LIBRAE*

PHILOSOPHICAL

TRANSACTIONS

OF THE

EOYAL SOCIETY OF LONDON

SERIES A

CONTAINING PAPERS OF A MATHEMATICAL OK PHYSICAL CHARACTER.

VOL. 191.

LONDON:

PRINTED BY HARRISON AND SONS, ST. MARTIN'S LANE, W.C.,
printers in ©rbiitarg to <$ tr Pajwig.

1898.

6?

v

CONTENTS.

(A)
VOL. 191.

Advertisement page vii

List of Institutions entitled to receive the Philosophical Transactions or Proceedings of the Royal

Society ix

I. Memoir on the Integration of Partial Differential Equations of the /Second Order

in Three Independent Variables when an Intermediary Integral does not exist
in general. By A. R. FORSYTH, F.R.S., Sadlerian Professor in the Univer-
sity of Cambridge page 1

II. On the Change of Absorption produced by Fluorescence. By JOHN BURKE,

M.A., Berkeley Fellow of the Owens College, Manchester. Communicated by
Professor ARTHUR SCHUSTER, F.R.S. 87

III. On the Occlusion of Hydrogen and Oxygen by Palladium. By LUDWIG MONO,

Ph.D., F.R.S., WILLIAM KAMSAY, Ph.D., F.R.S., and JOHN SHIELDS, D.Sc.,
Ph.D 105

IV. Comparative Photographic Spectra of Stars to the 3^ Magnitude. By FRANK

McCLEAN, F.R.S. ... 127

V. On the Application of Harmonic Analysis to the Dynamical Theory of the Tides.

— Part II. On the General Integration of LAPLACE'S Dynamical Equations.
By S. S. HOUGH, M.A., Fellow of St. John's College and Isaac Newton
Student in the University of Cambridge. Communicated by Professor G. H.

DARWIN, F.R.S. 139

a 2

VI. Electrification of Air, of Vapour of Water, and of Other Gases. By Lord

KELVIN, G.C.V.O., F.R.S., MAGNUS MACLEAN, D.Sc., F.R.S.E., and ALEX-
ANDER GALT, B.Sc., F.R.S.E. page 187

VII. Mathematical Contributions to the Theory of Evolution. — IV. On the Probable

Errors of Frequency Constants and on the Influence of Random Selection on
Variation and Correlation. By KARL PEARSOX, F.R.S., and L. N. G. FILON,
B.A., University College, London 229

VIII. A Compensated Interference Dilatometer. By A. E. TUTTON, Assoc. R.C.S.

Communicated by Capt. ABNEY, C.B., F.R.S. 313

IX. The Electric Conductivity of Nitric Acid. By V. H. VELEY, M.A., F.R.S., and

J. J. MANLEY, Daubeny Curator, Magdalen College, Oxford .... 365

X. On the Thermal Conductivities of Single and Mixed Solids and Liquids and their

Variation ivith Temperature. By CHARLES H. LEES, D.Sc., Assistant Lec-
turer in Physics in the Owens College. Communicated by Professor ARTHUR
SCHUSTER, F.R.S. 399

XL Experiments on Aneroid Barometers at Kew Observatory, and their Discussion.
By C. CHREE, Sc.D., LL.D., F.R.S. , Superintendent. Communicated by the
Author at the Request of the Kew Observatory Committee 441

XII. On the Heat Dissipated by a Platinum Surface at High Temperatures. By
J. E. PETAVEL, 1851 Exhibition Scholar. Communicated by Lord RAYLEIGH,
F.R.S. 501

Index to Volume 525

LIST OF ILLUSTRATIONS.

Plates 1 to 17. — Mr. F. MoCLEAN on the Comparative Photographic Spectra of
Stars to the 85- Magnitude.

Plates 18 to 23. — Mr. J. E. PETAVEL on the Heat Dissipated by a Platinum Surface
at High Temperatures.

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 time 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
the Society, to which they will always adhere, never to give their opinion, as a Body,

upon any subject, either of Nature or Art, that comes before them. And therefore the
thanks, 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.

1898.

LIST OF INSTITUTIONS ENTITLED TO RECEIVE THE PHILOSOPHICAL TRANSACTIONS OK

PROCEEDINGS OF THE ROYAL SOCIETY.

Institutions marked A are entitled to receive Philosophical Transactions, Series A, and Proceedings.
„ ,, B „ „ „ „ Series B, and Proceedings.

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

,, „ p „ „ Proceedings only.

America (Central).
Mexico.

p. Sociedad Cientifica "Antonio Alzate."
America (North). (See UNITED STATES and CANADA.)
America (South).
Buenos Ayres.

AB. Museo Nacional.
Caracas.

B. University Library.
Cordova.

AB. Academia Nacional de Ciencias.
Demerara.
p. Royal Agricultural and Commercial

Society, British Guiana.
La Plata.

B. Museo de La Plata.
Rio de Janeiro.

p. Observatorio.
Australia.
Adelaide.
p. Royal Society of South Australia.

Brisbane.

p. Royal Society of Queensland.
Melbourne.

p. Observatory.

p. Royal Society of Victoria.

AB. University Library.
Sydney.

Australian Museum.

Geological Survey.

Linnean Society of New South Wales.

Royal Society of New South Wales.

University Library.

P-

P-
P-

AB.
AB.

Austria.

Agram.

p. Jugoslavenska Akademija Znanosti i
jetnosti.

p. Societas Historico-Naturalis Croatica.
VOL. CXCI. — A.

Urn-

Austria (continued).
Briinn.

AB. Naturforschender Verein.
Gratz.

AB. Naturwissenschaftlicher Verein fur Steier-
mark.

Innsbruck.

AB. Das Ferdinandeum.

p. Naturwissenschaftlich - Medicinischcr

Verein.
Prague.

AB. Konigliche Bohmische Gesellschaft der

Wissenschaf ten .
Trieste.

B. Museo di Storia Naturale.
p. Societa Adriatica di Scienze Natural!.
Vienna.

p. Anthropologische Gesellscliaft.

Kaiserliche Akademie der Wissenschaf ten.
K.K. Geographische Gesellschaft.
K.K. Geologische Reichsanstalt.
K.K. Naturhistorisches Hof- Museum.
K.K. Zoologisch-Botanische Gesellschaft.
Oesterreichische Gesellschaft fur Meteoro-

logie.
Von Kuffner'sche Sternwarte.

AB.

P-

AB.

B,
B.
P-

d'Histoire Naturelle de

A.

Belgium.
Brussels.

B. Academic Roy ale de Medecine.
AB. Academic Royale des Sciences.
B. Musee Royal

Belgique.

p. Observatoire Royal.
p. Societe Beige de Geologic, de Paleonto-

•logie, et d'Hydrologie.
p. Societe Malacologiqne de Belgique.
Ghent.
AB. University.

Belgium (continued).
Liege.

AB. Societe des Sciences.

p. Societe Geologique de Belgique.
Louvain.

B. Laboratoire de Microscopie et de Biologie
Cellnlaire,

AB. Universite.
Canada.
Hamilton.

p. Hamilton Association.
Montreal.

AB. McGill University.

p. Natural History Society.
Ottawa.

AB. Geological Survey of Canada.

AB. Royal Society of Canada.
Toronto.

p. Astronomical and Physical Society.

p. Canadian Institute.

AB. University.

Cape of Good Hope.

A. Observatory.

AB. South African Library.
Ceylon.
Colombo.

B. Museum.
China.

Shanghai.

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

AB. Kongelige Danske Videnskabernes Selskab.
Egypt.

Alexandria.

AB. Bibliotheque Municipale.
England and Wales.
Aberystwith.

AB. University College.
Bangor.

AB. University College of North Wales.
Birmingham.

AB. Free Central Library.

AB. Mason College.

p. Philosophical Society.
Bolton.

p. Public Library.
Bristol.

p. Merchant Venturers' School.

AB. University College.
Cambridge.

AB. Philosophical Society.

p. Union Society.

England and Wales (continued) .
Cooper's Hill.

AB. Royal Indian Engineering College.

•
Dudley.

p. Dudley and Midland Geological and

Scientific Society.
Essex.

p. Essex Field Club.
Falmouth.

p. Royal Cornwall Polytechnic Society.
Greenwich.

A. Royal Observatory.
Kew.

B. Royal Gardens.
Leeds.

p. Philosophical Society.

AB. Yorkshire College.
Liverpool.

AB. Free Public Library.

p. Literary and Philosophical Society.

A. Observatory.

AB. University College.
London.

AB. Admiralty.

p. Anthropological Institute.

An. British Museum (Nat. Hist.).

AB. Chemical Society.

A. City and Guilds of London Institute.
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.

p. Institution of Electrical Engineers.

A. Institution of Mechanical Engineers.

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

AB. King's College.

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 Agricultural Society.

A. Royal Astronomical Society.

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

[ xi ]

England and Wales (continued).
London (continued).

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 Chirurgical Society.

p. Royal Meteorological Society.

p. Royal Microscopical Society.

p. Royal Statistical Society.

An. Royal United Service Institution.

AB. Society of Arts.

p. Society of Biblical Archaeology.

p. Society of Chemical Industry (London
Section).

p. Standard Weights and Measures Depart-
ment.

AB. The Queen's Library.

AB. The War Office.

AB. University College.

p. Victoria Institute.

B. Zoological Society.
Manchester.

AB. Free Library.

AB. Literary and Philosophical Society.

p. Geological Society.

AB. Owens 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.
Nottingham.

AB. Free Public Library.
Oxford.

p. Ashmolean Society.

AB. Radcliffe Library.

A. Radcliffe Observatory.
Penzance.

p. Geological Society of Cornwall.
Plymouth.

B. Marine Biological Association.
p. Plymouth Institution.

Richmond.

A. " Kew " Observatory.

England and Wales (continued).
Salford.

p. Royal Museum and Library.
Stonyhurst.

p. The College.
Swansea.

AB. Royal Institution.
Woolwich.

AB. Royal Artillery Library.
Finland.
Helsingfors.

p. Sociotas pro Fauna et Flora Fenuica.

AH. Societe des Sciences.
France.
Bordeaux.

p. Academic des Sciences.

l>. Faculte des Sciences.

p. Societe do Medecmc et de Chirurgie.

p. Societe des Sciences Physiques et

Naturelles.

Caen.

p. Soeiete Linm'emie de Normandie.
Cherbourg.

p. Societe des Sciences Naturelles.
Dijon.

p. Academic des Sciences.
Lille.

p. Faculte des Sciences.
Lyons.

AII. Academic des Sciences, Belles-Lcttreset Arts.

AB. TJniversit6.
Marseilles.

AII. Faculte des Sciences.
Montpellier.

All. Academic des Sciences et Lettres.

B. Faculte de Medecine.
Nantes.

p. Societe des Sciences Naturelles de I'Ouest

de la France.
Paris.

AR. Academic des Sciences de Flnstitnt.

p. Association Fran9aise pour I'Avancement
des Sciences.

p. Bureau des Longiiudes.

A. Bureau International des Poids et Mesures.

p. Commission des Annales des Fonts et
Chaussees.

p. Conservatoire des Arts et Metiers.

p. Cosmos (M. L'ABIIE VALETTE).

AB. Depot de la Marine.

AB. Ecole des Mines.

AH. Ecole Normale Superieure.

AB. Ecole Polytechnique.

AB. Faculte des Sciences de la Sorboime.

AB. Jardin des Plantes.

b 2

Prance (continued).
Paris (continued).
p. L'filectricien.

A. L'Observatoire.

p. Revue Scientifique (Mons. H. nu VARTGNY).

p. Societe de Biologie.

An. Societe d'Encouragemcnt pour I'lndustrie

Nationale.

An. Societe de Geographic.
p. Societe de Physique.

B. Societe Entomologique.
AB. Societe Geologique.

p. Societe Mathematique.
p. Societe Meteorologique de France.
Toulouse.

AH. Academie des Sciences.
A. Faculte des Sciences.
Germany.
Berlin.

A. Deutsche Chemischc Gesellschaft.

A. Die Sternwarte.

p. Gesellschaft fiir Krdkunde.

AB. Konigliche Preussische Akademie dcr

Wissenschaften.
A. Physikalische Gesellschaft.
Bonn.

AB. Universitat.
Bremen.

p. Naturwissenschaftlicher Verein.
Breslau.

p. Sclilesische Gesellschaft fiir Vaterlandischo

Kultur.
Brunswick.

p. Verein fiir Naturwissenschaft.
Carlsruhe. See Karlsruhe.
Charlottenburg.

A. Physikalisch-Technische Reichsanstalt.
Danzig.

AB. Naturforschende Gesellschaft.
Dresden.

p. Vereiu fiir Erdkunde.
Emdeu.

p. Naturforschende Gesellschaft.
Erlangen.

An. Physikalisch-Medicinische Societiit.
Frankfurt-am-Main.

AB. Senckenbergische Naturforschende Gesell-
schaft.

p. Zoologische Gesellschaft.
Frankfurt-am-Oder.

p. Natnrwissenschaftlicher Verein.
Freiburg-im-Breisgau.

AB. Universitat.
Giessen.

AB. Grossherzogliche Universitat.

Germany (continued).
Gorlitz.
p. Naturforschende Gesellschaft.

Gottingen.

AB. Konigliche Gesellschaft derWissenschaf ten.
Halle.

AB. Kaiserliche Leopoldino - Carolinische
Deutsche Akademie der Naturforscher.

p. Naturwissenschaftlicher Verein fiir Sach-

sen und Thllringen.
Hamburg.

p. Natnrhistorisches Museum.

AB. Naturwissenschaftlicher Verein.

Heidelberg.

p. Naturhistorisch-Medizinischer Verein.

AH. Uuiversitat.
Jena.

AB. Medicinisch-Naturwissenschaftliche Gesell-
schaft.
Karlsruhe.

A. Grossherzogliche Sternwarte.

p. Technische Hochschule.
Kiel.

p. Naturwissenschaftlicher Verein fiir
Schles wig-Holstein .

A. Sternwarte.

AB. Universitat.

Konigsberg.

AB. Konigliche Physikalisch - Okonomische

Gesellschaft.
Leipsic.

p. Annalen der Physik und Chemie.

AB. Konigliche Sachsische Gesellschaft der

Wissenschaften.
Magdeburg.

p. Naturwissenschaftlicher Verein.
Marburg.

AB. Universitat.
Munich.

AB. _K6nigliche Bayerische Akademie der
Wissenschaften .

p. Zeitschrift fiir Biologie.
MUnster.

AB. Konigliche Theologische und Philo-

sophische Akademie.
Potsdam.

A. Astrophysikalisches Observatorium.
Rostock.

AB. U niversitat.
Strasburg.

AB. Universitat.
Tubingen.

AB. Universitat.

[ xiii ]

Germany (continued).

Wtirzburg.

AB. Physikalisch-Mediciuische Gesellschaft.
Greece.

Athens.

A. National Observatory.
Holland. (See NETHERLANDS.)
Hungary.

Buda-pest.

p. Konigl. Ungarisclie Geologischo Anstalt.
AB. A Magyar Tudos Tarsasiig. Die Ungarische
Akademie der Wissenschaften.

Hermannstadt.

p. Siebenbiirgischer Verein fiiv die Natur-

wissenschaften.
Klausenburg.

AB. Az Erdelyi Muzeum. Das Sicbenbiirgische

Museum.
Schemnitz.
p. K. Ungarische Berg- und Forst-Akademie.

India.

Bombay.

AB. Elphinstone College.

p. Royal Asiatic Society (Bombay Branch).
Calcutta.

AB. Asiatic Society of Bengal.

AB. Geological Museum.

p. Great Trigonometrical Survey of India.

AB. Indian Museum.

p. The Meteorological Reporter to tlie

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.
AB. Royal Dublin Society.

AB. Royal Irish Academy.
Galway.

