
130 „ water,

into which was filtered slowly 85 c.c. of water containing 10 grams of sodium
formate. Two portions, each of 1 75 c.c., of this mixture were taken, placed in two
flasks, which were heated in a water bath side by side.

Mean temperature 79°'5. The observations are divided into sets of 50 c.c. each ;
the volumes collected in the two tubes were read simultaneously.

Mixture with glaaa doit

Intervals in
minute* for
each difference.

Mixture without gUu dntt.

ToUl.

Difference

Total.

Difference.

M,

10

c.c.
10

10-45

Mb
1-0

c.c.

1

20

10

9-47

3-0

2

30

10

7-85

5-0

2

40

10

7-95

8-5

3-5

50

10

8-02

125

4-0

10

10

7-95

5-5

5-5

20

10

7-13

12-5

7-0

30

10

7-05

18-0

5-5

40

10

7-28

24-5

6-5

50

10

743

31-5

7-0

10

10

7-55

7-5

7-5

20

10

778

15-0 7-5

30

10

7-92 22v>

7-5

50

20

14-37 :;- 7

16-2

10

10

7-5

8-0

8-0

20

10

7-87

16-5

8-5

80

10

7-83

24-5

8-0

40

10

7-4

32-5

8-0

50

10

7-7

40-5 8-0

10

10

733

8-5 8-5

20

10

7-5

16-5 8-0

30

10

7-«7

24-5

8-0

40

10

7-93

32-5

8-0

50

10

8-35

40-5

8-5

From the above results it is evident that the rate of evolution of gas from the
mixture with the glass dust is greater than that from the mixture without the
glass ; the difference between these rates diminishes as the change proceeds, owing,
partly, at least, to the greater diminution of mass in the former case than in the

MDCCCLXXXVni. — A. J M

266

MR. V. H. VELEY ON THE CONDITIONS OF THE

latter, for from the one 25 observations of 10 c.c. each were taken, out of a possible
165 ; from the other only 16 or 17, out of the same number.

Decomposition of Nitric Acid into Nitric Oxide and Water by means of Ferrous Sulphate.
6FeSO4 + 5H2SO4 + 2KNO3 = 3Fe2 (SO4)3 -4- 2KHSO4 + 4H20 + 2NO.

A convenient method of preparing pure nitric oxide gas consists in heating ferrous
sulphate and potassium nitrate with dilute sulphuric acid. A mixture was made up
as follows : — 6 grams of potassium nitrate and 36 grams ferrous sulphate were
dissolved in 100 c.c. water; the solution filtered to rid it of small quantities of a basic
iron sulphate, which invariably separated out ; to this solution was added gradually a
previously made and cooled mixture of 40 c.c. sulphuric acid and 20 c.c. water. The
twin U-tube method was used.

Effect of Pumice.

Temperature 34°'8-35°-2.

Intervals in minutes required for evolution of 10 c.c. nitric oxide.

Before introduction of pumice.

After introduction of pumice.

6-05 9-88
6-87 11-37
7-75

8-75

2-9
7-15
15-08

The introduction of the pumice produced a most violent effervescence ; some quantity
of the liquid frothed over. The above results show that, notwithstanding the loss of
mass due to this effervescence, the interval of time in which 10 c.c. of gas is given off
is suddenly reduced from 11''37 to 2'-9.

Effect of Graphite.

The experiment above was repeated, graphite being introduced instead of pumice ;
the phenomena observed were precisely similar.
Temperature 390'4-39°7.

Intervals in minutes.

Before introduction of graphite.

After introduction of graphite.

1-67
1-88
2-4
3-32

3-77
4-37
5-23
6-05

6-62
7-43

8-5
9-88

1-37
2-13

2-87
3-99

4-97
6-63
10-78
16-67

EVOLUTION OP OASES FROM HOMOGENEOUS LIQUIDS.

267

Thus, on introduction of the graphite the interval required for the evolution of
10 c.c. of gas is suddenly reduced from 9'"88 to 1''37, notwithstanding the loss of mass
due to a tumultuous effervescence of gas.

Effect of Barium Sulphate.

Some barium sulphate was prepared by precipitation of the chloride with sulphuric
acid, and washed carefully from all traces of chloride ; it was then ignited in the usual
manner.

Temperature 37°-G-380>0.

Interval! in minute*.

Before introduction of barium sulphate

After introduction of barium wulphtte.

4-43 11-9
6-88 30-03

5:08 22-58
10-0 351

The interval of time is here reduced from 30' to 5' by the presence of the precipi-
tated barium sulphate.

Decomposition of Ammonium Nitrite in Aqueous Solution into Nitrogen and Water.

NH^NOj = N8 + OH,.

The preparation of nitrogen gas by heating together concentrated aqueous solutions
of potassium nitrite and ammonium chloride is well known. The effect of finely
divided particles in promoting the evolution of nitrogen was studied as affording an
example of a case in which there is formed a gas, most sparingly soluble, and thus
not likely to be stored in any quantity in the solution of the salts undergoing decom-
position.

Effect of Silica.

A mixture was made of 40 c.c. of a concentrated solution of potassium nitrite and
120 c.c. of a cold saturated solution of ammonium chloride, containing about 33 per
cent, of the salt.

Variation of temperature, 570f7-57°'9.

Intervals in

minntes required for evolution of 10 c.c.

nitrogen.

Before introduction of silica.

After introduction of iilica.

5-92
6-03
6-13
5-95
6-13

6-62
6-8
7-18
7-77
8-55

8-0
43
872
9-72
10-55

12-02
128

2 M 2

268

MB. V. H. VELEY ON THE CONDITIONS OP THE

The interval of time is suddenly reduced to about one-third by the introduction of
the silica, as in some previous cases, but the rate of the decomposition quickly
resumes its former course.

Effect of Pumice.
This experiment was made with a similar mixture. Temperature, 58°-590>l.

Intervals in minutes.

Before introduction of pumice.

After introduction of pumice.

i

,

1 1

*

f

2-57

315

6-23 7-22

6-28

10-7

27

4-25

6-03 7-7

6-4

2-9

4-53

6-03 8-03

9-13

3-08

5-38

6-65

10-02

The interval of time is here reduced from 8'"03 to 6''28 by the introduction of the
pumice ; otherwise the results are comparable with those of the former experiment.

Decomposition of Ammonium Nitrate, into Nitrous Oxide and Steam.

NH4NO3 =

20H.

Owing to the solubility of nitrous oxide in water, the twin U-tube apparatus was
modified by the interposition between the flask and the U-tube of a bent tube,
provided with two bulbs at different levels, the form of which is described at length
in my paper " On the Rate of Decomposition of Ammonium Nitrate."* By its means
each measure of nitrous oxide was represented by the collection in the U-tube of its
corresponding volume of air. The manipulation and processes of purification of the
salt were the same as those adopted in the previous experiments. Such a quantity of
salt, about 200 grams, was taken that the loss of mass between any two consecutive
observations should be far too small to affect the results.

Effect of Pumice.

In this case the temperature of the pumice introduced was as nearly as possible
that of the fused salt, so as not to produce local solidification of the salt around the
pumice, and thus retard the decomposition.

Temperature, 182°'2-1820-.').

* ' Chem. Soc. Jonrn.,' 1883, p. 373.

INVOLUTION OF OASES FROM HOMOc KNK< H'S LIQUIDS.

269

Interval! in minute* required for evolution of 10 c.c. nitron* oxide.

Before introduction of pumice.

After introduction of pumice.

4-67
4-72

4-77
4-7

4-63 4-5
4-48
4-5
4-47

2-4

::•/
3-7

3-65
3-82
3-88
3-85

The interval is reduced from a constant value of 4''5 to 3'' 80 by the introduction
of the pumice.

Effect of Barium SulplwAe.

The experiment was repeated with precipitated barium sulphate. In order to
determine whether the change of atmosphere above the fused salt had any effect upon
the rate of the decomposition, the cork, with the thermometer, was taken out, and
the superincumbent atmosphere of nitrous oxide, steam, and vapours of the salt was
completely blown away. But on restoring the cork the interval required for the
evolution of the 10 c.c. of the nitrous oxide was the same as before, which proves
that the results obtained in the cases investigated above are not due to the necessary
partial change of atmosphere in the process of introducing the finely-divided material.
The intervals of time required for the unit of decomposition obtained after the altera-
tion of the atmosphere are given in the second column.

Variation of temperature, 191°'5-191°'8.

Intervals in minutes.

Before Introduction of barium sulphate.

After introduction of barium sulphate.

< / i

t

I

1

/

5-4 5-52 5-4

2-32

1-85

1-98

1-88

5-4 5-47 5-4

2-4

1-83

1-93

1-85

5-43 5-53

2-12

1-8

1-95

5-45

2-02

i-as

1-86

1-97

1-87

1-85

The results show that the constant interval is reduced from 5'*45 to 1''85, or, prac-
tically, to one-third, by the introduction of such a chemically inert substance as
barium sulphate.

Effect of Platinum Black.

A sample of platinum black was prepared by reducing the chloride with a slightly
alkaline solution of sodium formate ; the precipitated metal was washed free from all
traces of chlorine and partially dried over sulphuric acid in a vacuum ; it was thought

270

MR. V. H. VELEY ON THE CONDITIONS OF THE

advisable not to completely dry the tinely-clivided metal, so as to avoid a possible
absorption of oxygen.

Intervals in minutes.

Temperature, 200°-205°.
Before audition of platinum black.

Temperature, 182°-182°-5.
After addition of platinum black.

4-92 5-0
5-0 5-02
5-0 4-98
5-02 5-0

3-17 3' 17
3-2 3-13
3-2 3-17
3-13

The effect of platinum black is most striking; the interval of time is reduced
practically in the ratio of 5 : 3, even though the temperature was lowered 18°, in
order to obtain a rate of decomposition sufficiently slow for convenient observation.

Decomposition of Potassium Chlorate into the Perchlorate, Chloride, and Oxygen.

2KC1O3 = KC104 + KC1 + O2.

It is usually stated in the more recent text-books on chemistry (see page 259),
that finely divided chemically inert substances do not accelerate the evolution
of oxygen from potassium chlorate. To decide this point a quantity of potassium
chlorate was re-crystallised ; into two similar-shaped retorts were introduced
70 grams of the salt ; to one portion 7 gram of precipitated barium sulphate
was added. The retorts were placed side by side in a small square air bath, provided
with two holes in one side, through which the necks of the retorts passed. The air
bath was packed with asbestos, to distribute the heat of the lamp as uniformly as
possible. After the evolution of oxygen had commenced, but before systematic
observations were taken, 104 c.c. of gas were given off from the chlorate containing
the barium sulphate, but only 51 '2 c.c. from that without the sulphate. The
observations are divided into sets of 50 c.c. each, being the capacity of the two
graduated tubes which served to collect the oxygen. The time was noted at which
each measure of oxygen was evolved from the retort containing potassium chlorate
and barium sulphate, and simultaneously the volume which had been evolved from
the retort containing potassium chlorate only was read.

EVOLUTION OF OASKS 1 KOM HOMOGENEOUS LIQUIDS.

271

HATE of Evolution of Oxygen.

Mixture of 70 gram* pota-Mum
chlorate and

70 gram* of potassium chlorate.

•7 gram Iwrium nulpha(c.

Interval in minutes

for each difference.

Total relume.

Different.

Total Tolorait.

Difference.

C.O.

.-.-.

J

c.o.

C.C,

20

20

3-5

i-o

1-0

40

20

3-53

2-0

1-0

50

10

1-8

2-5

1-5

20

20

4-25

1-5

1-5

40

._.,,

4-6

2-5

1-0

50

10

3-15

3-5

1-0

10

10

5-68

1-5

1-5

20

10

5-32

4-0

2-5

30

10

6-85

7-0

3-0

40

10

633

10-0

3-0

50

10

7-43

12-5

25

10

10

5-83

2-0

2-0

20

10

7-03

5-0

3-0

30

10

6-13

7-5

2-5

40

10

6-18

10-0

2-5

50

10

135

3-5

From these results it is evident that the evolution of oxygen from the chlorate
with which the barium sulphate is mixed not only commenced at a lower temperature,
but also was more rapid throughout, though the difference between the two rates of
evolution diminishes as the reaction proceeds, owing to a greater loss of mass in the
one case than in the other. The whole course of the phenomenon is precisely
analogous to that observed in the case of the decomposition of sodium formate by
strong sulphuric acid in the presence or absence of glass dust.

In order to determine whether such a chemically inert substance as barium sulphate
is as effective a material as ntanganese peroxide in promoting the evolution of oxygen
at a temperature below the point of fusion of the chlorate, two portions of 20 grams
of the salt were taken. To the one was added '2 gram of barium sulphate ; to
the other, '2 gram of manganese peroxide, Mn60u, H80, prepared for a previous
investigation, and shown by spectroscopic analysis to be free from any considerable
quantities of the oxides of the alkalies and alkaline earths. Each of these mixtures
was placed in a piece of combustion tubing, the end of which was drawn out into a
capillary. The two tubes were heated side by side in an air bath.

272

MB. V. H. VELEY ON THE CONDITIONS OF THE

Mixture of 20 grama potassium
chlorate and

Mixture of 20 grams potassium
chlorate and

•2 gram manganese peroxide.

Intervals in minutes

•2 gram barium sulphate.

for each difference.

Total volume.

Difference.

Total volume.

Difference.

O.C.

10

CO

10

2:22

C.C.

i-o

c.c.

i-o

20

10

2-0

2-0

i-o

30

10

2-07

2-5

rs

40

10

2-43

3-0

•5

60

10

2-03

3-5

•5

10

10

1-13

•5

•5

20

10

•87

1-25 .

•75

30

10

•92

2-25

1-0

50

20

1-22

3-25

i-o

20

20

1-27

•5

•5

40

20

1-73

2-0

1-5

50

20

•92

3-5

1-0

20

20

2-58

•5

•5

40

20

4-58

1-5

1-0

These results show that, while the evolution of oxygen from the potassium chlorate
and manganese peroxide mixture is very rapid while the salt still remained unfused
that from the potassium chlorate and barium sulphate is exceedingly small under
the same conditions.

The results of the various experiments detailed above all point to the one conclu-
sion, that the rate of evolution of a gas formed by chemical change in an homogeneous
liquid is accelerated by the presence of finely divided particles which are chemically
inert. Thus, whether the gas is evolved from an aqueous solution or a salt in a
state of fusion, whether the gas is soluble in the liquid from which it is being
evolved or not, whether the chemical change from which the gas results takes
place at a temperature of 37° or 350°, whether the finely divided substance is
porous or not, the result is uniformly the same.

v

PAET II.

PHENOMENON OF INITIAL ACCELERATION AND EFFECT OF REDUCTION OF PRESSURE

ON THE RATE OF EVOLUTION OF GASES.

Before proceeding to discuss the effects produced by reduction of pressure on the
rate of evolution of gases from homogeneous liquids, it is desirable to call attention to
the phenomenon of initial acceleration. It was observed by Mr. HARCOUKT some
years ago, and confirmed by my experiments on the decomposition of ammonium
nitrate and on other chemical reactions detailed in the present communication, that

EVOLUTION OF GASES FROM HOMOGENEOUS LIQUIDS. 273

the rate of evolution of gas, external conditions remaining the same, is at first slow,
then gradually increases until it reaches a maximum, and for some time constant, rate,
from which point it decreases constantly with the diminution of the mass undergoing
decomposition. The curve, then, representing the velocity in terms of mass would
thus show a point of contrary curvature. This phenomenon of initial acceleration has
previously been observed in reactions occurring between two gases giving a third gas
as a resultant, as also between two liquids with liquid resultants. Thus BUNSEN and
ROSCOE in their photo-chemical researches* pointed out that the combination of
hydrogen with chlorine, under the influence of direct sunlight, starts at first slowly,
but quickly increases, reaching a maximum ; this phenomenon they called " photo-
chemical induction." To illustrate their results, they studied the rate of replacement
of hydrogen in tartaric acid by bromine, both the reacting substances being in dilute
solution ; this rate was found to increase slowly, reaching a maximum point. Again,
BERTHELOT and PEAN DE SAINT GiLLEst observed the same period of initial acceleration
in their studies on the rate of formation of ethereal salts, less marked in the etherifica-
tion of acetic acid by ethyl alcohol, but most marked in the case of valeric acid. On
this point they write as follows^ : — " Pour la concevoir il faut admettre une sorte
d'inertie, de resistance .-i vaincre qui retarde la combinaison dans les premiers instants,
et dont les effects compensent et au-del;\ pendant un certain temps ceux qui rdsultent
d'un e"tat de concentration plus grande dans le systeme. Cette acce'le'ration initielle
semble done constituer un caractere assez ge'ne'ral de ce genre de reactions." It is
apparently also common to the kind of reactions, here investigated, in which a gas is
formed from a liquid. But the intervals of time required for the unit of chemical
change do not, however, decrease in any regular uniformity during this period of
acceleration ; it is probable that the phenomenon is the result of several causes, the
peculiar effect of each of which cannot at present be determined by mathematical
analysis. In some cases it is due to a storage of gas within the liquid, particularly in
the formation of nitric oxide from nitric acid by means of ferrous sulphate, and in
others a delay is occasioned by the actual resolution of more complex into simpler
molecules, or even atoms, which in their turn react to form the gaseous product.
Thus, in the formation of nitrous oxide from ammonium nitrate, the delay may be
occasioned by the resolution of the molecule of the salt into ammonia and nitric acid,
as observed by BERTHELOT. In the case of the decomposition of potassium ferro-
cyanide by sulphuric acid the double salt at first splits up into the separate metallic
cyanides, which yield, by the action of the acid, hydrocyanic and subsequently
formic acid. This last is then decomposed into carbonic oxide and water. It would
appear that hydrocyanic acid is always an initial product in this decomposition.
Thus, to put the proposition in a general form, if in any chemical system a

• ' Phil. Trans.,' 1857, p. 355.

t ' Annales de Chimie,' [8], vol. 66, 1862, pp. 5-153.
t Ibid., pp. 26, 65.
MDCCCLXXXV1II. — A. 2 N

274

MR. V. H. VELEY ON THE CONDITIONS OF THE

compound abed decomposes initially into ab and cd, which in their turn form ac and

bd, thus

(i.) abed = ab -\- cd

(ii.) ab + cd = ac + bd,

then, if neither of these changes is instantaneous, the rate at which bd, the measured
compound, is produced would gradually increase until the velocity of change (ii.) is
equal to that of change (i.). The reverse case of (i.) may also occur, in which ab and
cd combine to form abed, which is again decomposed thus

(i.) ab + cd = abed
(ii.) abed = ac -\-bd.

The experiments illustrative of this phenomenon are given below for a variety of
chemical changes. In Part I. the intervals of time observed before the maximum and
constant rate was reached have in all cases been omitted in order to shorten the
tables. To make the results the more comparable, all the observations are reduced on
the supposition that the mass of the substances undergoing decomposition remained
constant.

Decomposition of Formic Acid into Carbonic Oxide and Water.

Constant mass of sodium formate = 5 grams, corresponding to 3'384 grams of
formic acid, and yielding a sufficient volume of carbonic oxide for 160 observations of
10 c.c. each. Temperature 80°'8 to 80°'9. The observations are given up to the
point of maximum and constant rate.

Intervals in minutes required for evolution of 10 c.c. carbonic oxide.

Reduced.

Not reduced.

18-98

9-47
6-81

5-92
5-56
5-17

4-74
4-51
4-27

6-53
5-67
5-33

5-30
5-33

To determine the effect produced by a temporary lowering of temperature, at the
conclusion of the observations recorded in the first column the mixture was cooled by
being placed into water at 20°, which caused the temperature to fall to 70°. The flask
was then restored to its position and quickly heated up to the former temperature,
80°'8 to 800>9, when observations were again made ; but, as the loss of gas in the
interval was not determined, the times recorded in the second part of the table are
not reduced. They show, however, that the phenomenon of initial acceleration
repeats itself,

EVOLUTION OP GASES PROM HOMOGENEOUS LIQUIDS.

275

Two other series of observations are given below, made under precisely the same
conditions of mass of salt, concentration of acid, and with slight differences of
temperature.

Interval* In minute* (reduced).

I.

Temperature 80 4-80-P.

II.

Temperature 80 7-80-fl.

13-13

8-02
6-11

5-18 4-18
4-50 402
4-28

10-62
7-15

r, M.J

5-26

5-03
4-93

Decomposition of Potassium Ferrocyanide by Sulphuric Add.
K4Fe(CN)6, SI^O + 6H2SO4 + 3H20 = 6CO + 2^80^ + 3(NHt)aSO4 + FeSO*.

Experiments made in the earlier stages of this research on the rate of decomposition
of potassium ferrocyanide with concentrated sulphuric acid in accordance with the
above equation also illustrate this phenomenon of initial acceleration. The twin (J-
tube method was used, but the standard quantity of air taken was such as would
correspond to the decomposition of '058 gram of the salt.

A mixture was made up as follows : —

21 grams crystallised ferrocyanide, corresponding to 860 observations.
420 . . . sulphuric acid.
27 „ water.

This mixture was heated until the salt had completely dissolved in the acid to
form a clear liquid, and then observations were commenced, little or no gas other
than small quantities of hydrocyanic acid being given off before the salt dissolved.

Temperature 150°'8-151°.

Intervals in minutes (reduced) required for evolution of
standard volume of carbonic oxide.

7-27
6-43
6-72
6-36

6-03
6-13
6-12

6-18
6-16
5-73
5-63

5-70
5-66

These results are precisely similar to those given above.

2 N 2

276 MB. V. H. VELEY ON THE CONDITIONS OP THE

In another set of experiments a mixture was made up as follows :-

31*5 grams potassium ferrocyanide.
420 „ sulphuric acid.

1-3

water.

Temperature 150°'8-151°-2.

Intervals in minutes (reduced).

13-67

9-54

6-83

12-36

8-33

6-44

10-48

7-94

6-01

9-74

7-30

5-84

Decomposition of Ammonium Nitrate.

In my former experiments on the decomposition of ammonium nitrate the same
phenomenon was observed ; in these the standard volume of air taken was equal to
the volume of nitrous oxide given off by the decomposition of '05 gram of the salt.

Temperature 210°-210°'5.

Intervals in minutes (reduced) required for evolution of standard

volume of nitrous oxide.

17-7
16-0
14-3
12-0

10-3
8-65
7-9
7-05

6-25
5-75
5-4
4-65

4-2
3-8
3-4
3-35

3-25
3-05
2-8
2-7

2-65
2-55
2-45
2-45

2-35
2-25
2-10
2-15

In other series of experiments a similar very gradual acceleration was observed ; of
these only one need be quoted in illustration.

Intervals in minutes (reduced).

6-57

4-07

2-96

2-87

2-50

5-20

3-82

3-11

2-70

2-49

4-83

3-58

3-01

2-68

4-45

3-42

2-99

2-59

Decomposition of Oxalic Acid by Strong Sulphuric Acid.
H2C304 - OH2 = CO2 + CO.

In the course of some experiments made to determine the effect produced by
alteration of pressure, the decomposition of oxalic acid into carbonic acid and oxide

EVOLUTION OP GA8B8 FROM HOMOGENEOUS LIQUIDS.

277

was selected as a case for investigation. The results obtained also illustrated this
phenomenon of initial acceleration. Owing to the solubility of carbonic acid in water
some modification of the apparatus was necessary to collect the mixed gases over
mercury.

The method adopted was as follows : — Two tubes were wired together above and
below, of which one was graduated in cubic centimetres, the other in millimetres. In
the former the mixed gases were collected, while the latter served to indicate the
pressure in millimetres of the mercury column corresponding to each reading of the
other tube, both standing in a small trough filled with mercury. Thus each uncor-
rected volume of 10 c.c. could be reduced in terms of standard pressure and tempera-
ture ; due allowance was also made for the tension of aqueous vapour, the tube which
served for the collection of the gases being moistened with water. 12 grams of
crystallised oxalic acid were dissolved in 120 c.c. sulphuric acid; after complete
dissolution of the oxalic acid observations were made ; these are divided into sets of
50 c.c. each.

Temperature 67°-4-670<6.

Intervals of time required for evolution of 10 c.c. of

carbonic oxide and acid.

I.

II.

14-0

7-36

12-6

7-50

10-31

7-13

8-12

6-36

7-83

6-27

In other series of experiments precisely similar results were obtained ; of these only
one need be quoted for illustration, made under conditions precisely similar to that
above, with the exception that the temperature was slightly higher, viz., 69°'0.

Intervals of time required for evolution of 10 c.c. of
the mixed gaseo.

I.

II.

9-5
8-12

1:71

6-20
5-94

^•39
5-20
5-08
5-03
5-03

In the above tables the Series II. is practically continuous with the Series I.,

278 MR. V. H. VELEY ON THE CONDITIONS OP THE

small quantity of the gases being lost between the last observation of I. and the first
of II. in the interval of time during which the collecting tube was re-filled with
mercury.

Decomposition of Nitric Acid into Nitric Oxide and Water by means of

Ferrous Sulphate.

In the formation of nitric oxide by heating together solutions of ferrous sulphate
and potassium nitrate with dilute sulphuric acid, this initial acceleration was not
observed under the conditions of the experiments. But when the pale green liquid
was warmed at about 40°, it gradually became darker by the solution therein of the
nitric oxide, a process which lasted often for half an hour or more. When the liquid
had become almost black, then, owing to some indeterminate cause, there was a
sudden outburst of gas, accompanied by an evolution of heat, the thermometer rising
one to two degrees. From this point the evolution of gas gradually decreased, at a
rate probably proportional to the diminution of mass.

Decomposition of Ammonium Nitrite in Aqueous Solution into Nitrogen and Water.

In the few experiments made in the course of the present investigation on the
above chemical change the phenomenon of initial acceleration was not observed, but
Mr. HABCOUBT has informed me that in his previous investigations, on a more extended
scale, he always observed the phenomenon, and showed that it could be reproduced
by temporary cooling of the liquid, as noted above in the case of the decomposition of
formic acid.

Production of Hydrogen from Zinc and Sulphuric Acid.
Zn + H2S04 = ZnS04 + H2.

Though the results obtained in measuring the rate of evolution of hydrogen from
zinc and sulphuric acid are not strictly comparable with the evolution of gas from a
homogeneous liquid, yet, as SPBING and AUBIN* have recently called attention in this
case also to an initial acceleration, which they call the " period of induction," it may
here be briefly considered.

In the present research, this phenomenon was observed in the course of experiments
made before the publication of SPBING and AUBIN'S paper. It is probably conditioned
by the mechanical adherence of bubbles of gas to the surface of the metal, which is
always apparent when sulphuric acid comes in contact with a regular surface of the
metal If these bubbles are removed as fast as they are formed, or if the metal is
continuously rolled about within the acid liquid, no such initial acceleration can be
observed.

» ' Annales de Chimie' [6], vol. 11, pp. 505-554.

EVOLUTION OF GASES FROM HOMOGENEOUS LIQUIDS. 279

Effect of Reduction of Pressure on the Rate of Evolution of Gases.

The retardation or promotion of certain chemical decompositions and combinations
by increase or decrease of pressure has from time to time been the subject of various
inquiries ; of these, some will be alluded to in the sequel. As the evolution of gas
from a liquid might apparently be modified by reduction of the superincumbent
atmosphere to which that liquid is subjected, as preventing a possible storage of
gas in its initial stage of formation, the effect produced by such a reduction of
pressure was considered a subject worthy of particular investigation.

The apparatus employed consisted mainly of three parts, a blown-out flask of stout
glass, a SPRENOEL pump with the usual gauge, and a tube serving for the collection of
known volumes of the gaseous product. Through the neck of the flask passed a
"["-piece of glass tubing ; the vertical part of this was closed by a rubber cork through
which passed the thermometer, while the horizontal part, of stout glass, enclosing a
capillary airway, was connected by thick-walled rubber tubing with the SPRENGEL
pump. The delivery tube of the pump, resting within a small crucible filled with
mercury, was slightly curved up in order to pass the gas into a stout piece of glass
tubing of rather larger diameter. To the upper end of this latter was sealed an
upright piece of capillary tubing, which served to deliver the gaseous product iuto
some form of collecting apparatus placed upon the working bench of the laboratory.
The mercury dropping from the pump overflowed from the crucible into a porcelain
dish, and was returned from time to time to the funnel, to which the rest of the
pump was connected by a stout piece of rubber tubing, which could be compressed to
a greater or less degree by means of a clamp. In order to study the rate of evolution
of gas at any desired pressure below that of the atmosphere, a small strip of paper was
gummed on the gauge, and the flow of mercury adjusted proportionally to the evolu-
tion of the gas, so that the level of the mercury within the gauge varied only a few
millimetres on either side of the paper strip marking the reduced pressure. After
a little practice this process was found to be easy. In order to test the tightness of
the various joints, which were all carefully lubricated, the whole apparatus was
exhausted to a few millimetres pressure ; if the mercury in the gauge did not rise
appreciably after the lapse of several hours, the whole being left generally over night,
the apparatus was considered sufficiently air-tight for observations lasting only a few
hours.

Decomposition of Oxalic Acid into Carbonic Acid and Oxide.

The method adopted for collecting known volumes of carbonic acid and oxide has
been described above. Experiments were at first made under a reduced pressure,
then the mixed gases allowed to fill up the vacuum, and observations commenced as

280

MR. V. H. VELEY ON THE CONDITIONS OF THE

before ; these are divided into sets of 50 c.c. each, there being a small loss of gas, as
observed above, before the last observation of each set and the first of the next.

Mixture used, 12 grams oxalic acid in 120 c.c. sulphuric acid.

Temperature 68°-9-690<0.

Interval of time for each 10 c.c. of the mixed gases.

Pressure 1C3 mm.

Press-lire 785 mm.

12-2

3-9 4-70

9-5

5-39

5-31

6-02

4-18 4'81

812

5-20

5-31

4-5

4'27

671

5-08

5-0

4-63

4-64

6-20

5'03

5-0

4-18

4-85

5-94

5-03

The results at the reduced pressure show an initial acceleration, then a constant,
followed by a gradually decreasing, rate ; when the mixed gases had filled up the
vacuum, which required a volume of 75 c.c., corresponding to a loss of 7'5 observa-
tions, the initial acceleration again repeats itself, followed by a constant rate practically
equal to that obtained at the point at which the pressure was increased.

These observations do not point to any marked difference in the rate of evolution
of gas at pressures below the atmospheric.

Another series of experiments was conducted under nearly identical conditions,
which point to the same general result.

Temperature 660'9-67°-0.

Intervals of time for each 10 c.c.

Pressure 132 mm.

Pressure

773 mm.

34-5

6-45 578

5-17

6-01

14-0

7-36

11-9

6-32 6-13

5-20

5-81

12-6

7-50

7-65

6-34 —

5-39

6-58

10-31

7-13

6-59

6-00

5-72

670

8-12

6-36

6-67

5-94

5-95

677

7-83

6-27

In the first set three observations were missed, owing to some irregularity in the
working of the SPBENGEL pump.

Decomposition of Formic Acid into Carbonic Oxide and Water.

A series of experiments was also conducted to determine the effect produced by
variation of pressure on this decomposition ; the method of working was the same as
that described above, but the twin U apparatus was used, the carbonic oxide being

EVOLUTION OF GASES FROM HOMOGENEOUS LIQUIDS.

281

collected over water ; the whole process waa therefore continuous, with the exception
of the interval during which the gas filled up the vacuum, or was pumped out while
the pressure was being reduced.

A. mixture was made up as follows : —

Sulphuric acid . . . . = 175 c.c.

Water = 58 „

Solution of sodium formate = 75 „

containing 10 grams of the salt in solution, which corresponds to 330 observations
of 10 c.c. each ; thus, the difference between successive observations caused by
diminution of mass and dilution of the sulphuric acid by the water formed in the
reaction is inappreciable.
Temperature 69°-690>2.

Interval of time required (or evolution of 10 c.c. of gw.

I.

IL

III.

At pressure of 152

mm.

At

prcfwure of 753 mm.

At pressure of 165

mm.

10-63

5-3

5-61

1572

7-22

7-26 7-87

2-85

7'13

7-37

673

5-43

5-85

10-33

7-18

7-33

5-3

7-08

7-52

6-4

57

6-12

9-38

7-33

7-17

6-27

7-25

6-03

5-58

6-22

9-15

7-23

7-03

677

7-03

6-22

89

7-40

7-27

6-90

7-47

The results set forth in the above table are practically continuous ; when it was
judged that the maximum and constant rate in Series I. was attained, the pressure
was suddenly increased by temporarily taking out the cork of the flask containing
the acid mixture, and the observations in Series IT. commenced ; the interval of time
between the last observation of I. and the first of II. was 12 minutes, but during this
time little, if any, gas was given off. Again, when in Series II. the maximum and
constant rate was reached, the pressure was reduced as rapidly as possible to the
former point by working the SPRENGEL, a process requiring about 8'*5, during which
the pumped-out gas was allowed to escape. The results show that at the reduced
pressure there is the usual period of initial acceleration ; on increasing the pressure
there is a retardation followed by an acceleration, but on again decreasing the
pressure the phenomenon is reversed, i.e., there is a period of initial retardation. The
intervals of time in the period of maximum rate of Series I. are slightly less than those
in Series IL, but these latter are equal to those in Series III., thus showing that
variations of pressure produce but little variation in the rate of evolution of gases
from liquids. It is also to be noted that, as regards the initial acceleration, the effect
of a sudden increase of pressure is precisely the same as that of a temporary lowering
of temperature.

MDCCCLXXXVIII. — A. 2 O

MR. V. H. VELEY ON THE CONDITIONS OF THE

In other series of experiments results precisely similar were obtained.
A mixture was made up of 202 c.c. sulphuric acid, 73 c.c. water, and 75 c.c. of
solution of sodium formate containing 10 grams of the salt. Temperature 64°'4-640<G.

Intervals of time required for evolution of 10 c.c. gas.

I.

At pressure of 763 mm.

II.
At pressure of 46 mm.

23-37
10-37
£-75
9-13

9-18 8-83
8-75 8-85
9-03 8-75
9-1

3-52
7-53
8-95
8-90

8-90
8-8
8-83
8-7
8-8

8-6

8-78

The observations in Series II. are in accordance with the preceding series, indicating
a temporary increase in rate followed by a period of retardation caused by reducing the
pressure. This is doubtless caused by the pumping out of some small quantity of gas
stored within the liquid, while the retardation followed by a period of acceleration in
Series I. is conversely in part due to some increase of storage within the liquid. The
carbonic oxide at the moment of its production from the formic and sulphuric acid is
somehow or another more soluble in sulphuric acid than ready-formed carbonic oxide,
which is practically insoluble.

Decomposition of Potassium Chlorate into the Perchlorate and Oxygen.

A comparative experiment was also made to study the effect of the reduction of
pressure on the evolution of oxygen from potassium chlorate, as presenting an example
of a fused salt. Into two pieces of combustion tubing were placed 15 grams
re-crystallised potassium chlorate ; their ends were subsequently drawn out, the one
into a capillary, the other so as to form a convenient joint with the SPRENGEL pump.
The two tubes were placed side by side in a small iron pot, filled with molten
potassium nitrate, heated by a BUNSEN burner. It was found possible to keep the
temperature of the molten nitrate fairly uniform. At first the rate was compared of
evolution of oxygen from both tubes, that from one passing directly into the collecting
tube, that from the other journeying by way of the SPRENGEL pump. As the rates
were equal, the pressure on one was reduced to 20 mm.

EVOLUTION OP GASES FROM HOMOGENEOUS LIQUIDS.

283

Volume of gu collected.

At prewnre of 705-4 mm.

At prc»urc of 788-4 mm.

ToUL

Difference.

ToUJ.

Differem*

c.c.
10

IT.

10

C.C.

8

.-.••
6

15

5

14-8

6-8

20

5

20

5-2

25

5

•Ji -

4-8

30

5

29-5

4-7

35

5

34-5

5-0

40

5

39-8

5-3

At pressure of 785-4 mm.

At preMore of 20 mm.

5 5

8

8

10

5

13-6

5-6

15

5

17

4-6

20

5

21-2

42

25

5

252

4-0

30

5

28-8

3-6

35

5

33-8

5

40

5

37

3-2

45

5

43

6

The results in the lower table show that on reducing the pressure there is a
temporary slight increase in the rate of evolution of gas, but afterwards the decomposi-
tion proceeds pari passu at the ordinary and at the reduced pressure.

Decomposition of Ammonium Nitrate into Nitrous Oxide and Steam.

The effect of variation of pressure on this decomposition was not completely investi-
gated ; one experiment was conducted with an apparatus similar to that described
above in the case of the decomposition of formic acid, but with a U-tube interposed
between the decomposition flask and the SPRENGEL, and surrounded by a freezing
mixture to condense the steam evolved. The fused salt having been heated under
a pressure of 20 mm., it was observed that a sudden increase of pressure, caused by
opening the decomposition flask by removal of the cork, practically stopped the
decomposition for some time ; thus, the phenomenon of the retardation, to be followed
by a period of acceleration, is much exaggerated.

Evolution of Hydrogen from Zinc and Sulphuric Acid.

Various experimenters have from time to time noticed the complete arrestation by
pressure of the displacement of acidic hydrogen by zinc and other metals. Thus, so
long ago as 1828, BABINET* writes, " Si on opere en vases clos lorsque le gaz (hydrogene)

* ' Annalcs de Chimic,' [2], vol. 37, p. 183.
2 o 2

284

MB. V. H. VELEY ON THE CONDITIONS OF THE

acquiert une force elastique suffisante, 1'action chimique s'arrete ; elle est suspendue
jusqu'au moment oil Ton donne issue au gaz comprimeV' BABINET states that decom-
position of zinc by sulphuric acid is stopped by a pressure of 13 atmospheres at a
temperature of 0°, and at 33 atmospheres at a temperature of 20° C. ; while LOTHAR
MEYER* observed that a pressure of 66 atmospheres is required. Experiments made
on this subject were more in accordance with the statement of BABINET ; to com-
pletely stop the reaction at 15° C. a pressure of 28 '4 atmospheres was required, this
being calculated from the volume of hydrogen which should be theoretically given off
by the observed loss in weight of the zinc. The superincumbent pressure was not
however equal to this, as a considerable volume of the hydrogen was dissolved in the
liquid, and was given off, on releasing the pressure, as carbonic acid from soda water.
This subject has also been examined by CAILLETET,! as also from an electric point
of view by F. J. SMITH. BEKETOFF^ also noticed that solutions of certain salts of
silver, such as the nitrate or the chloride, in ammonia, though not reduced by zinc and
an acid at ordinary pressures, were readily reduced when the materials were confined
in a sealed tube. Experiments were made to determine the effect produced by
decrease of pressure on the rate at which the hydrogen is given off.

I. Experiment on the Effect produced by Reduction of Pressure on the Rate of

Evolution of Hydrogen.

The method of experiment was the same as that described above ; the twin-U tube
method was adopted, 5 c.c. of dry air at 0° and 760 mm. being taken as the standard.

Weight of redistilled zinc =7 '089 grams; 55 c.c. of dilute sulphuric acid, con-
taining 2 grams replaceable hydrogen in 500 c.c., and thus equivalent to 7'15
grams zinc, were mixed with 450 c.c. water. At a pressure of 769 '5 mm., at first
no appreciable quantity of gas was given off; as soon as a few bubbles were being
evolved, observations were made in the usual manner. Temperature 26°'5-28°'7.

Intervals of time required for 5 c.c. of gas.

I.

At pressure of 769'5 mm.

II.

At pressure 112mm.

III.

At pressure of 7G9 5m.m.

IV.
At pressure of

112 ui.iii.

26-08 4-32 3-67

1-42 177

2-27 2-2

1-33

11-68 4-1 3-67

1-33 1-7

2-23 218

1-53

8-3 3-93

1-67 172

2-33 2-22

2-13

6-75 3-83

186 172

2-45 2-2

1-93

5-8 3-72

1-9 17

2-08 2'23

1-93

5-18 3-67

1-92

2-37

1-96

472 3-62

1-88

2-23

1-95

• ' POOOENDOEFF, Annalen,' vol. 104, p. 189. t ' Compt. Rend.,' vol. 68, 1869, p. 395.

J ' Compt. Rend.,' vol. 48, 1859, p. 442.

EVOLUTION OP OASES FROM HOMOGENEOUS LIQUIDS. 285

The rate of evolution of gas is at first very slow, and then quickly increases ;
on reducing the pressure (Series II.) there is a considerable increase in the rate; on
allowing the gas to fill up the vacuum the rate is slightly decreased, and on again
reducing the pressure the rate is again slightly increased.

Another series of experiments gave precisely similar results, which seems to indicate
that in this case the rate of evolution of gas is influenced to a slight degree by
reduction of pressure, a result which differs from those observed in previous cases of
the formation of a gas in a liquid But this result may be due either to an accelera-
tion of the chemical change taken in itself, or to the greater agitation of the liquid
by the formation of larger bubbles of gas, whereby the liquid already acted upon in
the immediate vicinity of the metal is more readily removed, and thus the metal comes
more intimately in contact with fresh layers of the acid.

From these results it is apparent that reduction of pressure from one to a fraction
of an atmosphere does not permanently alter the rate of evolution of a gas from a
liquid, but that an increase of pressure produces temporarily a retardation, or even
complete stoppage, of the evolution of gas, and, conversely, a decrease of pressure
produces an acceleration. The only apparent explanation of these phenomena seems
to lie in a storage, to some small extent, of the gas within the liquid, which is
increased or decreased temporarily by increase or decrease of pressure ; when matters
are so equalised that the possible storage of gas has reached its maximum, then the
evolution of gas would be the same, whatever be the pressure to which the liquid is
subjected. The effect produced by increasing the pressure is the same as that of
lowering the temperature.

It would seem, however, that such a gas as carbonic oxide at the moment of its
liberation in the so-called nascent state is of a kind different from the same gas ready-
formed, in that the one is susceptible of solution in sulphuric acid, while the other is
practically insoluble.

PART m.
THE RATE OF DECOMPOSITION OF FORMIC ACID INTO CARBONIC OXIDE AND WATER.

In the introductory sketch, remarks were made on the unsuitability of many
chemical changes leading to the formation of a gas for an investigation on the
variations in the velocity of such a change produced by alterations in the masses of
reacting substances, as also by changes in external conditions, such as temperature,
and material of containing vessel. It is difficult also to find cases in which are
fulfilled the conditions necessary for such an investigation, namely, that the masses of
all but one of the reacting substances should remain constant within small and
inappreciable limits. Another difficulty presents itself in the estimation of the
residue of unaltered material at the conclusion of any one of the observations, without

286 MR. V. H. VELEY ON THE CONDITIONS OF THE

producing complications in the chemical change itself. Further, the phenomenon of
initial acceleration, discussed fully in Part II., and so marked in all the chemical
changes investigated, introduces complications. The gas formed must not only be
practically insoluble in the liquid from which it is produced, but also in the liquid over
which it is collected. The only gases which fulfil the latter condition for the convenient
liquid, water, are nitrogen, nitric oxide, hydrogen, and carbonic oxide. Chemical
changes leading to the production of the last-named were more particularly inves-
tigated. The first trials were made on the rate of formation of carbonic oxide from
potassium ferrocyanide when dissolved in concentrated sulphuric acid; but these
experiments were soon discarded, for this reaction was found to be complicated initially
by the formation of hydrocyanic acid, and, subsequently, by that of sulphurous oxide.
The temperature at which this reaction takes place is also inconveniently high.

The decomposition of formic acid into carbonic oxide and water by means of
sulphuric acid was then selected as a suitable case for investigation. In preliminary
experiments it was ascertained that either from the acid or from its sodium salt the
volume of carbonic oxide required theoretically for the equation

HCOOH = CO + H20

was given off at conveniently low temperatures without any separation of carbonaceous
matter, and consequent reduction of the sulphuric acid. In fact the determination of
volume of carbonic oxide evolved affords a ready method for the estimation of the
quantity of formic acid or of a formate, and since these experiments this method has
been applied by Messrs. HABCOUBT and POULTON to determine the amount of this
acid in the secretion from Lepidopterous larvae.*

The twin U-tube arrangement was used, and, as explained above, 10 c.c. of dry air
at 0° C. and 760 mm. pressure was taken as the standard. Thus, each time that the
chemical operation represented by the equation

HCOOH = CO + H2O

was repeated '02063 gram formic acid disappeared, with production of 10 c.c. or
'01256 gram carbonic oxide and '00807 gram of water.

Obviously, for exact observations, it is required to fulfil two rather incompatible
conditions, (i.) that the sulphuric acid shall be sufficiently concentrated to effect the
decomposition, and (ii.) that it shall be sufficiently dilute for the amount of water
formed during the whole course of the operation not materially to dilute the acid.
Thus, the mass of dilute sulphuric acid must be relatively large, and that of the formic
acid relatively small.

* 'Brit. Assoc. Report,' 1887, p. 766.

EVOLUTION OF GASES FROM HOMOGENEODS LIQUIDS. 287

77(e Formic Acid or Formate used.

In tlie earlier experiments a solution of a known weight of sodium formate, either
•5 or TO gram, in a given volume of water was used; after the effect of the finely
<li vided particles had been observed, the solution was always filtered into the
sulphuric acid. Provided that the solution dropped through the filter at a sufficiently
slow rate, and the flask containing the acid was kept constantly stirred, the rise of
temperature caused by the admixture of the solution with the already previously
diluted acid was not great enough to produce any sensible evolution of gas. In later
experiments formic acid was used ; this was prepared according to the method
proposed by LORIN, viz., by the dry distillation of equi-molecular parts of dry sodium
formate and dehydrated oxalic acid. The acid thus obtained was redistilled, when it
boiled uniformly at 107°; it contained 76*42 per cent, formic acid. It is thus of the
same composition and boiling-point as the rather ill-defined hydrate HCOOH.OHg,
which boils at 107°'l and contains 77 per cent, of the anhydrous acid. Of this,
10 c.c. of a solution containing 3'08 grams of formic acid in 100 c.c. were taken
for each set of experiments.

The Sulphuric Acid.

Pure redistilled sulphuric acid was heated in quantities of 1 to 2 litres at a time
in a large flask until white fumes began to appear, so as to boil off any dissolved air,
and to completely destroy any oxidisable material, which might reduce in the course
of the investigations the sulphuric acid into sulphurous oxide. The acid was diluted
with a suitable proportion of water, the details of which will be given for each series
of observations ; and the amount of sulphuric acid per unit volume in these diluted
acids was determined either by taking its specific gravity or by precipitation with
barium chloride, or in most cases by both methods.

The apparatus (fig. 1) consisted of a flask (A) of about half a litre capacity, within
which was blown a bulb (B) of nearly a quarter of a litre capacity, the space between
them being completely filled with olive oil. The upper end of the larger flask was closed
with a caoutchouc plug (P), through the centre of which the neck of the bulb passed,
while through a side hole in the cork passed a smaller piece of tubing (T), bent twice
at right angles, the upper end of which was connected with a burette (Q) ; through
another side hole passed a piece of capillary tubing (F), bent twice in a U form. The
lower bend of the U was filled with mercury, while at the end furthest removed from
the flask was constructed in miniature a gas regulator (R) of the form originally devised
by Mr. HARCOURT. The neck of the bulb was closed with a caoutchouc plug, through
which the long-range thermometer passed, while a side piece of glass fused on served
to connect with the delivery tube. The whole arrangement was enclosed within a
double-cased air bath, not represented in the figure, heated by a gas flame issuing
from a small jet ; the flow of gas was regulated by a governor.

288

MR. V. H. VELEY ON THE CONDITIONS OF THE

Before a series of observations was commenced the above apparatus was heated to
a temperature slightly above that required, the olive oil expanding upwards through
the opened tap into the burette. The mixture of sulphuric and formic acids
previously made was heated independently in another flask to a temperature slightly

Fig. 1.

below that required ; the loss of carbonic oxide during this process was quite inappre-
ciable. The mixture was then poured into the bulb of the decomposition apparatus,
and its neck closed with the thermometer-bearing plug. When the temperature
appeared to be nearly constant, the tap of the burette was closed ; this caused the
oil to expand only in the U -tube (F), and to drive the thread of mercury down one

EVOLUTION OP GASES PROM HOMOGENEOUS LIQUIDS.

289

limb of the U and up the other until it reached the regulating apparatus, and thus
turned the gas jet down, the lowering of which caused the oil to contract until the
regulator was opened. This process repeated itself until an equilibrium of tempera-
ture was reached. By means of this apparatus the temperature could easily be
maintained within a twentieth of a degree on either side of that required. When the
temperature was constant observations were commenced, the volume of gas evolved
during the preliminary operations being collected in some measuring apparatus. At
the end of the observations the mixture was heated to 100° to 110° C., and the volume
of the gas given off measured, in order to determine the residue of undecomposed
formic acid, the necessary correction being applied for the volume of air displaced by
the expansion of the acid mixture. An account of the conditions of one experiment
will serve, mutatis mutandis, for each of the successive series. A mixture was
made up as follows :—

150 c.c. sulphuric acid;
60 c.c. water ;
10 c.c. water containing 1 gram of sodium formate.

The 1 gram of the salt yields 32 '9 observations of the standard unit of chemical
change of 10 c.c. dry carbonic oxide with the simultaneous formation of '2655 gram
of water, or a dilution at the completion of the reaction of '379 per cent. Some
quantity of gas was lost before observations were commenced, before a temperature
convenient for the purpose was attained ; the amount of residue at the conclusion
was determined as described above.

In the following Table the results obtained are set forth, in which r represents the
residue at the conclusion of each successive observation; 6— 0l the interval of time
in minutes and decimals required for that observation ; and T the sum of the several
values of 6—9^, or the time from the commencement.

SERIES I.
Temperature 73°7.

r

6- e,

T

15

25~-47

25-47

14

28-93

54-4

13

32-73

87-13

12

38-4

125-53

11

44-6

170-13

10

54-12

224-25

9

65-97

290-22

8

88-67

378-89

MIXXX2LXXXVIII. — A.

2 P

290 MR. V. H. VELET ON THE CONDITIONS OF THE

The relation between the values for r and T is expressible by the general formula

in which t is the interval of time elapsing from the moment at which, the conditions
remaining the same, r = oo and 6 — Ol = 0, and the actual commencement of the
observations ; C is a constant. In the above series the value for t is taken as 378',
and in the following Table are given the values for log (T + t), log r, and the sum of
their values : —

log (T + 0

logr

logC

2-606

1-176

3-782

2-636

1-146

3-782

2-667

1-114

3-781

2-702

1-079

3-781

2-739

1-041

3-780

2-780

I-

3-780

2-825

•954

3-779

2-879

•903

3-782

The values for the third column are as concordant as could be desired ; taking the
mean as 3'781, the observed and calculated values for T will be as follows :—

Intervals T.

Intervals T.

Observed.

Calculated.

Observed.

Calculated.

25-47
54-4
87-13
125-53

24-72
53-52
87-13
125-53

170-13
224-25
290-22
378-89

171-55
223-95
283-43
377-1

The greatest error between the observed and calculated values does not exceed
2 per cent., and is within the limits of experimental error. The curve representing the
intervals of time from the commencement of the set, in terms of the residues at each
of the several observations, is a portion of a hyperbola. It is apparently illustrative

of the law

dr_ r*

dT= " C '

which expresses the rate at which equivalent masses react upon each other ; in each
experiment 1/C is the amount of each unit mass, which reacts with the other per
unit time, when an unit mass of each substance is present. It would thus seem
probable that in the system of formic and sulphuric acids with water the chemical
change taking place does not merely consist in the removal of the elements of water

EVOLUTION OF GASES FKOM HOMOGENEOUS LIQUIDS.

291

from a molecule of formic acid, with production of carbonic oxide, but rather that one
molecule of formic acid acts upon another, probably with formation of an unstable
anhydride, which in its turn is decomposed into the second anhydride, or carbonic
oxide, and water. The reaction between permanganic and oxalic acids, investigated
by HARCOURT and ESSON,* affords an example of a case in which a hyperbolic curve
represents the course of a chemical change, while the etherification of acids by
alcohol, investigated by BERTHELOT,t is also of a similar nature. If the above
explanation of the chemical change occurring within the system is correct, the
decomposition of formic acid and its etherification by an alcohol would be chemical
changes analogous in kind. Thus the reactions can be represented as follows :—

(i.) HCOOH + C,H6OH = ^ J O + H2O ;
(ii.) HCOOH -f HCOOH = 0 + HaO '

though the product in the former case is stable in the presence of not too large a
quantity of water, while the product of the latter is unstable. The analogy of these
two changes is further borne out by the phenomenon of initial acceleration observable
in both cases.

Other series of experiments are given below in a series of Tables, in which are set
forth the conditions, the observed values for r, 6 — 0t, and T, the last being compared
with those calculated for certain values taken for t, and from the mean number for
logC.

SERIES II.

205 c.c. of 1 : 4 sulphuric acid,

10 c.c. of dilute formic acid, containing '308 gram of the acid, which gives
15 observations, yielding '12 gram water, or a dilution of '2 per cent.
Temperature 68°'9 — 69°'0.

r

0-0,

T observed.

T calculated,
« - 320, log C - 3 564.

107

22:82

22-82

22-82

97

3262

55-44

57-6

87

4478

100-22

100-22

77

58-33

158-55

156-44

67

76-2

234-75

227-02

57

95-8

330-5

334-6

47

131-08

461-6

461-6

• ' Phil. Trans.,' 1866, p. 201.
t ' Annales do Cbimie' [3], vol. 66, p. 113.
2 P 2

292

MR. V. H. VELEY ON THE CONDITIONS OF THE

SERIES III.
Mixture as in Series II. Temperature 75°'6.

r

6 -0,

T observed.

T calculated,
t - 248, log C = 3-445.

10-3

27!57

27-57

27-57

9-3

29-67

57-24

57-24

8-3

39-38

96-62

92-8

7-3

47-05

143-67

138-95

6-3

63-18

206-85

201-64

5-3

82

288-9

283-2

4-3

114

403

405-6

3-3

197

600

602

SERIES IV.
Mixture as above. Temperature 82°'l.

r

6-6,

T observed.

T calculated,
t = 250, log C = 3-502.

11-5

26-55

26-55

26-55

10-5

28-28

53-83

52-7

9-5

32-27

86-1

84-2

8-5

40-27

126-37

124-12

7-5

48

174-37

172-68

6-5

64-92

239-3

239-3

5-5

88-5

328-

327-8

4-5

127-4

455-2

456-3

SERIES V.
Mixture as above. Temperature 90°.

r

6-6,

T observed.

T calculated,
t = 180, log C = 3-234.

8-75

17-02

17-02

15-9

7-75

24-2

41-22

41-22

6-75

31-96

73-18

74-1

5-75

44-47

177-85

117-65

4-75

60-16

180-58

180-58

3-75

100-1

277-9

277-1

2-75

167-1

445

443-8

In the four preceding series the only known variable is the temperature ; but there
is not apparently any immediate correlation between the factor 1/C and the tempera-
ture, as the following figures show : —

EVOLUTION OF GASES FROM HOMOGENEOUS LIQUIDS.

Temperature.

I/O

69-2
75-6
82
90

•000272
•000357
•000315
•000583

It is probable that the factor 1/C is dependent not only upon the temperature,
conditions of concentration remaining the same, but also upon the variable and quite
unknown condition of the cleanliness of the containing vessel, whereby, in some
cases, more nuclei are presented for the evolution of the gas than in others. The
method of procedure after the residue had been determined was to siphon off the
acid as far as possible, and to keep closed the neck of the decomposition flask with the
thermometer-bearing cork between each series of observations ; but probably small
particles, not discernible by the eye, may have been introduced, producing a consider-
able acceleration in the rate of decomposition. This is rendered evident by the
following series of observations, made with a solution of sodium formate containing
particles of silica, which led, as mentioned above, to the investigation of the effect of
finely divided particles.

SERIES VL
Mixture as in Series I. Temperature 35°*9.

r

6-6l

T observed.

T calculated,
< = 885, log 0 = 8-778.

1

i

t

16-75

22-77

22-77

22-77

15-75

23-18

45-95

45-95

14-75

25-82

71-77

71-77

13-75

30-03

101-8

101-8

12-75

34-45

136-25

134-9

11-75

40-18

176-43

175-51

10-75

46-33

222-76

223-48

9-75

56-43

279-19

280-2

8-75

71-63

850-82

350-82

On comparison of the Series VI. with Series I., it is evident that the value for 1/C
in both cases is practically equal, viz., '000165 and '000162 ; but in Series I. the
reaction was effected at a temperature of 73°7, and in Series VI. at a temperature
of 350>9. But the law governing the rate of this chemical change is still valid in the
presence of these finely divided particles, the concordance between the observed and
calculated values for T being especially marked in this series.

It will be necessary to quote only one more series, in which observations were
carried on almost to the ultimate limit of the reaction.

294

MR. V. H. VELEY ON THE CONDITIONS OF THE

SERIES VII.

I 190 c.c. sulphuric acid
Conditions < 38'8 c.c. water.

5'0 c.c. water containing '5 gram sodium formate.
Temperature 6 8° '6.

r

e-et

T observed.

T calculated,
t=300, logC = 3634.

10-47

28-38

28-38

26-6

9-47

34-88

63-26

61-41

8-47

44-13

107-39

103-65

7-47

53-67

161-06

158-13

6-47

67-87

228-93

228-93

5-47

94-42

323-35

32518

4-47

131-53

454-88

465-6

3-47

219

673-88

663-8

2-47

392

1065-9

1092-2

1-47

960

2025-9

2025-9

With the exception of the last observation but one, the observed and calculated
results are very concordant ; thus, whether the time required for the evolution of the
10 c.c. of dry carbonic oxide be rather less than half an hour or sixteen hours, the law
governing this chemical reaction is still exemplified.

GENERAL CONCLUSIONS.

(i.) The rate of evolution of gases from homogeneous liquids is accelerated by the
presence of finely divided chemically inert particles, not only when the gas is merely
dissolved in the liquid but also when it is in a state of formation as a resultant of a
chemical reaction.

(ii.) In the earlier stages of a chemical reaction yielding a gas the phenomenon of
initial acceleration is observed, the rate of change increasing up to a maximum and
constant point, from which it decreases at a rate bearing some immediate relation to
the diminution of mass.

This phenomenon of initial acceleration repeats itself when the temperature is
lowered temporarily and then restored to its former point, or when the pressure
is increased suddenly.

(iii.) Reduction of pressure from one to a small fraction of an atmosphere causes
but little permanent variation in the rate of evolution of gas from a liquid ; though a
sudden increase of pressure stops temporarily the evolution of gas, and a decrease of
pressure produces temporarily an increase in the rate of evolution.

EVOLUTION OF OASES FROM HOMOGENEOUS LIQUIDS. 295

(iv.) Increase of pressure diminishes or even stops the evolution of a gas from a
liquid

The chemical changes in the case of homogeneous liquids herein studied are the
decomposition of formic acid into carbonic oxide and water ; of oxalic acid into
carbonic oxide, carbonic acid, and water ; of potassium ferrocyanide, forming carbonic
oxide and water ; of nitric acid into nitric oxide ; of ammonium nitrate into nitrous
oxide and steam ; of ammonium nitrite into nitrogen and water ; of potassium
chlorate into the perchlorate and oxygen.

The particular case of the formation of hydrogen from zinc and sulphuric acid
is also investigated.

The finely divided chemically inert substances include pumice, silica, graphite,
barium sulphate, and finely divided glass.

(v.) The particular case of the decomposition of formic acid into carbonic oxide and
water is also investigated ; the rate of change is shown to be directly proportional to
the mass of substance undergoing the change. The curve representing the interval
of time required for each unit of chemical change in terms of the mass present is
shown to be hyperbolic and illustrative of the law—

dr ,-

This indicates that the change results between equivalent masses, probably of
formic acid itself, the sulphuric acid merely serving to induce the reaction between
two molecules of the acid, precisely as it serves to induce the etherification of an
organic acid by an alcohol.

The observed values for the sum of the interval of time required for each unit of
decomposition taken as the standard are concordant with those calculated according
to the hypothesis adopted. So far as the investigation has been carried on no imme-
diate relation has been discovered between the rate of change and the temperature,
though the introduction of particles and the cleanliness, or otherwise, of the containing
vessel may have introduced errors not readily obviated.

In presence of finely divided particles, as silica, the law governing this chemical
change is still valid, though the change itself occurs at a much lower temperature.

In conclusion, I would again allude to my great obligations to Mr. VERNON
HARCOURT, and to Mr. W. ESSON, for help in the mathematical portion of Part III.,
to whom I would express my best thanks. I would also acknowledge, with thanks, the
receipt of a grant from the Government Grant Committee of the Royal Society, to
assist me in this investigation.

X. Chi the Induction of Electric Currents in Conducting Shells of Small Thickness.

By S. H. BUIIBURY, M. A. .formerly Fellow of St. John's College, Cambrulge.

Communicated by H. W. WATSON, D.Sc., F.R.S.

•

Received March 22,— Read May 3, 1888.

IK a conductor be placed in a magnetic field, and the field be made to vary, closed
electric currents will be induced on the surface or within the substance of the
conductor. If the conductor be a wire forming a closed curve in which the electric
current is regarded as having only one degree of freedom, the laws of induction in it
are well known.

The problem of induction of electric currents in solids or hollow shells has been
treated by several writers, generally with reference only to conductors of particular
shape, as spheres, infinite planes, cylinders, or ellipsoids, and with reference to special
variations of the external magnetic field.*

The object of this paper is to investigate a general theory applicable to surfaces of
any shape in presence of an external magnetic field varying in any manner.

It will be necessary to make use of one conclusion at which Professor LAMB arrives
in the second of the memoirs above referred to, namely, that the displacement currents,
which according to MAXWELL'S theory exist in the dielectric, have in all cases subject
to experiment no sensible influence in modifying the currents of conduction which
may be induced in metallic conductors. We may, therefore, treat the currents cf
conduction as the only possible currents.

Tlie Condition of Continuity.

Let u, v, w denote the components of electric current referred to unit of area.
Then, if there be no time variation of free electricity, the condition of continuity
requires that

du dv dw _

te + dy + 'd: =

* Professor NIVEK, ' Pbil. Trans.,' 1881, p. 307; Professor H. LAMB, 'Phil. Trans.,' 1883, p. 519, 1887,
p. 131; Professor LARMOE, 'Phil. Mag.,' January, 1884; Mr. O. HKAVISIDK, 'Phil. Mag.,' August,
September, 1886.

MDCOCLXXXVIII. — A. 2 Q 30.8.88

298 MR. S. H BURBURY ON THE INDUCTION OF ELECTRIC

at every point. We will suppose that a closed surface can be described completely
enclosing all the currents of the system. Then the stream lines must, in order that
the above condition may be everywhere fulfilled, be in closed curves, and the currents
are then said to be closed currents.

Of Current Sheets and Sliells and Superficial Currents.

2. Any surface to which the resultant currents or stream lines are everywhere
tangential is defined to be a current sheet. The space between any two current
sheets infinitely near each other may be defined to be a current shell. If a line be
drawn oh a current sheet perpendicular to the stream line at any point P, and dc be
any element of that line at P, and h be the distance at P between the two current
sheets forming the shell, then the ratio of the quantity of electricity which flows
through the area hdc in unit of time to dc is the superficial current,* or current
referred to unit of length, at P.

If the current sheet be a closed surface, S — 0, the stream lines form cloBsd curves
upon it, and no two of them intersect each other. Let there be given on S any system
of superficial currents. We can suppose the stream lines traced out on the surface.
Then the superficial current flowing along the belt between any two of them is
inversely proportional at any point in its course to the distance between the two
stream lines at the point.

Of the Current Function.

3. For any system of superficial currents which can be formed on a closed surface
S = 0 there exists a function <£, called the current function, such that the component
superficial currents at any point are

Ud& d<i>

= n-m " "•

-
dz ax

Wrf<l> j !/<)>

= m- -- I -r-

dx ay

where I, m, n afe the direction cosines of the normal to the surface. We will use

U, V, W for the superficial currents ; u, v, w for the currents referred to unit of area.

For there exists a function, <f>, of x, y, and z, which has any arbitrarily assigned

* It is usual to employ the term tuperficial current only in Cases Where the quantity of electricity in
question is infinitely great compared with h by analogy to the definition of a superficial distribution
of electricity in electrostatics. But the definition above given is unambiguous, and includes as a
particular case the case of a finite current in an infinitely thin shell.

CURKKNTS IN C( )N |)I< I IMi SHKLLS Of SMALL THICKNESS. 299

constant value along each stream line. Therefore there exists a function, <f>, which is
constant along each stream line, and such that d^dc is equal to the given superficial
current at every point, dc being an element of a line drawn on the surface at right
angles to the stream line. Then <£ is the required function.

For U, V, and W so defined are proportional to the direction cosines of the common
section of the surface S, and the surface $ = constant, that is, to the direction cosines
of the stream line.

Also

V + W = (.« + „•) + (P + n»

, dd> d<b ,, dd> d<b
— 2ln r

,
</: ax ay

^ d<f>

/d<t>\*

U) -

(*££.. MM

' \dv') ~\dv)

(if d<f>/dvf be the rate ot increase of <f> per unit of length of the normal to the surface
<ft = constant, and d<f>/dv be the rate of increase of <j> per unit of length of the normal
to the sheet)

fr

There may be an infinite variety of functions which satisfy the conditions for <f>,
but all of them give the same value for U, V, and W, If ^ be given, it completely
determines U, V, and W. Conversely, if U, V, and W be given at every point, they
completely determine the values of <f> on S subject to the addition of an arbitrary
constant.

Of the Currents per Unit of Area.

4. Let there be any finite space, and two functions S and $ such that within the
space

dS d<f> dS d<ft
dz dy dy dz

• JO J-t JQ J±

«> ii<p rto u<p

v = j~ j — T ~r

dx dz d* dx

JC3 11 JQ 1 ±

</h ntp rto n<b
dy dx rfjf dy

and hi + mv -|- mo = 0 at all points on the bounding surface.

2 Q 2

300 MH. S. II. BUUHURY ON THE INDUCTION OF ELECTRIC

Then u, v, and w, satisfying these conditions, may be the components of a system of
finite currents per unit of area within the space. For they satisfy the condition

du dv . din _
&c + dy~*~lL =

They also satisfy the conditions

rfS dS dS

u r + v ~j~ + w j — °»

dx dy dz

d<h d<f> dd>

u j + v j + w— = 0 ;
dx dy dz

and, therefore, in such a system any surface, S = constant, or (j) = constant, is a current
sheet.

If the surface S = constant be a closed surface within the space the currents upon
it are closed currents.

If we form a shell between two neighbouring surfaces, S = c and S = c + dc, the
superficial currents in that shell on S = c are determined by the current function <j> dc,

so that

TT , / d<j> <fcfr\

U = etc ( w -™ — m -- , &c.
\ dy dzj'

The currents per unit of area being assumed finite, the superficial currents are of
course infinitely small in an infinitely thin shell. If h be the thickness of the shell,

dS _ ndc dS __ mdc rfS _ Idc

d-~ ~A~ ' dy~ ~h ' ' dx~~dk'
And. therefore,

U = hu, V = hv, W = hw.

Of the Vector Potential of a System of Superficial Currents.

5. If S be any current sheet, <f> the current function, we have for the components
vector potential

7C3

- cZS
r

W

, ,
= \\(n ~ — m

d<)>\

~ — m J )
dy dz /

CURRENTS IN CONDUCTING SIIKI.I.S u| SMALL THICKNESS. 301

where r is the distance from a jx>int in the shell to the point at which F, G, H are
required. If the current sheet be a closed surface, or if it be a bounded surface,
and f/> be zero at the boundary, these expressions can be put in another form, as
follows : —

Applying STOKES'S theorem to any bounded surface S, and the function <j>/r, we
have

in which the integral on the left hand side is round the bounding curve. Therefore
for any closed surface, or auy unclosed surface, provided that <f> is continuous and
vanishes at the boundary,

and, therefore,

F is therefore a linear function of all the <£'s with coefficients functions of the
coordinates ; G and H have corresponding values. Given S and <j>, F, G, and H are
completely determinant, and are independent of h.

Corollary. — The vector potential due to any spherical current sheet is tangential
to every spherical surface concentric with the sheet, as shown by MAXWELL, § 671.

Of the Energy of a System of Current Sheets.

6. The electrokinetic energy of a system of currents over the surface or system of
surfaces S is

T = £ JJ (Fhu + Ghv + Uhw) dS

We can transform this by STOKES'S theorem in the same manner as we transformed the
integral

For, if the surfaces be closed, or if (ft be continuous and vanish at the boundary,

302 MR. S. H. BUKBURY ON THE INDUCTION OF ELECTRIC

or

Treating G<£ and H<£ in the same way, and arranging, we obtain

iff^J;/rfH dG\ I** ^\ fdG rf

T = i <£ <M3 -- ^H~wl -- T~) + n(~j -- ^
~JJ I \rt# («/ \02 a#/ \ dx dy

But

_ j _

dy dz' dz dx' dx dy

are respectively the x, y, and z components of magnetic force, or which is here the
same thing, magnetic induction. That is, if fl be the magnetic potential of the system,

rfH rfp- _ dn

dy dz d$ '

Hence

This value of T is unambiguous ; because, as is well known, dfl/dv is not discon-
tinuous at the sheet, even when the superficial currents are finite. To show this, it
is sufficient to take the tangent plane at any point for the plane of x, y. Then

<m _ dfl _ rfF dQ
dv dz dy dx

Now, F and G, or JJ (hu/r) dS and JJ (hv/r) dS, are the potentials of imaginary matter
distributed over § of surface densities hu and hv respectively. Therefore, dF/dy and
dG/dx, corresponding to tangential components of force, are continuous, although, if
hu be finite, dF/dz may be discontinuous, at the sheet.
Since

dn_/dH_dG
dv \dy dz

and

CUHRENTS IN CONDUCTIM! SIIKLLS O¥ SMALL THICKNESS. '••"'••

we see that T can be expressed as a quadratic function of all the <f>» with coefficients
functions of the coordinates.

Evidently, if we have two systems of currents on different sheets or on the same sheet,
their energy of mutual action is ^ JJ <f> dil <lv c/S = ^ JJ <f> dfl'fdv dS, where <j> and 11
relate to one system, and <f> and SI' to the other.

Comparison with Magnetic Shells.

7. If a system of magnetic shells be formed over the surface S, and <f> be the strength
at any point, regarded as positive when the positive face is outwards, the components
of vector potential of magnetic induction due to the system at any point not within
the substance of the shells are (MAXWELL, § 41G) —

They are, therefore, the same as the components of vector potential of the system
of electric currents over S, determined by <f> as current function.

It follows that the components of magnetic force or magnetic induction, namely,
dH/'dy — dG/dz, &c>,are at any point not within the substance of the shells the same
for the system of shells whose strength is <f> as for the system of currents whose
current function is <f> over the same surface ; or, as we may otherwise express it, the
magnetic potential due to the system of shells differs from that due to the system of
currents by some constant at all points external to the sheet, and by some, but not
necessarily the same, constant at all points within the sheet, the particular constants
depending on the definition we choose to adopt of the magnetic potential due to a
current shell.

8. Proposition. — There exists a determinate system of magnetic shells over any
closed surface, S, which has magnetic potential at each point on or within S equal
to that of any arbitrarily assigned external magnetic system.

For let P0 be the potential of the external system.

Let q be the density of a distribution of matter over S, the potential of which is
equal to P0 at all points on S, and, therefore, also at all points within 8.

Then q is possible and determinate by known theorems.

Let (f> be that function of x, y, and z of negative degree which satisfies the
conditions V2<£= 0 at all points outside of S, and d<f>/dv = q at all points on S,

304 Mil. S. H. BURBURY ON THE INDUCTION OF ELECTRIC

where dv \s an element of the normal to S, measured outwards. Then <f> is possible
and determinate by known theorems ; and <f> is the strength of the required magnetic
shell.

For let r be the distance of any point from an internal point O. Then at O the
potential of the system of magnetic shells whose strength is <f> is

But, by GREEN'S theorem, applied to the functions <j> and l/r and the infinite space
outside of S,

But V2l/r = 0, because 0 is within S, and V2<£ = 0 by definition at all external
points.
Therefore,

. : ,, •. , = P, , .I''..

Corollary. — There exists one determinate system of closed electric currents over

. any closed surface, S, whose magnetic potential, together with that of any arbitrarily

given external magnetic system, is constant at all points on or within S ; namely, the

system of currents whose current function is — <f>, where <j> is determined as in the

principal proposition.

9. The magnetic induction due to the combined systems is, therefore, zero at every
point within S. We will define the system of currents on S which has this property
to be the magnetic screen on S to the external system.

Evidently the proposition and its corollary will apply equally to a system of
magnetic shells or electric currents on S having at all points in external space the
same magnetic effect as that of a magnetic system wholly within S.

CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNESS. 305

Example of Magnetic Screen.

1 0. Let S be a sphere of radius a. Then P, the magnetic potential of the external
system, may, as regards its value on S, be expanded in a series of spherical surface
harmonics, including generally a constant term, namely,

Hence,

V, -2n_+J

— > i *» •"» * »

Then <f> is the solid harmonic

which satisfies the condition

-~ = -p = 7 at all points on S ;

and the value of <f> on S is

AO 1 2n + 1 . v

<p — ~~° . *• i •"-• i ••

4ir 4?r n + 1

The constant term — A0/4ir corresponds to a magnetic shell of uniform strength
over S, which gives constant potential AQ at all internal points, and zero at all
external points. It corresponds to no system of electric currents.