AN. Queen's College.

Italy.
Acireale.

p. Accademia di Scienze, Lettere ed Arti.
Bologna.

AB. Accademia delle Scienze dell' Istituto.
Catania.

AB. Accademia Gioonia di Scienze Natural!.
Florence.

p. Bibiioteca Nazionale Centrale.

AB. Museo Botanico.

p. Reale Istituto di Studi Superior!.
Genoa.

p. Societa Ligustica di Scienze Natural! i-

Geografiche.
Milan.

AB. Reale Istituto Lombardo di Scienze,
Lettere ed Arti.

AB. Societa Italiana di Scienze Natural!.
Modena.

p. Le Stazioni Sperimentali Agrarie Italianc.
Naples.

p. Societa di Naturalisti.

AR. Societa Reale, Accademia delle Scienze.

H. Stazione Zoologica (Dr. DOHRN).
Padua.

p. University.
Palermo.

A. Circolo Matematico.

AB. Consiglio di Perfezionamento (Societa di
Scienze Natural! ed Economiche).

A. Reale Osservatorio.
Pisa.

p. II Nuovo Cimento.

p. Societa Toscana di Scienze Natural!.

Rome.

p. Accademia Pontificia de' Nuovi Lincei.

p. Rassegna delle Scienze Geologiche in Italia.

A. Reale Ufficio Centrale di Meteorologia e di
Geodinamica, Collegio Romano.

AB. Reale Accademia dei Lincei.

p. R. Comitato Geologico d' Italia.

A. Specola Vaticana.

AB. Societa Italiana delle Scienze.
Siena.

p. Reale Accademia dei Fisiocritici.
Turin.

p. Laboratorio di Fisiologia.

AB. Reale Accademia delle Scienze.

Venice.

p. Ateneo Veueto.

AB. Reale Istituto Veneto di Scienze, Lettere
ed Arti.

Japan.

Toki6.

AD. Imperial University.
p. Asiatic Society of Japan.
Java.

Buitenzorg.

p. Jardin Botanique.
Luxembourg.
Luxembourg.
p. Societe dcs Sciences Naturelles.

Malta.

p. Public Library.
Mauritius,

p. Royal Society of Arts and Sciences.

Netherlands.

Amsterdam.

AB. Koninkli jke Akadcmie van Wetcnschappcn.
p. K. Zoologiscli Gcnootschap ' Natura Artis

Magistra.'
Delft.

p. Ecole Polyteclinique.
Haarlem.

AB. Hollandsche Maatscliappij der Weten-

schappcn.
p. Musee Teyler.
Leyden.

AB. University.
Rotterdam.

Ali. Bataafsch Genootscliap der Proefonder-

vindelijke Wijsbegeerte.
Utrecht.

AB. Provinciaal Genootscliap van Kunsten en

Wetenschappen.
New Brunswick.
St. John.

p. Natural History Society.
New Zealand.
Wellington.

AB. New Zealand Institute.
Norway.
Bergen.

AB. Bergenske Museum.
Christiania.

AB. Kongelige Norske Frederiks Universitet.
Tromsoe.

p. Museum.
Trondhjem.

AB. Kongelige Norske Videuskabers Selskab.
Nova Scotia.
Halifax.

p. Nova Scotian Institute of Science.
Windsor.
p. King's College Library.

Portugal.

Coimbra.

AB. Universidade.
Lisbon.

AB. Academia Real das Sciencias.

p. 8009:10 dos Trabal hos Geologicos de Portugal
Oporto.

p. Annaes de Sciencias Naturaos.
Russia.
Dorpat.

AB. Universite.
Irkutsk.

p. Societe Imperiale Russc de Geographic

(Section de la Siberio Orientale).
Kazan.

AB. Imporatorsky Kazansky Universitet.

p. Societe Physico-Mathematique.
Kharkoff.

p. Section Medicale de la Societe des Sciences
Experimeutales, Univcrsite de Kharkow.
Kieff.

p. Societe des Naturalistes.
Moscow.

AB. Le Musee Public.

B. Societe Imperiale des Naturalistes.
Odessa.

p. Societe des Naturalistes de la Nouvelle-

Russie.
Pulkowa.

A. Nikolai Hanpt-Sternwarte.
St. Petersburg.

AB. Academic Imperiale des Sciences.

B. Archives des Sciences Biologiques.
AB. Comite Geologique.

p. Compass Observatory.
A. Observatoire Physique Central.
Scotland.
Aberdeen.

AB. University.
Edinburgh.

p. Geological Society.

p. Royal College of Physicians (Research

Laboratory).

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

AB. Mitchell Free Library.
p. Natural History Society
p. Philosophical Society
Servia.
Belgrade.
p. Academie Royale de Serbie.

[ XV ]

Sicily. (See ITALY.)
Spain.

Cadiz.

A. Institute y Observatorio de Marina de San

Fernando.
Madi-id.

p. Comision del Mapa Geoldgico de Espana,

AB. Real Aeademia de Ciencias.
Sweden.
Gottenburg.

AB. Kongl . Vetenskaps och Vitterhets Samhalle.
Lund.

AB. Universitet.
Stockholm.

A. Acta Matliematica.

AB. Kongliga Svenska Vetenskaps-Akademie.

AB. Sveriges Geologiska Undersokning. •
Upsala.

AB. Universitet.
Switzerland.
Basel.

p. Natnrforschende Gesellschaft.
Bern.

AB. Allg. Schweizerische Gesellschaft.

p. Naturforschende Gesellschaft.
Geneva.

AB. Societe de Physique et d'Histoire Natnrelle.

AH. Institut National Gencvois.
Lausanne.

p. Societe Vaudoise des Sciences Naturelles.
Neuchatel.

p. Societe des Sciences Naturelles.
Zurich.

AB. Das Schweizerische Polytechnikum.

p. Natnrforschende Gesellschaft.

p. Sternwarte.
Tasmania.
Hobart.

p. Royal Society of Tasmania.
United States.
Albany.

AB. New York State Library.
Annapolis.

AB. Naval Academy.
Austin.

p. Texas Academy of Sciences.
Baltimore.

AB. Johns Hopkins University.
Berkeley.

p. University of California.

United States (continued).
Boston.

AB. American Academy of Sciences.
B. Boston Society of Natural History.

A. Technological Institute.
Brooktyn.

AB, Brooklyn Library.
Cambridge.

AB. Harvard University.

B. Museum of Comparative Zoology.

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.

p. Field Columbian Museum.

p. Journal of Comparative Neurology.
Davenport (Iowa).

p. Academy of Natural Sciences.

Ithaca (N.Y.).

p. Physical Review (Cornell University).
Madison.

p. Wisconsin Academy of Sciences.
Mount Hamilton (California).

A. Lick Observatory.
New Haven (Conn.).

AB. American Journal of Science.

AB. Connecticut Academy of Arts and Sciences.
New York.

p. American Geographical Society.

p. American Mathematical Society.

p. American Museum of Natural History.

p. New York Academy of Sciences.

p. New York Medical Journal.

p. School of Mines, Columbia College.
Philadelphia.

AB. Academy of Natural Sciences.

AF,. American Philosophical Society.

p. Franklin Institute.

p. Wagner Free Institute of Science.
Rochester (N.Y.).

p. Academy of Science.
St. Louis.

p. Academy of Science.
Salem (Mass.).

p. American Association for the Advance-
ment of Science.

AB. Essex Institute.
San Francisco.

AB. California Academy of Sciences,

r xvi J

United States (continued).
Washington.

AB. Patent Office.

AB. Smithsonian Institution.

AB. United States Coast Survey.

B. United States Commission of Fish and

Fisheries.
AB. United States Geological Survey.

United States (continued).
Washington (continued).

An. United States Naval Observatory.

p. United States Department of Agriculture.

A. United States Department of Agriculture

(Weather Bureau).
West Point (N.Y.)

AB. United States Military Academy.

PHILOSOPHICAL TRANSACTIONS.

I. Memoir on the Integration of Partial Differential Equations of the Second Order in
Three Independent Variables ivhen an Intermediary Integral does not exist in general.

By A. R, FOKSYTH, F.R.S., Sadler ion Professor in the. University of Cambridge.

Received November 23, — Read December 16, 1897.

THE general feature of most methods for the integration of partial differential
equations in two independent variables is, in some form or other, the construction
of a set of subsidiary equations in only a single independent variable ; and this
applies to all orders. In particular, for the first order in any number of variables
(not merely in two), the subsidiary system is a set of ordinary equations in a single
independent variable, containing as many equations as dependent variables to be
determined by that subsidiary system. For equations of the second order which
possess an intermediary integral, the best methods (that is, the most effective as giving
tests of existence) are those of BOOLE, modified and developed by IMSCHENETSKY, and
that of GOURSAT, initially based upon the theory of characteristics, but subsequently
brought into the form of Jacobian systems of simultaneous partial equations of the
first order. These methods are exceptions to the foregoing general statement. But
for equations of the second order or of higher orders, which involve two independent
variables and in no case possess an intermediary integral, the most general methods
are that of AMPERE and that of DARBOUX, with such modifications and reconstruction as
have been introduced by other writers ; and though in these developments partial dif-
ferential equations of the first order are introduced, stijl initially the subsidiary system
is in effect a system with one independent variable expressed and the other, suppressed
during the integration, playing a parametric part. In other words, the subsidiary
system practically has one independent variable fewer than the original equation.

In another paper* I have given a method for dealing with partial differential
equations of the second order in three variables when they possess an intermediary
integral ; and references will there be found to other writers upon the subject. My
aim in the present paper has been to obtain a method for partial differential equations
of the second order in three variables when, in general, they possess no intermediary
integral. The natural generalisation of the idea in DARBOUX'S method has been

* " Partial Differential Equations of the Second Order, involving Three Independent Variables and
possessing an Intermediary Integral," Camb. Phil. Trans., vol. xvi., 1898, pp. 191-218.

VOL. CXCI. — A. B 12.4.98

2 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

adopted, viz., the construction of subsidiary equations in which the number of
expressed independent variables is less by unity than the number in the original
equation; consequently the number is two. The subsidiary equations thus are a
set of simultaneous partial differential equations in two independent variables and
a number of dependent variables.

It then appears that such differential equations of the second order having no
intermediary integral divide themselves into two classes, discriminated by- the
distinction that, for the one class, what I have called the characteristic invariant, viz. :

_3F , 3F , 03F 3F 3F 3F
w2 r- -f pn -5- + q~ -57 — p 3 -- 2 &2 + a" = °>

1 3« L L 3/t 1 cb ± ay * of tic

can, qua function of p and q, be resolved into two linear equations, and. for the other
class, the characteristic invariant is irreducible.

The first section of this paper is devoted to the general theory of equations of the
second order, so as to construct systems of equations subsidiary to the integration ;
occasional paragraphs in the other sections develop the general theory in connection
with particular types of equations. The second section is devoted to the integration
of equations whose characteristic invariant is reducible : a method is devised whereby
the integration can be effected in those cases where the integral can be expressed in a
finite form without partial quadratures ; and various examples are given in elucidation
of detailed processes. The third section is devoted to the integration of equations
whose characteristic invariant is irreducible ; and the method is applied with
considerable detail to some of the equations that are important in mathematical
physics.

It should be added that the case of three independent variables has been selected
for detailed treatment, as being that of complexity next greater than the case of two
independent variables, the general theory of which is fairly complete. An inspection
of the results, as well as of the processes, will make it manifest that, for many of
them, generalisation to the case of n independent variables is immediate.*

SECTION I.
General Theory, -

1. Let the number of independent variables be three, and denote them by x, y, z.
Denote the dependent variable by •?;, and write

dv

•»- = n,

oz

dx

;>™2

= 1,
— ",

dv
dy

;>„.<!

= m,
= .&,

--''

* See a note at the end of the paper.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 3

Let any general differential equation of the second order be taken in the form
F (v, x, y, z, I, m, n, a, b, c, f, g, h) = 0.

When the proper value of v is substituted, this becomes an identity, so that, when
differentiated with regard to x, y, z, in succession, the results are identities. Hence,

writing

F, =

8F 8F_

, 9F

cv

, 3F 3F
; = -^ + -, n

or
y

8F
''SI

3F

' di

3F

9F 3F

^ + 9

8F

dm

,.8F

dm

en
,.BF

8F

have

... + F,r- = o

J

By CAUCHY'S theorem, a solution of F = 0 exists, determined by the values of r
and one of its derivatives, assigned for a relation between x, y, z. This implies that,
at all points on the surface represented by the relation, the values of v and, say, 8v/dx
are given ; and consequently the values of dv/dy and 80/82 are known at all points
on the surface.

Taking now v, I, m, n, as known on the surface, and denoting by p, q, the derivatives
of 2 with regard to x, y, along the surface, we have

dl = adx + hdy + gdz = (a + pg) dx -f- (h + qy) dy,

dm = hdx + bdy -\-fdz = (h + pf) dx -\- (b + qf) dy,

dn r= gdx -\- fdy + cdz = (g + pc) dx + (/+ ^c) dy,

so that, as Z, m, n, are known everywhere on the surface, the quantities

dl , dl

are known along the surface. These equations require the relation

B 2

4 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

dl dn __ tlm_ . dn

so that they determine five of the quantities a, b, c, /, g, h, in terms of the

remaining one, say

dn
J == </c»

dm dn „

dy *dy

_ dn

dl_ rf«, 2

rfwi rf«-

<M <"'/'

r?// f/a;

We also have

F (v, x, y, z, I, m, n, a, b, c, f, g, h) = 0,

so that, in general, there are six equations to determine the six derivatives of the
second order ; and if F is algebraical in a, b, c,f, yt h, there will be a limited number
of sets of values of these quantities, which can therefore be regarded as known along
the surface.

In the same way, the derivatives of higher order can be deduced everywhere on
the surface, and so, taking any point as an initial point, we have the values of all the
derivatives of v known there ; we then have a series in powers ofx — x0, y — y0, z — z0,
which, in CAUCHY'S theorem, is proved a converging series when oc0y^0 is an ordinary
point in space for the equation : and consequently we infer the existence of the
solution as established by CAUCHY'S theorem.*

This conclusion is justified only, however, if the equations do actually determine
sets of values of a, b, c,f, y, k. In the case where sets of values are not determined,
so that, e.g., the equation F = 0 becomes evanescent on the substitution of the values

* The most general form of the theorem may be stated as follows : —

If 0 (*, y, z) = 0 be an ordinary relation for the equation F = 0, that is, if it is not a solution of the
characteristic invariant equation, then a solution v of the equation F = 0 exists satisfying the
conditions : —

(i) v is equal to a given arbitrary function of x, y, e, everywhere along the surface 0 = 0;

(n) one of the derivatives of v is equal to a given arbitrary function of .r, y, z, everywhere along

the same surface ;
and a solution satisfying these conditions is uniquely determined by them.

See also a paper, Proc. Lond. Math. Soc., vol. xxix, 1898, pp. 5-13, in particular, p. 12.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 5

of a, 6, /, g, h, the preceding inference is not justified. Manifestly one such condition

will be

03F 3F _3F 3F 8F . 3F

but this is, of course, only one among a number of equations.

The method, practically contained in CAUCHY'S theorem, leads to a result only in
an infinite form ; moreover, it takes no account of the alternative when conditions are
not satisfied. We proceed, accordingly, to give another method, suggested by the
corresponding investigation (due to DARBOUX) for equations of the second order in
two variables.