Of a certain function called the Associated Function.

11. If S be any closed surface, and if P, Q, R be the components of a vector, such
that dP/dx + dQ/dy + dR/dz = 0 at all points within S, it follows that

Therefore there exists a determinate function, t/», of x, y, and 2, which satisfies the
conditions difi/dv = IP + mQ + 7iR at all points on S, and V2i/» = 0 at all points
within S.

I shall call this the associated function to P, Q, 11 for the surface S.

Evidently the vector whose components are P — d\l//dx, Q — d\fi/dy, R — d^i/dz is
tangential to S at every point, and forms closed curves within S. If the conditions
dP/dy = dQ/dx, &c., are satisfied at all points within S, then P — dty/dx, Q —

MDCCCLXXXVIII.— A. 2 R

30G MR. S. H. BURBUBY ON THE INDUCTION OF ELECTRIC

and R — d\j>/dz are severally zero at all points within S. For, since these conditions are
satisfied, there must exist a function x °f x> V> an(^ 2 8ucn that P = ty/dx, Q = dx/dy,
R = dx/dz at all points within S. And therefore P — d^t/dx = d(x — ifi/dx, &c. ;
and therefore d (x — $)/dv = 0 at all points on S, and V 2 (x — V*) = ° at all points
within S ; and therefore x — 'A = constant, and P — d^dx = 0, &c., at all points
within S.

12. If P, Q, R be the components of an electromotive force, and if S be a conductor,
then, whether the condition P — d^jdx = 0, &c., be satisfied or not, they will produce
on S a distribution of electricity having potential «/>. For let

and

p _^

*1 - 7 •

dx

Similarly,

M

Q = Qi + Q' = ^~ + Q',

R = RI -f- R' = -j- R ;

then the vector or electromotive force whose components are Px> Qu Rx is derived
from the potential $, and produces on S a distribution having potential vft. The vector
whose components are P', Q', R', or P — d^/dx, Q — d^jdy, R — d\jj/dz, forms closed
curves within or upon S, and cannot affect the potential.

13. Now let F0, G0, H0 be the components of vector potential due to a magnetic
system external to S. Let — 1/»0 be their associated function. Let the magnetic
screen to the external system be formed on S. Let F, G, H be the components of
vector potential due to it, and let — «/> be their associated function. Then for the
two systems together we have a vector whose components are F0 + F, &c., with
— (\IIQ + i//) for associated function ; and, since d (F0 + F)/cfy = d (G0 + G)/dx, &c.,
within S because of the screen, it follows that

O + G + j (i/»0 + iff) = 0 j> at all points within S.

ay

14. We are now in a position to consider the general problem of induction, when
electric conductors of any shape are in a magnetic field and the field is made to vary.
Electric currents are generated by induction in or on the surface of the conductors.

CURRENTS IN CONDUCTING SHELLS OP SMALL THICKNESS. 307

These induced currents will, in any observed case, rapidly decay by resistance, and of
course any calculations based on the hypothesis of there being no resistance cannot
express any actually observed phenomena. But as the currents vary from two causes,
(I) by induction, (2) by resistance, it is legitimate, for mathematical purposes, to
calculate the effect of each cause separately. With this object, we may, in determining
the law of formation of the induced currents, assume the resistance to be zero.

Let us take the case in which the conductors on which induced currents are to be
found are hollow conducting shells of any shape. Let their surfaces be denoted
byS.

15. In order that the application of LAGRANOE'S equations may be legitimate,
without introducing equations of condition, we must express the energy in terms of as
many variables, and no more, as there are degrees of freedom. Now the expression
2T = JJj (Fw -f- Gv + Rw)dx dy dz contains u, v, w as the variables, and F, G, H linear
functions of them. But u, v, w have to satisfy at each point two conditions, namely,
(1) the condition of continuity, (2) the condition lu -\- mv + nw = 0 at the surface of
the conductors. The number of variables u, v, w is greater than the number of degrees
of freedom.

Let us then take <f>, the current function, for independent variable, as it is subject
to no condition on any surface. Further, the given magnetic field either consists of,
or may be represented by, a system of current sheets, denoted by S0, on which the
current function is fa, and the magnetic potential due to it is fi0.

16. We may, therefore, without loss of generality, assume the given external
magnetic field to be of that character. Then the electrokinetic energy at any instant
due to the system of currents, as well original as induced, is

dv r dv

in which U is the magnetic potential of the induced currents, and the first integral is
over all the surfaces whereon <j>0 the current function is given, and the second over all
the conductors on which $ is to be determined by induction.

If the system have any other form of energy, as for instance, that of any statical
distribution, the expression for that energy cannot contain <f>.

If, therefore, the given external system vary continuously with the time, the
corresponding variation of the induced system is found by making

'" d<f>

or

7 (~T^ + r ) = 0

dt \ dv dv I
2R 2

308 MR. S. H. BURBURY ON THE INDUCTION OP ELECTRIC

at every point on each of the surfaces S ; that is

d /do,, dn\ _

dv\dt dt)~

at every point on each of the surfaces S.
But also

*•(£+£)=•,. ',..

at each point within S.

It follows that

df!0 da _

~dT + Ht'

or constant, at all points on or within S.

But dfl0/dt being given, there is one, and only one, determinate system of closed electric
currents on S which has this effect, namely, the particular system determined by the
method of (8) and which we called the magnetic screen. This, then, is the system of
currents which will be formed from instant to instant on the surfaces S in response to
the continuous variation of the external magnetic system. This result is stated by
MAXWELL, §§ 654, 655.

Case of a Solid Conductor.

. 17. I have assumed the conductors S to be hollow conducting shells. But if there
be within any of them any solid conductor, the proof shows that no closed electric
currents will, as the immediate effect of induction, be formed upon or within it,
because, as the immediate effect of induction, the magnetic force undergoes no change
within S. The outer shell S, with the induced currents upon it, acting for the instant
as a complete magnetic screen, completely shelters the enclosed solid from the direct
magnetic influence of the external system. As the superficial currents decay by
resistance, they cease to be a complete screen, and the interior solid becomes exposed,
in general very rapidly, to the influence of the external system. This effect we shall
have to consider later. But the immediate effect of induction is to produce only
superficial currents in the outer shell. And as this is true whatever be the form of the
enclosed solid, it is true if S consists of a solid conductor, instead of a hollow shell.

It may be said that we cannot conceive an electric current otherwise than as existing
in a conducting stratum of some finite thickness, nor as independent of resistance,
which our expressions hitherto obtained are. And questions may be raised concerning
the thickness of the solid actually occupied by the currents at any time during the
induction, that is the rate at which the currents penetrate the solid. We shall see
reasons later, see (31) post, for determining the rate of penetration in certain cases as
a function of the resistance of the material. In the meantime we may treat of the

CURRENTS IN CONDUCTING SHELLS OP SMALL THICKNW8. 309

currents as produced from instant to instant in a thin shell, whatever its thickness
may be.

T/ie Potential Induced on a Conductor.

18. Let now dF^dt, dG^dt, dH^dt denote the time variation of the components
of vector potential for the external system. Then — dF0/dt, — dGg/dt, — dllo/dt,
have an associated function, defined as in (11), which we will call i/»<,.

Similarly, if — dF/dt, — dG/dt, — dB./dt relate to the induced currents, they have
an associated function \j/.

The functions — dF^dt — dG^dt — dH^dt are the components of an electromotive
force, and, therefore, by (12), produce on the conductor a distribution having
potential «/»0.

Similarly the functions — dF/dt — dG/dt — dH/dt produce on the conductor a
distribution having potential 1(1. Initially on the formation of the induced currents
they satisfy the conditions

. it -f -£-£<*+*)=***

19. Hence we arrive at the conclusion that any variation of the magnetic field
outside of a conductor causes on the surface of the conductor —

(1.) A system of closed currents whose magnetic potential at the instant of their
creation is equal and opposite to the time variation of the magnetic potential of the
given system at all points on or within the conductor ; that is, a complete magnetic
screen.

(2.) It creates and maintains a difference of potential at different points on the
conductor, and this may be used to produce an electric current in a system connected
with the conductor.

20. The electrostatic distribution has energy, but such energy exists side by side
with the electrokinetic or magnetic energy of the closed currents, without (so to
speak) mixing. That is, there is no term involving products of U, V, W with
difi/dx, &c.

For, if we have any system whatever of closed currents within any closed surface S,
and in the field of a potential i|r,

fff {MS+ V^y + w*dz] tedydz = f | $(lu + mv + nw)dS

because we may take S so distant that lu + my -j- nw shall be zero everywhere
upon it.

310 MR. S. H. BURBTTRY ON THE INDUCTION OP ELECTRIC

The, Effect of Resistance.

21. If cr be the specific resistance per unit of area of the material of which a shell
is composed, the components of electromotive force must be, by OHM'S law, cru, cry, crw,
where u, v, w are the component currents per unit of area. That is, (a-/h) U, (<r/h) V,
and (<r/7i) W, where h is the thickness of the shell, and U, V, W the components of
superficial current. We see then that a-/h stands to U, V, W in the same relation as
cr to u, v, w. And the heat generated by resistance per unit of time and unit area of

the sheet is ha- (uz + v2 + «?) or | (U2 + V2 + W8).

22. Let dF/dt, dG/dt, and dfL/dt denote the time variations of vector potential.
Let y be the electrostatic potential. Then the law of decay of the system, if left

to itself to decay uninfluenced by induction, is expressed by the equations

<r TT dF d^f

<ru = - U = — — — —

ii dt dx

a- T, dG dy

crv =: — V =

h dt dy

& „. rfH dy , . ,

a-w = - W =—--— — . (A)

h dt dz

These conditions must be fulfilled with some value of y or other at every point
within the substance of the shell. If cr, the specific resistance, be constant, we obtain
by differentiation

i ,dv dw\
; ~*~ dy dz] ~

at every point within the substance of the shell.

23. If the currents decay as closed currents, their variations being given by that of
a current function, this requires vzy = 0 at every point within the substance of the
shell.

But we have also

7dF . dG dH , dy

at every point on the surface.

Therefore, if the currents decay as closed currents, S? is the associated function
above defined to dF/dt, dG/dt, dR/dt, which we denote by ty. The shell being of
very small thickness, it matters not whether the distribution whose potential is »/» be
on the inner or the outer surface. In either case dF/dt + d\(i/dx, &c., are the
tangential components of the time variation of vector potential.

Generally an arbitrarily given system of closed currents on a shell of given form

CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNB1B. 311

will not decay in this manner, unless the thickness of the shell be properly assigned
at every point, or unless <r be properly assigned. For if d<j>/dt, the time variation of
the current function, be given, dF/dt + d^/dx, and dG/dt + diji/dy, and dRjdt + dty/dz
are determinate. Now the equations A constitute two independent conditions to be
fulfilled at every point. If, therefore, <r/h be given, it is not generally possible to
satisfy the conditions A by any value of d<f»/dt.

The complete solution of any problem of this kind is the determination of d^/dt at
every point as a function of the time. That can be effected in special cases only.

Of Self-inductive Systems.

24. The class of cases most amenable to mathematical treatment is that in which
d<j>jdt •=• — K(j>, and, therefore,

dF dG rfH

where K is a constant, independent both of the time and of position on the surface.

In any such case, if Flt Glt Hj denote the initial values of F, G, H, then
F = F^""', G = G^""', H = Hje""', when the system decays in its own field.

The same must be the case with U, V, and W, and all linear functions of them, so
that U = U^e""', &c., and, ft being a linear function of U, V, and W, ft = ft^""'.

Also, since T is a quadratic function of U, V, W, we have dT/dt = — 2KrFl and
T = Tjc"2*', giving the rate at which heat is generated in the decaying system.

A system of currents in a shell which has this property shall be here defined to be
a self-inductive system. Professor LAMB, in his paper (' Phil. Trans.,' A., 1887, p. 131),
calls this mode of decay " the natural decay." If the system be left to itself to decay
in its own field, all the currents diminish proportionally, and the system varies in
intensity but not in form.

25. Let now x be the associated function to F, G, H, that is the function for which

% = /F + mG + nH on S,
av

and V4x = 0 within S.
Then evidently

<?>/r d-v <ty dy (f^lr d-v

J = * T~' j = K j ' J = " j '
'<•' ax ay ay ax dz

and our equations (A) become

<T F — dldx G - d/dy H —

h= U V W

312 MB. S. H. BURBURY ON THE INDUCTION OF ELECTRIC

Now if the current function <f> be given for a surface S, U, V, W, F, G, H, and x
are determinate.

Also the resultant of F — dyjdx, G — d-)(Jdy, H — dyjdz is necessarily in the
tangent plane. But it is not necessarily in the same direction in that plane with the
resultant current.

But the equations (B), or

F - dxldx G - dxldy _ H -
U V~ W

cannot be satisfied unless the resultant of F — dyjdx, &c., is in the same direction in
the tangent plane with the resultant current. They, therefore, express a necessary
condition which the current function, <£, must satisfy, in order that the system may be
capable of being made self-inductive. Evidently they express only one condition at
every point, that a line known to be in the tangent plane shall have a particular
direction in that plane. It may be put in the form of a partial differential equation
to be satisfied at every point on S, namely,

..

dxj dx \ dy j dy \ dz / dz

Since, if tj> be given, ^ is determinate, this is a partial differential equation in <j> only.
We may assume that, inasmuch as there are as many disposable quantities, namely,
the values of <f> at every point, as there are conditions to be fulfilled, there must for
every surface S be one or more solutions. As we shall see, if S be a sphere, and in
certain other special cases, there are many.

26. If this necessary condition for <f> be fulfilled, it makes the three quantities

F — dx/dx G — dx/dy H —

U V W

equal to each other at every point. But they will generally differ in values from
point to point on the surface.

Let each of them be denoted by Q. Then Q is a function depending on the form
of S and <f> and the position on the surface.

Then in order that the system, with <f> so chosen, may actually be self-inductive, we
must so regulate the thickness, h, of the shell, as that

at every point, K being a constant.

CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNESS. 313

Of course we may make cr vary instead of, or as well as, h. But it is most con-
venient to keep cr constant, or the shell of uniform material.

If h be so chosen with <r constant, it harmonises the equations (A)

(A);

dt dx d*

- V — _^£

h ' ' dt" dy

a <m_<ty

h ~ dt "" ./:

and the shell so formed is, with the given current function <f>, self -inductive.

It follows from the above that if a shell be self-inductive to a system of currents
denoted by U, V, W, then if the components of electromotive force due to the
external field be at every point on the surface proportional to the components of those
currents, they will induce in the shell a system of that type.

We see further that, if <£ do not satisfy the required condition, the shell cannot be
made self-inductive, however we may choose cr/h.

27. The constant K determines the rate of decay of the currents with the time.
Since K = cr/Qh, we see that K varies directly as <r, and inversely as h ; that is, if the
thickness of the shell be increased at every point in the same ratio, K is diminished in
that same ratio. Also K varies inversely as Q, and Q depends on the forms of S and <£.
As an example, if S be a sphere of radius a, and <f> a spherical surface harmonic of
order «, then, as is well known,

4ira 2n + 1 <r

(j = r— — r , and K = —7— - .
2n + I 4va h

28. If now a conducting shell S be placed in any varying magnetic field, and if the
system of currents induced on it by variation of the field be always self-inductive, the
state of the system at any time, t, can be found, if the law of time variation of the
external field be given.

For let fl0 be the magnetic potential of the external field, II that of the induced
system. Then we have
(1.) Due to induction alone

(2.) Due to resistance alone

rfft

Therefore, generally,

from which fl can be found as a function of t, if ddjdt be given

MDCCCLXXXVIIL — A. 2 S

314 MR. S. H. BURBUBY ON THE INDUCTION OF ELECTRIC

29. For example, let dd0/dt be constant as regards time, but have different values
at different points on the surface. At any point let dfljdt = — C. Then we have
dfl/dt + *fl = C to determine the value of fl at the point.

Whence O = - (I — e—1).

K ^

If now C be so great that we may make Kt infinitely small while Ct remains finite,
this represents the ideal case of a system of so called impulsive currents ; that is, finite
currents supposed to be created in an infinitely short time, and Ct represents the
impulse. In this case fl = Ct, and is independent of resistance.

If, on the other hand, we make Kt very great compared \vith unity, which we always
may do by sufficiently increasing the resistance or the time, the result becomes n = C/K;
that is, fl varies inversely as the resistance. This is a particular case of the result
obtained by Lord RAYLEIGH (" On Forced Harmonic Oscillations of Various Periods,"
'PhiL Mag.,' May, 188(5).

30. Again, let the external system vary according to a simple harmonic law, so that
fl0 = C cos \t, where C is constant as regards time, but a function of position in
space. Then our equation becomes, either on the surface or at any internal point,

. /-1\ • \

- 4- Kfl = (JX sin \t,
at

where C has different values according to the point selected, and X and K are
constant.

We may assume as a solution

fi = C' (cos \t + q sin \t).
Then

C'(K + q\) cos \t = (CX + C'X — KC'q) sin \t.

And, therefore,

X'

and

C'JX + ^J = -CX, or C'=-C.^p
And

X2 K .

11 ~ — L> -j — ~j (cos Xs — --• sm Xc)

CX
= — 3 — - j (X cos \t — K sin X<).

And, if

ft = — C sin a sin (X< — a),

CURRENTS IN CONDUCTING SHELLS OP SMALL THICKNESS. 315

and

ft0 + ft = C (cos \t — sin a sin Kt — a)

= C cos a cos \t — a. . . ....... . (D)

The internal field is therefore diminished in intensity in the proportion cos a : 1 ,
and retarded in phase by «/2ir of a complete period.

Thia result agrees with that obtained by Professor LARMOR in case of a spherical
sheet ('Phil. Mag.,' January, 1884).

31. The above results are obtained on the tacit hypothesis that the shell, what-
ever its thickness, is to be regarded for our purpose as a single shell in which all the
currents would decay pari passu, no allowance being made for variations along the
normal. On that hypothesis we may, if <r/h be finite, have finite superficial currents
in the shell ; and, as a consequence of their being finite, we have a finite difference
of phase, and intensity of the field diminished in a finite ratio, between the outer and
inner surfaces.

This method cannot give accurate results except in the case of very thin shells.
Another way of treating the subject would be to regard the shell as made up of
a number of separate layers or subsidiary shells, successively enclosing one another
and separated, suppose, by non-conducting surfaces. Then we might apply the
formula (D) to each separate layer, and finally proceeding to the limit, make all the
functions vary continuously throughout the thickness of the solid shell. We will
consider the question in this aspect later. In the meantime we will point out certain
consequences which follow from the formula (D), when <r/h becomes infinite, and the
superficial currents infinitely small, namely, since cot a = «/X, cos a = KJ^/(KZ -\- X2),
sin a = X/V/(KS + X2).

When a/A becomes infinite, K becomes infinite compared with X, Hence, cos a = 1
and sin a = a = X/K.

The formula (D) becomes then

ft = — Ca sin \t ;

or, since ft and a are infinitesimal, we may write

rfn n • w

— = — C sin \t.

da

Again,

X XQA
a= - = — - ;

and, writing dv, an element of the normal, for h,

''n=-CsinX^.
dv <r

282

31G MR, S. H. BURBURY ON THE INDUCTION OP ELECTRIC

Again, any given phase of the disturbance occurs at a later time in the inner than
in the outer field, the difference of time being dt = a/X. In the case we are now
treating a = X/K, and a/X = I/K = Q/arh. The ratio of h to this time, or cr/Q, is the
velocity with which the disturbance penetrates the shell. Since it is independent of X,
it must be the same for all systems of currents of the type <f> on the surface S.

We have thus obtained an answer to the question suggested in (17), so far as
regards a self-inductive system of currents. The velocity, namely, with which they
initially penetrate a solid, or, which is the same thing, the thickness of the stratum
which they may be supposed to occupy at a very short time after the commencement of
the induction, is proportioned to the thickness of the self-inductive shell at any point.

32. The energy dissipated in the shell per unit of time is 2»cT.

We see then that, comparing similar self-inductive systems with different values of
K, but the same mean energy, the heat generated on average per unit of time varies
as K, or, as this heat must all be drawn from the batteries of the primary system, the
cost of maintenance of the system varies as K.

Examples of Self-inductive Systems.

33. A spherical current sheet. (See the works cited above.)

Every spherical current sheet is self-inductive with ar/h constant, if <f> be a spherical
surface harmonic of any one order as A,YM. For, the sheet being spherical, i/» = 0 ;
and, by a known property of the sphere,

,, 4?ra TT F ' 4rra

r = L or — =

2n + 1 U 2n + 1

when a is the radius.
Similarly,

G 4?™ H

Y ftt+1' W" 2n + l'

»

The condition for (f> is then satisfied; and, as 47ra/(2n + 1) is constant, cr/h has
constant value over the surface, or the shell, if of uniform material, must in order that
the system may be self-inductive be of uniform thickness. In this case Q = 4vra/(2n + 1 )

and K = - • - , if h be the uniform thickness of the shell.

A— h

34. If S be a solid of revolution about the axis of z then any system of currents
on it, determined by an arbitrary function of z as current function, satisfies the
conditions F/U = G/V at every point with y = 0, H = 0, provided d<J>/dz be of the
same sign throughout S ; and, therefore, any such system may be made self-inductive
by suitably choosing a-/h.

For the lines of resultant current are circles round points in the axis as centres,
and the lines of vector potential are also circles round points in the axis. Therefore

CURRENTS IN CONDUCTING SHKLLS OP SMALL THICKNESS. 317

(ityjdz being always of the same sign) at any point in the surface the resultant of F
and G coincides in direction with that of U and V, and, therefore, E/U = G/V. Also
IF + mG + "H = 0 at every point, and, therefore, i/> — 0.

35. Again, if <f> be a function of z only, and if x, derived from it by the methods
above explained, do not contain z, the equations of condition reduce to two, namely :

and these must necessarily be satisfied at every point, because the resultant of U
and V is the intersection of the tangent plane with a plane parallel to that of xy.
And this line is also the resultant of F — dyjdx and G — d^dy. We can then
determine <r/h in terms of K. As an example, let us take for our sheet the ellipsoid

and make

<£ = Az

(see Professor LAMB'S paper, ' Phil. Trans.,' A., 1887, above referred to).
We have then

V»-A=,

w = o,

where ra is the perpendicular from the centre on the tangent plane. Also, at any
internal point,

yv
f~1 ^_ A "_

and at any external point,

dq dq_

„ y dP _ A*^"

1 = A, ^ 3— , G = — Aa -; -7- »

cr «OQ a" aoQ

<&* da*

where

dX
? =

and

r d\

q° ~ Jo x/{(«s + X) (6s + X) (c« +

318 MR. S. H. BURBURY ON THE INDUCTION OF ELECTRIC

and A. is the positive root of

y8 , * , .

8 ~r» ~

and therefore on the sheet

Now, x 'ia the function which satisfies d-^Jdv = ?F + wG = (Aj — A2)
on S, and V2^ — 0 within S.

That is, x = Gary, when C = (Aj — A2)/(a2 + Zr), and this is independent of z. We
can therefore make the shell self-inductive.

Again, on the sheet

- -- — , = — 477-U ;
dv dv

that is,

d&
and, therefore,

Aj = A . 4irab*c ~ .
Similarly,

A2 = A . 47ra36c ~ .

We have then to satisfy the equations

,dq<>y

TT . . „

r U s= - A -5 = K A 47ra&8c
A A fc2

or

If we now make a-jh = If (piss), where p is constant, we have, by differentiation,

A

which determines K in terms of (1) a, 6, c, (2) the absolute thickness of the shell,
(3) tr, the specific resistance of the material of which the shell is composed. It is
independent of the absolute value of the current function.

CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNESS. 319

Cnse of an Infinite Plane Sheet.

36. The case of an infinite plane sheet differs somewhat in its practical treatment
from that of a sphere, and requires independent investigation. It has been fully
treated by MAXWELL, MASOART and JOUBERT, and other writers. We here regard it
from a somewhat different point of view.

Let the plane be that of xy.

If then for any system of currents in it the condition F/V = G/V be satisfied, we
can always make the plane sheet self-inductive by suitably choosing cr/h.

For instance, let the system of currents in the plane be induced by the variation of
an infinitely small circular current i, parallel to the plane, and of radius a, and distant
z from the plane. In that case we see at once by symmetry that the induced currents
flow in circles round 0, the foot of the perpendicular from the centre of the circular
current on the plane. The same is the case with the vector potential of all the
currents ; and therefore the resultant of F and G coincides with the resultant current.
Also t/» = 0 in this case. We have then F/U = G/V at every point.

But

'7 =-2,17, f=-2,V.
dz dz

Therefore, in order that the sheet may be self-inductive, we must make

<r 2-irF 2-irG

h~~ ' * rfF : " * dG '
dz dz

But, if r be the distance of a point on the sheet from the circular current

*-1,
dy r'

G = — 2ai
and therefore

. d 1

G = — 2ai - - ;
dx r

d¥ .yz

— = — 6ai *T ,
dz r*

dG
^=

and

ff_ 2ir/c r*

A= T~ z '
or h CK. 1/r*.

That is the condition that the currents excited in a plane sheet under the influence
of an external field of the kind in question may be self-inductive.

320 MB. S. H. BURBUBY ON THE INDUCTION OF ELECTRIC

37. It is usual to treat the sheet with <r/h constant, and therefore without making
it self-inductive. We have generally at any point on the sheet

, .

dz dz

And, therefore, on the sheet

rf? _ JL 2K dG _ o^dG

dt" 2v dz ' dt ~~ 2v dz '
and

d dfl d_(<K* <H\

dt dv dt \dx dy)

d_dG d_d¥_
~ dx dt dy dt

a_ d /dG dY\ <r d dfl
2ir dz \dx dy) 2ir dz dv

We can now treat the problem directly by the method of (16).
For we have by induction in the absence of resistance

d dflo d dfl _
~&k dv ~*~ ~dt dv ~ '

if I10 be the magnetic potential of the field.

Therefore, generally, to determine the value of (I on the sheet,

o d^ dfl _ a_ d dfl
dt dv dt dv 2-jr dz dv

For example, if the external field is constant as regards time, and the sheet rotates
with uniform angular velocity o> round the axis of z,

and, when the motion has become steady,

_

dt dv ^Jd dv '

where 6 is the angle between a plane through the axis and the point considered, and
a fixed plane through the axis.

CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNESS. 321

And therefore we obtain a solution for steady motion in the form

L ffi> 4. <!B\ - ?- !L —

^te^dv ' ' dv) " 2-rrdz dv '
This is the problem of ARAOO'S disc. The result agrees with MAXWELL, § 668 (24).

Concerning Similar Shells Enclosing One Another.

38. Any such shell may be conceived as made up of a series of infinitely thin shells
successively enclosing one another. We will consider the case of similar shells, which
shall be defined as follows :—

Let S be a homogeneous function of x, y, and z of positive degree, the nth. Then
we may form about the origin a series of surfaces whose equations are S = c*, where c
denotes the linear dimensions.

We may call the shells concentric.

Points at which the same radius vector from the origin cuts the surfaces are
corresponding points.

The space between two neighbouring surfaces of the series is a shell. Two shells
are similar when dc varies as c ; that is, the thickness of the shell at corresponding
points varies as c.

If in two similar shells of a series all similarly situated the current at every point
in one is parallel to, and bears a given ratio to, the current at the corresponding point
in the other, the systems of currents are similar or corresponding.

The ratio last mentioned may be any power of c. If it be given, then <f> in the one
shell is equal to <£ at the corresponding point in the other multiplied by a power of c.

39. We can now compare the value of certain functions in corresponding systems of
currents.

Firstly,

Secondly,

_
Q = -- zr* - in all cases varies as the linear dimensions.

K = QT varies inversely as the square of the linear dimensions.

These results are independent of the form of S or <£, or the ratio between the
currents at corresponding points.

Thirdly, if we make

u, v, w vary as c",
then

U, V, W will vary as c"+1.

n and <f> will vary as c*+*.

MDOCCLXXXVIII. — A. 2 T

MR. S. H. BURBURY ON THE INDUCTION OP ELECTRIC

40. If any one of a series of similar shells Avith similar currents be self-inductive,
every one of the shells is self-inductive.

Tf all the shells within S be filled with similar currents, as above defined, the
components of current per unit of area must be

dS d^> dS d$>
dz dy dy dz

&C. = &c.

Let us now find a condition that, if a system of currents of the type <£ be generated
in the outer shell, any inner shell of the series, if a conductor and exposed to the
influence of the outer, shall have the corresponding system of currents of the same
type excited in it.

At any point on the outer shell S, since the shell is self-inductive, we have

/_ dy\ /dSdd> dSd(f>\

F — -j*) = oK = <r(-r:r — J~

\ dxj \dz dy dy dz/

where F is the component of vector potential of the currents in the outer shell S,
and x the associated function. Now, v2 (F — d\]dx) = 0 at all points within S. If,
therefore, v2« = 0, u being a harmonic of positive degree, at all points within S, it
follows that

Ac, dX\ fdSdd> dSdd>\

K F - -j* = <ru = o- (— -f- — — - H

\ ax i \dz dy dy dz/

at all points within S. The same is true of G — d\ldy if - ----- -, - is a solid

™ y dx dz dz dx

harmonic of positive degree, &c. Let S be any one of the shells within S, and
suppose it to become a conductor. If it be self-inductive to the system of currents
u, v, w, it follows that this system with reversed sign, and no other, will be excited in
it by induction when the corresponding system is excited in the shell S. For instance,
in the case of the ellipsoid above treated,

<}> = Az,

dSdQ _ dSd$ 0 Ay

dz dy dy dz ""6s

z z ""

and

\dz dy dy dz

If, therefore, the ellipsoid x2/az + yz/b2 + z2/c2 = 1 be divided into similar, similarly-
situated, and concentric ellipsoidal shells, each shell is, as we have proved above, self-

CURRENTS IN CONDUCTING SHELLS OP SMALL THICKNESS.' 323

inductive to a system of currents determined by <f> = Az. And, if this system be
excited in an outer shell of the series, it excites the corresponding system in an inner one.

Similarly, if - ^ — Z 5« ^c>» were spherical harmonica of negative degree, we

should prove that the system of currents of the type <f>, excited in an inner shell of
the series, would generate by induction a system of the corresponding type in any
outer one.

Of Shells of Finite Thickness.

41. If the superficial currents induced and maintained in a shell be very small, or
the currents per unit of area finite, the inductive effect of the shell itself is
inappreciable compared with that of the original or inducing system.

Suppose, then, we have a series of similar, similarly-situated, and concentric shells
successively enclosing one another, so as to form one solid shell of finite thickness.
Let each be separately self-inductive to the currents excited by the external field.
If the thickness of the solid shell, though finite, be small, we may, without great
error, neglect the inductive effect of any inner shell of the series of which it is
composed upon the outer ones.

Let then S be any shell of the series, fl the magnetic potential on the outer surface
of S due to the whole field outside of it ; then we shall have, by (30) and (31), if f!0
vary according to the simple harmonic law,

dtl . XQ . .,

— = — A — - sin \t, A being a constant ;

<>l' <T

and, if p be the phase,

•'/< _ XQ

dv <r

t

Let c denote the linear dimensions of any shell, and let c1( c.2 be the values of c at
the outer and inner surfaces respectively of the solid shell. Then

; dv

li = -- ,

and these equations become

dfl , dfl . XQA . ., A x • \,

- = h — = — A — sin A/ = — A - sin \t,

dc dv a K

dp^ X

dc K

or

fli — f!2 = AX [ - dc sin \t,

2 T 2

324 MB. S. H. BURBURY ON THE INDUCTION OF ELECTRIC CURRENTS.

and, since K varies inversely as c8,

flj — f!2 oc AX (cj3 — c.,3) sin \t,
Pi - P* <* Mci3 - C28)-

42. In any case of induction, if the primary or external system become after a
while constant, the induced currents will decay by resistance, and the electrostatic
charge, whose potential is \}>, will disappear also. Both the induced currents and the
electrostatic charge in disappearing generate heat in the conductor ; and this heat is
obtained at the expense of the chemical energy of the batteries of the primary
system. We know that the closed currents on the conductor, coming into existence
and decaying, cause on the whole more chemical energy to be spent in the batteries of
the primary system than is accounted for by heat generated in that system, the
excess being equal to the energy dissipated in the induced system of currents.
The energy of the electrostatic charge, if such exist, must also be obtained at the
expense of the batteries. We should then expect to find that charging a conductor
electrostatically in the neighbourhood of a closed battery circuit, or moving a charged
body in the neighbourhood of the circuit, tends to retard or accelerate the current ;
that is, to increase or diminish the chemical energy spent per unit of time in
maintaining the current constant.

[ 325 ]

XI. MiKjiu-tic Qualities of Nickel.

/t i/ J. A. Ewivo, F.R.S., Professor of Engineering in University College, Dundee,

and G. C. COWAN.

Received April 26,— Read May 17, 1888.

[PLATES 15, 16.]

ALTHOUGH determinations of the magnetic permeability of nickel have been made by
ROWLAND and others,* there appears to be no published investigation of the effect of
cyclic magnetising processes. The study of such processes is interesting not only in
its direct bearing on the relation of magnetisation to magnetising force, but indirectly
as yielding data from which one may calculate the dissipation of energy that occurs in
reversal or other variation of magnetism, in consequence of hysteresis in the relation
of magnetisation to magnetising force. Cyclic processes have been very fully
examined for various kinds of iron and steel,t and one object of the following
experiments was to obtain information of the same kind with regard to nickel.
Another object was to examine the effects of longitudinal stress on the magnetisation
of this metal in the same manner as they had been examined in iron by one of the
writers. J Sir WILLIAM THOMSON'S early results in this subject had shown that, when
subjected to longitudinal pull, nickel undergoes much change of magnetism, of a kind
opposite to that which ordinarily occurs in iron,§ and it seemed that a fuller
investigation of the effects of stress might be useful.

The experiments, with the exception of one group described at the end of this
paper, were made with specimens of nickel wire supplied by Messrs. JOHNSON and
MATTHEY. The wire was O'OGS cm. in diameter, and was supplied in what appeared
to be a hard-drawn state, in which its magnetic susceptibility was decidedly less than
when the wire was annealed. Its magnetic quality was examined both when in this
hard-drawn state and after annealing. The direct magnetometric method was
employed, in the same manner as in the experiments on iron referred to above. || A

* ROWLAND, ' Phil. Mag.,' Aug., 1873, and Nov., 1874 ; H. MEYER, ' WIEDEMANN, Annalen,' vol. 18,
p. 251.

f EWINO, ' Phil. Trans.,' 1885, p. 523; HOPKINSON, ibid., p. 455.

J EWINO, lof. cit., §§ 69-113.

§ Sir W. THOMSON, ' Phil. Trans.,' May, 1878 ; or ' Reprint of Papers,' vol. 2, p. 382.

|| EWINO, loc. cit., § 18.

30.10.88

326 PROFESSOR J. A. EWING AND MR. G. C. COWAN

straight piece of the wire, from 300 to 400 diameters long, was hung in a vertical
position with its upper end due east (magnetically) of a small mirror magnetometer.
Over the wire a tube was slipped on which the magnetising solenoid was wound, and
on the same tube there was another solenoid in which a constant current was kept
up, of just sufficient strength to neutralise the vertical component of the earth's
magnetic force. The field within the tube was therefore wholly due to the current in
the magnetising solenoid, and was exactly reversed when that current was reversed.
The strength of the magnetising current was adjusted by a slide resistance consisting
of a column of solution of sulphate of zinc, with two fixed blocks and one sliding block
of amalgamated zinc for terminals ; and a revolving commutator between the slide
and the solenoid allowed the current to be rapidly reversed. The slide was used to
vary the current slowly in studying the relation of magnetism to magnetising force,
and, in conjunction with the rapid commutator, it served also to demagnetise the
wire under examination by the method of reversals (that is, by numerous successive
reversals of a magnetising force which decreased slowly from a strong value to zero).
The magnetising current was taken from a secondary battery of ample capacity, and
was measured by a mirror galvanometer, calibrated to read the current in absolute
measure, which was included in the circuit of the magnetising solenoid. The current
which served to neutralise the vertical component of the earth's field was adjusted by
substituting for the nickel wire a wire of soft iron, and subjecting that to demagneti-
sation by reversals ; it was only when the earth's field was very exactly balanced that
this process gave complete demagnetisation. In all the results given below the
magnetising force $ and the intensity of magnetism 3 are expressed in absolute
c.g.s. units.

Cyclic Magnetisation of Nickel.

Fig. 1, Plate 15, exhibits the cyclic magnetisation of a piece of nickel wire,
0'068 cm. in diameter and 25'4 cm., or 374 diameters long, in the hard-drawn state,
as supplied by Messrs. JOHNSON and MATTHEY. Previous to this test the wire had
been strongly magnetised and demagnetised by reversals, and at the beginning there
was a little residue which this process had failed to wipe out. Nickel, like hard steel,
is much less easily demagnetised to a perfectly neutral state than soft iron. The
following are the observed values of f% and 3 during the first part of the experiment,
and the values of K, the susceptibility, which is

ON THE MAGNETIC QUALITIES OF NICKEL.

HABD-DRAWN Nickel Wire.

*

3

•

0

12

3-0

19

87

37

14-9

104

7-0

191

187

9-8

230

255

1M

37-5

341

9-1

64-0

378

7-0

754

404

5-4

104-4

420

4-0

After this the magnetising force was gradually removed and reversed, then re-
applied, then removed to zero, and then re-applied in the same direction as at first.
The curves in the figure show these subsequent actions sufficiently well to make it
unnecessary to quote the observed values of 3 and .$. After magnetising with
a force of 104 and getting an induced magnetism of 420, there was a residual
magnetism of 299, or 71 per cent, of the induced magnetism. It took a reversed
force of 18'5 to remove this. This is the quantity which HOPKINSON has called the
" coercive force." * The greatest susceptibility (K) was reached when $ was 24 and
3 was 270 ; its value is 11 '2, which gives 142 for the maximum permeability /A. The
energy dissipated through hysteresis in the large cycle, by double reversal of a
magnetising force of say 100 c.g.s. units ( — J 3 d $ ), was 25,400 ergs. In simple
removal and re-application of the force without change of sign there was but little
hysteresis, and the dissipation of energy in that process was relatively insignificant.
The greatest value of the magnetic induction SB ( = 4w3 + <&) reached in this
experiment was 5380.

The full lines in fig. 2 show in the same way the cyclic magnetisation of the same
piece of nickel wire after it had been annealed by heating to bright redness in the
flame of a Bunsen burner and cooling slowly in air. The greatest magnetisation now
reached under a force, .§, of 100 is barely so high as before, but the susceptibility at
earlier stages is much greater, and the coercive force and the dissipation of energy are
much less. The following values refer to the first part of the process ; here, as in the
former case, there was some initial magnetism which the process of demagnetising by
reversals (applied before these observations were taken) did not remove : —

* HOPKINSON, loc. tit., p. 460.

.328

PROFESSOR J. A. EWING AND MR. G. C. COWAN

ANNEALED Nickel Wire.

•6.

3.

f.

0

22

4-0

36

6-5

83

12-8

8-0

177

22-1

9-5

223

23-5

10-9

251

23-0

12-3

273 22-2

24-6

325 13-2

52-6

371 7-1

79-7 392 4-9

100-4

401 4-0

In this case the greatest susceptibility was 23'5, when $ was 9 '5. This gives 302
as the maximum value of /*.* These numbers are just about one-tenth of what they
would be for an iron wire annealed in the same way. The residual magnetism is 284,
or 71 per cent, of the induced, and the coercive force is 7 '5, or a good deal less than
half of what it was before the wire was annealed. The energy of the cycle of double
reversal of magnetism (J %d$) is 11,200 ergs.

The broken lines in fig. 2 show a magnetising cycle performed on another piece, cut
from the same coil of nickel wire, which, after being softened in a Bunsen flame, was
mechanically hardened by being stretched enough to give it some permanent extension
(namely, from a length of 26'6 cm. to 27 cm.). Here the maximum susceptibility is
only 8 '3, the coercive force is 18, and the highest value of 3, reached by applying a
force of 117, is only 340. The cycles for annealed and hardened nickel differ in much
the same way as the corresponding cycles for annealed and hardened iron.

Effects of Stress on the Magnetic Susceptibility and Retentiveness of Nickel.

The effects of stress, consisting of longitudinal pull, were examined (1) by magne-
tising wire from which a constant load of greater or less amount was hung, and (2)
by loading and unloading wire which was exposed to a constant magnetising force of
greater or less intensity. The same pieces of wire that served for the experiments in
cyclic magnetisation were tested for effects of stress, both before and after annealing,
and after being hardened by stretching.

While still in the hard-drawn original condition the wire of fig. 1 was tested for
induced and residual magnetism under varying magnetising forces : first, when not
exposed to stress, then when under a longitudinal pull of 2 kilogrammes, and lastly
when under a pull of 12 kilogrammes. The observations were made by applying
magnetising force, noting the induced value of 3, reducing the magnetising force to

* ROWLAND (loe. cit.) gives 302 as the maximum of fi in a specimen of cast nickel.

* 18 H 8., \. /?«/<• 17.

( 9j7im

r s s -*

s 6 o 'in f luaTpniknoyf jo

ON THE MAGNETIC QUALITIES OP NICKEL.

329

zero, and noting the residual value of 3 (which will be distinguished as 3r), then
applying a stronger magnetising force, and so on. By this means curves were drawn
showing the relation of the induced magnetism 3 to the force •§, and also the residual
magnetism 3r to the force •$, in each of the three conditions of stress. Before passing
from one to another load, the wire was demagnetised by reversals, and this process
was repeated after the new load had been put on. This precaution was taken because it
had been found in experiments on iron that the exact form which the curve of 3 and .§
took under any assigned load depended on whether the process of demagnetising
had or had not been performed after the load was applied.*

The results of these experiments with nickel wire in the hard-drawn state are
given below, and are shown in fig. 3. In the Table a column is added to show the
ratio of residual to induced magnetism.

HARD-DRAWN Nickel Wire.

No load.

Ratio.

2 kilos.

Ratio.

12 kilos.

Ratio.

•e

3

3r

3,/3

«

3

3,

3,/3

«

3

3,

3,/3

5-7

15

0

0

5-5

14

0

0

7-9

8

0

0

11-1

41

14

0-34

10-6

34

8

0-23

12

13

0

0

127

57

27

0-47

13-2

51

18

0-36

29

30

2

0-07

18-4

169

120

071

25-2

169

106

0-63

56

66

8

0-1-2

25-6

276

216

0-78

40-4

273

186

O-f.8

92

109

16

0-15

59-8

386

280

0-73

71-8

353

214

0-60

115

135

21

0-18

84

410

286

0-70

102

411

286

070

As the section of the wire was 0'363 sq. mm., each kilogramme of load corresponds
to a stress of 2'75 kilogrammes per sq. mm.

These results agree with Sir WILLIAM THOMSON'S experiments, which were made by
loading and unloading a nickel rod exposed to constant magnetising forces, in showing
that longitudinal pull reduces the magnetic susceptibility of nickel. The reduction is,
in fact, enormous even under so moderate a load as 1 2 kilos, (or 33 kilos, per sq. mm.),
a load well within the elastic limit of the wire.t Great as the effects of stress are on
the induced values of 3» they are still greater on the residual values, so much so that
a load of 12 kilos, may be said almost to do away with the retentiveness of the wire
with respect to such magnetising forces as the experiments deal with. In magnetisa-
tion by very low forces nickel, like iron, retains sensibly none of the induced magnetism
when the force is withdrawn, and one effect of longitudinal pull is to extend the range
of magnetising force for which this is true.

• Ewnra, loc. cit., §§ 96-105. .

t A specimen of this wire, which was loaded until it broke, showed little elongation until a load of
18 kilos, was reached. It broke with 23 kilos., after extending 9 per cent.
MDCCCLXXXVIII. A. 2 U

330

PROFESSOR J. A. EWING AND MR. G. C. COWAN

After the same piece of wire was annealed, a precisely similar set of experiments
was made, with results which are given below, and shown in fig. 4.

ANNEALED Nickel Wire.

No load.

Ratio.

2 kilos.

Ratio.

12 kilos.

Ratio.

c

a

3,

3r/3

f

3

3r

3r/3

£

3

3r

3r/3

5-3

29

9

0-31

3-8

17

5

0-30

9-2

3

0

0

8-2

84

52

0-62

11-0

83

48

0-58

15-1

9

0

0

11-1

162

118

0-73

16-9

140

86

0-fll

32-5

26

4

0-15

12-6

193

147

0-76

25-4

194

112

0-58

72

73

14

0-19

22-5

285

213

075

587

298

140

0-47

78

88

16

0-18

32-8

327

234

0-72

113-5

372

148

0-40

112

112

16

014

100

401

285

0-71

To exhibit more fully the effect which the presence of tensile stress has in reducing
the magnetic susceptibility of nickel, the same piece of wire (namely, that of the full-
line cycle in fig. 2) was again tested under various loads ranging from 0 to 12 kilos.
The results are shown in fig. 5, Plate 16, and from the curves drawn there the
following values of the maximum susceptibility K have been measured :—

Load in kilos.

Maximum

susceptibility K.

Total.

Per sq. mm.

0

0

15

2

5-5

9-1

4

11

4-5

6

16-5

2-6

8

22

1-9

10

27-5

1-5

12

33

0-95

The value of $ which corresponds to the maximum of susceptibility becomes higher
as the load is increased ; with 12 kilos., in fact, the maximum appears not to be reached
even when <§ is 115. By comparison with the full-line cycle of fig. 2, the no-load test
in this group shows that the wire had lost some of the susceptibility given by
annealing, probably because the load of 12 kilos, which had been applied after the
cycle of fig. 2 had been completed, and before these observations were made, had
produced a slight permanent hardening effect.

To examine the effects of applying and removing stress in a constant magnetic
field another piece of the same nickel wire was softened by heating, and was then

ON THE MAGNETIC QUALITIES OF NICKEL. 331

loaded up to 12 kilogrammes and unloaded, repeatedly, while a strong field was
maintained in action. Fig. 6 shows the resulting changes of 3» first when the
magnetic field was 6'9, then 2T8, then 53'5, and lastly 116. The dotted lines in the
same figure show the changes caused by loading and unloading on the residual
magnetism that was left after the strongest field (116 c.g.s.) had ceased to act.

The curves show that stress of pull, acting either on residual or on induced
magnetism in nickel, produces a large and continuous diminution of the magnetism,
and that cyclic variations of stress are attended by exceedingly little hysteresis in
the relation of magnetism to stress. The " off" curves lie distinctly below the " on "
curves, but only a little way below them. The hysteresis here is far less than in the
case of iron.

The same piece of wire was next hardened somewhat by stretching it till the original
length of 26 '6 cms. was changed to a length of 27 cms., and the process of loading
and unloading was repeated, this time up to 18 kilos. The results are shown for two
magnetic fields in fig. 7. In the stronger field the " on" and " off" curves so nearly ^
coincide that a single line only has been drawn. The piece of wire dealt with in this
experiment was the same, and in the same state, as the piece with which the cycle
shown by broken lines in fig. 2 was performed.

Once more a set of curves of 3 aud $ were taken with this piece (hardened by
stretching), under loads ranging from 0 to 18 kilos. These are given in fig. 8. They
show the same characteristics as those of earlier figures, with a still more striking
absence of susceptibility under the greater loads used here. With 18 kilos., for
instance, equivalent to about 50 kilos, per sq. mm., a force of 100 c.g.s. produced an
intensity of magnetisation amounting to barely 50. In the earlier curves of this
series the dotted lines show the magnetic changes that occurred as the magnetising
force was gradually withdrawn.

Finally, the initial parts of the curves in this group were examined by repeating
the earliest portion of each magnetisation with the wire placed much nearer to the
magnetometer. This was to determine whether there is in nickel any crossing of the
curves similar to the crossing that occurs in iron, in consequence of the " Villari
reversal " of the effects of stress.* Nothing of the kind was discovered in this metal.t
The results of this experiment are given in fig. 9, and from them one may find the
initial magnetic susceptibility, or ratio of 3 to Ǥ, at the very commencement of the
magnetising process. The first part of each curve is sensibly a straight line, until •§
reaches a value of about 5 c.g.s. units. In other words, for forces less than this the
susceptibility is as nearly as possible constant. When there was no load the initial

• Of. THOMSON, loc. eit.; EWINO, toe. cit.

t October 4, 1888. — It is, of course, possible that a crossing may take place at higher values of the
magnetic force than were reached in these experiments, bnt the analogy to iron points rather to a
crossing in the early portion of the curves, snch aa was looked for and not found in these experiments.
With regard to this point, see an experiment by THOMSON, ' Phil. Trans,' 1879, p. 83.

2 U 2

332 PROFESSOR J. A. EWING AND MR. G. C. COWAN.

susceptibility was 1'7. With each addition of load this was reduced, until it fell
with 18 kilos, to 07. These numbers should be compared with those given for iron
by Lord RAYLEIGH,* who has shown that under very feeble magnetising forces the
susceptibility of that metal has a finite and sensibly constant value. For one speci-
men of unannealed Swedish iron he gives 6 '4, and for another specimen 6 '8 as the
initial value of K, which is constant only up to a magnetising force of about 0'04 c.g.s.
units. Thus in nickel the initial susceptibility is much less than in iron (just as the
maximum susceptibility, at a later stage in the magnetising process, is much less), but
the range of magnetic force within which a sensibly constant value applies is
immensely greater.

Magnetisation of Impure Nickel.

A few supplementary experiments were made with a specimen of cheap commercial
nickel wire, 0'154 cm. in diameter, which was found to contain about 4 per cent, of
iron. A piece 41 cms. long was annealed and was subjected to cyclical magnetisation,
with results which are shown in fig. 10. The chief difference between this curve and
that of fig. 2 is the higher limit to which 3 tends in the present case, which is no
doubt to be ascribed to the presence of iron in this impure specimen. A set of
readings of residual magnetism were afterwards taken, and these, along with the ratio
of residual to induced magnetism, are also shown by curves in fig. 10. The maximum
ratio of residual to induced was 0'74. The dissipation of energy by hysteresis in the
cycle of fig. 10 was 12,600 ergs. The effects of stress on the magnetic qualities of
this wire were also examined, and were found to agree in all general features with
the effects observed in purer samples of nickel, which have been described above.

* ' Phil. Mag.,' March, 1887.

[ 333 ]

XII. Magnetic Qualities of Nickel (Supplementary Paper).
By J. A. EWTNG, F.R.S., Professor of Engineering in University College, Dundee.

Received Juno 14,— Read Jane 21, 1888.

[PLATE 17.]

THE present paper is a supplement to one with the same title, by the author and Mr.
G. C. COWAN, which was read before the Koyal Society on May 17 (p. 325, supra).
In that paper experiments were described in which the effects of stress, consisting of
longitudinal pull, on the magnetic permeability and retentiveness of nickel had been
examined, and it was shown that longitudinal pull had an immense influence in
reducing both induced and residual magnetism in nickel. It was, therefore, to be
expected (as Sir WILLIAM THOMSON pointed out in his first discussion of the effects
of stress on magnetic quality*) that longitudinal compression would make nickel more
susceptible of magnetisation, and more ready to retain magnetic polarity. Experi-
ments on the magnetisation of nickel under compression have now been carried out
under the author's directions by two of his students, Mr. W. Low and Mr. D. Low,
and the results are described below. Further experiments have also been made to
investigate the magnetisation of nickel, in very strong magnetic' fields, by the method
already used for iron by the author and Mr. W. Low,t and the results of these are
given at the end of this paper.

In dealing with the effects of tensile stress on magnetic quality, it is convenient to
test the metal in the form of a long wire, long enough to prevent the ends from
materially affecting the magnetic field throughout the main part of the length. But
in dealing with stress of compression this method of approximating to the condition
of endlessness is impracticable. Dr. HOPKINSON has shown that a short bar may be
brought to a condition of endlessness, suitable for the measurement of its magnetic
susceptibility, by sinking its ends in a massive yoke of iron, which affords an easy
path for the return of the lines of induction from end to end, outside the bar, and he
has made use of this plan in determining the form of magnetisation curves for various
• samples of iron and steel.J This method lends itself well to experiments on the

* ' Phil. Trans.,' 1878, or ' Reprint of Papers,' vol. 2, p. 368.

t " On the Magnetisation of Iron in Strong Fields" (1), 'Roy. Soc. Proc.,' vol. 42, p. 200; (2) 'Brit.
Assoc. Report,' 1887, p. 586.

J Hoi'KiNSO.N, " Magnetisation of Iron," ' Phil. Trans.,' 1885, p. 455. In Dr. HOPKINSON'S experiments

30.10.b8

334 PROFESSOR J. A. EWING ON THE MAGNETIC QUALITIES OF NICKEL.

influence of compressive stress, for it is easy to fix the lower end of the bar in the
yoke and apply weights directly, or by a lever, to the upper end. The arrangement
adopted in the present experiments is shown in fig. 11.* The sample under test was a

Tig. 11.

Arrangement for testing magnetisation of nickel under compression.

bar of nickel supplied by Messrs. JOHNSON and MATTHEY (which was found on analysis
to contain 075 per cent, of iron). It was 10 cms. long, and was turned to a diameter
of 0'656 cms. The yoke was of soft wrought iron, with a cross-section on either side
of 67 square cms. The lower end of the bar was supported in the yoke by resting on
the end of a screw-bolt ; on the upper end a short plunger of wrought iron pressed,
and through this the desired stress of compression was applied by means of a lever
(fig. 11). The clear length of the sample, within the yoke, was 5 cms. Over this there
was wound a magnetising solenoid of 250 turns, inside of which there was a small
induction coil wound close to the metal. The magnetisation was determined by
reversing the magnetising current while the induction coil was connected to a ballistic
galvanometer. To find the residual magnetism the magnetising current was broken
after reversal, and the effect of this break was subtracted from half the effect of the
reversal. In every case several reversals were made before a measurement was taken ;
and the process of demagnetising by reversals t was resorted to whenever it was
necessary to get rid of residual effects of previous magnetisation.

In the first place, the nickel bar was examined in the rather hard rolled or drawn
state in which it was supplied, by applying magnetising forces which were raised, step

the bar was cut at the middle of its length, to allow an induction coil to be slipped out. This must have
had the effect of making his measurements of susceptibility and retentiveness somewhat lower than they
would have been had the bar been continuous. In the present experiments there was no joint in the bar
itself, but there were, of course, joints between the ends of the bar and the yoke. These must have had
some influence of the same kind, though less in amount, from the fact that the bar's ends were sunk a
good way into the yoke, to give a large surface of contact.

* The figures are numbered consecutively with those in the former paper (' Phil. Trans.,' 1888, A.,
p. 325).

t ' Phil. Trans.,' 1885, p. 539.

PROFESSOR J. A. EWINO ON THE MAGNETIC QUALITIES OP NICKEL. 335

by step, to about 150 c.g.s. units. Then the bar was demagnetised, a load was
applied to the lever producing a stress of compression, and under this the bar was
again magnetised. The process was repeated under one and another of a series of
loads, the greatest of which produced a compressive stress of 20 kilos, per sq. mm.

The results of this group of experiments are shown in fig. 12, Plate 17, in the form
of curves connecting 3 (the intensity of magnetism) with $ (the magnetising force), for
each of the following states of stress : — 0, T9, 3'5, G'8, 10, 13'3, and 19'8 kilogrammes
per square millimetre. It will be noticed that the effect of compressive stress in
augmenting the magnetic susceptibility of nickel is no less remarkable than the effect
of tensile stress was shown (in the former paper) to be in reducing the susceptibility.
The influence of stress is especially noticeable in the neighbourhood of the bend, or
what WIEDEMANN calls the " Wendepunct " of the curves. This is well shown by
the following Table, which gives the maximum value of the magnetic susceptibility *c
for each state of stress : —

Intensity of strew

Maximum

of compression.

soaeeptibility.

Kilos, per sq. mm.

*.

0

5-6

1-9

6-9

35

8-4

6-8

12-2

10

16-8

13-3

20-3

19-8

29

Concurrently with these observations another group was taken to determine the
influence which the presence of these stresses of compression during magnetisation
had on the amount of residual magnetism held by the metal when the magnetising
current was broken at each stage in the process, the stress being maintained constant
while the magnetising current was made and broken. Curves of the residual magne-
tism (3r) in its relation to <§ are given in fig. 13, for the same set of loads as the
curves of induced magnetism in fig. 12 refer to. They show that a state of compres-
sion during the application and removal of magnetising force augments the residual
magnetism even more than it augments the induced magnetism. In other words, the
ratio of residual to induced magnetism is increased by the presence of compressive
stress. It was found, in the former paper, that tensile stress reduced this ratio, so
much, indeed, that under a strong pull there was scarcely any retentiveness left.
Here, under compression, we have the opposite effect : there is enormous retentiveness
when the stress is considerable. With no load the maximum value in the ratio of
residual to induced magnetism is 0'56 ; with a compressive stress of 10 kilos, per sq.
mm. it is 0'91 ; with one of 19'8 kilos, per sq. mm. it reaches the astonishing value
of 0-96.

336 PROFESSOR J. A. EWING ON THE MAGNETIC QUALITIES OF NICKEL.

By way of showing more fully the influence of stress in facilitating the magnetisa-
tion of nickel we may draw, in ROWLAND'S manner, curves connecting the permeability
yx with the induction 93. This is done in fig. 14 for three states of stress : (l) no load,
(2) a compression of 10 kilos, per sq. mm., (3) a compression of 19'8 kilos, per sq.
mm. In the first the maximum permeability is only 71, in the second it is 212, and
in the third it is 357. The points directly found from the observed values of 93 and •§
are marked by dots upon these curves.

The nickel bar was then softened by heating it to redness in a charcoal fire, and
allowing it to cool slowly ; and experiments similar to the foregoing were made with
it in the annealed state. The relation of induced and residual magnetism to magne-
tising force was examined while the annealed bar was in three states of stress : (1)
under no IcTad ; (2) under a compressive stress of 3 '5 kilos, per sq. mm.; (3) under
a compressive stress of 6 '8 kilos, per sq. mm. The stress was 'not increased beyond
this for fear of hardening the bar by producing permanent set. Figs. 15 and 16
show the results of this group of tests. In fig. 15 the induced values of 3 are shown
by full lines and the residual values by broken lines, in relation to $. In fig. 16 the
permeability p. is shown in relation to the induction 93. It will be noticed by com-
paring figs. 15 and 16 with figs. 12 and 14 that the effect of annealing this bar is (as
with the nickel wire used in former experiments) to increase the permeability at early
stages of the magnetising process, but to reduce it at later stages, and to reduce the
highest value to which the magnetism of the metal was raised. Fig. 1 6 shows that
the curve of p. and 93 for nickel suffers the same kind of inflection as the corre-
sponding curve for iron* when the magnetisation is pushed to high values.

The ratio of residual to induced magnetism in this annealed nickel bar (as in all
former samples both of iron and of nickel) passes a maximum in the neighbourhood of
the " Wendepunct." The values of this maximum are 0'84, 0*88, and 0'91 for the
three states of stress examined, namely, no load, 3 '5 kilos, per sq. mm. and 6 '8 kilos,
per sq. mm. respectively.

A further experiment was made on the annealed bar to determine the initial value
of the magnetic susceptibility under very feeble magnetising forces. For this purpose
a new induction coil was wound on the bar, with many more turns than the former
coil, and the early part of the curve of 3 and Ǥ was examined by the ballistic method
as before. Fig. 1 7 shows the results for the two conditions of no load and a com-
pressive stress of about 5 kilos, per sq. mm. With this, as with the nickel wire
tested in earlier experiments, the curve of magnetisation is at first a sensibly straight
line with a definite inclination. The initial value of the susceptibility is from 2 to
2 '5, and the initial permeability is therefore 25 or 30. The initial permeability
determined by this experiment has been utilised in plotting the point where the
curve p. and 5B in fig. 16 cuts the axis of /*.

It is interesting to notice that the residual magnetism after a very weak field has
* 'Phil. Trans.,' 1885, p. 574; 'Roy. Soc. Proc.,' vol. 42, p. 208.

PROFESSOR J. A. EWING ON THE MAGNETIC QUALITIES OP NICKEL. 337

been applied is sensibly nil. It was only when the value of Jp was raised to about
2 c.g.s. units that any trace of residual magnetism could be detected with certainty,
although by that time the induced magnetism had become great enough to allow
a tenth of it, or less, to have been determined without difficulty. As to the induced
magnetism, it is to be noticed that the presence of stress has much less effect on
the initial value of the permeability than it has when a later stage in the process of
magnetisation is reached.

The experiments were completed by examining the magnetisation of nickel in very
intense fields, by means of the " isthmus " method.* The same nickel bar was fitted
with conical expanding end pieces of soft wrought iron, into which the ends of the
bar were sunk, the whole forming a built-up bobbin with a short narrow neck of
nickel, and the diameter of the neck was turned down to 0'399 cm. On the neck
two induction coils were wound, one close to the metal and the other a little way out,
so as to enclose an annular space, in which the field was measured by observing the
difference in the inductive effects on the two coils. The bobbin was placed between
the pole-pieces of the large electro-magnet of the Edinburgh University Laboratory,
and its magnetism was measured by suddenly withdrawing it from between the poles,
while the magnet was more or less strongly excited. Afterwards readings of the
residual magnetism (which of course did not show itself when the bobbin was with-
drawn from the field) were taken by removing one of the conical end pieces and
slipping off the induction coil.

Measurements were made in fields ranging from 3450 c.g.s. to 13,000 c.g.s.
Allowing for residual magnetism these produced values of the induction 9), which
ranged from 9850 to 19,800. Treating the magnetising field, which was measured
outside the metal, as equal to the magnetising field within the metal itself (an
assumption not far from true), we may calculate the permeability p. and the intensity
of magnetisation 3- The permeability ranged from 2'9 with a field of 3450 down
to 1'5 with the highest field that was applied. The values of 3 fluctuated irregularly
between about 480 and 540, but showed no distinct progressive change either in the
way of increase or decrease as the field was strengthened. The mean of 3 in six
determinations was 515, and this may be taken as fairly representing the limiting or
saturation value of the intensity of magnetism for the particular specimen of annealed
nickel dealt with. Saturation was practically reached at the lowest field, viz., 3450.
Of the whole quantity, 515 for 3. the residual part was 160, and this also was
sensibly constant throughout the range of these experiments.

Roy. Soc. Proc,' vol. 42, p. 200.

MDCCCLXXXVIII.— A. 2 X

9 21

L/ c?<

Roberts -Austerv

.DIAGRAM I .TENSILE ST R ENGTH AND. ATOMIC VOLUMES

- — A

+s./

.

DIAGRAM

_ ELONGATION AND ATOMIC VOLUMES

•

18

ATOMIC VOLUMES

[ 33U ]

XIII. On certain Mechanical Properties of Metals considered in relation to the

Periodic Law.

By W. CHANDLER ROBERTS-AUSTEN, F.R.S., Professor of Metallurgy in the Normal
School of Science and Royal School of Mines, South Kensington, Chemist and
Assayer of the Royal Mint.

Received and Read March 15, 1888.

[PLATE 18.]

THE influence exerted by a small quantity of metallic or other impurity on a mass of
metal is shown by a remarkable series of phenomena the nature of which has hitherto
been but little studied, although the effect produced by the presence of such added
matter is widely recognised by metallurgists. There are many cases in which a small
quantity of impurity has so entirely altered the appearance and the physical properties
of a metal as to lead, in the absence of other evidence, to its being mistaken for a
distinct elemental substance. The valuable mechanical properties conferred upon
metals by associating them with small, but definite, amounts of other metals constitute
the main reason why metals devoted to industrial use are seldom employed in a state
of purity. A familiar instance of the influence of a small quantity of a metalloid on
a mass of metal is presented by the extraordinary change in the properties of pure
iron which attends the introduction into the metal of a small quantity of carbon.
There is no fact in metallurgy of which the importance is more widely recognised, and
when BERGMAN,* in 1781, experimentally demonstrated that the differences between
pure iron, steel, and cast-iron depend on the presence or absence of carbon, he expressed
his astonishment at the smallness of the amount of carbon capable of producing such
effects, and he stated that the explanation of the phenomenon presented a " difficulty
of difficulties " ; and the problem lias certainly not been solved in the century which has
elapsed since BERGMAN wrote.

In other directions the evidence as to the importance of the action of traces of
impurity is just as strong. This is indicated by the fact, referred to by Sir HCSSEY
ViviAN.t that " one thousandth part of antimony converts first-rate ' best selected '

* 'De Analysi Ferri, Opnscnla Physica et Chemica,' by TOKBER.N BERGMAN, vol. 8, 1783: or French
translation (from the Swedish) : ' Analyse du Fer,' by M. GKIU.NOH. Paris, 1783.
t Lecture delivered at Swansea in 1J-80.

2X2 30.10.88

340 PROFESSOR W. C. ROBERTS-AUSTEN ON MECHANICAL PROPERTIES

copper into the worst conceivable," and by the observation of Mr. PREECE,* that " a sub-
marine cable made of the copper of to-day," the necessity for employing pure metal being
recognised, "will carry twice the number of messages that a similar cable of copper would
in 1858," when less importance was attached to the presence of foreign matter in the
copper. It may be well to refer to a but little known case in which the change in the
structure of a metal produced by the presence of a minute quantity of foreign matter
becomes at once evident by comparing the fractured surfaces of the pure and impure
masses. Bismuth, when pure, has a fracture which shows large brilliant mirror-like
crystalline planes ; but, if only the 101Q0th part of tellurium be present, the fracture is,
as a specimen submitted to the Society showed, entirely different, being minutely
crystalline and lighter in colour than pure bismuth.

The mode of action of these small quantities of impurity is still very obscure, but
it should be remembered that Professor W. SPRING, of Lie"ge, has recently given
evidence t in favour of the view that molecular polymerization may take place even
in a solidified alloy, and MATTHIESSEN,J in a classical series of researches on the
electrical resistance of alloys, communicated to this Society nearly thirty years since,
was led to the view that in many cases the constituent metals of alloys exist in the
form of allo tropic modifications, the quantities of the metal producing a rapid decre-
ment in conductivity being too small to enable the effect to be explained by attributing
it to the formation of chemical compounds.

In the present paper, attention is directed to the way in which the tenacity and
extensibility of metals may be affected by small quantities of metals and metalloids,
with the view of showing that the relations between these small quantities of the
elements and the masses of metal in which they are hidden are under the control of
the Law of Periodicity, which, as originally expressed, states that " the properties of
the elements are a periodic function of their atomic weights." CARNELLEY§ has set
forth at some length the reasons for supplementing the law as follows : — " The pro-
perties of the compounds of the elements are a periodic function of the atomic weights
of their constituent elements " ; and the question arises, may the law be so extended
as to govern the relations between the constituent metals of alloys, in which, as is well
known, the atomic proportions are often far from simple.

The influence of a small quantity of one metal on another is so marked that it
appeared well to approach the consideration of the problem by investigating the
nature of the change so effected in the mechanical properties of metals. Gold was
the metal selected as a basis for the experiments, mainly because it can be more
readily brought to a high degree of purity than any other metal : the accuracy of the
results of the experiments are not likely to be disturbed by the oxidation of the gold

* ' Instit. Civil Engineers Trans.,' vol. 75, part 1, 1883.

t ' Bull, de 1'Acad. Roy. de Belgique,' voL 11, 1886.

J ' Phil. Trans.,' vol. 150, I860, p. 85 ; and vol. 154, 1864, p. 167.

§ ' Phil. Mag.,' vol. 8, 1879, p. 368.

OF METALS CONS I DKHKD IN RELATION* TO THE PERIODIC LAW. 341

or by the presence of occluded gases : it possesses considerable ductility and tenacity ;
and the amount of the metallic or other impurity added to the precious metal can be
determined with rigorous accuracy. With the exception of iron, gold has received
more attention than any other metal in relation to the effects of impurities upon it,
and much information upon the subject is scattered through the works of the older
chemists ; but the first systematic experiments were made, by the direction of
the Lords of the Committee of Privy Council, by Mr. HATCHETT in 1803, who
endeavoured to ascertain " the chemical effects produced on gold by different metallic
substances when employed in certain proportions as alloys." He obtained results of
great interest, which were communicated to the Royal Society* ; but in his time the
importance of submitting metals to mechanical tests was not appreciated ; his
observations were thus mainly directed to ascertaining whether gold is rendered hard
and brittle by the presence of foreign metals. The gold he employed was only of
commercial purity, and he specially examined the effect of impurities on the standard
gold used for coinage, which contains 916'7 parts of pure gold and 83 '3 of copper per
1000 parts. He showed, by means of bending and hammering the gold, that small
quantities of certain metals render it very brittle, and he concluded that " the
different metallic substances which have been employed in the present experiments
appear to affect gold nearly in the following decreasing order : —

1. Bismuth.

2. Lead. > These are nearly equal in effect.

3. Antimony.

4. Arsenic.

5. Zinc.

6. Cobalt.

7. Manganese.

8. Nickel.

9. Tin.

10. Iron.

11. Platinum.

12. Copper.

13. Silver."

In 1886, in a lecture delivered at the Royal Institution.t I pointed out that
standard gold breaks with a load of 18 tons to the square inch, and elongates 34 per
cent, before breaking. If the standard gold has only ao*o6th part of lead added to it,
it becomes very fragile, and breaks, as is indicated by the following diagram, with a
stress of about 5£ tons to the square inch, instead of the 18 tons borne originally.
It is remarkable that -r^oth part of lead added to gold does not appear to diminish

• ' Phil. Trans.,' 1803, Part 1, p. 43.

t RoBKBTS-AcsTKN, ' Roy. Inst. Proc.,' 1886 ; and ' Engineering,' May 28, 1886.

342 PROFESSOR W. C. ROBERTS-AUSTEN ON MECHANICAL PROPERTIES

Percentage of lead.

its tenacity more than -y</b~oth part. Professor KENNEDY has kindly made further
experiments on pure gold to which yg-oth part of lead had been added ; and his results
substantially confirm those just referred to, for he found that the breaking load of
the contaminated gold was 1'84 ton per square inch, and the elongation unmeasurable.

In the experiments which follow, pure gold was in all cases melted under a pure
form of charcoal ; a certain amount of the metallic impurity to be added was carefully
weighed, on a delicate assay-balance, and tightly wrapped in pure gold foil. This
little packet was held in a charred splinter of wood, and rapidly submerged in the
molten gold. It was found better not to stir the gold, but to thoroughly mix the
contents of the crucible by giving it a swinging motion. The gold was then poured
through an atmosphere of coal gas into an iron mould lined with lamp-black, and the
resulting bars had the following dimensions : — 3'5 inches length, 0-295 inch breadth,
0'205 inch thickness, the sectional area of the bars being therefore 0'0605 square
inch. The length varied in some cases. The results of analyses of different parts of
the bars showed that the mixing had rarely been defective, but there was evidence
in some cases that liquation had disturbed the homogeneity of the metal, though not
to any serious extent.

The test-pieces were made in the Assay Department of the Royal Mint, by
Mr. GROVES, whose careful manipulation and great experience in melting the precious
metals were of much service to me.

In the case of very volatile metals, such as potassium and cadmium, a rich alloy
with gold was first prepared by fusion in an atmosphere of pure hydrogen. The alloy
so formed was analysed, and the necessary amount of it added to pure gold in the
manner already described so as to give approximately y0- per cent, of foreign metal
in the solidified mass. It may here be pointed out that the purity of the gold

OF METALS CONSIDERED IN RELATION TO THE PERIODIC LAW. Ml

employed had been well established by careful comparison with gold purified by
myself in 1873* for use as Trial-plates in connection with the coinage of this country ,t
the purity of which has been recognised by no less an authority than M. STAS. The
amount of foreign matter added to the gold could therefore be readily ascertained
with minute accuracy by the ordinary method of assay, except in the case of metals
of the platinum group. It should be observed, however, that the method of assay,
which consists in eliminating impurities and in comparing the weight of the purified
gold with that of the portion of metal taken for assay, does not enable the assayer to
distinguish whether an impurity is metallic or a metallic oxide. However carefully
the experiments were conducted, it was at times found impossible to prevent the small
amount of added metal from being oxidised to a certain extent during the casting of
the bars. The error thus occasioned is believed to be but small, and was proved not
to be serious in the case of those metals whose oxides can be reduced by hydrogen,
by laminating or breaking in fragments the portion of metal to be submitted to
•Bay and heating it to bright redness in a stream of pure and dry hydrogen.

The testing-machine used in the following experiments belongs to the Metallurgical
Laboratory of the Royal School of Mines, and is of the form devised by Professor
GOLLNEB, and used by himself at Prague, and by Professor BOCK at Leoben. It is a
double lever vertical machine, adapted only for testing short pieces of metal, and
working up to a stress of 20 tons.J The pieces of metal tested were not provided, as
they should have been, with enlarged ends, but they nevertheless seldom broke within
the jaws of the machine, and when they did the result was rejected.

The purest gold attainable has a tenacity of 7*00 tons per square inch, and an
elongation of 30 '8 per cent. Professor KENNEDY found that a less pure sample,
which contained 999'87 parts of fine gold in 1000, broke with a load of 6'29 tons per
square inch, and elongated 18 '5 per cent, before breaking.

In selecting tenacity as the property to be tested with a view to ascertain the effect
of the added matter, the following considerations presented themselves : —

Professor SPRING has built up alloys by compressing the powders of the constituent
metals ; and, by pointing to the evidence of molecular mobility in solid alloys, he has
done much to show the close connection which exists between cohesion and chemical
affinity. KAOUL PICTET has concluded that there is an intimate relation between the
melting-points of metals and the lengths of their molecular oscillations, the length of
the oscillations diminishing as the melting-point increases, and, as CABNELLEY has
pointed out, " we should expect that those metals which have the highest melting-
points would also be the most tenacious." It is known that the melting-points of
metals are altered by the presence of small quantities of foreign matter; their

* « Chem. Soc. Jonra.,' vol. 27, 1874, p. 197.
t Cantor Lectures, ' Soc. Arts Journ.,' vol. 32, 1884.

J This machine is described by Professor KENNEDY, F.R.S., ' Instit. Civil Engineers Proc.,' vol. 88»
1886-87, part 2.

344

PROFESSOR W. C. ROBERTS-AUSTEN ON MECHANICAL PROPKRTIKS

cohesion is also thereby altered : the degree of cohesion may thus be investigated
either by change of temperature or by mechanical stress, being, as it is, some function
of each of these. It might have been well to ascertain the amount of change in the
melting-point of gold produced by the presence of different elements in small quantity,
but, unfortunately .slight variations in high melting-points are very difficult to determine
with any approach to accuracy, and it appeared to be better to ascertain the effect of
metallic and other impurity on the cohesion of the gold as indicated by the amount of
force externally applied in an ordinary testing-machine, and in that way to ascertain
whether the effect of added metals is amenable to any known law.

In the following experiments only the purest gold I could prepare was employed.
It broke with a load of 7 '00 tons per square inch, and elongated 30'8 per cent, (on
3 inches) before it fractured, as recorded in the diagrams, Plate 18, Nos. I. and II.,
subsequently referred to. The effect on the tenacity of gold produced by adding
to it about -fa per cent, of various metals and metalloids is shown in the following
Table, in which the results are arranged according to the tensile strengths : —

Name of added
element.

Tensile strength.

Elongation, per
cent, (on 3 inches).

Impurity, per cent.

Atomic volume of
impurity.

Reduction of area
at fracture, per cent.

Tons per sq. in.

Potassium

Less than 0"5

Not perceptible

Less than 0'2

45-1

Nil.

Bismuth .

0-5 (about)

>»

0-210

20-9

||

Tellurium

3-88

»

0-186

20-5

))

Lead

4-17

4-9

0-240

18-0

Very slight.

Thallium .

6-21

8-6

0-193

17-2

15

Tin . . .

6-21

12-3

0-196

16-2

Not measured.

Antimony

6-0 (about)

qy-

0-203

17-9

54

Cadmium .

6-88

44-0

0-202

12-9

See note t

Silver . .

7-10

33-3

0-200

10-1

»

Palladium

7-10

32-6

0-205

9-4

75

Zinc . .

7-54

28-4

0-205

9-1

74

Rhodium .

776

25-0

0-21 (about)

8-4

See note f

Manganese

7-99

297

0-207

6-8

60

Indium

7-99

26-5

0-290

15-3

72

Copper

8-22

43-5

0-193

7-0

See note f

Lithium .

8-87

21-0

0-201

11-8

60

Aluminium

8-87

25-5

0-186

10-6*

46

The figures given in the fourth column show that there is some divergence in the
amounts of impurity added to the gold. That this is not of much importance in
these preliminary experiments may be inferred from the fact, already mentioned (on
p. 342), that the -^o^th part of lead appears to produce nearly the same diminution in
tenacity as the 3-Qijth part ; a mere trace of certain metals, moreover, will greatly
diminish the tenacity of gold.

[* This is the value given by LOTHAR MEYER. MALLETT'S determination of the density of pure
aluminium would give 10'45.

t These test-pieces drew out after the manner of pitch, that is, as a viscous solid. — August 4, 1888.]

OF MKTALS CONSII>I:KI:I> IN UKI.ATION TO TIII-: I-KRIODIC LAW. 345

The results are also graphically represented in the accompanying diagrams (Plate 18,
Nos. I. and II. The tests were made with great care by Dr. E. J. BALL, Assistant in
the Metallurgical Laboratory of the Royal School of Mines.

It will be evident, from the figures given in the Table, that certain mechanical
properties of gold are greatly affected by the addition, in small quantities, of
potassium, bismuth, tellurium, and lead, while other metals, such for instance as
silver and palladium, hardly produce an alteration. The change in the structure of
the precious metal is, in some cases, very remarkable, as drawings submitted to the
Society showed. Pure gold has a silky fracture, while gold containing the f^th
part of lead, tellurium, bismuth, or antimony shows a well-developed crystalline
structure, the crystalline planes diverging from a line in the centre of the fractured
bar. The character of the fracture does not appear to be closely, related to that of
the added metal, as lead, thallium, and indium, which produce marked crystalline
structure in gold, are, amongst metals, almost colloidal when pure. In these cases,
then, the influence exerted by the added impurity can hardly be considered to be only
due to a power to develop crystalline form. The question now arises, does this power
to produce fragility correspond with any other property of metals in accordance with
which they may be classified ?

The facts represented in the Periodic Law were, in 1879, graphically represented
by LOTHAR MKYER in his well-known curve of the elements. By adopting atomic
weights as abscissze and atomic volumes as ordinates, he showed that the elements can
be arranged in a curve resembling a series of loops, the highest points of which are
occupied by caesium, rubidium, potassium, sodium, and lithium, while the metals which
are most useful for industrial purposes occupy the lower portions of the several loops.

An examination of the results obtained in my experiments, so far as they have yet
been carried, shows that not a single metal or metalloid which occupies a position at
the base of either of the loops of LOTHAR MEYER'S curve diminishes the tenacity of
gold. On the other hand, the fact is clearly brought out that metals which do
render the gold fragile all occupy high positions in MEYER'S curve. This would appear
to show that there is some relation between the influence exerted by the metallic
impurities and either their atomic weights or their atomic volumes. It seems hardly
probable that it is due to atomic weight, because copper, with an atomic weight of
63'2, has nearly the same influence on the tenacity of pure gold as rhodium, with an
atomic weight of 104, or as aluminium, the atomic weight of which is 27. The atomic
volume is of course obtained by dividing the atomic weight by the specific gravity of
the element, and it at once becomes evident, from the tabulated results and from the
diagrams which graphically represent them, that the metals which diminish the tenacity
and extensibility of gold have high atomic volumes, while those which increase these
properties have either the same atomic volume as gold or a lower one. Further,
silver has the same atomic volume as gold, 10'2, and its presence in small quantity
has very little influence, one way or the other, on the tenacity or extensibility of the
MBOOCLXXXVITI. — A. 2 Y

PROFESSOR W. C1. ROBERTS-AUSTEN ON MECHANICAL PROPERTIES

metal. When the metals are ranged in order of atomic volumes, potassium, which
renders gold very brittle, assumes the position to which its very high atomic volume
of 45 entitles it. Aluminium, indium, and lithium occupy somewhat abnormal positions
on the diagram, for they possess high atomic volumes and yet they appear to increase
the tenacity of gold, although they reduce its capability of being elongated.

The influence of cadmium in increasing the extensibility is very remarkable.
Arsenic, again, has a higher atomic volume than gold, and should therefore render
gold somewhat fragile. Several experiments were made with it, and the bars proved
to be very fragile, but the results are not embodied in the table, as the bars did not
appear to be uniform in composition. The influence of zirconium is also noteworthy.
A fine specimen of crystalline zirconium was obtained from Messrs. HOPKIX and
WILLIAMS, but the metal appears to unite itself with gold with great difficulty.
When wrapped in the foil and added to gold, purposely kept considerably above its
melting-point, the foil melted and released the zirconium, most of which fell, through
the molten metal, to the bottom of the crucible, and remained there when the gold
was poured out. After several attempts, an amount of material, which subsequently
proved, on assay, to be about 0'2 per cent., was alloyed with the gold, and a perfectly
sound bar obtained, which appeared to have extraordinary strength, for it broke with
a load of 12 tons per square inch, pure gold breaking at 7 tons. Its fracture was
remarkably close-grained, and it elongated 12 per cent, before breaking. If
subsequent experiments should confirm this high tenacity, the result would be
opposed to the view set forth in this paper, as zirconium has a high atomic volume,
and should diminish the tenacity of gold.

It may be added that it was useless to employ anything but chemically pure gold,
and the supply available only amounted to 40 ounces. As the preparation of gold of
high purity occupies a considerable amount of time, it was considered best to publish
the results already obtained. The effects of 0'2 per cent, of nickel, cobalt, iron, and
platinum, which occupy very low positions on MEYER'S curve, have severally been tried
with standard gold, and do not appear to reduce either its tenacity or extensibility, and
there is no reason to assume that they will behave differently in the case of pure gold.

Allusion has already been made to the close connection which exists between the
tenacity of metals and their melting-points, and CARNELLEY has pointed out that
the melting-points are inversely as the atomic volumes, " the only important excep-
tions to the rule being arsenic, selenium, tin, antimony, tellurium, thallium, lead, and
bismuth." It can hardly be a matter of chance that, as my experiments prove, all
these elements (with the exception of selenium, about the effect of which I am at
present uncertain) diminish the coherence of gold, and there are but few others that
do so — a fact which is alone sufficient to point to there being some connection between
the action of minute quantities of impurities and the Periodic Law.

It would be difficult to suggest any explanation as to the mode of action of the
various elements until the influence of each element in small but varying qunntities,

OF METALS CONSIDEKED IN RELATION TO THE PERIODIC LA.W. 347

both singly and in association, has been investigated. Questions of much industrial
interest present themselves, especially iu connection with iron; with regard to this
metal, the evidence as to the action of other elements upon it would appear to tend in
the same direction as in the case of gold, although the question is greatly complicated
by the relations of iron to oxygen, and by the presence of occluded gases. It may
be sufficient for the present to point out that the atomic volume of iron is 7P2 ; carbon,
the atomic volume of which is small (4'0), when present in quantities varying from
0'2 to 1 per cent, increases its tenacity. Silicon, notwithstanding that it has a larger
atomic volume (ll'l) than iron, apparently increases its tenacity, although little can
as yet be said as to its influence in very small quantities. The same observation
applies to small quantities of manganese. This metal has an atomic volume of 6*8,
and when present in very large quantities, 12 to 15 per cent., confers great extensi-
bility on iron. Sulphur and phosphorus, on the other hand, have large atomic volumes,
15'1 and 13'2 respectively, and both these elements have, as is well known, a preju-
dicial effect on the qualities of iron.

It should not be forgotten that the knowledge of the effect produced on metals by
small quantities of added matter has had a remarkable effect on the development of
chemistry, mainly by sustaining the belief of the early chemists in the possibility of
"ennobling" base metals or "degrading" precious ones. This is specially evident
from the writings of GEBER, BIWNGUCCIO, GELLERT, and ROBERT BOYLE ; and it is
hardly strange that, in the absence of a knowledge of analysis, they should have
believed in the efficacy of a transmuting agent, when it is remembered that in the
specimens submitted to the Society the presence of yioth part of such metals as lead,
bismuth, and potassium has entirely altered the appearance of the fractured surfaces
of pure gold.

ADDENDUM,

August 1, 1888. .

The test-pieces were all cast in the same mould, and their sectional area was about
0'06 of a square inch ; the sixth column of the Table, p. 344, gives the reduction per
cent, in sectional area of many of the test pieces at the point of fracture, so far as it
was possible to measure them, but the irregular nature of the fractured surface
rendered the measurements for the most part untrustworthy, and it would therefore
be of but little use to plot these data on a curve. The behaviour of several of the
test-pieces under longitudinal stress resembled that of a viscous solid, and in such
pieces the fracture was wedge-shaped, with a more or less sharp edge, the section
remaining rectangular.

In some cases, notably in that of the test-pieces containing palladium and lithium,
tin- fractures revejilcd tin- presence <>f a minute cavity, which, doubtless, determined

2 Y 2

348 PROFESSOR W. C. ROBERTS-AUSTEN ON MKCHANICAL PROPKKTIKS

the point of fracture, and, to some extent, therefore, affected the tenacity. With
reference to this point, it is worthy of remark that Professor BAUSOHINGER, of
Munich, has demonstrated, by the aid of a remarkable series of test-pieces of iron
and steel exhibited at the Nuremberg Exhibition of 1882, that the presence of a
minute defect at the point at which fracture ultimately takes place, while not greatly
affecting the tensile strength of the test-piece, may nevertheless prevent the metal
from contracting to so small an area as would have been the case if the metal had
been perfectly sound. The elongation of the test-pieces (given in the third column
of the Table, p. 344) and the atomic volume of the added impurity are plotted in
diagram No. II., Plate 18, which agrees closely with diagram No. I., representing tensile
strength and atomic volume. Cadmium exhibits marked irregularity in both diagrams,
but the only striking difference between the two diagrams is presented by tellurium and
bismuth, the former of which seems to be more prejudicial to the elongation of gold
than to its tenacity. It may be added that some of the metals, zinc and rhodium
for instance, although possessing smaller atomic volumes than gold, appear to diminish
its elongation while they increase the tenacity of the precious metal. This diminu-
tion, though not very marked, causes an irregularity in the portion of the diagram
occupied by metals with smaller atomic volumes than gold.

Mode of Purifying the Gold employed in the Experiments.

The gold employed in the foregoing experiments was purified by a method which
was adopted, after much careful consideration, by the author of this paper in the
preparation of the " Trial Plate " of gold which, by the direction of the Lords
Commissioners of H.M.'s Treasury, was to supplement the "Standard Trial Plate," the
use of which, for verifying the composition of the coinage, has been prescribed by
law since the 17th year of King Edward IV. The purity of the gold so prepared
has been recognised by M. STAS, and Mr. LOCKYER has also satisfied himself of its
high degree of purity by a comparison of photographs of its spectrum. The gold,
having been used in the Assay Office of the Mint throughout a long period of years,
had already been purified many times, and the only metallic impurities liable to have
become associated with it were silver, platinum, and lead ; and those only in very
minute quantities. This gold was dissolved in nitre-hydrochloric acid, the excess of
acid being driven off by slow evaporation. Platinum and the allied metals were
carefully sought for, but were not detected, and the chloride of gold was then
dissolved in a large quantity of distilled water, so that each gallon contained about
one ounce of metal. This solution was allowed to rest for three weeks, when the
finely-divided chloride of silver was separated by careful decantation of the
supernatant solution. The last traces of chloride of silver are only thrown out of
solution when chloride of gold is rendered very dilute. A warm solution of oxalic
acid was then added, to precipitate the gold ; the first and last portions of the gold

OK MKTAI.S CONSIDKRKI) IN HKLATION TO TIIK I'KRIODIC LAW. 349

precipitated were rejected, the middle portion being carefully washed with hot hydro-
chloric acid of sp. gr. 1*1, afterwards melted, with the addition of bisulphate of
potash, in a clay crucible and cast in a stone mould.

There are other methods of obtaining pure gold, which are, in some respects, more
simple, and the best of these is, perhaps, that which involves the precipitation of
gold from its chloride by the passage of a stream of pure sulphurous anhydride.
The author believes, however, that the method above described is the most trust-
worthy, a view which is confirmed by Messrs. HOFMANN and KRUSS, who, in a recent
paper,* state the results of submitting to a careful examination certain methods
employed for separating gold from other metals, and they conclude that oxalic acid
is the best reagent for separating gold from platinum, which was the metal the
presence of which the author was most anxious to avoid. In discussing the results
obtained by Messrs. HOFMANN and KRUSS, Mr. W. BETTELt points out that for large
quantities of gold sulphurous anhydride is not a suitable precipitant.

• « LIEBIO'S Annalen,' vol. 238, 1887, p. 66.
t ' Chemical News,' vol. 56, 1887, p. 133.

C7 O

/%//. 7'ni/w. UWH. A. /'/ate 19.

Fig.

West,N«wmaji iC? li Ui .

[ 351 ]

XIV. On the Specific Resistance of Mercury.

By R. T. GLAZEBROOK, M.A., F.R.S., Fellow of Trinity College, and T. C. FITZPATRICK,
B.A., Fallow of Christ's College, Demonstrators o.t the Cavendish Laboratory,
Cambridge.

Received Jane 19,— Read Jane 21, 1888.

[PLATE 19.]

OF late years several determinations of the electrical resistance of mercury have been
made, and the differences between the results arrived at have been greater than would
be expected at first sight from the nature of the observations involved. The results
of the experiments have been expressed either in terms of the ohm (109 absolute
C.G.S. units) or of the B.A. unit, which, according to the determinations of Lord
RAYLEIGH and one of the authors of this paper (R. T. G.), is equal to '98667 ohm.

Iii the case of Lord RAYLEIGH'S observations, a direct comparison was made between
the mercury unit and the original B.A. standards. Other observers have constructed
copies of their mercury resistances in German-silver wire, which have been com-
pared with the B.A. standards at the Cavendish Laboratory by one of us, or have
compared their tubes directly with copies in platinum-silver wire of the B.A. units
which have been sent from Cambridge after careful testing. The result of these
various comparisons of recent years is as follows, and may conveniently be put in
tabular form, giving the value in B.A. units of the resistance of a column of mercury
1 metre long, 1 square millimetre in cross section, at 0° Centigrade.

This is done in Table I., which also gives the value as found by various observers of
the resistance of I ohm expressed in centimetres of mercury at 0° C.

28.11.88

352

MESSRS. R. T. GLAZEBROOK AND T. C. FITZPATRICK

TABLE I.

Observer.

Date.

Value of 1 metre of
mercury in B.A. U.

Value of Ohm in centi-
metre of mercury.

Lord RAYLEIGH and Mrs. SIDGWICK*
MASCART, NKKVILLE, and BeNOirt-
STEECKERJ

1883
1884

|s*:>

•95412
•95374
•95334

106-23
106-33

L. LORENIZ§

l ss:>

•95388

105-93

Row LAND ||

1887

•95349

106-32

KOHLKAUSCH^f

1887

•95331

106-32

GLAZEBROOK and FITZPATRICK . .

WUILLEUMIBR**

Ihss

I sss

•95352
•95355

106-29
106-27

Since the original standards of the British Association are at Cambridge, it is
possible to compare there the mercury unit and the B. A. unit directly ; and, on
hearing the results of Professor ROWLAND'S careful investigations, communicated to
the British Association at Manchester, it was thought advisable by several members
of the Electrical Standards Committee to repeat the experiments at the Cavendish
Laboratory. This seemed the more desirable, as the number expressing the B.A, unit
in terms of the ohm is given by Professor ROWLAND as '98644 ; and this agrees much
more closely with the results of the similar observations at Cambridge, viz., '98667,
than do the corresponding values of the mercury unit in terms of the B.A. unit, viz.,
•95349 (ROWLAND), and -95412 (RAYLEIGH).

It may be useful to state clearly what is meant by the B.A. unit. In 1864 Messrs.
MATTHIESSEN and HOCKIN constructed a number of coils of various materials to
represent at certain specified temperatures resistances of 109 C.G.S. units of resistance
as determined by the Electrical Standards Committee.

Eight of these coils (two being of platinum-iridium, two of platinum, three of
platinum-silver, and one of gold-silver) have been retained in the possession of the
Committee, while copies have been distributed to other electricians.

The temperatures at which six of these coils are equal to each other, and to one
B.A. unit, are given in the B.A. Report for 1867. Since that date the coils have been
repeatedly compared among themselves, and also with others of the original copies ;
and, with one exception, the apparent changes in their relative values, if any have
occurred, are exceedingly small, and could be accounted for by the supposition that
the temperature of the coil at the time of observation was unceitain to about '1° C.
Since, then, it is exceedingly improbable that all these coils of such different materials

• " On the Specific Resistance of Mercury," ' Phil. Trans.,' 1883.
t ' Journal de Physique,' June, 1884.
J ' WIEDKMANN, Annalen,' vol. 25, 1885.
§ ' WIEDEMANN, Annalen,' vol. 25, 1885.
|| Communicated to (he British Association, 1887.

f ' Abhandl. der k. Bayer. Akad. d?r Wissenschaften,' II. ClaRse, vol. 16, Abth. iii.
** ' Comptes Rendus,' June 4, 1888.

ON THE SPECIFIC RESISTANCE OF MERCUR5T. 353

should have changed by exactly the same amount during the last twenty-one years, it
is inferred that no change has taken place in them, and the B.A. unit is defined as
the mean of the values of the six coils at the temperatures at which they were said
by HOCKIN (B.A. Report, 1867) to be correct. It was used in this sense by Loi-d
I! \vi.i:n;ii in his electrical papers (' Phil. Trans.,' 1881, &c.), and this is the meaning
attached to it in the various reports of the Electrical Standards Committee since the
year 1882.

The method employed in making the observations differed but little from that given in
Lord RAYLEIGH'S paper. The resistance at 0° Cent, of the column of mercury filling the
tube is determined, in B.A. units ; the length, L, of the column of mercury is measured ;
its mean cross section is found by measuring at a known temperature the length, I, of a
column nearly filling the tube, and then finding the mass of mercury in the column.
The mean cross section thus found needs correction for irregularities in the tube, and
these are obtained by the ordinary process of calibration. The formula, as given in
MAXWELL'S 'Electricity and Magnetism' (vol. 1, § 362), requires a small correction,
for the fact that the length of the column used to determine the cross section does not
quite fill the tube.

Let * be the cross section at a distance x from one end. Let X be the length of a
thread of mercury, which is passed along the tube, when its middle point is at
a distance x from one end. Then, assuming the cross section to be constant over
the length X, we have s = C/X, where C is the constant volume occupied by the
thread ; hence, if n be the number of points at whioh X is measured and p the density
of mercury in grammes per c.c.,

Again, let ^ be the average cross section of the tube over the portion (L — I) at the
end which is not occupied by the mercury used to find the average cross section, and
let s be the average cross section of the rest of the tube.

Then,

,.;=_' ::. ••; :;;•-.?*©; ••• ...... ••<»

and, if r be the resistance of a column of mercury 1 metre long, 1 square mm. in
section, at 0° Cent., R the measured resistance of the tube,

therefore,

MDCCCLXXXVIH. — A. • 2 Z

354 MESSRS. R. T. GLAZEBROOK AND T. C. KIT/I'M KICK

so that, if we write p. for

we obtain

~ W* I1 + L (y^"1)} (4)

and

r =

The term depending oil (L — l)fL is, of course, extremely small, but in some of the
tubes employed it exercised a sensible effect on the result.

In addition to the corrections necessary to reduce the results to the standard
temperature 0° C., the lengths L and I require corrections of importance.

The extremities of the tube opened into two large ebonite cups which were filled
with mercury, and the observed resistance R includes that of the mercury in these
cups which is situated just beyond the ends of the tube. Lord RAYLEIGH has shown
that, on the assumption that the diameter of the mercury column in the cups is
infinitely large compared with that of the tubes, the correction required would be
equivalent to adding to the length of the tube '82 of the diameter. The experiments
of MASCART, NERVILLE, and BENOTT, ' Re'sume' des Experiences sur la Determination
do 1'Ohm,' Paris, Gauthier-Villars, 1884, have justified this theoretical conclusion.

To make more certain on this point, we decided to repeat MASCART'S experiments.
A third cup was made, of the same size as the two terminals which are described later,
differing from them only in having two openings, such as c, Plate 19, fig. 2. A tube,
95 '8 cm. long, and about 1 B.A. unit in resistance, was taken. Its resistance when
filled with mercury in the usual way was found. Its diameter, as found from its length
and resistance, was 1'08 mm. ; and therefore the theoreticfil correction for the two
ends is equivalent to an addition to the length of '886 mm., or to the resistance of
'00092 B.A. unit. The tube was then cut in two pieces, and an end was inserted in
each of the two openings in the third mercury cup. This was filled with mercury,
corked up, a thermometer passing through the cork, and replaced in the trough with
the usual packing of ice. When the whole had cooled down its resistance was again
measured and found to have increased by '00089 B.A. unit. This increase is due to
the resistance of the two cut ends, and the difference in the observed and the theoretical
values is within the errors of the determination.

Another determination was made with a tube of about twice the cross section. In
this case the theoretical correction was equivalent to an increase of resistance of
'00061 B.A. unit, while the observed was '00059. It will be noticed that in both
cases the observed value was less than the theoretical. This result was also found to
be the case by LORENZ, who gives as the factor in the correction deduced from his
experiments, the value '82 — '35 d-Jd^, d± being the diameter of the bore, and <!., the

ON THK SI'KCII'IC IM'SISTANVI: OK MKltCi'KY. 355

outer diameter of the tube. In the case of the last tube mentioned above we had
<fj = I'G, d* = 6 mm., and LORENZ'S term '35 rf,/r/., hss the value "09, so that, according
to him, f..r this tube, the correction should be 73 d}. Our experiments would make
(his to be too small, though there is some evidence for a rather smaller coefficient
th.m '82. At the same time, it is hardly sufficient to justify any change, and we shall
therefore add to the observed length of the tube a quantity 8L equal to '82 of the
ili.nneter of the tube.

The correction to the length / is not quite so simple. It arises from the fact that
the ends of the column are not plane surfaces at right angles to its length, but
portions of a curved surface which is spherical only if the tube, which was placed in a
horizontal position, be so narrow that the effect of gravity may be neglected. The
length I is the extreme length of the mercury column measured from end to end, ninl
the volume found from it is therefore too great by the amount contained between the
mercury and two vertical planes touching the mercury column at the extremity of
each meniscus respectively.

This volume may be expressed in the form 7ra2o7, where a is the radius of the tube
and 8J a correction to be subtracted from the length. In cases in which the end of
the mercury is spherical the calculation of 67 is simple.

Let ACB, fig. 1, represent a section of the mercury meniscus by a vertical plane
through the axis of the tube ; let CD be the axis of the tube, meeting AB at right
angles in D ; let the angles of contact at A and B, between the mercury and the
glass of the tube, be 6 ; and let O be the centre and b the radius of the mercury
bubble.
Then

OA = b, DA = a,
and

Angle DAO = 0.

The length of the mercury column was determined by reading microscopes, as will
be explained below, and in all cases the readings for both A and C were taken so that

2 z 2

356

M !•:>>!;< i;. r. <;i,\/i:i:i;iH)K A\I» T. c. KIT/.I-ATUICK

the distance CD was measured. This distance should, of course, be constant for any
given tube, and experiment showed that it was nearly so. Let us put CD = c.
Then we have

• a — 25 _ b ~ '
~ OA ~ b '

AD a

~ + e

Hence,

Now, a is known with sufficient accuracy from the length and mass of the mercury
column, so that the above equation gives us 6.
The following Table gives the values observed : —

TABLE II.

Tube.

Xo. of observation*.

. c in cm.

a in cm.

A

VI.

18

•026

•059 4-2-30

VIII.

12

•029

•062

40-50

II.

8

•025

•i 155

40-20

IV.

•2

•024

•067

44-20

The mean value of 0, allowing for the number of observations in each tube, is
4l°'45. The angle of contact between mercury and glass is usually given as 42°, so
that the agreement is very good.

Now, the volume of the spherical segment ACB is easily seen to be

while that of the cylinder on AB as base and of height CD is

TTO?
COS

(1 - COS 6}.

The correction, therefore, to be subtracted for each end of the tube from the whole
volume tra"l is the difference between these two, and this

Tra8
3 cos3 6

(1-3 sin2 B + 2 sin3 0} ;

therefore,

ON TIIK SI'KCII-IC 1JKS1STANCK OK MKRCURY. 357

If we pnt. 0 = 4l°'45, aa found above, then

8/ = -41! X a,

and this is the value used for the tubes VI., VIII., and II. in the Tables which follow.
In the case of the wider tubes, it was not at first so clear what the amount of this
correction ought to be. If ACB, fig. 2, again represent the meniscus, it was evident
that B was not vertically below A, and, further, that the extreme point C did not lie
on the axis of the tube.

Let AE be vertical through *A, and FC'CG a vertical touching the end of the
meniscus, and let DC' be the axis of the tube.

Then, in the case of one tube of 1'9 mm. in diameter, we found that

AF = -46 mm.,
CC'= -11 inm.,
BG = '34 mm.,

approximately. Observations on other tubes gave somewhat similar values, though
the difference between BG and AF was not always so great as in the above. The
exact calculation of the volume between the plane through FG and the meniscus
ACB is not possible. We have calculated the correction on the assumption that it is
the same as that for a spherical surface through AC'E ; the effect of gravity has been
to draw this surface down into the position ACB, and it is assumed that the volume
ACBE is approximately the same as AC'E.

Taking the values given above, we find on this assumption that

8/ = '46 X a.

It seemed desirable, however, to verify this result by direct experiment, and this
we did, following the method adopted by Lord RAYLEIOH. Ebonite plugs were
turned which exactly fitted the ends of the tubes, and these plugs were inserted and
pressed up against the ends of the mercury columns so as to flatten them, and the

358 MESSRS. R. T. GLAZEBROOK AND T. C. FIT/PATRICK

length of the column was measured with the reading microscopes. The ebonite plugs
were then removed, and the full length of the mercury column measured. The
difference between these two gives us $1 directly ; for one tube 1'9 mm. in diameter
the mean of a number of determinations which were in fair agreement gave

81 = -45 mm. ;
and for this tube we have, therefore,

81 = '47 X a.

For a tube such as those used for the half units, for which the diameter was 1'57 mm.,

we found

81 = '35 mm.,

and this gives

SI = '4.1 X ".

It is, therefore, clear that for these tubes we may, without serious error, use the
value given by the above theory, viz.,

81 = -4(5 x ".

and this has been done in the calculations.
Thus, the equation to determine r becomes

WR x 10*

r =

pp. (L + SL) (I - SI

and, as will be seen when the values of the various quantities involved are introduced,
this may be written

WR x 10* T, SL , SI

In this expression temperature corrections are necessary to p, L, aud /, while the
weight W will require reducing to its value in vacua.
Let

t'° = temperature at which the length L is measured ;
b = coefficient of linear expansion of measuring rod = '000017 ;
t = temperature at which the thread of length / is measured ;
p* = density of mercury at 0° = 13 "5 957 grammes per c.c. ;
y* = coefficient of expansion of mercury = '000182 ;
g = coefficient of cubical expansion of glass = '000025.

* These values are taken from the ' Travanx et Memoires da Bureau International des Poids et
Mesnres,' vol. 2. See ' Nature ' for April 3, 1884.

ON THI-: SI'KCIFIC RKSisi'AXCE OF MERCURY 359

Hence,

Volume of thread at 0° = W//j.

Volume of thread at «° = VV(1"<"y<) •

P

W (I 4- «yf )

Mean section of tube at t° = -

pi (1 4- W)

\\- /i . *\
Mean section of tube at 0° =

Length of tube at 0° = ^ '* •

(1 ~r 391')

Hence, the value of W/plL corrected for temperature is

_W (1 + yt) (1 + Iff?) .

PlL(l 4-W

and, if W0 be the weight in vacua, a- the density of air, and p of the brass weights
used,

1:1 ' W-W ,-

.

= W (1 — -000062],

taking dry air at 10°, the mean temperature of the weighings, and putting p' = S'l.
Hence, finally we get, introducing all the corrections,

{ 1 - "' + «f - -000002 - ^ ( ' - l)
[ L t L \/w, /

' ' H-(r-§r/-/>)<-(6-i^K; .... (s)

r -

and, if we put in numerical values for the one unit tubes, we have

»1M j t _ .,, «. + .42 - .000002 - '-if-' (1 - l)

p/i /L 1 L n / L V«! /

r =

«!
+ -000149 1 — -000009*'! ' • • • • (y)

while, for the other tubes, the third term is '46a/?. We may write this

r = 1-^(l+A), ........ (10)

where A is a small fraction, being the sum of all the correcting terms with their
proper signs.

The methods used in finding L and I differed but little from those employed by
Lord RAYLKIQH. The tubes were supplied by Messrs. POWELL and Sox, and a
number were roughly calibrated. The best of these were selected and were cut so as

360 MESSRS. R. T. GLAZEBROOK AND T. C. F1TZPATRICK

to have resistances of very approximately 1 B.A. unit, 1 legal ohm, ^ B.A. unit, and
•£ B.A. unit respectively. The ends of the tubes were ground square by placing each
in a groove in a piece of hard wood and allowing the end to pass through a hole
which it accurately fitted in a brass plate fixed at right angles to the groove.

The tube was made to rotate slowly round its axis, and the end ground by emery
on a copper plate.

To find the value of the length (L), two small rectangular pieces of brass were
used. These were carefully squared, and had a fine X engraved on a small piece of
white metal inserted in the centre of one of the faces of each block.

The tube was placed under two reading microscopes, which could be adjusted
longitudinally by micrometer sci'ews, graduated to '00002 inch, and one of the
rectangular pieces of brass was brought up to each end, and adjusted so that one
edge touched the tube as nearly as could be along the horizontal diameter while the
cross mark on the brass lay on the axis of the tube produced. The microscopes were
then focuseed on the crosses, and several readings taken as the tube was turned
round on its axis; except in the case of one tube (No. III.), no appreciable
difference was observed on turning the tubes round, and in the case of III. the
difference did not amount to '004 centimetre.

The tube and brass pieces were then removed and their places taken by the
standard metre of the Cavendish Laboratory.

This is a bronze rod 1 metre long, divided into decimetres ; the first decimetre is
divided into millimetres. The length of the rod, as determined at the Standards Office
of the Board of Trade by comparison with the standard metre, S.S., is 99'995 cm.
i '001 at 0° C., while its coefficient of expansion is '000017 per 1° C.

The first decimetre and the millimetre subdivisions have no appreciable error.

For the lengths greater than one metre a second bar, made by the Cambridge
Scientific Instrument Company, was used. This was 30 cm. in length, and was
divided into decimetres, which were compared with those of the standard.

The two bars were placed end to end in a long wooden box, which they fitted
tightly, and the distance between the last division of the standard and the first
division of the bar measured with the microscopes.

The temperatures were read by a thermometer laid on the scale or against the
glass tube.

The length thus found was that of the glass tube, together with the sum of the
distances between the cross marks on the brass blocks and their edges which were in
contact with the tube. This distance was carefully determined by aid of the reading
microscopes and the standard metre.

The values found for this correction differed at most by '005 cm., and the obser-
vations on the length of the tubes, which were repeated in each case three or four
times on separate occasions, agreed to about the same amount. Thus, the error in the
length of the tubes is probably in no case greater than '002 cm. in a length of about
100 cm.

ON THE SI'KCIKIC IIKSISTANCB OF MERCURY.

301

The micrometer screws were only used to measure very small lengths, never so
great as '1 cm., and were tested and found without any error which could be
appreciable.

The tubes were cleaned by passing through them in succession nitric acid, caustic
potash, and distilled water, the last being repeated three times, alcohol, and finally
ether, redistilled for the purpose. These were followed by air dried with chloride of
calcium and passed through cotton wool. While the dry air was being passed
through the tubes were heated with a spirit lamp, and then allowed to cool.

In several cases, following a suggestion of Professor ROWLAND'S, a plug of cotton
wool was pushed through the tube with a wire, in order to loosen any small particles
of dust which might adhere to the sides.

To calibrate the tubes, a short thread of mercury was inserted and moved into the
various positions required ; its length X being measured with the reading microscopes.

Table III. gives one set of readings for each tube. For the tubes VI., VIII., V.,
IX., the calibration observations were repeated, using threads of different lengths.

TABLE III.

VI.

VIII.

II.

V.

IX.

I.

III.

1-453

1-5654

1-5512

1-5464

1-3172

1-225

1-4334

1-439

1-5576

1-5860

1-5386

1-3220

1-219

1-4146

1-482

1-5452

1-6122

1-5386

1-3270

1-212

1-4130

1-432

1-5426

i'6400

1-5374

1-3310

1-208

1-4168

1-432

1-5452

1-6712

1-5428

13282

1-210

1-4234

1-430

1-5470

1-6742

1-5428

1-3234

1-214

1-4110

1-420

1-5470

1-6680

1-5560

1-3202

1-214

1-4010

1-416

1-5380

1-6588

1-5610

1-3170

1-212

1-3844

1-423

1-5360

1-6732

1-5648

1-3156

1-214

1-3670

1-422

1-5396

1-6688

1-5730

1-3216

1-218

1-3656

1-426

15322

1-6844

1-5882

1-3226

1-225

1-3690

1-429

1-5192

1-6908

1-5840

1-3234

1-224

1-3706

1-430

1-5188

1-6956

1-5758

1-3210

1-222

1-3706

1-443

1-5178

1-6802

1-5746

1-3270

1-221

1-3706

1-453

1-5292

1-6832

1-5784

1-3284

1-219

1-8720

1-458

1-5370

1-6782

1-5824

1-3356

1-214

1-3686

1-458

1-5446

1-6716

1-5786

1-3412

1-212

1-3576

1-4G4

1-5480

1-6558

1-5786

1-3430

1-211

1-3552

1-464

1-5576

1-6676

1-5704

1-3436

1-210

1-3552

1-464

1-5576

1-6788

1-5664

1-3426

1-209

1-3676

1-466

1-5532

1-6772

15674

1-3470

1-212

1-3660

1-467

1-5532

1-6720

1-5704

1-3500

1206

1-3750

1-478

1-5460

1-6612

1-5720

1-3554

1-208

1-3906

1-484

1-5292

1-5756

1-3540

1-208

1-4006

1-489

1-5166

1-5756

1-3524

1-205

1-4122

1-498

1-5050

1-5692

1-3530

1-206

1-4094

1-507

1-4768

1-5528

1-3530

1-3976

1-512

1-4584

1-5420

1-3556

1-3882

1-504

1-4430

1-3504

1-8910

1-511

1-4360

1-3440

1-518

1-4316

1-3356

1-4316

1-3240

1-4280

1-3240

MDCCCLXXXVIII. — A.

3 A

362

MESSRS. R. T. GLAZEBROOK AND T. C. FITZPATRTCK

The mean length of the threads and the corresponding values of p. are given in
Table IV.

The agreement in the two values is sufficiently satisfactory.

TABLE IV.

Mean length of

Mean length of

No. of tube.

mercury column,

c-

mercury column,

f-

Value of it used.

in inches.

in inches.

VI.

1-459

1-00044

1-228

1-00046

1-00045

VIII.

1-593

1-00079

1-436

1-00078

1-00079

II.

1-661

1-00045

• •

f t

1-00045

V.

1-564

1-00016

1-484

1-00015

1-00016

IX.

1725

1-00009

1-335

1-00012

1-00010

I.

1-214

1-00002

. ,

,

1-00002

III.

1-386

1-00032

• •

1-00032

To find the average cross section of the tube a thread of mercury almost filling the
tube was used. In nearly all cases this was the same thread as was used in finding
the resistance. The extreme length (/) of the thread between its curved ends was
found with the reading microscopes and the standard scale, while the curvature of the
ends was found by reading the distance between the end of the meniscus and the
point in which the mercury touched the glass. For a given tube this distance did
not vary very greatly in most of the observations, and, with one exception, VI. (2),
the mean value has been taken in calculating the correction 81. The method of
determining this correction has already been given. The temperature was observed
by means of a thermometer laid alongside the tube, which was generally left in
position for some hours before observations commenced. The microscopes and ther-
mometer were then read at intervals of about 15 minutes, and, when two or more
consecutive values for both were found to be the same, it was supposed that the
temperature of the mercury was that given by the thermometer. This was verified
in several instances by taking the temperature of the mercury after it had been
allowed to run from the tube into the small crucible in which it was weighed.

As has already been stated, the mercury thread did not entirely fill the tube, and,
in consequence, a small correction was needed. The amount of this correction is

; — 1 ), s being the mean cross section, and ^ the mean of the cross sections

at the ends.

To determine the ratio s/51 a point was found on the tube from the calibration
experiments at which the actual cross section was equal to the mean value. A short
thread of mercury, some 3 to 4 mm. in length, was introduced, and its length
measured when its middle point was at the point of mean cross section : let this
length be I. The thread was then moved to one end of the tube and its length

ON THE SPECIFIC RESISTANCE OF MERCURY.

in various poMtiuns close up to the end. The same process was repeated
at the other end : let the mean of the lengths thus found be ^. Then, if we suppose
that the curvature of the meniscus did not alter (and experiment showed that this
assumption is nearly true), we have very approximately the equality

Thus, in tube VI., which is slightly smaller at both ends than it is in its central
part, the value of 7 is 3'63 mm., while the length of the thread was the same at each
end, and was equal to 378 mm. For this tube the mean value of (L — J)/L was
•003G7, and the average value of the correction '000150. It is larger in the case of
this tube than with any other used, for ija most the effect of one end was opposite to
that of the other.

Table V. gives the values of the lengths of the threads at the two ends respectively,
the mean of these two or llt and of / for the various tubes.

TABLE V.

Tuba

Length at one end.

Length at second end.

A-

I.

VI.

3-78

3-78

3-78

363

VIII.

4-72

4-33

4-53

4-58

JI.

8-68

8-87

3-75

3-87

V.

3-90

3-89

3-90

3-94

IX.

3-48

353

3-51

345

I.

Tube

uniform

III.

5-68

5-56

5-62

5-5C

The mercury was weighed, using a balance by OERTLINO and the weights employed
by Lord RAYLEIGH ; these were compared with each other and with a set of weights
which Mr. SHAW had previously compared with the standard 500-gramme weight of
the Laboratory. A small correction of about '1 milligramme on 10 grammes was
found, and has been introduced, but it is too small to affect our results.

The mercury was weighed in both pans of the balance, and the weighings repeated
on two or more different days.

In the electrical measurements the tubes were compared directly by CAREY
FOSTER'S method with the B.A. standards, using the bridge designed by Dr. FLEMING
which was employed by Lord RAYLEIGH. The tubes had been so adjusted that the
difference between them amounted only to a few centimetres, at most 70 divisions of
the bridge wire. We were thus independent of variations in the resistance of the
wire due to temperature changes, and the value of the bridge wire division was taken
as -0000498 B.A. unit.

The ends of the mercury tube were connected, in a manner to be described shortly,

3 A 2

3G4

MESSRS. B. T. GLAZEBROOK AND T. C. FITZPATRICK

to the bridge by copper rods, and rods of the same material and almost the same
resistance were used to connect the standard coils. It was hoped in this manner to
compensate the effect produced on the resistance of these rods by changes of tempera-
ture in the room. Since the difference between the two sets of rods was only
equivalent to one bridge wire division, this was completely secured.

The coils used were the following : — For the tubes VI. and VIII. the standard F
was employed ; for V. and IX. the standards F and G in multiple arc ; and for I. and
III. F, G, and Flat in multiple arc.

In the case of the legal ohm, tube II., a coil of 100 B.A. units, ELLIOTT No. 68,
was placed in multiple arc with the tube, and the difference between the combination
and F was found in the usual way. The temperature of the water baths in which
the coils were placed was taken with a thermometer which had been compared at
Kew, and the necessary corrections applied. The temperature of the baths and of the
room in which the experiments were made never differed greatly from 10°. The
following Table VI. gives the values of the coils at 10°, with their temperature
coefficients. The value of F and G are taken from Dr. FLEMING'S chart. The differ-
ence between the two coils at the time of the observations was determined and found
to agree exactly with that given by the chart. In the case of Flat, which is not one
of the six coils mentioned on p. 352, repeated observation during the last two years
has shown that it is now slightly lower relatively to the others than when examined
by Dr. FLEMING ; the change is not greater than '0001 B. A., unit, and is probably due
to a slight imperfection in the insulation. We have taken the value relative to F and
G given by our own observations ; as Flat is only used in multiple arc with F and G
for the tubes I. and III., any uncertainty in its value is divided by nine in the result,
and the error introduced is too small to trouble us.

The coil of 100 B.A. units is one of the standards of the Association, and, like the
other coils, is of platinum-silver wire.

TABLE VI

Coil.

Value at 10".

Coefficient.

F

•99807

•000272

G

•99778

•000263

Flat

•99857

•000277

No. 68

99'847

•0270

For the one unit tubes the terminals of F dipped into mercury cups on ebonite,
which were connected to the bridge by the copper rods above mentioned. When the
coil G was used its terminals dipped into the same cups, and for the one-third unit
tubes I. and III., the coil Flat dipped into two other cups connected with the first
two by thick pieces of copper. The resistance of these connexions was determined

ON THE Kl'KCIFKJ RESISTANCE OF MERCURY. 365

by finding the difference between Flat and F directly, and then when Flat was con-
nected to the bridge by these copper pieces. In this way we found the resistance of
the connexions to be '00136 B.A. unit. The temperature was about the same as
that at which the connexions were used. In the values of Flat given in Table VIII.
the resistance of these connexions has been included.

In all cases the temperature of the room was almost exactly the same as that of
the wuter baths.

One of the ebonite cups into which the ends of the tubes opened is shown in figs. I
and 2, Plate 19, which is drawn to scale full size. In their design two points mainly
were attended to. The first was, that it should be possible to reduce the mercury in
them very nearly to 0° C. ; the second, that there should be no contact between
copper and mercury, for BENOIT has shown that the conductivity of mercury is in a
very short time appreciably increased by contact with copper.

The glass tube passes through an india-rubber cork, which fits into the terminal at
c, G! ; the tube was usually adjusted so that its end was flush with the inner surface
of the terminal.

Mercury was then poured into the cup and allowed to run slowly through the tube
into the second terminal until each was about two-thirds full. The top shown in
fig. 1 was then placed over the terminal and secured by four small screw-bolts passing
through the flange a, a. When these were screwed down the terminals were com-
pletely water-tight, and could be left covered with melting ice or water for days
without leakage.

The top consists of a flat plate of ebonite, with four holes to receive the bolts.
Through this plate two ebonite tubes, dd, ee (figs. 1 and 3) pass. A hollow platinum
cup, /, about 3 '5 cm. long by rather more than 1 cm. in diameter, is secured firmly
into the tube d, d, and a thick piece of copper rod, g, fits the interior of the cup
tightly, any interstices between the two being filled with mercury ; the surface of
the copper rod was well amalgamated. This copper rod, g, is brazed to the copper
rods, g ', which form the connexion with the bridge. Pieces of india-rubber tubing,
hh, kk, about 10 cm. long, are fastened over the upper ends of the tubes, dd, ee ;
the connexion to the bridge passes through a cork, which closes the upper end of
the tube, hh, and the junction is made water-tight with marine glue. A thermo-
meter, t, graduated to fifths of a degree Centigrade, passes through ee and the
india-rubber tube, kk, which fits it closely, and gives the temperature of the mercury
in the terminal.

In taking the observations the ice was packed closely round the terminals up to
the tops of the tubes, h and k, so that the copper rods were surrounded by ice for
12 or 14 cm. above the level of the mercury. Contact between the copper and the
mercury in the terminals was thus established through the platinum cup, f\ the
surface of this is about 12 sq. cm. This surface was amalgamated in the following
manner : — The cups were platinised by electrolysis from a solution of platinic

866 MESSES. B. T. GLAZEBROOK AND T. C. F1TZPATKICK

chloride in nitric acid. On immersing the cups in mercury, after heating to drive off
traces of the acid, amalgamation readily took place.

After this process thoroughly good contact between the platinum and the mercury
was secured. To test this, the two platinum cups were placed in the same vessel of
mercury, and the ends of the copper rods connected with the bridga In this position
the resistance of the connexion was measured, and gave the same value before and
after the experiments, while no appreciable change could be noticed on taking one of
the platinum cups out of the mercury and again replacing it. When only about one-
third of each platinum cup was in the mercury, the resistance was increased by about
•00004 B.A. unit. In use care was taken to place sufficient mercury in the terminals
to cover the cups entirely. The following additional experiment shows the goodness of
the contact : The platinum cups were placed in the mercury, and the resistance
measured as described ; then the copper rods, g, were removed from the interior of the
cups and placed in the same vessel of mercury, the other ends of the copper rods
being in connexion with the bridge, and the resistance was again measured. No
difference between these two measurements could be detected.

Thus, the contact through the platinum was practically as good as if the copper
rods, g, had dipped directly into the mercury in the terminals.

Tables VII. and VIII. give details as to the mercury employed. The general
method of treatment was as follows : — Mercury from the ordinary stock in the
Laboratory was treated with nitric acid and potash, and then distilled in vacuo in the
Laboratory still. Ihis mercury after being once distilled was again mixed with nitric
acid, being allowed to stand overnight in contact with it. After this it was heated
with caustic potash, and then well washed and dried by being strongly heated, and
finally it was passed through a second still, newly set up for the purpose, in which
only mercury which had been previously distilled and treated as above was ever placed.

ON THE SPECIFIC RESISTANCE OP MERCURT.

1
•3

4

^ S

T2

I I I I I I I I I

i - -i i-

— j —

r r s

•-'f,

-i .•:

888888 888^8 8

•2

0 <N >« <N oj do o

-

r s r

I t S I

CO

S I S

>I GC CO ^^ »-« O
-l :: -l -l -i -l

<M <N <M <M Ol OOOO

«O i/5 i5 ip >h "O Cl 30 C5 Oi C5 OIOIOIO)

«b «b ?b «b «b «b o o o o o eb cb co eo

_ ,H _ _i ,_( ^H Ol Ol Ol Ol Ol i-l--!-!^

i-HO!WiO«O(X) i-l Ol CO Ui IN. ^-1 Ol CO -f

?^a,

a 00
Ol "

8 s r

"2-p j ja

M

»

71 ~l

- ~

Ol O> Ol CO i

fference
ecn »tan-
l :in.l R.

il

0>

•

• ? i I r. i -

5883 8

i *O »O SO *O

> CO CO CO OO

~ ". '.

IJJ

Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi Oi

•8*

Date of
lectrical
servation.

i«-« ip C* <M •* t> »H •«* 00 »O

>JJ| f-4 I-OM f"4 P4 Clf-H^HfH^H ^H^f^HrH

O O> ~ «/i Ol

55

CO

O CO O Oi •

00 »O CO O

01 Ci ^H ^H

i Ol OT '31 Oi O O ^r- <

> Ol Cl Oi O* O O 'C* <

) Oi Oi Oi Oi O O Oi *

to

CO

SO >C O

CJ -• —i

Ci 00 00

OiOr^CO^f <N ^ •— t '
^-iOt>»OO O1O1CC'

^•H 'V^ T* ^^ ^~* ' —.1.1 'V^ ^^ WV V» .'W Vrf

888888 88888 S3SS

t — ^ft if »O O ^^ O O^ CO 1C CO t*«" i^ w

it^r^O*-*^ SD (^ w 01 £ Oi-^-^^7

} t*^ t^ OO OO 00 00 I1* t^ t^ '30 t^ CD 00 00

i O> Oi Oi Oi Oi O5 Oi Oi Oi Oi O* Oi Ov O«

i Oi Oi Oi Oi Oi Oi Oi O. Oi Oi Oi Oi Oi Oi

« r^ 00 CJC5 >O
0 ^H do do 61 0 o

•c •* •«? —i

t 6, s :. r

I-H ei eo •*

> s =

•p J3 ja *O na js tJ X X "g .

'" gjgl*^

a c •" •_

01 CO"*®

368 MESSRS. R. T. GLAZEBROOK AND T. C. FITZPATRICK

Tables VJI. and VIII., which contain the results of the observations, do not require
much explanation. The one unit tubes are given in Table VII. In tube VI. filling
2, the curvature of the ends of the mercury column was much less than in any other
case, and the correction 81/1 has, therefore, been calculated specially for this tube
The result is clearly too high, but there is no reason in the details of the measure-
ments for omitting it. It may be noticed that the results of the 4th and 7th fillings
are not given in the Table. These will be referred to again shortly ; they were fillings
of a special character, and it seemed best to treat them separately.

The extreme difference between any two fillings is '000135, and the difference
between the mean result and the extreme is about half of this. In some of the
columns the figures have been given to six places ; the only object in this is to secure
accuracy in the fifth figure in the final result.

In the Table for tube VIII., the fourth and sixth fillings are omitted for a similar
reason to that given above. The extreme difference between any two fillings is about
the same as in the case of VI. The mercury used in VIII. 5 was taken directly
from the iron bottle in which it was supplied by Messrs. TUBES and WILKINS, and
was not distilled in the Laboratory. In the first filling of VIII., and the first and
third fillings of VI., the filling from which the cross section is determined was different
from that used to determine the resistance. This was due to the fact that at first
the necessity of having the tube almost completely full when finding I was not fully
appreciated, and too much mercury was allowed to escape from the ends in removing
the terminals. In the other cases in this table the two fillings were the same. The
mean value for tube II. is reduced by the result of the 2nd filling, which is clearly too
low, but the observations themselves do not give us any reason for rejecting it. All
the fillings for this tube have been included.

The mean value of r found from these three tubes is '95354 B.A. unit, and the
greatest difference between any two observations is '000241 B.A. unit. The average
error independent of sign is '00005.

If, however, we rejected the results of VI. 2 and II. 2, the mean would hardly be
affected, but the mean error would be reduced to '000037.

We may, therefore, fairly put as the result of the observations on the one unit
tubes r = '95354 ± '00004. This is, it will be observed, the value given by
tube VIII. The value for VI. is raised unduly by the observation VI. 2 ; that for
II. is unduly lowered by the result of II. 2.

ON THE SPECIFir RESIST A XC'K «K MKRCURY.

l-H

a

jo-

CO 00 '.•» 00 OS >r>

•66 66. 66 6 •-"

85

70

•000402

•0011:':'

,- — • .* «

-
-

n

S3

559

8 ' 8 8 8

) 1-1 "*> «O CD (

» l-H OS OO l-H I
I CO «O CO OO <

00

88 88

CO <O

°™ K 6

CO

i>.qo
°obob 6 6s

O CO
OJ t>

B

s

'

OS

P

8 s

S 2

2-9

0-9

*n||» jo 'OK

i

1 ••

r.

^5 00 Oi ^f

CO CO CO CO

V> <M OO rH rH l-H ~H
^H —I i-H r-l OS OS OO

ts. ?O ^O ^O (N O
-* "5 00 CO ^H t^

0>OS S

.'« - -
•N -N ^H I-H IN <_ .

CO CO CO CO CO CO

ujj «

-t

^H IN I-H <N

Q t. ; t4 .* • s

M1MV( |. \\\V|||. A.

3 B

II

OS

i

. r.

111

ij.g

• - ~

ti _n

Is

" '

••i.wid pua
jo 'dmaj,

C0-«f> COCO — CO CO-*

-

32

IS

8

<N 65 TO

9 88

.: •-

g

!§

s£

CO CO

ll

ill"1

— r
.- .-.
co »i

I-H ^H

ss

)FH 0000

II II

"-•

!•>. to O5 '

OJ CO CO I

33 3:

jo -dmaj,

> •* -*00
><N <N — •

> pH ^H CO

: : 66 6-H

s

E "

SS 2^ 2?

US OS OS WS

/ —

$i

jo -draaj,

OOO5

66

,,.-„

•5

r

O - O =

00

> 00 TO <

jo

M 's< «^ * S s
^^ *^

•sll

jr — — .

• «

•» o

I

370 MESSRS. R. T. GLAZEBROOK AND T. C. FITZPATRICK

Table VIII. gives the results of the observations on the one-half and one-third unit
tubes. One other filling of I., which is not given, was taken, but it was clear that
some accidental error had been made in the measurement of I or of W, for the result
differed from the others by over one in a thousand. This is the only filling in the
whole series which has been entirely rejected. The result of V. 1 is clearly too low.

If we give the result of all the fillings in Table VIII. equal weight, we obtain
as the mean value r = '95342 B.A. unit, and the average error is '000029.

The difference between this and the result derived from the one unit tubes is, at
first sight, considerable, but it must be remembered that for the tubes I. and III. an
error in the measurement of resistance of '00003 or one thirty-thousandth of a B.A.
unit gives rise to an error of one ten-thousandth in the result. This, of course, is an
extremely small quantity. Moreover, the small uncertainty which attaches to the
corrections SL and 81 would produce a larger effect in these large tubes, and our
observations tend to show that the coefficient "82 in the value for SL is possibly
rather too great.

It will be noticed, however, that the values of R, the actual observed resistance of
a tube, differ among themselves by extremely small quantities.

In taking a final mean, it was clearly unfair to weight the observations equally, and
we came to the conclusion that the probable accuracy was roughly inversely propor-
tional to the area of the cross section. We have, therefore, attached the weights 3,
2, and 1 to the results from the one unit, half unit, and third unit tubes respectively,
and arrive at the final result that the resistance of a column of mercury 1 metre long,
1 square millimetre in section, at 0° is

•95352 B.A. unit.

If we give equal weights to all the observations, the result will be '95351, so that
the effect of the weighting is hardly appreciable.

It remains to consider the four special fillings of tubes VI. and VIII., which have
been omitted. In two of these, VI. 4 and VIII. 4, the attempt was made to fill the
tube quite full, when measuring I ; a very small bubble of mercury was left protruding
from each end of the tube when it was placed under the reading microscopes, and
then flat pieces of brass were brought up simultaneously against the ends, there being
a layer of thin paper between the brass and the mercury. It was hoped in this way
to squeeze out the superfluous mercury and leave a column with flat ends exactly
filling the tube. It seems probable in the case of VI. 4 that this was successfully
accomplished, for the value of r found from the experiment is '95354 B.A. unit.

With VIII. 4, however, it was clear, on looking through the microscopes at the
mercury column, that the ends in contact with the paper were slightly curved, and
this was still more obvious when the brass and paper were removed. The resulting
value of r is accordingly too low, being '95342 B.A. unit.

In the case of the two fillings, VI. 7 and VIII. 6, mercury was used which had

ON THE SPECIFIC RESISTANCE OF MERCURY.

been passed once through the still at the University Chemical Laboratory, and then
treated with nitric acid. It was clear, from the appearance of the mercury, that it
was impure ; but it was thought of interest to determine a value for mercury in the
purifying of which no special trouble had been taken. The impurity shows itself at
once in the results, for, while the mean value of R for VI. is '99996, for this filling
R = -99989; while for VIII. the resistance of the filling is '99990 B.A. unit, against a
mean of 1 '00006. The corresponding values for r are '95348 and '95329.

This mercury was then treated with nitric acid, &c., redistilled in our own still, and
used again in the fillings VI. 8 and VIII. 7 ; the values of R were '99996 and
1-00004, and of r '95360 and '95361 respectively.

Thus, the impurity has been clearly removed by the distillation and acid treatment.

In some cases a tube was filled on one afternoon, and its resistance determined. The
whole was then allowed to stand over night, being re-pucked in ice in the morning,
and the resistance again measured, but no appreciable change was noted. Thus, for
VI., on January 9, the value '99994 was found, while the same filling, re-packed on
January 10, after the mercury had stood for 16 or 18 hours in contact with the
platinum of the terminals, gave R = '99996 ; the difference is within the temperature
errors of the coils.

Some experiments were made on the effect of known impurities on the mercury in
altering its resistance. In one case, about one two-thousandth part of zinc filings was
added to the mercury. On mixing, the surface of the mercury was made foul ; the
mercury was then passed through a filter paper and used in VL , but the effect on the
resistance was not appreciable. It is probable, of course, that the filtering had
removed a large portion of the zinc, but the experiment gives some idea of the
amount of impurity which the resistance measures will detect.

In another filling of VL, a mixture of mercury with a small percentage of tin was
used. The resistance was much too small to be measured on the bridge — the bridge
wire has a resistance of '05 B.A. unit approximately. This mercury was treated in
the usual way, distilled in the Laboratory still, and then in our own, and on being
again used gave as the value for R '99991 B.A. unit. Thus our treatment was
sufficient to remove the tin from the mercury.

Some observations were also made on the change of resistance of mercury with
temperature.

Thus, on January 5, tube II. was placed in a trough in water at about the tempera-
ture of the room, and its resistance measured. The tube was then packed in ice and
measured. Similar observations were made with tube III., and the results are given
in the Table IX. Other observations confirmed the results there given.

3 B 2

372

MESSRS. R. T. GLAZEBROOK AND T. C. F1TZPATRICK

TABLE IX.

Date.

Tube.

Temperature.

Value.

Value at 0*.

Coefficient

January 3 ...
„ 9 ...

II.

III.

9-5
10-5

1-02028
•33297

1-01186
•32993

•000878
•000875

We conclude finally from the experiments that the value of

r is '95352 B.A. unit.

If we take as the value of the B.A. unit the mean of those found at the Cavendish
Laboratory, we have

1 B.A. unit = -98667 Ohm,
we find that

r = 94081 Ohm,

or 1 ohm is equal to resistance at 0° C. of a column of mercury 1 square millimetre in
area and

106*29 centimetres in length.

These values agree closely with those communicated at Manchester to the British
Association by Professor ROWLAND, viz. :

r = -95349 B.A. unit.
1 ohm = 106 '32 centimetres of mercury at 0° C.

The value of r in B.A. units does not differ greatly from that found at Wurzburg by
STRECKER and by KOHLRAUSCH ;* the difference, however, is greater than can be
accounted for by error of experiment, but is, I think, capable of easy explanation.

STRECKER'S comparison of his mercury tubes with one of the B. A. units sent from
Cambridge was made at a temperature of from 9C to 10°"5 (' WIEDEMANN, Annalen,'
vol. 25, p. 482). The resistance of the mercury at this temperature was reduced to
0° by means of his own formula (loc. cit., p. 474), which gives a mean coefficient up to
10° of '000909. Now, this is a larger value than is given by any other observer, as is
shown in the following Table : —

* See the Table on page 352.

ON THE SPECIFIC IIKS1SI \.NVl-; Of MKUrrUY.

373

TAI-.I.I: X.

STBKCKIR*

SlIMKNSf
LoKENZj

LENZ§
BBNOIT ||

GLAZIBBOOK** ......

GLAZEBKOOK and FITZPATBICK ft •

Mean

Avenge coefficient up to 10°.

•000909
•000865
•000901
•000884
•000877
•000861
•000861
•000876

•000879

Thus, STRECKER'S value is higher than the mean by '00003, and if we were to
reduce his observations from 10° to 0°, using the mean coefficient given above, we
should obtain the value for r, '95362.

This value may possibly be a little too high, but at any rate the reasoning is suffi-
cient to show that the difference may easily depend on a small error in the tempera-
ture coefficient.

The same reasoning will apply to KOHLRAUSCH'S results, for his comparisons
between the mercury tubes and the wire standards were usually made at temperatures
differing from 0°, and were reduced to 0° by the use of STRECKER'S formula.

The value of the ohm in centimetres of mercury at 0°, as given by KOHLRAUSCH in
a letter to R. T. G., February 16, 1888, is 106'32, agreeing exactly with ROWLAND.

MASCART, NERVILLE, and BENOIT found a value for r which is as much above our
value as KOHLRAUSCH is below it. At the same time, their value of the B.A. unit in
ohms is less than ours, leading to the result that the value of the ohm in centimetres
of mercury is 106 '33 ; or, again, the same value as ROWLAND'S. This might seem to
show that there was some small change in the B.A. unit used by MASCART between
the time it was compared at Cambridge and the date of their observations.

[Quite recently, June 4, 1888, M. WUILLEUMIER communicated to the Academy of
Sciences at Paris the results of some experiments by LIPPMANN'S method, which give
the value 106 '27.]

Thus we may conclude that the experiments of MASCART, STRECKER, ROWLAND,
KOHLRAUSCH, WUILLEUMIER, and ourselves are in fairly close agreement, and that

* ' WIKDEIIANN, Annalen,' vol. 25, p. 475.

t ' Electrotechn. Zeitschr.,' vol. 3, 1882, p. 408.

J ' WIEDEMANN, Annalen,' vol. 25, p. 11.

§ ' Etudes Electrometrologiques,' vol. 2, 1884.

|| " Resume d'cxperiencea sur la determination de 1'ohra," 'Journal de Physique,' 1884, p. 230.

f ' Phil. Trans.,' 1883, p. 185.

*• ' Phil. Mag.,' October, 1885, p. 352.

ft Supra, p. 372.

374 MESSRS. R. T. GLAZEBROOK AND T. C. FITZPATRICK

the value of the ohm expressed in centimetres of mercury at 0° does not differ from
IOG'31 by more than '02 of a centimetre, or two in ten thousand.

From this result the values found by LORENZ and Lord RAYLEIGH differ appre-
ciably. With regard to LORENZ'S value, we may notice that the comparison between
his tubes and the B.A. unit was very far from being direct.

«/

The tubes were compared with a Siemens' unit issued by SIEMENS and HALSKE, and
this with a copy of the B.A. units sent to LORENZ by Lord RAYLEIGH. The tempera-
ture coefficients of the two coils and of the mercury required to be known, and
corrections introduced. The final value found by LORENZ for the ohm in centimetres
of mercury is 105 '93.

In this determination, tubes were used 1 metre long, and 1 , 2, and 3 centimetres in
diameter, and some part of the large difference may possibly be due to the fact that
the lines of flow of the current near the ends of the tubes can hardly have been
cylindrical.

No such explanation, however, can be offered of the difference between Lord
RAYLEIGH'S result and our own. His comparisons were direct, and the results of
the observations on the various tubes employed are extremely concordant ; the tubes
actually used by him have since been broken, but the end pieces are still at the
Laboratory. We thought it was worth while to fit up one of our tubes, No. VI., with
his end pieces, and find its resistance. In this way, we were able to eliminate any
error which might have occurred in the resistance of connexions, as new connexions
were made for the purpose and had their resistance specially determined.

The experiment was made on May 19th, and the value found, when the temperature
in the terminals was about 3°, was

1 -00000 B. A. unit.

The mean value for VI. previously found, the temperature in the end pieces being
lc'4 C., was

•99996 B.A. unit.

Thus, this experiment fully confirms the value we had already used, and shows that
no error can have been introduced by the connexions. Lord RAYLEIGH has himself
pointed out that the fact that the temperature of the mercury in his terminal cups
was from 5° to 6° C. would lead to an over-estimate of the value of r, and he con-
cludes that this over-estimate may in his case have been as much as '00008.

In the course of our observations we had several times determined the resistance of
a tube as the mercury in the end pieces cooled down from 9° or 10°, the temperature
of the room, to the temperature at which it was finally steady, which was on the
average about 1°'4 C.

Table XI. gives the results of these determinations for VI. Each horizontal line
refers to the same filling. Taking the resistance when the mercury in the end pieces
was at 2° as 1 B.A. unit, the Table gives the temperatures at which the resistances
were measui-ed, and the increase of the resistance of the tube up to the temperature
in question.

ON THK SPECIFIC RESISTANCE OP MERCURY.

TABLE XL

875

Temperature

2°'8

4°-5

- s

In -I

Increase of resistance above
that at 2° ...

•00004

•00010

•00035

•00043

Temperature

2°-9

4°-6

9°

Increase of resistance above
that at 2° ....

•00003

•00010

•00035

Temperature

.; -,;

Increase of resistance above
that at 2°

•00020

,; 4

Increase of resistance above
that at 2°

•00025

4°

5°-5

Increase of resistance above
that at 2°

•00010

•00020

pni
w*

376 ON THE SPECIFIC RESISTANCE OF MERCURY.

The results of the Table are represented graphically in fig. 3, in which the abscissse
represent temperature and the ordinates resistance, the temperature being that
indicated by the thermometers in the cups. The tube was of course packed in the ice
during these observations. It would appear from the curve that our own observations
may be slightly too high, possibly as much as '00004, through the temperature in the
cups being on the average 1°'4 0. instead of 0°, while at 6° an error of about '00024
might be introduced. This error is equivalent to that caused by the whole tube being
at 0°'3 instead of at 0°, or by about 5 per cent, of the tube being at the temperature
of the mercury in the terminals. In tube VI. some 6 or 7 per cent, of the tube was
within the corks used to close the terminals. It may be noticed that the observations
on tube VI. given in the last line of Table XI. were made with Lord RAYLEIGH'S
terminals. We thus infer that, while the fact that in Lord RAYLEIGH'S experiments
the terminals were at 5° or 6° may explain a small part of the difference between our
results, reducing his by about '00024, it cannot possibly account for the whole,
amounting as it does to '00060, and we must look in some other direction for the
explanation.

[ 377 J

XV. Invariants, Covariants, and Quotient- Derivatives associated with Linear

Differential Equations.

By A. R. FORSYTH, M.A., F.R.S., Fellow of Trinity Colleyp, Cambridge.

Received January 7,— Read January 12, 1888.

THE present Memoir deals with a set of invariants and covariants of linear differential
equations of general order. The set is proved to be complete, that is to say, every
covariantive function of the same type can be expressed as a function of the members
of the set, the only operations necessary for this expression being purely algebraical
operations. The transformations, to which the differential equations are subjected,
are supposed to be the most general consistent with the maintenance of their order
and their linear character ; they are, linear transformation of the dependent variable
and arbitrary transformations of the independent variable. The covariantive property
of the functions considered is constituted by the condition that, when the same
functions are formed for the transformed equation, they are equal to the functions for
the original equation, save as to a factor of the form (dz/dxy, where z and x are the
two independent variables.

The memoir, with the exception of a single and rather important digression, is
occupied solely with investigations of the forms of the functions, of their interdepen-
dence, and of methods of construction. The earlier part deals chiefly with the
synthetic derivation of the functions, the later part with their analytic derivation.
Tables of the functions have not been calculated ; in most cases the expressions of
the functions are given in their forms as associated with the differential equation
when it is taken in an implicitly general canonical form, and only in very few cases
are functions given in connexion with an explicitly general form. Within these
limits the subject of the memoir has been strictly confined ; there is not, for instance,
any attempt at classification of differential equations of the same order as discrimi-
nated by forms and values of invariants or covariants.

The contents of the memoir are as follows : —

The first section gives references to previous writers on the subject, viz., COCKLE,
LAGUERRE, BRIOSCHI, MALET, and HALPHEN ; and, in particular, some of the results
obtained by HALPHEN in his well-known essay and in a subsequent memoir are stated.
It appears that previous results are confined to invariants, and that, with the
exception of two special invariants of the general equation, the invariants obtained
are not derived for equations of order higher than the fourth. In order to connect

MDCCCLXXXVm. — A. 3 C 9.11.88

378 MB. A. R. FORSYTE ON INVARIANTS, CO VARIANTS, AND QUOTIENT-

my results with those previously obtained, there is given at the end of the section a
very short statement of the kinds of covariantive functions which are here introduced.

In the second section there are given the general relations between the coefficients
of a linear equation before and after it is subjected to the most general transforma-
tion. From these relations the value of the invariant ®3 is deduced ; a method is
indicated which leads to the values of ©4, ©5, ®6, 67 ; and it is proved that, for the
first general form of differential equation adopted, there are n — 2 fundamental
invariants, each of which consists of two parts : — (i.) a part linear in the coefficients
and their derivatives ; (ii.) a part, not linear, every term of which contains at least
one factor which is either the algebraic coefficient of the term next but one below the
highest in order in the differential equation, or is a derivative of that algebraic
coefficient. A canonical form of the differential equation is adopted, the reduction to
which is possible by the solution of an equation of the second order ; for this canonical
form the second part of each of the fundamental invariants vanishes. Finally, the
expression of these invariants in their canonical form is given.

In the third section two processes of deducing invariants from those already found
are obtained, called the quadriderivative and the Jacobian ; and it is proved that all
the algebraically independent invariants which can be deduced by these processes may
be arranged in classes according to their degrees in the coefficients of the differential
equation. The first class is constituted by the n — 2 priminvariants of the second
section ; the second class contains n — 2 quadriderivatives of these priminvariants
and n — 3 independent Jacobians ; and each succeeding class contains n — 2 proper
invariants. In the course of the section several propositions are proved which lead to
this selection of proper invariants.

In the fourth section it is shown, by the application of CLEBSCH'S theorems as to
the classes of variables which arise in connexion with the concomitants of algebraical
quantics in any number of variables, that there are in all n — 2 dependent variables,
associated with the original dependent variable of the differential equation, and
distinct in character from one another. The complete set of n — 1 dependent
variables are subject to similar linear transformations ; and at the end of the section
some properties of the linear equations satisfied by them are inferred.

In the fifth section the quadriderivative and Jacobian processes are applied to the
dependent variables, original and associate, which possess the invariantive property ;
and it is proved that there are two classes of independent co variants, viz., those
which involve each one dependent variable and its derivatives only, and those which
are Jacobians of a single invariant and each of the dependent variables in turn. A
limitation on the former class, according as they are considered associated with a
differential equation or a differential quantic, is pointed out ; and a symbolical
differential expression is obtained for each of the proper derived invariants and
derived covariants.

In the sixth section some illustrations of the theorems already proved are given,

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS 379

by applying them to equations of the lowest orders. When they are applied to the
equation of the second order, they give the theorems already obtained by RUMMER
and SCHWARZ. When they are applied to the equation of the third order, the
canonical form of which is binomial and which has a single primin variant, the adjoint
equation is derived ; and the case of a vanishing priminvariant is discussed from two
points of view. The quotient-equation of the cubic, that is, the differential equation
satisfied by the quotient of two linearly independent solutions of the cubic, is worked
out ; and the primitive of the cubic is deduced from a supposed knowledge of two
special solutions of this quotient-equation, in a form which is the analogue of the
corresponding results for the quadratic. In this connexion the cubic quotient-
derivative occurs, corresponding to the Schwarzian derivative ; it is one of a series of
similar functions. For the equation of the fourth order two canonical forms are given,
one being the special case of the general canonical form, the other being a more direct
analogue of the canonical form of an algebraic binary quartic. The quotient-equation
is deduced and some properties are proved ; and the quartic quotient-derivative is
obtained. Finally, the two associate equations of the quartic are given ; and there
is a verification that all the priminvariants (and hence all the concomitants) of these
associate equations are expressible in terms of the invariants (and hence of the
covariants) of the quartic.