2. The principle underlying DARBOUX'S method is, in effect, similar to that which
underlies AMPERE'S ; but as DARBOUX'S method most easily admits of application to
equations in which the derivatives of highest order occur linearly, it is customary to
form derivatives of a given equation, in order to secure that the equations discussed
shall possess this property. If however the equation be already in a linear form, a
mere generalisation of AMPERE'S method can first be tried, for the number of
subsidiary equations is considerably smaller than in the other method, and conse-
quently the integrations (if they can be performed) are correspondingly easier.* It
will be sufficient for the present purpose to consider a particular example, say

When we substitute for b, f, g, h, we find

dm , , , . dn , dn dm . , N

^+(l»-ff-l)^ + S-S + (fli-M + fir-Jp)o-0;

the equation now must not determine the value of' c, and so we must have

dm . -,\dn dn dm

_ + (p_g.. !)_ + --- =

and there is also the identical condition

dm dn dl dn

_ fy\ _ ___ ^ _ I *V _ - . - I I

da: " dy dy T-dy.

From the first of these, it follows that either

p — g = 0,
or

* A discussion of the subsidiary system, and of the relation of its integral to the solution of the
original equation, will be found later, in §§ 27-29.

6 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

When p - q = 0, then z is a function of x + y. Also the second equation becomes

rf / \ "• i \

- (m — n) = -~(m — n),
dy^ A» •

so that m — n is a function of x + y. We infer, from the general considerations

adduced, that we may take

,n — n = <& (x + y, z),

where * is an arbitrary function.

When q 4 1 = 0, then y + z is a function of x. Also the second equation is

dm dm dn dn __

" '

which, by means of the identical condition, can be transformed to

dl dm_
-dy+dy-0'

so that I — in is a function of x, and the corresponding inference is that

I - m = ® (x,y + z),

where 0 is an arbitrary function.

Each of these is an intermediary integral, and the integration can be completed.*
3. Passing now to the generalisation of DARBOUX'S method, we change the variables

from x, y, z, to x, y, u, where u is a function of x, y, z, as yet undetermined, so that

also z is a function of x, y, u, not yet determined. For the consequent variations of

2 when x, y, u, vary, write

dz dz

and when x, y, u, are the variables, denote the variations of the other quantities by d.
Thus

dv
dx

= 1 + np,

dv
dy

= m + n<}>

dv
du

dz

= n— ,
du

dl

dl

dl

dz

dx

= a + gp,

dy

= h+ fjq,

du

= v^>

dm
"dx

= h + fp,

dm
dy

= 6 +Jfc

dm
du

, dz

~J "du'

dn

dn

dn

dz

dx

= g + cp,

dn

= / + cy,

du

du '

The example is discussed (and the solution completed) in another connection in § 9.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.
Derivatives of the third order will be required. Write

8V

7o =

* = 8/8^ '

80 =

a**

8\-

a., =

"»= ^ '

so that

|- /30 cZ?/ -f- «t ofe
cZ/i = ft cZx + y0 dy + ft cZz
cZgr = «j cZa1 + /3j dy + «., 02;
rZZ> = 7o J.T + 80 dy + 7l <fe
cZ/ = /3j cZa; + 7l cZ# + y83 1?2
cZc = a.2 dx -\- yS2 dy + a3 cZs

Now, from the equation F = 0, we have, on supposing a proper value of v
substituted, an identity, which thus admits of being differentiated and leading to
identities. Thus

d¥ 3F 8F SF 3F 3F

cTT + 3^ "« + 0A A) + 07 a' + si- y« + a/

or say

X + A«0 + H^ + Gai + Byo + F^8,
and also

Y + AA, + Hyo + G/S! + B80 + F7l + Cft, = 0

Z + A«i + H& + G«2 + Bri + V/3, + Ca3 = 0

+ a- a'-' = Ol

C« = 0

Consider the forms taken by these equations when the new variables arc used.
We have

dc

= a

rf/i. _ ('/'
rfy "~ *^rt "1"' f;,4

ds
1 du '

dff o df/
dy dn

dz

db s db

dy du

dr.

~— y\

1 da

df a df

— = y, + p.ifl. -,—
dy dot

dc dc
dy 3 du

dz

= «o , •

3 du

6

PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

The following relations, free from derivatives of the third order and not involving
derivatives with regard to u, subsist among the derivatives of a, b, c, f, g, h, viz. :

dh da
dx dy

dq

db dh _ df df I

d£-^ = *P-0rf = P;£-2±: >•

df dij _
dx ~dy ~ P'2+

Further, from those equations, we have

dy

dr, dc

— P i " — 1 T"
(HI ax

' '1
<i <1

p Ida dh

dh,
dx

(hi

pq

A, = -

'1

p dx

dh
dx'

pq (/// '

7o = —

1

7

db dk

Other expressions are obtainable, but they are equivalent to this set in virtue of
the three relations above given. We take the value" of ft to be

dh dz

du ' du

Substituting these values in the three equations, we find
_ A — 4- X -1- A — -- A £ fda *V • ~ dh

: _ i n J _ "" n

«y+ ' i Ty = Q>

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

_ 0 A + Y + H f + B f - B -M? - *) + F -I./* _ ^
1 p */ <fy p \dx dy! p \tbt dy/

: f

n da;

o i 7 i A \ i r^ •' T> • \

1^1 ,~^ •" ' /. \ ^7l/ ^., / „ J-. "1~ ,- ' _7.. " ' _J.. I

where

+ c — — — c -- d;l = o

^ dr J pq dy ~

A = Apc + H^^r + Bq" — Gp — Fr/ + C ;

and in each of these ft has the above-mentioned value.

4. There are two considerations, initially distinct, but found in the course of the
argument to be concurrent, which enable us to obtain a certain set of subsidiary
equations : they correspond to the two modes of obtaining the subsidiary equations
in AMPERE'S method of solving equations of the second order in two independent
variables.

According to the first of them, we note that the new variable u is as yet limited
by no conditions ; it has hitherto remained arbitrary. Suppose it chosen so that
A = 0, that is,

+ Hjyq + Eq- - Gp - Ft/ + O = 0.

Then the term in jSj disappears from the three equations ; and these (after soint-
reductions in which A = 0 is used as well as the identical relations affecting the
derivatives of a, b. c,f, y, h) take the forms

dy

Gf 4-F^O >.
dx d

=0

dy

According to the second of the considerations indicated, we assume that the
new variable u, which has been adopted, is an argument in an arbitrary function
that occurs in the solution. Then fi^ will, through the term dh/du in its value
dh/du -r- dz/du, introduce a triple differentiation with regard to u beyond any
differentiation that occurs in the integral equations, while no one of the other terms in
any of the equations will introduce more than a corresponding double differentiation
with regard to u. Assuming the integral to be of such a form that these differen-
tiations give rise to derivatives of the arbitrary function, it follows* that ft will

* Provided always that the number of derivatives of the arbitrary function in question, as occurring
VOL. CXCI. — A. C

10 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

contain a derivative with regard to u of the arbitrary function in question, of
higher order than any other term in any of the equations. Now the equations must
be satisfied identically when the value of v is substituted in them ; hence the term in
/8j must disappear in and by itself in each case, that is, we have

the same conclusion as before. The remaining parts of the equations must also
vanish ; their forms are already given.

5. The quantities, which have to be determined for the present purpose, are
a, b, c, f, g, h, I, m, n, v, z, viz., eleven in all. They are functions of x, y, u.
Omitting those equations in which derivatives with regard to u occur, the eleven
quantities are to be functions of x and y. Constants that arise in the integration
are constant because the variation of u does not appear explicitly ; that is to say,
the constants are functions of u.

The equations for the determination of the eleven unknowns are partial differential
equations of the first order ; their aggregate is constituted as follows :

First, for the equations defining quantities, we have

dv 1 dv

— — = I -j- np, — := m -\- no,

ftX CLlI

dl dl

~r~ — a + gp, ~r = h + w.

dx dy • yy.'

dm , dm

— = ll + fp, -— :

dx dy

dn dn

- = q + Cf>. - :

dx dy

eight in all.

But there are certain relations among derivatives that must be satisfied. We
have

d /dv \ d /dv \
dy \dx j dx \dy /
that is,

or, since

dp d fdz\ d fdz\ da

j — — I — ) = ~ I — »

dy dy \dxj dx \dy J dx

we have

in the solution, is finite. If the solution is not expressible in finite terms, the inference is not neces-
sarily jnstified in the present connection ; we should then fall back upon the first of the two arguments.
An example will be found in §§ 41-43.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 11

a relation that is satisfied identically in virtue of the defining equations. As it
is deduced from two of these equations, and it is satisfied identically in virtue of
others, the inference is that the set of equations must consequently be reduced
by one -in number when only those which are independent are to be retained.
Treating the other three in the same manner, we find

dk 'dff da dff

db df dh df

do; " dx dy " dy f '

df dc dff dc

dx dj~ dy dy J

equivalent to the identities obtained in § 3. As these are deduced from the eight
defining equations, they are satisfied in virtue of those eight ; they do not constitute
any addition to the aggregate. Seven, therefore, is the number in this class.
The remainder are the equations characteristic of F = 0, viz., they are

A=0, f=0, r, = 0, £=0,

being four in all. It thus appears that the tale of independent partial differential
equations in the system is eleven, being the same as the number of quantities to be
determined.

It is to be noted — the verification is simple — that the original equation

F = 0

is an integral of these eleven simultaneous equations. Hence, for their effective
solution, other ten integrals would be required if further considerations cannot be
introduced ; but it will appear from examples that this can be done, having the effect
of appreciably shortening the process of integration.

6. One generalisation is immediately suggested by the results obtained. In solving
equations of the second order in two independent variables, the subsidiary system is
composed of a set of simultaneous equations involving one independent variable in
effect ; and the preceding investigation shows that, for equations of the second order
in three independent variables, a subsidiary system can be constructed in the form of
a set of simultaneous equations involving two independent variables in effect. It is
thus suggested — and it is easy to see that the suggestion can be established definitely
— that, for an equation of the second order involving n independent variables, a subsi-
diary system can be constructed in the form of a set of simultaneous partial differential
equations of the first order involving in effect n — 1 independent variables and a
number of dependent variables, this system being subsidiary to the integration of the
proposed equation.

c 2

12 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

7. The first of the equations, A = 0, belonging specially to the postulated equation
V = 0, can be expressed in a different form. The quantity z is regarded as a function
of x, y, u ; and p, q, denote the values of dz/dx, dz/dy, respectively, when u is con-
sidered constant. Let the equation connecting z, x, y, u. be given (or taken) in the
form

M = u (x, y, z) ;

then we have

CM 3« ou CM

=r- + p K- = 0, 3- + Q a~ = 0.

ex L 0j ay * cz

Substituting for p and q, the equation A = 0 becomes

_L H a. TJ _u r a. v — j. r •

+ rl 5- -. -- \- r> 3- -\- IT 3- -5- -f- r 5— 2 T ^ I a

ox oy \ dy / dx az oy dr \v

which, after the preceding explanations in § 4, is an equation satisfied by an argument
of an arbitrary function in the integral of the differential equation

It is not difficult to prove that this equation is invariantive for all changes of the
independent variables. For suppose them changed according to the transformations

x = £ (x, y, 2) ,
y' — rj (x, 11, z) ,
z' = £ (x, y, z) ;

and let £,, £•/,£;,... denote derivatives of f , . . . while I', m', n', a' . . . denote deri-
vatives of v with regard to the new variables. Then

I, m, n = ( &, >?.,, ^. % I', tn', n') ;
Vy> ^y fey

and

a = (<*',l>',c',S'ig',hr
h = («.', 6', C', /', g,', /,'

the terms represented by -f ... being terms involving the derivatives I', m, n', of
the first order only. If, then, the differential equation

F (a, b, c,f, y, h, I, m, n, x, ij, z) = 0
becomes

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 13

after these substitutions are made, we have

A' := A&5 + UU
H' = 2Ag^ + H

+ G (&,.- + &,) + F
and so on ; so that, if

u (x, t/, z) = u (at', y, s),

we have

'

A/ i + H' T + • • • = A r + H v - + • • •

\ 3f / of 3ij \ 9.'- / oa; o//

on substitution and collection of terms. The equation may therefore be called the
characteristic, invariant of the original differential equation.

8. A method for integrating partial differential equations of the second order,
when they possess an intermediary integral, has been given by me elsewhere ; its
aim is the actual derivation of the intermediary integral which, being of the first
order, can be regarded as soluble. The preceding method makes no assumption as
to the existence of an intermediary integral, and indeed is entirely independent of
that existence ; so that it can be applied not merely to that former class, but also
to equations that do not satisfy the preliminary conditions for the possession of an
intermediary integral.

SECTION II.
Equations having a Resoluble Characteristic Invariant.

9. As a first example (which, it will be seen, possesses intermediary integrals),

consider the equation

b=j-g+h.
The characteristic equation is

— pq + <f — p + q = 0,
which can be resolved into the two equations

The other three equations, deduced as in §§ 3, 4, are easily found to be

tlh , dh . dy , \ dy r

1_ p- — ^~ (P — 1 — *•) T~ ^

dx dy dx dy

dx dy do: dy

df df , dc ,x'fc

~"4-- — h K(P "•" <7 ~" U T" = *-

dx dy dx ay

14 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF
and we have the relations of identity, viz. ,

dh da dg dg

dx dy dy dx

db dh df df

dx dy * dy dx

df dg dc dc_

dx dy dy dx
Take first the form

showing that z is any function of x + ,'/• The three deduced equations then become

so that h — g, b — f,c — f, are functions of x -\- y.
Hence, as, by the original equation, we must have

we take, as integrals of the differential equations of the present type,

h - g = Ft (x + y, z),

where F1 and F2 are arbitrary. Hence also

~- (m — n) = h — g =

(fYl — Ttf\ — b — f —

oy

a ,

y, z),

Fx (a; + y, 2),
Fj (x + y, z),

It therefore follows that

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER,
and consequently there exists a function, say 4> (x -f y, z), such that

15

and therefore we have

m — n = <E> (cc + y, z),

where * is an arbitrary function. We might proceed from this equation to the
primitive.

Next, take the relation

2 + 1 = 0,

deduced from the characteristic ; this shows that z + y is a function of x. The three
deduced equations now are

dh ! dh dy dg

— _L db _L ^ _L 4£

df df dc dc

dx dy dx dy

=

J

and inserting the value q = — 1 in the three relations of identity, they become

dh da dg dg ~\

dx dy dy dx

. nA >

dx dy " P dy ilr, ''

Combining these, we find

df dg dc dc

dx dy dy dx

dh da^
dy dy

db dh^
Ty " 'dy

dy dy
so that h — a, b — h, f — g, are functions of x. Hence, as we must have

f- g = b- h
by the original equation we take for an integral equation, as in the former case,

a — h — G2 (x, y + z),

16 PROFESSOR A. B. FORSYTE ON THE INTEGRATION OF

where G! and G2 are arbitrary. But

a- (m — 1) = l> — a = — G2 (x, y + z),
ox

r*

y- (TO — l) = b — h= Gj (x, y -f- z),
^-(m-f) = f-g= G, (x, y -1- z),

P*,, \ /.' V? I \ * J

so that we have

3,r ?y 3.t

consequently there exists a function, say — B (x, y + z), such that

8B d®

U2= +

and we then have

,

(jr., = -t- ri — -^

da; r)//

/ — m — B (x, y + s).

We may proceed from this equation to the primitive.
As two distinct intermediary integrals, viz.:

/ — m = B (x, y + 2),
w, — )( = <!> (.r -(- ?/, 2),

have been obtained, it is worth noticing that they can be treated simultaneously, for
they verify identically the JACOBI-POISSON condition of coexistence. If we introduce
two new functions, 0 and $, defined by the equations

30 M

9 (£•*)= -•

so that 0 and <£ are arbitrary functions, then the simultaneous integral is easily
obtained, say by MAYER'S method,* in the form

v = 6 (x +y,z) + <t>(x,y + z),

which is the general primitive.

10. "We next proceed to an example in which the given equation does not possess
an intermediary integral. It is not difficult to construct differential equations

* Math. Ann., vol. 5 (1872), pp. 460-466: see also my ' Theory of Differential Equations,' Part I.,
§§ 41-43.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 17

of the second order in three independent variables, possessing a general primitive
involving a couple of arbitrary functions of two arguments, but not possessing an
intermediary integral.
Let

8^ <£ (u, v) = & L, gM gy e£ (M> v) = <£12) ^ <£ («, r) = £,,,

and so on for higher derivatives and for other functions. Take two functions

and consider the equation

v = $ + p<l>i + (r<j>, + e + X0, + p.0,,

as one from which <£ and # are to be eliminated by means of derivatives of order not
higher than two ; the quantities p, cr, X, p, being denned as

We have

Cfr X = ax

l>\y + c'^, /A = a'a;

For second derivatives, it is unnecessary to form expressions for l> and c, for the
latter gives the only equation which contains <£222, and the former the only one which
contains #22i, so that, when elimination is to be performed, these equations would be
ignored. We therefore take

+ er«h13 + (1 + 2a) 8n
k = 1 « & + a' + & >

y) 0n + («'

0-^122

+ 0' + y) 013

VOL. CXCI. — A.

18 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

What is required is the elimination of the functional forms from these equations,
if this be possible.

Manifestly all the third derivatives of <£ and 0 disappear in the combination
a + /— g — h ; in fact,

If, by means of the expressions for £, w, », it be possible to eliminate 6 and (/>, we
must have a relation of the form

« +/- 9 ~ h + & + Vn + & = °>

where £, r;, £, do not involve 6 or <£.

In order that the terms in ^ and <£3 may disappear, we have

'rt + (1 + c',) g= 0;

that those in <j>n, ^>]3, <^23, may disappear, we have

= 0, - (a\ -- 6'j) + <?£ = 0,

that those in 0L and 0.2 may disappear, we have

(1 + a) f + #, + (1 + y) ^ = 0, «'| + (l + ^8') , + y'^ = 0 ;

and, finally, that those in 0n, 012, ^22, may disappear, we have

- y + M?+ £) = o, - (*' - y') + ^ = 0,

- (« - r) + («' - y') + /* (f + £) + M = o.

These equations are to be satisfied simultaneously.

Using the last set of three, we have, on substituting in the third from the first and
second,

and similarly, from the first set of three,

(p + cr
We accordingly take

DIFFERENTIAL EQUATIONS OP THE SECOND ORDER. 19

With this valuo, the first set gives

P
that is,

(a\ - &',) (arx + Ifl + c,2) = (at - 6j) (a\x + b\y + c\~).

Since x, y, z, are independent variables, this can be satisfied only if

a\ l\ c\

"i^^i" >Say'
so that

a- = kp,

Similarly, the second set gives

a — 7 a' — 7'

, — ,

X ^

leading to

«' /^ 7'

« = > •= v = K> sa>''

so that

^ = K\.

The first set of equations can now be replaced by

f + r) + C = 0,

«! — 6j — /D^ == 0,

f + 17 + «L£ + &ti? + c^ = 0,

^ + * K£ + 6,17 + cT0 = o.

Hence k = — 1, and so cr — — p ; also

so that

g <L_ _r_

Similarly, by the other set, we find «= — '], and so /x, = — X ; and

As the values of ^, •>;, £, must be the same in the two determinations, and as the
variables x, y, z, are independent, we have

D2

20 PROFESSOR A. B. FORSYTE ON THE INTEGRATION OF

whence

p = p\, ft = a + |, y = a + ~ ,
so that

A = a (.x + y + z) + ly + 2^ 2.

It is easy to verify that, when p is distinct from 0 and co , there is no intermediary
integral, that is, no relation between /, m, n, involving only one of the arbitrary
functions 0 and <. We have

and then the differential equation is

It has no intermediary integral. Its primitive is

v = <(> + e -i- \ {p (^ - ^) + ^ -- ^},

where

^=^ + ^2), 5 = (9(a; + 2,?/);

and X has the above value, and p is neither 0 nor oo .

11. Now take a particular case, so as to illustrate the method of integration.
Let

a — 0, p — 1 ;
then

* = i(yt*).

The difierential equation is

/, , , 11 — m — n

a + f.-c,-h+ =0;

U T ~

and it is required to obtain the primitive

" = 4> + 0+i(2/ + *)W>1-<fe + 01-02),
where

^ = ^ (a; + y, z), 6 = 6 (x + z, y).

For the differential equation thus postulated, the characteristic equation is

p2-pq+p-q = 0j

that is,

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 21

We therefore have two solutions. The first is

p -q = 0,
leading to

z = function of x + y ;
the second is

P + 1 • = 0,
leading to

y — function of x + z.

The other three equations, particular to the equation under consideration, are
da^ dg_ dh_ dy <kl . d<l , 2a — h — g

dx r dx dx ' *' dy "~ dx T ify + y + s = ".

^ _ » ^ - — ' « #. ._ (1L , 4£ , 2A - 5 - / _ 2/ - m - M

die -^ rfo dx, r * dy dc ' ~ dy ' y + z (y + z)- '

dx dx d.v •* dy dx di/ i/ + z (// + ;p)3

that is,

' * ^ ' ~^T^ =°'

2h. -I - f 21 - m — '„

dy)

m

, ^ tf = °'

Taking first the case

p + 1 - 0,
we have

But

2« _ h - ff - 2g +/+ c = ( j£ + p ^) (21 - m - n) = ^ (2l - m - n),

and

d 1

dxy.+ z~ (y + 0)3 (y + z)* '

so that

d . , ,. d /2l — m — n\

-7- (a — h — g + /) + -T- I - - ) s= 0.

fte v ^ \ y + z I

We thus recover the differential equation, which is an integral of the system ; the

22 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

arbitrary function, which would arise through the integration, is definite : we have,

in fact,

21 - m-n
a-h-r/ + f + -— —- =0.

Using this integral, the second equation becomes (on the elimination of 21 — m — n)

d .. , . a + //—& — a
- (h — 0)4-- ~ = 0.

dy v y + z

Combining this with the first equation, we have

d . , a — 111 + I

— (a - 2/i + 1) + - = 0,

dx v y + z

so that, as p -f- 1 = 0, we have

dx \ y + .-:

Since p -{-1 = 0 implies that z -\- x is SL function of y, we infer that

n — 2h + 1) , „ „

- = arb. in. ol 11,

y + -

when

-/ -{- x = arb. fn. of y ;

and consequently an integral that can be associated with the original equation is
given by

a - Ik + 1,

y + "•

, y),

where 6 is an arbitrary function.

Taking next the case p = q, the alternative that arises out of the characteristic
equation, we have the three other equations the same as before. It is now necessary
to take account of the three equations of identity, which, when the relation p = q is
used, are of the form

dh da /dg dy \

dx dy ' ~ P \dy dx J '

dx dy ~ " \dy dx I '

df da Idc dc

- — — — = n I

da; dy f \dy dx

so that, eliminating the terms in p from the three equations, we have

DIFFERENTIAL EQUATIONS OP THE SECOND ORDER. 23

d , . d . 2a — h — ri

— (a — (J) — (a — (!) + =0

d,c v dy v y + ^

_rf. /i ^\ tl ,1, f\ , 2/t-i-/ 2l-m-n

Eliminating the term in 21 — m — n from the second and the third of thesu by
means of the original differential equation, we obtain the modified equations in the
forms

A (h _/} _ A. (h _f} + 1± l=Ar^ = 0

clx v dy^ J ' y + z

d , . d , •. . n — k + f/ — c

— (a — c) -- :- (a — c) + - = 0.

dx vy dy w y + s

By the first of the former and the second of the latter, we find

Now, as j> = f/, 2 behaves like a constant under the operation c//<7^ — djdtj ; hence
we have

77 ; ) l ~ i = ® '

when

^ - ^ , - 0

7 » / * "•

«« «y /

Consequently

a — -2;/ H- c , .. „

— = arb. in. of x + ?/,

when

2 = arb. fn. of x -\- y;

and we therefore infer that

a — '2(i + c

•y +

. , >

= <h(jc -{• 11, z)

is an integral that can be associated with the original differentia] equation, <l> being
an arbitrary function.

In order to proceed to the primitive, we take fii'st

a — '2y + c = (y + z)tl> (x + y, z),
and introduce a new arbitrary function f, defined by

9 =Jm ~ 3/i u + 3/122 ~~ Jw-2>

24 PROFESSOR A. R. FORSYTE ON THK INTEGRATION OF

so that

\&c 9z / ' ]1

where / is an arbitrary function of x + y, «• Hence

3* 92

where G is arbitrary so far as this equation is concerned.

Similarly, introducing a new arbitrary function g, defined by

the equation

a - 2k + b =

leads to the equation

a - 2ft + I - ( *- - j-
\cte oy

Hence

r/,2) + ^ - <j, + F (x + y, z),

where F is arbitrary so far as this equation is concerned.

In order that the two equations, giving the values of I — n and I — m respectively,
may coexist, they must satisfy the PoissoN-JAOOBi condition (U, V) = 0, which,
when developed, gives

/1-/2+G + S'i-5'8 + F = 0;

so that, taking account of the arbitrary character of the functions, we have

F=-C/i-/l). G a -(ft -ft),

Thus

I - n = (y + z) (fn - 2/13
l-m = (y + z) (gu - 2gn

It is easy to verify, not merely that these equations coexist, but also that each of
them satisfies the differential equation ; but neither is an intermediary integral in
the customary sense, for each of them includes two arbitrary functions of two
arguments.

The equations are of the first order ; it is easy to obtain the primitive in the form

v = 2/+ 2<j + (y + z)(fl -f, + (Jl - ft),

where/j =/(# -f y, z) and g = y (x + z, y) are arbitrary functions, and flt fz, yL, y.,,
are their respective first derivatives.

DIFFERENTIAL EQUATIONS OP THE SECOND ORDER. 25

1 2. It will be noticed that in these examples the equation A = 0 (which is of the
second degree in p and q) is resoluble, so that it can be replaced by two linear
equations, and that the latter have, in turn, been combined with the other equations
of the system. Now, these equations are of LAGBANGE'S linear form, and their
integral is such that some combination 6 of variables can be an arbitrary function
of some other combination ^. Further, it has appeared that the integral of the
subsidiary system (other than the original equation) is such as to make some
combination i/> of the variable quantities a fxinctional combination of 6 or <f) at the
same time that 0 and <£ are functionally related, so that, as the functional forms are
arbitrary, we infer that

t = v(0, #,

where V is arbitrary, is an equation that can coexist with the original equation.
Hence it is to be inferred that when A = 0 is a resoluble equation, that is, can be
resolved into two equations linear in p and q, arbitrary functions of two arguments
occur in the most general integral equivalent of the original equation,

13. The converse also is true, viz., if an integral relation involve at least one
arbitrary function of a couple of distinct, arguments and be equivalent to a partial
differential equation of the second order, and not to an equation of order lower than
the second freed from arbitrary functional forms, then the, characteristic invariant
equation can be resolved into two linear equations. (The number of independent
variables is, of course, presumed to be three.)

Let £ and 77 be two independent functions of x, ?/, z, so that not more than one of
the three quantities

i^y — *).,£;„ %,n-, — &,„ &>?.r ~ "»?.:£>•,

can vanish. As regards the arbitrary function of £ and rj, let it occur in the
integral equation in the form

where <f> denotes the derivative of the arbitrary function of highest order occurring
in <s>. Then we have

a£

a = || {<£n£.3 + 2<£lzf,77.c + <£227?,3} + derivatives of <£ of lower order,

VOL. CXCI. — A. R

26 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

and so for the others. Now the integral equation is, by hypothesis, equivalent to a
partial differential equation of the second order, say to

F (a, 1), c, f, g, h, I, m, n, v, x, y, z) = 0 ;

hence when these values are substituted the equation is to be satisfied, and accord-
ingly the terms involving the various combinations of the arbitrary functions must
disappear. Thus the highest power of </>n — or what in effect is the same thing, the

highest power of <£u »— — must disappear of itself, and therefore
£83F 3F , ,83F 3F , 3F 3F

or, with the former notation,

A£.3 + H&& 4-

38
But the term involving the highest power of <£]2 >.- , which will be of the same

3®

degree as the highest power of <£u „ , , must also disappear of itself; and this gives

rise to the equation

2Af^. + H (i,% + ^.,.) + G (^ + &,,) + 2B^y + F (^.. -f ^,) + 2C^S - 0 ;

and likewise tlie term involving tho highest power of <f>.,a ^ must disappear,
leading to the equation

AT,.5 + HT,^ + G^T,, + Br,/ + F^, + Cr,* = 0.

From these we have

, + 1 Hr,y + J Gij.-)2 = (i H2 - AB) T,/ + 2 (i GH - £ AF) 77^ -f ;(i G3 - AC) ^

-f 1 H^ + 1 Gfr) (AT;, + 1 Ht;, + 1 Gr?,)

= (i H» - AB) ^ + (i GH - i AF) (^ + f „,) + (i G2 - AC) fas ;

so that, squaring the last, subtracting the product of the first two, reducing, and
removing the factor A, we have

(Discrt. of A) (£#.. - ^'f = 0.
Similarly, by taking modifications of the first equation in the form

or in the umn

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 27

and removing factors B and C respectively, we find

(Discrt. of A) (£,„ - f^.)J = °,
(Discrt. ofA)(^y-^)3 = 0.

Now it bus been seen that not more than one of the three quantities

can vanish. Consequently

the Discriminant of A = 0,

in other words, the equation A = 0, that is

A/3 + Hpq + B</ - Gq - F/> -1- C - 0,

can be resolved into two equations linear in ±> and (/. This establishes the
proposition.

But if <£, instead of being a function of two arguments £ and TJ, were a function
of only a single argument u, then instead of three equations we infer only one
equation of the type

AM,S + Hu.^ + BM/ + Gu,n, + Fyy'.- + C»,c = 0,

and its resolubility cannot be established. (It is, of course, not the case that it
is not resoluble in particular cases ; it is not resoluble in general.) Hence what, (It/1
equation A = 0 cannot be resolved into two equations, linear in p and q, »v infer
that the arbitrary functions which occur in the integral equivalent are functions of
only a single argument.

14. In the case when A = 0 is resoluble into two linear equations, and when the
other equations possess integrable combinations, a method can be constructed for
obtaining those combinations. Thus take the example considered in § 11, where the
deduction of the combinations is fortuitous in the sense that no indication of the kind
of combination is given. Let

8 (a, b, c,f, (j, h, I, m, n, v, x, >j, z) = 0
be an integrable combination ; that is, we must have

r-°. 'r = °-

ax ay

Either (i) one of them, or (ii) a linear cross between them, or (iii) both of them, must
be satisfied in virtue of the set of equations.

E 2

28 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OV

Let us consider, first, the case when

p + 1 = 0

so that q remains arbitrary. Evidently

£=0

<fy

will not be the equation to be satisfied ; so that the effective combination must arise
in the form

when p + I = Q. In other words, we must have the equation
30 ?-0 , 26 M fc6 cd v6 Bt)

a7 + a aT + h^i + <J& ~ (j~: + 9 a +f a- + ' ^

r^ ili_ ,c6_db_ ,dO_<hi,rO_<y_, M_ % , o0_ <lh_
+ Sa (b:. + 36 «>...- + 8c (f.« + a/ <lx + % <Jt- + 8/6 rf^! ' ' '

(where the value p = — 1 has been inserted), satisfied in virtue of the system of
subsidiary equations. The relations of identity all involve q, and therefore cannot
be useful for the purpose. Hence the above equation must be a linear combination of

da <U< i Y -

~T~ ~ T + •». = 0

ax ax

KM ax

ill/ Jf
d^"^+ /l = ° ^

where

-y -n — ^ — .'/

A. = ~

;'/ + s

-. '2h — b —f '11 — •«;. —

y + a (y + ^)

7 _ — —

y + z (y+ s)3

these being the subsidiary equations particular to the present case when the value
j-> = — 1 is inserted.