The seventh section is really a digression from the main subject of the memoir ;
some of the properties of the quotient-derivatives of odd order are therein investigated,
the two principal relations being that which is consequent on the general quotient
transformation of the dependent and the independent variables, and that which gives
the homographic transformation of both variables. These quotient-derivatives have
some connexion with reciprocants ; but, on account of the restriction on the subject
of the memoir, there is here no investigation of that connexion. Quotient-derivatives
of even order are obtained from different forms of linear equations ; and a relation
between the two kinds of derivatives is indicated.

The eighth and last section is mainly devoted to a proof of the functional com-
pleteness of the concomitants of the second, third, and fifth sections. There is a
homographic transformation of the independent variable, which changes one canonical
form into another ; and the method of infinitesimal variation is used in connexion
with this transformation to obtain the characteristic linear partial differential equations
satisfied by any concomitant. They are found to be two in number ; one of them is
an equation which determines the form of a concomitant, the other determines the
index of the concomitant when its form is known. These characteristic equations
are first applied to deduce the covariants which involve the original variable,
and next to deduce the invariants derived from 83 ; and simplified forms of the
invariants and covariants of higher grade are obtained. Finally, there is given a
general proof, founded on the theory of linear partial differential equations,* that

* This method has already been applied by Mr. HAMMOND to the corresponding proposition in the
theory of binary quantics ; see ' Amor. Jonrn. Math.,' vol. 5, 1882, pp. 218-227.

3 C 2

380 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOT1ENT-

every concomitant can be expressed as an algebraical function of the concomitants
which have already been obtained, and that their aggregate is therefore functionally
complete.

The following is a tabular index of the contents of the paragraphs of the Memoir: —

SECT. I. — 1-3. Analogy between transformations of algebraical equations and of differential equations.

4. Results of MALET and COCKLE on sem in variants of differential equations.

5. Invariants given by LAGUERRE and BRIOSCHI.

6, 7. HALPHEN'S results in his memoirs of 1880 and 1883.

8. Difference of canonical form chosen by HALPHEN from that in this memoir.
9, 10. Statement of the aggregate of invariantive functions.
SECT. II. — 11, 12. General transformation of a linear differential equation.

13. The invariant due to LAGUERRE and BRIOSCHI.

14. Definition of index of invariant.

15-17. Dimension-number and homogeneity of invariants.

18. Proper invariants.

19, 20. Method of infinitesimal variation; modification of relations of § 11.
21-25. The invariants 63, 64, 96, eg, 67.

26. Number of non-composite linear independent invariants.
27, 28. Form of the linear invariants.
29, 30. Canonical form of differential equation.

31. General value of Qa for this canonical form.

32. Summary of results of section.

SECT. III. — 33, 34. Quadrinvariants ; the quadriderivative process.
35, 36. Jacobians ; aggregate of proper qnadrinvariants.

37. Invariants of the qnartic.
38-40. Proper cubinvariants.
41—43 ; 44. Proper quartinvariants.

45. Proposition relating to Jacobians.

46. General conclusion as to quartinvariants.

47. Proper quintinvariants.

48. General propositions.

49. Aggregate of derived invariants.

50. Semi-canonical form of invariants.

51. Finality of results.

SECT. IV. — 52. LAGRANGE'S " equation adjointe " of the equation of order n.
53, 54. Sets of variables subject to same linear transformation.

55. CLEBSCH'S theorem on classes of variables in algebraical quantics.

56. Application of CLEBSCH'S theorem to sets of variables.

57, 58. Selection of those variables which are also subject to functional transformation.

59. Dependent variables associate with original dependent variable.

60. Digression on invariants.

61, 62. Inferences as to the variables and the linear equations satisfied by them.
SECT. V. — 63-65. Derived identical covariants in the original variable.

66. Limitation on their number when associated with a differential equation.

67. Restricted limitation on their number when associated with a differential quantic.

68. Derived identical covariants in the associate variables.

69. Mixed Jacobians.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 381

SICT. V. — 70. General propositions relating to dependence of mixed covarianta.
(cont.) 71. Inference as to covariants which involve only the original of the dependent variables.
72, 73. Mixed covariants involving the associate variables.

74. Aggregate of covarianta.
75-78. Limitation on number of identical covariants for qoartic and qnintic.

79. Symbolical expression for Jacobian derivatives.
SECT. VI. — 80. Application to equation of second order.

81. Reduction of cubic to canonical form.

82. The " adjoint " equation of the cubic.
83, 84. Simple integrable forms of the cubic.

85. Quotient-equation for the cubic.

86. Primitive of the cubic when two particular solutions of its quotient-equation are

known.

87. Case of vanishing priminvariant ; cubic quotient-derivative.

88. Particular integrable case of cubic.

89. Primitive of cubic when only one solution of quotient-equation is known.

90. Special equation in Schwarzian derivatives.

91. Reduction of quartic to general canonical form.
92, 93. Alternative canonical form of the quartic.

94. Quotient-equation for the quartic.

95. Primitive of the quotient-equation when three solutions are known.

96. Quartic quotient-derivative.

97. Primitive of qnartic when three solutions of its quotient-equation are known.
98-100. Case of vanishing invariants.

101. Solution of quartic indicated when only one solution of its quartic equation is known.

102. Adjoint-equation of the qnartic.

103. Self-adjoint associate equation of the qnartic.

104. Form of this equation when the priminvariant O3 of the quartic vanishes, with a

particular example. .

105. General form of the self-adjoint associate equation.

106. Determinantal relation among the self-complementary variables.

107. Verification of the theorem that all the priminvariants of the self-adjoint associate

equation are invariants of the original qnartic.

108. Other associate equations.

109. Theorems relating to the general equation of order n.
SKCT. VII. — 110. Quotient-derivatives of successive odd orders.

111. Quotient-derivatives and reciprocants.
112, 113. Law of transformation of vanishing quotient-derivatives ; converse not necessarily

true.
114-117. Transformation of quotient-derivatives for homographic variation of independent

variable.

118. Illustration of limitation on converse of § 112.

119. Derivation of quotient-derivatives of even order.

120. The hyper-linear quotient-derivative.

121. The hyper-quadratic quotient-derivative.

122. Scheme of quotient-derivatives of successive odd orders.

123. Relation between derivatives of odd and of even orders.

SKCT. VIII. — 124. Reproduction of canonical form of differential equation for homographio transforma-
tion of the independent variable.

382 MR. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-
SECT. VIII. — 125, 126. Reduction of this transformation by method of infinitesimal variation.
(cent.) 127-129. Characteristic form-equation and index-equation satisfied by concomitants.

130. Deductions from these characteristic equations.

131. Application to identical covariants.

132, 133. Simplification of "proper" identical covariants.

134. Application to derived invariants.

135, 136. Application to the Jacobian of a priminvariant and a derived invariant
associated with that priminvariant.

137. Proof that the set of concomitants obtained in §§ II., III., V. are functionally

complete.

138. General limitation in the number of identical covariants associated with a

differential equation.

SECTION I.

HISTORICAL INTRODUCTION.

1. Similarity in properties of differential equations and of algebraical equations
has long been of great value, both in the development of the theory and in the
indication of methods of practical solution of the former equations. In recent years a
great extension of this similarity has been made by the discovery of certain functions
associated with linear differential equations which are analogous to the invariants of
algebraical quantics ; and, principally owing to the investigations of M. HALPHEN,
this extension has had an important influence on the theory of cubic and quartic
equations and on the recognition of fresh integrable forms of equations.

2. The most general modification of the form of an algebraical equation, without
causing any change in its order, is that which arises by the application of TSCHIRN-
HAUSEN'S transformation ; the effect of it is that, by the satisfaction of certain
subsidiary equations, the coefficients of terms in the transformed equation are
evanescent, and these terms are therefore annihilated. There exist in the trans-
forming relation a number of constants, taken in the first instance to be arbitrary,
and subsequently determined by the subsidiary equations, which, however, do not in
cases of high order always admit of possible algebraical solution ; and the two
simplest cases are those in which the transforming relation is lineo-linear and lineo-
quadratic.

Now, in the case of linear differential equations, transformed without change of
order, there is an exact analogue of the lineo-linear relation just mentioned, whereby
the term involving the differential coefficient of order next to the highest is made to
disappear. If x and y denote the independent and the dependent variables respec-
tively, the relation is of the form

y = uf(x) = u\,

where u is a new dependent variable and X is determined by an equation of the first

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 383

order. The analogue of the lineo-quadratic relatiou apparently does not exist ; but
another equally effective transformation of the differential equation is possible, being
that whereby the independent variable is changed. And, by a proper transformation
to a new variable z, concurrently with the former change of the dependent variable,
though with a different multiplier f (x), it is possible to remove the terms which
involve the two differential coefficients of order next below the highest which occurs.
This has been known for some time, having been pointed out, first apparently (in
1876) by COCKLE, and afterwards (in 1879), independently, by LAQUERRE.

3. Here would seem to be the limit in this regard to the analogy between alge-
braical and differential equations ; but within the limit there are striking properties
in common. It is well known that when the proper lineo-linear transformation is
applied to an algebraical equation so as to remove the term next to the highest, the
remaining coefficients are the algebraical coefficients of the leading terms of HERMITE'S
covariants associated with the quantic which is the sinister of the equation ; and
these coefficients are therefore seminvariants. An exactly similar property holds for
differential equations, but its full recognition has only been gradual The following
are, so far as I can discover, the chief references to this part of the subject, and,
though a chronological order is avoided, they will serve to indicate the development.

4. In a memoir entitled " On a Class of Invariants," * Professor MALET obtained,
and applied to the solution of special questions connected with the cubic and quartic,
two classes of seminvariants of differential equations ; one of these is invariantive
for change of the dependent variable, the other for change of the independent
variable. And though, to obtain the form of the latter he has used the two kinds of
transformation successively, he has not apparently obtained in a direct form functions
which possess the invariantive property for both transformations. Soon after the
appearance of this paper, and in connexion with it, Mr. HARLEY t proved that
Professor MALET had been anticipated by Sir JAMES COCKLE, who had in several
memoirs (exact references are given by Mr. HARLEY) given in forms, sometimes
explicit and sometimes implicit, the leading results obtained by Professor MALET
relating to the seminvariants of the two classes. At the end of his paper Mr. HARLEY
states that, in a recent letter, Sir JAMES COCKLE had suggested the possibility of
forming " ultra-critical" functions, i.e., functions invariantive for both transformations
effected concurrently.

5. In this last suggestion, which is not stated to have been worked out to a definite
issue, Sir JAMBS COCKLE has been anticipated by M. LAOUERRE, who in two notes J
gave what is here called the fundamental invariant of the cubic, but without any

* • Phil Trans.,' 1882, pp. 751-776.

t " Professor MALET'S Classes of Invariants identified with Sir JAMES COCKLE'S CriticoidB," 'Roy. Soc.
Proc.,' vol. 38, 1884, pp. 44-57.

J " Sur les equations differentielles lineaires du troisieme ordre," ' Comptes Rendns,' vol. 88, 1879,
pp. 116-119 : " Sur quelqnes invariants des equations differentielles lineaires," ibid., pp. 224-227.

384 MR. A. R. FORSYTE ON INVARIANTS, CO VARIANTS, AND QUOTIENT-

indication of his method of obtaining it ; he also gave the first of the two classes of
semin variants. Almost immediately after the appearance of these notes Professor
BRIOSCHI communicated in a letter * to M. LAGUERRE a method of obtaining the
invariantive results and of extending them, which, applied to the cubic and quartic,
led to explicit expressions for the invariants of both equations ; and the invariantive
property of the functions is constituted by the relation that if <j) (p, dp/dx, . . .) be
the function for the original equation with coefficients p, and <f> (q, dq/dz, . . .) be
the same function for the transformed equation with coefficients q, an equation of the
form

(!)"*<*§• •.•>-*<*£->

is satisfied. There is a premature conclusion as to the permanence of form of these
functions for equations of all orders, the corrected expression of which is given later
in the present memoir (§ 28).

6. The two notes of M. LAGUERRE and the letter of Professor BRIOSCHI are the
suggestive starting point of M. HALPHEN'S investigations in invariants, which occupy
part of his extremely valuable memoir.t So far as the invariants, qua theory of
forms, are concerned, the leading investigations are contained in the third chapter.
He there points out the functional identity of the invariants of LAGUERRE and
BRIOSCHI with functions previously (in July, 1878) obtained by himself |; the
connexion between absolute and relative invariants is derived a priori; and the
necessary limitation on the form of invariants arising from homogeneity in weight is
deduced. A method is indicated, potentially suitable for the formation of invariants,
by connecting the general linear equation with the linear equation of the second
order ; the fundamental invariant of weight 3 — the same as for the cubic — is derived
and its permanence of form for equations of all orders is pointed out ; but, except this
and the invariant of weight 4 for the quartic, no others are calculated. In fact, the
method involves extremely difficult analysis for any but the simplest cases ; and even
for the invariant of weight 3 an invariantive property of LAGRANGE'S " Equation
adjoin te" is used in addition. The rest of the memoir is devoted to the application
of these results. For this purpose, the author takes his general differential equation
in a definite canonical form so chosen that the term of order next to the highest does
not appear and the invariant of weight 3 is unity — two relations which suffice to
determine the new independent variable and the multiplier of the dependent variable.
The applications, leading to most important deductions, chiefly concern the general

* " Sur les Equations differentielles lineaires," ' Bulletin de la Society Mathemat. de France,' vol. 7,
1879, pp. 105-108.

t " Memoire sur la reduction des Equations differeutielles lineaires anx formes integrables." ' Memoires
des Savants fitrangers,' vol. 28, No. 1, 301 pp. (Grand Prix des Sciences Mathe'matiques, annee 1880;
published 1882).

t In his Doctor's Thesis ' Sur les invariants differentials.' Paris, 1878.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 385

cubic and a limited form of the quartic for which the invariant of weight 3 vanishes
identically, and in the case of which the new independent variable is determined by
taking the fundamental invariant of weight 4 to be unity ; in the notation of this
memoir such a quartic would assume the binomial form

7. In a subsequent memoir,* M. HALPHEN considers the general quartic and its
invariants, which he identifies (p. 330, I.e.) with the differential invariants of tortuous
curves. And he deduces (p. 339, I.e.) from the two fundamental invariants
(v, «7 = B3, 84, in my notation to a numerical factor pres) the series of successive
invariants, which are the successive " Jacobian derivatives " herein obtained.

The following investigations were completed before I knew any of the details of
this last- quoted memoir by M. HALPHEN, my starting point having been M. BRIOSCHI'S
letter ; and, though the results relating to the form of the Jacobian series for the
quartic are thus anticipated by four years, it does not seem necessary to modify the
investigations which relate to the equation of general order n possessing n — 2
fundamental invariants.

8. The great advantage of the canonical form chosen by M. HALPHEN is that a
given equation can be reduced to it by means of differential equations of soluble
form of the first order only — that is, their dependent variables can be explicitly deter-
mined as functions of the independent variable, though the functions may not be
evaluable in known forms ; but there is an attendant disadvantage from the point of
view of the invariants that their expressions, even for the canonical form, remain
complicated. In preference to M. HALPHEN'S canonical form I choose that from
which the two terms of order next to the highest are absent, and the reduction to
which is always possible by the solution of a linear differential equation of the second
order. The great advantage of this, as the canonical form, is that, when the invariants
— at first called fundamental and subsequently priminvariants on account of the
property about to be mentioned — are constructed for this form, they are purely linear
functions of its coefficients and their derivatives, with the further essential property
that the expression of each is independent of the order of the equation, so that, in
fact, each is an invariant of every equation of order not less than its index.

9. The number of these priminvariants is n — 2 ; from them there are constructed
the series of what have been called derived invariants, which include Jacobian and
quadriderivative functions ; and in the aggregate only those are retained which are
proper or non-composite. All these functions are entitled invariants.

There is then indicated a set of dependent variables associated with the dependent
variable of the given equation, the last one of which set is the variable in the

• " Snr les invariants des equations diflterentiellea linruircs du qnatri&me ordre." ' Acta Math.,' vol. 3,
1883. pp. 325-380.

MDCCCLXXXVIir. — A. 3 D

386 MR. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

" Equation adjointe " of LAGUANGE ; they are all transformable by a substitution
similar to that which transforms the original dependent variable, viz., multiplication
by some power of dz/dx ; and they possess the property that all combinations of them,
similar to those by which they are constructed, are expressible explicitly in terms of
variables of the set. From this set of dependent variables there are deduced functions
of them and their differential coefficients possessing the invariantive property ; and,
again, from the aggregate all composite functions are excluded. These functions are
entitled identical covariants.

Finally there is obtained a third class of functions possessing the invariantive
property, and involving in their expressions the dependent variables and the
coefficients of the differential equation ; and those functions are excluded from the
aggregate, which can be algebraically compounded by means of functions occurring
earlier in the class, of invariants, and of identical covariants. These functions are
entitled mixed covariants.

For purposes of simplicity and of distinction between these classes of functions
there is an advantage in considering, as the ground form, a differential quantic
(being the sinister of the differential equation) rather than the differential equation
itself; for, in the case of identical covariants of order equal to and greater than that
of the equation satisfied by the variable in question, they can by means of the
equation be changed into mixed covariants. It is necessary to mention this both
here and later when the functions occur ; but, beyond this mention, further notice is
not taken of the possible fusion of the two classes of functions.

10. The general aggregate of concomitants of the differential equation is taken as
including these three classes of functions, and later in the memoir it is shown to be
complete ; and the expression of every function is only implicitly general, that is, it is
given in connexion with the canonical form of the equation. A few of the prim-
invariants of lowest index are given for a semi-canonical form, but these are the only
exceptions. Again, my aim has been the investigation of invariautive forms from the
purely algebraical or functional point of view, and not from the geometrical ; I have
nowhere in this part adverted to SYLVESTER'S Reciprocants. The identity of some
classes of the latter with HALPHEN'S Differential Invariants is known*, and thus the
three species of covariantive functions constituted by Differential Invariants, Recipro-
cants, and Invariants of Differential Quantics have known points of connexion.
The discovery of further relations between them would be of great interest and
value.

* SYLVESTER, " On the Method of Reciprocants as containing an exhaustive theory of the Singularities
of Curves," « Nature,' vol. 33, 1886, p. 227.

I>i;ilIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 387

SECTION II.

PRIMINVARIANTS OP A LINEAR DIFFERENTIAL EQUATION.
Transformation of the Differential Equating

11. The general linear differential equation of the 71th order is of the form

rf-Y rf-'Y «(«-!) rf-Y

+ ~¥T ~ *

where RQ, Rlf R2, . . . are functions of x ; by the substitution

and subsequent division throughout by R0, it is changed to

where P2, P3, . . . are seminvariants of the former equation.

Similarly, an equation which determines another dependent variable u as a function
of another independent variable z may be written in the form

n\ „ d*~*u nl

Suppose now that these dependent variables are so connected that the relation

y = «x . . ......... (i)

is satisfied, X being some function of x. In order that (L) may be transformable into
(ii.), z must be some function of x; and when this is the case there will be a number of
equations, evidently u in number, connecting X, z, x, and the two sets of coefficients
P and Q, which may be obtained as follows. The actual substitution of «X for y
in (1) gives

rf«M dX (fr-Ht n! <iPX (fr-*u

= dj? + " dx <k»-f "*~ 2 ! n - 2 ! rfx« <**•-» T. ', ! .'

"-«« , tt rfX rf»-»

-x + ^-2)rf

3 D 2

388 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

In this equation it is necessary to change the independent variable from z to x, in
order to compare it with (ii). We have*

_ ^.

•",-! •! dr"
where

(Jm

A.Mt, = Limit, when p = 0, of -— [<j> (x -f- p) — <f> (x) }'
and

(2).

It at once follows that, writing m ! Cw>, for A«t, and denoting differential coefficients
sometimes by dashes (chiefly when there are powers higher than unity) and sometimes
bj Roman numeral indices, we have

CMi, = ==* = coefficient of pm in {<f> (x + p) - $ (x)}'

ItV i

i.e., in (p^ + ^p^ + i.p^ +...)' ...... (3).

Substituting now in the semi-transformed equation, we immediately find the
coefficient of d'u/dz' -f- s ! to be

ro! d"-*X

2ln-2l Z «-2.' «-3.' ' ' ' sin -s -21

r = n-> n\ ft = n-r-,

'

- r -

^

rr0 if,

with the symbolical interpretations P0 = 1, Pj = 0. But the present form of the
equation must be effectively the same as (ii.), and the coefficients of corresponding
derivatives of u must therefore be proportional to one another. In the transformed
equation the coefficient of frujdz" -£• n 1 is (here s = n, so that r = 0 and t = 0 are
the only values for terms in the summation) A^X, or the coefficient of d"u/dz* is
A»_,X/n ! ; that is, it is Xz'*. Hence we have

TO

* ScHLailiLCH, ' Vorlesungen iiber einzelne Theile der hoheren Analysis' (3e Auflage, 1879), p. 5.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 389
or, what is the same thing,

XQ.-. ,„ _ '-;-

»-« r.o i.o r

Now, let

W* = 5? + 2!0-2!P' rf**-'

. .. . .- . , . . .' (4)
symbolically, then the coefficient of

n — r— tl
in the foregoing expression is

r

r!«I 3?

the summation extending to those values of r and t that leave r + t the same
throughout, that is, the coefficient is

r + t'.
and, therefore,

J -'. _ *

n-s! ,,0 n-0\e\

When * is changed into n — * and the quantities C are introduced from (3), this
takes the form

If it were desirable the summation on the right-hand side might be extended to the
value 6 = n, for C,^,,. vanishes if m < m',

12. The only place where, as yet, the zero value -of Pl has been admitted is in the
definition of W,, but no essential use of that value has been made ; now, however, it
will be found that the removal of the terms involving Pt and Ql from (i.) and (ii.)
materially simplifies the analysis. Writing (iii.) in detail for the lowest values
of s and using the condition Qt = 0, we have in succession

(5)-,

(6)',

and so on, the number of equations being «.

390 MR. A. R. FORSYTE ON INVARfANTS, COVARIANTS, AND QUOTIEXT-

13. Tlie first invariant. — In the particular cases of n = 3 and n = 4, Biuosciii lias
shown (/. c. § 5) that there is a function of the coefficients such that

remarking that the invariantive forms remain the same for differential equations of
higher order ; and HALPHEN has, for the general equation, obtained this invariant by
another process. Before passing to more general investigations it is easy to see that
this result follows from equations (5)', (6)', (7)' ; and the deduction of it requires
modifications of those equations which are subsequently of great use. We have,
from (4),

W0 = X , W1 = X';
and from (3)

p — l(n _1\»'«-ZZ" P — z'«-l

\J»>n-\ — IV* Llz > ^«— 1, «— 1- »

while, generally,

n — z'» •

^m, m — z »

so that (5)' now is

0 = 2'X' + i (w - 1) Xz" . . . •* . v . , (5),

an integrated form of which will be subsequently taken. Writing with BRIOSCHI

we have

2" =2'Z,

21U = Z' (Z' + Z2),

X' = -i(n-l)XZ,

X" = -i(«-l)X{Z'-i(H-i)Z2},
Xui = - ^ (n - 1)X,{Z" - f (n - 1)ZZ' + i(n - 1)2Z3}.
Again,

C., ._» = A (n - 2) 2'-* {4»'^ + 3 (n - 3) 2"8}

= h (n - 2) 2'"~2 { 4Z/ + (3» - 5) Z2} ;

C..!, »_2 = i (n - 2) 2'"-32" = 1 (n - 2) 2'»-*Z,
by means of which (6)' changes to

£ Xz'2Q2 = -3\- (n - 2) (4Z' + (3»» - 5) Z2} + i (n - 2) ZX' + i (X" + P2X),
which, on substitution for X" and X', reduces to

2Z' - Z2 = - (P, - Q/2) ._.,.,., .(<p) _. (G).

DERIVATIVES ASSOCIATED WITH LINEAR DI I I KIIENTIAL EQUATIONS. 391

Similarly

C,, ,_8 = A (« ~ 3) «'-• (22^- + 4 (n - 4) *W + (n - 4) (n - 5) *"»}

= A (» ~ 3) *'"~8 (2Z* + (4n - 10) ZZ' + (n» - 5n + 6) Z8} ;
C._u ._, = A (n - 3) z'-5 {4zV« + 3 (n - 4) 2"3}
= aL4 (" ~ 3) 2'-8 {4Z' + (3u - 8) Z-} ;
C._8, „_, = $ (n - 3) «'-* 2" = fc (n - 3) z'-8 Z.

By means of these (7)' changes to

£ Xz-'Qa = ^ (n - 3) X {2Z" + (4n - 10) ZZ' + (n* - 5n + 6) Z8}

+ A (» - 3 ) { 4Z' + ( 3 n - 8 ) Z2 } X' + £ . £ ( n - 3 ) Z (\" + Pa\)

reducing on substitution for X'", X", X' to

Equations (6) and (7) agree with the equations given by BRIOSCHI for the case of
the cubic (n = 3) and that of the quartic (n = 4) ; the elimination of Z between them
is in process precisely similar to that in those special cases, and it leads to the result

14. It appears from this investigation and from the results of BRIOSCHI and
HALPHEN that there are rational integral functions of the coefficients of the diffe-
rential equation and their derivatives such that, when the same function is formed
for the transformed differential equation, the two functions are equal save as to an
integral positive power of z. These functions are called invariants ; the exponent of
the power of 2' may be allied the index of the invariant

Dimension- Number ; Homogeneity.

15. The index of an invariant can easily be settled by the following considerations.
We can assign to each coefficient of the differential equation a certain number, called
for this purpose its dimension-number, suggested by the similarity with the theory of
dimensions of homogeneous functions.* For the present the dependent variable y will

* This process is practically identical with M. HALFBKN'S assignation of weight ; see above, Historical
Introduction, § 6.

392 MH. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-

not have a definite number assigned to it, but will have associated with it an arbitrary
number m ; to dyjdx we assign a number m — 1, to d^y/dy? a number m — 2, and so
on up to dnyjdx*, to which TO — n is assigned. Now, if to Pr we assign a number — r,
the number assignable to Pr d"~r y/dx*~r is m — (n — r) + ( — ?•) = m — n, and is,
therefore, the same for all values of r, i e., for all terms in the differential equation ;
and, consistently with these arrangements, the number to be assigned to dfPr/dxf
is — p — r.

In exactly the same way and by the same rules we can similarly assign numbers to
the coefficients Q and derivatives of these coefficients, and, as before, leave the number
assigned to the dependent yariable arbitrary.

16. If now an invariantive function of the kind spoken of in the last paragraph be
denoted by 0 (x) when formed from the coefficients P, and therefore by 0 (z) when
formed from the coefficients Q, the invariantive relation is of the form

e (z) zv = 0 (x). ,:::,

An equation of this form can exist only if

(i.) Every term on one side has. according to the foregoing assignation of numbers,
one and the same dimension-number, and similarly for every term on the
other side ; and
(ii.) The two sides have the same dimension-number for the variable x, and the

same dimension-number for the variable z.

The first of these conditions requires a certain kind of homogeneity in the function
0, examples of which will immediately be given ; the second of the conditions
determines the index //,. For let — cr be the dimension-number of 0 (x), which may*
be written 0, (x) ; then — cr is also the dimension-number of ©„ (z), these two
numbers being respective multiples of the units implicitly assigned to x and z respec-
tively. Consistently with the assignation of dimension-numbers, the quantity z must
be considered as having a number -|- 1 assigned to it in virtue of its dependence on z
and a number — 1 assigned to it in virtue of its dependence on x. Hence the
z-dimension-number of 0,, (z) zv is p. — cr, and its x-dimension-number is — /A, while
the corresponding numbers of 0, (x) are respectively 0 and — cr. The second of the

conditions requires

p. - cr = 0,

both of which are satisfied by /* = cr. Hence the invariantive functions are such that

where 0, (x} is a function of the coefficients P and their derivatives such that every
term in the function has one and the same dimension-number — cr.

17. The following examples will illustrate these general explanations. The quan-

DERIVATIVES ASSOCIATED WITH LINEAR DIFFKKKXTIAL EQUATIONS. 898

titiea which have a dimension-number — 3 are P3 and dP»/dx, and therefore the
general form of 83 is

(the coefficients A, ... being constants throughout) ; those which have a dimension-
nuraber — 4 are Pi? dP^/dx, d^P^/dx"-, P./, and therefore the general form of »t is

those which have a dimension-number — 5 are P5, dPJdx, d^P^/dx2,
PSP3, and P3 dPcJdx, and therefore the general form of B5 is

RjJ PSJ r'j FPP FPS-
Brf* + '~dJ+D~dJ+*'P*P3+ P81ZT'

and so on.

18. It is evident that the product of two functions 6,, 8,, is a function of the type
6,+,.. A. composite function of this kind, resoluble into the product of two functions
with lower indices, will not be considered as properly associated with the dimension-
number — (<r + <r'). The functions will be supposed ranged in order with increasing
index, and a composite function may thus be considered as included in the aggregate
of earlier functions. The method of determination of the invariants 8,, will appear to
be practically founded on the solution of a partial differential equation of the first
order, as is usuil with all invariantive functions of any nature ; and, as would be
expected when the most general possible form of Sf is adopted so as to determine the
assumed constants, composite functions of index p will occur associated with undeter-
minable arbitrary constants. For simplicity of calculation, it would therefore appear
desirable to exclude from 6, all terms which occur disjunctively in the aggregate
^<r®,>-»» but owing to the form of the implicit partial differential equation to be
satisfied this is not completely possible ; what proves to be possible, as will be seen
later, for the adequate determination of a non-composite function 8, is that terms

and terms, of course, of the dimension-number — p, involving as factors either P2 or
some derivative of Pa or combinations of them, alone need be considered.

We now proceed to what is practically the formation of the partial differential
equation, deriving it by a generalisation of the ordinary method of infinitesimal
variation which is used to obtain the characteristic differential equations satisfied by
concomitants of algebraical qualities. The general characteristic equation is not
explicitly given on account of its complicated form ; it is implicitly given in all the
particular cases, and its principal use is to obtain the numerical coefficients of the
different functions 8.

MDCCCLXXXVIII.— A. 3 E

394 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT

Modification of the Coefficient Relations (III.).
19. By integration of equation (5), we have

Xz' i(»-D = constant;

since the equations (iii) are homogeneous in the dimensions of X, this constant may
be taken of any arbitrary value other than zero, and so we may write

(iv).

With the form of equation (ii) adopted, this is the only equation which helps to
determine X and z ; and therefore we may consider z as arbitrary and, when an arbitrary
value is assigned, X is determinate. We therefore assume

2 = a; + e/t, ........... (9)

where c, an infinitesimal constant, is to be considered so small that squares and higher
powers may be neglected, and p. is an arbitrary non-constant function of a;*. From this
it at once follows that .

z' = 1 + £/, .......... (10)

and, for values of k greater than unity,

while from (10) and (iv) it follows that

X=l-i(n-l)v', .... . . . . (11)

and

d*-*\ . .

__=_i(n_1)

Hence

W0 = X = 1 - i(n - 1) V'; Wj =

and by (4), for values of r greater than unity,

; W, = Pr - i(n - 1)6 (l, 0, Po, .

= Pr-|(n-l)eTr
say. Also by (3) we have

* The functions are shown by this process to be invariants only for an infinitesimal, but otherwise
perfectly general, transformation ; but the immediate purpose is to obtain the numerical coefficients and
not to prove the property of general invariance, which, otherwise known, could be derived by the
principle of cumulative variations.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 395

C... = coefficient of p~ in (p? + i, pV + $, />VU + . . . y

{p + «(W' + i PY + i, pV" + ...))'

and therefore

C

while for values of m greater than s

When all these values are substituted in (iii.), it becomes

(n-l)e *-• P,

and, therefore, dividing each side by the coefficient of Q, and retaining only first
powers of e, we have

= P,(i- V)

after a slight reduction. This equation is true for the values s = 2, 3, . . . , n ; and
particular cases, to be used immediately, are

Qs = P8 (1 - 3cM') - 3^UPS - i (n + 1) e^,

Q4 = P4 (1 - 4C/x') - 6eM«P3 - (n + 5) ^P, - ^ (n + 1) «/*',

Q6 = P6 (1 - 5£/A«) - 1 Oe/x«P4 - |(n + 7) V«P8 - | (n + 3) eM"P2 - | (n

Q, = P6 (I - 6CAt') - 15e/A«P8 - | (n + 9) V»P4 - 5 (n +

Q7 = P7 (1 - 7,/x1) - 21 VP, - S (»+ 1 1)V«P, - V (n f

- V (n + 3) €M'P3 - 7 (n + 2) e^P2 - f (n + 1 )
3 £ 2

390 MR, A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

20. But for our present purpose we require, not merely the expressions (12) for the
coefficients Q in terms of P, but also expressions for the derivatives of different orders
of the quantities Q. Writing (12) in the form

so that we may, to the order of email quantities retained, differentiate <£, with regard
to z or x indifferently, we have

dsf ~ ' dsf C daf '
But as before (§ 1 L ) we have

JrV m = r T} /7»>P

"> ri ^ -Pr.M <* * »

<fcf "~ m = i m! da?" '
where

B,t,

«. . e , • f dx 1 , d*x 1

= coefficient of Pr in \PJZ + 2iP*& + V.

Now we have

dx 1
- = j

that is,

cfa; „

3?" 77;

to the first order of small quantities, and similarly for the differential coefficients ol
higher orders ; whence

-^ = coefficient of pr in {p — c (/D/A* + -j, p2^ + ^, p3/iui +•••)}"»

and, therefore, •

7T == l ~

while, for values of r greater than m,

Br _ em

rl r — m + 1! dxr~m+l

When these values are substituted, the equation for drP,/dzr becomes

d'V, "' = '-' r! d*P, d

, _ , » ,

' ~r^'~'

dsf ' 'df m= , 7/1 — 1 ! r - m + 1!

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 397
and, therefore,

Jr(\
ll W»

~ " ' ~ ~ ., m-l!r-m

It will be noticed that the coefficient of the first term is 1 — (r + *) e/i1. The
quantity d'Qt/dzr, which has a dimension number — (r + *), will be multiplied by z'1"*"'
in the invariantive equation, i.e., by 1 + (r + s) e/*1 ; and the first term in the product
of the two will thus be d'YJdyf. The correctness of this result furnishes a slight
indirect verification of (13).

As particular examples of (13), corresponding to those of (12), we have

* (I - V) - 3e/x«P3 - 8f (,»•?,) i (n + 1)

398 MR. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

-

f 4 f1

(» + 5)

- e

'+ 3) ^

+ 5 (n + 4) ^P3 + | (3n + 7) /*'P8} - A (n + 1) e/.TiU.

21. We now proceed to construct the functions ®4, 05, ©fi, @7 ; the value of 03 has
ahready (§ 13) been found, and may be taken in the form

;'V<i+..>f-i'M- e,= Ps-|. •;:^-'-.-^-:'/ . (H)

Calculation of the Invariants ®3, <s>4, @5, ®6, 07.

22. The invariant 04.

The most general form possible is

and the invariantive equation is

0,(a;) = ^
= (1

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 399
Now

V)

and the values of the remaining quantities have already been given ; when they are
substituted in the equation and the factor — e is removed, the equation becomes

0 = A {e/iT, + (n + 5)/t"P§ + A(n + 1) ft'} + £D(n + i)M»p§

which is satisfied identically, provided

B=-2A,

Hence, taking A to be unity, we have

e4 = P4-2f-' + f^-|^P88, ..... (t5)

dx * da? * n+ 1

which practically agrees with BRIOSCHI'S function y (I. c., p. 107) for the value n = 4,
the order of the equation in connexion with which the function is obtained.

23. The invariant 05.

The most general form possible is

Proceeding exactly as in the last case, it is easily found that the conditions
necessary for the identical satisfaction of the invariantive relation are

B=- | A,
C= -VLA>

D=- | A,

'=

And, therefore, taking A to be unity as before, we have

400 MB. A. R. FOBSYTH ON INVARIANTS, COVARIANT3, AND QUOTIENT-

24. 77ie invariant 6fi.

The most general form possible is

Proceeding as before, it is found that the necessary conditions are

. _B ___ 3C_.3D^ 14E_ n+1 F n + 1 G re + 1 7J

= " = ~ ~ ~

— 3 - 5 5 3n + 7 5 ~ 3» + 7 10 ~ 14?i + 31 10 '

3K + L = 0,

. 7n_+_8
n + 1

R _ so A 7n3 + 28n + 25
(n + I)8

These conditions leave two constants undetermined ; they may be taken to be A and
L. When the values of the others are substituted, it appears that the part of the
function involving L can be expressed in the form

that is, a composite invariant of index 6. As no new function is thereby determined,
we omit it (§ 18) by making L zero ; and then, taking A to be unity, we have

-p OB, 104 53 B. go

~ r« ~ * dx ~ 3 <fe* " 8 do* " 14 da* " 7

25. TAe invariant 07.

Proceeding as in the last three cases from the most general form, we are led to a
number of relations among the assumed constants which leave two of these constants
undetermined. When all the other constants have their values in terms of these two
substituted, it is found that the aggregate of the terms involving one of the constants

DKKIVATIVKS ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 401

M can be expressed in the form MBse4, that is, a composite invariant of index 7. As
this is not a new function, it is omitted, as in the case of 88 ; and the quantity which
involves the other constant is then

3*44f d»P,/ rfl>,\ ,rfP,rfP,l, p21155n* + 6048* + 6909 ,

H"^r 5iT^r ~aa(»+ijF

Invariants such as these which have one part linear in the coefficients of the
differential equation will, for brevity, be called linear invariants.

General Form of Linear Invariants ; Canonical Form of Equation.

26. The last few results suggest a general deduction, which can be derived directly
from the equations (iii.), as to the general form of linear invariants. From those
equations we have

\-'»Q W W

*~ vfr _ "« /-i i YY'-\ p i

• - . ^H— n n—i ~T~ _ 11 '-'«— «+!> *— < "T" • • •

so that,

X (z''Q. - P4) = sP^\' + |« (?i - s) '- P,.^ + terms involving P,_j, P^3, . . . ;

whence, by (8),

*"Q. - P. = *P~ i ( i ( n - s) Z - i ( n - 1 ) Z } + . . .

= — ^s (s — 1) P,_! Z + terms involving P._2, P,_s- • • •
Tims,

z'^Q,-! — P^! —• function involving P^y P^_3, ....
and, therefore,

_ =1 + w'^iQ^jZ = function involving P^.., P._8, . . .
ax

Combining these two, we have
>> {Q- ~ i (« - 1) d^:1} - {P. - i (* - l)d-^r} = ^notion involving P._2,

z

MDOCCLXXXMII. — A. 3 F

402 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

Proceeding in this way and remembering the analogy of the simpler cases of s, we
should be able to gradually reduce the highest index which occurs on the right-hand
side ; but the terms will become more complicated, owing to the successive differen-
tiations that take place. Now the form of the result to which we are to attain is
known, being an invariantive relation ; the successive operations earned out in the
way indicated always decrease the highest index on the right-hand side and will
never re-introduce a coefficient P already eliminated ; hence, the result attainable from
the foregoing starting point is unique, and we are therefore led to the conclusion : —
There is only a single non-composite linear independent invariant for each index from
3 to n, and, therefore, there are in all n — 2 linear invariants.

27. But, again, as the successive steps in the gradual reduction are taken, differential
coefficients of Z of various orders will enter as factors with P,_2, P,_3, . . . and
differential coefficients of these, the function on the right-hand side being always an
integral function of Z and its derivatives. Now, for the latter, we could substitute
from equations like (6) and (7), and others which have not been given ; but every
derivative of Z can be obtained from (6) and from equations deduced from it alone by
differentiation. In that case there would be introduced into the terms containing
P,_2, • • • and Z and its derivatives factors of the form P2 or powers of P2 or deriva-
tives of P2, or combinations of these ; with the result that, when all the operations
are completed so as to leave the invariantive equation, the non-linear terms in 0 (x)
will each contain at least one factor which is either P2 or a power of P2 or a differential
coefficient of P2. Hence each of the non-composite linear independent invariants
consists of two parts :

(a) A part which is linear in the coefficients P and differential coefficients of these
quantities P, each term having the proper dimension -number ;

(b) A part which is of the second and higher degrees in these quantities, each term
having the proper dimension-number, and every term having at least one factor which
is either P2 or some derivative of P2.

These general conclusions are evidently satisfied in the case of the linear invariants
already obtained.

28. It will be proved immediately that the numerical coefficients iu the linear
part are independent of n, the order of the equation ; those of the non-linear part
are not independent of n, as may be seen from the special cases already discussed.
Hence the linear part is the same for equations of all orders not less than the index
of the invariant, but the non-linear part varies from one equation to another ; and
therefore BRIOSCHI'S remark made d propos of 03 for the cubic and quartio and of €)+
for the quartic "que ces formes invariantives restent les memes pour les Equations
differentielles d'ordre supdrieur " applies only to the linear part.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 403

Canonical Form of Differential Etptation and of the Invariants.

29. The conclusion as to the general form of the invariants suggests that a form of
the differential equation might be adopted which would give to tlie invariants an
expression considerably simplified. If it were possible to take Qe to be zero (and this
possibility will be proved immediately), then, when the function 6, (z) is formed from
the coefficients of the transformed equation (ii.), the non-linear part of the function
will cease to appear because every term in this part contains a vanishing factor ; and
the part that remains constitutes the whole of the invariant. Hence, for the differential
equation, thus transformed, the invariant is a purely linear function of its coefficients,
and, in this linear form, the invariant is determinate wheu once the numerical coeffi-
cients are obtained.

30. In order to obtain the transformed differential equation, the invariants for
which have this simple form, we return to the original equations of transformation.
By them, considered as applied to equation (i.), there were two quantities at our
disposal, viz., X and 2. One relation between them has already been assumed in
deducing the equation (5) or (iv.) ; and any other may be taken to completely deter-
mine them, provided it does not violate that already adopted. Such a relation which
is permissible is to suppose the quantities X and z, already subject to (iv.), to be quan-
tities which will make the coefficient of ^""^/t/z""2 in (ii.) zero, that is, make Q3 = 0.
Hence, by (6), we must have

••"""'' : '"V""1' 2Z' = z' + ;rTTp" "•••'• ' : "••• :

where, by (8),

*" 7 2 V

7'- "^rix'

If we write

the equation which determines Z is transformed into

and then we may write

X=^-1, z'=0-2 ........ (20).

Hence, by the solution of a linear differential equation of the second order, as (19),
the two terms of orders next below the highest can be removed from a linear differeu-
tial equation of any order.* This modified form may be called the canonical form of
the differential equation.

* This result has already been referred to, as a general statement (§ 2 ) ; the exact references ore —
COCKLE, ' Quarterly Journal of Mathematics,' vol. 14, 1876, p. 34C, for the cubic;
LAGUKBRK, ' CompHs Rendus,' vol. 88, 1879, p. 226, for the general equation.

3 F 2

404 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-
Since 6 = z'~l, it follows that

which can immediately be verified, [z, x] being the Schwarzian derivative

Determination of the Linear Invariant of Index cr in its Canonical Form.

31. The difference between the linear part of 6ff(z) for a differential equation with
a non-evanescent Q2 coefficient and the whole function 0,, (z) for the equation with an
evanescent Q.2 coefficient lies, not in any difference between the two sets of numerical
coefficients, for passage from the former ©„. to the latter is effected merely by making
Q3 zero, but in the condition, that for the former &ff the independent variable z was
not determined and so could have an arbitrary value assigned, while for the latter this
independent variable is completely determined. In order, therefore, to obtain the
invariant Q^ in its canonical form it will be sufficient to determine the linear part of
6, in its uncanonical form, for which we adopt the same process as in the particular
cases 04, 65, 06, Q7 ; an arbitrary value is assigned to z, nearly equal to x, and the
coefficients of the linear part are determined, the remainder of the terms not being
necessary for a knowledge of the canonical form. To this we pass by retaining the
linear part alone, and the independent variable must then be considered determinate.

We assume

"~
+ ( — l)a~2 B,_2 _2ii + a part which vanishes with Q.2.

For the determination of the ratios of the constants B0, Blf . . . , it is sufficient to
consider the terms involving /A" which occur when the invariantive equation

is transformed by means of the equations (10), (12), (13). From the last two these
terms can at once be selected with the following results : —

(i.) In Qff the term involving p." is

_^eo-(0-_l)/i"P_1;

(ii.) In dQv_l/dz the terms involving /tu are

-e(<r-l)A4"Pff_1-l£(cr-l)(cr-2)^<^; , . -

* See my ' Differential Equations,' p. 92.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 405
(iii.) In d'Qi/dz', where s > 2 < <r — 1 and r +* = <r, the terms involving /*" are

(iv.) In rf'~!Q2/j2*~8 the term involving ft" is

When these quantities are substituted and the coefficients of corresponding terms
are compared, the satisfaction of the equation requires the following relations : —

iB0o-(o--l) = B^cr-l)

iB1(cr-l)(«r-2) = iB3{2(2o--3)}
i B2(o- - 2) (o- - 3) = i B3 {3(2o- - 4)}

B,_3 3 . 2 = i B,_2 {(<r - 2) (o- + 1)}.

The last equation determines B,_2, the coefficient of rf*~2 Qo/Jz*"2, which is not
required for our present purpose, because this term, though linear, will not occur in
the canonical form. The other equations give

B! = ±0- B0,

2 (2<r - 3)

---
'"' 2 . 3 (2<r - 3) (2<r - 4) *'

4 ~~

2 . 3. 4 (20 - 3) (2<r - 4) (2<r - 5) 1J

*~8 ~"

2 . 3 . 4 . . . (<r - 3) (2<r - 3) (2«r - 4) . . . (<r + 2) l-

Hence, taking B0 to be unity and passing to the canonical form, we obtain the non-
composite linear invariant of index <r in the form

r.-V. . . . w,

406 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-
where aj = 1 and, for values of r greater than 1,

(a- - 1) (<r - 2)«Q - 3)« . . . (<r - r + l)»(<r - r)
"r= 2.3...r(2<r-3)(2<r-4)...(2<r-r-l)

By means of this method it would be possible to determine many, if not all, of the
coefficients of the general linear invariant in its uncanonical form ; this investigation,
however, must for the present be deferred.

32. The general results obtained thus far may be stated as follows : —

When the linear differential equation

d'y , r;B _ »' p d*-'y _ 0

«faf "•",., r In- r! r d&-'~ :a

has its dependent variable y transformed to u by the equation

y — u\

and its independent variable changed from x to z, where z and A (which is a function
of x) are determined by the equations

' >=«•-'. !=»"• •• ••.-•

the transformed equation in u is the canonical form

d"u r = * nl ~. d*~ru

d& r^3 rln — r' dz"~r

The coefficients P and Q of these equations are so connected that there exist n — 2
algebraically independent functions ©„(#) of the coefficients P and their derivatives
which are such that, when the same function ©,(2) is formed of the coefficients Q and
their derivatives, the equation

is identically satisfied. The possible values of a- are 3, 4, 5, ...,«; the function
©X2) i8

r = »-S

where ali(r is unity, and for the remainder of the co-efficients a

_ (<r - 1) (<r - 2)* (ff - 3)2 . . . (a - r + I)8 (<r - r)
av'ff~ 2. 3 r(2<r - 3) (2<r - 4) . . . (2<r - r - 1) '

KKIUVATIVKS ASSOCIATED WITH LINEAR DIFFKKKNTI \ L EQUATIONS. 407

so that B,(z) is independent of n ; and therefore any invariant of an equation in its
canonical form is also an invariant of all equations of higher orders when in their
canonical forms.

SECTION III.
DERIVED INVARIANTS OF A LINEAR DIFFERENTIAL EQUATION.

Quadrinvariant*.

33. We are now in a position to construct an infinite number of invariants which
are linearly independent (and also algebraically independent) of one another ; they
are, in the first instance, functionally derived from the n — 2 invariants obtained in
the last section, which may, therefore, be called the fundamental invariants or the
priminvariants of the equation. The method of obtaining the first set of n — 2 new
invariants is that adopted by BRIOSCHI for the cubic.

With the notation already adopted, we have for the general priminvariant

and thence by (8), after taking logarithmic differentials,

and, therefore, also

<rZ'= £ (log e.(x)} - z*' £ {log e,(*M - *'2 £ {log «.(*)! -

Substituting in (6) and writing

*. (*) = 2o- £ {log 9, (x) } - [£ {log 8, (x)} J-^ Pz ,

we have

*"<!>, (2) = *,(*),

80 that <!>„ is a new invariant with index 2 derived from the priminvariant 0, ; and
from every priminvariant such an invariant can be derived. Now, when the equation
is taken in the canonical form, the value of *, (z) is

*. = 2<r £ (logec(2)} - [^ {loge.(z)}J

408 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-
and, therefore, we may take a new invariant

;- (2o- +l)e^2 ....... (vi.)

as the derived invariant in its canonical form. The index of 0,^ is evidently 2(<r + 1) ;
the number of these derived invariants is n — 2 : and it will be convenient to call this
function of 6ffi sometimes the quadriderivative function, sometimes the quadrinvariant,
associated with 0,.

34. All these quadriderivative functions, derived from non-composite invariants, are
algebraically independent of one another ; but there is no a priori necessity that the
quadriderivative function of a composite invariant should be a composite function.
Let such an one be

then the function obtained from it by the foregoing process is

*pil = 2 (x + p.) <DP <*>; - (2\ + 2/t + 1)

so that

Hence, ^>PII cannot be composed from the invariants ah-eady obtained ; but, if we
choose to introduce a new invariant

the index of which is unity, then <£PI, is composite.

35. This new invariant can be otherwise derived. For ©£/©£ is an absolute invariant
(of index zero), and therefore

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 409
whence, after logarithmic differentiation,

and, therefore, the foregoing quantity is, as stated, an invariant of index unity.
Taking it in an integral form, we have as a new invariant

e*.M,i = /*e,,el-xeA«M, -. . . . . . . . (vii.)

of index X -f- p. -\- 1 ; from the similarity of its form with the Jacobian of two binary
qualities, it will be called the Jacobian of 8A and 8,. As we have already seen,
quadriderivatives of composite functions are composite functions when Jacobians are
retained.

36. The number of Jacobians which we need to retain as independent of one
another is diminished by the two following results : —

(A) If either of two functions be composite, their Jacobian is composite. For,
taking

*, = e^e,, (x + P = <r),

we have

<*>„„. , = M ex - (x + P) 8,8,8,,

= ®fif, M, 1 "f- *M*A, * 1.

a composite function.

(B) Of the \(n — 2)(n — 3) Jacobians 8^, derived from the primin variants
only n — 3 are algebraically independent ; for between the three which are
derivable from any three priminvariants 0^, 6,,, 8, we evidently have a
relation

Hence the independent Jacobians, as well as the quadriderivative functions 8ffi „ are
all of the second degree in the coefficients of the differential equation, and may
therefore be called quadrinvariants. And from the foregoing it follows that the
aggregate of " proper," i.e., non-composite, quadrinvariants is composed of two sets,
which are

(i) the n — 2 quadriderivative functions 8,,, given by (vi.) ;

(ii) the £ (n — 2) (n — 3) Jacobians 8^, given by (vii.), of which only n — 3
are algebraically independent of one another.

37. The simplest example that occurs is in the case of the quartic equation of
which the canonical form is

MDCCCLXXXVIII. — A. 3 G

410 MR. A. B. FORSTTH ON INVARIANTS, CO VARIANTS, AND QUOTIENT-
There are two primin variants, viz. : —

©3 =

and there are three proper quadrinvariants, viz. : —

e-u = 6Q3Q"3 - 70?,

«*i = 8 (Q, - 2Q'3) (Q"4 - 2Q'"3) - 9 (Q'4 -2Q"3)',

63.4,1 =

= 4Q4Q'3 - 3Q3Q'4 + 6Q3Q'3 - 8Q?..

And if we choose we can replace any one of these by a linear combination which
includes that one ; thus we could replace @3li,i by ©s,*,! — BM, the value of which is

dz 3 dz

Independently of the special application to the deduction of quadrinvariants, the
preceding analysis shows that, when a number of invariants are given, there are two
methods of forming new invariants, viz., the quadriderivative process and the Jacobian
process.

Cubinvariants.

38. We now proceed to apply these methods to obtain the proper invariants of the
third degree. The quadriderivative process will not produce any invariants of this
degree when applied to any of the invariants already obtained ; and, therefore, all
that remains for us to do, remembering proposition (A) of § 36, is to form the
Jacobians of the priminvariants with the proper quadrinvariants.

39. First, the Jacobian of any priminvariant with a proper quadrinvarianl which
is itself a Jacobian is a composite function.* For, if J denote the Jacobian of 0P and
0Xi (l> u we have

But

* This is the exact parallel of a well-known proposition in the theory of algebraical forms; see
CLEBSCH'S ' Theorie der binaren algebraischen Formen,' p. 117.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 411
and therefore

I®*--1,, « H"'

•'1^

HA«^

Hence it follows the Jacobian under consideration can be constructed from prim-
invariants and proper quadrinvariants ; it is, therefore, a composite function, and
must be omitted from the aggregate of proper cubinvariants.

40. There thus remain only the Jacobians of the priminvariants with the quadri-
derivative quadrinvariants, and of these the total number is (n — 2)8. But, denoting
the Jacobian of 8,( , and 8X by ¥,? A, we have

*„,, = X8A8;,, - 2(0- + 1)8,,,
%. M = ft 6,e;, , - 2 (a- + i) 8,, ,

so that

A - *e**r = 2 «r + i e,

and, therefore, when any invariant ¥,iX is considered as given, any other of this type,
derived through 8,f t and so involving the same <r, can be expressed in terms of ¥r, x
and of invariants of earlier classes ; and hence out of the n — 2 functions derived
through 6,, , it is necessary to retain only one of them. This being so, it appears
natural to retain that function, which has X = or, and is the Jacobian of 6,. i and of
the priminvariunt 6,, with which 6,,, , is associated. Denoting it by 8,, 2, we have

;,i-(2o-+2)e,,1e;, ...... (viii.)

with index 3cr + 3 ; the number of these functions is n — 2, and their aggregate con-
stitutes the aggregate of independent proper cubinvariants. But it should be
remembered that there are (n — 2) (n — 3) other proper cubinvariants, which for their
expression require some one at least of this aggregate.

The proper cubin variant 8,i2 will be said to be associated with 8^

3 a 2

412 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIKNT-

Quartinvariants.

41. We now pass to the consideration of the invariants of the fourth degree which
can be obtained by the methods hitherto adopted.

The most obvious instances of composite quartinvariants are those constituted by
(i) products of priminvariants by cubinvariants proper and composite, and (ii) products
and powers of the second degree of proper quadrinvariauts ; while proper quartin-
variants are to be sought among

(a) Jacobians of priminvariants with proper cubinvariants, the Jacobians of primin-
variants with composite cubiuvariants being composite functions, by (A) $3G;
(/3) Jacobians of proper quadrinvariauts with proper quadrinvariants ;
(y) Quadriderivative functions of quadrinvariants, proper and composite.

These must be considered in turn.

42. First, for (a) ; we denote the Jacobian of ®a<2 and 0A by X,^, so that

Xff,A = A0X e:.s - (3o- + 3) e,,8 el,
x^ = ^e, e;.2 - (3o- + 3) e,,8 0; ;

and, therefore,

xex X,,, - fie,, x,,, = (3o- + 3) 0,,2 e,,Mil.

Hence it follows that, when one of the functions X,iX is known, all the other quartin-
variant Jacobians derived through the same cubinvariant are expressible in terms of
that one function and of invariants of lower degree ; and, therefore, as in § 40, the
(n — 2)2 invariants of this type can be resolved into n — 2 classes, in each of which
classes only one function need be retained. As before, we cboose from the class
derived through ©^ that function which is the Jacobian of®, and ©ffj2; and, denoting
it by ©^g, we have

©,,3 = o-0, ©:,2 - (3o- + 3)0ff,2©;, ....... (ix.)

a proper quartinvariant with index 4o- + 4. The number of these proper quartin-
variants is 71 — 2 ; and, in particular, the invariant 0ff|3 will be said to be associated
with the priminvariant 0^.

When 0^3 is expressed in terms of 0, and its derivatives alone, a simpler invariant
can be obtained by taking a linear combination of ©^3 and 0*a; there is, however,
no apparent advantage at present in taking such a combination as a canonical form,
and there is the present disadvantage of destroying the law of formation. The modifi-
cations will be indicated later (§ 134).

43. Second, for (ft) ; there are three cases which occur, viz. : —

(a) The combination of a Jacobian ®x,M,i with a Jacobian QP>VI l ;

(b) „ „ „ ®A,*I with a quadriderivative 0,, j ;

(c) „ „ quadriderivative ©,,! „ „ ®p, ^

DERIVATIVES ASSOCIATED WITH LIKK.Yft DIFFERENTIAL EQUATIONS. 413
Now, in $ 39 we have seen that the Jacobian J of «. and 8^, is given by

a result enunciated for the case in which 8, (there 8P) is supposed a priminvariant,
though, in the proof, no such limitation was introduced. This may be applied to the
consideration of (a) by writing 8, = 8P, „, l ; the functions 8^ ., l and 8Mt ,t l are then
cuhinvariants (composite, moreover), and therefore J can be expressed in terms of
invariants of the first three classes. Thus from (a) no proper quartinvariants arise.

The same formula may be applied to the consideration of (b) by writing 8, = 8p , , ;
the functions 8Aiir)] and 6^»,i are again cubinvariants (composite, moreover, if X and p
differ from p) and so J can be expressed in terms of invariants of the first three
classes. Thus from (b) no proper quartinvariants arise.

These two results can also be deduced as follows. For (a) we take

j = (x + M + 1) BA,M. , e;,,, , - (p + a- + 1) 8P,,, , «;,„, „

and a cubinvariant

V = (x + M + i) e^e; - pe.e;,,,, ;

from which

Now the right-hand side is the product of a quadrinvariant and a cubinvariant, and
therefore J is composite. Similarly, for (6) we take

and

e,, 9 = pe,ep, l - (zp + 2) ep> , e; ;

from which

The right-hand side, as before, is the product of a quadrinvariant and a cubinvariant ;
and therefore Jl is composite.

The last method may be applied to (c) also, and leads to a similar result ; for, taking

. ,2 - (2<r + 2) e^.e:,

we have

<re,P - (Ip + 2) 0P, ,9,., = (2<r + 2) e,,, {(2p + 2) 8,. ,8^ - (78,8,, ,},

i.e., the product of a quadrinvariant and a cubinvariant. Hence P is composite, and
therefore from (c) no proper quartinvariants arise.

414 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-

Combining these results, we see that the class (/8) furnishes no proper quart-
invariants.

44. Before passing on to the class (y), it may be remarked that the results of the
last article are particular examples of a more general proposition, viz. :—

The Jacobian of an invariant of degree m and an invariant of degree n, where m
and n are greater than unity, is a composite invariant, that is, it can be
expressed in terms of invariants the degree of each of which is less than
m -}- n.

For, calling the two invariants <£„ and ¥„ (of indices p. and v respectively), their
Jacobian J, and K the Jacobian of 0, and ¥„ — an invariant of degree n + 1 and
therefore of degree less than m + n — we have

J =

K =

and therefore

that is, equal to the product of an invariant of degree n and an invariant of degree
m + 1 ; and therefore J is expressible in terms of invariants all of degree less than
its own.

45. Third, for (y) ; since it follows from § 34 that the quadriderivative function of a
composite function is itself composite, provided the proper Jacobians of composing
functions be considered as a prior class, we see that the quadriderivative of a
composite quadrin variant is a composite quartinvariant ; and, therefore, any proper
quartinvariants that occur in the present class will enter as either

(a) the quadriderivative of a quadrinvariant of the type 0^ i ; or
(") » » » » » ®x, Mil-

Denoting the function in (a) by P, we have

P = (4o- + 4) ©„. ,6';, , - (4cr + 5) 0?, ,.
Also

e,. 2 = oex, , - (20- + 2) e,, , 0:,

0,, 3 = (T0X, 2 - (3o- + 3) 6,, , 0; ;

and it is not difficult to prove that

<r20*P = 4(o-+ l)0,,10,,a+ 4 (0- + 1)20^ - (4o-+ 5)0^,

so that P is composite, for it can be algebraically expressed in terms of invariants of
the first three degrees. Thus among the functions (a) there will be no proper
quartinvariant.

liKIUVATIVES ASSOCIATED WITH LINEAR DIFFKUKNTIAL EQUATIONS. 415
For (b), one of the simplest methods is to introduce the function

and then, by § 34, it follows that the quadriderivative of BJ^I is composite if that
of 4>Ai). be composite, for the Jacobian of BAB(t and 4>A>(1 is expressible in terms of
invariants of the first three degrees. Since **,„ is of index unity, its quadri-

derivative T is

T =

Now (§ 39) the Jacobian of a priminvariant and Q^^i can be expressed in terms of
invariants of the first two degrees, and, therefore, the Jacobian P of a priminvariant
and to^t can be expressed in terms of invariants of the first two degrees; con-
sequently, the Jacobian Q of a priminvariant and of P can be expressed in terms of
invariants of the first three degrees at most. But

Q = (rB,F - (<r + 2) PB;

= o^X,. - 3oB,B>^ + **,, {"BX - (<r + 2)B'r2j ,
so that

Q + ***
Hence,

2Q*,, + **, M B*. ! -

and therefore T is composite. Hence also, the quadriderivative of Bx>(1|1 is composite,
so that there is no proper quartinvariant among the functions (6).

Combining our results, we see that the class (y) furnishes no proper quartinvariants.

46. The general conclusion in regard to proper quartin variants is therefore the
following : —

There are n — 2 independent and proper quartinvariants, and these are given by
(ix.); all other quartinvariants derived by these methods are either composed of
invariants of the three farmer classes, or, if proper, can be expressed in terms of in-
variants of the three former classes and of one or more of the n — 2 independent proper
quartinvariants.

Invariants of Higher Degrees.

47. The investigation of the proper quiritinvariants proceeds on similar lines to
that for the quartinvariants. It is easy to see that, in forming the Jacobians of BCi ,

* It may be remarked, as worthy of note, that T -f- 2*!A,^ is the Schwarzian derivative with regard to
* of the absolute invariant log (6Aue,,~l>).

416 MR. A. R. FORSYTH ON" INVARIANTS, COVARIANTS, AND QUOT1ENT-

and 0X for all values of cr and X, the same limitations on the mutual independence of
the (n — 2)2 functions so derived exist as in §§ 40 and 42 ; and hence of this type
there are n — 2 proper and independent quintinvariants given by

ri,e', ....... (x)

the remaining proper quintinvariants being expressible in terms of the functions 6^4
and of invariants of the first four classes.

48. By means of some of the results obtained we can show that all the invariants,
obtainable by any of the methods hitherto used or by any combination of them, are
expressible in terms of these different classes in succession of n — 2 proper invariants
associated with the n — 2 priminvariants. For

(i) These proper invariants of any class are obtained by forming the fitting
Jacobians of the proper invariants of the class next preceding and the priminvariants ;

(ii) By proposition (A) of § 36 and the theorem of § 44, it follows that all other
Jacobians are composite ;

(iii) By the analysis of § 45 it follows that the quadriderivative of any Jacobian
is composite, if we retain as representative invariants the successive Jacobians of
proper invariants. Now, after ©„,!, all the proper invariants ©^.j, ©,,,3, . . . are
Jacobians, and therefore quadriderivative functions formed from them are composite,
•A result already proved in § 45 for ©^ , ; and thus the quadriderivative operation
applied to any proper invariant will produce only a composite invariant.

(iv) It is easy to see that, if we take any proper invariant <I> of a class higher than
the first and from it, considered as a fundamental invariant, construct the same
functions as ©„_ 2, ©,, 3, ... are of ©„, all the resulting invariants will be composite.
For, considering in particular the cubiderivative function of <J> corresponding to ®a<z, it
will be the Jacobian of the invariant <£> and of the quadriderivative of that invariant ;
this quadriderative will in general come under the he;id of those considered in (iii),
and therefore will be composite ; but in any case the theorem of § 44 shows that the
function will be composite, since 3> is of a degree higher than the first. Similarly for
all the other functions.

It therefore follows that the operations, similar to those whereby the invariants
®<r, i. ®<r,2> • • • are constructed from ©„, only lead to composite invariants when
applied to proper invariants of any class beyond the first, and that the only operation
which can lead to proper invariants is the Jacobian, and even that operation onlv
produces proper invariants of any degree when applied to the n — 2 invariants ©,, and
the respective proper invariants of the preceding degree associated with ©r.

49. The general conclusion as to the derived invariants is as follows : —

It is convenient to range the derived invariants in classes ; all the invariants in any
one class are, when the differential equation is taken in its canonical form, homogeneous
in the coefficients Q of the equation and their derivatives ; and the degree of any

DERIVATIVES ASSOCIATED WITH LINEAR D1K1 IIIJKM I Ah EQUATIONS. 417

class is taken to be the common degree of all the invariants of the class. In each
class the invariants are of two kinds, viz., composite, these invariants being expressible
in terms of invuriunt.s of earlier classes; and proper, these not being expressible in
such terms. The number of proper invariants in any class above the second is (n — 2)8;
but only n — 2 of this number are quite independent of one another, and the remain-
ing (n — 2) (n — 3) proper invariants of the class can be expressed in terms of one
(or more) of the independent proper invariants and oT invariants of lower classes.
And the following are the proper invariants of the classes in succession : —

First, the priminvariants 68, 64, . . . , 6,, . . . , B», each of which is linear in the
coefficients of the differential equation, supposed reduced to its canonical form, and
their derivatives ; the index of each invariant is the same as its subscript number ;

Second, (i) the quadriderivative functions

«„. , = 2(7^8",- (2cr + l)e'J,

which are n — 2 in number (o- = 3, 4 ..... n) and are independent of one another ;
the index of e,,, is 2tr + 2 ; and (ii) the £(n — 2) (n — 3) Jacobians

of index X -j- p. -f 1 (X, p. = 3, 4, . . . , n), but only n — 3 of these are independent,
and the remainder can be expressed in terms of these n — 3, properly chosen, and of
priminvariants. The two kinds of proper invariants in this class are algebraically
independent of one another ;

Third, there are n — 2 independent cubinvariants given by

6,. , = <**,&„. , - (2<r + 2) e,, ,6',

of index 30- + 3, and there are (n — 2) (n — 3) proper cubinvariants dependent on the
foregoing n — 2 ;

Fourth, there are n — 2 independent quartinvariants given by

e,, , = crf^e',, 2 - (3o- + 3) e,, .e',

of index 4<r + 4, and there are (n —2) (n — 3) proper but dependent quartinvariants ;
and, generally, the rth class contains n — 2 independent proper invariants given by

,,r-2 - (r - 1) (a- + l) 0,,r_8e', . . . , . (xi.)

of index ?• (a- + 1), and also (n — 2) (n — 3) proper but dependent invariants.

And all the invariants of the 7-th class, for every value of ?•, are of degree r in the
coefficients of the differential equation and their derivatives.

50. In this connexion two points remain to be noticed. It has already (§ 7) been
remarked that M. HALPHEN has, for the quartic, derived a series of invariants from

MDCCCLXXXVIH. - A. 3 H

418 MB. A. R. FORSYTH ON INVARIANTS, COVARTANTS, AND QUOTIENT-

the invariants @3 and 04 by a process which is effectively a continued repetition of
the Jacohian process ; and he has* two derived invariants, A, which is practically
BKIOSCHI'S quadriderivative of 63> and 8, practically the same function of @4. He
also (1. c., p. 339) forms Jacobians, which can be expressed in terms of functions
B3i r (in the notation of the present memoir) ; and these together constitute his
aggregate of invariants for the quartic.

Lastly, the important simplification of the forms of the invariants due to the
reduction of the equation to its canonical form has been repeatedly remarked in the
preceding paragraphs ; it is, in fact, owing to this that the foregoing classification has
proved practicable. When, however, the differential equation is not assumed to be
thus reduced, a change necessarily takes place in the explicit forms of all the in-
variants ; thus, for instance, in the case of a non-evanescent coefficient Q2, it is not
difficult to verify that

from which a non-canonical form of ©„_ l — the value of &ff being supposed known — is
at once apparent. But into the expressions of those proper invariants which are
Jacobians the coefficient Q2 does not explicitly enter until substitution begins to be
made for the invariants in this Jacobian form.

Finality of the Results.

51. The results so far obtained, though very general, have not been shown to be
exclusively so. It has been proved that all the linear invariants which exist are
included in the set of priminvariants ; and that all the invariants derived from them
by the given methods can be expressed in terms of the proper invariants of the classes
as arranged. But no proof has been given that, for degrees higher than the first, any
invariant possible can be deduced by the methods used, or that any invariant can be
expressed in terms of the assigned invariants. Until one of these two propositions
(or some equivalent proposition) is established, we are not in a position to declare that
all possible invariants of the differential equation can be expressed in terms of the
given invariants.

The consideration of this question will be deferred until Section VIII., where the
investigation will include not merely the invariants, but other invariantive functions
yet to be obtained.

* ' Acta Math.,' vol. 3, pp. 335 and 341 respectively.

DERIVATIVES ASSOCIATED WITH LINEAR DIF1 I.HKM IAI, IMITATIONS. 419

SECTION IV.

ASSOCIATE EQUATIONS AND DEPENDENT VARIABLES.
LAQRANUE'S " Equation adjointe."

52. It was proved by LAORANGE,* that in connexion with every linear differential
equation there exists another linear equation of the same order, and that a know-
ledge of the primitive of either is sufficient to lead to the primitive of the other. Let
y\,y*, • • • , y« be n special and linearly independent solutions of the equation

then

,1.1"

v •=• v, =

<«-*>

y - 1 •

y— i

y— i

is an integrating factor. For, since y\,y$. • • • >y»-\ satisfy the equation separately,
the 7i — l quantities R can be found in terms of them ; and, when these values of K
are substituted and the equation is then multiplied by v, it takes the form

,/.,•

y . y .

(n - 1) (n - 2)

y._3, y.-2,

y

y «-•:»

= 0.

But an integrating factor of the equation satisfies the relation

* ' Miscellanea Taurinensia,' vol. 3, 1762 ; ' OZnvres,' vol. 1, p. 471.—" Solution dc differcnts problt-nu s
de calcul integral."

8 H 4

420 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

as can be seen at once by multiplying the foregoing equation by v and integrating by
parts. This is LAGRANGE'S associate equation (" Equation adjointe ") ; it is of the
same order as the original equation ; and its special and independent integrals may be
taken to be the n determinants each of (n — I)3 constituents given by

(n-2) (n-2) (n -

y. , y._i , y—s

(n-3) fn-3 (it-

y/ •}/ -J/

y« , y.i-\ , ?/*-2

(n-2)
/I
(n-J)

It is well-known that the Lagrangian associate of the Lagrangian associate is the
original equation ; it is evident that, if either be in its canonical form, the other is so
also. It will now appear that the equation is only one of a set of equations, and its
variable only one of a set of dependent variables, associated with the original equation.

Sets of Variables subject to the. same Linear Transformation; Algebraical Combination.

53. The n special integrals y constitute a fundamental system of integrals, and
each of the members Y1, Y2, . . . , Y, of any other fundamental system is a linear
function of the former set, so that in effect a change from one fundamental system
to another is only a linear transformation of the dependent variables concerned.
(There is here no question of the necessary modifications of fundamental systems
owing to the presence of "singular" values of the independent variable). This
transformation may be represented by

!,¥„...,¥.) = (MX ?„?*

where M is a constant matrix with a non-vanishing determinant. But this applies
not only to the dependent variables, but also to their derivatives of all orders, so that
we have

x (tSu (tyf I \ w*v dx rt*

for any value of r. And, if we retain this equation for values 0, 1, 2, ...,«.— I of
the index r, we shall have in all n sets of variables subject to the same linear trans-
formation ; and these variables are linearly independent of one another, since for the
satisfaction of the differential equation we need the nth differential coefficients of the
quantities y, which have been specially excluded.

DKIUYATIVKS ASSOCIATED WITH LINEAR DIFFKUKNTIAL EQUATIONS. 421

54. Since the n quantities y are linearly independent of one another, they may
be looked upon as the coordinates of a point in a manifoldness of n — I dimensions ;
and, if we assume the same linear independence of the derivatives of all the orders
up to the (n — l)th inclusive (which is equivalent to an assumption that no linear
function of the quantities y with constant coefficients is equal to a rational integral
algebraical function of order less than n — I — an assumption justifiable with general
coefficients, though not necessarily so in any particular case), then each of the n — 1
sets of derivatives, each set being constituted by those of the same order, may be
looked upon as the coordinates of a point in a manifoldness of n — 1 dimensions.
And, since the law of linear transformation is the same for all the sets, all these
points may be taken as belonging to the same manifoldness. There are thus n different
and independent sets of cogredient variables connected with the single manifoldness
of n — 1 dimensions.