When therefore we substitute

da _ cUi_ <tt_ (U^ dj_ da

dx " dc ' ~ A> dx ~ dx + Y) dx ~ fa + A

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 29

in the equation derived from 0, the latter should become an identity ; that is, we
should have

30 30 ,od 30 [M 30 ,.30 30 \ , 30 fdli

~ = — h # E^- + / 5 — h c »- + K~ (i

on \oz y ol .am on / act \dx

30 <fo 30 30 </</ 30 ffy 30

satisfied independently of the values of derivatives of h, c, y, with regard to x. It
therefore follows that, if a function 0 of the suggested type should exist, it must

satisfy the system of equations

30_

3r" ~ °'

30 30

¥ + *7 - °'

30 30 30

and

30 30^ 30_ .. 30_ v 30 30

„, 30 „ 30 ,, 30

and the number of functionally independent solutions of these homogeneous
simultaneous equations is the number of integrable combinations of the subsidiary
system.

This system of partial differential equations of the first order must lie rendered
complete by associating with it the JACOBi-PoissON conditions. This complete
system, obtained by the regular processes, is without difficulty proved to be
equivalent to

"3/6 =r 3«" al'

, o$_ 3_0_ a0_ _ _l _ o6_ _ l_ 30 . _ /9^_ __ 30

s 3i ~" ~ 3^1 ~~ "" 3/i "~ // + ~ 3/ ~ ,y + a 3# \3« '~ db
and

30 30_ 2/ - M - n 30 2a - 3/t. + & +/-.'/ 30 _

a« "8« ' "(y + s)"3 S« ' y + s ob '

the latter being the modification of the last of the four initial equations.

The complete system thus contains nine equations : it involves thirteen variables,
viz., a, b, c,f, <j, h, I, m, n, v, x, y, z ; and consequently it possesses four functionally

30 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

independent solutions. These can be obtained, by any of the regular methods, in

the form

2a — 3h + b + f — y 21 — in — n

— —

But the last is zero, owing to the original differential equation ; and by using this
imposed restriction, the second becomes

a - ->h + I

.'/ + *

Consequently the most general solution of the system is

a -2

*

where <1> is arbitrary ; an equivalent of this is

a - -2h + I a ,

,, + , =e (- + *'> y)>

where d is arbitrary.

Similarly a new relation between derivatives of the second order can be deduced
by taking the alternative solution p — q = 0 of the characteristic equation.

The rest of the solution proceeds as before when once the system of partial
differential equations satisfied by

$ = 9- (a, b, e,f, <j, /,, I, m, n, v, x, y, z) = 0
is obtained. Now, when p = <j, the relations of identity are

ill^ da Idij dij

d,c d;/ ~~ * \di/ ili

(tl^ dh^ _ Idf df

iLv " dy ~ ^ (dy " da '

df djf _ fdc, (k\
d.v ~ dy ~~~ P \dy " dx)'

These can be used to eliminate p from the equations particular to the present case,
and the latter then become

da_ da /djL <lg\ -i

dx dy \dx~-~dy) + ^

ML M (df_ df\

dx dy "\dx " dy)^ ' U

djl dy_ /dc_
dx " dy "\dx "

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER,
which hold whatever be the value of p. But from 3 = 0 we have
33 . 33 /7 . v , 93 / , 93 n ... 33 .

+ - <* + w) + (o + a?) + (/< +fp) + (j/ + ep)

...+

dc

= o,

and

33 33

37 + 37

33

33

33

33 rfrt 33*

3ff f/?/ 3r- <•///

For our immediate purpose, p = q ; hence, subtracting, we hav

33 33 . 33

3^ - 37 + 37

33

3/

33

37

_

31

which, being free from p and q, must be satisfied in virtue of the above three
equations. This being the case, it must, when we substitute for

da da

dti dk

dc

df.

•Iff '

be satisfied independently of the values of

tlh dk

(1L .. (]L
d:>; fiii '

df/
dz

Assigning the necessary conditions, we find

-
36

3/ ' ' 3A '

33 33 33 _
3« 3</ 3c " " '

33 33 , 33

a^ ~ + 37

33

33

371

33

33

33

32 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

The integration of these equations can, as already stated, be effected in the same
way as for the preceding part of the solution of the original equation ; the most
general solution of the system is found to be

a — 2f/ + c _ ,/,, \

15. That the method just expounded is not restricted to individual instances of
equations, for which A = 0 is resoluble, can be seen as follows.
We consider, more generally, the case when

Apz + H/></ + B</2 — Gp — F<7 + C = 0

is resoluble into two linear equations. We have

(A/) + $Ilq - iG)2 = (±H* - AB) (f + (AF - £GH) q + ,}G2 - AC.

Let

1H- - AB = &\

AF - £GH = - 20ty ;
then, since

(III- - AB) (}G- - AC) = i (AF - iGH)-,
we have

iG3 - AC = 6'^,
and the equation is

that is, we have the two equations

(iH -e)q- (JG - ^) = o

where

6* = iH2 - AB
Taking the former of the two equations, viz.,

we seek to obtain combinations of the three equations of identity with the three
equations particular to the present case. The first equations of each of these sets, as
given in §§ 4, 5, are

da dJt^ dg d<j

dy ~ da; """ P rfy ~ ? ~dx ~

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 33

Multiply the latter by £H — 0, and add it to the former. In the resulting equa-
tion, the coefficient of dg/dx is

which is

the coefficient of dgjdy is

- Hp - B2 + (1H - 6) p + F = - f(iH + 0)p + Eg] + F

But this coefficient, multiplied by A,

= AF -

and therefore the terms, involving derivatives off/, are

apj

(r, U +

A

The terms involving derivatives of a are

A A^. 2 J"1 - 6 d

A ., . +

\fb: A dy

and those involving derivatives of h are

i_ m _L -R /i TT _i_ m a.

") ~T -- r & 7~ = (i-tl + ") \ ~i -- ---- A~

' dx dy ' \dx A

The equation is now in its simplest form. The other pairs may be treated in the
same way ; and thus, corresponding to the equation

we have a system of three subsidiary equations free from p and q in the form

Y + A8/i + (£H + 6) Bb + (|G + ^) 8/= 0

Z H

where

3= ^

dx A

VOL. CXCI. — A. F

PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

If

u (a, b, c,f, g, It, I, m, n, v, x, y, z) — 0

be an integrable combination of these equations, then we must have

tin da
3« dx

da

du dh

• \ i olt i

-f ?ip) + -^ (a

3« r/A 3?«

a// </// a//

.', H - e

on

£• (g
a»

= 0,

,,

(h

Multiply the latter by '- — — , and add to the former; where p and q occur in

the result, it is in the form

.' H - 0 } G - 00

and so the equation is

da

36

CK,

Cli -,

£ 8c
AH -

A

(•<> 5 , 3// » a«

57 o/ + -5- So + --T-
fV 09 oA

+

A

i G -
A

3, n

j;II- 0
A

1H — ^ 3«
-—
A oy

iQ-
A

This must be satisfied in virtue of the three preceding equations and independently
of the actual values of 8«, . . ., Bf. Substitute for S«, 86, Sc ; then the coefficients
of Sy, ?>g, 8/i, and the term independent of these must vanish. We thus obtain four
linear homogeneous partial differential equations, viz.,

A 8a A H + 0 36

3(6 I G + 00 3«. A 3»

3« A 3a -.V G + 00 oc

: a/ " ' "i H + 0 36 " " J~G + 00 ac

i H + 0 I G + 00

, 7 a« 3w ,

+ 4 x— + « x, +

dn

+

7 — p

A \ t)/

iG — 00/

"-

w , « «

+ wi XT + ^ ^7 + ^ ^— +

'

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 35

with

0 = Ap + (fcH - 0) r/ - (iG - ty).

This system of four equations must be rendered complete by constructing the
additional equations that arise out of the JACOBi-PoissoN conditions. If, when
complete, the system contains n equations, then it possesses 13 — n functionally
independent solutions. Among these must be included (i) the original differential
equation

F (a, b, c,f, g, k, I, m, n, v, x, y, z) = 0 ;

(ii) the two distinct integrals of

dx^ _ dy dz

~A ' ~ -iH -6 ~~ i-G - 6ty '
say these are f, yj.

Putting these on one side, there are thus 10 — n new functionally independent
solutions. A not uncommon case is n = 9, when there is one new solution, say u.
Then we have

u = i/, (£ 17),

where $ is an arbitrary functional form ; and this equation coexists with the original
equation

F— 0.

16. Thus far we have considered only one of the two equations into A = 0 is
resoluble. When we consider the other equation, viz..

Ap + (|H + 0) q - (1Q + 0$ = 0,

the sole difference in the general analysis is manifestly a change in the sign of 0 ; and
we therefore obtain the corresponding system of linear homogeneous partial differ-
ential equations, determining an integral combination (if any), by changing the sign
of 9 in the preceding system. The method of integration is the same as before.

It may happen that neither of these two systems possesses a solution distinct from
the differential equation. If, however, either (or both) should possess such a solution,
then n must be less than 10, and certain conditions — viz., those in order that the
system when complete should contain not more than nine equations — must be satisfied.
These are the conditions in order that one equation — or two equations, if the result
hold for both systems — of the second order involving an arbitrary function of two
arguments should be associable with the given equation.

And it should be noted that the characteristic invariant of an equation associable
with the given equation is satisfied by that linear equation in the characteristic
invariant of the given equation which is used to derive the new equation. The
result is general, and the proof of the general result is immediate.

F 2

36 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

17. Now it may happen that the simultaneous system of equations admits of no
new common solution in either case ; the inference then is that no equation of the
second order containing a single arbitrary function can be associated with, or is
compatible with, the given differential equation. But it may then be that some new
equation of the third order — new, that is, in the sense that it is not one of the
immediate derivatives of the given equation — containing an arbitrary function can be
associated with the given equation ; and this may occur with each of the linear
factors of A = 0. And so on, precisely as in DAKBOUX'S method for dealing with
partial differential equations in two independent variables ; we seek to obtain one
equation (or, it may be, two equations) of finite order which ai'e compatible with the
given equation, contain one arbitrary function, and are not mere derivatives from
that given equation.

We have been proceeding on the supposition that the equation possesses no
intermediary integral. If no other equation of finite order is compatible with
the given equation,* then the method ceases to be effective. In that case, the only
result generally attainable seems at present to be that which occurs in the establish-
ment of CATJCHY'S existence-theorem ; the integral certainly contains two arbitrary
functions, but its expression (in the form of a converging series) is not finite.

18. Suppose that the conditions for the existence of a new common solution
are satisfied for neither of the systems in §§ 15, 16, so that no new equation of the
second order, containing only a single arbitrary function, is compatible with the
given equation. We proceed to construct the system of subsidiary equations which
determine an equation (if any) of the third order containing only one arbitrary
function, and compatible with the given equation

F (a, b, c,f, g, h, I, m, n, v, x, y, z) = 0.

On account of this equation, we have three derived equations of the third order,
viz., with the former notation

X + Aa0 + H/30 + Gax + By0 + Fft + C«2 = 0"

Y + A/30 + H7o + Gft + B80 + F7l + C& = 0
Z + Aai + Hft + G«3 + B7l + F& + C«3 = 0 .

and the new equation (if any) must be compatible with these.

19. The process is an amplification of that used in § 2. When the proper value
of v is substituted in F == 0, the latter becomes an identity, so that, when it is

* A simple instance is given by

y + «

where X is a positive constant other than a.n integer.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

37

differentiated with regard to the independent variables, the results are identities. By
hypothesis, no new equation is derivable when first derivatives are formed : we
therefore form the derivatives of the second order, being six in all ; viz., they are

dot?

= 0,

-
dydz~ '

dz da;

= 0,

dx dy

= 0,
= 0,

equations which contain derivatives of v of order 4. Let these fifteen derivatives be
denoted by s1} s2, . . . , sl5, their definitions being given by the scheme

dx + dy

do.i

r

dy,

dz

rs

»o

?'io

*u

'14-

'13

12

Further, let (XX) denote the part of *y^ which is free from derivatives of the fourth

order, (XY) the corresponding part of j—^, (XZ) that of ^-^, and so on. Then the
six equations are

(XX) + Arx + Hr2 + Gr, + Br, + Frs + Cr(,
(XY) + Ar2 + Hr, + G»-B + Br7 + Fr8 + Gr,
(XZ) + AJ-S + Hr5 + GrG +Br8 + Fr0 + Cr]0

(YY) + Ar,
(YZ) + A»-8
(ZZ) + Ar6

Hr + Gr

Hr8

Hr

Gr

10

Brn
Br12

Br13

Fris +

18

Fr

Cr =

ls

0 ^

0
0
0
0
0

As before, let the variables be changed from x, y, z, to x, y, «, where « is a function of
x, y, z, as yet undetermined, whence also z is a function of x, y, u. For the

38

PROFESSOR A. R, FORSYTH ON THE INTEGRATION OF

consequent variations of z when x, y, u, vary, we write p and q for dz/dx, dz/dy,
respectively; and we adopt the notation of £ 3. The new expressions for the
variations of v, I, in, n, a, b, c, f, g, h, are given in § 3 ; those for the variations of
a0, al9 «2, «31 A,, ft, &, y0, yi, 80, are given by the equations

da}

*g

— rw ~r

'-dx = >''•> + ''^'

= » 7 + 'UP,

da.?.
~

dy

daz
dy

dy

dy

du

du

da.2
du

du
dz_
du

dz

10

du

dy

_
du

rf^

H 1

du

_ rfg

dz
fa

dz

V J

The following relations subsist among the derivatives of «0, . . . , 80, free from
derivatives of order 4 and from derivatives with regard to u, viz.,

dx

~dx.

dx

dy

da^

~dy

rlfti J- rJfti

dy
da.

dy
da,

da}

dx
da;

dx

dx

dx

dy " dy

d^ dfiz

dy •* dy

dy

dx

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

39

These six identical relations reduce the twenty equations in the two foregoing
columns to fourteen independent equations ; hence the fifteen derivatives r can be
expressed in terms of one of them, say r5, and of derivatives of «0, . . , , S0 ; and the
value of r-a is dfi^du -r- dz/clu. These expressions for the other fourteen, in terms
of rB, are

— — - ( p -. l — (I —
1 (1 \ dy dx

, dfta
2+ dy

7 /

T

r q-
5 P

l_
?5 2>

'-P

'p dx

10

_ oj _

P

dy

13

J- _U J- / ^' ^ -4- - ^

- r5 - - 3 ( ^ '- ^ rf " ' ^ tlx

15

5 2? '/• dx p dx

11 7 Q 1 ,1 Q

1 Up, 1 rtp.) J

T- —5 -f" "^ — " "~7^ +

20. When these values are substituted in the six equations, and terms are collected,
it appears that the terms in r5 in the six equations are

A A A A? A , A

respectively. For reasons which, being the same as before, need not be repeated

40 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

here, the term in r& is made to disappear from each of the equations, and thus

we have

Ap2 + Hpq + B</ - Gp - F<7 + C = 0,

the characteristic invariant. (We hence notice that this is a particular illustration of
the remark in § 16, viz., if two equations are compatible with one another, their
characteristic invariants are either the same or, being resoluble, have at least one
factor common.)

The six equations, after some reductions of an easy character, take the form

-

dx dx / \ dx dy i \dy * dy / dx Ay

dy j dx dy

For the aggregate of differential equations in the system, we have one pair

do dv

- = I + np, ~

do dv

= I + np, ~ = m + nq ;

three pairs, one of which is

dl dl

te=a + W> d,,=h + 9P->
six pairs, one of which is

du da

^==«0 + «ip, ^ = ^o + «i?;

one characteristic equation,

A = 0;
and the above six equations.