55. In the theory of the concomitants of algebraical quantics of any order in the
variables of a manifoldness of n — 1 dimensions, it is necessary to consider all tho
possible classes of variables which can enter into the expressions of these con-
comitants. CLEBSCH * has proved that there are in all n — 1 different classes of
variables which thus need to be considered, and that, if »„ xs ..... x, ; ylt yz, . . . , y,;
Zj, z2, . . . , 2, ; ... be n sets of cogredient variables, the several classes are constituted
by minors of varying orders of the determinant (itself an identical covariant)

Z2, . . . , Z,

those of one class being minors of one and the same order. The variables of any class
are linearly, but not algebraically, independent of one another, except in the case of
the first class, constituted by minors of order unity, and the last class, constituted by
minors of order n — 1 (the complementaries of those of the first class), in each of which
classes the n variables are quite independent of one another. And all similar combina-
tions of variables are expressible in terms of variables actually included in the classes.
56. In connexion with our differential equation we have obtained n different and
algebraically independent sets of cogredient variables ; the functional derivation of
the sets, one from another in succession, by the process of differentiation has been
excluded from any interference with their algebraical independence. We already
have one class of variables, viz., ylt y.2) . . ., y, analogous to the first class of algebraical
variables, and another class of variables, viz.,vl(v2,. . . , vm analogous to the (n — l)th
class of algebraical variables ; and the relation

* " Ueber cine FundamentalaufRabe der Invariantentheorie," ' Gottingen, Abhandlungun,' vol. 17, 1872.

422 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

which is satisfied, is precisely the same as the corresponding relation between the
similar variables helping to define the higher class (CLEBSCH, 1. c., p. 4). Hence,
from the point of view of purely algebraical forms, we infer that the suitable
algebraical combinations of the sets of variables, which have arisen in connexion with
the differential equation, are the minors of varying orders of the determinant

A =

!/i .2/2 .

y\

y,,

(n-D

2/2 ,

y*

which determinant, as we know, is a non-evanescent constant ; and these variables
may be ranged in classes, which for the present may be called linear, bilinear, tri-
linear, ....

Algebraical Combinations functionally Invariantive.

57. Now, after having obtained the merely algebraical result, it is necessary to take
account of functional dependence of the sets due to differential derivation. In the
case of the algebraical quantics, it is a matter of indifference which set of minors of
the first order be taken to constitute the first class of variables, which set of minors of
the second order be taken to constitute the second class, and so on ; thus for the
second class the same kind of variable is obtained by taking the (xy] minors, as by
taking the (yz) minors, or the (xz) minors. But a difference arises in the case of the
variables occurring in connexion with the differential equation. There are n sets of
linear variables distinct in character from one another ; for y\, i/2, . . . , y'n are special
integrals of an equation quite different from the original equation, though they are
subject to the same law of linear transformation as ylt y2, . . . , yH. There are
^n (n — 1) sets of bilinear variables distinct in character ; thus

y i» y\

y i. y\

y i. y\

are three distinct variables of this class, subject to the same law of linear transforma-
tion ; and so on for the higher classes.

58. Most of these, however, will be excluded. These forms of variables have been
suggested in connexion with the theory of linear transformations, for which trans-
formations algebraical concomitants involving them are covariantive. The invariants

DKKIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 423

considered in the earlier part of this memoir have possessed their invariantive pro-
perty for functional transformation ; and, therefore, if forms involving the dependent
variables are to be included in an aggregate of concomitants together with the
invariants, these forms must have the same invariantive property for such functional
transformation. In this aggregate of concomitants the variables themselves will be
included ; and, therefore, we must select from the foregoing algebraical combinations
those which have the invariantive property of reproducing themselves, save as to a
power of z', after transformation.

Of the n sets of linear variables constituted by the several sets of n quantities y,
n quantities y", and so on, only the first set has the property of being reproduced by
the new variable, save as to a power of ^ ; and we already know that, if u be the new
dependent variable, then the relation is, by (iv.),

y = Mz'-4(<|-i> . . (xii.).

Of the £n(n — 1) sets of bilinear variables, each set containing ^n(n — 1) variables,
only a single one has the invariantive property of self-reproduction, save as to a
power of z ; and this single one is the set constituted by the £n (» — 1) variables of
the type

This statement, which leads to the retention of the single set and the exclusion of
all the remainder, can be at once verified by making substitutions of the type (xii.) ;
and the result of the substitution on the typical variable of the present class is that,
if <a denote the original bilinear variable and v± the transformed bilinear variable

du.

then we have

dz'

= XV v, = v,z'-<"-"=

(xiii.).

Similarly of the \n (n— 1) (n — 2) sets of trilinear variables, each set being consti-
tuted by corresponding minors of the third order, there is only one set of which each
variable has the functional invariantive property ; and a typical variable of the set to
be retained is

y" yr> yr
y," y,', y,

424 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-
The relation of transformation is

t, = \^z']+zv = v 2'~*!("~3> (xiv )

where v3 is the corresponding transformed trilinear variable.

And in general of the n\/p\ n—pl sets of ^-linear variables, each set being
constituted by corresponding minors of the pth order, there is only one which has
variables possessed of the functional invariantive property, and of this set a typical
variable is

' Vi, y\, 2/1". •• • . i/i0'"0 •
11 11 ' 11 " 11 (p~i)

2/2> 2/2 > 2/2 > • • • > 2/2

<p ~

where as before y denotes dy/dx. If vp denote the same jp-linear variable associated
with the transformed equation, the law of transformation is

+ ~ + ---

v

(xv.).

The last set of variables is that for which p = n — 1 ; and the typical variable of
the set is the variable of LAGRANGE'S " Equation adjointe."

Associate Dependent Variables.

59. Hence there are, in all, n — 1 sets of variables ; all the variables in any one
set are particular and linearly independent solutions of a differential equation the
dependent variable of which is a typical variable of the set. Hence, connected with
the given differential equation, there are n — 2 other differential equations ; these
may be called the associate equations. The n — 2 new dependent variables, derived
by definite laws of formation, may be called the associate dependent variables , and,
calling them in turns the associate variables of the first, second, . . . , (n — 2)th rank,
the differential equation of which the dependent variable is the associate of the
(p — l)th rank is linear and of order nl/pln—p\. For the functional transfor-
mation of the original dependent variable given by (xii.) the law of transformation of
the associate variable of the (p — l)th rank is given by (xv.); and, if we call two
ranks complementary when the sum of their orders is n — 2, then associate variables
of complementary rank are transformed by the same relation, since for such variables
the index of the factor power of z' has the same value.

The associate dependent variables may therefore be ranged in pairs of complementary

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 425

rank ; in the case when n is even there is one dependent variable of self-complementary
r;ink. Each pair has the index of the factor power of z different from that for any
other pair. The simplest case of this arrangement is that which combines in a pair
the original variable y and the variable tp_l of the "adjoint " equation of LAORANOE ;
and the two dependent variables have the same functional transformation. Since tf_}
is thus a covariant for the transformation, HALPHEJJ infers that the invariants of the
equation satisfied by tf_l are invariants of the original equation, and he has used this
proposition to construct 63 (see § G).

Reference to Invariants.

GO. Before proceeding further with the associate variables, there is one point which
may be considered conveniently here. The quantities y,, y,, . . ., y,, from which the
associate variables are constructed, are, all of them, covariants with the same index ;
and, in particular, the associate of the first rank is in each case the Jacobian of two
of these covariants. Now, when we were considering invariants we were led to new
invariants by forming the Jacobian of any two ; and thus there is suggested a new
means of forming invariants, if, e.g., for three, which by involution to suitable powers
can be made to have the same index, we form the same function as the associates of
the second rank are for the special values of the original variable. It may, however,
be easily proved that such invariants are composite. For let <I>, ¥, X be three such
invariants with the same index 6 (e.g., they might be ej", 0£\ 6J", and 8 = \fip) ;
then the new function

n = *, ¥, x

4>' ¥' X'

<&'', ¥", x"

Let [*] denote the quadriderivative of «I> ; then

and similarly for the others, so that after substitution for *", ¥", X" we have

But the determinant on the right-hand side is

X
X'

*'

MDCCCLXXXVIII. — A.

3 I

426 MB. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

and hence II can be expressed in terms of the invariants which have been retained as
fundamental and of proper derived invariants. Thus, in the case of the invariantive
combination suggested by the associate variable of the second rank, only composite
invariants are obtained ; the result is general for all the invariantive combinations
thus suggested.

Combinations of the Associate Variables.

61. Consider now the complete set of dependent variables, viz., the original
variable y and the n — 2 associate variables, as a fundamental set. Any one of them
satisfies a linear differential equation of determinate order ; and with it, as an original
dependent variable, there will be associated a number of new dependent variables, in
number 2 less than the order of the equation, and functionally derived from it in the
same manner as the preceding have been derived from y.

Taking a few simple cases, consider first that of the associate of the first rank
and let

^12 = 2/i2/2 - 2/22/1 ; VM = yzy\ -

so that v12, v34,, vB6, are particular solutions of the equation whose dependent variable
tz is the associate of the first rank. One of the set of variables, associate of the first
rank with £2, is

2/i2/2 - 2/2/K 2/1/2 - 2/2/1

2/32/4 - 2/42/3, 2/3/4 - 2/4/3

= 2/2

2/1 > ?/s , 2/4

2/3 >

y\,

2/a > 2/3 , 2/4

2/2, 2/3, 2/4

and is, therefore, expressible in terms of associates of the original variable y. Again,
one of the set of variables, associate of the second rank with t2, is

r

'66

'34»

12,

S/lS/2 - 2/22/1, 2/32/4 - 2/42/3, 2/52/6 - !/62/5

2/1/2 - 2/2/1, 2/3/4 - 2/4/3> 2/5/0 ~ 2/6/5

2/1/2 -

. 2/82/U4 - 2/4/3, 2/6/6 - 2/62/U5

2/12/2 - 2/22/1, 2/32/4 - 2/4/3, 2/52/6 - 2/62/5
2/1/2 - 2/2/1, 2/3/4 - 2/4/3, 2/5/6 ~ 2/6/6
2A/2 - 2/2/1, 2/3/4 - 2/4/3, 2/6/6 ~ 2/6/5

- 2/22/3 V1466 ~

56

2/22/4 ^1356 ~

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 427
where

y,. y». y*> y.
tf?> y*»» yV> y*.
y",. y",. A y.

If y^ 77 f. v

and similarly for v^ ; hence it is expressible in terms of associates of the original
variable y.

As a last example, consider the set of variables associate of the first rank with ta ;
one of these variables may with the foregoing notation be written —

vm*m-*rnv*»

and this easily proved to be

again expressible in terms of associates of the variable y.

62. From these particular results and the preceding investigations the following
inferences may be drawn : —

(1.) The system of associate variables, constituted by y, tz, tz, . . . , f»_i, is
functionally complete ; that is to say, the variables in the systems associate with
any one of them (derived from that one by the functional operations of the type
which led to y, t^ t3, . . . ) are, qua variables, expressible in terms of combinations of
particular associates of the original variable y ; in the formation of these combinations
it may be necessary to introduce functions of the coefficients of the original equation.
Hence, as typical dependent variables, the associates of the variables associate with y
may be looked upon as expressible in terms of the variables associate with y, the
necessary combinations of which are only multiplicative and additive ; and they
therefore introduce no new associate variables.

(2.) Invariants of associate equations are all of them invariants of the original
equation ; the complete converse of this may not be affirmed.

(3.) Differential equations in associate variables of complementary rank are
mutually " adjoint."

The last inference is suggested by the following considerations : — When we
construct the equation adjoint to the differential equation, of which the dependent
variable is tp and order n \jp 1 n — p !, the process can be performed in a manner
similar to LAGRANGE'S adopted in § 52. The integrating factor, which is the
dependent variable of the adjoint equation required, can be constructed as a
functional determinant of special solutions t of the equation, in number one less than
the order of the equation. For the special integrating factor, corresponding to that
of § 52, let the particular t solution omitted from the determinantal expression be the
functional determinant of yl9 y& . . . , yr When, in the integrating factor, determinant
substitution takes place for the particular solutions in terms of the quantities y, it
appears that the determinant is of dimensions

3 I 2

428 MB. A. E. FORSYTH ON INVARIANTS, COVARTANTS, AND QUOTIENT-

1

_

p — II n —p\

in each of the quantities ylt y2, . . . , yM> and of an additional unit dimension in each of
the quantities yp+i, yf + $, • • • , y»- Since the functional determinant of ylt y2 ..... y,
is a constant, the part of the factor dependent on the [n — 1 ! -f- p — 1 ! n — pi} — 1
dimensions in the quantities ylt y.2, . . . , yn may be expected to be a constant ; the
later part may be expected to be a functional determinant of yp+l, yp + 2> • • • > U*
This last is a special value of the dependent variable of complementary rank, and
is the conjugate of the dependent variable tp_l omitted in the construction of the
integrating factor. Hence it may be expected that the variable of the differential
equation, adjoint to that in tf, shall have as its variable tn_p.

I do not propose to attempt to give here, however, a rigid investigation of the
inferences just suggested.

The equation of lowest order for which an adjoint exists is the cubic ; after the
formation of this adjoint equation, which will be effected later (§ 82) in connexion
with the investigation of some questions about the cubic, the identity of the
covariants of the two equations will be evident. Similarly for the case of the quartic
(§§ 102-107).

SECTION V.
IDENTICAL AND MIXED COVARIANTS.

63. In the last section a set of n — 1 dependent variables y, t2, t& . . . , ta-l has
been obtained which are algebraically independent of one another, and each of which
possesses the same invariant! ve property as the fundamental .invariants ; and, just as
was the case with the invariants, we can, by using the methods employed in
Section III., deduce other covariants from each of these dependent variables alone,
from combinations of them with one another, and from combinations of them with the
invariants. As it is desired to retain only those functions which are not composite, a
selection must be made as before. The forms of the functions will be destitute of one
of the characteristics of the invariants ; their indices depend on the order of the diffe-
rential equation, and the number expressing this order enters into the numerical
coefficients, so that these new covariantive functions vary from one equation to
another.

Identical Covariants in the Original Variable.

64. In this class are included all those functions possessing the invariantive
property, and involving the dependent variables alone or their derivatives, but not
the coefficients of the differential equation, when taken in its canonical form ; on
which account they may be called identical, or absolute. Beginning with the original
dependent variable, we have

UERIVATIVKS ASSOCIATED WITU LINEAR DIFFERENTIAL EQUATIONS. 429

so that we may consider u as a covariant of index — i (» — 1). Proceeding in the
same manner as in § 33, it is easily found that

which at once gives a new invariantive form. When the transformed equation is in
its canonical form, so that Q2 vanishes, the new covariant is

with index 2 ; or we may write

U8 = (n - 1) uti" - (n - 2) t*'» . . . . . . . . (xvi.)

of index 3 — n. This is the quadriderivative of u..

65. There are thus two co variants u and U2 ; from them as fundamental co variants
we can deduce the series of successive Jacobiana. Thus the covariant next in degree
is (§40)

or say it is

U8 = (n - 1) uU'2 - 2 (n - 3) Uau' ........ (xvii.)

with index 3 — n — ^ (n — 1) + I = — f (n — 3). The next is

U4= (n — l)tiU's — 3(n — 3) U,u' ....... (xviii.)

with index — f (74 — 3) — £ (n — 1) + 1 = — 2 (n — 3) ; and so on. And the rth
covariaut in the complete succession is

Ur = (ft - 1 ) u Ur'_! - (r -l)(n- 3) Ur_! u' . . . (xix.)

with index — ^ r(n — 1). By means of the propositions used for the invariants it is
easy to see that this series constitutes the aggregate of proper covariants involving u
alone ; for all others, obtained by combinations and by the application of the functional
operations to combinations other than those which give results (xvi.)-(xix.), are, by
those propositions, proved to be composite.

430 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

66. The number of terms in this succession of identical covariants is given by values
of?' from 2 to n — 1, so that the succession includes, besides u, the n — 2 terms U2>
U3, . . ., !!„_!. When the value n is assigned to r, the resulting covariant is one
which involves d'u/dz", and which can, therefore, by means of the differential equation
satisfied by u, be transformed so as no longer to involve this differential coefficient.
It will then involve derivatives of u of order less than n, and also the seminvariant
coefficients Q ; such a covariant will be called a mixed covariant, because its expression
depends partly on the variable and partly on the coefficients. Further, every succes-
sive covariant derived by the Jacobian process can be similarly transformed, and will
then become a mixed covariant. There will be some limitation on the number of

ndependent covariants of the mixed type thus obtained ; for the elements, so far as
concerns the dependent variable, are only n in number, being u, u', . . ., u<Jt~^\ and
elimination of these quantities among more than n mixed covariants will lead to
relations involving covariants and functions of the coefficients of the differential
equation only. From the fact that the quantities, which occur in the result of the
elimination, and are not functions of the coefficients, are covariantive, it is a priori
probable that such functions of the coefficients as enter are invariants of the differen-
tial equation, or combine with the variables to constitute mixed covariants.
For example, in the case of the cubic equation, which is

-^ -f &su = 0
in its canonical form, and has @3 for its priminvariant, it is easy to show in general

U3 = (n - I)2 wV" -3(n-l)(n-3) uu'u" + 2 (n - 2) (n - 3) w'3,

so that, when n = 3,

U3 = (3 — l)a uzu" = — 4 w8@3 ;

and there are, therefore, no proper identical covariants involving u alone, except u and
U2 for the cubic. The corresponding investigations, which must be deferred until the
mixed covariants are obtained, are given in § 76 for the quartic, in § 77 for the quintic ;
and the general investigation for an equation of any order is indicated in § 138.

67. In the case when there is given, not a differential equation, but a differential
quantic of the form

l n - r\

and we are seeking the identical covariants for the transformation which changes this
quantic to

n\ fr-u

*

\ -

r n

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 431

save as to a power of z, the number of the covariants of this type is (with the
reservation of § 77) unlimited ; and in the case when the second form is the canonical
form, so that Qj and Q3 vanish, all the covariants thus obtained for this form of
quantic are purely identical, that is, they do not involve the coefficients Q.

Identical Covariants in the Associate Variables.

68. Two kinds of identical covariants are possible. First, there are those which
involve only a single one of the set of dependent variables ; and the aggregate of
these, for each of the associate variables, is similar to that just given for the original
dependent variable. Second, there are those which involve more than a single one of
the set of dependent variables, and which, therefore, may be called simultaneous ; but
it will appear (see § 72) that this class need not be retained, for they can all be
derived from proper invariants and covariants by purely algebraical processes of
multiplication and the like. Hence the former class alone requires to be retained.

To find all the covariants depending on the associate variable of general rankp — 1,
the process is the same as before ; we take the variable in connexion with the normal
form of the fundamental differential equation and, denoting it as in (xv.) by vp with
index — %p (n — p), we find a quadriderivative function

which is covariantive, or say

with index — (np — p3 — 2). When the series of Jacobians of vp and the functions
V are formed in succession, they are found to be

t;',> . . . . (xxi.)

and the general term in the succession is

vf. . . . (xxiii.)

The index of the covariant VA, is — £s (np — p2 — 2). The number of covariants in
the succession is (with a reservation similar to that in § 77, post) infinite when the
associate vp is regarded as the variable of an unretained associated differential quantic ;

432 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

n!

it is finite, with value - , — 1, when the associate vp is regarded as the variable

pi n —pi

of the associate differential equation.

The foregoing aggregate includes all proper covariants which involve vf alone in
their expression ; this result is derivable from the propositions which were proved
in Section III., and may be verified separately for the covariants. Thus, for instance,
if t^T denote the Jacobian of ~Vpif and Vft,, it is easy to show that

p(n — p)vfli = r(np — p* — 2) "V^rV^.+i — s(np — p* — 2) Vpi,Vpjr+1 ,
whence it follows that T is composite.

Mixed Covariants in the Original Dependent Variable.

69. By this title invariantive functions are indicated into whose expression there
enter the dependent variable or variables and the coefficients of the original differential
equation. One method of obtaining them is that adopted in § 35, viz., to combine
the variables and the invariants in such a form as to be absolutely invariantive, and
from this form derive a relative invariant which is practically a Jacobian.

Beginning with those which involve only a single dependent variable and taking u
first, we have ®l~luZir an absolute covariant, so that

0J| - ' (z) %2<r = 0;;- ' (a) y2* ;

from which it follows, by taking logarithmic differentials, that

0^ «'

is a covariant of index unity, or say

, (xxiv.)

with index a- + 1 — i (n — !)•

70. The following propositions enable us to select the non-composite mixed
covariants : —

(i.) It is evident that, if 0, be a composite invariant, then Q, (M)J will be a composite
covariant ; hence we need only consider such functions as are derived from proper
invariants.

With every proper invariant there is associated a proper mixed covariant of the first
order in u ; but, when one of these proper mixed covariants is considered as given, all

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 433

the others can be expressed in terms of that one and of invariants. For from the
proper invariant Bp there is derived the proper mixed covariant

so that

which verifies the statement.

(ii) When the successive identical covariants Us, U3, U+, . . . are taken instead of
•M, new covariants are obtained by forming the Jacobians of these covariants U and
the invariants Q. All such covariants are composite. For, taking UA with index /*,
equal to — ^X (n — 1), and denoting the Jacobiau of UA and O, by J, we have

J = /JJ.e', - oe,U'A :

and we have

so that

(n - 1) uJ + (re,UA + 1 = /xUA {(n -

whence J is a composite covariant. Hence this class of covarianta must not be
included in the aggregate of proper covariants. (See also § 39.)

(iii) It is unnecessary to form the series of successive Jacobians from u and Q, (t<)j
as fundamental covariants ; for the Jacobian of u and 0, (u}^ is composite, and, there-
fore, all subsequent Jacobians are composite. To prove the statement, denoting this
Jacobian by 0, (w)2, we have

= 2o-(2o- - n + 3) w/?6,+ (n - 1) (4o- + 2) uu'%'. + (n - I)2 u

+ 2o-(n-

after substitution. But
so that

= (« - I)2 (2o- -f 1 ) u2e'2, + 402 (2tr - n + 3) w^*

+ (n - 1) 4<r (2<r + 1) uu'e^e', + 4^ (n - 1 )
= 4a«e^(n - 1) uu" - (n - 2) «'*}

+ (2<r + 1) { (n - I)2 u2Q'2, + 4<r (» - 1

and, therefore, 6, (n)z is composite. (See also § 39.)

MDCCCLXXXVIII. — A. 3 K

434 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

(iv) Similarly the Jacobian of 0<r(u)l and any invariant Hp of index p is composite;
for, denoting it by V, it is easy to show that

2p0p^(n)2 -f (n - 1) teV = {2cr + 2 - n - 1} 0,(u)l0p(u\,

whence it follows that V is composite.

In the same way it may be proved that the Jacobian of 6, (u)l and 0f (u)t is com-
posite. (See also §43.)

71. The general result, therefore, is that all the covariants, which can be obtained
by the methods used, are expressible in terms of the identical covariants u, U2, U3, . . .
and of the mixed covariants of the first order 6, (u)l ; all of these are proper, i.e., they
cannot be expressed in terms of invariants and covariants of earlier rank, but all the
mixed covariants can be expressed in terms of any one of them and of invariants.

Mixed Covariants in the Associate Variables.

•• • . • • ,' f __ £>

72. The aggregate of mixed covariants, which involve in their expressions only a
single associate variable, is for each associate composed similarly to the corresponding
aggregate in the original variable ; and all the covariants, which can be obtained by
the methods employed, can be expressed in terms of the identical covariants
"Up, Vj,i8, Vpi3, . . . , and of mixed covariants

6. (vp)} = 2er@yp + p(n-p) vf#.

of the first order. These mixed covariants are proper, but they can all be expressed
in terms of any one of them, and of invariants.

By retaining as proper covariants one at least of these mixed covariants of the first
order in each of the associate dependent variables, we are enabled to dispense with
the simultaneous identical covariants (§ 68) as being composite. For the simplest
simultaneous identical covariant is the Jacobian of two of the dependent variables,
6ay u and vp ; and it is easily proved that

(n - 1) u0, (vp\ -p(n- p) vpe, (u), = 2<r€>(r{(n -l)w'f-p (n - p) vfv'},

so that this Jacobian is composite.

The application of the analysis of § 60 (which shows that the invariant function
obtained by constructing a function for invariants, similar to them in the same way as
vp is to u) to covariantive combinations of more than two of the associate variables
taken simultaneously shows that such combinations can be expressed in terms of the
covariants already obtained, and are therefore composite.

73. It has been shown, in (iii) of § 70, that successive Jucobians of d.(u), and u are

DERIVAT1VKS ASSOCIATED WITH I.INKAU DIFFKKKNTIAL EQUATIONS. 435

composite ; the same holds of those formed with 0, (u\ and vr For, denoting the
first of such Jacobians by T, we have

whence by means of the expression for 0,(u).2, in (iii) of § 70, which is a composite
covariant, it follows that

= {cr + 1 - *(» - 1) W)i[(» - l)ut/, -p(n -J») vO

But the simultaneous identical covariant on the right hand side is composite ; hence
T is composite. So for the others in succession.

Lastly, as in (iv.), the Jacobian of any two mixed covariants of the first order in
any variables is composite. For taking

it is easy to show that

(„ _ i)MW+ {p+ 1 - \p(n -p)}0,(vf)l0.(u)3
= 2<r- n

It has just been proved that the second factor on the right hand side is composite ;
and therefore W is composite.

It follows, from all these results and the propositions proved in § 48, that all the
simultaneous identical covariants are composite.

74. The general conclusion as to the aggregate of covariantive concomitants is
thus : —

The aggregate of proper concomitants associated with a differential quantic or a
differential equation is composed of three classes —

(A.) INVARIANTS, being functions of the coefficients of the quantic or equation ;

(B.) IDENTICAL COVARIANTS, being (i.) functions of the dependent variable and
its derivatives (which when of sufficiently high order change into mixed
covariants if associated with a differential equation) ; and (ii.) functions of
the associate dependent variables and their derivatives ; but any function
involving more than one dependent variable is composite ;

(C.) MIXED COVARIANTS, being functions of the dependent variables, original
and associate (but not involving more than one dependent variable), and
of the invariants and their derivatives.

3 K 2

436 MR. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTJENT-

And when the complete set of non-composite invariants, and the complete set of
non-composite identical covariants in each of the dependent variables are retained, the
independent non-composite mixed covariants consist only of Jacobians of the first
order of any one invariant, and each of the dependent variables in turn.

Limitation on Number of Identical Covariants.

75. The following gives the limitation on the number of proper identical covariants
in the original variable when the equation is a quartic ; and when there is given a
quantic, not an equation, of the fourth order the reservation mentioned in § 67 is here
indicated.

For the general equation the first few identical covariants in their present forms are: —

U2 = (» - 1) tm" - (n - 2) u*

U3 = (u - I)2 uW - 3 (n - 1) (n - 3) uu'u* + 2 (n - 2) (u - 3) u'3

U4 = (n — I)3 uW — 4 (n — I)9 (n — 4) u2u'ua — 3 (n — I)8 (n — 3) wV2

+ 12 (n — 1) (n — 3)2 ira'2 t*" — 6 (n — 2) (n - 3)2 u'*

U5 = (n — 1)* ttV - 5 (n — I)3 (n — 5) usu'u* - 2 (« - I)3 (5n — 17) wV'u"1
+ 4 (n — I)2 (5n2 — 36n + 67) wW" + 6 (n — I)2 (n — 3) (5n — 17) u*u'u"2

- 60 (n - 1) (n - 3)3 uu'3u" + 24 (n - 2) (n - 3)3 u'6.

76. Taking first the case of the quartic equation

we have as covariants of this equation

Ua = 3tmu — 2w'2,
= 9ttV

v - 27Msw"8

TJ3 = 9ttV — 9uu'uH

from which

U4 + 3U22 =

whence from the differential equation

U+ + 3U22 = — 108 Q3wV - 27 Q,>i*.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS 437

Hence

U4 + 3U,8 + 27u4e4 = - 54w8 (uQ's + 2«'Q3),

= -18u8(3uQ'3 + 6«'Q3)
= -18«3 (3u8's + 6t*'e3),

BO that U4 is expressible in terras of invariants and covariants already retained. And
this relation between the invariants and covariants, viz.,

U4 + 3U/ + 27^ + 18u303 (M)! = 0.

is practically the same as the differential equation, which may thus be considered as
replaced by a relation between its invariants and covariants.

77. Taking now the case of the quantic of the fourth order, viz.,

(which we are entitled to include among the aggregate of invariants and covariants,
its index being £), we find, just as in the case of the equation,

being £), we find, just as in the case of the equation,

18w30 u =

so that U4 can be expressed in terms of the invariants and covariants and of <J>V If,
then, 4>4 be included as a fundamental covariant, and, in consequence of this inclusion,
all the proper derivatives from it be also included, then we have U4 and all subse-
quent identical covariants expressible in terms of the covariants of the system thus.
increased. But if, on the other hand, the quantic (and derivatives from it) be not
included, then the number of the identical proper covariants may be taken as
unlimited ; and 4>4 and all its derivatives are composite in terms of the invariants and
covariants. This is the reservation referred to in § 67.
78. Taking, as a last example, the quantic, viz.,

(with 4>6 = 0 for the equation), we have

4ttV - 30MU' V
U8 = 4utta — 3u'»,
3 = 4w2ul" - 6t*uV + 3u'»,

438 MB. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-
and therefore

Tfcu6

so that

3 = loQ3t*V

by (16) in its canonical form. And by (15) we have

Q4 = 04 + 2Q'3)

and

Q3 = 03

in the canonical form of 0, so that

Hence

«4*6 ~

= 10w3

by (iv.) and (xxiv.) ; and, therefore,

7@'32) + 1OM*«'0'3 + 1%803 (U2
3 + J# («20'32

From the existence of this covariant relation, inferences as to the number of identical
co variants may be derived similar to those made in the case of the quartic.

Symbolical Expressions for Successive Jacobian Derivatives.

79. A very simple symbolical form can be given to the covariants, obtained by
continued application of the Jacobian process from two fundamental concomitants.*

* See also HALPHEN, ' Acta Math.,' vol. 3, p. 333.

DERIVATIVES ASSOCIATED WITH LINEAR 1)1 Fi I III NTI AL KQ NATIONS. 439

First for the case of the derived invariants, we may take (xi.) as n representative,
viz.: —

To transform this we write —

e
•

BO that

Hence,

O
If then we write —

this equation comes to be

•'. //^^r .

and therefore

or by re-substituting we have

*"'

'"«

_

the symbolical form for the derived invariants.

Similarly, for the identical covariants, it may be proved that

and that

Corresponding expressions may easily be found for the mixed covarianta.

410 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-
SECTION VI.

APPLICATION TO EQUATIONS OF LOWEST ORDERS.
Equation of the Second Order.

80. For the equation of the second order there are no invariants. So far as
concerns the reduction of the equation to a normal form, it is at once evident that, by
a literal application of the result in § 30, the equation would be reduced to the form

by the solution of a linear equation of the second order. There is thus no simplifica-
tion or advantage in the reduction, for the original equation of the second order might
as well be solved, the subsidiary equation being, in fact, identical with the original:*

But it is interesting to notice bow the well-known theory of the solution of the
equation of the second order is contained in the general results. In the case of
n = 2, we have, by (iv.),

By the transformation y = \u the equation

is transformed to

provided (21) z be determined by the equation

{*, x] = 2P2.

The two independent solutions of the transformed equation may be taken to be
1 and zf; and hence the two solutions of the y-equation are 2'"* and 22'"*. And z is
now the quotient of two solutions of the original equation.t

f}. ,.',,.;.>, •!•_•. vim •:> ;;.,f 3,,.,' ,/i.tZv ;.. ".v ^.O

* This result may be compared with the result of applying TSCHIRNHAUNEN'S transformation to the
general algebraical quadratic equation.
f See my ' Differential Equations,' p. 92.

DKKIVATJVES ASSOCIATED WJTH LINEAR DIFFERENTIAL EQUATIONS. 441

/•''/nation of the Third On/' r.

81. The general results obtained in § 30 show that, by the solution of

K*}=*P*
the equation

is transformed to

where
and

= 0,

2"e=P8-i^:

and, if we write z = 0~2, the equation determining 0 is

(22)

The form (22) is the canonical form of the cubic.

82. First, if the solution of (22) be known, then that of

cPv
dz*

(23)

can be derived from it, and conversely. For let M,, ?tj, u3 be three special and linearly
independent solutions of (22); then we have

= A,

where A is a determinate constant. Introducing a new quantity v3, defined by the
equation

VS = MjW!2

we have

= — 6

MIXV« lAXXVIII. A.

3 L

442 MR. A. R. PORSYTH ON INVARIANTS, COVAR1ANTS, AND QUOTIENT-

Hence three linearly independent solutions of (23) are u2u[s — usul2, usu\ — «i«'3, and
Wjtt'z — u.2u\, say vlt v2, v3 respectively. This proves the first part of the proposition ;
and for the converse we have

= ui v\u + v*u + vsu — Mi viui + viui + vaus
= A.ul,

so that, if the solution of (23) be known, then that of (22) can be derived.

It is evident from the method of formation of (23) that it is the " adjoint " of (22),
see § 52 ; the fundamental invariant is the same for the two equations, the change of
sign not affecting the invariantive property. We thus have a verification of the
proposition (2) of § 62.

83. Second, one immediately integrable form of the equation (22) occurs when
6 = C2~8, c being a constant, for the primitive is

u = AjZ"" + A22"* + A82»"

where mlt «i2, ma are the roots of the equation m (m — 1) (ra — 2) = c. Another
occurs when © = cz~*, in which case the primitive is expressible in terms of BESSEL'S
functions.*

84. A third case, mentioned by BRIOSCHI (1. c., § 5), occurs when 0 vanishes ; we
may then take

M! = 1, % = 2, % = 22,
so that

and therefore

?i _ y*

y* y$

or y^ = I//, which is practically equivalent to a general quadratic relation

(*Xyi» y» y*f = °-

Since 0 vanishes, we have for the uncanonical equation 2P8 = 3 dP^/dx ; and, there-
fore, three linearly independent integrals of

dy 3 dP3

*LOMMEL, 'Mathcmat. Annalen,' vol. 2, pp. 624-635, but without any notice of the adjoint relation
between the equations of odd order considered.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 443
being z"1, z'~lz, z'~*z*, are given by

P, P\e-*dx, ffi W-'dx}*,
wliere 0 is determined by the equation

Between any three linearly independent integrals there subsists a homogeneous
quadratic relation.

The Quotient-Equation for the Ciibic.

85. By this is to be understood the differential equation satisfied by the quotient of
two solutions of (22). Since every solution of the fundamental equation implicitly
contains, in linear and homogeneous form, three arbitrary constants, such a quotient
will implicitly contain five (= 6 — 1) independent arbitrary constants ; and the
differential equation which it satisfies will therefore be of the fifth order.

Let «j and «2 be any two solutions and s their quotient, so that

Then, by (22), we have

= t< *.

and, therefore,

0 = Wja1" + Sites'1 + Stt"^.

When this equation is differentiated and substitution is made for «"', it follows thut

and another differentiation and substitution give

When MI, tt'i, w11! are eliminated between these three equations, we have

_ = <>, ... (24)
- Ss'e, 4s"1, 6s"

3s", 3*1

the equation required, evidently of the fifth order.

3 L 2

444 MR. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-
86. Similarly, had we taken

ttg = U^,

it would have appeared that the differential equation satisfied by o- is the same as
(24). Hence we derive the conclusion that, if a- and T be special solutions of (24),
the primitive of it is

A + B<r + CT
= A' + B'«r + C'r '

Now, if we consider these two special solutions cr and T to be known, we have

0 = u^ + 37/V' + 3«'X,
0 = wj r1" + 3u\ T" + 3u\ T1 ;
and, therefore,

O^T* -

so that

u\ (a*^ — ^^) = constant.

Hence we may take

(o-V

- 0*1*)-*,

as three special linearly independent solutions of (22) ; they constitute a fundamental
system of integrals, and any other integral can be expressed in terms of them.

87. It is not uninteresting to see how from these forms the case considered in § 84
may be deduced ; we then have

r=cr>,

so that

r1 = 20-cr1, T" = 20-0^ + 2o-' \

whence, by neglecting a factor — 2~l, which may be absorbed in the quantities u, the
three special solutions are

1 a- a3

Ul=-, uz = -> u3 = --
Taking u± = I/a; we have

the substitution of which in

0 = Mjtr1" + Sw'jO* +

DKIUVATIVES ASSOCIATED WITH l.INHAR DIFr'KKKNTIAL EQUATIONS. 445

gives

0= -2{<r, z}, i.e., {<r, z] = 0.

As we are seeking the values of special integrals, it will suffice to obtain them in as
simple a form as possible. Since <r may not be a constant, we therefore take a- = z,
and the corresponding values of u are

and consequently 6 = 0, which agrees with the former result.* In this case the
quotient-equation is

ST, 5**, 10s"1
s*, 4slu, 6s"
«"•, 3s", 3s1

= 0

(25),

and the primitive of this equation is

s =

A + Rg + Cc*

A' + Wz + CTc

(26).

The function on the left-hand side of (25) will be called the quotient-derivative
associated with the cubic, or, more shortly, the cubic quotient-derivative ; the corre-
sponding function for the equation of the second order, viz.,

s"1, 3s"
s", 2s1

which is 2s'2 {s, z}, being a multiple of the Schwarzian derivative, may be called the
quadratic quotient-derivative. The consideration of these derivatives, and of others
of higher order, will be resumed later ; but it may be mentioned that, if 0 denote the
Schwarzian derivative {s, z}, ¥a (= 2s'2 ff), and T3 respectively the quadratic and the
cubic quotient-derivatives, then

and the equation (24) is

¥ - 27s'3 e2 -

"¥2 - 54s'V + OOs'sV").

88. A particular case referred to by MAi,ETt is at once reducible to one of the cases
considered in § 83 ; for, supposing s = (az + b)/(cz -f- d) so that 6 vanishes, we have

and, therefore,

'sul = 3s"*,
= 2susm.

* The result would similarly follow, if a were taken in its general form (at + 6)/(c* + d).
t ' Phil. Trans.,' 1882, p. 759.

446 MR. A. R, FORSYTH ON INVARIANTS, CO VARIANTS, AND QUOTIENT-
Thus the quotient-equation becomes

27s/8e2 = ISes'sV = 27s"3«,
and therefore, if 0 be not zero, it is given by

. - W-

m\fj.m

(cz

which is practically the first integrable case of § 83.

89. The integration of the original differential equation, as given in § 86, depends
on the supposed knowledge of two special solutions of the equation (24); and the
formulae are, for the cubic, the analogues of those quoted in § 80 for the quadratic.
It is not, however, necessary to suppose two special solutions known in order to
obtain the primitive ; this primitive can be derived from a knowledge of a single
special solution cr. For we have

0 = Mjo-"' + 3u\<rli + a1 3u\,

0 = «j (a-" — 3o-'0) + 4w!1o-m -f 2o-u 3u\,
and therefore

4o-i«r"i - 6<r"2

whence we may infer

V
or

«1 =
and

and an expression for us can be deduced by the ordinary method, for two particular
solutions of the cubic are known.

Similarly, from the single solution T of the quotient-equation we should have

and an expression for w2 can be deduced by the ordinary method.
90. In connexion with the equation

regarded as an equation of the second order determining u^ a result, which is rather
curious from the analytical point of view, can be obtained. Denoting by p the

DDIUVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 447

quotient of two solutions of this equation of the second order, and taking one of these
solutions to be «j, we have, by the application of a well-known formula,

dz "itjV
But, by the result quoted in § 80, we have

_iii J

« t ^ v i **

so that a combination of the results obtained gives the solution of the equation in p,
which is

Equation of the Fourth Order.
91. The general results obtained in § 30 show that, by the solution of

{*,*}=*P»

the equation

% + «?.% + «r.2 + *.»-o

is transformed to

£+4Q,t+Q.u = 0 ....... (27),

where

z/ly = u,
and

. ..

2 dz

and, if we write 2' = 0~*t the equation determining 6 is

The form (27) is a canonical form of the quartic in conformity with the general
canonical form ; and the quartic can be reduced to this form by the solution of a
linear equation of the second order.

448 MB. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-

92. But the equation (27) is not the only simpler form, consistent with complete
generality, to which the original equation can be reduced ; the following transforma-
tion presents a close analogy with the reduction of an algebraical binary quartic to its
canonical form. From previous investigations we know that, when the substitution
z y = u is applied to the original equation, it becomes

where, if Z denote z"/z', we have

Z" - 3ZZ' + Z3 =

dT

(these three being equivalent to two independent equations), and

_ A ^-?i __ Ai p

fi » a O t; -1- 9

.

25

The quantity z is at our disposal ; and, if we choose it, not as before in a way to
make Ra vanish, but so as to make R3 vanish, then the differential equation is

(28)

which is an alternative canonical form of the general quartic. The equation which
determines z is then

Z" - 3ZZ' + Z8 = | P3 - ^ P3Z,

or, writing Z = — v'/v, so that vz' = 1, this is

a linear cubic with its priminvariant = £ @3. Hence, by the solution of a linear cubic,
the general quartic can be reduced to the canonical form (28) ; the new independent
variable z is \dx/v, where v is any integral of this cubic equation ; and the coefficients
R4 of the canonical form are then given by the equations

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 449

r -I/*)'

or

=r» = (^'-*P,)v3

and

and the dependent variables are connected by the relation

y = ttv'.

93. There are many special cases of these forms depending on simpler analysis ;
thus, one of such cases is that wherein the priminvariant 63 vanishes, and then the
form (27) comes to be binomial, while in (28) the coefficient R^ is constant.

The two forms (27) and (28) are practically the alternative normal forms of the
quartic ; it is not possible by this method to reduce the general equation to the
binomial form

for such a reduction requires that the coefficients of cPu/dz?, d?u/dzz, du/dz shall all
vanish — three conditions which cannot, in general, be satisfied by proper determina-
tion of the multiplier X and the independent variable z. In the case of all the forms
which have been chosen the general assumption has been made that it is desirable to
remove from the equation the term of order next to the highest ; for any equation, in
which this might not be done, other forms could be obtained, but the analysis of
Section II. shows that those forms adopted have the advantage of being most easily
obtained. It may be remarked that, for the reduction of any equation to the
canonical form adopted, the subsidiary equations are all of order less than that of the
equation to be transformed.

The Quotient- Equation for the Quartic.

94. The differential equation satisfied by the quotient of two solutions of the
quartic must be of order 7 (= 2.4 — 1), since each of the solutions contains implicitly
four constants in linear and homogeneous form.

Taking ux and «3 as two particular solutions of the equation in its canonical form,
and denoting their quotient by //,, we have

"2 = %/* 5

proceeding as in the corresponding case for the cubic, the following equation is
obtained, viz. : —

MDOCCLXXXVIII. — A. 3 M

450 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QTJOTIENT-

0 = 4«' V + Cu V + 4«*V" + u, (p.* + 4Q3/,'),

and thence, by continued differentiation and substitution,

0 = 10M1 V + 10w V»+ u\ W - 12Q^') + U + 4

0 = 20t*Vu + ""i (1 V -

0 =

u

+ u\ 6/1' - 8 (QS/A') - 4Q,>«

- + 4 |3 (Q3/A') - 4 £ (Q^1
u\ J21/t' - 20 £ (Qs/,1) - 40ft"Q3 -
) - 8 |(Q4/,') - 40 |(Q3/,") - lOQ^t" - 800,/t"1}
) - 4 , (Q^«) - 10 (Q,/) -

(29).

The determinantal equation which results from the elimination, between these four
equations, of the four quantities ult u\, u\, u[\ is the equation required ; it is
evidently of the seventh order.

95. Had the initial quotient relation been taken us = ujj, the equation in p would
have been the same as the equation in \L ; and similarly for an initial relation
w4 = Uj\. Hence it is to be inferred that, if X, cr, p be three particular solutions of
the //.-equation, its primitive is

A + EX + Co- + Dp
ti~ A'+B'X+CV + D'/j'

96. In particular, if in the original equation Q3 =0, Q4 = 0, so that the two
priminvariants vanish, the equation which determines \L is

, 21/A 35^ =0

(30),

the left-hand side of which may be called the quartic quotient-derivative. Special
solutions of the original differential equation are now

so that

M! =1, u2 = z, u3 — z2, t*4
[L — z, p = z2, X = z8 ;
and, therefore, the primitive of the equation (30) is

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 451

A + B* + C*« + Da"

•-•" (31)-

The generalisation to the case of the equation of order n is so obvious as to render it
unnecessary to give the forms of the equations explicitly.

97. Suppose now that three solutions X, a-, p of the quotient-equation equivalent
to (29) are given, as in § 95 ; then we have, by the first of those equations,

0 = 1*! (XlT + 4Q3X') + 4M'1X1" + GttV + 4ululX1,
0 = it, (<r* + 4Q3<r1) + 4ttl1aJB + Bu^o* + 4u'V,
0 = «, (p* + 4<V) + 4W'1/>m + 6t*V + 4«V>

and therefore

Xlr + 4QSX1, X", X1
<r"+4Q3cr', (T11, cr1

or, what is the same thing,

0 = «,

X", X", X1
or", o* cr1

X"1, Xu, X'
o» a», o-'
Pm> P", P1

Xm, X", X1
cr1", (7", tr1

A P", P'

Hence, writing

we have

so that we may take

A= X1U, X", X1

Pm» P". P'
«j* A =: constant,

(32);

and the primitive of the general equation is

u = (A + BX + Co- + D/>) A-*.

It is evident that no one of the quantities X, cr, p may be constant, nor may any two
of them have a constant ratio.

98. It has already appeared, in § 96, that, if the two priminvariants vanish, then
relations

3 M 2

452 MR. A. R. FORSYTE ON INVARIANTS, COVARTANTS, AND QUOTIENT-

for the transformed, and therefore

Va _ .Vs _ y*

y\ y* y»

for the untransformed, equations hold. (It should be remarked that these are not the
most general pair of quadratic relations ; in fact, interpreted geometrically, they
represent a pair of quadrics which by their intersection determine a tortuous cubic.)

We now proceed to prove the converse — that, if two quadratic relations of the
foregoing type hold, then the priminvariants vanish.

Taking the four solutions of the equation in the form

M! = A-', u.2 = XA-», u3 = O-A-', M4 = />A-J,

the relations given are equivalent to the new relations

o- = X2, /> = X3.
When these values are substituted in A, it becomes

A= X"1, X", X'

2XXm + GX'X11, 2XX" + 2X'2, 2XX1

3X2Xm + ISXX'X" + GX'3, 3X2XU + 6XX'2, 3X2X'
= - 12X'6.

Since any constant factor may be absorbed into the particular solutions ult u%, Wg, u^,
we may take

«! = X'-«.
Again we have

0 = MI (X1T + 4Q3X') + 4«'1Xm + Gw^X" + 4wlil1Xi,
0 = M! (a** + 4Q3o-') + 4tt11ojii + GM^CT" + 4Mili1oJ.

When in the latter we substitute cr = X2, and from the resulting equation we
subtract the former, multiplied by 2X, the new equation is

0 = Ul (8\l\m + 6X"2) + ±u\ . GX'X" + 6u\ . 2X'2,

which, by the substitution of the value of ult changes to

0 = - 10X'» (X, z],

or, since X' is not zero, we have

(X, z} = 0.

We therefore take X = z ; the four solutions become 1, z, z2, z3 ; hence Q3 and Q4 are
both zero, and the priminvariants vanish.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 453

99. In the case of the alternative normal form (28) for the quartic, the quotient-
equation is, as before, of the seventh order ; and, if X, <r, p be three special solutions of
it, we have

0 = HI (6IV + /*") + u\ (12M'R* + V") + 6u>" + 4tt'V

for IJL = X, a; p, and therefore

u1

4X111 + 12X%, X", X1
+ 12^ er", (r1

X"+GR3X", X", X1

p" +

or, since these determinants are independent of R^ we have a result the same in form
as before

= 0,

X1", X", X1
<r»', <r», cr'

= constant.

The results for this normal form, which correspond to those given in §§ 95, 97, are the
same as are there given.

100. For both forms it appears that, when the two primin variants vanish, four
solutions are given by u = 1, z, z9, z3 ; hence the primitive of the equation

the priminvariants of which vanish, is

y=ffi[A.+ Eie-*dx + C{ie-*dx}i+ {\0-3dx}*]
where 6 is determined by

101. But, as in the case of the cubic, it was not necessary to know more than a
single solution of the quotient-equation in order to obtain more than one solution of
the original equation, so in the case of the quartic the knowledge of a single special
solution of the quotient-equation, not a constant, is sufficient to give two special
solutions. For, if /* be such as to satisfy the quotient-equation, we can from the
first three equations of (29) find the value of t*Vwi explicitly and thence w, ; the value
of M2 is then known being ILUV Similarly, from a knowledge of two solutions of the

454 MB. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

quotient-equation, three solutions of the differential equation can be explicitly obtained,
and the fourth can be derived.

Associate Equations of the Qicartic.

102. The associate equations are two in number. One of these has a dependent
variable of the type

t— ult u.2, «3 . /, . . . ...... • (33),

u\,

u\,

of which there are four distinct values, so that the equation is of the fourth order.
The equation is, in fact, by § 52,

=-«z«M

dz

that is,

The priminvariants of this are
(a) -Q3,

08) Q* - 4

dz

dz

or, since change of sign does not affect the invariantive character, the invariants for
the adjoint equation are the same as for the original equation.

103. The other of the associate equations has a dependent variable of the type

v =

u\,

(35),

of which there are six linearly distinct values, connected, however, by a permanent
bilinear relation

= 0.

For the variable given by (35) and the quantities u as satisfying (27), it is easy to

prove that

vui
and thence that

(^ + 4Q3v)u - 4Q^ - 2vQ'4 = - 8Q3 (u>'2 - u\u\\

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 455

104. In the case when the invariant 93( = Q3) vanishes, so that the quartic is
canonically binomial, the equation in v is linear and of the fifth order only,* being

dv

and there is, therefore, a linear homogeneous relation among the six quantities v.
The constants in this relation depend partly on the choice of the fundamental
system of integrals, partly on the invariant Qt ; e.g., for the equation

d*u , c

^ ?
we may take

M1 = Z"', U2 = Z-, U, = Z-, W4=Z"S

where

2m, = 3 - {5 - 4 (1 - c)»}», 2^ = 3 - {5 + 4 (1 - c)»}»,

2ro2=3+{5-4(l-c)»}', 3m4=3 + {5 + 4(1 - c)»}*,
the indices mlt m2, m^ m4 all being roots of

m (m — 1 ) (m — 2) (m — 3) + c = 0 ;
and the linear v relation is then

Multiplying the equation by v, it can be integrated once, with the result

d*v dv cPv

where A is a determinate constant. This constant depends, like those before, partly
on the choice of fundamental integrals and partly on the invariant 04 ; and it changes
from one quantity v to another. Recurring to the particular example, we have

vu = {5 — 4 (1 — c)»}»28 = 0z*t
say ; and, substituting, we find

20* = 2c0* + A
• HALPHKN, ' Acta Math.,' vol. 8, p. 329.

4 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

or

Similarly, we should find

l -c),
= 10 (1 — c) — 8 (1 — c)'.

AS4=10(l-c) + 8(l -c)».

105. In the case of the general quartic (for which Q3 does not vanish) the differen-
tial equation for v is

dz

= 4

40,*)

or, expanded and rearranged, it is

When the covariants 03 and 64 are introduced, this is

106. A first inference from this equation (36) is that

v*

= B®

8>

where B is a determinate constant depending on the selection of the original funda
mental system of u integrals.
107. Next, the substitution

changes (36) into

V =

(37)

dz

= 0 . . (38),

where, after long and laborious analysis depending largely upon continued application
of the theorems of Section III. as to the values of the successive derived invariants,
it may be proved that

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATION?. 457

81 '"' i
» = Tir»

. 1 *V« + 3&n

— Ins (_,

H' 1

85 = J 64 ej — 6'4 + ^ {^fy e3i4 + ?£j e'sBs.3 + T^T

^6 = ^~« if^HS e3,5 + TlVW ^3,3^3, 1 + ?7l 8/3e3,+ + T8 3

Let the priminvariants of this associate sextic be denoted by 4>3, <J>4, 4>6) *6 ; and
let the Jacobian 4e46'3 — 8836^ — a proper covariant of the quartic — be denoted by
"V. Then for 4>3 we have

,1 5J i i ?a i

" TO e,s " " « e,s

i _ QS.» i i 8'sQ.i.i i B».a + SQ'gB,,!
* 191 0s :«> (.) (_) •

that is, $3 is an invariant of the original quartic. Again, we have by (15) of § 22 the
invariant *4 given by

for in the present case n = 6 ; and it is not difficult to prove that, when the foregoing
values of S are substituted, the value of *4 is

e3.3-¥eS)l2)-Ae4 ...... (40).

I give below the values of *5 and <J>a, founded on (16) and (17) of §§23, 24; the
analysis is long for each of them, but, as it is of a character precisely similar to that
for 4>3, it is not reproduced here. The value of <I>6 is

' + i .... (41),

and the value of <!>„ is

MlKVrl. XXXVIII. —A. 3 N

458 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-

— Ci 8 11934

1 '.:

79 ®3.5~i~Ta865~9

:; :t V '«,

T8» e»

, ,

~

+ i
TTS^T

iu

1333

335 8' 4

72

— Ta

(42).

These values show that all the priminvariants (and therefore all the derived
invariants) of the associate sextic are included in the invariants of the original
quartic ; and since the variable of the sextic is covariantive, and is included among
the covariants of the given equation, it follows that all the covariants, identical and
mixed, of the associate sextic are composed of covariants and invariants of the original
quartic. Hence, the theorems of § 62 are verified for the linear quartic.

108. There are many other equations possessing covariantive properties similar to
those in the associate variables ; among such equations are those, for instance, which
have their dependent variables composed of one or more than one of the aggregate
of dependent variables, original and associate. Thus the equation, which has
for its dependent variable the square of the dependent variable of the equation of
order n, is of order %n (n + 1), and all its invariants are invariants of the original
equation ; and the reduction of such an equation, when obtained, to its canonical form
will be very similar to the reduction to its canonical form of the associate equation
which has, for its dependent variable, the variable associate of the first rank of the
equation of order w -f 1. Thus, for instance, if we write t = u~ where

= 0

it is easy to prove that the equation in t is

dz* v ' ' 0 dz

and the verification that the priminvariants (and therefore all the concomitants) of
this equation are included among the invariants of the quartic would proceed on lines
very similar to those of the verification for the quartic.

109. For the general differential equation of order n, the equation satisfied by the
quotient of two solutions is of order 2n — 1 ; a knowledge of n — 1 special solutions
Xj, X2, . . . , X,(_! gives the primitive in the form

. . _ A0 +
"

+ . . . +

BO
and leads to the derivation of n particular solutions of the original differential equation

in the form

_j _i _i _i_

where

t-l) x(»-2)

.-1)

,-2)

\ (« - i) \(.n- V

A ,.A ......

n — I ' • — 1 '

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 459
And in the case when

the degenerate form of A is easily proved to be

(n- 1)1

so that we may take

u —

« —

As in §§ 87, 98, it would be proved that Xx = z and that all the primin variants vanish.
It is not proposed to consider here what are the possible methods of forming for
the general equation of order n the associate equations in the different variables,
nor therefore to verify, as was done in § 107 for the quartic, the general theorems
enunciated in § 62.

SECTION VII.
QUOTIENT-DERIVATIVES.
The Derivatives of Odd Order.
110. Writing down the series of quantities with binomial numerical coefficients

*'

, *

.

a"

, u

>

, 3*"

, 3*'

, *

*"

> ^"

, Gs"

, 4^

«T

, 5«*

, !<*•

, lOa"

, 5^

, *

.•*.

, 6*'

, 21«T

, 20alu
, 35fP

, 35*"1

, GJ , s
, 21s" , 7s' , *

>

, 8s
, 9**"

, 28*^
, 36*""

, 56*T

, 70*"
, 126»T

, 56«Hi , 28A 8*' , s
, 126s", 84.sm, 36s", 9s1, s

and forming determinant squares as above, viz , the first element of the first line only;
the first two elements of the two lines after the first one ; the first three elements of

3 N 2

4GO MR. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

the three lines after the first two ; the first four elements of the four lines after the
first three ; and so on, we obtain the series of quotient-derivatives connected with the
linear equations of successive orders.

The first* of these functions, viz., s1, is the linear quotient-derivative ; in conformity
with the notation used for the remaining functions, it will be denoted by [s, z]^
Then

0,21 = 0 . . .'. . ;. .... ..... .,,, (43)

is the differential equation satisfied by the quotient of two solutions of the equation

dit

dz = °*

and the primitive of the equation (43) is

where A and B are constants.

The second of these functions, viz. : —

A

(43')

s", 2s1
sm, 35"

is the quadratic quotient-derivative (§ 87); it will be denoted by [s, z]2. Then

0. *1 = 0 (44)

is the differential equation satisfied by the quotient of two solutions of the equation

= 0,

and the primitive of the equation (44) is

'

B

(44'),

where the coefficients A and B are constants.
The third of these functions, viz. : —

*"', 3s", 3*'
s'v, 4s"1, 6s"
s", 5slT,

This function is inserted merely to make the enumeration complete.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 461
is the cubic quotient-derivative (§ 87) ; it will be denoted by [*, z]s. Then

= 0

is the differential equation satisfied by the quotient of two solutions of the equation

./'/'

and the primitive of the equation (45) is

A i A f i A -rS

AO ' «M* • '"'r / A c\\

*= U0 + KiZ+BS ' •;.-.' ' ' ' (45>'

where the quantities A and B are constants.

Similarly, in general, the nth of these functions is the ntic quotient-derivative,
which will be denoted by [s, z]«. Then

• . r [>,4. = 0 (46)

is the differential equation satisfied by the quotient of two solutions of the equation

and the primitive of the equation (46) is

_ AO + Atz + Agg* + . . . -I- A,,! f i .g|x

• • * •* . Tk . -n _a . . *» «*~1 ••*••. ^*U jt

where the quantities A and B are constants.* The equation (46) is of order 2n-l, of
course non-linear, though it is of the first degree ; its primitive (461) involves effectively
2n-l arbitrary independent constants.

111. In the case of the quadratic derivative, the primitive (44') of the equation (44),
obtained by equating the derivative to zero, is symmetrical qua function of the
variables in 5 and z. Regarded in this light, the variables in the equation may be
interchanged, so that the equation

[>,2]2 = 0
implies the equation

[z, «1 = 0,

• This result was given by CAPT. MxcMAHON in A note, unknown to me at the time of reading of this
memoir, in the ' Philosophical Magazine ' for June, 1887, p. 542.

462 MR. A. E. FOESYTH ON INVAEIANTS, COVAEIANTS, AND QUOTIENT-
and the one derivative is a factor of the other ; hi fact, we have the relation

On account of this property the function [s, z\2 is called by SYLVESTER a reciprocant.

In the case of derivatives associated with equations of order higher than the second,
the primitive of the differential equation, which is obtained by equating the derivative
to zero, is not symmetrical in regard to the dependent and independent variables ;
they may not therefore be interchanged, and hence these derivatives are not recipro-
cants of any of the known types. It is elsewhere * shown that the connexion between
the two classes of functions is constituted by the property that the quotient-derivatives
are combinations of homographic reciprocants, such combinations being, however,
illegitimate for the preservation of reciprocal invariance.

Transformation of the Derivatives.

112. By means, however, of the primitives of the derivative equations, relations
are easily obtained which suggest some of the transformations of the derivatives.
For, taking the most general change possible, viz., of both the dependent and the
independent variables, suppose (i) that $ and z are connected by the equivalent
relations (46) and (46'), (ii) that cr and s are connected by the equivalent relations

[<r, a], = 0
and

_ C0 +cy + ... + c»_is--i

= -'

and (iii) that z and x are connected by the equivalent relations

[z,xl=o
and

Then the algebraical relation between cr and x is

-

= H0 + Klz + .".r

where

/3-l = (m-l)(n

and the differential relation is consequently

[cr, X\ = 0.

*" Homographic Invariants and Quotient-Derivatives." 'Messenger of Mathematics,' vol. 17,1888,
pp. 154-192.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 463
Hence we have the result that, if

[or, *]„ = 0, [a, *]. = 0, [z, x]f = 0,
then

[<r, x\ = 0,
where

113. But, on the other hand, while the algebraical relation between <r and x
involves the proper number of arbitrary constants, they are not in general equi-
valent to 2p — I independent constants ; for, by the method of construction of cr,
all the constants which enter into its expression are composed of the other
(2m — 1) + (2« — 1) -f (2p — 1) independent arbitrary constants, a number in
general less than 2p — 1. There is therefore not, in general, a justification for an
extension of the result so as to include its converses in the form that, if any three of
the derivative equations be satisfied, the fourth is satisfied ; and it is only when there
are certain coefficient-limitations on the form of <r as an algebraical function of x that
the converse can be asserted. An illustration will be given in § 118.

114. The simplest case which occurs is that in which m = 2 and p = 2; for
then p = p, and the deduction to be made is that, if

[«, *1 = 0,
then

+ d' gz +

where a, b, . . ., h are constants. The converse is also tme ; for in homographic
transformations an interchange of the transformed variables leaves the functional
character of the transformation unaltered. Since then these homographic transfor-
mations do not alter the order of the derivative equation, we are led to investigate
the modification caused by them in the derivative itself.

Considering then, the ntic derivative [s, z]g, let us find the effect of a homographic
change of the independent variable given by

* ••

*= — ^ = a;(z)
in CAYLEY'S notation. Now, as in §11, we have

dz" " rml r\ daf
where

464 . MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

y = coefficient of p» in [x(z + p) - x(z)}r

(eh-fg)p

/ _ i y- r

ffl-1!

m_r!r_l!

+

so that, writing
we have
Hence

m!m-l! ^_r
' r — 1 ! m — r!

115. The method of reduction of the determinant transformed by the substitution
of this last relation is conveniently indicated by the reduction of the cubic derivative.
Denoting ds/dz, d2s/dz*, ... as before by s1, .s", . . . and dsjdx, d2s/da?, . . . by sta % . . .,
we have

[s,z]3=

30s,

- 12P<f>sm + 36^^%

+ 60^} ,

Multiply the second and third columns by Xj and X2 respectively and add to the first,
choosing Xj and X2 so that su and s, no longer occur in the first constituent of that
column ; it will be found that su and st have disappeared from the other constituents.
The value of ^ is 2<f>, of Xj is 2<£2. Multiply the third column by X3 and add to the
second, choosing X3 so that s{ no longer occurs in the first constituent of the new
second column ; it will be found that, for the value of 2<f> of Xg, s, has disappeared
altogether from the second column ; and we have

[*»«]» =

30s,

06sT -

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS 465

Treating the rows of the new determinant in the same way as the columns of the
old were treated, we find

I*, *1 =

, 30s*,,, 30*.
0**,», 40s*,,,, 60**,,
50**,T,

In the right-hand side a factor 0s can be taken from the first column, 0* from the
second, 0 from the third ; and then 0° from the first row, 01 from the second, 0s from
the third, giving as the power of 0 the sum

so that

0, z}, = 0* [s, a;},,
or

1 16. The result of the reduction of the ntic derivative is

The method is similar to that used for the cubic derivative. Thus the numerical
factors which determine the algebraical multiples of the second, third, fourth, . . .
columns, to be added to the first in order to remove all differential coefficients of order
lower than d's/dxf, are respectively n — 1, (n — 1) (n — 2), (n — 1) (n — 2) (n — 3), . . . ;
the numerical factors which determine the algebraical multiples of the third, fourth,
fifth, . . . columns, to be added to the second in order to remove all differen-
tial coefficients of order lower than d*~l8/daf~l, are respectively (2 1/1 !} (n — 2),
{31/2! 11} (n — 2) (n — 3), {41/311!} (n — 2) (n — 3) (n — 4), . . . ; the numerical
factors which determine the algebraical multiples of the fourth, fifth, sixth, . . . columns,
to be added to the third in order to remove all differential coefficient* of order
lower than &-**/daf-*, are respectively [3 1/2 1} (n - 3), {4 1/2 I 2 !} (n — 3) (n — 4),
{ 5 1/3 1 2 1} (n — 3) (n — 4) (n — 5), . . . ; the corresponding multipliers for the modifica-
tion of the fourth column are

Jj (»-*). jTij (« - 4) (» - 5), ^ (n - 4) (« - 5) (« - 6), ...;
and so on.

MDCCCLXXXVIII. — A. 3 O

46G MB. A. K. FORSYTH ON INVARIANTS, CO.VARIANTS, AND QUOTIENT-
117. By somewhat similar work it may be proved that

. _ <'"' I , 2l /49\

~ +<f)*'lS'Z]>l

and the combination of (48) and (49) gives

. _ (fld-bey> (

«

«±/l . _ (fld-bey> (ffz_+ XT r -, , ,

' ffz + 4 ~ (eh -/,,)•• (a, + d)»- L

which is the general formula of transformation for the simultaneous homographic
transformation of the dependent and the independent variables.

118. The following simple case will sufficiently serve to illustrate the kind of
limitation, which prevents the converse of the proposition of § 112 from being, in
general, true. From the general proposition it follows that if

[<r, s]s = 0 and [s, x]% = 0,

then

[o-, x\ = 0.

The question then arises : What are the conditions to be satisfied in order that

[>, x\ - 0
may be a necessary consequence of

\cr, s]3 = 0 and [cr, x]3 = 0 ?

Taking the two latter as given, we may replace them by an integral algebraical
equation :

as8 + 2bs + c Aa? + 2Ba; + C

2B'a; + C'

(50),

the two fractions being the values of er, corresponding to the two derivative equations.
And, if it is to be necessary that

|>, a:]2 = 0,

then this algebraical equation (50) must be equivalent to one or more equations of the
form

aa;

8 '

(51)

Hence, when the value of s given by (51) is substituted in (50), it must become an
identity ; the conditions for which are

DERIVATIVES ASSOCIATED WITH LINEAR DIFFKKKNTIAL EQUATIONS. 4G7

aa* + 26ay + cy2 = XA "I a? a* + 26'ay + c'y2 = X A' 1

aa£ -f 6 (J3y + aS) + cyS = XB I , a'aft + 26'()8y + a8) + c'yS = XB- L
a£* + 2b/38 + cS2 = XC J a'02 + 26'08 + c'S2 = Xtf J

six equations, apparently, and really five equations involving the ratios of the four
quantities a, ft, y, 8, so that two conditions must be satisfied among the constants of
equation (50). We at once find

X2 (AC - Bz) = (aS - £y)2 (ac - 6s),
Xs (A'C - B's) = («8 - £y)2 (aV - &"),
X2 (AC' + A'C - 2BB') = (a8 - j8y)2 (ac' -f a'c - 266'),

and therefore the two necessary conditions are

AC-BE AC7 4- A'C - 2BB' A'C' -B'»

ac - V at + a'c - 2W a'<f - i" '

Assuming these conditions to be satisfied and denoting the common value of the three
functions by I*2, we have

ecS - /8y = XR

To find the value of the ratios a : ft : y : 8 we write

y = a6, 8 = fa a = #/,,

so that 6, <f>, \ji are the quantities to be determined. The first of them can at once be
obtained from

a + 2M + effi A

and the second from

a + 26ft + ctf C

a' + 2V$~+c'<t>3 '' ' C' '

From the first three equations we have for any value of £

»(«+ W + 26(a+68)(y + £8) + c(r + £S)2:

It follows that

cXP2 = Atf2 - 2Ba/3 + Ca2 ;

and similarly from the second three that,

c'XP2 = A'09 - 2B'«£ + C'a8.

[It may be remarked that these are the types of the equations which would have
been obtained if substituting for x in terms of s from (51) had taken place in (50)].

3o 2

468 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-
Hence i/» is determined by

A - 2I + (2 e

_
A' - 2B> + C ~ c''

It may be noticed, though the fact is not directly connected with the present inves-
tigation, that the equation (50) is, if rendered a non-fractional equation, apparently
the most general quadrate-quadratic relation between s and x. But, as a matter of
fact, in order that the most general quadrate-quadratic relation of the form

c0) + s (c^x2 + 26^-f cj + (a.,x2 + 2&,x +c-2) = 0
may be expressible in the form (50), the condition

Oy,

must be satisfied. The proof of this is easy, as is likewise the verification that the
coefficients of the non-fractional equivalent of (50) satisfy the condition.

Derivatives of Even Order.

119. All the derivatives which have hitherto occurred have had the order of the
highest differential coefficient of the dependent variable entering into their expression
an odd integer, and the reason of this is that the dependent variable is the quotient of
two solutions of a linear differential equation having its right-hand member zero, so
that each solution contains implicitly in homogeneous form n arbitrary independent
constants, and the quotient of the two therefore implicitly contains 2/i — 1 arbitrary
independent constants. Hence the differential equation satisfied by the quotient is of
order 2n — 1.

But, if we take the quotient of two solutions of the equation

d'y

(where x ls n°t zero), these solutions are no longer linearly homogeneous in the n
implicit constants, and the quotient will therefore contain implicitly In independent
arbitrary constants. Hence the quotient- equations will be of even order; and like-
wibe the quotient-derivatives, if they exist.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 469
By the transformations of § 1 1 the foregoing differential equation becomes

h . . . = v,

mm

where

and, in order to make the term in d*~^ujd^~} disappear, the relation Xz'4u~u = 1 has
been adopted ; hence we have

The variable z is at our disposal ; and, though in the general theory a choice of z
fundamentally more effective than the following can be made (as was done in §§ 29, 30),
yet, our present aim being the deduction of the quotient-derivatives, we shall here
assume that z is so chosen as to make V a constant a, a choice which appears to render
most simple the required deduction. We then have

and the equation takes the form

Let ft be the quotient of two solutions, say «t and w8, of this differential equation,
so that

t/1 = P + A1U)+AaU8+ ... 4 A.U.,

u, = P + Bx Ut + B, U2 + . . . 4- B. U.,
w, = «j 11.

Then the differential equation satisfied by p. is of order 2» ; and it can be obtained in
a manner similar to that employed in §§ 85, 94.

The quotient-derivatives will be obtained for correspondingly limited forms of
differential equations, viz., those in which the left-hand side is constituted by a single
term, which is that of highest order in the differential coefficient.

120. Example I. — For the equation of the first order

we have, since "_. — " ,//, the equation

a = IJM 4

or

0 = (/x — I)a4-

470 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-

Differentiation of this gives

0 = 2/*'
hence

0 ^= IL ""

V, /*"

or, if we take a new variable s = /u, — t , this may be written

0 =

(52).

Now from the original equation we have at once

x = az + B,
2 = 02 + C,

and therefore

B + az

(52') ;

this relation, in which A and B (and from the point of view of (52) a also) are
arbitrary constants, is the primitive of the equation (52). It evidently contains two
independent arbitrary constants.

The linear derivative was s1, being connected with the linear equation du/dz = 0 ;
the new derivative

which is connected with the less simple form of linear equation, will be called the
hyperlinear derivative.

121. Example II. — For the equation of the second order

we at once have, by double differentiation of the equation u.2 = u^, the relation

Successive differentiations of this and substitution for v}\ give

0 = 3/t'a + 3 A'i + (/'

0 = Q^a + ( V" - 2/t'Q.) «', + (^ -

HKIUVATIVES ASSOCIATED WITH UNKAR DIFKKRRXTIAL EQUATIONS. 471
Hence the equation satisfied by /i is

. V
, V

- 5/t"Q3 -

, 8ft1
, 6/x"

= 0-

The quotient-derivative for the present case — it will be culled the hyperquadratic
derivative — is obtained by selecting from the left-hand side of the quotient-equation
the terms independent of Q8 and Q'a and by writing * for p, — 1 ; thus it is

«"', 3*", 3s1
6s"

and then

= 0

(53)

is the quotient-equation when Q2 is zero. But in that case

u, = B, + Bl2 + &2»,

where a = 2&, so that

A +

C + D* + 65s

(53'),

where, from the point of view of (53), A, B, C, D, b are arbitrary constants, is the
primitive of (53), containing four independent arbitrary constants.

122. Proceeding in this manner we obtain for similar linear equations of successive
order a series of derivatives in each of which the order of the highest differential
coefficient entering is an even integer ; and their form is indicated in the following
scheme, similar to that of § 110. The hyperlinear derivative is obtained by forming
the indicated determinant from the first two elements of the first two rows ; the hyper-
quadratic derivative is similarly obtained from the first three elements of the three
rows after the first ; the hypercubic derivative similarly from the first four elements of

472 MB. A. R. FORSYTH ON INVARIANTS, COVARTANTS, AND QUOTIENT-

*

s" , 2*'

s'" , 3s" ,
6* 4s'"

ST , 5s* , 10sm , 10s"
s* , 6.sT , 1 5s* , 20.sm

, 5s1 , s ,
, 15s", 6s1,

the four rows next after the first two ; and so on. And the primitive of the
hyper-ntic derivative equation

[(«, z)}, = 0 . . mv (54)

is

_ A0

s -^—

B0 + BjZ + . . . + B^ Z"-1 + C2»

(54'),

where from the point of view of the derivative equation (54) the constants A, B, c
are arbitrary.

Relation between the Derivatives of Even and of Odd Order.

123. In the integration of the derivative equations the following connexion between
the two sets of derivatives is of interest. Let

be the equation in the (n + l)tic derivative of odd order, and

the equation in the hyper-wtic derivative of even order ; their primitives are of the
form

AQ + AI& + ... 4-

and

B

_ CQ + (V + . . . + C.-^-1

D

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 473

respectively, where all the constants are arbitrary. Hence, from the point of view of
an integral equation, we may write

where E, B, = A,, and so E, is an arbitrary constant. It therefore follows that

[(*.-E.,2)].= 0
is a general first integral of

[«« <Ui = 0.
Again

from the point of view of an integral equation, the value of Frt being D^/Cg, so that it
is an arbitrary constant ; hence

=o

«V z ._t

is a general first integral of

[(o--» *)]. = 0.

Combining these results, we see that

is a general second integral of

[*., *]. = 0,
where

1

r. + fltv

Similarly

0— 2, «]— a = 0

is a general fourth integral of the same equation, where

1

«. = E. +

F _L 2J3 , •

and so on.

Similar results are obtainable for the equation which involves the derivative of even
order.

MDCCCLXXXVI1I. — A. 3 P

474 MR. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-
SECTION VIII.

CHARACTERISTIC EQUATIONS SATISFIED BY CANONICAL FORMS or INVARIANTS

AND COVARIANTS.

Reproduction of Canonical Form.
124. When the differential equation of order n is taken in its canonical form

d*u n\ _ d*~su

d* ~*~ Sln-Sl^d*^*'

and is transformed so as to have a new dependent variable rj and a new independent
variable £ then, from the investigation in §12, it follows that, if we take

the new equation will be without the term in d*~lri/dl;a~} ; and, from the investigation
in §30, it follows that, if £ be determined by the equation

{£ z] = o, •

the new equation will be without the term in d*~2^/c?^"~2, that is, the new equation is
in its canonical form. The last equation gives

t az + b . .

^=^M ' •''•'•"• • - ' <56)>

where a, 6, c, d are the constants ; and the equations (55) and (56) give the relations
by which a canonical form of differential equation can be transformed into a canonical
form.

125. As we are proceeding to investigate, by the method of infinitesimal variation,
the partial differential equations which are satisfied by the concomitants in their normal
forms, it will be convenient to adopt the process of §19 and make £ nearly equal to z.
Thus, taking in (56) the determining conditions b = 0, a = d, c = — \ ed, where c is
infinitesimal so that its square may be neglected, we have

r = c,

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 475

and all higher derivatives are zero to the order of small quantities retained. We now
have

f * i / 1 \ 7 / t\t7\

Now, by §114, it follows that

d* __r^m/ iV--' mlm — l\

dsf* rmi r!r — l!m — rl

for the relation (56), where, in the present case,

- _ ad — be

~~ (d + a)1 ~ r ** '
and

Hence, to the order of small quantities retained, it is necessary to consider on the
right-hand side of the transforming formula only the terms arising from r == m, and
r = m — 1 ; and thus

~ + lm(m-l)€ —. {
Applying these equivalent operators to (57), we have

....... (58).

Similarly, if vf be the associate variable of rank p — 1 and index — ^p (n >— p)
and if t]p be the same transformed associate, we have

(59).
Again, if 6,. be an invariant of index p., and if 4>M be its transformed value, so that

3 P 2

476 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-
we have

. . (60).

126. Now, in discussing invariants and covariants in their canonical forms as solu-
tions of partial differential equations, we may, so far as they are functions of the
coefficients Q of the original differential equation, cease to consider them as explicit
functions of these quantities, and can consider them as functions of the priminvariants
and of differential coefficients of the priminvariants ; for each of the coefficients Q
can be expressed uniquely in this last form. Thus we have

Qs = ®s>

and so on.

Form-Equation and Index-Equation of a Concomitant.

127. We may therefore define the most general covariant possible when in its
canonical form as a function of (i) the dependent variables, original and associate,
and of their differential coefficients, and of (ii) the priminvariants and their diffe-
rential coefficients, which is such that, when the same function is formed for the trans-
formed differential equation in its canonical form, the relation

u(m)

t

is satisfied, X being the index, and the bracketted numerical exponents denoting
differentiation of corresponding order with regard to the respective independent
variables.

128. As an example, sufficiently indicative of the general case, consider identical
covariants which are functions of u and its derivatives alone, so that we may write

* (u, «...) =

?, V>V, ••• )•

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 477

Substituting for u, u\ «",... from (58), and expanding with a retention of terms up
to the first order, we have, as the additive part of <f> which is given by those terms
arising in connexion with t*("), •

Combining these and comparing the two sides, we find that the finite term on each
side is <f> ; and the remaining conditions therefore are

Reverting to the original variables u and z, we may write these equations in the forms

(n - 2m - 1) «W2 + 2<r<£ = 0

(62).

The latter of these two equations determines the form of a covariant <f> ; the former
determines its index <r.

129. The process of obtaining the differential equations satisfied by the function <f>
of (61) is similar to the foregoing; and the result of the work is that the general
concomitant <j> of index X satisfies the equation

+T S [(« + ,.) e»a?*u,l . (xxvl),

M = » ,=o|_ t"'<' J

which may be called the index-equation, and also satisfies the equation

p = »- 1 r-nl/p! n-p! -1

P

=T s

^ = 1 I = I

which may be called the form-equation.

The equations (62) are at once seen to be particular cases of (xxvi.) and (xxvii.)
for concomitants <£, which involve u and its derivatives alone. For the identical

478 MR. A. a. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

covariants in each of the associate variables there are pairs of equations exactly
similar to (62) ; and the equations which determine the invariants are

T 2

130. The index-equation involves operations the only effect of the application of
which is a change in the numerical coefficients of the various terms in the concomitant
to which they are applied. The form-equation involves operations which replace any
derivative of an element of the function by the derivative of that element of order
next lower ; and, if the aggregate of the orders of the various derivatives entering
into the composition of any term be called the grade of that term, the effect of the
operations in the form-equation is to replace such a term by a set of terms of grade
less by unity.

From the facts that both the characteristic equations satisfied by a concomitant
are linear and that the algebraico-differential operations which occur in them leave a
term unaltered in order in the variables and degree in the invariants, coupled with
the preceding conclusion as to the modification of the grade of the term, we can
derive the inference that every concomitant, if not irreducible, can be resolved into
sums and products of irreducible concomitants each of which has the property of
being an aggregate of terms such that, for the aggregate, the orders of the different
terms in the dependent variables are separately the same throughout, the degree in
any invariant is the same throughout, the dimension-number for every term is the
same, and the grade of every term is the same. For instance,

is a concomitant of index 2p. + 2 — (n — 1) — p (n — p) ; the different terms are
resoluble into products of concomitants each of which has the preceding properties.
Hence for every irreducible concomitant there are three kinds of numbers which are
characteristic, viz., the separate orders in the different dependent variables, the
separate degrees in the different invariants, and the grade of the concomitant ; and a
knowledge of these numbers gives the dimension-number, and thence the index, of
the concomitant.

Applications of the Differential Equations.

131. Example I. — The identical covariants which are functions of u.

In order to obtain all such identical covariants, it is necessary to obtain the most

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 479

general solution of the equations (62). For this purpose, proceeding by the ordinary
method, we have to obtain a series of integrals of the subsidiary equations

d±_du_ du' du" du'"

0 = : 0 " (n- l)u~ 2(»-2)tt'~ 3(n-3)«*

Now integrals of these are

A = «,

B = (n - 1) Au" - (n - 2) u'\

so that by the theory of partial differential equations the most general solution of the
form-equation in (62) is

<f> = function of u, (n — 1) uu" — (n — 2) w'8, ...

The number of independent integrals of the subsidiary equations necessary for the
construction of this most general solution is the same as the highest order of differ-
entiation that occurs ; each of the integrals when freed, by means of preceding integrals,
from all hut one of its arbitrary constants itself furnishes a solution of the form-
equation — a conclusion from the ordinary theory of partial equations of this type.
With each new derivative of u of higher order supposed to occur in the concomitant,
there is a new subsidiary equation ; and consequently a single new integral is necessary,
which must of course include in its expression this new derivative. The earlier
investigations show how to derive such a function ; for, by taking the Jacobian of u
and the derived covariant involving what has hitherto been the derivative of highest
order, we obtain a function which involves the new derivative, is invariantive, and so
will furnish the new integral of the subsidiary equations.

It thus appears that any identical covariant which involves at the highest the wth
derivative of u can be expressed in the form

<M«, U2, Us, . ...U.);

and, as this result is true for all the values of m that can occur, we derive the conclusion
that the series of successive covariants already given is a complete series, that is, any
identical covariant can be expressed as an algebraical function of terms of the series.
132. These fundamental functions of the series which come after U8 are not, how-
ever, in their simplest form ; they can be replaced by others, necessarily their
algebraical equivalent and involving the proper derivatives of u, but of lower order
in the variables. In fact, if the grade of the fundamental covariant be an even
integer and equal to 2r, the covariant may be taken in the form

<k,=wu«'>-a1«V2'-1> + aju'V*-2'-. . . + (- l/a^u*— » + ...+(-

480 ME. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOT1ENT-

the subslitution of this quantity, which must be a solution of the differential form-
equation, leads to the condition

a. =

s(n-s)
so that

_ 2r! n — 2r + a — 1 ! n - s — 1 !
2r-sln- 2r -llsln-V. '

since «0 is unity ; and this is true for s = 1, 2, . . . , r, so that ^ is determinate and
can replace U2r. For instance,

and this is functionally the same as U4 reduced, for it is easy to verify that

The index of <j>.y is evidently 2r — (n — 1), which should, therefore, be the value
of cr in the index-equation of <f>.2r ; and the substitution of <f>2r and comparison of
coefficients of (— 1)' a,u(>)u(2r '~ l) gives

2<r -f n — 2s — 1 + n — 2 (2r — s) — 1 = 0,

which is true for all values of s, so that with this value of o- the index-equation is
satisfied.

133. But when the fundamental covariant is to involve an odd derivative of u as
its highest, so that the grade is to be an odd integer, say 2?- + 1 (which is the case
with U^^, we may not take ^-^i to be of order in u so low as the second ; for, with
an arrangement of terms similar to that in <j>y) the last of them would be of the type
M(rV + 1). When substitution takes place in the form-equation, this term gives rise to
a term {u(r)\* which will not occur in connexion with any other term in <£2r4 j, and,
therefore, for the satisfaction of the equation, would have a vanishing numerical
coefficient. The other numerical coefficients would similarly vanish, and the assumed
form of <f>y+i would be evanescent.

The simplest form of ^+1 1K one which is of the third order in u, being a numerical
multiple of the Jacobian of u and <}>2r ; we take as this form

n - 2r — 1 ,
= M- 2 U,.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 481

This is an invariant, and eo is a solution cf the equation (as will be verified immediately
in connexion with a cognate case) ; and it involves the (2r -j- l)th derivation oft*.

As the function next in succession beyond ^>sr+l we take <(*& + & which has already
been found. It is not difficult to see that

3n - 4r — 5 .

differs from u*<f>2r+z by a resoluble function,

Replacing now the quantities U^, U8, ... by the functions <f>, we can enunciate the
result of § 131 in the form : —

Every identical covariant, which is a function of « and its derivatives alone,
can be expressed as an algebraical function of u, U2, U8, $4, ^5, ...
134. Example II. — The derived invariants which are functions 0/"83.
The form-equation for these invariants «/» is

* - 1

and in order to obtain the most general solution of this equation it is necessary to
obtain a proper number of integrals of the associated subsidiary equations

0 ~ 1.66, 2.70', 3.88T,

Integrals of these involving derivatives of B3 in successive orders are

A = 63,

B = A6'8 - * e'.»,

C = A!B"3 - 4A636"3 + ^ e'3»,
D = A6*3 - G6S6"3 + ^ eV,

When we proceed to construct the general solution of the form-equation by modifying
these integrals so that each may contain only a single constant, the right-hand sides
are the successive invariants derived from 83, or are algebraically equivalent to them ;
and thus the required general value of «/» is

= function of 63,

j, .,,

The derived invariants, which arise in successive formation after B^j, are not in their
simplest forms ; they can be reduced in a manner similar to that adopted for the

MDCCCLXX XVIII. — A. 3 Q

482 MR. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT-

reduction of the identical covariants. They form, however, a complete series of
functions, that is, any invariant which is a function of 03 and derivatives of 03 can
be expressed algebraically in terms of the elements of the complete series.

As in the preceding case of identical covariants, the 2rth derived invariant is of even
grade; and the invariant 03i2r can be replaced by i/>3i2r (which is functionally equivalent
to it), where

the coefficients alf «2, . . . , ar being given by the equation

2r! 2r+5! 5!
2r-sl 2r-s+5\ s + 5l s!

a. =

The (2r + l)th derived invariant is of odd grade; and the simplest functional
equivalent is an invariant of the third degree in ®3 given by

^asr + i = es^'s,«r — i(2r + 6) e'3t/»3,2r.

Similarly for the derived invariants which are functions of 0^ and its derivatives
alone. The simplified functional equivalent of 6M]2, is

*„,* = eve;2" - ftew-" + ... + (_ ly&e^e^-' +... + (_ info

where

2r! 2/*+2r-l! 2/t-l!
Pi —

2r — sl 2/t + 2?- — s — 1! 2/t + «— 1! s! '
and the corresponding simplified functional equivalent of @M>2r + 1 is

' r -\- ft ,

So far as regards the index-equation, the first of (63), for these functions, we at once
have, after substitution, the value 2/i + 2r for X in connexion with «/v,or, and the value
3/x, + 2r + 1 for X in connexion with Vv^+i •

It has been assumed in both of these examples that the Jacobian is an invariant ;
it is interesting to verify this in connexion with the differential equationa

135. Example III. — The Jacobian of a derived invariant and the priminvariant.

Let <f> be a derived invariant of 6M, and therefore a function of 0^, and its diffe-
rential coefficients alone ; let p be the degree of <f> in 0M, and let v be its grade. Then
the index X of <f> is

X = p.p -f v.

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 483
Let AM denote the operator

of the form-equution, so that

4^=0 .......... (64);

and let VM denote the operator of the index-equation

»=0

so that

VM<£ = A<£ .......... (65).

Since <f> is homogeneous of degree p in the quantities 6, we have

When this is multiplied by /x, and subtracted from (64), then

so that, if we denote by !!„ the operator

"• = ?,">•' 3e<?'

we have

n^="<A .......... (66),

which may be called the grade-equation of <f>.

186. Denoting by i/> the Jacobian of 6M and <j>, we have

The index of ^ is p. + 1 + ^» 8° that it has to be shown that if/ satisfies the two
equations

and it follows from these that

n^ = (•> +
3 Q 2

484 MR. A. E. FORSYTE ON INVARIANTS, CO VARIANTS, AND QUOTIENT-
For the first of these equations, we have

*' = e; a0M + 8* a0; + e* 90; + • • • '

and therefore

+ e* A" a<§; + 8* A" 30^ + e* A" a0;

But, by (64),

A^=0,
and, therefore,

and so on. Hence*

= 2/te, - + 2 (p. + i) e; + 2 o* + 2) 0; + . . .

= 2VM<£ = 2A<£ ............... (G7).

We now have

and

A MeM = 0 = A ^, A X = 2/i0M) A ^' =

so that

A^ = 0,

and »/» therefore satisfies the form-equation.
Again,

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 485

Now, by (65),

V^ = ty,

so that

J&- ( \ ty- v a*_

v" ae<r> "**' a) ae<r» = ae<r"

and, therefore, for all values of <r,

Hence,

................ (68).

We now have

and i/» therefore satisfies the index-equation.

The fact that the invariant i|> satisfies the form-equation is the justification of the
statement made earlier (§ 131), that the application of the Jacobian operation enables
us to obtain the successive integrals of the subsidiary equations necessary for the
construction of the general solution.

Functional Completeness of the Set of Concomitants.

137. A set of concomitants will be considered functionally complete when any
concomitant whatever can be expressed as an algebraical function of members of the
set ; and this we shall prove to hold of the aggregate of invariants and covariants
which have been obtained in Sections II., III., V.

Let a concomitant $ have as elements entering into its expression u, u\ uu .....
UM, where r is less than n ; vf) v1,,. . . . , v0"^, for values 2, 3, . . . , n — 1 of p, where

486 MB. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-

rp is less than nl/pl n — p\ ; 8M, B',,, . . . , 6(VM, for values 3, 4, . . . , n of /*, where
there is no restriction on the value of s^. Such a concomitant <f> satisfies a form-
equation given by (xxvii.) ; the equation (xxvi.) is satisfied when the form of <f> is
known, the index alone being therefrom determined.

Now, in the form-equation the number of partial differential coefficients of $
(including dfydu, . . ., 9<£/3vp, . . . , 9<£/96M, . . . , which do not explicitly occur, but
which may be considered as present with zero algebraical coefficients) is

r 4- 1 from the terms implying differentiation dependent on u,
rp + 1 . vp,

Hence the total number of partial differential coefficients of <f> is

»•+ 1 + "T1 (TV + 1) + Tfo + 1);

p=2

all the partial differential coefficients in the form-equation, with regard to quantities
other than those supposed to occur in <f>, vanish.

In order to obtain the most general solution possible as a value of <£ involving the
quantities which occur, we form, according to the usual rule, the necessary subsidiary
equations by means of fractions involving differentials ; the number of these fractions,
excluding the fraction d<f>/0, is the same as the number of partial differential
coefficients of <4 in the linear equation, and, therefore, the number of independent
subsidiary equations, being one less than the number of fractions, is

N = r + "T'^-f 1) + T(<v + 1)

p=Z M=«

p — n — 1 fi = n

= r -f 2n — 4 -}- 2 rp + 2 SM.

p = 2 fi = »

To construct the general function <f> we therefore require N independent integrals of
these subsidiary equations.

Now of invariantive functions, which have the properties of being independent of
one another and of involving in the aggregate all the specified quantities and
individually at least one of the quantities, and from each of which, on account of
these properties, independent integrals of the present subsidiary equations can be
constructed after the manner of §§ 131 and 134, we have the following : —

DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 487

(i) The r identical covarianta in u, given by u, Uz, U3, . . ., Ur (or their

functional equivalents, reduced as in §§ 132, 133).
(ii) The rf identical covariants in vp, given by vp, V,jZ, V^,, . . ., V^ (or their

similarly reduced functional equivalents). This is the case for each of the

associate variables vz, vs, . . ., v,-i«
(iii) The «„ derived invariants involving 0. alone and given by «„, 8^j, O^j,

<#V,4, • • •, &.*• This is the case for each of the priminvariants 68, Bv

• • •> 8«'
(iv) The mutually independent bilinear Jacobians ; as a set of algebraically

independent functions, retained after the indications of §§ 36, 72, we may

take the Jacobian of 68 with each of the quantities u, v3 v,_l, Sv

. . ., e,. The total number of these is 1 -f (n — 2) + (n — 3), t.e., it is

2n — 4.

Hence the total number of algebraically independent concomitants, involving the
specified quantities and obtained by our earlier methods, is

pmn — 1 P ' »

= r + 2 rf + 2 *M + 2» — 4

jj=S M = J

= N;

and from each of them an integral can be constructed, which is independent of all the
other integrals.

From the first three of the classes we have already had examples of the method of
construction of integrals ; as an example of the last class, we may take the subsidiary

equation

du

(n— l)tt 2/X6,.

Previous integrals are u = A, 8^ = B, so that an integral of the equation which
appears is

2/iBw' + (n — 1) A8^ = C,
that is,

2/x8^«' -f (n — 1) uS'^ = C,

or, what is the same thing,

It may also be remarked that, while the class (i) of functions constitutes the set of
integrals derived from the u fractions alone in the subsidiary equations, the class (ii)
constitutes the separate sets from the fractions in each of the other associate
variables taken individually and alone, and the class (iii) constitutes the set from the

488 MR. A. E. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT-

fractions in the priminvariants alone, the part played by the class (iv) is in making
equal to one another the individxial fractions in these three principal sets.

We thus have, by means of the functions previously obtained, the full number of
subsidiary integrals necessary to construct the most general solution of the form-
equation ; and it follows that any concomitant can be algebraically expressed in terms
of the concomitants previously given.

Hence the aggregate of the concomitants (or their simplified algebraical equivalents),
obtainable by the quadriderivative and Jacobian operations from the priminvariants
and the dependent variables, is functionally complete.

Limitation in the Number of Identical Covariants.

138. This for particular cases has already (§§ 76-78) been indicated; without
entering at present on the details of the general case, it will be sufficient to obtain
the general result, which, by means of the result of § 1 32, can be simplified. In fact,
IT,, of grade n, can be replaced by a function the first term of which is either UUM or
uzu(*\ according as the grade n is even or odd ; and our present purpose will be
effected by showing that USUM, which will include both cases, is covariantive. For,
since the differential equation

«<">+ V - Qr t*'"-r) = 0

r = » r\ n — rl ^

is permanently true, we shall have

a rv" _?!— n (»-r>

a covariant, if U~UM be a covariant. Now, as has been implicitly proved in the last
paragraph, this covariant is expressible in terms of invariants and covariants already
obtained, the identical covariant of highest grade in such an expression being U,_3 ;
and the expression is therefore an equivalent for USUM. On the other hand, viewed
as an identical covariant, UZUM differs from !!„ (or «UB, in the case of n even) by an
aggregate of terms each of which can be resolved into factors of lower grade ; and
therefore, since the aggregate is covariantive, on the hypothesis of the covariantive
property of U*UM, it is expressible in terms of identical covariants of lower grade. A
comparison of the two expressions thus obtained for U?UM gives U» in terms of
covariants of Jower grade, so that U» is reducible ; and all succeeding identical
covariants are also reducible.

DKHIVATIVKS ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 489

It IB necessary, then, to show that U*UM is covariantive ; if it be so, it must have
its index equal to n — |(n — 1 ) = £ (3 — n), and so the relation

„«„« =

must be satisfied. Now, by (58), we have

u*«{l + i(

and therefore, by (57),

usu<"> =

/rft\ (S-«)

="••' (2) .

showing the covariantive nature of the function.

A similar conclusion as to limitation of number holds with regard to the identical
covariants in the associate variables.

MDCCCI.XXXVIII.— A. 3R

[ 491 ]

XVI. Tlie Smnll Free Vibrations and Deformation of a Tlnn Elastic Shell.

By A. E. H. LOVE, B.A., Fellow of St. John's College, Cambridge.

Communicated by Professor G. H. DARWIN, F.R.S.

Received January 19.— Road February 9, 1888.

CONTENTS.

PAOI.

§ 1. Historical introduction. — Poissox ; KIKCHHOFF'S first theory of plates; KiRCHBorr's

second theory ; BOUSSISESQ ; DE ST. VENANT 491

§ 2. Theory of the present paper for thin shells ' . 496

§ 3. Internal strain in an element of the shell 499

§ 4. Geometrical theory of small deformation of extensible surfaces 605

§ 5. Equations of motion and boundary-conditions 512

§6. Possibility of certain modes of vibration .VJ'»

§ 7. Vibrations of spherical shell 527

§ 8. Vibrations of cylindrical shell 538

§9. Summary of results 543

§ 1. — Historical Introduction.

I PROPOSE, in the first place, to give a brief account of the principal theories of the
vibrations and flexure of a thin elastic plate hitherto put forward, and afterwards to
apply the method of one of them to the case when the plate in its natural state has
finite curvature.

Passing over the early attempts of Mdlle. SOPHIE GERMAIN, the first mathematician
who succeeded in obtaining a theory of the flexure of a thin plane plate was POISSON.
In his memoir* he obtains the differential equation for the deflection of the plate,
which is generally admitted, and certain boundary-conditions, which have met with
less general acceptance. The idea of POISSON'S method may be simply stated. The
plate being very thin, we may expand all the functions which occur in the equations
of equilibrium and boundary -conditions in powers of the variable expressing the
distance of a particle from the middle-surface in the natural state, then, taking only
the terms up to the third order, we obtain the differential equations for the determi-
nation of the displacements which are generally admitted. The meaning of POISSON'S
boundary-conditions is as follows t; — Suppose the plate to form part of an infinite

" Mi'inoiro sur 1'Eqnilibre et le Mouvement des Corps elnstiqnes," ' Paris A cad. Mem.,' 1829.
t Cf. THOMSON and T.ur, • Natural Philosophy,' part 2, pp. 188-9

3 R 2 2f,.11.«8

492 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

plate, and to be held in its actual position, partly by the forces directly applied to its
mass, and partly by the action of the remainder of the plate exerted across the
boundary ; if the plate be now cut out, it will be necessary, in order to hold it in the
same configuration, to apply at every point of its edge a distribution of force and
couple identical with that exerted by the remainder before the plate was cut out.
Now, it has been shown by KIKCHHOFF * that these equations express too much, and
that it is not generally possible to satisfy them ; but the method proposed by
THOMSON and TAIT! gives a rational explanation of KIRCHHOFF'S union of two of
POISSON'S boundary-conditions in one, and renders his theory complete. However,
the objection raised by DE ST. VENANT| to the fundamental assumption that the
stresses and strains in an element can be expanded in integral powers of the distance
from the middle-surface, seems to require a different theory.

The next epoch in the theory of plates is marked by KIRCHHOFF'S memoir just
referred to. The method rests on two assumptions, viz. : (1) Every straight line of
the plate which was originally perpendicular to the plane bounding surfaces remains
straight after the deformation, and perpendicular to the surfaces which were originally
parallel to the plane bounding surfaces ; (2) all the elements of the middle- surface
(i.e., the surface which in the natural state was midway between the plane parallel
bounding surfaces) remain unstretched. Both these assumptions may be shown to be
approximately true in the cases of flexure and transverse vibration, but, as assump-
tions, they appear unwarrantable. In this memoir of KIRCHHOFF'S the union of two
of POISSON'S boundary-conditions in one was first effected, the method employed to
obtain the equations being that of virtual work. The theory of this memoir will be
referred to as KIRCHHOFF'S " first theory."

KIRCHHOFF § has given a general method for the treatment of elastic bodies, some
of whose dimensions are indefinitely small in comparison with others. In this method
we consider, in the first place, the equilibrium of an element of the body all whose
dimensions are of the same order as the indefinitely small dimensions. When we
know the potential energy due to the internal strain of such an element, we obtain
by integration over1 the remaining dimensions the whole potential energy due to the
elastic strain of the body. Then, taking into account all the forces which act on the
body, we can form the equation of virtual work, which will lead directly to the
differential equations and boundary-conditions of our problem.

In KIRCHHOFF'S method it appears that, to a first approximation, the bodily forces
produce displacements which are negligible compared with those produced by the
surface-tractions exerted upon the element by contiguous elements, and that, to the

* "Ueberdas Gleichgewicht und die Bewegung einer elastischen Scbeibe," ' CBELLK, .Tourn. Math.,'
vol. 40.

t Loc. cit., pp. 190-1.

I Translation of CLEBSCH'S 'Elasticitat,' Note on § 73, p. 725.

§ ' Vorlesungen Uber Mathematische Physik,' pp. 406 et seq.

AND DEFORMATION OF A THIN KLASTIC SHELL. 493

same order of approximation, the displacements, when divided by finite quantities of
one dimension in length, are negligible compared with the strains.

The application of this method to the theory of plates nppears to have been first
made by GEURING, a pupil of KIRCUHOFF'S, at the latter *s request ; and the results
will be found in Kii:< •HIIOKF'S thirtieth lecture, and in CLEBSCH'S ' Theorie der
Elasticitiit fester Ktirper,' §§ (54 et seq. We shall call the theory thus deduced KIRCH-
HOFF'S "second theory." POISSON and KIRCHHOFF had both arrived at the equations
S, T, R = 0,* which express that the traction exerted on an element of a surface
normal to the middle-surface of the plate is everywhere tangential to the middle-
surface. These equations are fundamental in KIRCHHOFF'S second theory. This
appears to lie at the root of the objection raised by DE ST. VfiNANxt to this theory, as
it is stated by him that S and T, if they exist, may produce important effects,
especially when the material of the plate is not isotropic.

It seems unnecessary to explain in detail THOMSON and TAIT'S treatment of the
problem. We need only note here that the equations S, T, II = 0 are a basis for this
theory also.

\_AddedJuly, 1888. — An important inference from the method is that a line of
particles initially normal to the middle-surface is approximately normal to this surface
after strain. This is expressed by the vanishing of the shears a and b, as given by
equations (11) infra. This conclusion is intimately bound up with the conclusion that
S and T vanish. At the edge of the plate S and T may have given values which do
not vanish, and the approximate perpendicularity of line-elements originally perpen-
dicular to the middle-surface will here break down. The transition from a state of
things in which S and T exist at the edge to one in which they vanish, on a surface
parallel to the edge and very near to it, is illustrated by the discussion in THOMSON
and TAIT'S 'Natural Philosophy,' §§721-729. The conclusion seems to be that
KIRCHHOFF'S general method for the treatment of elastic bodies, some of whose
dimensions are indefinitely small in comparison with others, cannot be applied to the
elements. situated very near to the edge of a plate, as the strain is not produced in
these by the action of contiguous elements. We may, nevertheless, regard it as
giving correctly, not only the potential energy due to the strain of an element at a
distance from the edge, but also the whole potential energy arising from the strain in
all the elements. It will thus lead us to the right differential equations of motion ,or
equilibrium and boundary-conditions.]

The theory of the flexure of an elastic plate has been placed in a much clearer light
by the researches of BOUSSINESQ, who has treated the subject in a masterly manner
in two memoirs. In the first of thesej he has certainly proved that S = 0, T = 0,
R = 0 is an approximation to the actual state of stress within an element of the

* I use THOMSON and TAIT'S notation for the stresses, strains, and elastic constant*.
t Translation of CLKBSCH'S ' Elasticitat,' p. 691.
J ' LIOUVILLE, Journal do Math.,' 1871.

494 MR. A. E. H. LOVK ON THE SMALL FRKK VIBRATIONS

plate ; and he says that R = 0 to a higher degree of approximation than S and T.
Taking 2h for the thickness of the plate, and the plane of xy for the middle -surface
in the natural state, we have, on integration, with reference to 2,

hfUS

dc dy

» + ?+.£)*

a.« ci/j

Assuming that the bodily forces are not such that if- applied to a body of finite
size they would produce deformations indefinitely great compared with those produced
in the plate, and that P, Q, U do not vary rapidly from one element to another, we
see that S, T, R are small compared with P, Q, U. BOUSSINESQ proceeds to express
three of the strains in terms of the rest by means of the relations S, T, R = 0, as
was done in KIRCHHOFF'S second theory ; then, by means of these approximate values,
he finds S, T to a higher order, and on substituting in the general equations of equili-
brium obtains the well-known equation for the deflection of the plate. The method
of securing the union of two of POISSON'S boundary-conditions in one is the same as
that previously given by THOMSON and TAIT.

BOUSSENESQ returned to the subject in 1879, in a second memoir published in
' LrouviLLE's Journal.' Apparently dissatisfied with the assumptions S, T = 0, he
proposed to consider the subject in the following manner. Let the plate be divided
into similar elementary rectangular prisms, whereof the linear dimensions are all
comparable, and suppose these prisms bounded by the plane surfaces of the plate, and
by pairs of parallel planes at right angles to these surfaces. Two neighbouring
prisms must always be in nearly the same condition as regards strain, except in the
case of prisms situated near the edge. Hence, generally, the component stresses will
be approximately the same at all points on the same surface parallel to the middle-
surface, and not infinitely near the edge of the plate. Hence, in this kind of
equilibrium, the stresses will be approximately independent of the position on the
middle-surface of the centre of the element. This is precisely KIRCHHOFF'S result*
deduced from the kinematics of the system, and it appears certainly true when the
plate is very thin. BOUSSINESQ wishes his theory to apply to plates of small finite
thickness, and he proposes to replace the equations just found by the following

T. 8T 8f 8S 8S ' ,,,. ~ TT,

' Vorlesungen,' p. 453, remark on equations (8).

AND DEFOK.MATIoN i»K A THIN ELASTIC SHELL. Iti

these suppositions are more general than those of the former paper, and enable the
author to take account better of the effects of aeolotropy of the material of the plate.

DE ST. VENAXT* obtains the equations for flexure on the assumptions (I) that
R = 0, (2) that the middle-surface of the plate is bent without stretching, so that
the extension of any line-element through a point distant z from the middle surface
and parallel thereto is z/p, where p is the radius of curvature of the normal section of
the middle-surface through a line parallel to the element. From these suppositions,
of which the first is justified in the manner of BOUSSINESQ'S memoirs, the ordinary
equations are deduced and extended to the case of eeolotroptc plates. From the
inapplicability of the second of these suppositions to the case when the plate is
initially curvedt we may be justified in denying it the right to be a foundation for
the theory.

The question between the methods of KIUOHHOFF'S second theory and BOUSSINESQ'S
memoirs may be taken to be that of the degree of approximation obtainable by the
former. It seems to be established that the terms which occur in CLEBSCH'S equations J
are correct to the order of approximation adopted ; but the question arises whether, if
it were desirable to obtain a higher degree of approximation in the equations, this
could be effected by means of KIRCHHOFF'S second theory ; and it appears that, so
long as the equations S, T = 0 are retained with R = 0 for the purpose of giving
three of the strains in terms of the rest, this question must be answered in the
negative. It must be observed that KIRCHHOFF only uses these equations for this
purpose, just as BOUSSINESQ does in his first memoir, while the equations and con-
ditions are found by applying the principle of virtual work.

In a recent paper§ I have proposed a modification of KIRCHHOFF'S second theory,
with the view of showing how his kinematical equations, whose accuracy has been
disputed by BOUSSINESQ, can be made exact. The equations referred to are those
unnumbered on page 452 of the ' Vorlesungen.' In these certain differential
coefficients are introduced, and afterwards neglected as small ; and BOUSSINESQ has
contended that they should be retained. In the paper referred to I have endeavoured
to show that these differential coefficients have no meaning so long as we are treating
the equilibrium of an elementary portion of the plate, all whose dimensions are of
the same order as the thickness, so that the equations can be made exact by simply
omitting these differential coefficients. As will hereafter appear, KIRCHHOFF'S process
applies directly to the theory of a thin elastic shell, and the modification proposed in
the theory of plates has place equally in that of shells. This will be fully explained
in the sequel (Art. 2).

* Translation of CLKBSCH. Note to § 73.
t This will be proved in the sequel.
J ' Elasticitat,' pp. 306, 307, equations (105) and (106).

§ " Note on KIRCHHOFF'B Theory of the Deformation of Elastic Plates," ' Cambridge Phil. Soc. Proc.,'
vol. 6, 1887.

496 MR. A. E. II. LOVE ON THE SMALL FREE VIBRATIONS

§2. Theory of Shells.

In this paper the potential energy of deformation of an isotropic elastic shell is
investigated by the same method as that employed by KIKCHHOFF in his discussion of
the vibrations of a plane plate.* The shell is supposed to be bounded by two surfaces
parallel to its middle-surface, and is deformed in any arbitrary manner. The expres-
sion given by KIRCHHOFF for the energy of the plate per unit area of its middle-
surface is

KA» + ft« + 2Pl* 4- f ~ (q, - Pa)

l + & fa -f

•where 2h is the thickness of the plate, K the rigidity, and 0/(l -f #) = <r/(l — cr), o-
being the ratio of linear lateral contraction to linear longitudinal extension of the
material ; o-j, <r.2 are the extensions of two line-elements of the middle-surface initially
at right angles, and T the complement of the angle between them after strain ;
?i> P& Pi are quantities defining the curvature of the middle-surface after strain, viz. :—

p.2 — q\= sum of principal curvatures,
— (p2 ql + pf) = measure of curvature ;

so that, if pl} p.2 be the principal radii of curvature after strain, the first term of the
above reduces to

l + e

A similar expression to that given by KIECHHOFF is obtained below in the case of the
shell initially curved ; but here the quantities qlt p2, pt are replaced by the difference of
their values in the strained and unstrained states, a result which might have been
anticipated from the remarks made by KIRCHHOFF (' Vorlesungen,' p. 413) on the strain
of a rod initially curved, since the strain of an element is a linear function of these
quantities.

We wish to obtain equations of motion and boundary-conditions in terms of the
displacements of a point on the middle-surface of the shell, these being reckoned
parallel to the lines of curvature and perpendicular to the tangent plane at the point.
For this purpose it is necessary to express all the quantities which occur in the
potential-energy-function in terms of these displacements. As the geometrical theory
of the deformation of extensible surfaces appears not to have been hitherto made out,

- * Called above " KIRCHHOKF'S second tlioorv."
•f- • Vorlesungen,' p. 454.

AND DEFORMATION OP A THIN ELASTIC SHKI.L. 497

it was necessary to give the elements of such a theory for small deformations. The
expressions obtained for the principal radii of curvature show that the potential energy
due to bending is never the same quadratic function of the changes of principal
curvature as for a plane plate, except in the single case where the middle-surface is a
sphere and unstretchc'l.

The general variational equation of motion is developed in the foi-m of surface and
line integrals, and the equations reduce to those of CLEBSCH * in the case of a plane
plate. The terms herein which depend on externally applied forces are obtained
directly, without the use of the arbitrary multipliers which render the calculations of
CLEBSCH so tedious, and without the necessity which he finds for correcting an
" error " t as regards the distribution of force at the edge, thus avoiding some of the
criticisms of DE ST. VENANT.J

We know that when a plane plate vibrates the transverse displacement is indepen-
dent of the displacements parallel to the plane of the plate ; and when the transverse
vibrations alone are taking place no line on the ujiddle-snrface is altered in length.
I discuss the question whether vibrations of the shell are possible in which this last
condition holds good, and show that it leads to three partial differential equations
giving the displacements as functions of the position of a point on the middle-surface,
and that these equations are not in general of a sufficiently high order to admit of
solutions which shall also satisfy the conditions which hold at a free edge. This
result is quite independent of the theory adopted, as the equations of inextensibility
are in the most general case a system of the third order, while the boundary-conditions
are four in number. It would, of course, be possible to find a system of forces applied
to the boundary which could artificially maintain this kind of vibration. It appears,
then, that the term of the potential eneigy which depends on the bending, which is
multiplied by As, is small compared with the term depending on the stretching, which
is multiplied by h ; and, in order to obtain the limiting form of the theory when h = 0,
we may form approximate equations of equilibrium and motion and boundary-con-
ditions by omitting the term in ha. Having formed these equations, I proceed to
discuss the question whether the shell can execute vibrations in which there shall be
no tangential displacement, and it is shown that this requires both the principal
radii of curvature of the middle-surface to be constant at every point. The
frequencies of the purely radial vibrations of a sphere and an infinitely long circular
cylinder are given ; the displacement is a simple harmonic function of the time, and
is the same at all points of the sphere or cylinder. The formula for the frequency
admits of independent verification. Another general result deduced from the
approximate equations is that any shell whose middle-surface is a surface of revolution

• « Elasticitat,' pp. 306, 307; Equations (105), (106).
t Ibid., p. 284.

J Translation of CLEBSCH, p. 691. The method of CLEBSCH is styled " ohucure, indirecte, fort
cmnpliqueW

MDCCOLXXXVITT. — A. 3 S

498 MR. A. B. H. LOVE ON THE SMALL FREE VIBRATIONS

can execute purely tangential vibrations such that every point moves perpendicularly
to the meridian through it, and the displacement is symmetrical about the axis of
revolution.

The special problem of the vibrations of a spherical shell has been discussed by
Lord RAYLEIGH.* In his paper it is assumed that no line on the middle-surface is
altered in length ; the boundary-conditions are not considered. The form of the
potential energy taken is a quadratic function of the changes of principal curvature 01
the middle-surface, and this I have proved to be in this case the true form in Art. 7.
The assumption of inextensibility does in this case lead to expressions for the dis-
placements which cannot satisfy the boundary-conditions which hold at a free edge.

The method developed in this paper is applied to the problem, and the approximate
equations integrated. The solution comes out in tesseral harmonics with fractional
or imaginary indices, and the frequency is givefa by a transcendental equation ; in
case the shell be hemispherical this equation is simplified, and to express the sym-
metrical vibrations only the ordinary zonal harmonics with real integral indices are
lequired, and the frequency equation can be solved.

As a further example of the application of the method to small vibrations I have
discussed the vibrations of a cylindrical shell. The displacement of a point on the
middle-surface is expressed by simple harmonic functions of the cylindrical coordinates
of the point. In the case of the symmetrical vibrations the frequency equation is
easily solved.

AKON has applied the method of CLEBSCH to the problem of shells. In his memoir t
a point on the middle-surface of the shell is considered as defined by two parameters,
as in GAUSS'S theory of the curvature of surfaces ; the displacements are referred to an
arbitrary system of fixed axes ; and the expressions found for them are the same as
those in Art. 4 of this paper, but the work contains a small error (see note to
Art. 4). Free use is made of arbitrary multipliers in order to obtain the equations
of equilibrium referred to the fixed axes. As these are in a very unmanageable shape,
a method of forming equations referred to moving axes is indicated ; the equations are
first obtained with reference to fixed axes, and it is proposed to transform these. The
transformation is not effected, but some reductions are made with a view to it
(pp. 169 et seq.). In these reductions all effects due to the change of direction of the
axes as we go from point to point on the middle-surface are neglected, so that the
results are erroneous (see note to Art. 6).

A theory of the vibrations of a shell whose middle-surface is a surface of revolution
has been given by MATHIEU.^ The method is similar to that employed by POISSON
for the plate, viz., taking y = 0 for the middle-surface, all the quantities which occur

* " On the Infinitebimal Bending of Surfaces of Revolution," ' London Math. Soc. Proc.,' vol. 17, 1882.
f "Das Gleichgewicht und die Bewegung einer nnendlich diinnen beliebig gekrUmmten elastischen
Schale." ' CRKLLE, Journ. Math.,' vol. 78, 1874, p. 138.

J " Memoire sur le Monvement vibratoire des Cloches," ' Journ. de 1'ficole Poly tech n.,' cahier 51 (1883) .

AND DEFORMATION OF A THIN KLASTIC SHELL. 4S)'J

ore expanded in powers of y, and approximate equations taken. These equations are
included in those given in the present paper for shells whose middle-surface is any
whatever. MATHIEU gives for the special case some of the theorems on purely
normal and purely tangential vibrations here proved (see notes to Art. 13). The
solution for spherical shells is given in his paper. The introduction of the generalised
tesseral harmonic into this solution enables us to recognise that a certain type ot
vibration given by MATHIEU cannot exist (see note to Art. 18). The objections raised
by DE ST. VENANT to POISSON'S method for plates seem to lie equally against its
extension to sheila

§ 3. Internal Strain in an Element of the Shell.

1. Suppose the lines of curvature on the middle-surface of the shell to be drawn ;
let these be the curves a = const., ft = const. ; then any point on the middle-surface
is given by its a, ft. At each intersection of a curve a with a curve ft suppose the
normal to the middle-surface drawn and lengths h marked off upon it inwards and
outwards from the surface, the loci of the extremities of these lines are two surfaces
parallel to the middle-surface. If we suppose the space between these surfaces filled
with isotropic elastic material we obtain the elastic solid shell which we wish to treat.

Let the middle-surface be covered with a network of the lines a = const., ft = const,
at distances from each other comparable with the thickness of the shell, and suppose
the normals drawn as above described at all the points of these curves. The shell
will thus be divided into a great number of elementary prisms ; and, according to
KIRCHHOFF'S general method, we must first discuss the equilibrium of one of these
elementary prisms.

Let a, ft bo the parameters of the centre P of one of these elementary prisms before
strain. Imagine three line-elements of the shell (1, 2, 3) to proceed from P, the
elements (1) and (2) being along the lines ft, a. through P, and (3) along the normal
at P to the middle-surface. Then after strain these lines are not in general co-
orthogonal, but by means of them we can construct a system of rectangular axes to
which we can refer points in the prism whose centre is P. Thus, P is to be the
origin, the axis of x is to lie along the line-element (1), and the plane of x, y is to contain
the line-elements (1) and (2); then the line-element (2) will make an indefinitely
small angle with the axis y, and the line-element (3) will make an indefinitely small
angle with the axis z.

By means of the lines of curvature and the middle-surface we can construct a
system of orthogonal surfaces (a, ft, y), so that we may use the formulae of orthogonal
coordinates with reference to a, ft, y.

We write for the distance between two near points —

382

500

MH. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

2. The point P is defined before the strain by its a, ft, and lies on a certain
surface y = 0 (the middle-surface). The prism whose centre is P is held in equi-
librium by the action of adjacent prisms, and its parts are not in the same configura-
tion as that in which they would be found if this prism were separated from the rest
of the shell and left to itself.* Now, if this portion were isolated from the action of
neighbouring portions, any point of it (Q) would take a certain position defined by
the intersection of three surfaces of the family (a, ft, y), which we may take to be
a -|- p, ft + q, r. Hence, when this prism is subject to the action of neighbouring
prisms the position of Q will be given with reference to the (x, y, z) axes at
P by pjh^ + u0, q/h.z + % r/ha -f w0, and after the strain is effected it will be given
by P/^i + u'> 9/^2 + v'> r/hs + w' referred to the axes of (x, y, z) defined in Art. 1.
The component displacements (ult vlt w^) of Q are u — u0, v' — v0, w — W0.

Consider a system of rectangular axes fixed in space, and after strain let £, 77 £, be
the coordinates of P referred to this system, and let the directions of the (x, y, z) axes
be connected with those of the fixed (£, 17, £) axes by the scheme —

X

y

?n

ra3

Then, after strain the coordinates of Q are

0. >

These expressions are functions of a + p, ft + q, r ; and, hence, for each of them we
have 3/t)a = 9/9p and 3/3/8 = 3/3g. In forming these differential coefficients it is
important to observe that «', v', w have no differential coefficients with respect to a, ft.
Throughout the space within which u', v, w exist, viz., the range of values of p, q, r,
which correspond to points within the elementary prism treated, a, ft do not vary.
In his theory KIRCHHOFF first introduces the differential coefficients analogous to

* This remark was made by ARON, in his memoir in BORCHARDT'S (CKELLK'S) ' Journal,' vol. 78, p. 138.

AND DEFORMATION OF A THIN ELASTIC SIIKI.I,

501

du'/dtt . . ., and afterwards neglects them as small. So that the equations (6) and (7)
to be obtained below are unaffected by the modification of the theory here proposed.
Equating the differential coefficients of (1) with respect to a and p, we get

a

i . a

a/i,

a /i

i a«

at/ a«/

and, similarly, by differentiating with respect to ft and q.

3. Now, taking the set of three equations above written, multiply them by ll} mu nl
and add, then by 12, mit n8 and add, then by lz, m^, n^ and add, and repeat the process
on the second set ; the six resulting equations may be written

A,

r >

(2)

and

*i

. . (3)

In these crlf o-., are the extensions of the line-elements (1), (2), and m is the sine of
the angle the axis y makes with the line-element (2) after strain, so that, if (L2, Ms, Ng)
be the direction cosines of the line-element (2) after strain referred to the fixed axes
of (6 >?, 0,

502

MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

M2(l + o-2)
M2 = 7n2 +

(1 4- } = h -- I

?-B.(l+..)-&\- W

f 2V o/9

2 ™" 2 * 1 -/

Also

'

', = », (J, | + ^

n

', = ». (J, | + -, I-

>• (5)

According to the general principles of KIECHHOFF'S method, we may for a first
approximation omit the u, v, w which occur in equations (2) and (3), thus
re-writing : —

B./ am *\ T',

g— = <i5^(^}~ hk r + h* P p

srG'

a /i

a

a^
a /i

a m v, Kfi

r " P ~ *

Since we must have

we find

a2 M/
^ ,

a* «'

7> =
W'

1/L\ T' =A^ 1/1') v = -*'.

• • (8)

Let Ku Aj,
let K\ — K1

I, K2, A2, T2 be the values of K'I} X'lf T',, /c'2, X'2) r'2 before strain, and
Klt \\ — Ax = Xj, T j — Tz = T^ and similarly for the others, then

I = — A2, «! = — X2) T! = T2 = 0.

AND DEFORMATION OP A THIN ELASTIC SHELL. 503

In (6) and (7) suppose u' = «0, v = r0, w' = w0 ; then

? = p L (£) + ?! (!,) + 4 r> ™th five other

subtracting these from (6) and (7), we find

A

A,A, 3y ~ A, A,*,

/o\

4. These are simply the conditions of continuity of the mass of the shell when
deformed. To obtain the forms of ult vlt wl from them we shall have to introduce
stress-conditions. As the quantities in (9) are small, it will be sufficient to omit
products, and so form equations of equilibrium of the element referred to the
orthogonal coordinates ( p, q, r) as if we were referring to fixed axes at P.

If A, B, C be three functions of r to be determined, we have

Ilence, for the six components of strain, and for the cubical dilatation 8,

A>i /• tCa f C/\_/

To determine A, B, C, we have the stress-equaticns

504

MH. A. E. H. LOVE OX THE SMALL FREE VIBRATIONS

3 3 3 "I

^ =- (TO — n 8 + 2ne) + hz ^ (nc) + hs ^ (nb) = 0,

h^ . (?ic) + h2 _- (m — n 8 + 2n/) + hs ^- (no) =0, \
hi *-» (nb) + h* ^ (na) + ^3 5; (m-n 8 + 2n.9) = 0,

where m = k -\-
Hence,

and
Thus,

k being the modulus of compression, and n that of rigidity.

= ' = '

\

- *)

— n ica — X, r3
T+ n V ^

If there be no surface-tractions on the surfaces initially parallel to the middle-
surface, viz. , r = ± A3A, then A = 0, and B = 0, and also at the surfaces

(m — n) 8 + 2ng = 0,

so that

Thus,

u =

Ag m + n'

— K

o-2

J

Hence,

* Expressions equivalent to these have been given by ARON, but his work contains an error. His
equations (7, a), (7, b), p. 145, are strictly analogous to equations (6) and (7) above, but the terms in

p ~-(t~) • • are all omitted. The test * — ^- = ~ — ~— is not applied ; if it had been, there would
0* \."i/ dp Og v' Sq Op v'

have resulted equations which in my notatiou are T\ = 0, T'S = 0, but the values of t\, T'S are calculated
subsequently by the method of Art. 7, and are the same as those given in equations (8).

AND DEFORMATION OP A THIN ELASTIC SHELL. 505

a = 0, 6 = 0, c = nr — 2/c, ,

and the potential energy per unit volume is
- «)82 + 2n(e« + /2 + .92) + » («s

-f- a tenn in 2,
where z is written for r/A8.

where z is written for r/A8.

Multiplying this expression by dz, and integrating from & to — h, the term in z
disappears, and we find for the potential energy per unit area

= § *A> + v -i- 2^ +

or

. (12)

The term containing h3 is the term depending on the bending, and the term con-
taining h is the term depending on the stretching of the middle-surface. We shall
hereafter denote by Wp W2 the expressions

/ x\o.m + W/x 0\

§ 4. Geometrical Theory of Small Deformation of Extensible Surfaces.

5. We have now, by means of equations (4) and (5), to express the potential energy
in terms of the displacement of a point on the middle-surface.

Let «, v, w denote the displacements of the point P on the middle-surface, u being

MOCOCLXXXV1II. — A. H T

oOfi

MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

parallel to ft = const., v to a = const., and w along the normal outwards, and let 3/3ns
denote differentiation along the normal.

The square of the length of a line joining two near points of the surface before
strain is

and the square of the length of the line joining the same two points after strain is

i

o

neglecting small quantities of a higher order. But this same square of the new length
of the line is

where 8 (l//^), 8(l/h2) are the increments of l/hlt l/h.2 produced by strain, so that

l

*

S ( , = A.M^-

and similarly for &2, also

and so for d (h.zv).

In the two expressions for the square of the new length of the line we may equate
coefficients of (dctf, (df$f, and (da. d/B), and omit powers of u, v, w, or their
differential coefficients above the first ; thus

a /i

w

a .

where we have written

in accordance with LAMP'S result (' Lecons sur les Coordonnees Curvilignes,' p. 51),
viz., pj, p.2 are the principal radii of curvature of the middle-surface before strain,
reckoned inwards.

AND DEFORMATION OF A THIN KI.ASTIC SHELL.

507

6. To express K.,, A,, *, in the same way, we suppose a system of fixed axes at P,
whose directions coincide with those of the (x, y, z) axes at P before strain. The
coordinates referred to P of a point near P on the deformed surface are

S£ =

.
*

u 803,

8*7 = + dv -
8£ = dw — u

where 80,, 802, 803, are the elementary rotations of the axes (x, y, z) about themselves,
when the origin is changed from a, ft to a + 8a, ft -|- 8/8, viz. : —

80, = - .£- f )dp = - ~ </&

"** \ A 1 A O

i

(14)

'• J

Hence, by equations (4),

substituting for o-j, o-2, and neglecting small quantities of the second order, we find

whence,

dw u

(15)

3T 2

508 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

These are the llt mlt nt . . . , referred to axes at each point determined by three
certain line-elements at the point, if 8 denote the change in the value of these as we
go from any point P to a near point on the middle-surface, then referred to the fixed
axes at P, we have

£\I £\7

8/j = r-1 da. -j- -^ dfl — m, 80,j -f- n, 80.,, and so on ;

vet op

so that

f-\ \h ^ _ ^ ft M .fL /* \l_i_ A/^ 9Jf_ u\\

, ra r, a» , . a /i\i , a mi

8m, = rfa [£ |^ a- - ^ V a^ (,j} - A2 ^ ^ j J

, ra / dio «i i

g^ = da. U~ 1 % a -- - \ — T~
|_3a 1 3a pj l,ft

J0ra r, a» tti i r, a» a /i

+ c?y8 UT; Mi a --- \ — T — l ^i 5 -- *i»8« ^

M La/9 1 s* h/ SAI aa ' s^v

in the same way

j fa fi ^ i j. 9 /
8La = aa r- \ ho ^- — /i^oW «-

aa2a3 l s«A

SM = - da

K IT

- — hJt.2v 5- ( —

•

We may form the K'Z, \\, K\ from these, for m^a, n^ are small quantities of the
second order, and ls — «* — > Z3 -* - are also of the second order; hence, to the first
order, using equations (5) we obtain

AND DEFORMATION OP A THIN ELASTIC SHELL.

a m 3

50'J

. r /, 3u> u\. 3 /i\

= *• L- (*' £ - s) *« * U)

The relation X' = — K' reduces to

(16)

and each of these expressions vanishes (L.AM6, p. 80) ; thus, this condition is fulfilled
identically.* Using these relations, we find

m. 9/9 3/9 9a p4 9« pi 9/9

X _;. «j.*
AI « H AI

3/3 3/3 i

.
AZ 3/8* H ^2 3/9 3/9

/i\ *• ?»

\hj ~ Pi 8/9

(18)

7. The quantities defined by equations (5) have been calculated directly ; we wish
to obtain an interpretation in terms of quantities defining the curvature of the
middle-surface after strain.

• This may be taken as a verification in some degree of the preceding work. In endeavonring to
form equations referred to the above set of moving axes, ARON neglects the f-0^ fO^ I0t and deduces
values of X'g, t\ (my notation), which do not satisfy the relation Va + «', = 0 (see the memoir above
qnoted, pp. 169 et .-•••/.). In consequence, he is obliged to make an assumption that 3 (t'/ij"1) / cU is a small
quantity of the second order.

If the relations (17) had not been known, the theory of deformation would prove them.

510 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

First, suppose we are dealing with an inextensible surface, then

8£ 89 8?

, a? ,817 a?

By equations (5), since Z,Z3 + mjrag + n^ = 0, and ya + m27n3 + nsn3 = 0,

X' - - K 3 * P(l>g) *£ _i_

h "2 9 «/9 8«3 + 9

' . ISA a _ L _ ,

1 2 "r "•"

_ _

9/3 "r a («/8> a« ay9 "•" a («/s) a« a/9 J '

1 » [8 («/8) 8«8£ 8 («/8) aa8yS a (a/3) 3a 8/9 '

/ A 7, 3

ft re

(a/9) 8/S3 "" 8 (a/3) 8/3* 8 (

Hence, taking the notation of SALMON'S ' Geometry of Three Dimensions/ chapter 1 2,
section 4, we have

, >

*!» V F = ic',, hrh* G = K2, V*a E' = -
and the equation for the principal radii of curvature is*

so that, if p'i, p'2 be the roots of this equation,

' X' - 1. .1

'- P\ P\

P iPs

Also

*2 ^i + ^i2 = K'z ^'i + 'f'l2 + K2 Aj — Aj K'2 — K2

Plp'a Pi Pa Pi Pa
* SALMON, p. 346. I have changed the sign of p so that the roots shall be the />, and />, of Art. 5.

AND DEFORMATION OF A THIN ELASTIC SHELL. 511

In the case of the sphere, this is

— - - — i + ~ ( ~r + -7 ) » where a is the radius,
P i P t a a\P\ Ptl

or

for any other inextensible surface this will not be the case.

Now, suppose the surface extensible, and consider (a, ft) as two parameters defining
a point on the deformed surface ; in this view they will not be orthogonal parameters,
and we find

i

— -(1+
or

F = -TT to the first order ;
*A

so

l-2<r, 1-2,7,

~V~' ~V '

Again,

A, 3f
- 1 + <r, 8. '

with similar expressions for mlt nt.

To find /c'2, X'u K\ from the definitions in equations (5), we notice that the terms in

"s/t\ rl / h \

ai \i + oj ' a0 \i + o-J

will always be multiplied by terms of the form

Now

11 \ XT -»r ^1 ^»

«, = K«2 - «,«,) = »hNs - niM, == ^ j^

and similarly for wi3, 7t3 ; thus, the differential coefficients of ^/(l + o-,), ^'(1 + <r«)
will be multiplied by factors which vanish identically, viz., they are of the form

, ,

a« a (a/8) "•" a« a (a/3) a« a («/3) •

Hence,

+ ff^ **

V*.

512 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

The equation for the radii of curvature being

(E> + EV) (G> + GV) - (F> + FV)2 = 0,
where

V = v/(EG - F*) = 1~f1,~g> nearly,

A|A|

we find that to the first order

I=-<'!(1+,1 + ^) = i+^-«s.

We have already found expressions for K.Z, X1( KJ in terms of the displacements;
hence, we have found expressions for the new principal curvatures and the position of
the new principal planes, in terms of the displacements, for the position of these
planes depends only on F' or on KJ ; we have also found the interpretation of the
*.,, X1? jq in terms of the quantities defining the curvature and the extension.

In the case of an inextensible sphere, the potential energy due to bending is

For any other surface, whether extensible or not, this will not be the case. If the
middle-surface were unextended, the above would be right to small quantities of the
first order, but we always require the potential energy correct to small quantities of
the second order.

§ 5. Equations of Motion and Boundary-Conditions.

8. Following KIRCHHOFF'S method, we are going to apply the principle of virtual
work to obtain the differential equations of motion and equilibrium, and the boundary-
conditions.

Let Xj, Yj, Zj be the components of the bodily force per unit mass parallel to the
lines of curvature ft = const., a = const., and perpendicular to the tangent plane to
the middle-surface, acting at any point Q of the shell. Let QP be perpendicular to
the middle-surface before strain, and let 13, ms, «3 be the direction cosines of QP after
strain referred to axes at P, as in Artt. 1, 6 ; if u, v, w be the displacements of P,
and 2 the distance PQ, then, when a small variation in the configuration is made, the
displacements of Q will be found from equations (1), dropping the p, q, to be

AND DEFORMATION OF A THIN ELASTIC SHKI.I, 513

8« + 2 8/3
8v + 2 S»ns
810 -|- 2 8n3.

Let A,, BI, C, be the components of the system of forces per unit area applied to
the edge of the shell, and holding it in its actual configuration. The systems of
forces X,, Yt, Z, acting at all points of a line through P perpendicular to the middle-
surface, and the similar (Alt Bu C^ system, will each reduce to a resultant force and
couple.

The resultant of the Xu Y^ Z, system is a force at P whose components are

f* f* f*

X, dz, Y = I Y! dz, Zl dz per unit area,

and a couple whose components are

f* f*

L = — J YI 2 dz, M = + I Xl z dz, 0 per unit area.

The resultant of the At, B,, C, system is a force at the point P in which the
middle-surface cuts the edge, whose components are

A = j A! dz, B = j BI dz, C = \ C^z per unit length of the curve in

which the middle-surface cuts the edge, and a couple whose components are

f* /•*

Bj z dz, V = + I A! z dz, 0 per unit length of the same curve.

The general variational equation of motion is

0 = - J|[ [X, (Su + 2 8ZS) -f Y, (Sv + 2 8m3) + Zl (8w + 2 8n3)] dS dz
- f f [A! (Su + 2 8/8) + B, (8v + 2 Sw8) + G! (8«; + 2 8ns)] ds dz
~ f f 8WX SS + 2>iA Jf SWZ dS

MDCCC'LXXXVIII. — A. 3 U

514 MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

where dS is an element of area of the middle-surface and ds an element of arc of the
edge.

Observing that by equations (15) 8ns = 0, and integrating with respect to z from h
to — h, we get the equation

- j](X Su + Y Sv + Z 8w) dS - j](M 8/8 - L 8m3) rfS

- |( A 8u + B 8v + C Siv) ds - f (V S13 — U Sm3) ds

j d$ + 2nh 8W2

9, (10)

which contains in itself all the equations and conditions of the problem.

All the double integrals which occur in this equation can be expressed, partly as
surface-integrals over the middle-surface, and partly as line-integrals round the edge,
by means of the theorem,

A, ..... (20)

where the first integration extends to all values of (a, ft) which correspond to points
on a surface having s for an edge, and X, y. are the cosines of the angles which the
normal to the edge drawn on the surface and produced outwards makes with the
directions of the lines ft = const., a = const, at the edge.

To prove this theorem,* let a line of curvature a = const, meet the edge in an even
number of points, and let X1} X2, ... be the values of X at these points, then

80

The partial integrations will be effected by means of the relations

a / aa^ ax

ax

* (7/. MAXWELL, 'Electricity and Maj^netism,' Art. 21. This theorem is otherwise proved by A RON.

ANI> I >K FORMATION OF A THIN HLASTIC SHELL.
In evaluating the line-integrals we shall use the formula—

d-\ •> a 33

d.o o ,o,vo

, a
**%=

in which cfo is the element of the normal to the edge drawn as above stated.
9. From equations (15) we have —

,

515

(22)

(23)

so that

ff/wsi TC \JQ fffw/ ) 9&r _L **\ ,( , dto> , 8r\"l
j j (M 8/3 - L 8ms) rfS = j] [M (- /h ^ + J - L (- h, ^ + -JJ

f (L/i - MX) 8u> ds. . f -r . . , . '. - '. . (24)

Again,

- 8» - (XU + pV} Sw - (XV-

t, . . . . (25)

by integration by parts.
Again,

f f SWi dS

ff/o

= 1 J (2K* -

<1«<lfl . ff/n

M. + 1 J (2Xl '

3 u 2

516 MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

Thus,

8 f 1 3Aa / m - n . \ 1 3 r 1 3A, / . TO - « \ 1

— 55 (1-55 (2*ii — - xi r + 5- i r 2^ 2Xi — -«»)r

d/3 |_A! d/3 \ m J/J d* lA8« \ TO 2/J

3 9 1 9 8

8 /!\/^ m-w M

iasfrilSX, — K o } ^
8)3 I 3/3Vli/\ TO /J

__ M aA8\ _a /«, a&,\
a/s

n r a\ _0_ M aA8\ _a /«, a&,\ i -i
9«/ "~ 9« v^ ay3/j J

ff^ 7«S f1 8 /!\/o «-».\ 1 a fl/. m-n

— rfa dp 811 \ - K- T- 2*2 -- - Xj + -3- ^- (2X, -- , -- K2

JJ LPl8a\A2/V »» / Pi3a t^s V m

« 3 /«i\.«i

m —

TO + 71

f\i j I hi/n\ m—n \ 3 Sw 3 f A, / . m — n \1 ,

XAjda — p 2X, -- -*«)-5 -- a" it (2^i -- -KoHSw
J |_ A»\ TO V 3« 8a [Ap\ m 2/J

ldAl/«\ TO — M \

W — T ^ !(2X1 -- ~K2 )
A2 8a \ m •*/

TO
/e, -, 2 Sh,

-
8w

f d.[ - ^ Sr-1-

J L 7iip

AND DEFORMATION OF A TI1IN ELASTIC SHKI.L. 517

Using (22), we find for the line-integral part,

— n . \ ., /_, m — n \ m + n

28*,

+J **%,[-

where, by integration by parts, the second terra becomes

Again,

f(8W,dS=ff2( ^
JJ JJ \m-(-n

ff0/ 2"t . «-« \s Aerf/9 ff

-f 2 - <r2 + o"i do'a , , +

JJ \m + n m + n 7 /^A, JJ

'

a /i\/ 2m

a /i

ffjjotT/2"1 . m — n, \ 2 ,/2m ,m-n \2 "I

\\dadp6w (- -o-j+ - <r2 r-r --- M- - crg + - o-j J-r-:

JJ |_\wi4n ' «T +*•/%*, ft ' \«» -f « 'TO + nV^A,/)^

2 / 2m .m — n \ . 7C>2/2wi

(28)

518 MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

where the line-integral part is

fck|2X(- -0"! + - - o-2 ) + ^rss I 8w + f ds \2p. ( - <r2 + " * ^ } + ACT I 8v.
J \m + n m + n 7 J \?» + n m + n 7

Again,

= ff f— i— irlrs+ i i^8«^a^ + ff f- r^-^SiH- r is^&>&i«^8

JJ \ n,^plo»vCi "i^sPi d^/ JJ \ Ajpjd/Sdr l^h^p^of/

• I 5~ I ~~ T~ 5 5*i ~t~ I 512 ) "r 5o I ™~ IT 2 \o 37a i Oii ) "® <*a ^P
JJ [_d« \ Aj da dr A2p! df*/ dp \ n^opct'' p3 of/ J

. . (29)

10. Collecting the terms, we have the differential equations of motion
m p_ _3_ AN /

' Lrl \ 2/ \

— n

ft

3- !*•

i T

2nh — 2-^- U- ,

d*[\hj\m+n * m + n

3 /1W 2m , m-n \ 3 /w\l

5~ (r )(- -<*s + - -o-i — ^-i-^ (n =°

Oa \A2/ \m + » ' TO + 71 V X 3/3 W/J

f ^L . . 4

L M8 "*" a

L JL'W
1P8 3/3 a2

4 j,, m Pi 3 /1\/0.
f 7i/i8 - - ^ — 2X,

m + n [j>2 3/3 kj/ \

TO — «.

- K.
m

o _\ «

h/>8a/9UA2/fa~ *»

\m 4- n

r 3fl/2m ,m-« \1

— 2^^7-1- -0-0+- - <r, if

L 3/3 1^ \m -f w r TO + n l/J

. 3 /1\/ 2m . m — » \ 3/ra\"l ,„.,

25s(r) (- -0-1 + - -- o-o — A2 =- — „ ) = o, ..... (31)

' 2 z

AND DEFORMATION OP A THIN KLASTIC SHKI.I.

519

•

3/3 *

m + n J f c") /«, 9A

m 13-3/3 3> U 3.

3 /«, 3AA1 1
" 8. \A, 3/3 / J J

2m

/oo\

• • (32)

The first terms in these equations reduce to those in CLEBSCH'S equations
(' Elasticitat,' pp. 306-307) in case the shell becomes a plane plate.

The second terms (in ph?) arise from the " rotatory inertia."

The third terms (in h?n) arise from the term W, in the potential energy, and depend
on the bending ; the fourth terms (in 2«A) arise from the term W2, and depend on the
stretching.

11. The boundary-conditions are

.!•:. ..'.• ft

V

-

p

ro 1

X

m + n

TO-H \ 0»i-fn
— -»f2 — 2- put|1
m

(33)

520 MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

3*10 1 3V /, 3*10

- ' +

, _ l - r • ~

3/3 [A, \ m l/\ rA,3/3\ m

3 fAi /.., m — 7i \"1 1 3A, / , m— n
- -L-1 ( 2X, -- -K, H — r~ !(2X. -- -
alAj\ m f/J As 3a V m

3 / 1 \ / m -

XV - /xU + I nh* -- p* 2*3 - - X, - X2X1 - K> + 2 V *i = 0- (35)

r v/

TO

The first terms in each of these equations are the same as those in CLEBSCH'S
equations, pp. 306, 307.

The couple — [XU + /tV] is that called by DE ST. VENANT the moment of torsion ;
the couple XV — /j,U is that called by him the moment of flexure, and their axes are
the normal and tangent to the edge respectively. The former of these may be con-
sidered as arising from a distribution of force in lines normal to the middle-surface
and in the edge ; the difference of the forces in consecutive elements gives rise to a
resultant force normal to the middle-surface which coalesces with C. This is the
explanation of the union of two of the boundary-conditions given by POISSON in one.

We are going to apply the equations just developed to determine the small free
vibrations of the shell. The terms depending on the rotatory inertia will be
neglected.

§ 6. Possibility of Certain Modes of Vibration.

1 2. Now let us suppose, if possible, that the shell vibrates in such a manner that
no line on the middle-surface is altered in length. This requires that <rlt cr2, CT be all
zero. Thus, from equations (13) we derive

, 3w . , , 3 /

AND DEFORMATION OF A THIN ELASTIC SHELL. 521

These are three partial differential equations to determine the forms of u, v, w; and,
if either u and to or v and w be eliminated, they will in general lead to an equation
of the third order to determine v or u. When one of these is determined, the rest
are known. But at the edge we have to satisfy four boundary-conditions, and this
will not be generally possible with solutions of a system of equations such as the
above.

13. Since <rlt o-2, CT may not in general be regarded as of a higher order of small
quantities than *2, X^ Klt it follows that the term in Wz in the potential energy
which contains h as a factor is very great compared with the term in Wt which
contains h3, and we may form approximate equations of vibration and boundary-
conditions by omitting the latter term.

The equations of motion thus formed are —

*» , L »T » a f 1 / 2m
0 = vi + M* ~ — 2 a~ 1 T -

& PL °* IA \m + n

8/l\/2wi, . m-n

2m . m — n \ , 9 / w \"

l >

— n \ 1 2m ni—n

And the boundary-conditions are —

. / 2m .771 — TO \

2X - - <r, + - - <r2 -f /ACT = 0,
Vm + n r7/n-n V

/2m m — n \,x „

2/x - <T8 -f - o-j + XCT = 0.

\m + n m + n 7

(1.) Let us examine the possibility* of purely normal vibrations.
Since u = 0, v = 0, the equations of motion become simply

' '' , r. n 2771 /I 1 2<7 \ /qo\

37. + 2- -{— t + ri-K: )w = o, ..... (38)

d<8 /) TO + n \pl* p? ptfiJ

-where <r = (m — n) /2m is the ratio of linear lateral contraction to linear longitudinal
extension of the material of the shell.

* MATHICI: convince!* Limself of the impossibility by general reasoning.
MDOCCLXXXVIII. — A. 3 X

522 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

In order that all parts of the system may be in the same phase, it is necessary th.it
1/pj2 + l/p^ + 2<r/pip.2 = const, all over the surface.

Again, in the u, v equations we must pick out the terms containing w, and, observing
that w is independent of a, ft, we may write them—

3* IA \Pi Pa /I d* \hj \p3 pJ

Thus,
and,

But, by equations (17),

' ' - ' 6 ~ L)IS
Substituting, we get

= 0

i g Y • • :^>'v- • :- (39)

A| as \p, "*~ PO J = ' J

So that 1/pj + l/p2 =r const, all over the surface.

The two conditions of possibility of normal vibrations show that the middle-surface
must have both its principal radii of curvature constant at every point. These
conditions are satisfied by the sphere, the circular cylinder, and the plane.

Again, if the surface be bounded by an edge, we have, since vr=-Q, \(l/pl + cr/p^= 0,
/* (Vpa + °"//°i) = 0 J these can coexist for all values of X, fi if tr3 — 1=0, and

I/Pi = ± I/ft.

To make n positive, or the material resist distortion, we must have ^ — <r positive,
so that a- cannot be = 1 ; the equation cr = — 1 makes n = 3m = 3k + n, so that
k =. 0 or the material of the shell would offer no resistance to compression ; thus, the
equations above written cannot coexist for all values of X, /A, and hence one of the two
X, /x, must be zero, and one of the two equations l/p2 + <rfp\ = 0 and l/pt + cr/p.z = 0,
must hold at the edge. These conditions cannot be satisfied on a sphere or cylinder.

> m:i ORMATION OP A THIX ELASTIC SHELL. 523

The complete spherical shell may execute purely radial vibrations, and the frequency

IS

1

(l-o-)aV
where a is the radius.*

The indefinitely long circular cylinder may also execute purely radial vibrations
with a frequency

1

a being the radius.

Observing that the more accurate equations of motion and boundary-conditions,
which contain the terms in h3, will in all such terms have only differential coefficients
of to with respect to a or ft, the above theory is seen to hold also if these more
accurate equations be considered.

(2.) Again, consider the possibility t of purely tangential vibrations, the edge being
a line of curvature.

Since w = 0, the third of equations (36) gives

(o-j + o-trj/p! + (o-2 + ovj/pz = 0

at all points of the surface.

Now, the boundary-conditions at a = const, are

and with two functions u, v it will not generally be possible to satisfy these con-
ditions.

If, however, the surface be of revolution, and ft be the longitude, then
dh^/dft = 0, and all the conditions can be satisfied by taking

(1) u = 0, -I

9a L at all points of the surface,

<2>a£=:0 J

r\

(3) ;r (Agv) = 0 at the edge ;

* [In the paper as read, this result was verified by reference to a question set in the Mathematical
Tripos, part III., 1885. It has since been pointed out to me that it coincides with the formula given by
LAMB in 'London Math. Soc. Proc.,' vol. 14, p. 50. — July, 1888.]

t MATUIEU deduced the possibility of some purely tangential vibrations from his differential equations.