The three equations of § 18 become three integrable combinations of the above six
with the foregoing equations in the six pairs.
The number of quantities to be determined is

10 quantities . . . . «0 ..... S0 ;

6 ,, .... a, ...,/*;

. . . . /, m, n ;

1 quantity . . . . v ;

1 „ . . z;

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

41

or 2 1 in all. There are certain relations among the differential equations ; and
further, four integrable combinations of the new system are known to exist, viz.,
the initial equation F = 0 and the three equations derived from it. What is
wanted is, if existing, a new integrable combination.

21. Thus far the analysis applies whether the characteristic invariant is or is not
reducible to two linear equations. Suppose now that, as in § 15, it vanishes, in
virtue of one or other of the two equations

= 0

Ap

0)q- (iG

and consider these in turn, in the same manner as before.

Then, by combining the equations of identity with the equations particular to

F = 0, we have

(XX) + AS«0 + (HI + &) 8& + (AG + 6ty) Sat == 0 •]
(XY) + A8& + (ill + 9) 37u + (i G + 0<f>) Sft == 0 '
(XZ) + ASai + (ill + 6) S/3, + (r^G + <ty) 8«3 == 0
(YY) + ASyu + (J H + 0) 8S0 + (J-G + 0</>) 37] = 0
(YZ) + A8& + (4H + 0} S7l + (£G + 6$) S& = 0 \ ;

J

(ZZ) -f A8«3 + (^H + 0) S& + (^G + 04.) Say = 0
as a modified form of the equations for

Ap + (Ul - 0) <L - (1G - 6$) = 0 ;

and in this form
Let now

s d . %H-0 d

8 = <>., + A dy

E (a0, . . ., 80, . . ., h, I, m, n, v, x, y, ;) = 0
be an integrable combination, so that we must have

BE da SE

*•* J~J t& Wfl I «-A I

5 — V" i 2 3~ , ~r

r\*4 ft & f> ft ft ) '

( OCy tCrf- (-.[t/ tltt-

, ,, . v L

+ v (* + np) + p- + p ^, = 0,
do cs

, OPJ

S "^r~

Multiply the first by A, the second by ^H — 0, and add ; then using the equation
connecting p and q only, we find

A3 + A + (4H - 0) + (4G - W - 0,

where

VOL. CXOJ. — A.

42

Ite

D

PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

a_ 7 A • A • /. JL • , !

JB 9e 9/ • 8w 8w

+

3,3-3

^7 + " s~ -r / -r

'

en

+ n + A + §o - + n + A

a*

])

n 87

8

8.

Now this is to be satisfied in virtue of the preceding six equations and indepen-
dently of the particular values of 8«0, . . ., 880, belonging to any integral of the
original equation. Hence the equation must be expressible in the form

At {(XX) + ...}+ X, {(XY) + ...}+ X3 {(XZ) + . . .}
+ \, {(YY) +...}+ Afi {(YZ) + ...}+ A6 {(ZZ) +...},

where A1} . . ., AG, are indeterminate multipliers; the conditions sufficient and neces-
sary for this are

0) - -

- 0,

G +

(J G +

- A( J-G +

- (iG +
+ G

•37- +

o«0

- + A2 (AG
+ tf)

- - A^ - 0,

and

_8E_

8a0

t-')9f|-&
-ff- + A-g-

+ ^) ?r ~ i

C5Cj

^T^

oB

= 0,

^_ _Sl-.-0

/3J 5i> Wl

8E

. A _8E_
^

(YZ)

8E_1
8«3 J

8E

3«3

= 0.

•22. This system must be rendered complete by the addition of the JACOBI-POISSON

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, 43

conditions. If, when complete, the system contains N equations, then it possesses
23 — N functionally independent solutions. Among these are to be included—

(i) The original differential equation F = 0 ;

(ii) The three derivatives of F = 0 with regard to x, y, z, respectively ;

(iii) The two distinct integrals of

d.v _ ily <7z

A : = |H"-~0 = HI -. 00 '
say these are f, t].

Putting these on one side, there are thus 17 — N new functionally independent
solutions, so that N must be not greater than 1G in order that the method may be
effective. If, when N = 1 G, the solution is u, then

U = * (£ 7,),

where i// is arbitrary, is an equation of the third order that can be associated with
the given equation.

The same process, with corresponding results when the appropriate conditions are
satisfied, is adopted for the alternative linear equation

arising out of the reducible characteristic invariant.

23. An example in which no equation of the first order involving only one arbitrary
function, or no equation of the second order involving only one arbitrary function,
can be associated with a given equation of the second order, is furnished by

The general primitive is

• ,, = F + G + £(y + z) [F, - F, + G! - G,}

+ TV (U + *Y (Fn ~ 2Fie + F,c + Gn - 2G13 + G22},
where

F = -F(x+y,z), G = G(x+z,y),

and the subscripts 1, 2, denote derivation with respect to the first and the second of
the arguments in the respective cases. The associable equations are of the third
order at lowest ; and they are

«„ - 3& + 3y0 - S0 = (y + z) * (x + z, y),

«o - 3ai + 3a2 — as — (u + z) y (x + y> 2)«

where $ and "^ are arbitrary.

In a similar way in part, and by induction in part, it may be proved that the

integral of

, ,. . i 21 — m — n

a-h-g+f+I - y + z = 0,

G 2

44 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

where I is a positive integer, can be expressed in finite terms. To express the

integral, let

F («, 0) = F, where a = x + y, /£? = z,

G («', /3') = G, where «' = .r + «, P = y \

~\ ? P

and denote by A the operation g~ ~~ 90 > by A' tne operation ^, — ~^> , so that

8G £G A'T<-8-G 9
~" -'*"

Then the value of r is

v = F + G

+ A (y/ + z) (AF + A'G)

+ , (y -r- :)' (A'F + A"G).

24. But it is necessary to take account of Avhat has been achieved when one
equation or when two equations (say of the second order) have been obtained
compatible with the given equation and involving each one arbitrary function.
The method adopted in § 11 to pass to the primitive has manifestly no element of
generality.

Now the three equations are not sufficient to express a, 7>, c, f, g, h, in terms
of I, m, n, and the variables ; but they frequently will serve to express groups
of combinations of a, b, c, /, g, It, in terms of those quantities. Thus the three
equations in § 11 suggest combinations a — h, h — b, g — y (which are the derivatives
of I — m), and a — g, h — f, g — c (which are the derivatives of I — n). This,
however, is only a slight modification of the former method ; it, again, has no element
of generality.

Another plan would be to differentiate the three equation^ up to any order with
the hope of determining all the derivatives of the highest order that occur in
terms of derivatives of lower order. If this were possible, substitution in the
equations of differential elements such as

dl = adx + hdy + gdz

and successive integration would ultimately lead to v. It appears in general, how-
ever, that relations of interdependence among the equations prevents them from

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, 45

being adequate for the purpose at any stage; the relations are, in fact, satisfied
conditions of compatibility. This method is, therefore, ineffective.

An effective method can, however, be obtained as follows. Restricting ourselves
for the moment to the equation of the second order with two compatible equations
also of that order'" — the restriction is made only to simplify the explanations — we
have v as expressible in terms of two arbitrary functions. Hence each of the
quantities /, m, n (and therefore any combination of v, I, m, n), can be expressed
in terms of two arbitrary functions. Now in one of the compatible equations we
have one arbitrary function which is to be identified with one of the arbitrary
functions in v ; hence it is to be expected that a proper combination of v, I, m, n,
is an intermediary integral of that equation involving a new arbitrary function, which
must be identified with the other of the arbitrary functions in v.

Similarly for the other of the compatible equations, there is an intermediary
integral involving the two arbitrary functions. The conditions of coexistence of the
two intermediary integrals must be assigned ; it will appear that, if the conditions
are not satisfied identically, they provide the means of identification of the various
arbitrary functions.

It is to be observed tha.t the intermediary integral or integrals thus obtained
cannot be regarded as intermediary integrals of the original equation in the ordinary
sense of the phrase, for each of them involves two arbitrary functions. But they
are intermediary for the respective compatible equations : each of them involves one
arbitrary function more than occurs in the compatible equation. The result mani-
festly does not imply that the original equation possesses any intermediary integral ;
in fact, the assumption throughout our investigations has been that no proper
intermediary integral exists.

25. A methodt has been given elsewhere for constructing the intermediary integral.
In effect, it amounts to the use of the conditions which must be satisfied in order
that the derivatives

I- num + gna 4- ux =
hui 4- bua 4 fit,, + ?'y = 0
gut + /»„, 4- «<« 4 w- = 0

from the supposed integral

« (Z, m, n, v, .r, y, z) = 0

shall cause the compatible equation

© (a, . . . , /(., /, m, n, v, x, y, 2) = 0
to be satisfied without regard to the values of the differential coefficients of v of

* The explanations will be seen to apply, mutatis mutandis, to other cases of the second order, and
indeed to cases of any order, when compatible equations are known.
f In the memoir cited in the introductory remarks).

46 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

the second order. These conditions are the simultaneous partial differential equations
of the first order determining u.

Thus, dealing with the case of § 11, when the compatible equations of the second

order are

, . 21 — m — n

a 4- f — a — h — — - ,

II + z

a - 2g + c = (y + z}<j> (x + y, z),

we know that the first has no intermediary integral. As regards an intermediary for
the second, substituting for a, b, <?, from the derivatives of the ?<-equation, the result

- 2h - H'-k

'/ "/ '"/ / \ "„.

must, qti.d equation iny, g, h, be evanescent ; hence we find

(u,,, -\- u/)2 = 0, v,, = 0, — -' — - - = (y + z) 0 •

'I' I V'M

that is, the equations for u are

en
da

(ili

^ = 0,

(ill, cu cu , . en n

•5 -- — + (I — m] 2 - (y + z) „- 6> = 0.
c.i' c// ' (.:/• ' cm

Tlie system is a complete system ; hence it possesses four functionally independent
solutions. "Writing

^ = ,r/ni — 3gn.2 + 3f/,« — #,3.,,

these four solutions can be expressed in the form

z,x+y,
l-m-^ + z) (gn - 2gu + ry33) - ^ + ^3,

and another involving v, which would require either the arbitrary constant or the
arbitrary functional form to which the third would be equated. (The fourth is, in fact,
a primitive of the compatible equation under discussion, though it is not necessarily
the common primitive of the three simultaneous equations.) We thus infer that

} (gn - 2(/J3 + gK) - (Jl + ff, = $ (x _|_ y>
is an intermediary of the second of the equations.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 47

In a similar manner, by writing

it can be shown that

3 = X

is an intermediary of the third of the equations. When the conditions of coexistence
of these two are assigned, they determine the arbitrary functions t/> "nd x m tlie
forms

so that we have

Z - TO = (;y + 2) (</n - 2</13

I - n = (y + z) (/u - 2/ia +/«) + /; -/2 - .9l + c,,

and the primitive can be obtained by the customary process, leading to the form
v = 2

where ./'=./' (sc + y, s) and </ = r/ (,« + ~', Z/) are arbitrary functions.

26. If one or both of the equations compatible with the original equation were of
the third order, we should then seek an equation of the second order involving
one arbitrary function more than that equation of the third order ; and \ve should
proceed in a manner similar to that of the preceding plan, the conditions of coexist-
ence of the different equations furnishing the means of identification or comparison
of the arbitrary functions that occur.

If there be no equation of the third order, we should similarly proceed to obtain
possible equations of the fourth order, if any; and so on with the orders in succession.
The method is one of general application if equations of any order compatible with
the original equation exist.

SECTION III.
Equations having an irresoluUe characteristic invariant.

'27. The investigations contained in the preceding sections of this paper have
referred for the most part to those equations

F (a, b, c,f, a, h, I, w, n, v, x, y, z) = 0,
whose characteristic invariant

oaF , SF , ,?F 3F 8F 3F

r-& +MV; + (r % -**j - qy + a =

is resoluble into equations that are linear in p and q. Those contained in the present
section refer to equations whose characteristic invariant is irresoluble.

48

PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

In the first section (§ 2), a generalisation of AMPERE'S method was dealt with very
briefly, partly because that method and DARBOUX'S method apply most effectively
to equations in which (with few exceptions) the derivatives of the highest order
occur linearly ; and, of the two, it is DARBOUX'S method which can be more effectively
applied to other equations. The fact that the characteristic invariant was resoluble
proved of material importance in the general theory.

It is to be remarked, however, that some of the equations which occur most
frequently in mathematical physics, for example

8e

3,6i3 8>/3 c;'~ ot

8-V c-r 8V 0 8-V

-\ ~~; ~r~ ^ , •+" ^ T — C* ~

8.-;3 ^ ?/ tV ct- '

the latter two being, for purposes of application, made to depend upon the equation

32,. ^i,. PJ,.

"^ O I ^ ) |~ ^ o ~™ ~~ ~~* K V»

<.:,* d/ ±-

belong to the class which have their characteristic invariant not resoluble, and at the
same time are linear in the derivatives of the highest orders that occur'. Accordingly
both AMPERE'S method and DARBOUX'S method generalised can be applied to such

equations.

Moreover, the generalisation of AMPERE'S method can also be applied to equations
of the form

of

wher

e = a, h, y,
ft, (>, f,
1 ff, f, c,

and the quantities 6, A, . . ., H, AJ; . . ., Hj, U, do not involve derivatives of the
second order. For, when the equation is transformed by the relations of § 1, it takes
the form

J + cl = 0,

where I = 0 is the characteristic equation ; in other words, taking account of 1 = 0,
we must associate J = 0 with it as an equivalent to the postulated equation.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

49

28. We consequently begin with the generalisation of AMPERE'S method. Let the
variables be changed from x, y, z, to x, y, u, where u is a function of x, y, z, as yet
undetermined, so that 2 is a function of x, y, u, as yet also undetermined. With the
notation previously adopted, we have

dv

dv

dv

dz

dx

= 1 -

r np,

- = m + na,

dy

du

= n

du

dl

M -14-

dl

dz

dx

= a ~

\-9P>

'¥ "" (jq>

du

= 0

du

> :

dm
dx

-h-

VJP,

dm , ,
dy ' '"/ft

dm

= f

dsi

du

dn

rf;t

dn

dz

dx

= 9 -\

h cp,

7- = / + C<1>

dy J

du

= c

du

and therefore, from the equations involving derivatives with regard to x and y alone,
it follows that

dl dn

a = j, - P T,: + lrc>

dx

ihn dn „

b—- q ,- + q-c

dy HI/

dn

dm

-

dx

dl

so that we have

dn

7- + Pflc

dy

dn

\

dm

du (U

du

which is the condition in order that the necessary relation

dij \ do; / ~~ dx \ dy
be satisfied.

When the postulated equation of the second order is such that, on the substitution
of the foregoing values for a, b,f, y, h, it has a linear form in c, let it be

J + cT = 0.

Suppose that the variable u (or z as a function of x, y, u) is determined so that
- . 1 = 0,

which, after the earlier explanations, is the characteristic invariant ; then we
have also

J= 0.

VOL. CXCI. — A. H

50 PROFESSOR A. R, FORSYTE ON THE INTEGRATION OF

The system of equations now is

1=0, J = 0

dm dn _ dl dn

dx " dy ' ~ dy " dx
do dv

involving the quantities I, in, n, v, z, as functions of x and y. In all the derivatives
here contained, u is parametric ; and consequently all the constants that arise in the
integration are constants on this supposition : in other words, all of them are functions
of u. Consequently, when the integrals of the system are obtained, one constant (at
choice) can be taken to be u ; all the other constants are then functions of u,
arbitrary so far as the system is concerned ; and any arbitrary function of x and y
that occurs is also (possibly) a function of u. In order to determine the limitations
on the arbitrary functions, the equation

dv dz

— = n —

du du

must also be satisfied ; this equation will usually give relations among the arbitrary
functional forms, or will determine one of them.