3 X '2

524 ME. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

and the equation of motion is

Hence a general theorem : — For any surface of revolution there exists a system of
symmetrical vibrations, in which every element moves perpendicular to the meridian
plane through a distance which is the same for all points on a parallel of latitude, and
the frequency of such vibrations depends only on the rigidity of the material, while
the ratios of the intervals are independent of the material. These are the only
purely tangential vibrations of which the shell is capable.

14. Let us examine more minutely the question whether a spherical shell can
vibrate in such a manner that no line on the middle-surface is altered in length.

Taking a = 9, /? = <f>, the colatitude and longitude of a point on the middle-surface,
and a the radius, ^ = I/a, ha = l/(a sin 0), thus

= Be + w'

i ^\

= W + U COt 6 + - ^ 57 '

sin0 d<f>

(41)

ttCT =

a2** = - ; ^ + cot^ ^ — ^ , ^ — wcot 0,

d»

a2*, = -

0 d

cos a ow

. . (42)

^

Suppose (T!, cr2, TT all zero, then

and

. - a /
U-

These are the conditions given by Lord RAYLEIGH, and they show that u cosec 0 and
v cosec 6 are conjugate solutions of the equation

AND DEFORMATION OP A THIN ELASTIC SHELL.

(43)

and w is given by the equation

Substituting from o-lf <r.2, CT = 0, we find

a«

ir

Now,
Thus,

a\«

i •'/<•

__

sin8 5

i ar. a/. a a\«/ « \i

= - a ^ si" ^ »m ^ 5^ • "/i
sin4 0 80 L \ Sff/ \sin ^/J

= ^ 8m P

sin

-a

~ 81U P COS

^" i
-3 + M

*

Hence, *2 = Xj.*

The boundary-conditions arising from the terms in §w, &v in Art. 11 are now

2X(1 -O-)KJ — 2/t(l -<r)»c1 = 0,
cr)^ — 2X(1 — <T)K! = 0;

since X2 -f- p? = 1, and £ — <r is positive, the only way of satisfying these equations
is to take ^ = 0, *2 = 0 at the edge.
Now,

^* ^ ^ / * ~ •

.2 «r« , 3« j d / 1 < '

And, as shown by Lord RAYLEIGH,

• «• « r fl"

u = 2 sin ^ A, tan* -

v= 's'sin^lA^tan'-

• »2 L -J§in<*

u? = ' 2* (s + cos 0) |A, tan' -

• This might have been written down at once by the aid of GAUSS'S deformation theorem.

(44)

5-2G MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

From (43)

• a 5 f • a 3 / u Yl , 1 9V
sin 0 gj [sin B ^(^-0)\ + ao 3? = ° 5

put

then

a

sin 6 tan' - = uu,

+ «0 = cot 6 ~£ + u0 (s* cosec2 0 - cot2 0),

and

sec" r>

7 UVV . /

0 - » t fl _L * / t /) _i_ s

do 0 \ sin I

to-n _

so that

tan ^

cPun s + cos 0

9ins<T '

cPun . dun f 2s cos 0 l '£ cos* (/ 1 . s* + s cos 0 I s

, u

0 .dun_ f 2s cos Q 1 2 cos2 0~] g3 + s cos 0

" d0 '- U°S [f sin» ^ ' " sin 0 " siii" ^ J + "sin^^~ "°

hence,

« = oo r ff "1

a2/c2= S (s3 - s) A, tan'- cosec2 0\

« = 2 L * J

Again,

d f \ dun\ d fcos ^ + s

~" 8 W°

_ (cos 0 + s)2 /2s cos 0 1_ 2 cos2 <?\ ^ — j

sin3 0 V sin3 & sin 0 siu* 0 J sin3 0

hence,

t = oo r ^ *icos«^

a2Kj = S (s8 — s) A, tan* - cosec2 6 ,

so that *2 an(l *i cannot both vanish all along any curve drawn on the middle-surface,
unless the A vanish, which gives no displacement.

We have shown explicitly in this particular case that the assumption that no line
on the middle-surface is altered in length does lead to expressions for the displace-
ments which cannot satisfy the boundary-conditions which hold at a free edge.

AND DEFORMATION <>i A THIN KLABTIC SHELL.

.VJ7

§7. Vibrations of Spherical Sin //.

15. Let us now apply the equations of Art. (13) to the discussion of the vibrations of
a spherical shell ; we have

u*p . /)3H» 3 / 2m ff, , m — n <T|

m + n kf m + n

- 2

2m

m-n

L-

* T»

I -/-">

'^WV

/»5p . -,3*1; _3/2m <r. , m — n <7,\

sin 0 _— := 2 ^— I — •" ? +

n . ot* o<f> \m 4- n Aj m + n AI/

_23/IWJ^
9^ \*i/ \m +

. /,

- sm . , = — 2 a on 0

n 3r TO

S\S

3 T» — n

m-n

(45)

In these we are to substitute for hlt h2 their values

ht — I/a, ^2 = l/(a sin

and for o-j, <r2, er their values from

3u .

a / « \ . ./,s/f\

= 5- T—^ -f Sin 0 53 1 T— a
tty \Bu\ 6] off \3in 6]

Let us take u, v, w as functions of < to be proportional to e*', then the period is
2ir/2> ; also take p*a?p = fl*8, where K is a number, then we have the three equations —

n

| + w)
d& /

<~

. '"

(46)

^(^+Mcot0 + 8-^|+2M>).

n8 0) = 0, (47)
(48)

528 MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

The first two of these are —

i 3m — n\/&u, ,a^"\ , ndw3m — n / , 3m —

+

m + n

&u, ,a^"\ , ndw3m — n / , 3m — n\

5^ + cot 0-^+2^- - — 1 + - ~ ) W cot2 8

tiffi dOJ 30 m + n \ m + n/

, / 3m — n\ / 3m_— _n\ cos 0 30 . 3m — w 1 &v 1 3*w _ ,

" m + n)* ~~m + n)sirf6d4> + m + n still 30 ty + tirfB ty* = '

i (&° , i a dv\ , a a\ 2 3m — n dw cos 6 3u / 3m — n\

+ U^ + cot d .- + v (2 — cosec2 ^) + ^— 3 - - «- -f- - -7,^7(2+- -)

> < 0- 361 sin 0 m + n c<f> sin2 0 d<£ \ m + n ]

1 / 3m — w\3^ 3m — n 1 9%
rsh?0l/ "l^T^/a^H'^T+^sTn^S^ =

Substituting from (48), these are —

2 cos 0 3w ,

- + -=0, . (51)

, .

-^ + = 0, (52)

and, writing

, ....... (53)

m + n

(48) becomes

** /^* i A /3 1 d"\ /c ,\

-«; = c^+Wcot^ + -1^. • ...... (54)

Substicucing for w, we find

a = 0, (55)

J f fl _L _-

H ^ +

= 0. (56)

Since u, v, w must be the same for </> + 2n as for ^., we may put

w a cos s(f>, v oc sin s<^>, w a cos s<£,
where s is an integer.

AND DEFORMATION OF A THIN KLASTIC SHKI.l. 529

Then, for u, v, w as functions of 6 it is convenient to take equations (51), (54), and
(56), which become

du , . .,-, 2«cos0 . if dw . _.

]u---V+-- = 0, . (57)

+ cot0^ + [2+K*— {I +s2(l + c)}cosec!0]r

}, (58)

dff3

. scos 6 sc du

Bin 0

(59)

Differentiating (59) witli respect to 6, dividing by c and subtracting from (57),

0

/o .2 «rcos^ * dv

(2 + ^ .-

Write u sin ^ = U, v sin 0 = V, thus,

f [(2 + <•> sin« • - ,'] ^, = . - . + 9in« »;.. . (60)

and (59) becomes

rfU «V

We are going to substitute from (60) in (58) and (61) ; the result will enable us to
eliminate V, and obtain an equation for w.
We have

*» /, 1\ adw , Od\

— - 1 1 + - 1 sm* 0 -- + s . sin 0 -
TT — 2 \ c) <16 d0 .

(2 + ««) sin8 0 - «*

therefore

rfU ««/ 1\ sin'g f • A**, adw 2 (2 -f ««) sin* g cos g

= ~1 h-»811 + "--*

pPV flrfV 2(2

"1 "

Substituting in (61), we have, on multiplying by cosec 0[(2 -f- *cj)sin8 6 — a2]8,

MDOOCLXXXVIII.— A. 3 Y

530 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

+ cot0[(2 + ic*)sins 6 - 3^] +

- 4(2 + K2) sin2 0 - s3] ^ + 4(2 + K2) sin2 0 + .s2] cot 0^

=0. . : : . . . - . .-, (62)

Now, substituting for dU/d0 from (61) and for U from (60), (58) becomes

sin

^/ 1\ . Zfidw dV

sc {# . s \ 28COS0~2\l rc)B °dO~ S_d0

™ ~ " * ' 2 2

~Bin*0\2c ~smB " sin* 0 (2 + K2) sin2 Q -

or, multiplying by sin 0[(2 + /c2) sin2 6 — s2],

- - s.w f(2 + K2) sin2 0 - s2] + *2s 1+ ^ sin 0eos 0Ta = 0.
2 L\ / J \ /; / do

Multiply this by s and add it to (62), thus,

or

dw

/fto\

(63)

Also, between (60) and (61), eliminate V, then

.-in^. (64)

The equations we have to satisfy are (61), (63), and (64). Writing p instead of cos 6,
these become

AND DEFORMATION OP A THIN ELASTIC SHELL. 531

• <65>
(66>

. (67)

Of these (67) gives V when U and w are known. The solution of (66) consists of
two parts — one, the complementary function which satisfies (66) when w = 0 ; the
other, the particular integral which satisfies (66) when w is a solution of (65). We
may show first that this particular integral is proportional to ( 1 — jt2) (div/dp) \
take it to be X (1 — /**) (dwjd^}.

For, writing (65) in the form

a.va 2. dw

~ &*? ~ 2/t(l - /i«) -

and difierentiating, we have

and the left-hand side is found by using (66) to be

so that X(l — /x2) (dw/dn) ^ a particular integral of (66), if

*cY2cX = (2 + jc«)/(l -f c) = 2 + K* - K2/2X,
which are both satisfied by

Thus,

is a particular integral of (66).

3 Y 2

532

MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

16. We have now to consider the complementary functions.
Ih equations (65), (60), write

(2 + *

1),

. .

then these will be the equations of tesseral harmonics of orders a, ft respectively.

Calling T^ (/*) the solution which does not become infinite for p. = 1 , we have

U =

\A(1- 2) A fTo( \>

To find V we have

so that
Hence,

V = (i _ ^j {T« 0*)}

» 0*).

— j

cos ,

(69)

17. Properties of 1™ (/*).
The differential equation is

(70).

and for any value of a, real or imaginary, this is satisfied by the integral

f {/* — cos <£ v/(/x.2 — 1)}" cos stj> d$.*

Jn

Also, if we put

* HEINE, ' Handbuch der Kugelf anctionen,' pp. 225 st seq.

AND DEFORMATION OF A THIN ELASTIC SHELL. 533

P. (jt) IB a solution of the differential equation of zonal harmonics

a(a+l)P = 0,,V«-W^ . (71)
and tliis is satisfied by the integral

This form would not be adapted for arithmetical work if a were imaginary.

If a be imaginary, then will «(«+ 1) — — (i + <f)t where q is real. If q be
integral, (71) is the equation of MEHLER'S ' Kegelfunctionen ' ; and it is shown by
NEUMANN* that this equation is satisfied by the integral

i:

cos </<£ d<f>

o ./(/* + cosh <f>) '

and this is finite when /t = 1, but infinite when /*= — !; the form of demonstration
adopted holds equally when q is not integral.

In general, writing — o» = a (a + 1), and changing the independent variable to
2 = (l — /*)/2, the equation for P becomes

^£, 1-2* df •• P_

rfz» ^(l -Z) <fe "~Z(1 -2)

so that one solution for P is the hypergeometric series F(a', ft', y, z) where oi -\-ft =1,
a'/J' = a>, y •=. 1 ; and this is finite for 2 = 0 or /* = 1, so that

/l-M\a
--

which converges for all real values of p. between + 1 and — 1, but diverges for
M=-l.

In our equations the quantity ft is always real ; the quantity a may be complex of
the form — £ + iq ; in any case we have always a solution of our equations in series or
definite integrals.

18. Supposing Ti'^/x), Tj,f)(/i) known, we shall be able to write down the values of
<ru o-j, OT ; and then, supposing the surface bounded by a small circle /* = const., we
have for the boundary-conditions

• "Ueber die Mehler'schen Kogclfunctioncn," ' Mathcmat. Annden,' vol. 18, 1881.

534

MR, A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

for some fixed value of /i.

Returning to our equations (69), we have, omitting the <f> factors,

sv

-

, 9 / V

SU

hence,

or

(73)

omitting «^ and t factors, and writing T0 and T^ for TJ,f)(/i), T^(/i).

Substituting in the boundary-equations (72), we have, on elimination of the ratio
A : B, the frequency-equation

; . . (74)

AND DEFORMATION OF A THIN ELASTIC SHELL,
and, if p. = 0 at the edge, or the shell be hemispherical, this is

-*)'' . (75)

In the case of the symmetrical vibrations s = 0 and the expressions found involve
indeterminates. In any other case the above expressions show that it will not be
possible for the motion to be purely tangential, since, for this, A = 0, and we should
have to make dTft/dfi. = 0, T,, = 0 for some value of p..*

19. In the case of the symmetrical vibrations we have to put M, f», w independent
of <£ in (54), (55), (56) ; this gives

=s

„ j n -' A l " * I

«*

-

du

j

From which

2e

where /8, a have the same meaning as before.
Hence,

I

ao-j = A I —

The boundary-equations (72) become

(76)

(77)

(78)

.... (79)

* Using only the differential equations, MATHIEC supposed that there c mid be onaymmetrical
tangential vibrations.

536 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

which can be satisfied either by

•B^O, and l+(l + o-)«(a+l)P.-(l-o-),i --"=<>, (80)

rt/4

or by

tJP
A = 0, and y8(/3+ l)P,-2/ = 0 ... (81)

This gives two types of motion.

In the first the motion is partly tangential and partly radial. Since Pa (/*) cannot
have equal roots, u and w cannot vanish together, or there are no lines of no displace-
ment. The displacement is purely radial along the lines dP» (p.)/dp, = 0, and
purely tangential along the lines Pa (p.) = 0. The ratios of the frequencies of the
component vibrations of this type depend on cr, i.e., on the material of the shell.

In the second type the motion is purely tangential, every point moving through a
distance along the parallel to the edge through it, which is the same at all points of
the parallel. The lines dPft(p.)/dp. = 0 are nodal. The ratios of the frequencies of
the component vibrations are independent of the material of the shell.

20. For a hemispherical bowl /u, = 0 at the edge.

(1.) In the motions of the first type Pa(p-) is to vanish with p, ; hence, a. is an odd
integer, or, i being any integer, we have

(2 + *2)/(l + c) = (2i + 1) (2t + 2) = 01 say,
where

c [K» - 4 (1 + cr)/(l - cr)] =««(! + <r)/(l - cr).
This gives

*4(1 -cr)-2K2(l +3cr + ft))+ 4 (co - 2) (1 -f cr) = 0 ; . . •' . (82)

this equation has always real roots.

If K;2, K? be the roots, and pit p\ the corresponding values of p, according to the
formula pza2p = n/c2, then

(83)

- /*2) {P2(-
o afl

17= 0,

f~V(l -^)(;

L

,• = o
To get arithmetical results, let us choose cr = ^ ; the equation for K2 becomes

+ 2)-2} =0,
and KJ, K'J are given by the table :—

AM) DEFORMATION OP A THIN ELASTIC

5:17

i

0

1

2

3

4

6{(l + (,'+l)(2i+l)}

m

42

96

174

276

8{(2.- + l)(2.'+2)-2}

0

80

224

432

704

«»

12

f40
i 2

48 ± v/(2080)

87±v/(7137)

138 + ^(18340)

«i

3-464 . . .

6 324 . . .

9-676 . . .

13-095 . . .

lfi-505.^.

«'i

0

1-414 . . .

1-635 . . .

1-587 . . .

1604...

The tones of the second series are all near together ; those of the first are separated
by intervals rather less than for a harmonic scale.

(2.) In the motions of the second type A = 0, and Pp (/*) vanishes with /* ; hence ft
is an odd number, and

2) = w. (84)

If p"i be the value of p corresponding to K",-, we have

and

V = 0, w — 0,

v = s " [v/0 -

and K'\ is given by the table : —

. . . . (85)

t

I

2

3

4

5

*i -a

10

28

54

88

130

«"i

3-158

5-291

7-347

9-380

11-401

«"<:«",

I

1-673

2-323

2-966

3-605 . . .

These intervals are nearly fifths.

MiN-n I.X\X\ 111.- A.

3z

538 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

§ 8. Vibrations of Cylindrical Shell.

21. As a further example, suppose the middle-surface of the shell cylindrical ; and,
to fix ideas, suppose there is a rigid disc at one end, and at a distance c from it a free
edge bounded by a circle.

Let a be the radius of the circular section of the cylinder, and a, z, </> cylindrical
coordinates of a point on the middle-surface, the origin being at the centre of the
rigid disc.

In the equations of motion of Art. 13, we have to put

hl=l, h.2 = I/a, l/Pl = 0, I/A, = I/a.

Taking u, v, w proportional to &>*, and *3?i = p°azp, these equations become

3%** ^ 3*K 3m — TO /f)2?« . 1 3'0 \ . „ m — n 1 8w

a,* T „* u ~r ,.2 a .5 T

m + n \02J a oz dq>/ m + n a

a~¥ + ~ V + " - -f- - -7- '- H — - ) 4- 4 - - = 0 (87)

ck2 a2 a3 3$2 m -f- n \as 3<£2 a vzo<j>/ m + n a? c<f> •

H? 4m 1 / . 3» \ . „ »i — TO 1 3w

-w=- - ( w + ^- )+ 2 - -5-. (88)

a* m + TO a2 \ o0/ m + TO a 82

Put 4mj8/(m + n) = /c2 — 4w/(w + n), then (88) is
and (86), (87) give

4:WZ- G U , /C .X (J"U OtTt ~~ ft/ J. C/"y . ^ 77t ' u, I \y v wwi _

» 5I"rn ~r To ' ( ^, 52 "T ^^ o^ / == ^>

„_ _i „ I 2 I 2 Ci J.2 I „ , i ., « d M &JL ' £>

7/t "t~ »*• cs tt a O(p T/I' ~r ?& a oz dp flo ?/t

93v «2 4w 1 92v 3m — w 1 92w 1 4m / 3'3w , 32v \

4. I -L. I -— /« I \ — fi

^ o i o " i o K io "i *> "i i o/^ I "^ r> *N » r n 10 / — v>

O-2 fl Til -\- H d VW W "- r'?r''" /».*H iw. -4- T). \ n?r ri/ri r)/r»- /

or

1 99

AND DEFORMATION OF A THIN ELASTIC SHELL. 539

Let u a cos s<f>, v a sin s<j> ; then fur w and r as functions of z we have the equations

Let u^, u2* be the roots of the quadratic (95), then

u = cos *<£ e'f' [ Pl cos utz + P2 cos /ijZ — P^ sin
v = sin s<> e'^Q sin uz + Q sin uz + Q' cos

— P'2 sin
+ Q'2 cos

so that

= cos s
+ cos .s

sn
j) cos
3 z 2

sn
cos)

- _

_ + I), - C - = 0,
where

A =

rf»

........ (93)

c- '-i 4. 2* l +

-a(} rl-<r 0

b.«k-#i±*-

«f\ /8 l-<r/ J

To satisfy these, take

t* = P cos 02 ] if = — f sin u2 ]

J- or }• ...... (94)

v=Q8in/i2j v = Q cos uz I

Then

P (B - A/i2) + QC/t = 0,
( ,. . PCu-f

F

whence

(B - Au») (D - u«) - C3^ = 0.

This is a quadratic in u8, viz. :

A/i*-(B + AD + C2)/A2 + BD = 0; ..... (95)
and we have

Q = C-P'' Q' = C.P'.. ..... (96)

. (97)

. (98)

540 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

Calculating from these, we find, dropping the time-factor,

erj = — cos s<j> [ftjPj sin n^z + /AC PC sin p..zz + f^P^ cos p.tz + f^P^ cos /A2z],

K, \ / s \ -i

/*iQi — '- pi cos /^z + (/i2Qa -- P2 cos /tgz
I \ / J

— sin 6-<£ I //*1Q/, — P'j sin /t^ + 2Q'2 - P'2 sin

J

If there is a rigid disc at z = 0, then v and w vanish with z, so that

........ (100)

The first of these is, by (96),

\ +

a T-»
/tf — D

so that (100) can only be satisfied by P'1? P2, both zero, and consequently Q'j, Q'2,
both zero, unless we take /u,t2 = /i32 and Q\ + Q'2 = 0.

If pf = /i22, we have P^ = ^ P'3 and Q\ = — Q'2> so that the terms in u, v, w,
o-j, cr2, IB- which contain P'j, P'2) Q'l5 Q'2 all vanish identically.

It follows that to satisfy the conditions at z = 0 we must drop out the P', Q' terms.

The boundary-conditions at '» = c are—

^ = 0*
where we have to take only the part in Pj, P2, Q1; Q3 and to write

Mi — D M.>* — I)

Hence, we have

a* <rs8 + l C 1 . , c* <rs8 + l C

and

AND DEFORMATION OP A THIX ELASTIC SHELL. 541

Eliminating P,, P2, we get

a* <r«/9 + l C

Ml»C \ / . <r* <« £ + 1 C

or

sin

From (95), A (n* - D) (p.* - D) = AD2 - D (B + AD + C2) + BD = - DC2 ;
substituting and re-arranging, we find

- A

7 %+ ' +CD! - -

this equation gives the frequency.

22. In the case of the symmetrical vibrations, * = 0, and we have

and P2 = 0, Q1 = 0, but Q2 is finite. Thus, the equation just written involves some
induterminates.

We take the solutions

u = Pj cos p.]Z,

r = Q2 sin /ijZ.

The conditions m = 0, <rl + tro-., = 0 reduce to

( I + ^ J /tjpj sin /it c = 0, Q^fig cos fi2 c = 0 ;

hence, either

Q., = 0, and sin /i, c = 0,

or

P, = 0, and cos fa c = 0.

542 MR. A. E. H. LOVE ON THK SMALL FREE VI1WATIONS

This gives two types of motion.

In the first, the motion is partly tangential and partly radial ; the expressions for the
displacements are c

u = 2 P, cos-- «v/

i=l C

v = 0,

(103)

— <r a-iri ^ . iirz ,

' j

•n

where the equation for K is — c2 = iV, or

(1 - <r) - 2 _ .., 2 , ,

- -

and p2a2/> = w*2, i being any integer.

The displacement is, for each normal type of vibration, wholly tangential along
the circles sin iirz/c = 0, and wholly radial along the circles cos iirz/c = 0 ; there are
no points or lines of no displacement. The frequency depends on the length and
radius of the shell, and the ratios of the intervals for consecutive tones depends on cr,
i.e., on the material of the shell.

In the motions of the second type the displacement is purely tangential, and is
expressed by

10 =0,
where the equation for the frequency is

> = °° •' /' 4- 1 — -

v= 2 Q,-8in- e'f*, }. ...... (105)

i = 0 If 9

J

or

4/>,2 = (2i+ I)27r2n/c2p ........ (106)

In this case the circles sin (2i + t) 7r2/2c = 0, are nodal lines. The frequency
varies inversely as the length of the cylinder, and the intervals between consecutive
tones are independent of the material of the shell.

Note. — July, 1888. — In the paper as read some examples were next given of the
application of the method to problems of equilibrium. These are now withdrawn, as
of little physical interest, and not directly relevant to the subject of the paper (see
Summary).

AND DEFORMATION OF A THIN ELASTIC SHELL. 543

§ 9. Summary.*

This paper is really an attempt to construct a theory of the vibrations of bells. In
any actual bell complications will arise, which have been omitted in this discussion,
partly from variations of the thickness in different parts, and partly from the
want of isotropy in the material. We can hardly expect a metal which has been
subjected to the process of bell-manufacture to be other than very seolotropic, while it
is notorious that bells are usually thickest at the rim. The difficulty of the problem
in its general form seems to make it advisable to begin with the limiting case of an
indefinitely thin perfectly isotropic shell, whose thickness is everywhere constant, and
so small compared with its linear dimensions, that powers of it above the first may be
neglected in mathematical expressions, which contain the first and higher powers
multiplied by quantities of the same order of magnitude.

Of previous theoretical work we have examples in Lord RAYLEIOH'S ' Theory of
Sound,' and in his paper on the " Bending of Surfaces of Revolution," in ARON'S and
MATHIEU'S memoirs, and in IBBETSON'S treatise on the Mathematical Theory of
Elasticity. In the ' Theory of Sound ' Lord RAYLEIGH treats the vibrations of a
thin ring or infinite cylinder of matter, supposed to be deformed in such a way that
the motion is in one plane and the elements remain unextended, and remarks that at
the time of publication this was the nearest approximation to a theoretical treatment
of bells. He afterwards applies his theory of the bending of surfaces to obtain a more
exact analytical method of treating the problem, but his disregard of the boundary-
conditions which hold at a free edge appears to vitiate this theory. AROX can hardly
be said to have attained a theory of bells, and the interest of his memoir is mainly
mathematical ; his inaccuracies have been already referred to. I have also previously
referred to the objection which lies against MATUIEU'S method of treatment ; this and
the complexity and difficulty of some of his analysis seem to render a new method
desirable. I shall have to refer to IBBETSON later.

The theory here put forward rests on the form of the function expressing the
potential-energy of deformation per unit area of the middle- surface of the shell.
Supposing that the surface is stretched and has its curvature changed, we find that
the energy consists of two terms. One of these contains only the functions defining
the stretching, while the other contains also those defining the bending of the middle-
surface. The modulus of stretching is proportional to the thickness, while the
modulus of bending is proportional to its cube. Unless, therefore, the functions
expressing the stretching, viz., the extensions and shear of rectangular line-elements
of the middle-surface, are of a higher order of small quantities than those defining
the bending, viz., the changes of the principal curvatures and of the directions of the
principal planes, the vibrations depend on the term which involves the stretching, and
not on that which involves the bending. Now, it seems to have been universally

• Partly rewritten, July, 1888.

544 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS

assumed by English writers that the reverse of this is the case, viz., that the vibrations
take place in such a way that no line on the middle-surface is altered in length. This
will be borne out by a reference to Lord KAYLEIGH and IBBETSON. The theory of
the present paper rests on the fact that the functions expressing the stretching and
those expressing the changes in magnitude and direction of curvature are of the
same order of small quantities. This is proved in the following way : — The potential
energy consists of two parts ; one, Q2, proportional to the thickness h ; and the other,
Qj, proportional to hs. The first is expressed in terms of the stretching, and the
second in terms of the bending of the middle-surface. Some previous theories have
proceeded as if Ql only occurred. If this were the case, we ought to get an approxi-
mation by supposing that Q%/h = 0. This is equivalent to assuming that there is no
stretching of the middle-surface. We should therefore get an approximation by
supposing the surface inextensible to the first order. The stretching and the bending
are expressed, to the first order, by linear functions of certain differential coefficients
of the displacements. Our supposed method of getting an approximation is then to
make the functions expressing the stretching vanish. Now, I have shown that the
functions expressing the displacement are thus, to a certain extent, determined, and
that iu such a way that the boundary-conditions cannot be satisfied. The boundary-
conditions referred to are the exact conditions found by retaining the complete
expression for the potential energy. Tt is inferred that the functions expressing the
stretching cannot be taken equal to zero for an approximation ; or, in other words,
• small compared with those expressing the bending ; and, thus, QJh3 and Q2/7i, are of
the same order of magnitude. The conclusion that Qj is small compared with Q2
seems inevitable.

The argument breaks down for a plane plate through the vanishing of the curvatures ;
Ql is then alone of importance. In the case of an open shell or bowl whose linear
dimension is small compared with its radius of curvature, and large compared with its
thickness, both terms are important. When this is so, we get a class of cases for
which the linear dimensions concerned are of three different orders of magnitude, and
this case will not come under the method of the present paper. It may be compared
with the problem of the watch-spring mentioned in THOMSON and TAIT'S ' Natural
Philosophy,' Part 2, p. 264, which stands between a bar and a plate. The very open
shell or bowl stands in the same way between a plate and what I have called a shell.

The theory of this paper proceeds as if Q2 alone occurred. It is to be regarded as
the limiting form for indefinitely thin shells. A complete theory of bells, even when
regarded as uniformly thick and isotropic, could only be obtained by using the exact
equations formed by retaining both terms of the potential energy.

Again, English writers have assumed that the potential energy, which they suppose
to depend only on the bending, will be the same quadratic function of the changes of
principal curvature as it is for a plane plate. The same authorities as before may
be quoted, and we may also refer to a question set in the Mathematical Tripos,

AND DEFORMATION OK A THIN KLASTIC SMKLL. •' ' '

.T:iini:iry 18th, morn., 1*7S, c|iir-ti<>n 77. To test this assumption involved the investi-
gation of Artt. 7. .-. an. I die n-Milt is tli.it it is only in the case of a sphere supposed
imstretched that the potential i-nrr^y lias this form. This is the case treated by
Lord liAYi.KKiii, hut his method still fails, for a complete sphere cannot be bent
without stretching, while, if the sphere be incomplete, the conditions which hold at a
free edge cannot be satisfied ; this is explicitly proved in Art. 14.

A general result is derived from the consideration of the functions expressing the
kinetic and potential energies, Q2 only being retained. Both these functions are
proportional to the thickness of the shell, and thus the periods of vibration are inde-
pendent of the thickness. That this result holds for a complete thin spherical shell
vibrating in any manner has been demonstrated by LAMB (' London Math. Soc. Proc.,'
vol. 14, 1882, p. 52). His equations (7) and (9) when reduced are independent of the
thickness.

Two general results are obtained without solution from the equations of motion.
The first is, that vibrations involving displacement along the normal only are impos-
sible except in the cases of the plane, complete sphere, and infinitely long circular
cylinder. IBBETSON'S treatment of the problem appears to assume (I) inextensibility,
(2) the incorrect formula for the energy, (3) normal displacements. The other result
is that any surface of revolution can execute purely tangential vibrations which are
symmetrical with respect to the axis of revolution, and in which the motion is purely
torsional, or perpendicular to the planes through the axis. These must not be
confounded with the familiar vibrations of finger-bowls, which are most probably a
type with two nodal meridians.*

The theory of the vibrations of a thin spherical shell bounded by a small circle is
an interesting example of the general theory of vibrations of an elastic solid. In an
infinite solid there are two types of vibratory motion, the longitudinal and the
distortional, both of which are propagated as waves. In a bounded solid this state of
things is modified by reflexions at the bounding-surfaces, so that the purely longitu-
dinal and purely tangential waves do not in general exist separately. Again, in all
cases of displacement in one direction only, as in the vibrations of strings, bare, and
plates, there may be displacements in different directions which are independent of
each other, with their corresponding nodal lines or points. This also is modified in
the general solid. The types of vibration, for example, of a portion of a spherical
shell bounded by a small circle are partially made out in this essay. One immediate
result is that there are in general no nodal lines, properly so called. Tn any type the
displacement along the parallels vanishes at one set of meridians ; the other displace-
ments vanish together at another set of meridians. These sets are ranged at equal
intervals round the sphere. There appears to be good reason to suppose generally
that the corresponding proposition will not obtain with reference to nodal parallels.
The establishment of the fact would require a solution of the general frequency cqua-

• RATI.EIOH, « Sound,' vol. 1, Art, 234.

MDCCCLXXXVIII. — A. 4 A

546 MR. LOVE ON THE VIBRATIONS, KTC., OF A THIN ELASTIC SHELL.

tion, and this I have not been able to effect. One case, however, is readily solved,
and that is where the displacement is symmetrical with respect to the pole of the
sphere. It appears here that the vibrations divide themselves into two types, one
purely tangential with displacement along the parallels, the other partly radial and
partly consisting of displacements along the meridians. There are no nodal meridians.
In the purely tangential vibrations there exists a series of nodal parallels, whose
number corresponds to the type of vibration. The intervals for the various tones are
each of them nearly a fifth. In the partly radial vibrations the radial displacement
vanishes at one set of small circles, and the tangential displacement at another set.
The number and position of the nodal circles for the purely tangential vibration
coincide exactly with the number and position of the circles along which the
tangential displacement vanishes in the corresponding partly radial mode. The
vibrations of the two types belong to different normal modes of vibration, and have
different frequencies. If we like to extend the meaning of " nodal lines," so as to
include the small circles just referred to, then we may state another result in the
form that for partly radial vibrations there are two periods and modes of vibration
which have the same set of " nodal lines." The tones of one of these sets are all
very near together ; those of the other set are separated by intervals nearly the same
as for a harmonic scale.

A discussion of the vibrations of an elastic shell in the form of a circular cylinder
closed at one end by a rigid disc perpendicular to its axis leads to similar conclusions
as to types of vibration and their definition by nodal lines.

It is unfortunate that solutions of the frequency equation for the case of two
"nodal" meridians dividing the shell into four equal portions could not be obtained,
as these probably include the gravest mode of vibration of which the shell is capable.
The tones of the symmetrical vibrations discussed are very high, and the theory in its
present state cannot easily be tested by experiment. There is, however, one result
which would seem to admit of practical verification, viz., it is found that, for similar
thin shells, the frequency is independent of the thickness, and varies inversely as the
linear dimension.

[ 547 ]

XVII. Colour Photometry. — Part II. The Measurement of Reflected Colours.
By Captain ABNEY, C.B., R.E., F.R.S., and Major-General TESTING, R.E., F.R.S.

Received May 3,— Read May 31, 1888.

[PLATES 20-23.]

§ XXVII. Old Method of Measurement*

IN our first paper on this subject we have shown how the luminosity of the spectra of
various sources of light can be measured ; and the present paper is an extension of the
subject, dealing with the measurement of the light reflected from bodies in terms of the
colours of the spectrum of the light illuminating them. By the method which we
adopted in the first part of " Colour Photometry " this can be effected, and, indeed, we
carried that out in several instances. The method then employed was very simple. If
we wished to measure the illuminating value of the spectrum of light reflected from a
metal, we placed it at an angle in front of the slit of the spectroscope, so as to reflect
the light from the crater of the positive pole of the electric light through the photometer,
and measured the luminosity of each part of the spectrum thus formed by the method
we indicated in our paper. Again, in experimenting with GORHAM'S discs, such as
MAXWELL employed, where it became necessary to determine the light reflected from
the different coloured papers or cards used in the discs, the plan first adopted was to
replace the receiving shadow screen of zinc oxide (see § VI) by the coloured papers,
and again to make a luminosity measurement. This plan answered its purpose, but it
was rather laborious. When two or three colours are combined by rotation to form a
grey, and black and white sectors are combined to match that grey, in order to
ascertain the total luminosity of each colour, the angular value of the sectors being
known, it is necessary to refer the luminosity to that of some standard reflecting
surface, which is naturally a white one. As the comparison light is coloured by
falling on coloured paper, the value of the spectrum reflected from such paper could
not by this first method be directly compared with that reflected from the white
screen. In the case of a coloured screen, the curve of spectrum luminosity would
therefore have to be reduced to that in which the comparison light was white. This
difficulty was surmounted by making half the receiving screen white and half of the

* Tho numbering of the paragraphs and figures in this paper is a continuation of that of Part I. —
Bnkcrian Lecture. ' Phil. Trans.,' 1886.

4 A 2 22.12.88

548 CAPTAIN ABNEY AND MAJOR-GENERAL FEST1NU

colour whose luminosity was to be measured, illuminating the shadow of the rod
thrown on the coloured paper by the spectrum colour, and that thrown on the white
card by the white light reflected from the surface of the first prism (§ XXVI). This
did away with any reduction or calculation ; but still an objection remained, as, for
definite comparison, it was almost necessary that the same observer should always
make the measurement.

§ XXVIII. Revised Method.

One of us having to measure the colour of various water-colour pigments for a
Government enquiry on their fading, it became important to introduce some other plan
by which the same end could be attained. Various artifices were tried, but finally we
came to the conclusion that a spectrum photometer was necessary, and on these lines
the following various modifications of our original apparatus were devised by one of us
(Captain ABNEY) : — The collimator, prisms, and camera were at first kept as in the
colour photometer ; but for the camera lens was substituted a lens divided into equal
segments, which could be centrally separated, as in a heliometer. The light coming
through the last prism fell as a square patch on this divided lens, and the two segments
were separated so that two spectra fell on the focussing screen, one above the other.
A slit in a card was then passed across this double spectrum, and any required ray

Fig. 13.

i

-^sf

was isolated. P is a right-angled prism attached by a rod to the top half of the slit
so as to reflect the ray from the top spectrum to one side, whilst the ray of the same
colour from the bottom spectrum traversed the slit unimpeded and fell on the lens Lm,
forming a patch of monochromatic light on the screen. The ray which was reflected
by P was again reflected by a mirror M2, and fell on another lens LIV, by which a similar
patch of monochromatic light could be made to fall over the patch formed by Lm. Each
of these monochromatic rays cast a shadow of a rod, placed in front of the receiving
screen, and the shadow cast by each spectrum was illuminated by light of the same
colour coming from the other. To measure the value of a coloured paper, the screen
was made half with a white card and half with the coloured paper, as in the figure.
The shadows were made to touch at the intersection of the card and coloured
paper. In front of the light which illuminated the shadow cast on the white card was
placed a motor rotating movable sectors, as described in our paper recently read before

u.\ tiM.'M |{ I'llMl'o.MKl |;y 541)

the Royal Society.* Two methods presented themselves of equalising the illumhmt !• >n
of the shadows — first, by moving the slit across t IK.- spectrum whilst the sectors rotated
\\ ith a fixed aperture ; or, secondly, by placing the slit at known places in the spectrum
and equalising the illumination of the shadows by altering the aperture of the sectors.
In cases where the absorption of the spectrum by the colour increased rapidly, the
first method was most convenient, but where the absorption was very gradual, the
latter method was found to be most accurate, and was most usually adopted.

*

Fig. 14.

This plan of producing the two spectra, at first sight, seemed everything that could
be wished, but a difficulty occurred which rendered a further modification advisable,
for it was found that the two spectra were not of proportionate intensity throughout.
This was discovered in following out the necessary order of experiment ; which was,
first, to compare the luminosity of the spectrum on the coloured paper with that on
the white card, and then to compare the values of the two spectra to one another
by throwing both shadows on white card. If the spectra were of proportionate
intensity throughout, it should only have been necessary to measure the relative
values of any one ray, and the same ratio ought to have been obtained for any other.
When trying this, however, it was found that if the two shadows were cast by rays
in the red end of the spectrum, there was decreasing value in one of the spectra
towards the violet. In fact, in the extreme violet the spectrum of one was only
about three-fourths as bright as that of the other. The cause of this difference
became apparent when examining the matter. The half lens which focussed one
spectrum received the rays which had passed through the thinnest part of the
prisms, whilst that focussing the other had passed through the thickest parts. The
difference in the ratio of the brightness at different parts of the spectrum was
traced to the different amounts of light absorbed by the different thickness of
glass traversed. Any slight shift in the position of the line of separation of the lens
altered the nitio of absorption, and, as in some cases such a shift could not well be
avoided, the method, though practicable, was scarcely practical. It became evident,
then, that some means must be adopted of forming each spectrum with the light
which had traversed the same thickness of glass.

• " Photometry of the Glow Lamp," ' Roy. Soc. Proc.,' vol. 43

550 CAPTAIN AHNKY AND MA.IOK-GKNKRAL FEHT1NG

$ XXIX. Apparatus jui«U ;i .l</«/>teil.

After many experiments, it was determined to fix a double-image prism behind the
collimating lens. This double-image prism of Iceland spar was made by Mr. A. HILGER,
with his usual ability, and was so adjusted that when the central half of the collimator
slit was used the two spectra, while of the same length, were separated by one-eighth
of an inch on the focussing screen, the ordinary camera lens being employed. The
reflecting apparatus was also slightly altered by substituting for the fixed reflector
a second right-angled prism attached to the card so as to reflect the light through
the second lens L,v. There was a great advantage in this, for with the fixed reflector
the colour patch travelled across that formed by the direct beam, and thus the same
parts of the image of the prism's face were not always superposed. The plan of
attaching the reflector to the slit card got over this difficulty, and rendered the

measurements more accurate.

Fig. 15.

§ XXX. Adjustment of the Instrument.

The adjustment of the instrument, when using the double-image prism, required
care, and the following plan was adopted. The whole slit of the collimator was
illuminated by light from the arc in which lithium and sodium were vaporised. The
two spectra now overlapped, since the separation of one-eighth of an inch was only
obtained when the slit was one-fourth of an inch in height. The bright lines of the
lithium in the two spectra were then made to coincide by turning the double-image
prism ; the central portion of the slit in the collimator was then used, and the slit in
the card passed through the two spectra. If the collimator slit was properly adjusted
m the vertical and a bright line in one spectrum traversed the centre of, say, the top
part of the aperture in the card, the same bright line in the other spectrum ought to
traverse the centre of the bottom part of the aperture. If this were not so the colli-
mator was readjusted, and the same operation gone through. To make doubly certain
that the adjustment was correct, the direct and reflected rays from different parts of
the continuous spectrum of the positive pole were made to form superposed patches
on white card, and shadows of a rod were cast by each so as to touch. The rotating
sector was placed in front of the brightest, and the illumination of the two equalised.
If the same aperture of sector equalised the illumination throughout the spectrum
the adjustment was considered as complete, if not, a new adjustment was made till
such was the case. It was found in practice that a very good adjustment could be

ON COLOUR PHOTOMETRY. 551

mode by noting if the colours of the two shadows were of exactly the same hue,
more especially in the transition between orange and green, and in that from the
blue-green to blue. The slightest departure from true adjustment invariably showed
itself in these two parts of the spectrum. We should not hesitate to adjust the
instrument by this means alone, though in all the measures taken the comparison of
the two spectra on white card was invariably made.

§ XXXI. Exclusion of Extraneous Light.

There is another point of special importance to be attended to, viz., the exclusion of
all extraneous light from the receiving screen. If two shadows are to be compared
together, when the whole of the screen is white or of the same colour, the admission
of extraneous light is not detrimental ; but, if one shadow falls on a white ground and
the other on what in white light is a coloured ground, it is absolutely necessary to
keep the screen free from all light except that forming the shadows.

It was curious to note the change in colour produced on the coloured half of the
screen when illuminated partially by a portion of the spectrum weak in luminosity, and
partially by weak white light. It was absolutely impossible to match the colours,
when even a very small percentage of white light fell on the screen. The whole
apparatus was placed in a darkened room, the electric light being in a lantern.
Extraneous light was excluded by placing the screen at the end of a box 18 inches
wide, 12 inches deep, and 2 feet long, the interior being blackened. A white card
placed at the end of the box was then invisible when the electric light was burning
and the slit in the card was placed beyond the limit of the spectrum.

§ XXXII. Width of Slit Employed.

The great point in measuring accurately was to adjust the luminosity so that it was
of such brightness that the eye could readily distinguish any small difference in the
brightness of the illuminated shadows. This was effected by altering the width of
the slit in the collimator from time to time. When the brightest part of the spectrum
was under measurement, the width was about t^th of an inch, and, when the least
luminous parts, it was opened to about -j^-th of an inch. The slit in the card remained
invariable, being about -^jth of an inch in width. The screen was placed 3 feet from
the slit card.

§ XXXIII. Experiments with Emerald Green, Vermilion, and Ultramarine.

The first experiments were conducted to ascertain the composition of the grey light
given by a set of discs of emerald green, vermilion, and French ultramarine. Discs of
these colours, 6 inches in diameter, were prepared ; and a larger pair of black and
white discs arranged on the same axis. The sectors of the three colours and of the

552 CAPTAIN ARNKY AND MA.TOR-GKNKRAL !• Ks'l'l M ;

black and white were altered by trial to match when rotated in the patch of white
light formed by the recombination of the spectrum. When these same coloured discs
were rotated in day light or gas light they, as was to be expected, no longer formed
a grey, but had a predominant tint of green or red ; and the illuminating value differed
from that of the black and white discs. The alteration in hue and luminosity in
passing from one source of light to another, showed the necessity of using the same
source for making the match and for measuring the luminosity of the colours. In
adjusting the apparatus as explained above, with both parts of the receiving screen
white, we found that the rotating sectors had to be set with an aperture of 69° in
order to get a balance throughout the spectrum. Measurements were then made
throughout the spectrum of the intensity of light reflected from each of the coloured
cards, the aperture of the rotating sectors at each part giving the relative amount of
light reflected, the maximum value being 69°.

§ XXXIV. Calculations of Luminosity.

The mean angular values of the coloured cards in the rotating disc which matched
the white and black disc were as follows : — Emerald green, 133°7 ; vermilion, 96°'6 ;
French ultramarine, 1290>7. In order, therefore, to get the comparative amount of
light reflected from each coloured sector in the disc in terms of that reflected from the
emerald green, the readings of the red card were reduced in the ratio of 96'6/1337,
or multiplied by '722, and those of the blue by 1297/1337, or '97, those of the
green card being unaltered. From these figures the curves on fig. 16 were plotted ;
the straight line at 69 being taken to represent the amount of light at each part of
the spectrum which was reflected from a white card sector of 133°7, the ratio of the
ordinates of the other curves to 69 would indicate the proportion of each ray reflected
from the coloured sector as compared with that from a white sector of 133°7.

From these curves the luminosity curves in Plate 20, fig. 17 were constructed.
The outer curve is the normal curve of white light, as given iu Part I. of " Colour
Photometry ' (§ VIII. and fig. 3), the scale of the spectrum being the same. The curves
for the colours were then made, their ordinates bearing the same proportion of those of
the outer curve that those of the curves in Plate 20, fig. 16 bear to 69.

Abney <fc Feeting

I'lul iron*. 18K8. A /'/„/,

Abrvey

Phil.

AftKnfrufnn jo

•Jl

nV

. • ,

Abney k Feeting.

1 H 88. A . PluU li 2

,

I

(N

Abwy k, Festing.

Phil.

Fig 30

.s lH88.A.P/«te 23.

t

«ooo no mo mo teoo iso ao isc tooo ZM soo ?;o van

Fig. 31.

«000 Z50 SOO HO MOO 25O MO 7iO OOOO ISO SOO VO WOO

Fig 32

«000 KO SCO 7!0 SOCO MO SOO OO HXU 250 SOO

ON COLorU I'HOTO.MKTKY.

553

Emerald Green. Sector taken as unity.

Original reading!.

Reduced from plotted cunre.

l.uminomtj cunrea.

Scale number.

Reading.

Scale number.

Reading.

While.

Emerald green.

42-80

5-0

43-00

50

2-0

•10

43-30

5-0

44-00

5-0

L6-0

120

44-30

65

45-00

8-0

65-0

45-32

95

45-50

11-0

84-0

13-40

45-83

14-5

45-80

13-0

90-0

17-00

46-85

23-0

46-00

14-0

94-0 18-90

47-36

27-0

46-35

17-5

97-0

24-60

47-87

34-0

46-85

23-0

99-5

33-20

48-38

40-0

47-10

25-0

100-0

36-20

48-89

43-5

47-35

27-5

99-5

39-60

47-50

30-0

99-0

48-00

49-45

49-0

47-85

34-0

..,-,,

48-30

49-91

50-0

48-90

44-0 82-0

52-30

50-42

52-0

49-40

49-0 70-0 50-00

50-93

50-0

49-90

51-0 48-0

35-50

51-44

48-0

50-40

52-0

31-0

88-30

51-95

46-0

50-90

50-0

20-5

14-80

52-46

41-0

51-90

46-0

10-0

6-60

53-48

34-0

52-90

38-0

5-5

8-00

54-50

24-0

54-00

28-0

3-5

1-40

55-26

20-0

54-50

24-0

2-8

•80

57-05

15-0

55-00

21-0

2-1

•60

58-07

15-0

57-00

15-0

•9

•20

58-00

15-0

•6

•13

Vermilion. Sector = '722 of Green.

Original reading!.

Reduced from curve

Reading
reduced by
728.

Luminotitjr cunrea.

Scale number. Reading.

Scale number.

Reading.

While. Vermilion.

42-80 40-0

43-00

45-0

36-0

2-0

•80

43-30 50-0

43-50

55-0

39-0

5-0

43-80

60-0

44-00

61-0

44-0

16-0 10-10

44-30

65-0

44-50

66-0

47-5

37-0

25-50

44-80

65-0

45-00

66-0

47-5

65-0

44-70

45-30

62-5

45-20

66-0

47-5

74-0

51-00

45-80

co-o

45-50

65-0

47-0

84-0

57-00

46-35

45-0

45-80

60-0

43-0

90-0

56-00

46-85

29-0

46-00

55-0

100

'.Kl-5 53-00

47-60

18-0

4635 45-0

325

97-0 45-70

47-35

14-0

46-85

29-0

21-0

99-5 30-00

47-85

9-5

47-10

21-0

16-0

100-0

23-20

49-50

7-0

47-35 14-0

9-0

99-5

13-00

52-90

4-0

47-85 9-0

65

98-0

9-20

57-00

4-0

48-90

7-U

5-0

82-0

6-00

58-50

4-0

49-40

6-0

4-5

70-0

4-50

49-90

5-5

4-0

48-0

2-80

50-90

5-0

3-5

20-5

1-00

51-90

4-0

3-0

10-0

•40

52-90

4-0

3-0

5-5

•20

54-00

4-0

3-0

3-5

•15

55-00

4-0

30

21

•10

57-00

4-0

30

•9

•07

58-00

4-0

3-0

•6

•03

Mix (VI. \\.\\ 111. A.

4 B

554

CAPTAIN ABNEY AND MAJOR-GENERAL FESTLNC
French Ultramarine. Sector = '97 of Emerald Green.

Original readings.

' Reduced from plotted curve.

Reduced to
•97 reading.

Luminosity.

Scale number.

Reading.

Scale number.

Reading.

White.

Blue.

42-80

3-0

43-00

3-5

3-4

20

•10

43-30

4-0

44-00

6-0

5-9

16-0

1-40

43-80

6-0

45-00

4-0

3-9

65-0

3-80

44-30

5-5

45-50

4-0

3-9

84-0

5-00

44-80

4-0

45-80

4-0

3-9

90-0

5-30

48-90

5-0

46-00

4-0

3-9

93-5

5-50

49-40

6-5

46-35

4-0

3-9

97-0

5-70

49-90

9-0

46-85

4-0

3-9

99-5

5-80

50-40

11-0

47-10

4-5

4-2

100-0

6-10

50-95

14-0

47-35

4-5

4-4

99-5

6-40

51-45

17-0

47-85

5-0

4-9

98-0

7-00

51-95

20-0

48-90

5-5

5-4

82-0

6-30

53-00

24-0

49-40

6-5

6-1

70-0

6-10

54-00

27-0

49-90

9-0

8-7

48-0

6-00

55-00

26-0

50-90

14-0

13-5

20-5

4-00

56-00

26-0

51-90

20-0

19-5

10-0

2-80

58-00

26-0

52-90

24-0

23-3

5-5

1-85

54-00

26-0

25-0

3-5

1-30

55-00

26-0

25-0

2-1

•80

57-00

26-0

25-0

•9

58-00

26-0

25-0

•6

•20

The above tables give the figures on which the curves were based. Column I.
gives the spectrum scale, and Column II. the original readings. Columns III. and
IV. give the adopted readings at the different scale numbers. Column V. gives, in
the cases of red and blue, the ordinates reduced in proportion to the angles of the
sectors, as explained above. The last column but one gives the ordinates of the curve
of luminosity of the light reflected from white card, which is the outer curve in fig. 5
(Part I.). The last column gives the ordinates of the luminosity curves of the colours.

Reverting to fig. 17, if the area of the outer curve, which is 534, represents the total
amount of light reflected from a white card sector of 133°'7, the areas of the curves for
the green, red, and blue, which are respectively 221, 156'3, and 47'9, represent the
amount of light from the coloured sectors; and the total of these, or 425 '2, represents
the amount of light reflected from the disc made up of the coloured sectors.

Now the black and white disc which, when rotated, matched the coloured disc, con-
sisted of black and white in the proportions of 278 : 82, and we find that the black
paper reflects 8 '3 3 per cent, of white light ; the proportion of white to the whole disc
was therefore (82 + '0833 X 278, or) 105 to 360, the amount of light reflected from
the disc should therefore be represented by 105/1337 of 534, or 419'4, a result coin-
ciding within the limits of error of observation with that just obtained for the
coloured disc.

ON COLOUR PHOTOMETRY.

Inteniitiet.

l.uuiincMiticft.

number.

Emerald

L' r • • : , .

Yormilion.

French
ultra-
marine
blue.

Sum of
Inteusitiea.

Sum
x-37.

Emerald
green.

Vermilion.

ru™.

• •
blue.

Sum of
lumlno-
•itiec

Sum
x-»7.

43-00

5-0

36-0

34

11 t

16-4

i

•80

•10

1 ...

1 •

44-00

5-0

45-0

5-9

55-9

20-7

1-20

10-10

1-40

1-270

4-70

45-00

8-0

47-5

3-9

59-4

22-0

44-70

3-80

5370

19-90

45-50

11-0

47-0

3-9

61-9

22-9

13-50 o7'00

MO

75-50

97 M

45-80

13-0

43-0

3-9

:.:••'.»

22-1

17-00 56-00

5-30

78.

29-00

46-00

14-0

40-0

39

57-9

21-5

18-90 53-00

5-50

77-40

46-35

17-5

::.'

19-4 24-60 45-70

5'70

70-00

28-10

46-85

23-0

21-0

3-9

47-9

17-7 JM-20

30-00

5-80

•oO

47-10

25-0

16-0

4-2

45-2

L6-7

:w-20

23-20

6-10

65-50

24-30

47-35

27-5

9-0

4-4

40-9

15-1

39-60

13-00

6-40

59-20

81-90

47-85

34-0

0-5

4-9

45-4

16-8

48-30

9-20

7-00

64-50

23-80

48-90

44-0

5-0

5-4

54-4

20-1

(WO

6-30

04-60

23-90

40-40

49-0

4-5

6-1

59-6

22-0

50-00

4-50

6-10

60-60

22-40

49-90

51-0

4-0

8-7

G3-7

35-50

6-00

44-30

16 10

50-90

50-0

3-5

13-5

67-0

24-8

L4-80

1-00

4-00

19-80

51-90

46-0

3-0

19-5

25-4

6-60

•4'J

2-80

9-80

3-60

52-90

88-0

3-0

•_':;::

64-3

23-8

8 • •

•20

1-85

5-00

L-80

54-00

28-0

3-0

25-0

56-0

208

1-40

•15

1-30

2-85

1-05

55-00

21-0

80

25-0

49-0

18-1

•60

•10

1-50

-00

57-00

15-0

3-0

25-0

43-0

15-9

•20

•04

•80

•64

-J.I

58-00

15-0

3-0

25-0

43-0

15-9

•13

•03

•20

•:;(-,

•15

The above table gives a summation of the ordinates at each point of the scale of the
three coloured curves; in addition, the sixth and eleventh columns give the sum of the
ordinates multiplied by '37, which is the ratio between 133'7 and 360; and the
resulting curves, which show graphically the combination producing the grey light,
are No. IV. of figs. 16 and 17 (Plate 20).

For convenience of comparison, we have reduced the luminosity curves to the
normal scale of wave-lengths, as shown in fig. 18, which has been made from the
following table : —

550

CAPTAIN ABNEY AND MAJOR-GENERAL FESTINO

Luminosity Curves reduced to the Normal Scale.

S,aV

number.

W»ve-
lenfrth.

Ordinatea of luminosity curve*.

I.

II.

III.

IV.

I,.

II,.

III,.

43-00

699

•OS

•43

•05

-.53

44-00

662

•75

6-30

•87

7-92

•04

•37

•05

44-50

645

2-00

1420

1-40

17-60

1-00

45-00

629

3-80

32-50

2-76

39-06

•32

2-75

•23

45-20

623

6-20

38-00

3-20

47-40

4-40

45-50

614

10-50

44-50

3-90

58-20

1-88

7-90

•70

45-80

606

13-80

45-50

4-30

63-60

3-33

11-00

1-04

46-00

601

15-80

44-30

4-60

60-70

5-10

14-20

1-49

46-35

593

21-30

3970

5-00

6600

9-90

18-20

2-28

46-85

580

30-80

27-80

5-40

74-00

20-40

18-60

5-35

47-10

574

35-00

22-30 5-90

63-20

25-00

15-90

4-22

47-35

569

39-00

12-80 6-30

58-10

29-50

9-75

4-77

47-50

566

43-00

10-10 6-60

59-70

33-50

47-85

560

50-00

9-50

7-20

66-50

51-70

7-80

5-95

48-90

540

60-00

6-90

7-30

74-20

53-50

6-10

6-48

49-40

530

61-00

5-50

7-50

74-00

57-40

5-10

6-85

49-90

521

45-50

3-60

770

56-80

45-50

3-60

7-70

50-40

512

31-50

2-10

6-70

40-30

50-90

503

20-00

1-40

5-70

27-10

51-90

488

10-30

•60

4-40

15-30

52-90

475

5-10

•30

3-10

8-50

54-00

464

2-50

•27

2-30

5-07

54-50

459

1-50

•23

1-90

3-63

55-00

454

1-20

•2

1-50

2-90

57-00

436

•44

58-00

428

•30

§ XXXV. Testing the Accuracy of the Measurements.

Let us now consider fig. 16. The space between the horizontal line at 69 and that
at the bottom of the diagram may be taken to represent white light, and that between
any other two horizontal lines would, in the same way, represent degraded white or
grey.

Evidently also the space between the combination curve No. IV. and the bottom line
would represent grey, for it has been shown that, if all parts of the spectrum be com-
bined in proportion to the ordinates of that curve, the result is grey. If now a
horizontal line be drawn tangential to the highest point of the curve, the space
between that and the bottom line would represent grey ; and, as that between the
curve and the bottom line represents grey, the difference of these two or the space
between the curve and the tangential line must also represent grey : that is to say,
grey should result from the combination of all parts of the spectrum in the proportions
of the ordinates lying between the tangent and the curve.

This consideration suggested a good test for the accuracy of the method employed
in the measurement of the colours, and of the proportion in which they should be
combined to produce white (or grey). If the rays of the spectrum itself could be
taken in the proportions in which they are reflected from any pigment, say emerald
green, and recombined, the resulting light should be of the same colour as that
reflected from the pigment. This was proved to be so in the following way : — A card

ON COLOUR PHOTOMETRY. 557

disc, about 24 inches in diameter, was tuken, and commencing about 4 inches from its
centre the scale of the spectrum was laid off on a radius. Concentric circles were
drawn through the different points of the scale, and, from the centre as origin, angles
were laid off proportionate to the height of the ordinates of the emerald green curve
(fig. 16) ; the intersections of the lines forming these angles with the circles through
the corresponding abscissae gave a figure which was cut out of the disc (Plate 23,
fig. 19). The latter being rotated in a proper position in front of the spectrum
allowed the proper proportions of the different rays of the spectrum to pass through,
and then, being recombined on the screen, produced the colour of the pigment. The
identity of the colour with emerald green was proved by reflecting white light on to
a square of paper coloured with that pigment placed near the colour patch. A similar
mask (fig. 20) being cut out to correspond to the ordinates of the curve No. IV.
taken above the tangent to its lowest point produced a grey patch on the screen.

(In the prismatic spectrum the rays are more or less curved ; but, as their curvature
will not correspond at all points to that of the disc, it is necessary, in order to obtain
a correct result, to reduce the breadth of the spectrum by shortening the collimator
slit.)

A further proof of the accuracy of the measurements was also adopted. The three
discs were rotated with a larger white * disc on the same axis ; a card, having the
curvature of the outside circle of the coloured discs and a breadth the same as that
of the rod usually employed, replaced the rod, and, as before, the rays from the
two spectra cast shadows, one on the white rotating disc and the other on the
rotating coloured sectors, which, it should be recollected, to the eye gave a grey.
Measurements were made as before, and the readings being reduced proportionally to
the white which would have been present when the black sector was rotated with the
white sector, the curve No. V. in fig. 16 was obtained. At first it would appear that
the brightness of the mixture was too great ; but, as a matter of fact, it was not, for
the white card employed in the experiment was slightly greyer than that used in the
white disc which, when rotated with the black, gave the grey. It will be noticed
that this curve is practically parallel to that obtained by the summation of the three
luminosities ; this appears to confirm its correctness. If any curve give* a grey, then
any other curve parallel to it will do the same.

§ XXXVI. Experiments with Yellow and Blue Discs.

There are two favourite colours which are often used in class demonstrations to
show the formation of a grey on their rotation in proper proportions. One is pale
yellow chrome, and the other a French blue, also very pale. Discs were prepared
with these colours, and the grey produced matched by black and white. The colours
were subsequently measured by the plan already described ; the following tables give
the results of the measurements

• The beam lighting this liad passed through the rotating sectors black and white sectors, therefore,
could not be used, since scintillation was produced.

558

CAPTAIN ABNEY AND MAJOR-GENERAL TESTING
Chrome Yellow.

Original readings.

Reduced from plotted curve.

Luminosity.

Scale number.

Beading.

Scale number.

Height.

White curve.

Yellow curve.

42-80

60-0

43-0

61-0

2-0

1-70

43-80

61-0

43-5

61-0

5-0

4-40

44-30

63-0

44-0

62-0

16-0

14-40

44-80

63-5

44-5

63-0

370

33-80

45-30

63-5

45-0

63-5 65-0

60-00

45-80

63-0

45-5

63-5 84-0

77-30

46-85

61-0

46-0

62-5 93-5

84-00

47-35

63-0

46-5

60-5 98-5

88-50

47-85

66-0

47-0

61-5 100-0

92-00

48-90

63-0

47-5

64-0

100-0

92-70

49-90

55-0

48-0

66-0

95-0

91-00

50-95

47-0

48-5

65-0

88-0

82-90

51-95

39-0

49-0

61-5

80-0

71-20

52-45

35-0

49-5

58-0

67-0

56-30

52-95

30-0

50-0

54-5

43-0

33-90

54-00

23-0

51-0

46-5 19-5

13-10

55-00

21-0

52-0

38-5

9-5

5-30

56-00

21-0

53-0

30-0

5-5

2-40

57-00

21-0

54-0

23-0

3-5

1-10

58-00

21-0

56-0

21-0

1-5

•45

58-0

21-0

•6

•18

Area = 464

Light Blue.

Original readings.

Reduced from curve.

Height x 1-13.

Luminosity.

Scale number.

Keading.

Scale number.

Height.

White curve.

Blue curve.

42-80

13-0

43-0

13-0

14-5

2-0

•40

43-80

13-0

43-5

13-0

14-5

5-0

1-00

44-80

10-0

44-0

12-0

13-6

16-0

3-15

45-80

8-0

44-5

11-0

12-4

37-0

6-40

46-85

9-0

45-0

9-5

10-7

65-0

10-00

47-35

10-0

45-5

. 8-5

9-6

84-0

11-60

48-40

' 11-5

46-0

8-0

9-0

93-5

12-10

49-40

15-0

46-5

8-5

9-6

98-5

13-60

49-90

17-0

47-0

9-0

10-2

100-0

14-65

50-40

2H), •

47-5

10-0

11-3

100-0

16-20

50-95

25-0

48-0

11-0

12-4

95-0 17-10

51-95

30-5

48-5

12-0

13-6

88-0 17-20

53-00

34-0

49-0

13-5

15-2

80-0 17-80

53-50

36-0

49-5

15-5

17-5 67-0

16-90

54-00

88-0

50-0 18-5

20-9 43-0

14-00

55-00

38-0

51-0

25-0

28-2 19-5

8-00

56-00

37-0

52-0

30-5

34-5 9-5

4-60

58-00

36-0

53-0

34-5

39-0 5-5

3-10

54-0

38-0

42-9 3-5

2-20

56-0

38-0

42-9 1-5

•90

58-0

36-0

40-7

•6

•85

Area = 92-8

ON COLOUR PHOTOMETKY.

559

Sole
number.

Intend tie*.

Lumlnodtiei

Chroma
yellow.

Light blue.

8am.

8am

x-47.

Chrome
yellow.

Light blue.

Sum.

Sam
x-«T.

r.«.

61-0

14-5

75-5

35-48

1-70

•40

2-10

1-00

43-5

61-0

14-5

75-5

35-48

4-40

1-00

5-40

2-54

44-0

62-0

13-6

75-6

35-58

14-40

315

17-55

>-j:,

44-5

6*0

12-4

75-4

::. J-

33-80

6-40

40-20

17-10

45-0

63-5

107

74-2 34-87

60-00

10-00

70-00

32-90

46-5

63-5

96

73-1 34-35

77-30

11-60

88-90

41-80

46-0

tl-J-;,

9-0

715

33-60

84-00

12-10

96-10

45-20

46-5

60-5

9-6

70-1

32-95

88-50

13-60

102-10

4794

47-0

61-5

10-2

71-5

33-60

92-00

1465

106-60

50-10

47-5

64-0

113

75-3

35-18

92-70

16-20

108-90

51-18

48-0

66-0

124

78-4

36-85

91-00

17-10

10810

50-81

485

65-0

13-6 *

78-6

37-05

82-90

17-20

100-10

47-04

49-0

61-5

15-2

76-7

36-05

71-20

17-80

89-00

41-83

49-5

58-0

17-5

75-5

35-48

56-30

16-90

73-20

3440

50-0

54-5

20-9

75-4

35-28

33-90

14-00 47-90

22-51

51-0

46-5

28-2

747

35-11

13-10

8-00

21-10

9-92

52-0

38-5

345

73-0

34-31

5-30

4-60

9-90

4-65

53-0

30-0

39-0

69-0

32-43

2-40

3-10

2-50

1 17

54-0

23-0

42-9

65-9

30-91

1-10

2-20

:;:;..

1-55

56-0

21-0

42-9

63-9

30-03

•45

•90

1-35

•66

58-0

21-0

407

Cl-7

29-00

•18

•35

•53

•26

Area

= 261-7

In this case the chrome yellow was taken as the standard. The chrome yellow
required 169° of the disc, and the blue 191°; hence, the ordinates of the blue were
multiplied by 1'13. Curves I. and II. in tig. 21 (Plate 21) give graphically the
intensities of these two colours, as also Curve III. the sum of the intensities. It will
be seen that there is a deficiency in the yellow and in the blue and violet, which
together will give a grey, as indicated before. Fig. 22 shows the luminosity of the
colours in the spectrum of the light from the positive pole of the electric light.
Coming to the question of the total luminosity of the two sectors, we have the area
of the chrome yellow = 464, whilst that of the blue = 93. The sum of the two is
557. As the angular value of the yellow sector is 109, this value has to be reduced
by 169/360 = '47, and is 261 7.

The angular value of the white sector used in the match discs was 158 '5. As the
black reflected '0833 of white light, the total value of the rotated white was
(158-5 + iJOl'5 X '0833, or) 175'4. The area of the curve of luminosity of the white
being 534, the luminosity of the grey was 175 -4/360 X 534 = 260'4, a value very
near to that found as that of the rotating coloured sectors. The fact that we are
only, in this case, dealing with two colours, and that these colours are fairly luminous,
makes the calculated and observed values of the greys in the two discs less liable
to differ than when the colours are more in number and of less luminouty.

560 CAPTAIN ABNEY AND MAJOR-GENERAL FESTING

§ XXXVII. Photograplis of the Rotating Discs.

The next experiment made was to ascertain if the effect of equality of grey
produced on the eye when the coloured and black and white sectors were rotated
would be shown if they were photographed together.

The above figure shows the impression produced on a photographic plate by
rotating green, red, and blue sectors, which were kept of the same angular value
respectively as given in § XXXIII., as were also the white and black sectors. They
were illuminated by the electric light, and photographed on a bromo-iodide plate. It
will be seen that there is a falling off of luminosity in the combination of the coloured
sectors, an effect which might have been predicted from the Curve V. in fig. 16, in
which there is a falling off of intensity in the violet. This shows itself in the photo-
graph, since the plate is but little sensitive below F towards the red.

§ XXXVIII. Matches by Colour-blind People.

This experiment suggested that it would be of great interest to try what results
would be obtained by a colour-blind person using the same three sectors. R., who
had so kindly helped us before (see " Colour Photometry," Part L, § XVI.), again
came to our aid and made observations. He was totally deficient in the perception of
red, and mistook the vermilion disc for dark green when we showed it to him. The
total absence of red perception in him enabled him to match green and blue rotating
sectors against black and white sectors. He obtained a balance when the blue sector
was 115, the green 245, and the white sector when corrected 134'8. Since the

ON COLOUR PHOTOMETRY

561

vermilion appeared to him as dark green, the only effect of introducing the red sector
into the coloured disc was to lower the total luminosity, and to diminish the quantity
of green necessary to produce with the blue a balance against the black and whit*
discs. Thus, with 126° of the vermilion card, 103° blue and 131° green matched
99°'5 white; and, with 188° vermilion, it required 110° blue and 62° green to match
91°-5 white.

In our fonner paper we gave R.'s spectrum curve, which indicated the proportion of
light which he receives from each part of the spectrum, as compared with normal
sight. Fig. 24 shows the curves of luminosity of the light reflected from the coloured
sectors, and the same reduced so as to correspond to R.'s sight are indicated by dotted
lines. The next table gives the numerical results of this reduction.

Scale.

I,.

",

III,.

44-00

•07

•r.

•08

44-50

1-5

45-00

44

3-8

•32

45-20

5-9

45-50

2-40

10-1

•89

45-80

4-10

13-5

1-28

46-00

6-10

17-0

1-78

4635

11-40

21-0

2-62

44-85

22-00

20-0

3-85

47-10

26-00

165

4-38

47-35

30-00

99

4-86

47-50

33-50

47-85

35-00

7'5

5-07

48-90

46-50

5-3

5-60

49-40

47-00

4-U

5-70

49-90

35-50

8-8

6-00

The areas of the reduced curves are — green 172, vermilion GO, and ultramarine 36.
The angular values of the sectors, it will be remembered, were 133'7, 96'6, and 1297.
Taking the areas to represent values of luminosity, R.'s values per 1° of sector are —
emerald green T286, vermilion '621, French ultramarine blue '278. The area of the
normal curve for white, as stated above, is 534, representing 133°'7. The area of
R.'s curve is 343, giving a value of 2'566 for 1°.

Applying these values to the observations, we find a very close correspondence when
the two colours were used, but not quite so close when the red was introduced. The
angle of white in R.'s observations was 135°7, which, multiplied by 2'566, gives 346
as the value of the luminosity. The value of the blue luminosity is 115° X '278 or
32, of the green 245° X T286 or 315 ; and these added together make 347, which is
very close to the value obtained for the white.

In the second observation similar calculations will make the value of the white 255,

MDCCVLXX X VIII. — A.

4 c

562 CAPTAIN ABNEY AND MAJOR-GENERAL TESTING

and of the coloured sector 275. In the third observation they are 235 and 247
respectively. (The dotted lines in fig. 18 show the luminosity curves of the different
colours for R. on the normal scale of wave-lengths.)

§ XXXIX. Comparisons by Gas Light.

Our observations were extended to gas light comparisons. Our first endeavour was
to obtain a comparison between the intensity of the crater of the positive pole of the
electric light and gas. In order to do this we used a collimator, as described in
No. 232 of the ' Proceedings of the Royal Society,' 1884. On one slit the light of
the positive pole was focussed by means of a lens, whose aperture was reduced to
about 1 mm. in diameter ; on the other was focussed the brightest part of the flame
of gas in an ARGAND burner. The spectra from the two sources appeared on the
focussing screen of a camera, one above the other, and just touching. A card with a
slit was passed through the spectra to isolate any part required. The two spectra
were viewed by a RAMSDEN eye-piece, and the intensity of the electric light reduced
by means of the movable rotating sectors we have already described till equality was
established. In the brightest parts of the spectrum the light was too intense to be
readily compared ; so, in order to diminish the brightness, a photographic plate, on
which an even grey tint had been produced by development, was interposed between
the eye and the eye-piece. The light would then be sufficiently reduced in
intensity to allow fairly accurate and concordant measures to be made. It may be
here remarked that a series of three tints was prepared, from a very light grey, which
cut off about l/4th of the light, to one which cut off 19/20ths, and according to the
brightness of the part of the spectra under measurement, so was the darkness of the
interposing glass increased. There is an intensity in each case which gives the greatest
facility of accurate measurement, and this we endeavoured to obtain. (For the sake
of convenience, in the following table the ordinates of the luminosity curve of the
light reflected from white card have been increased in such a proportion that the total
luminosity from the gas light is equal to the total luminosity found for the electric
light. The curves of the colours have been calculated on the assumption that, as
before, the white sector had an angular value of 133'7.)

ON COI.ol I: IMIOTOMETRT.

Gas Light.

Luml,. entity.

I.umincxitv to a colour-blind penoo.

Intermit}-

u com-

Mi

i nd to

electric
light

WUta

card,
133'-7.

Emerald
green,
148°.

Vermilion,

iir.

French
ultra-
mariue,
»8°.

Emerald
green,
148°.

Vermilion,
11»'".

Fivnch
ultra-
marine,
88°.

White
card,

m 7.