29. The relations thus obtained constitute an integral of the equation. For
suppose that in the expression for v we consider u eliminated in favour of z ; then

7 OP , Be 7 Bv /' , 7 dz 7 \

dv = -5- occ + -5- cty + -5- » aoj + o ay + -j" du .

dt; By J Bx \f ' dii, J

But also

;
dv =

whence

do ,1

-7- dx
dx

do ,
^'J^

dv
1 + du

dv

Bv dz

du

Bz du

dv

Bv

Bo

do

and therefore, comparing these with the equations of the system leading to the
integral form, it follows that

ov Bo

m = 3- > n = - -

cy dz

Next, take a quantity c such that

dn dz

du ~~ ~du

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 51

Since

dv dv dz

— = I -J- np, — = n -
ax du du

are satisfied by the integral relations, we have

d d f dz\

— (( + np) = — (n - - ,
du ^ dx\ du]

and therefore

(]L _i_ — dn d*
du " dn dx du

the terms in -~- cancelling ; consequently

dl, fdn \ dz dz

— = — PC — = 7 — - •
du \(lx J du J du

Similarly, from

dv dv dz

— = m + nq, — - = n — »
dy dn du

we find

dm . dz

du ^ du

the quantities g and f which occur here being those which formally occur in the
derivatives of I, m, n, with regard to x and y.
Again, we have

- 3?; . dl , dl . dl

= d =- =. dl =• — dx + -7- dii + -7-

3,r rf.« rf y du

whence

^

du

and therefore, comparing with the former equations, we have

3% , d2v (?v

= a^' Al = cte3y' ^"

Similarly we find

7 _ 3^ ,._ ^«_ _

^ ^~* ^ o, ^/' -^ ^ O~ J ^

3/ o^«

Now, when we take the combination

H 2

52 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

J + cl = 0,

and eliminate the derivatives of I, m, n, with regard to x and y, we have the original

equation

F = 0,

or the original equation is satisfied in virtue of the integral system. But. from this
integral system, the value of v is such that

and therefore v is an integral of the partial differential equation. We consequently
have the theorem —

When an equation F = 0 of the second order is transformed into J + cl = 0 by
means of the equations

dl

dm dn

p —

ax *• ay

dl dn

and when, in the integral equivalent of the simultaneous system

I = 0, J = 0 "|

dm dn dl 'dn

dx " dy dy dx

dv dv

— = I + np, — = m + no

dx dy

all the arbitrary constants are made functions of a parameter u, and the arbitrary
functions of x and y are also made functions of u, subject to the equation

dv dz

— = n — ,
du du

(which, in fact, will generally determine either an arbitrary function or relations
among the arbitrary functions), then the value of v thus obtained is an integral of the
original equation F = 0.

30. The integration of the equation F = 0 is thus made to depend upon the
integration of a simultaneous system involving fewer independent variables, and

DIFFERENTIAL EQUATIONS OF THE SECOND OEDER. 53

upon the subsequent determination of the arbitrary functions in the integral equiva-
lent of the system. A question therefore arises as to how an integral equivalent
can be obtained. At first sight it seems that, as the number of equations (being
five) is equal to the number of unknowns (I, m, n, ?>, ?) to be determined, HAMBURGER'S
method* might be applied to our special instance, though not when the Dumber of
independent variables in the original equation is more than three. But, as a matter
of f;ict, one of the equations of the system is a functional consequence of two others ;
viz., the equation

dm dn dl dn

dx dy "~ dy " dx

is a functional consequence of

dv dv

d~V=l + n?> Jy =m+ ^

It thus follows that there are only four equations independent of one another
involving the five variables ; consequently HAMBURGER'S method does not apply. On
the other hand, the inference is that, as the equations are fewer in number by unity
than the number of variables to be determined, one arbitrary element must exist
in any general integral equivalent. This arbitrary element and other arbitrary
functional forms, by the foregoing theory, are determined by means of the equation

dv dz

-— = n — - >

nil. du

so far as they can be made determinate.

It is therefore necessary to seek for some integral combination of the subsidiary
system, apparently without at present having any perfectly general process of
constructing such a solution. It may, however, be pointed out that, as there are
four independent equations involving five quantities, they can be used to determine
four of them in terms of the remaining one or, more symmetrically when this is
possible, to express all five of them in terms of some variable. When such expres-
sions have been obtained, they are to be substituted in

dv dz

T~ = n T '
du du

the full solution of the resulting form of which equation will then serve to determine
the quantities.

We proceed to consider one or two examples in connection with the foregoing
theory and explanations, dealing particularly with well-known equations.

* CEELLE, t. Ixxxi. (1876), pp. 243-281 ; ib., t. xciii. (1882), pp. 188-214, the number of indepen-
dent variables being two, and the number of equations being equal to the number of dependent
variables.

54

PEOFESSOE A. E. FOESYTH ON THE INTEGEATION OF

Application to V2v = 0.

31. When the method is applied to the potential equation, which is

a + b + c = 0

with the present notation, the substitution of values (say of a and b) is required to
lead to a result evanescent so far as the determination of coefficients of the second
order is concerned. The substitution gives

til

so that we must have

(In dm dn

~ + ~ -- * ~ + c

P~ + 22 + 1 = 0,

m rf/i

, -- (7i-
ay 1 ay

rW <7?i . rfm

-- PT~ —

tl>: L ax

and the differentiations with regard to x and to y in these relations are effected on
the supposition that the unexpressed variable u is constant.
The subsidiary simultaneous system thus is

— o

dl dn dm dn

dx " dx dy * dy

dl dn dm dn

dv

— = t + np

dv

— = m + nq

When HAMBURGER'S method, as expounded in the second of his memoirs already
quoted (§ 30), is applied to this system, it is found that the algebraical equations for
the determination of the subsidiary multipliers are inconsistent with one another
unless all the multipliers are zero ; there is then a null result. Accordingly integrable
combinations must be obtained otherwise.

Now the general solution of the equation

is given by

P3 + q" + 1 = 0

p = constant, q = constant,
z — px — qy =. constant,

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 55

these constants occurring in association with u constant. We may, therefore, assume

2 = u + xp (u) + yq (u) ,

where p and q are arbitrary functions of u, subject solely to the condition

P* + </ +1 = 0.

Because the differentiations with regard to x and to y are effected on the hypothesis
that u is constant, the other equation can be taken in the form

d » d

- (I — np) + -- (in — no) = 0.
ax x dy v

so that a function £ of x and y (and possibly also involving u) exists such that

I — np = — i "i — nq = — — •
dy dx

Moreover, we have

dv dv

t + np = — > r/i + nq =• — >
dx dy

consequently

dv d% do r'£

— + —
dy dx

From the last two equations, it follows that

. J dx
and thence that

Consequently a function w of x and // (and possibly also involving u) exists such

that

. £ dw

pv + tf = - — ,

whence

duo dw

56 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

Substituting these values, we have

d*w

d-w

M= PjZ+^tedj-Pdf

2m = —

2n =

dii?

d-iv
~dx2

2P^.

d-w'

dy*

dho

dx dy

dy\

Thus far account of variations with regard to x and to y has been taken. But,
as regards variations of u, we have

dz

that is,

d I dw

dv

-j- = n
du a tt,

dw
dy

-w d-w

where A denotes 1 + xp' + yq. This is the equation of limitation upon the form
of iv ; if its general integral were known, the general value of v could be deduced.

2. Before proceeding with the consideration of this result, it is worth noting the
relation of the equations

; (>S M

l-np= J-, m - nq = -

dx '

to the original equation. Because

it follows that

where A denotes
so that

= u + xp

1. Sit, 1 (hi-
p dx q 3//

z

1 + xp + yq,

r\fc

aT

dy

?f

m-nq= - ± -p

G/v

- np = aT

dy

when in £ substitution for u is made in terms of x, y, z. From the former we have

a — gp — np' ^- = g-J- +
9 -Cp- np' £ =

,
+

, 8(5

? dx

a? ,

dydz

and from the latter

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 57

,3it *- *- °" *-

o - /<? - «? 5- = ~

,i* ,u

f—cq — nq 5- = — •5-3 j» -rf p- »' ^— .

* 82 3*& -^ 3z2 3z •* 3s

Multiplying the second of these equations by p, the fourth by q, and then adding all
four, the function $ is eliminated, and we have

a + b + c = 0.

33. Again, the general theory in a preceding Section (see particularly § 4)
immediately suggests that -u occurs as an argument of an arbitrary function. This
being so, let the variables be considered as transformed to x, y, u, where u now is
known, and effect the transformation directly upon the equation

a + I + c — 0.
We have, for any function P,

3P , 3P 7 3P j 7T3
ir- ax + ~ ay + -5- dz = dc
ox oy on

dP j f/l> rfP

= -:- aiC + T~ «V + V^ »M

dx dy du

dP dT dP 1

so that

3 d j> <1
8b; r6; A (f« '

_3 rf g d

dy dy A rfw '

1 r?
A rfi7'

We thus have

a? = d& ~ 2 A"

^^ d»f 2^ ^, „
A^ rftts - du \ A2 A» ^ h

- 4- - to" 4- W")

3 1 ^ h W '

A%d«~ A2*t2 dw I A A3

whence, adding and remembering that a +b-\- c = 0, it follows that

d»» rf2» 2

\-QT~ 1 = 0.

A *

VOL. CXCI. — A.

58 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

It is immediately obvious that one solution of this equation (and therefore of the
original equation] is given by equating v to any arbitrary function of u, a result that
admits of simple verification.*

34. But it is not at first sight clear how this solution connects itself with the
general solution indicated in § 31 ; the connection can be made as follows.

The general value of v is

dw dio

if this is to be an arbitrary function of u, say /(«), as for the solution under considera-
tion, we must have

ihu chv . ft \

and consequently

w — ^ f(u) -f G (u, p'x + q'y),

where, so far as concerns this relation, G is any arbitrary function of both its argu-
ments. Writing

this is

&' /» / \ | /"N / \

w = — j (U) + U- (u, r)),
so that

dy? clij-

— • /)~i fTf

' "" " / / ^ O >

' Gin"

Now ,

p* + q* + i = 0,

* This result was published in the 'Messenger of Mathematics,' vol. xxvii. (1898), pp. 99-118, in a
short paper entitled " New Solutions of some of the Partial Differential Equations of Mathematical
Physics." The form was altered from that in the text, so that it might be symmetric in the variables,
and the theorem was given as follows : —

If p, q, r, be three arbitrary functions of u such that

p* + f + ra~ - 0,
and if u be determined as a function of x, y, z, by the equation

au = xp + yq + zr,

where a is any constant, also if v denote any arbitrary function of u, then v satisfies LAPLACE'S equation

V3D = 0.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 59

so that

pp + qq = o,

say

q -p

and so

d-w _ J_ cPw _ J^ d\o

' ~ ~

q- dx? ' py dxdy

Consequently

Now we should have

dy rfe .

— = n — = (1 + rj) n,

(lu du ' '

that is,

from which the form of G is given by

G = A (u) + >?B (u) - |/' (u) [(1 + ,) (log (I + ,) - I}],

A and B being arbitrary functions.

The form of G is, however, not so important for the present purpose as is the value
of a'Gar2. We deduce

1/93 _ — _

O V r\ n —

1 +*?'

and consequently

£ ?» M /' (it)

p q — 1 1 + a?/ + ?/<?'

being the proper values of I, m, n, as given by

v=f(u)

z = u -\- xp -\- yq

The particular solution is thus seen to be included in the general solution defined
by the equations of §31.

35. It is worth inquiring whether, in the notation of the preceding articles, there
is any solution of the potential equation which is a function of u and rj alone, say

v = F (u, 17),
I 2

60 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

other than the solution v = function of u alone. When the variables are taken to
be x, y, u, the equation to be satisfied by v is

d*v . d*v 2 I dto , d*v \ _ o

cly? dyz A \ dx du dy dut

where

A = 1 + xp + yq = 1 + 17.

Now

so that, as
we have

dv r f „ . ,A r

^ = a, + to + y? > ^ '

also

^ ,ar
^ :~-P a, '

and therefore

Hence the equation becomes
and therefore

2 er

~r ^ ^ — 0.

1 + r, dr,

where ^ and «/» are arbitrary functions. We thus have the theorem* —

* This result can also be expressed in the form symmetrical as regards the three variables. When
thus modified, we have the theorem —

If Pt ?> '''> 6e three arbitrary functions of u such that

pa~ + <f + r-2 = 0,
and if u be determined as a function of x, y, z, by the equation

au = xp + yq + zr,
where a is any constant, then, writing

a — xp — yq — zr
where F and G are arbitrary functions, v satisfies LAPLACE'S equation

V2« = 0.
This theorem also was stated in the paper referred to (§ 33, note).

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 01

If2>(u) and q(u) denote any functions ofu satisfying the equation

p* + ,/ + i = o,
and ifube defined by the equation

z = u + xp (u) + yq (w),
then

where <f> and t// are arbitrary functions, satisfies LAPLACE'S equation

V3w = 0.

36. The solution just obtained can, like the solution of § 33, be connected with
the general solution. Owing to the linear form of all the equations and of the
expression for u, it will be sufficient, in the first instance, to take the part

.

-

1 + r, " 1 +

say ; because the term (f> (u) in v is, in effect, identified by the preceding case. We

thus must have

d'lv dw ir

the most general solution of which is

« = nh;J + H (* *)•

where, so far as concerns this relation, H is any arbitrary function of both its
arguments. We have

dhu _2 __ j/£ 2a y>;2-f ,3 83H

~d# = ~ (1 + 17)* ~p~ "~ (1 + nT P ~ P " &)*• '

_J __ 2lt, 2x PV± . Ov —

~ ~ 2 r r p v

/a
' q

and therefore, after some simple reductions on substituting in the expressions of § 31,
we have

62
But

so that
and then, as
Ave have

PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

dw dw

P~" "CL

dae

d I dw

dw

1 d I dw die \ . I d*w

l *> + ff — — o

ra J «)/ / * \ rfa;3

t/ / \

» 82H -i

7

an equation to determine H.

Substituting the value of 32H/c?72 in the above expressions for I, m, n, we find

(1

^ +

+

1

?>i = —

which are the proper values of I, m, n, as given by

i + .^' + H'

z = u + xp + yq

The value of H is not required for the preceding investigation ; but it can be
actually deduced from the preceding equation. We have

also

so that

Thus the equation for H becomes

_r «
^

(i

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 63

the term involving x explicitly vanishes, as it should, and therefore we have

JU.*- -S— J&4..-JL_d,

(1 + „)» V (1 + tf p f- (1 + ,)3 q' *

2 ?" 2

Hence

the explicit value of IT, where A and B are arbitrary functions ; and thus the value
of iv is known. But, as already remarked, the value of 32H/3»^2 is sufficient for the
identification of the solution.

37. After the preceding investigation, it is natural to inquire whether a solution
more general than that which is expressible in terms of u and tj, and the form of
which has been suggested by the theory, can be obtained directly from the original
differential equation.

When the variables are taken to be x, y, u, where

z = xp + yq +

0 _ ,3 I 3 I j f"

|? and 2 being functions of u, the equation « + b -f c = 0 becomes

- _j_ — — Ip — \- q - — — ) = 0,

where

•»7 = xp' -\- yq.

Let £ be defined by the equation

£ = »/' + yq",

so that M, >?, C» are three variables functionally equivalent to x, y, z; and let us
inquire what solutions of the form

are possessed by the foregoing equation. We have

p' = qd, q' = — pQ,

p" = qff - p&\ q" = -p0' -

so that

64 PEOFESSOR A. R, FORSYTE ON THE INTEGRATION OF

p* + j* = - &\
p'p" + q'q" = - 60',
p"* + q"* = - ff* - 6\
pp" + qq" = P,

'" + qq"1 = see',

quantities which occur in the substitution. Now

^ __ v> <?? _i_ p" £* ,

dx " Srj " 3f

d*v ., 32F , 32F

— • />Y* , 1T */.>i ^i

7 .> ™~ IJ o o r *'/' A' ^ ^i*

//o>a -*• *-i«» -*• •* *-Wi r\r

similarly

and therefore

<P» '2?