43-0

lOO'O

4-CO

•25

2-25

•16

0

0

0

0

44-0

82-0

2-48

23-30

1-88

•15

142

•11

1-80

44-5

740

62-70

6-60

52-80

3-13

•39

3'70

•22

435

45-0

67-0

99-00

8-80

84-30

4-17

•75

7-15

•.;;,

- 10

455

61-0

117-00

20-60

97-10

4-99

3-68

17-20

•89

10-80

46-0

5(5-0

118-50

26-80

83-10

5-04

B-80

25-80

1-56

41-80

465

50-5

113-00

.1 ;•>

59-00

4-72

17-40

29-90

2-40

57-50

47-0

45-5

104-00

40-20

88-90

4-52

28-10

2320

3-20

73-00

47-5

42-0

96-00

45-50

12-9o

4-52

35-60

10-10

3-50

7500

48-0

38-2

88-< I

48-30

8-60

4-25

39-50

7-20

360

69-50

48-5

35-0

70-20

45-00

6-70

3-70

38-40

5-70

3-20

60-00

49-0

32-5

42-70

5-20

3-30

:•- 60

4-70

3-00

53-50

49-5

30-0

45-70

35-60

3-65

3-00

33-90

3-50

2-85

43-50

50-0

27-30

•_•:;•:,"

2-05

2-75

2350

205

2-75

27-30

510

-:;:,

10-50

8-35

•67

1-54

835

•67

1-54

10-50

52-0

20-2

4-30

3-20

•28

•88

3-20

•23

•88

4-:$u

53-0

17-5

2-20

1-34

•11

•52

1-34

•11

•62

2-20

54-0

15-0

121

•53

•06

•33

•53

•00

•83

1-21

55-0

12-7

•75

•20

•16

•20

•16

•75

56-0

107

•37

Areas

534-00

199-00

238-00

27-60

143-40

71-50

16-00

282-00

\

The second column of the above table gives the comparative intensity of the gas
light we used, taking the intensity of all the rays in the electric light as 100 (see
Plate 22, fig. 25). The rapid loss of intensity towards the blue evidently would
much modify the quantities of red, blue, and green necessary to match the black and
white sectors when compared in light of this description. The other columns of the
table give the ordinates of the calculated luminosity curves of fig. 26.

The three sectors were matched as before by us in this light, and it was found that
148° green, 119° red, and 93° blue were required to balance 116° of white.

The area of the curve for the white representing a sector of 133°'7, as before, is 534
that for 116° would therefore be 116/1337 of 534, or 464.

The areas of the curves of the three coloured sectors on the above proportion are
as follows: — Emerald green 199, vermilion 238, French ultramarine 27 '6, making a
total of 464-6.

We also tested R. by gas light with the same three coloured discs. The red being
the same as the green to him, we varied the red at pleasure, and got the following
results :—

4 c 2

564 CAPTAIN ABNET AND MAJOR-GENERAL TESTING

With green 62, blue 110, red 180, he matched 92 white.

His areas for the above would have been 60, 19, 112 respectively, making a total of
191. The area for the white would be 194.

Again, with green 138, blue 94, red 128, he required white 10G. The areas would
have been as follows : 133'.}, 16, and 77 respectively, or a total of luminosity 226'5.
The luminosity of the white used was 223.

Again, with blue and green alone, he required 288 green and 72 blue to match 141
white. The areas of the above were 281 and 12 '4 respectively, or a total luminosity
to him of 29 3 '4. The luminosity of the white used was 29 7 '4.

§ XL. Comparison of Gas Light, Sky Light, and the Electric Light.

Fig. 24 (Plate 2) shows the proportions of the different rays in sky light, gas light,
and the electric (crater) light, the last being taken as the standard of comparison.
The sky measured, it may be stated, was a fairly blue sky and not very pale. The
light diffused through a cloud on a cloudy day in April we have found to be almost
exactly similar to the light of the electric arc, and, in fact, is degraded sun light.

We have shown how the three coloured sectors vary in proportion to form a grey
when examined by electric light and by gas light. The variation would be even greater
when sky light of the blue shown was employed. On a cloudy day, however, the
proportions of each colour would be approximately the same as found for the electric
light. It is evident that for quantitative measures for testing light no reliance
can be placed on the results, unless light of an uniform character be always employed.
Day light, being composed of variable amounts of sun light and sky light, should, in all
cases, be avoided.

§ XLI. Reflection of Light from Metals.

As a matter of curiosity, we wished also to determine the intensities of the different
rays reflected from some of the metals which were coloured. We took ordinary
polished copper, such as is supplied in commerce for etching purposes ; a piece of the
same copper highly burnished ; a piece of highly burnished brass ; and a piece of highly
burnished gold. Fig. 27 (Plate 2 1) and the following tables give the result. In the
figure the continuous lines show the luminosity curves, and the dotted lines the
intensity curves.

ON COLOUR PHOTOMETRY.
Highly Polished Copper.

560

Original routing*.

Reduced from plotted curre.

l.illiiin...iM.

Scale number.

Reading.

Scale number.

Height

II right »hcn
while - 100.

Height of

white curre

Height of
copper curre.

42-80

51-0

UK)

51-0

73-9

2-0

i r,

43-30

51-0

43-5

51-0

739

5-0

364

44-30

51-0

44-0

51-0

rs-9

16-0

11-70

44-80

51-0

44-5

51-0

739

370

25-80

45-85

50-0

! . • '

51-0

65-0

48-00

46-85

47-0

455

50-5

7:: J

84-0

63-30

47-85

41-0

16-0

497

7:21

935

f.720

4890

37-0

46-5

48-0

69'rt

08-8

68-50

49-90

35-0

47-0

46-0

86-7

100-0

66-60

50-95

34-0

47-5

;;•>

99-0

63-10

51-95

33-5

18-0

41-0

59-5

05 •

.-.••.40

53-00

33-0

48-5

56-8

88-0

50-00

54-50

320

490

365

52-9

80-0

IJ30

57-05

30-0

49-5

35-5

514

67-0

3450

58-05

30-0

50-0

35-0

50-7

43-0

21-80

51-0

34-0

49-3

19-5

960

52-0

335

48-6

95

4-70

53-0

330

48-0

5-5

2-60

54-0

32-5

47-3

3-5

1-68

56-0

31-0

1 1 '.'

1-5

•66

58-0

30-0

43-5

•6

•26

Burnished Copper.

Original reading*.

Reduced from plotted curve.

Luminosity.

Scale number.

Reading.

Scale number. Height. ^.^

Height of
white curre.

Height of
copper cunre.

4220

80-0

43-0 75-0

94-0

2-0

1-9

4325

72-0

43-5 72-0

5-0

44-00

60-0

44-0 62-0

77-5

16-0

124

44-80

53-0

44-o

56-0

70-0

37-0

25-9

45-30

47-0

45-0

50-0

625

65-0

40-6

45-80

43-0

45-5

45-0

56-2

-l"

47-7

46-60

37-5

46-0

41-0

51-2

93-5

47-3

47-50

29-5

46-5

37-0

462

98-5

45-5

48-50

25-0

47-0

32-5

40-5

100-0

40-6

48-70

24-0

47-5

29-5

37-0

99-0

86-S

49-40

220

48-0

27-0

337

95-0

32-1

52-00

17-0

48-5

25-0

31-0

88-0

26-9

54-60

14-5

49-0

23-5

29-5

80-0

23-5

56-20

14-0

49-5

21-5

27-2

67-0

1--

58-70

14-0

50-0

21-0

25-5

43-0

11-3

51-0

19-0

23-0

195

4-6

:.•_>..

17-0

21-2

9-5

2-0

54-0

15-0

18-7

3-6

•7

56-0

14-0

17-5

1-5

3

58-0

14-0

.«

•6

•1

566

CAPTAIN ABNKY AND MAJOR-GENERAL TESTING
Highly Polished Brass.

Original readings.

Reduced from plotted curve.

Luminosity.

Scale number.

Reading.

Scale number.

Height.

Height when
white - 100.

Height of
white curve.

Height of
bnu» curve.

43-30

57-0

43-0

57-0

82-60

2-0

1-60

43-80

57-0

43-5

57-0

82-60

5-0

3-80

44-30

57-0

44-0

57-0

82-60

16-0

13-00

44-80

56-0

44-5

57-0

82-60

37-0

30-10

45-60

54-5

45-0

56-0

81-20

65-0

52-80

46-85

55-5

45-5

56-0

78-30

84-0

65-70

47-85

53-5

46-0

55-0

79-75

93-5

74-50

48-85

52-5

46-5

55-5

80-47

98-5

79-20

49-90

49-0

47-0

55-0

79-75

100-0

79-70

50-95

46-0

48-0

54-0

74-30

95-0

74-30

51-95

420

48-5

53-0

76-90

88-0

67-60

52-95

39-0

49-0

52-0

75-45

80-0

60-30

54-00

370

49-5

51-0

74-00

67-0

49-50

55-50

34-0

50-0

49-0

71-10

43-0

30-50

57-00

30-5

51-0

45-0

65-20

19-5

12-70

5800

28-5

52-0

41-5

60-20

9-5

5-70

53-0

39-0

56-50

5-5

3-10

54-0

37-0

53-60

3-5

1-90

56-0

32-0

46-40

1-5

•68

58-0

28-0

40-60

•6

•24

i

Gold.

Original readings.

Luminosity curves.

Scale number.

Reading.

Height of
white curve.

Height of
gold curve.

435

29-0

5-0

1-45

44-0

35-0

16-0

7-60

44-5

40-0

37-0

14-80

45-0

44-0

65-0

28-60

45-5

46-5

80-0

37-20

46-0

48-5

93-5

42-80

46-5

50-0

98-5

49-20

47-0

51-0

100-0

51-00

47-5

51-0

99-0

50-50

48-0

51-0

95-0

48-50

48-5

50-5

88-0

44-40

49-0

49-5

80-0

39-60

49-5

48-0

67-0

32-20

50-0

46-0

43-0

19-90

51-0

41-0

19-5

8-00

52-0

35-0

9-5

4-30

53-0

31-0

5-5

1-70

54-0

27-0

3-5

•61

55-0

24-0

2-1

•49

56-0

21-5

1-5

•32

58-0

16-0

•6

•09

ON COLOUR PHOTOMKTRY. 567

The intensities were obtained by substituting for the first small glass prism in front
of the slit apiece of the metal whose reflection was to be examined. The two images
were received on a white screen and equality established, as before described. A very
interesting confirmation of the accuracy of measuring the illuminating value of
different light was found in the case of burnished copper. In 1886 the " luminosity
curve " of the spectrum of the light reflected from copper was obtained, the piece of
copper used being part of the same plate from which the intensity curve was obtained
in our recent experiments. The dots close to the curve of luminosity derived from
the intensity curves show the agreement of the results obtained by two very different
methods. In the same year the luminosity curve of emerald green was made by our
former plan, and on reducing the curve in the proper proportion, so that the maximum
coincided with the maximum of the luminosity curve obtained from the comparison of
the intensity of light reflected from the emerald green with that from a white surface,
it was found that the agreement was extremely close. These coincidences confirm the
y accuracy and value of our method of measuring the luminosity of light of different
colours.

§ XLII. Comparison of Reflection from, unth Transmission through, a Pigment. <

We wished also to ascertain whether the light reflected from a pigment was
identical with that transmitted through the same. In some cases, when there is
quasi-metallic reflection, in parts this cannot be the case ; but in ordinary colours this
reflection is not present. We tried several, and came to the conclusion that the
transmitted and reflected lights are of the same character. Fig. 28 gives the result of
one such experiment made with Prussian blue. The transmitted light was measured
by placing in the lower spectrum a thin film of gelatine impregnated with the pigment.
The curves of the figure were constructed from the following table.

568

CAPTAIN ABNEY AND MAJOB-QBNEBA.L TESTING

Transmitted light.

Reflected light

Scale number.

Reading.

Reduced to
white = 100.

Reading
white = 100.

Calculated
intensity k'^-s
white.

43

4-0

4-6

23

1-8

44

5-0

5-7

24

3-6

45

6-0

6-9

25

5-4

46

10-0

11-5

:>

10-9

47

16-5

19-0

33

19-0

48

25-5

29-3

38

29-9

49

34-0

39-1

44

398

50

42-5

48-9

50

50-7

51

51-0

58-6

56

61-5

52

60-0

69-0

62

724

63

66-5

76-5

65

77-8

54

68-0

78-2

66

79-6

55

69-0

79-3

66

79-6

56

69-0

793

65

77-8

57

70-0

80-5

64

76-0

58

70-0

80-5

64

76-0

A certain amount of white light being reflected from the coloured surface, a cor-
rection is necessary for this, and the last column was derived from the preceding one
by deducting 22 (the amount of white light present) and multiplying by T81.

§ XLIII. Intensity Curves of Coloured Pigments.

Figs. 29 (Plate 22) and 30, 31, 32 (Plate 23) give the curves of some of the many
colours which have been measured by the method described in this paper. These
curves are plotted to the normal scale of wave-lengths, and the following tables
give the readings at the different wave-lengths.

ON COLO IK riluTO.MKTKY.

569

Vennlllon.

Carmine.

Venetian red

Men-uric iodide.

Indian red.

Scarlet lake.

4350= 6-75

4300 = 37-00

4200 = 22-5

4300= 3-0

4200 = 25-7

4250 = 36-5

4800= 675

4500 = 38-50

4300 = 25-0

4500= 2-0

4300 = 27-0

4400 = 34-0

5000= 7-50

4550 = 39-00

4400 = 27-7

4600= 4-0

4500 = 29 5

4500= :

5300= 9-50

4700 = 36-50

45(0 = 29-5 4750= 5'8

4750 =* 30-5

4600 = 317

5500 = 11-50

4800 = 34-50

4550 = 30-0 4900= 4'2

4900 = 30-0

4700 = 30-5

5600 = 15-00

5000 = 32-00

4750 = 30-0 4950 = 3'5

5000 = 29-3

4800 = 28-2

5750 = 31-50

5100 = 31-00

5000 = 30-5 5000= 37

5150 = 29-0

5000 = 25-5

5800 = 40-00

5200 = 31-50

5200 = 31-5 i 5200= 5'5

5300 = 30-0

5150 = 25-0

5900 = 59-00

5300 = 33-00

5300 = 32-5

5400= 6-8

5400 = 31-5

5200 = -J , -J

6000 = 78-00

5380 = 34-50

5400 = 34-5

5500= 8-0

5500 = 33-5

5300 = 26-0

6200 = 97-00

5450 = 33-50

5500 = 39-5

5700 = 18-0

5600 = 36-3

5500 = 29-5

6500 = 94-50

5550 = 33-25

5600 = 47-0

5750 = 22-0

5750 = 42-5

5600 = 34 0

6600 = 90-50

5600 = 34-00

5700 = 58-0

5800 = 29-5

5800 = 48-5

5700 = 41-0

6750 = 84-00

5700 = 41-00

5750 = 61-0

5900 = 54-5

5900 = 52-0

5800 = 50-5

7000 = 72-50

5800 = 50-00

5800 = 65-0

6000 = 66-5

6000 = 58-5

5900 = 62-0

5900 = 59-00

5900 = 72-0

6100 = 70-0

6100 = 63-2

6000 = 76-0

6000 = 67-00

6000 = 78-5

6200=71-7

6200 = 66-5

6100 = 82-0

6100 = 70-50

6100 = 83-5

6300 = 73-5

6250 = 67-5

6200 = 82-0

6200 = 72-50

6200 = 87-5

6400 = 74-5

6400 = 70-6

6300 = 81-5

6300 = 74-00

6300 = 90-5

6500 = 75-4

6500 = 72-5

6400 = 80-6

6400 = 75-00

6400 = 92-2

6600 = 75-8

6990 = 76-0

6500 = 80-0

6500 = 76-00

6500 = 93-3

6750 = 76-3

6800 = 77-3

6600 = 75-00

6620 = 94-0

7000 = 76-3

6800 = 72-00

\

Emerald green.

Chronic grom.

4200 = 25-00

4250 = 16-0

4300 = 25-75

4400 = 17-6

4500 = 29-00

4600 = 20-2

4600 = 34-50

4800 = 22-6

4700 = 49-00

4900 = 26-0

4800 = 60-00

5000 = 30-5

4900 = 68-00

5100 = 36-0

5000 = 72-50

5200 = 42-0

5125 = 75-00

5300 = 44-6

5200 = 73-50

5310 = 45-0

5300 = 69-50

5400 = 43-5

5400 = 63 00

5500 = 40-5

5500 = 55-50

5600 = 35-0

5600 = 47-50

5700 = 22-0

5700 = 39-70

5750 = 20-5

5800 = 32-00

5850 = 19-8

5900 = 26-00

6000 = 20-5

6000 = 20-50

6200 = 24-0

6100 = 16-50

6300 = 26-0

6200 = 13-40

6450 = 32-0

6300 = 10-00

6400 = 9-00

6500 = 8-00

6600 = 7-60

6750 = 7-00

7000 = 6-50

MIKCCLXXXVIH. — A.

4 I)

570

C.M'T. ABNEY AND MAJ.-GEN. TESTING ON COLOUR PHOTOMETRY.

French ultramarine.

Prussian blue.

French blue (pale).

Cobalt blue.

4350 = 38-00

4200 = 60-r,o

4250 = 40-00

4350 = 44-0

4600 = 38-00

4500 = 67-00

4400 = 47-00

4500 = 51-7

4750 = 35-00

4600 = 68-00

4500 = 54-00

4600 = 53-3

4900 = 27-00

4800 = 64-00

4600 = 61-00

4700 = 54-0

5000 = 21-50

5000 = 59-00

4625 = 61-25

4800 = 53-6

5200 = 12-50

5250 = 49-50

4700 = 60-50

4900 = 50-5

5500 = 8-00

5500 = 41-75

4750 = 58-50

5000 = 41-5

5750 = 7-00

5800 = 34-50

4900 = 48-00

5100 = 29-0

6300 = 6-85

6000 = 30-00

5000 = 43-70

5200 = 20-0

6500 = 7-50

6300 = 25-50

5100 = 40-20

5300 = 15-0

6620 = 8-50

6500 = 23-50

5200 = 36-50

5400 = 11-5

7000 = 6-50

6800 = 22-00

5400 = 31-00

5500 = 8-5

7000 = 21-00

5750 = 24-30

5600 = 6-5

5850 = 24-00

5750 = 6-0

6000 = 25-20

6000 = 7-5

6200 = 28-20

6100 = 10-5

6300 = 30-20

6200 = 14-0

6400 = 32-70

6300 = 18-5

6500 = 34-50

6400 = 23-5

6600 = 36-50

6500 = 28-5

6700 = 37-50

6600 = 34-5

6700 = 40-0

Chrome yellow.

Aureolin.

Cadmium yellow.

Yellow ochre.

Gamboge.

4350 = 30-0

4250 = 15-0

4200 = 21-5

4375 = 15-50

4275 = 15-0

4500 = 30-7

4350 = 13-0

4400 = 22-5

4500 = 21-55

4300 = 14-5

4600 = 32-5

4400 = 14-0

4500 = 23-5

4550 = 21-60

4375 = 13-7

4700 = 37-5

4500 = 15-7

4700 = 28-0

4600 = 21-00

4500 = 16-0

4750 = 44-5

4600 = 16-2

4800 = 32-0

4750 = 21-70

4550 = 17-5

4800 = 51-5

4700 = 19-0

4900 = 37-0

4800 = 23-00

4600 = 18-0

4900 = 61-5

4800 = 25-0

5000 = 42-0

4900 = 25-50

4650 = 18-5

5000 = 68-5

4900 = 33-0

5100 = 49-5

5000 = 29-30

4750 = 21-5

5100 = 76-5

5000 = 42-0

5200 = 57-5

5100 = 34-00

4900 = 28-5

5200 = 80-5

5100 = 53-0

5300 = 65-0

5200 = 40-50

5000 = 36-5

5300 = 86-5

5200 = 65-0

5400 = 73-0

5300 = 49-00

5200 = 53-5

5400 = 91-5

5300 = 75-5

5500 = 80-5

5400 = 59-00

5250 = 57-5

5500 = 95-0

5400 = 83-0

5600 = 88-5

5500 = 67-50

5300 = 60-5

5550 = 96-0

5500 = 89-0

5700 = 91-5

5600 = 75-50

5400 = 67-5

5600 = 94-5

5600 = 92-7

5800 = 93-5

5750 = 79-50

5500 = 73-0

5750 = 89-5

5700 = 93-5

5900 = 94-6

5800 = 80-00

5700 = 75-5

5870 = 87-0

5800 = 94-0

6000 = 95-0

5875 = 79-00

5900 = 78-5

6000 = 89-5

6000 = 93-5

6100 = 94-0

5900 = 77-00

6000 = 80-0

6100 = 91-0

6150 = 93-0

6200 = 92-0

6000 = 77-00

6200 = 81-7

6200 = 92-0

6300 = 93-5

6300 = 89-5

6250 = 77-50

6350 -= 82-0

6300 = 92-0

6400 = 94-0

6400 = 86-7

6300 = 77-50

6400 = 91-5

6500 = 94-6

6500 = 84-0

6400 = 77-50

6500 = 91-0

6620 = 96-0

6750 = 77-0

6500 = 77-50

6700 = 89-5

7000 = 87-7

[ 571 ]

XVIII. Combiustion in Dined Oxyycn.

By H. BREKETON BAKKK, M.A., Didwich College, late Scholar of Bnlliol College.

Oxford.

Communicated by Professor H. B. DIXON, F.R.S.
Received July 4,— Read November 15, 1888.

THE chemical changes occurring in combustion hold such an important position, not
only in the history and advance of chemical science, but also in the applications of
science to industry, that special interest is attached to the discovery of the nature
and order of the chemical changes involved. The phenomena presented by the
oxidation of carbon, sulphur, and phosphorus have been studied by chemists as
typical of those which are found in other processes of burning. With the object of
determining the conditions necessary for the oxidation of these three substances, I
began in 1884 an investigation, the results of which are described in the following
paper. That water vapour might play an important part in such actions seemed very
probable. The interesting facts brought to light by my former tutor, Professor H. B.
DIXON, with regard to the oxidation of carbon monoxide (' Phil. Trans.,' 1884) made
it seem likely that water vapour might exert as strong an influence on other combus-
tions as he has shown it does on that of carbon monoxide. It was suspected some
years ago that the combustion of carbon is affected by the presence or absence of
moisture. In 1871 M. DUBKUNFAUT read a paper before the Academic des Seirnces
describing some experiments bearing on this point. He had performed combustions of
sugar-charcoal in oxygen dried by strong sulphuric acid, and found that the carbon did
not undergo combustion as readily as it did if the oxygen was moist. He ascribed
the incompleteness of the oxidation of the carbon to the presence of moisture which
was inaccessible to our reagents, that is, moisture which did not cause a weighed
sulphuric acid tube through which it was passed, to increase in weight. A few weeks
later M. DUMAS, who had determined the equivalent of carbon by burning a weighed
quantity of pure graphite in pure oxygen, repeated the experiments, and, by using a
large quantity of the partially dried oxygen, succeeded in burning the whole of a
small quantity of graphite.

In my first experiments on the combustion of carbon in oxygen,* wood charcoal was
employed. It was freed from hydrogen by heating in a current of chlorine for several
hours. It was placed in glass tubes, into which phosphorus pentoxide had been

• ' Chem. Soo. Journ.', 1885, Traru. p. 349.

4 D ~t 12.1

572

MR. 11. HHKHKTO.N I'-AKKU OX I'OilliUSTlON IN DRIED OXYGEN.

previously introduced, the two substances being separated by a disc of platinum foil.
The latter did not fit so closely as to prevent the free diffusion of the gaseous contents
of the tubes. Oxygen dried by sulphuric acid was then passed through the tubes,
which were then sealed at both ends. Experiments have shown that the long contact
of phosphorus pentoxide with gases has an immeasurably greater drying effect than
the mere passage of the gases through tubes containing that substance. After
standing for some days each of these tubes was heated, side by side with a similar
tubs containing charcoal in moist oxygen, by the flame of a large Bunsen burner.
The moist carbon always burnt with the scintillation characteristic of such a combus-
tion, but the dry carbon remained apparently unaltered. On analysis, however, it
was found that a certain quantity of carbon had been burnt. The following Table
gives the results of the analysis of the gaseous contents of the tubes, after the carbon
had been heated to redness in them for about two minutes.

I.

II.

III.

IV.

V.

VI.

Dried

Dried

Dried

Dried

Dried

Dried

Wet.

for

Wet.

for

Wet.

for

Wet.

for

Wet.

for

Wet.

for

1 week.

2 weeks.

4 weeks.

8 weeks.

12 weeks.

16 weeks.

Carbon dioxide .

60-1

15-4

51-0

19'0

45-3

14-1

23-3

12-5

68-8

15-8

52-4

178

Carbon monoxide

22-2

20-6

31-2

14-8

32-6

27-8

60-0

27-5

23-2

242

25-2

16-5

Oxygen ....

41-3

t t

46-1

t f

28-2

t t

39'0

33-3

. .

45-0

Nitrogen . . .

27-6

22 -8

17-7

200

221

29-8

16-6

21-0

17-8

26-8

22-3

20-t>

It will be noticed that in four out of these six experiments more than half the
oxygen was left uncombined with the carbon, whilst none remained in the wet tubes.
The comparatively large quantities of carbon monoxide produced are striking when we
consider that the temperature was not high, and that the reduction of carbon dioxide
by carbon is never rapid. These results formed the starting point of an investigation
in which the precautions taken to ensure the purity and dryness of the substances
used were made much more elaborate.

The carbon used in the later experiments was in the form of charcoal. It was
prepared by heating sugar in a silver dish to a bright red heat. The porous mass was
broken up and finely powdered. It was transferred to a combustion tube, and heated
to redness for three days in a current of pure chlorine dried by sulphuric acid, in
order to eliminate hydrogen from the powder. The charcoal was washed by decanta-
tion with distilled water until the washings gave no opalescence with silver nitrate.
To free it from occluded hydrochloric acid, it was placed in a hard glass tube sealed at
one end, while at the other end was placed a stick of solid potash. The tube was
then exhausted by a Sprengel pump, and the end containing the carbon was heated
to dull redness for several days, the vacuum being constantly maintained. Hydro-
chloric acid gas was evolved, partly absorbed by the potash, but at first passing over
in such quantities that it could be collected from the air-pump. After this treatment

MR. H. BRKKETON BAKER ON COMBUSTION IN DRIED OXYGEN. 573

the carbon was considered sufficiently pure for the experiments. It was kept in a
small stoppered bottle, but, as this involved exposure to a small quantity of air, it was
thought advisable before the carbon was used for the preparation of each experiment
that the portion taken should be heated to redness in a vacuum. This served not
only to dry it, but also to free it from occluded oxides of carbon which are products of
its slow combustion in air at ordinary temperatures.

The oxygen was prepared by heating pure potassium chlorate, and was stored in
gas pipettes over previously boiled mercury. In the gas were placed plugs of phos-
phorus pentoxide. These plugs were made thus : — A piece of glass tubing, 1 cm. in
diameter, and a longer piece of glass rod, exactly fitting it, were heated to dry them
thoroughly. The tube while still warm was pushed to the bottom of the bottle
containing the pentoxide, and the portion of the latter so introduced was compressed
by forcing the glass rod into the tube like a piston. Both tube and rod were then
removed from the bottle, with the plug almost entirely protected from the air. The
tube was then quickly plunged under the mercury, through the tubulure of the
pipette, and the piston pressed down. The plug was thus pushed out of the tube,
and rose through the mercury into the oxygen which was to be dried. This plan of
dealing with the phosphorus pentoxide, in the form of a compact cylinder, was
extremely useful when it was necessary to introduce it into glass tubes which were to
be drawn out in the blow-pipe flame. If only a small quantity of the oxide adheres
to the portion of the glass to be heated, it diminishes the ductility to a remarkable
degree, and renders it extremely brittle.

Carbon heated in contact with Platinum.

A mixture was made of platinum black (previously heated to redness in a vacuum)
and the pure charcoal. A hard glass tube was slightly contracted in the middle, and
heated in a current of dried air to dull redness, to get rid of the moisture which
clings so tenaciously to glass. On one side of the constriction was introduced the
mixture of platinum and carbon, and on the other plugs of phosphorus pentoxide.
Oxygen was passed through the tube and the ends sealed. The oxygen was left
drying for six weeks. At the end of this time the part of the tube containing the
mixture of carbon and platinum was heated to redness for three minutes with no sign
of visible combustion. Analysis showed that the tube now contained—

Carbon dioxide ... 27 per cent.
Oxygen 73 „

A second experiment, in which the oxygen was dried for three weeks, and the
mixture of carbon and platinum was heated for four minutes, gave —

Carbon dioxide ... 34 per cent.
Oxygen 66 „

574 MR. II. HUKKKTON BAKER ON COMBUSTION IN DI'.IKI) OXYtiKX.

The quantities of carbon burnt in these two experiments do not differ greatly from
the quantity of carbon burnt under similar conditions when no platinum was present.
There is, however, this important difference between the two cases, that in the
presence of platinum only carbon dioxide is produced.* In the absence of platinum
varying quantities of carbon monoxide were formed, generally exceeding in amount
the carbon dioxide produced at the same time.

A third experiment was made with carbon in contact with platinum. The
charcoal was in the form of a rod, about 4 cm. in length. It was enclosed in a coil of
fine platinum wire, which had been previously freed from hydrogen by heating to
redness in oxygen. The ends of the coil were attached to thicker platinum wires,
which were sealed into two pieces of quill tubing. These passed through two holes in
an india-rubber stopper which fitted into the neck of a small flask (fig. 1). The flask

*. 1-

contained pure oxygen, dried by plugs of phosphorus pentoxide placed in the flask.
A small quantity of mercury covered the stopper, to prevent diffusion of water vapour
through its mass. The gas was left drying for fourteen days.

The platinum coil was heated by an electric current to a bright red heat. The
carbon did not catch fire, and the glow disappeared directly the current was stopped.
With a stronger current, the platinum was heated nearly to whiteness ; the carbon
showed visible combustion for two seconds, after which the glow died out. On
analysing the contents of the globe after the experiment, more than half the gas was
found to be oxygen.

A similar experiment was tried with a rod of carbon in moist oxygen. On raising
the temperature to dull redness by means of the heated wire, the carbon caught fire,
and, although the current was immediately broken, there was no cessation of the
combustion until the rod was entirely .consumed.

These results point to the conclusion that the contact of platinum does not cause
the union of dry carbon with dry oxygen, but only influences the products of
combustion.

* It has been proved that dry carbon monoxide and oxygen unite readily without flame in presence of
red-hot platinum (DixoN and LOWE, " Chem. Soc. Joarn.', 1885, Trans, p. o75).

MR, H. BRERETON BAKER ON COMBUSTION IN DRIED OXYGEN. 575

Tlie Products of Combustion of Charcoal «,,.

In the course of the experiments on the combustion of carbon in dried oxygen, :i
gas-bolder was constructed to contain oxygen dried by phosphorus pentoxide. The
only liquid which could be used for driving out the gas was mercury, and, as the amount
of gas required was about eight litres, its use would have been inconvenient. The
following apparatus was fitted up.

Fig. 2.

A large bottle was graduated and fitted with an india-rubber stopper, through
which a large tap-funnel was passed. Through another hole in the stopper a narrow
upright tube was fixed. This passed to the top of an inverted two-litre flask, also
provided with an india-rubber stopper. A bent tube also passing through this stopper
was connected with a second flask arranged in the same way as the first. A third
flask, similarly fitted, completed the gas-holder. It resembled three inverted wash-
bottles, with the exit tube of one connected with the pressure tube of the next. The
whole, including the large bottle, was heated for several hours by Bunsen burners
playing on the sides, while a current of air, dried by sulphuric acid, was drawn
through the apparatus.

In order that no moisture should be given off from the india-rubber stoppers in the
three flasks, each was covered with a little mercury previously boiled. On the surface
of the mercury in each flask five or six plugs of phosphorus pentoxide were placed.
A current of oxygen was passed through six long, nearly horizontal, tubes filled with
concentrated sulphuric acid, and then through the apparatus described above. After
passing for an hour a sample of the gas was collected as it issued from the last flask.
It was found to be pure oxygen. A hard glass tube containing dried charcoal was
drawn out at both ends. One end, bent upwards, was passed through the opening in

576 MR. H. BRERETON BAKER ON COMBUSTION IN UR[ED OXYGEN.

the stopper of the third flask ; the other end was sealed. A small quantity of boiled
sulphuric acid was poured into the bottle through the stoppered funnel, so that the
pressure of the gas during its drying might be greater than the atmospheric pressure.
The gas was thus dried, partly by phosphorus pentoxide, and partly by sulphuric acid.
Since the connecting tubes were narrow, only a slow diffusion of slightly moist gas from
the bottle containing strong sulphuric acid could take place into the flasks. The gas
was thus thoroughly dried in the second and third flasks. The oxygen used in the
experiment was taken from this third flask, though the pressure used to drive it out
was derived from the sulphuric acid poured into the large bottle. The appai-atus was
left to dry for four days.

Experiment I. — The sealed end of the tube having been broken off under dried
mercury, the carbon was heated to dull redness in a stream of the dried gas. At
first it glowed a little. The current was stopped. A blue flame was seen to run
along the tube from the heated point to the end. This could only be due to carbon
monoxide.

Experiment II. — As it was evident that either the charcoal or the glass was not
quite dry, the tube, still connected with the oxygen flask, was sealed at its open end,
and allowed to dry for another week, gentle heat being occasionally applied. At the
end of this time the sealed end of the tube was broken under mercury, previously
dried. The part containing the charcoal was heated as before with a Bunsen burner.
This time no glowing was seen, although 200 c.c. of oxygen was rapidly passed over
the red-hot charcoal. After this a sample of 27'5 c.c. of the gas was collected. It
was analysed and found to contain —

c.c. Per cent.

Carbon dioxide 1 '4 5'0

Carbon monoxide ll'O 40 '0

Oxygen 15'1 54'9

27-5 99-9

Experiment III. — The apparatus was allowed to stand for another week, and then
the charcoal was raised to bright redness with a gas blow-pipe. No visible combustion
occurred ; 29 '6 c.c. of gas was collected. The analysis gave—

c.c. Per cent.

Carbon dioxide '6 2'2

Carbon monoxide 11 '8 39 "5

Oxygen 17'2 58'1

29-6 99-8

These analyses are very striking, showing that a large quantity of carbon monoxide
is produced when carbon is heated in a current of oxygen, even when the oxygen is
in excess. They seem to point to the conclusion that carbon burns first to carbon

MR. H. BRERETON BAKER ON COMBUSTION IN DRIED OXY<

577

monoxide, and that this, unless prevented by drynees, as in this case, or by some
other influence, then produces carbon dioxide.

This conclusion is strengthened by the fact that when a piece of charcoal is heated
in a rapid stream of ordinary oxygen a long flame is seen to issue from the glowing
solid in the direction of the current of gas. This is probably the second stage in the
burning of the charcoal.

It has been shown (C. J. BAKER, ' Chem. Soc. Journ.,' 1887) that a small quantity
of carbon monoxide is produced by the slow absorption of oxygen by charcoal. This
was given out by heating to 45° in vacuo. The amount was not more than 5 c.c.,
given out by heating half a gram of charcoal for two hours. Though such a small
amount was not likely to vitiate an experiment made at more than atmospheric
pressure, and lasting not more than two minutes, it was thought advisable to use
precautions to obviate any possibility of error from this source. The following method
.was then tried.

The oxygen was contained in a gas pipette with a stop-cock. Phosphorus pentoxide
was introduced, and the gas was left drying for a week. The aperture of the pipette
was connected by a piece of dried and paraffined india-rubber tubing with the tube
containing the carbon. In this india-rubber tube phosphorus pentoxide was placed.
The carbon had been heated to redness in vacuo to free it from any carbon monoxide
which might have been occluded. The tube containing it was drawn out in the
middle, and phosphorus pentoxide placed in the part remote from the pipette. To
prevent the formation of carbon monoxide in the charcoal while the oxygen was
drying, the tube was exhausted as completely as possible and the free end drawn out
to a point and sealed.

Fig. 3.

The oxygen was to be driven out by mercury which had been boiled and cooled in
an atmosphere dried by sulphuric acid. After a week's drying the gas was admitted
to the vacuous tube. The sealed tip was broken and connected with a set of GEISSLER'S
potash bulbs. These would absorb the carbon dioxide produced. Next in series was
another set of weighed potash bulbs which would indicate whether all the carbon
dioxide had been absorbed by the first set. Connected with these bulbs was a hard
glass tube containing copper, oxide heated to redness in a small gas furnace. In this

MDCCCLXXXVI1I. — A. 4 E

578

MR. H. BRERETON BAKER ON COMBUSTION IN DRIED OXYGEN.

tube any carbon monoxide which was produced in the combustion would be converted
into carbon dioxide. Next in the series was a third set of potash bulbs to absorb this
carbon dioxide. The calcium chloride tube of this third set of bulbs had a conducting
tube attached to it which dipped under mercury contained in a trough. Here the
residual gas was collected.

The carbon was raised to a red heat by a Bunsen burner. No visible combustion
took place. Oxygen was passed over it at the rate of about 20 bubbles a minute.
The gas collected over the mercury was at first a mixture of oxygen and nitrogen.
The latter was derived from the air contained, at the beginning of the experiment, in
the bulbs and the copper oxide tube. The gas collected towards the end of the
experiment was pure oxygen.

When all the oxygen had been driven over the charcoal, the potash bulbs were
disconnected, filled with air, and weighed. The second set, which were used to test
the efficacy of the first set, had not increased in weight. From the increase in weight
of the first set, the weight of carbon dioxide produced in the combustion was found.
The increase in weight of the third set gave the weight of carbon dioxide produced
by the oxidation of the carbon monoxide formed in the combustion. The analysis of
the gases so found yields the following results : —

Carbon dioxide .... 1*7
Carbon monoxide . . . 27'8
Oxygen (by difference) . 70 '5

lOO'O

These striking results are confirmed by the analyses of the gases contained in sealed
tubes after charcoal had been heated in dried oxygen, mixed with nitrogen.

Dried for 1 week.

Dried for 4 weeks.

Dried for 8 weeks.

Carbon dioxide . . .
Carbon monoxide . .
Oxveen .

15-4
20-6
41-3

141

27-8
28-2

12-5
27-5
39-0

Nitrogen

•

22-6

29-8

21-0

That this carbon monoxide was not produced by the reduction of carbon dioxide by
charcoal was shown by a series of three experiments, described later.

It was thought that, if a rapid current of air were passed over charcoal heated to a
temperature below the ignition point* of carbon monoxide, the carbon monoxide, if
produced, might be swept away from the area of combustion before it was oxidized.

The air used in the experiment was freed from water vapour by passing through

* MM. MALLARD and LE CHATELIER found that a mixture of carbon monoxide and oxygen ignited at
abont 650° C., though gradual union took place at 400°. (' Annales dcs Mines,' vol. 4, 1883.)

MR. H MKKKKTON I:\KKU <>.\ O>MWSTK>N'

DUIKD OXYGEN.

a tube of fused calcium chloride, and from carbon dioxide by a wash-bottle of strong
potash. The purified charcoal was placed in a thick, hard glass tube (fig. 4) : with
this was connected an apparatus for analysing the gases by weight tiimil.-ir to that
used in the last experiment. Instead of the conducting tube dipping under mercury,
a wash-bottle, not shown in the diagram, was connected with the last set of potash
bulbs. This contained a solution of pyrogallic acid. A small tube of potash solution
was placed upright in the bottle. By tilting the flask, and so upsetting the potash
into the pyrogallic acid, it could be seen at any moment of the experiment whether
all the oxygen was being used up by the carbon.

The carbon dioxide produced by the combustion is absorbed in bulbs A. Bulbs 11 did not increase- in
weight, showing that the absorption in A was complete. Bulbs C absorb the carbon dioxide formed
by the carbon monoxide produced in the combustion.

The air was drawn through the whole apparatus by means of a water pump. Tin-
rate was kept constant, so that 30 bubbles per minute passed through the bulbs.

Experiment I. — The temperature of the carbon was about 500°. It was regulated
so that some lead chloride placed in a similar tube by the side of the carbon tube was
just melted.

Percentage of carbon dioxide produced . . . 88'2
,, carbon monoxide 117

Experiment II. — Temperature of carbon rather lower.

Carbon dioxide

Carbon monoxide

99-9

82-1

17'8

99-9
Experiment III. — Temperature of carbon about 400° (m.p. of lead iodide).

Carbon dioxide .
Carbon monoxide

63'24
3676

100-00

4 F. •:

580 MR. H. BRERETON BAKER ON COMBUSTION IN DRIED OXYGEN.

Experiment IV. — The carbon was placed in a U-tube, which was heated in the
vapour of boiling sulphur (440°).

Carbon dioxide 57'28

Carbon monoxide 4272

100-00
Experiment V. — Temperature of carbon 440°.

Carbon dioxide 60 '8 G

Carbon monoxide 39 '14

100-00

Experiment VI.— At 440°.

Carbon dioxide 46'8 1

Carbon monoxide 53" 19

100-00

In the sixth experiment the rate of passage of the gas was increased to 40
bubbles a minute.

Experiment VII. — Instead of air a mixture of oxygen and nitrogen containing
15'4 per cent, of oxygen was used. It was contained in a large bell-jar dipping in a
large and deep pneumatic trough (fig. 4). Temperature of the carbon 440°.

Percentage of carbon dioxide = 33*4
,, carbon monoxide = 66 "5

99-9

Thus carbon monoxide is always produced when carbon is heated in such mixtures of
oxygen and nitrogen at temperatures below 500°. It might, however, be due to the
reaction

CO2 + C = 2CO
instead of

2C + O2 = 2CO.

This supposition was proved to be untenable by the following experiments : —
Experiment I. — Pure carbon dioxide was passed very slowly over carbon heated to

the melting-point of lead chloride (500°). The gas was then collected over mercury

in a gas pipette. 278 c.c. of gas was collected in three hours, so that about 1'5 c.c.

of gas passed over the charcoal in a minute, This collected gas was entirely absorbed

by caustic potash.

Experiment II. — At the same temperature 150 c.c. of carbon dioxide was passed

over the heated charcoal at the rate of 1 c.c. a minute. It was entirely absorbed by

caustic potash.

MR. H. BRKBETON BAKER ON COMBUSTION IN DRIED OXYGMN 581

Experiment III. — The carbon dioxide was passed over charcoal at 440°, the gas
being afterwards passed through two sets of potash bulbs, over red-hot copper oxide,
and through a third set of potash bulbs to analyse the gases produced. The first set
gave an increase in weight of '4548 gram, the second set no increase, and the third
set no increase. The experiment lasted four hours. Therefore at these temperatures
carbon dioxide is not decomposed by carbon, and the carbon monoxide in the former
experiments must have been formed by direct union of carbon and oxygen.

The question now arises, Does the burning of carbon under ordinary circumstances
take place in two stages ? Is carbon monoxide first produced, and this by further
oxidation transformed into the dioxide ? The problem seems incapable of direct
solution. I venture to advance the following considerations with regard to it : —

I. It has been proved by direct experiment that more carbon monoxide is burnt by
air at 500° than at 440°. We find, in the results described above, that when carbon
is burnt in air at 440°, more carbon monoxide appears in the products of combustion

t than at 500°, amounting in one case to 53 per cent.

II. It has been proved, also, that when the oxygen is diluted with a larger quantity
of nitrogen than is contained in air, less carbon monoxide is oxidised, though the
oxygen is present in excess. In the experiment already described, by diminishing
the percentage of oxygen from 21 to 15, we get a larger quantity, 66 per cent., of
carbon monoxide in the products of combustion.

III. Lastly, by drying the oxygen in which the carbon is heated, the percentage of
carbon monoxide is largely increased, amounting in one case to 94 '7 per cent, of the
products of combustion.

And since it is found that, when the conditions of the experiment are made more
and more unfavourable for the oxidation of carbon monoxide, (1) by lowering the
temperature, (2) by decreasing the proportion of oxygen to nitrogen, (3) by drying
the oxygen, we get more and more carbon monoxide produced, are we not justified in
assuming that the combustion of carbon first produces this lower oxide ?

Combustion of Sulphur in Oxygen.

In the preliminary experiments on the combustion of sulphur in dried oxygen, the
sulphur used was purified by repeated sublimations.

Experiment I. — The sulphur was placed at one end of a tube filled with oxygen,
phosphorus pentoxide being introduced at the other to dry the gas. The tube was
sealed up and left drying for five days.

The tube was heated by an Argand burner, side by side with a similar tube
containing sulphur in moist oxygen. The sulphur began to melt at the same moment
in both tubes. Soon afterwards there was a sudden explosion in the moist tube, and
a few seconds later a small blue flame appeared on the surface of the dry sulphur.
This continued for a short time and was then extinguished On analysing the gases
in the two tubes, oxygen was found in the free state in the dry tube, but not in the
moist tube.

582 MB H. BRKRKTON BAKER ON COMBUSTION IN DRIED OXYGEN.

Sulphur was then further purified as follows : — Some powdered sulphur was
resublimed, and as it might contain hydrogen, or some compound of hydrogen, it
was melted in a slow stream of sulphur chloride vapour, and distilled several times in
an atmosphere of this substance. Some hydrochloric acid gas was produced, showing
that this precaution was not needless. One end of the tube was sealed and the other
bent and fixed into one neck of a dry WOULFF'S bottle. The other neck was con-
nected with a water pump which maintained a nearly complete vacuum in the
apparatus. The sulphur was kept at a temperature of 150°-180° by an Argand
burner. In this way all the sulphur chloride was got rid of.

Experiment II. — A tube was sealed up containing this purified sulphur in oxygen,
and was left drying for five days as before. It was heated over an Argand lamp with
a comparison tube containing some of the purified sulphur in moist oxygen.

The moist sulphur was seen to burn with a sudden flash, while the dry sulphur was
distilled several times backwards and forwards in the tube without any visible
combustion. Analysis of the contents of the dry tube showed that only -p/th of the
oxygen had been converted into sulphur dioxide.

Combustion of Carbon Bisulphide in Oxygen.

The union of these two substances as brought about by an electric spark is well
known to be extremely energetic. It seemed of interest to investigate whether
dryness has the same effect on the burning of carbon bisulphide as it has on the
burning of each of its constituents when heated separately in oxygen.

Purification of Carbon Bisulphide. — About 50 c.c. of the commercial substance
were distilled five times from white wax, and in this way the liquid lost most of
its unpleasant odour. To get rid of hydrogen compounds, as far as possible, it was
sealed with a small quantity of chloride of sulphur in thick glass tubes and heated to
180° in an air bath for several days. The contents of the tube were distilled, the
first portion only which distilled over at 46° being taken. This was shaken with pure
mercury at intervals for several days until the metal remained bright after several
hours' contact. The purified liquid was again distilled and sealed up with some
phosphorus pentoxide. This drying agent seems to have no action on the bisulphide.

5 cubic centimetres of the vapour were collected over dried mercury, and to this
vapour was added three times its volume of dried oxygen. A plug of phosphorus
pentoxide was introduced, and the tube was left for twelve days. At the end of that
time a spark of the smallest possible length from a small coil with one bichromate cell
was passed through the mixture. The mixture exploded with a bright flash, leaving
a deposit of sulphur on the sides of the tube.

A second experiment was tried in the same way, the mixture being allowed to dry
for six weeks. The result was the same.

Two more experiments were made. Long tubes were carefully dried ; phosphorus

MR. H. BREHETON BAKER ON COMBUSTION IN DRIED OXYGEN. 583

peutoxide was introduced at one end and sealed bulbs of purified carbon bisulphide
at the other. A stream of dried oxygen WHS passed through the tubes and the ends
sealed. The bulbs of carbon bisulphide were then broken, and the vapour, mixed
with oxygen, was allowed to dry over phosphorus pentoxide for two weeks. Each
tube was then heated over an Argand burner with a comparison tube containing
the same gases in a moist state. Though the wet tubes exploded first in every case,
the dry ones did so a few seconds later, and apparently with equal force.

Combustion of Ordinary Phosphorus in Oxygen.

Commercial stick phosphorus was melted under a solution of potassium bichromate
in dilute sulphuric acid, dried and heated to 60° in a sealed tube with phosphorus
trichloride. The tube was allowed to project some 15 centimetres out of the air bath,
so that the sublimed phosphorus could condense in the cool part, but it was so inclined
that any chloride which distilled over should run back into the heated part, which
contained the melted phosphorus, mixed with phosphorus pentoxide. In three weeks
the greater part of the phosphorus was deposited in the cold part of the tube. The
iridescent crystals were melted and allowed to run into the cooler end of the tube,
which was drawn out to a diameter of about 5 mm. This part was then sealed off.

The tube of purified phosphorus was placed with some phosphorus pentoxide in a
tube of hard glass bent at right angles, and previously heated to redness. One end
of this was connected with the stoppered neck of a Bunsen's gas pipette by a joint of
dried india-rubber. The pipette contained oxygen which had been drying for eight
days over phosphorus pentoxide. The tube was exhausted as completely as possible
by a Sprengel pump with two fall tubes, and the end connected with the pump
sealed. The tubulure at the bottom of the pipette was connected with a flexible
india-rubber tube, dried as far as possible, which was connected with a resei-voir of
dried mercury. The mercury reservoir could be raised or lowered according as a high or
low pressure was desired (fig. 5). After the whole apparatus had been allowed to stand
for a week to allow the small residue of air in the vacuous tube to dry, the following
experiment was carried on in a dark room. The oxygen was admitted to the vacuous
tube, and the tube containing the phosphorus was broken. On the pressure being
diminished, a very faint luminosity was seen on the phosphorus, which flickered for
several hours, the interval between the extinguishing and the reappearance of the
light being about a second and a half. This flickering was not due to variations of
pressure caused by the shaking of the mercury in the reservoir, as the experiment
was carried on in a cellar with a stone floor, on a solid wooden bench screwed to the
ground. The tap was then turned off and the tube allowed to dry for three days
more. At the end of this time the luminosity had disappeared, nor did it show itself
though the pressure was varied in every possible way. The phosphorus was then
melted by heating the tube with a spirit lamp, and still no luminosity appeared. It

MR. H. BRERETON BAKER ON COMBUSTION IN DRIED OXYGEN.

was then heated until the phosphorus boiled. A faint glow appeared for an instant
and then vanished.

Fig. 5.

The india-rubber tube in connection with the pipette was then clamped. The
phosphorus tube thus closed was removed from the pipette. The air was squeezed
out of the free half-inch of india-rubber tubing, and a little water allowed to enter.
Brilliant luminosity at once flashed out, and without any heating the phosphorus burst
into a vivid white flame.

Ordinary phosphorus therefore does not burn in dry oxygen, though its temperature
be raised to its boiling point (290°).

Combustion of Amorphous Phosphorus in Oxygen.

This substance was one of the first experimented upon, and, though it was by no
means pure, it gave indications of a very different behaviour when heated in moist and
in dry oxygen.

It was purified in the following way : — Commercial amorphous phosphorus was
washed with water and dried by a current of air at 150° in a glass tube. One end of
the tube was then sealed and the other connected with a mercury pump. The tube
was exhausted and heated to 240°. A large quantity of gas was given off. No less
than 75 c.c. were evolved from 5 grams of phosphorus. On allowing a bubble to
escape into the air it caught fire. It had a strong smell of phosphine.

MR. H. BRERETON BAKER ON COMBUSTION IN DRIED OXYGEN. 585

When no more gas was evolved, the tube was removed from the pump and sufficient
phosphorus trichloride added to cover the phosphorus. The chloride was boiled and
the tube sealed. It was heated in an air bath to 200° for three days. Any traces of
phosphine or hydrogen would attack the chloride of phosphorus with fonnation of
hydrochloric acid and free phosphorus. The tube was opened and the phosphorus
chloride distilled off. To get rid of hydrochloric acid, the phosphorus was heated in a
vacuous tube which had a piece of solid potash at the other end.

To get rid of any traces of ordinary phosphorus which might have been formed in
these processes, a stream of purified air was drawn over the purified substance heated
to 100° for two days.

Tubes were constructed like those used for carbon, with plugs of phosphorus pent-
oxide at one end and purified amorphous phosphorus at the other. These tubes were
each heated over an Argand burner with a similar tube containing the same phos-
phorus in moist oxygen. In six of them, which had been drying from 2-5 weeks, no
visible combustion could be observed, though the temperature was raised sufficiently
nigh to distil the phosphorus. It condensed in yellow globules which could again be
distilled, confirming the last results obtained on the influence of moisture on
the combustion of ordinary phosphorus. After the experiments the gases were
analysed, and found to consist of pure oxygen. In one experiment the tube had been
standing for two months. The phosphorus pentoxide was very moist, and it was
thought probable that the gas would not be sufficiently dry to prevent combustion.
The end of the tube containing the phosphorus was heated over a £-inch flame of an
Argand burner, side by side with a comparison tube of phosphorus in moist oxygen.
In a little tune the moist phosphorus burnt with a bright white light. The dry
phosphorus began to distil slowly. When the heavy white vapour reached the moist
oxide of phosphorus it burnt with a green flame quite slowly, the flume repeatedly
going out and being re-kindled. The flame never moved to any other point, but
burnt in this position until all the oxygen was used up.

These experiments show that phosphorus, both in the crystalline and the amorphous
state, does not undergo combustion when heated in dry oxygen.

Tlie Temperature required for the Combustion of Amorphous Phosplwrus.

It was frequently noticed that amorphous phosphorus burnt readily at 360°, the
boiling-point of mercury. In order to see if a lower temperature would suffice, the
following experiment was performed : — In the horizontal part of a long tube, bent at
right angles, a small quantity of pure amorphous phosphorus was placed. The vertical
part of the tube was open, and dipped into a trough of mercury. After the tube had
been filled with moist oxygen, the other end which contained the phosphorus was
sealed. This end was inserted in the hole in the side of the air bath. The tempera-
ture was kept constant, at 260°. After ten minutes the mercury had risen 5 centi-

MDCCCLXXXVIII. — A. 4 P

MR. H. HRKRETON Ii AKKR ON COMBUSTION IN DRIED OXYGKN.

metres in tlie tube. The combustion went on gradually, diminishing slowly in rapidity,
and was complete in four hours. The presence of the phosphorus pentoxide produced
tends to dry the ga.s, which would, I think, account for the diminution in the rate of
combustion.

A similar experiment was made with moist oxygen at a temperature of 70°. No
diminution in volume of the oxygen was noticed, though the tube was heated for
three days continuously.

The temperature of the bath was raised to 100°, and kept constant for twenty-
three days. During this long heating the combustion was proceeding very gradually,
and at the end of the lime half the oxygen was used up. The combustion of amor-
phous phosphorus, therefore, at 100° is very slow.

To show the difference which dryness makes in the rate of combustion, two similar
tubes were heated to 100° in the same air bath. One contained phosphorus in oxygen
which had been previously dried by a plug of phosphorus pentoxide for three days.
The oxygen in the other was kept saturated with moisture by a drop of water floating
on the mercury within the tube. After being heated to 100° for seven days both
tubes were examined. In the dry tube the mercury had risen 8 mm. ; in the wet
tube it had risen 150 mm.

They were heated another seven days. The combustion in the wet tube was
complete ; in the dry tube the mercury had only risen 2 mm. In the former experi-
ments, therefore, the phosphorus pentoxide produced does diminish the rate of
combustion.

The luminosity of phosphorus is extinguished if a trace of turpentine vapour be
present. A drop of turpentine was allowed to float on the surface of the mercury in
a tube containing amorphous phosphorus in moist oxygen. It was heated to 100° as
before. The combustion was perceptibly slower than before, though the oxygen was
kept saturated with moisture. It went on gradually, however, until after seven days
the mercury had risen 100 mm.

Amorphous phosphorus, as has been stated by many observers, undergoes an
extremely slow combustion when allowed to stand in air. After having been kept for
a year in a stoppered tube, a specimen of the pure substance was found to be quite
moist, and, when it was washed with pure water, the washings gave the reactions of
phosphoric acid.

The Conversion of Amorphoiis Phosphorus into Yellow Phosphorus.

In one experiment on the combustion of amorphous phosphorus in oxygen, I wished
to be quite sure that the temperature to which the wet and dry tubes were heated
was the same. Two such tubes were prepared, bent in the middle at a right angle.
The phosphorus pentoxide in the dry tube was placed in one arm of the bent tube, the
phosphorus being in the other. The ends of the two tubes containing the phosphorus

Ml{. II. MIIKUKTOX IIAKKK ON C( iM UfSTION IN DRIED i)\\

587

were passed through the cork of a large l>oiling tube containing mercury. A long
tube open at both ends and passing through the cork served to condense the mercury

vapour.

Fig. 6.

The mercury was boiled. After a few minutes the wet phosphorus burnt, the dry
phosphorus showing no change. Further than that, though kept in the boiling
mercury for four hours, it was apparently unaltered. We should have expected that
it would have been, partly at all events, transformed into the ordinary modification.
Is it possible that the change from one modification to the other is affected by
dryness?

To answer this question the following experiment was undertaken : — Two tubes
containing pure amorphous phosphorus in (a) moist and (/>) dry nitrogen were heated
in the vapour of boiling mercury for four hours. The phosphorus in both tubes was
apparently unaltered. No sublimate of ordinary phosphorus could be seen on the cold
parts of either tube. When the same tubes were heated in sulphur vapour at 440°,
the production of ordinary phosphorus was quickly noticed in both tubes. Dryness
does not apparently affect the temperature of the change.

Two tubes were prepared bent at right angles. A small quantity of amorphous
phosphorus was placed in each, and one was filled with pure nitrogen, the other with
pure oxygen, both gases being saturated with aqueous vapour. The ends of the tubes
were then sealed. In order to heat the phosphorus contained in them, the limbs
containing the phosphorus were placed in an air bath, holes being cut in its sides for
their reception. The air bath was provided with a PAGE'S regulator and the tempera-
ture was kept constant at 260°. The other limbs of the tubes were outside the air
bath and were kept cool (fig. 7).

After six hours the tubes were removed. No perceptible change had taken place in
the nitrogen tube ; a small quantity of ordinary phosphorus appeared on the cold jmrt
of the oxygen tube.

4 F 2

588 MR. H. BRERETON BAKER ON COMBUSTION IN DRIED OXYGEN.

When the tip of the oxygen tube was broken under mercury the liquid rushed in
and filled the tube, showing that combustion was complete. The nitrogen tube was
taken into a dark room, and the points of both ends broken. A current of air was
drawn through it with the mouth. No taste or smell was perceptible, and after this
treatment no luminosity appeared in the tube. A trace of ordinary phosphorus can
be detected in this way. We must conclude, therefore, that at this temperature of
260° no change takes place when amorphous phosphorus is heated in a sealed tube
containing nitrogen.

LEMOINE ('Deutsch. Chem. Gesell. Berichte,' 1867) has shown that the conversion
of amorphous phosphorus into the ordinary modification is prevented if there is a
considerable tension of phosphorus vapour in the tube. Though it was scarcely
probable that this would exist in the tubes above described, where no taste of
phosphorus was found in the gas, it was thought advisable to perform two experi-
ments without sealing the tubes in which the phosphorus was heated. Nitrogen was

Fig. 7.

prepared in quantity by exposing the air in large bell-jars to the slow action of sticks
of phosphorus. When the sticks were no longer luminous, the nitrogen was considered
ready for use. In order that no vapour of phosphorus should accompany the nitrogen,
the gas was passed through a tube containing red-hot copper oxide. It then bubbled
through a solution of alkaline pyrogallate to take out the last traces of oxygen, and
was then passed over pure amorphous phosphorus contained in a long tube. To
prevent access of air a wash bottle of water was connected with the end of this tube.
The tube itself passed through an air bath which was fitted with a PAGE'S regulator.

In the first experiment the air bath was kept constantly heated to 265° for three
hours, a slow current of nitrogen being passed through the tube. No deposit of
ordinary phosphorus was found on the cold part of the tube. In the second experi-
ment the phosphorus was kept for five hours at a temperature of 278°, but not only
was no deposit of phosphorus produced, but, after cooling, no smell or taste of ordinary
phosphorus could be detected by drawing air through the tube. It is proved, then,

MR H. BRERETON BAKER ON COMBUSTION IN DRIED OXYOBt 589

that amorphous phosphorus combines with oxygen at a lower temperature than that
at which it is changed to the ordinary modification when heated in an inert gas. The
statement which is ordinarily made that amorphous phosphorus, when heated to 260°,
is converted into the yellow variety and if this is done in air or oxygen it then
catches fire, is incorrect. Amorphous phosphorus must be considered as undergoing
a true combustion, very slow at ordinary temperatures, slow at 100°, and quick at some
temperature just below 260°.

Combustion of Tellurium in Oxygen.

The tellurium was purified by dissolving it in fuming sulphuric acid. The deep red
solution was precipitated by the addition of water. It was thought that the possi-
bility of occluded hydrogen being present would be obviated in this way. As has
been shown, hydrogen has a great influence in bringing about combustion in dried
gases, and its elimination was made a principal object in these purifications. The
precipitated tellurium was washed with distilled water several times, and dried by
heating in vacuo over phosphorus pentoxide.

It was placed in hard glass tubes in oxygen dried (1) for two, (2) for three weeks,
but in both cases, when heated over an Argand burner with comparison tubes filled
with moist oxygen, combustion appeared to take place with as much readiness as was
the case when moisture was present.

Combustion of Selenium.

Selenium was purified by several sublimations, at first in vaciu>, then in selenium
chloride vapour. It was then freed from all traces of the chloride by repeated subli-
mations in vacuo, pieces of solid potash being present.

Tubes were prepared containing this purified selenium in pure oxygen. In some of
them the oxygen was dried by phosphorus pentoxide, while in the others the oxygen
was saturated with aqueous vapour.

Experiment I. — A tube containing selenium in oxygen, which had been standing
in contact with phosphorus pentoxide for two weeks, was heated with a similar tube
containing selenium in moist oxygen. The dry selenium began to burn at the same
moment as the moist, with a slightly less intense flame. On analysis the oxygen was
found to be totally used up in both tubes.

Experiment II.— The tube containing selenium in oxygen, dried by phosphorus
pentoxide, was allowed to stand for two months. It was supported above an Argand
burner with a tube containing selenium in moist oxygen. No difference could be
observed between the combustion of the moist and dry substances.
Selenium, therefore, when purified in this way, burns in dry oxygen.

590 MR. H. BRERETON BAKER ON COMBUSTION IN DRIED OXYGK.N

Combustion of Boron in Oxygen.

In these experiments the boron used was given to me by Mr. FRANCIS JONES, of
Manchester. It had been prepared by him in his researches on boron hydride. As
hydrogen might be occluded in the substance, it was heated in a tube of hard glass
connected with a Sprengel pump About twelve times its volume of gas was evolved,
which burnt in air with a green flame. It was probably boron hydride. The boron
was then sealed in a tube with oxygen, and left drying for a week. It burnt readily
on heating. As all the hydrogen might not have been removed, it was placed in a
tube, and heated for three days in an air bath at 200° in boron chloride vapour. A
large quantity of hydrochloric acid was produced, and it was only after treating it in
this manner three times that the boron chloride did not evolve hydrochloric acid gas.
After the hydrogen had thus been eliminated, the boron was heated in dried oxygen,
and found not to undergo combustion at the bright red heat of the blow-pipe flame.
The end of the glass tube in this experiment was bent and made to dip under mercury,
to prevent the glass from blowing out under the pressure of the heated oxygen.

Combustion of Arsenic in Oxygen.

Commercial arsenic was mixed with purified charcoal and heated in a vacuous glass
tube. The crystalline sublimate was heated in a sealed tube with arsenic chloride
vapour. The arsenic was then distilled again in a vacuous tube, at one end of which
was a piece of solid potash to absorb hydrochloric acid.

Two experiments were made with this substance. In the first the oxygen was
dried over phosphorus pentoxide for one, in the second for two months. The arsenic
was found to burn as readily when dry as when moist, and when the ends of the
tubes in which the experiments were done were broken under mercury, the mercury
filled them entirely, showing that the oxygen was used up in both cases.

Arsenic, therefore, when purified in this way burns in dry oxygen.

Combustion of Antimony in Oxygen.

This element was prepared by heating tartar-emetic with charcoal. The metal so
obtained was powdered, and placed in a hard glass tube, which was then filled with
chlorine. The tube was then heated to 200° in an air bath. The process was
repeated twice. The antimony so obtained was freed from hydrochloric acid by
heating in vacuo over potash. On being heated in oxygen which had been dried for
six weeks, it was found to burn readily. Two other experiments were done, in one of
which the antimony was heated in oxygen which had been dried over phosphorus
pentoxide for four months but neither showed any difference in behaviour between
the moist and dry gas.

MR, H. BRKRETON ItAKKR ON COMBUSTION IN DRIED OXYGEN. 591

Geneiftl Conclusions.

1. Pure charcoal, heated in oxygen dried by phosphorus pentoxide, does not burn
with a flame. Partial combustion, however, goes on, both carbon monoxide and
carbon dioxide being formed.

2. If the charcoal is mixed with platinum black, and heated in the same way in
oxygen, about the same amount of charcoal is burnt. Carbon dioxide is, however, the
only product.

3. When charcoal burns in oxygen, its combustion probably goes on in two stages.
It forms, first, carbon monoxide, and, if circumstances are favourable, this undergoes
further oxidation to the dioxide.

4. Sulphur, boron, amorphous and ordinary phosphorus do not burn in dried
oxygen. Ordinary phosphorus does not even show luminosity, at any pressure, in
dried oxygen.

5. Amorphous phosphorus is not converted into ordinary phosphorus when heated
in nitrogen to 278° C. This substance undergoes true combustion when heated to
260° in moist air or oxygen, without any previous change to the crystalline variety.

6. Selenium, tellurium, arsenic, and antimony show no difference in their combus-
tion, whether the oxygen be moist or dry.

I am indebted to Professor H. B. DIXON, F.R.S., for much valuable advice and
encouragement during the progress of this investigation.

INDEX

TO THB

PHILOSOPHICAL TRANSACTIONS (A)

FOR THE YEAR 1888.

A.

AHNBY (W. DB W.) and FESTINO (E. R.). Colour Photometry.— Part TI. The Measurement of
Reflected Colours, 547.

B.

BAKEB (H. B.). Combustion in Dried Oxygen, 571.

BiSSET (A. B.). On the Motion of a Sphere in a Viscous Liquid, 43.

BIDWELL (S.). On the Changes produced by Magnetisation in the Dimensions of Rings and Rods of

Iron and of some other Metals, 205.
BUBBUBY (S. H.). On the Induction of Electric Currents in Conducting Shells of Small Thickness, 297.

C.

Cobalt and nickel, on the ultra-violet spectra of, 231 (see LIVEI.VU and DEWAB).

Colour photometry . — Part II. The measurement of reflected colours, 547 (see ABNET and FESTINO).

Colours, the measurement of reflected, 547 (see ABNET and FKSTINO).

Combustion in dried oxygen, 571 (see BAKEB).

I'orariants, invariant!, and quotient -derivatives associated with linear differential equations, 377 (a

FOBSYTH).
COWAN (G. C.) (see EWIMI and COWAN).

MDCCCI.XXXVIII. — A. 4 O

594 INDEX.

D.

DEWAB (J.) (see LIVEINO and DEWAR).
Dew-point instrument*, 73 (see SHAW).
Diameters of a plane cubic, on the, 151 (see WALKER).

Differential (linear) equations, invariants, covariante, and quotient-derivatives associated with, 377 (see
FORSYTE).

E.

Elasticity and the internal friction of metals, the effect of magnetisation on the, 1 (see TOMLINSON).
Electric currents in conducting shells of small thickness, on the induction of, 297 (see BURBURT).
Evolution of gases from homogeneous liquids, the conditions of the, 257 (see VELET) .
EWINO (J. A.). Magnetic Qualities of Nickel (Supplementary Paper), 333.
EWING (J. A.) and COWAN (G. C.). Magnetic Qualities of Nickel, 325.

F.

FESTING (E. E.)(see ABNEY and FESTING).
FITZPATRICK (T. C.) (see GLAZEBROOK and FITZPATRICK).

FOBSYTH (A. R.). Invariants, Covariants, and Quotient-Derivatives associated with Linear Differential
Equations, 377.

G.

Oases, the conditions of the evolution of, from homogeneous liquids, 257 (see VELEY).
GLAZEBROOK (E. T.) and FITZPATRICK (T. C.). On the Specific Resistance of Mercury, 351.

H.

Hamilton's Numbers, on. — Part II., 65 (see SYLVESTER and HAMMOND).
HAMMOND (J.) (see SYLVESTER and HAMMOND).
Hygrometric methods, report on, 73 (see SHAW).

I.

Induction of electric curretitg in conducting shells of small thicknesr, on the, 297 (see BURBURY).
Invariants, covariants, and quotient-derivatives associated with linear differential equations, 377 (see

FORSYTE).
Iron, changes produced by mr.gnetisation in the dimensions of rings and rods of, 205 (see BIDWELL).

INDKX

Linear differential equations, invariant*, covariants, and quotient-derivative* associated with, 377 (Me

FOBSYTH).

LIVEIXU (G. D.) and DEWAB (J.). On the Spectrom of the Oxy-hydrogeu Flame, 27.
LivEixt; (G. D.) and DEWAB (J.). On the Ultra- Violet Spectra of the Elements.— Part III. Cobalt and

Nickel, 231.
Lovi (A. B. H.). The Small Free Vibrations and Deformation of a Thin Elastic Shell, 491.

M.

Magnetic qualities of nickel, 325 (see EWJNQ and COWAH).

Magnetic qualities of nickel (supplementary paper), 333 (see Ewixo).

Magnetisation, on the changes produced by, in the dimensions of rings and rods of iron and of some other

metals, 205 (see BIDWELI.).

Magnetisation, the effect of, on the elasticity and the internal friction of metals, 1 (see TOMLIXBOK).
Mechanical properties of metals considered in relation to the periodic law, on certain, 339 (nee ROBERTS-

AUSTIN).

Mercury, on the specific resistance of, 351 (see GLAZIBBOOK and FITZPATRICK).
Metalt, changes produced by magnetisation in the dimensions of rings and rods of iron and of some

other, 205 (see BIDWEI.L).

Metals, effect of magnetisation on the elasticity and the internal friction of, 1 (see TOMLIXBON).
Mftals, on certain mechanical properties of, considered in relation to the periodic law, 839 (see ROBKBTB-

AUSTEN).
Motion of a sphere in a viscous liquid, 43 (see BASSBT).

N.

Nickel, magnetic qualities of, 325 (see EWINO and COWAN).

Nickel, magnetic qualities of (supplementary paper), 333 (see Ewixo).

NicM, on the ultra-violet spectra of cobalt and, 231 (see LITEINO and DIWAK).

O.

Oxygm, combustion in dried, 571 (see BAKKK).

Oxy-hydrogenjlame, on the spectrum of the, 27 (see LIVIINU and DEWAK).

P.

Periodic law, on certain mechanical properties of metals considered in relation to the, H30 (see ROBIBTS-

ACSIEN).

Photometry, colour.— Part II., 547 (see ABNKV and FKSTIXO).

Physical properties of matter, the influence of stress and strain on the, 1 (see TOMLISSON).
Plane cubic, ou the diameters of a, 151 (see WALKEB).

INDKX.

Q.

Qiintient-derivativef, invariants, covariants, and, associated with linear differential equations, 377 (see
FORSTTH).

R.

Reflected colours, the measurement of, 547 (see ABNEY and FESTIX<.)-

ROBERTS- AUSTEN (W. C.). On certain Mechanical Properties of Metals considered in relation to tin-
Periodic Law, 339.

S.

SHAW (W. N.). Report on Hygrometric Methods : First Part, including the Saturation Method and ilu-

Chemical Method, and Dew-point Instruments, 73.

Shell, the small free vibrations and deformation of a thin elastic, 491 (see LOVE).
Specific resistance of mercury, on the, 851 (See GLAZEBROOK and FITZPATRICK).
Spectra of the elements, on the ultra-violet. — Part III., 231 (see LIVEING and DEWAR).
Spectrum of the oxy -hydrogen flame, 27 (see LIVEING and DEWAR).
Sphere in a viscous liquid, on the motion of a, 43 (see BASSET).

Stress and strain, the influence of, on the physical properties of matter, 1 (see TOMLINSON).
SYLVESTER (J. J.) and HAMMOND (J.) On Hamilton's Numbers. — Part II., 65.

T.

TOMLINSON (H.). The Influence of Stress and Strain on the Physical Properties of Matter. — Part I.,
Elasticity (continued). — The Effect of Magnetisation on the Elasticity and the Internal Friction
of Metals, 1.

U.
Ultra-violet spectra of the elements, on the. — Part III., 231 (see LIVEING and DEWAB).

V.

VELEY (V. H.). The Conditions of the Evolution of Gases from Homogeneous Liquids, 257.
Vibrations and deformation of a thin elastic shell, the small free, 491 (see LOVE).
Viscous liquid, on the motion of a sphere in a, 43 (see BASSET).

W.

WALKER (J. J.). On the Diameters of a Plane Cubic, 151.

HARRISON AND SONS, PRINTERS IN ORDINARY TO UER MAJESTY, ST. MAKn.s'- |,.\.\K.

o

Q

a

L82
v.179
cop. 2

Ho/ax society of London

Philosophical transactions,
Series A: Mathematical and
physical sciences.

A

Sci.

Seruk

PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET

UNIVERSITY OF TORONTO LIBRARY