<fy» ~ 9 ft

dy* r dy*' W fy 3?

Again,

so that, as

— ^— _1_ / ?J- _L / '" _J_ "'\ ^

we have

n^ I ,y /32 __ |_ /D2 _ [^ /32j

./ ^, ,;^, "r* 1 ,j>.,7,,, " n,. :u* "i " ^.. ' ^ •

Taking account of the values of r) and £, it is not difficult to prove that
& (zp'" + yq'") = -n (00" - 30'* - 0*) +

consequently

a°-F

=

aaF ari
a?"1" aT? j

and the equation satisfied by F becomes

+ 300' — + {T, (00" - 30'2 - 0*) + 300'a

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, 65

We can at once infer the result of § 35 as to 'the solutions which, being of the
suggested form, explicitly involve u and 17 only ; but other solutions in a finite form
do not suggest themselves.

38. It need only be remarked that, with the slightest change of notation, all the
preceding results can be associated also with the equation

* I ""T n ~~ ^ ^KTT '

,v~ cy* ct-

where c is a constant.

Application to V2y + /cr = 0.

39. Consider next, but more briefly, the equation

where K is a constant ; in our notation, it may be written

« + l> + c + K-V = 0.
The characteristic invariant equation is

the same as for V~v = 0. Substituting for a and b their values as given in § 1, it is
seen that, on account of the characteristic equation, the term in c disappears from

the result ; and we have

T dl dn din dn • 9 o

~" d,c * dx dj " d]i

The subsidiary system thus is

1 = 0, J = 0 "1

dm dn <U__ (In

d;<; dj/ dy dv

dv , dv_ _ . (

the third of which is an analytical consequence of the fourth and the fifth. As i
the preceding example, the general solution of I = 0 is

z = u + xp (u) + yq (u),
where p and q are functions subject to the relation

and in the subsidiary system the differentiations with regard to x and y are partial

VOL. CXCI. — A. K

6G

PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

on the supposition that u is the third variable : that is, in these partial differentia-
tions in the subsidiary system, u does not vary. Consequently J = 0 is expressible
in the form

JL(J _ np) + ^(m- nq) + K*V = 0 ;

and therefore

— (1 — np} + i r (m ~ ncl) = — f 2 — - = — K- (I + np}
ilc~ v dx dy v ax

dx dy

so that

d~ dr

- (I - np) + --„ (m - nq) = - KS ^ = - /c2 (m -{- nq) I

ft

K~ m = » i — r "r <7 jl — !?*

/ V <fa r^/y J </y-

Operate on the first equation with d*/dyz + K~, on the second with dzjdx dy, and
subtract ; then, dividing by K", we have

f-

~ PK'

Similarly, operating on the first equation with d*/dx dy, on the second with
d*/dx* -\- K1, subtracting, and dividing by K2, we have

« + I o + *2 ) m = I — <7 -':' . + 1p - - + '77-, — <7

- dip I \ 2 (/.';- A das dy * dy~ i

It therefore follows that some function w of x, y, and u exists such that

d-w d-w d-w

= P -pr + 2? — p — —

X (/,•!!- J (/,'.' rt?/ -1 (?W3

dho , d-w

m = — q -— • -f 2p j-j- + 5 — — ^/c2w i. ;

* rfa^ •* r/A'rf?/ J r7w3 *

but, so far as the subsidiary system is concerned, w is perfectly arbitrary — a result in
accordance with the general explanation of § 30, there being only four functionally
independent equations for the preliminary determination of five quantities. Also

_

m + nq = 2 (p -r-r- -f q -— =
\ dxdy * dif

do
-

dv

—
dy

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 67

hence

2 / dw , dw\

v — (P ~T + 2 T" '
V (fo J rfy /

no arbitrary function of u needing to be added, for it can be considered as accounted
for in the arbitrary function w.

But now, regarding variations of u, we must have the equation

do d?

— = n —
du, du

satisfied ; that is, the function w must satisfy the equation

, d, I dw , dw\ . fiPw . d-w .

7 T5 KW

ay ) \(b:2 dy-

where A denotes 1 + xp' + y<2- As in the former example, this equation imposes
the limitation upon the arbitrariness of w; when its general integral is known, the
most general value of v can be deduced.

40. It is an inference from the general theory that the quantity u, determined as
a function of x, y, z, by the equations

z = u + xp(u]

is an argument of the arbitrary functions that occur in the solution of the equation
V3v -f- K2v = 0. We therefore transform the variables from x, y, z, to x, y, u, where
u now is known ; and the result, obtained by analysis similar to that in § 33, is

d-o d-v -l

Manifestly a function of u alone is not a solution of this equation ; but, on the
analogy of § 35, we are led to consider what solutions (if any) of the form

are possessed by the equation, 17 having its former value xp' + yq. When this value
is substituted, the equation takes the form

where
Hence

K 2

G8 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

so that, as 6 is a function of u only, we have

where A and B are independent of 17, that is, are arbitrary functions of u. We thus
have the theorem*-

Jfp (?{) awJ (j (u) denote any functions of u satisfying the equation

^ + ga + 1 = 0,

«/tJ //'« be defined us a function of x, y, :., ly the equation

z = u + xj>(u) + yq(u],
also if v denote

»/; 6c(V/ arbitrary functions of u, then v satisfies the equation

V:v + K--V = 0.

Tlie connection between this solution and the general solution indicated in § 39
can be established as in the corresponding case of the potential equation (§§ 34, 36).

,. . f)-6-' , 3-y 3f

Application to ^~ \ 4- „— = u, ^— .
* * ox3 c>f- ot

-11. The j)recediug examples have led, in each case, to a solution which (though
not the most general) was expressible in a finite form. We now take one other

example, viz.,

yl + I + c = 0,

which can be regarded as the equation for the variable conduction of heat in two

* This result can also be expressed (see the paper already quoted, § 33, note) in a form symmetrical
as regards the three variables, as follows : —

If 2>, q, r, le tliree arbitrary functions of u such that

and if u la determined as a function of x, y, z, by the equation

au = xp («) + yq («) + zr (u),
a being any constant, also if v denote

0 (ii) e>« to' + 111' + s'-') ( i>'- + 'i'1 + i-'*f "" + \i, (u) e ~ « to' +

« — xp' — yq — zr

where 0 and ty denote arbitrary functions of u, then v satisfies the equation

V2t> + A = 0.

G9

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

dimensions ; it will be seen that the general solution is not expressible in a finite
form.

The characteristic equation is

say q = — i ; and then u is determined by

z + iij = xp (u) -f u,
where p (n) is arbitrary. The subsidiary system is

'. ' ri + £ + i£-o ">

-y 7
— = Z + np,

We easily iind

.

, + y j~ ) »» + * 7-, — yp

- ' (/,'' ^///- * (/

ly

'

0

I

and so we have

. dw
2'l -- — ypw

d

n =

d-tv dw

~df " y d^

. d~w du: . <

= l -^ -yp^-'^;

l=2i^77,+P

dxdy <'>/••

where w is an arbitrary function of x, y, and u. But we also must have the equation

do d:.

•- =n —
ait du

satisfied ; that is, we must satisfy the equation

dy du

(ho
yp -
Ir du

, . ,, x> [ dho dw\

ywp — — i { 1 + xp (u)} \ -j-^ + y — } .

' ' ' [ d>f ' dx J

If a general value of w can be deduced, then the general value of c could be
constructed, and conversely.

42. Taking z + iy = *-, z — l>j = s, the equation can be written in the form

70 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

a solution of which is given by

x 8*F

--

v = F (*, * ) - • 4 -- 7 - -y, 2! Ss3 fc/* ~

where F is any arbitrary function of s and s'. Now the solution, which appears to
be most frequently useful, of the original equation is obtained by taking

v = Ye-*",
where ^ is constant, and V is independent of x. In this case,

and the equation for V is

To identify the two forms, we must have

TT /,. c.'\ - _1_ - ~

r 1 6, i ) — j/ — ^ ^ , "p 0 . I ~~

7 ds os 2 ' \ 7

= V - xp. V + ^V - . . . ;

consequently

V = F (*, 6-')

_ /ji_\" 3s"? f

'

for all values of n. All the conditions are satisfied in virtue of

8'V

3s3/ = iwV;

and consequently the more general solution above obtained includes the less general
solution customarily used when the arbitrariness of F is made subject to the equation

This equation is of LAPLACE'S linear form with equal invariants, which, moreover,
are constants ; hence the whole series* of derived invariants is unlimited in number,
and the number of derivatives of an arbitrary function of s and the number of deri-
vatives of an arbitrary function of s', that occur in the most general solution, are
unlimited in each case : that is, the solution is not expressible in finite terms.

* DAKBOUX, ' Tlieorie Generale des Surfaces,' t. ii.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 71

43. To see how this solution is included in the general solution of § 41, it is neces-
sary to determine a function w such that

2i -- -- ypw = v — (T4y *«"»7 F (s, s') = 3> (s, s'),

say. Now, for the derivatives with regard to y, s is parametric, being xp -J- u ; and
s' = s — 2iy. We thus find on integration

<£ being arbitrary, and s parametric in the integral. Evaluating the integral through
integration by parts, and dividing by the exponential factor on the left-hand side, we
find

w = e *'ypyd>(x. u) — --•<<& (s, s

r \ i „„

say, so that u^ is expressed in terms of x, y, 11, and w.2 in terms of s, s, x.

Denoting derivatives of w2 with regard to s, s', x, by 8/Ss, S/S.s-', S/8x, respectively,

we have

fl i S_ . B_\ . $_ . /_S_ _, 3_\ _ _ _4 &;_

I cj y I I £ ^ /^ I £ ., I o / I "" cv 1 cj y )

this last change arising from the fact that every term in w.2 satisfies the transformed
equation because 3? satisfies it ;

Consequently

OS iy O-S (

. ^.-a
^ ?./ )

^ _i_ §i .

j?3 " " 7 S*7^-

Further,

dw,

y^ =

so that

72 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

d-iv dw

.,

4 T -

-l^/ rty 2 B.\ /S 8W . H\

= « (y £ - hWJ - ( & + ^J ^ - 4 -^ j

But

yji Bs' *f[>
and therefore1

Now we must li.ive

^?;

-
(In

that is,

(! + "30 (f

and so

But, because r = ff>, .<? = 2 -f ?)/, *' — z -- iy, we have

that is, the foregoing value of n is the s-derivative of v. Further, comparing the
values of n, we find

*v/ - ——

whence

<f> == € "A. (Uj,

A being an arbitrary function ; hence

the explicit value of w.

Next, for the value of I, we have

from the value of to, just obtained ; and

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 73

<Pu>t > o- div>a S-«'3 . ^w S^. 16

= 4 ^ ( - *-—} w

O# 06' V *Y 0-S' ' *

\ / /

o* 08f <y

4 £-<l>

c- ^ /
7 ox ox

Hence

, _ d-w . d-w 4 o-<l> 2<1> B/'

I — P ", ., ~r 2< , 7 — — - ;r~s~; == -, = ~i .

• </?/- ax (I i] 7 ox dx ox ox

or the value of I is the as-derivative of v.

Similarly for the value of m. The solution is thus seen to be included in the form
given in § 41 ; moreover, the value of iv obtained in the preceding investigation is a
solution of the equation there required to be satisfied,

44. It has been seen (§ 27) that the equations to which the method involving
derivatives of v, I, m, n, alone can be applied — which method has been indicated as
the generalisation of AMPERE'S for the case of two independent variables — belong to
a distinctly limited class. Moreover, even for equations in this class, it may happen
that an integrable combination of the subsidiary system cannot be constructed, or
that the quasi-general process sketched in § 28 is not effective. Consequently it
becomes necessary to have some other method ; and this is provided by what has been
indicated as the generalisation of DARBOUX'S method for the case of two independent
variables.

In the discussion, we shall consider only the case when it proves necessary to
construct the first derivatives of an equation F = 0 ; but the explanations apply,
mutatis mutandis, to other cases when second derivatives of F = 0 and derivatives
of higher orders should be constructed. Examples of the latter cases, when the
characteristic invariant is resoluble, have already been given in Section II.

45. The original equation is

F (a, 1), c, f, g, h, I, m, n, r, x, y, z) = 0.
The characteristic invariant is

Ayr + H/K/ -f- Br/ - GI> - Yq + C = 0,
and it is supposed to be irreducible. The subsidiary equations are

y I \<ly L dy / dx

'll'-L _ J-L] . H(^- - p'~\ + Bl^' -^Wft*

5 dx J \dx dy J \dy

VOL. CXCI. — A. L

74 PROFESSOR A. E. FORSYTE ON THE INTEGRATION 0V

and equations of definition and derivation are

dv 1
- = 1 + np,

dv
dy

dl

dl

dx

dy

dm j ,,

dm

\u ~ l + ^'

dy

dn

du

= m + nq
= h + (J<1

dv___ <h_
du du

du

= 'J

du

dm , dz_
du du

dn

dz

du ~ du

So far as concerns derivatives with regard to x and y, there appear to be twelve
equations, viz., the characteristic invariant, the three subsidiary equations deduced

by taking

— DF

DF

and the eight equations of definition and derivation ; and the number of quantities
to be obtained from the subsidiary system is eleven, viz., a, b, c,/, g, h, I, m, 11, v, z.
On the other hand, F = 0 is an integral of the system, and it is a persistent relation ;
consequently one of the set of three equations can be regarded as depending on the
other two. Again, from the equations

~=l + np, f~ = m +

we have

identically satisfied in virtue of the values of dl/dy, dm/dx, dn/dx, dn/'dy, given by
the other equations of the set of eight ; hence one of these can be regarded as
dependent functionally on the rest, and the set of eight are therefore equivalent to
seven only.

The whole set of equations in the subsidiary system, involving derivatives with
regard to x and y, thus contains tea independent equations ; and eleven quantities
are to be obtained. Hence it may be possible to express ten of these in terms of
one of them, or to express the whole eleven in terms of a single new quantity,
arbitrary so far as the set is concerned ; its arbitrariness will then be limited so that
it shall satisfy the subsidiary equations involving derivatives with regard to u.

46. The relations thus obtained, as satisfying all the subsidiary equations, con-
stitute an integral of the equation.

This result is established by an argument similar to that in § 29 for the case when
the subsidiary system is simpler ; accordingly here it will not be repeated.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

Application to Vzv = 0.

47. The investigations as to the potential equation, given in § 31, did not require
the consideration of derivatives of «, b, c,f, g, h ; but the example can be used to
indicate the method of proceeding when such derivatives must be taken into account,
as in the preceding theory.

For the equation

a -f b + c = 0,

the special subsidiary equations involving derivatives of a, b, c, with regard to

x and y, are

da dn dh da

-; p v + -r- - q T~ = 0

dx d.i' dy dy

dh df db df
p -i- _j_ q --- = 0

dx flu di/ dy

dq df df

-Z- — in — -|- ~^—

d.r r dx di/

dc

tlie characteristic equation is
and the relations of identity are

p* + <f +1 = 0;

d<l <l'i d(/

<lh

<i(L

(h,

df db

df

df

J

But the latter can be replaced by the equations of definition, viz.,

dl

dl

together with

T~ = c
dx

T yi'>

dy-

die J dv,

dm ,

' +Jp,

dm i \ f
lh/~ +-/(/'

dm ,. d~
du ' du

»>

dn

dn ,

dn dz

+ cp,

-<y--=t + <•<!,

du dit, _,

dv

dv

dv dz

I -f np,

- = m + n2,

du ~ du '

it is, indeed, from the first two columns of these equations of definition that the
relations of identity are deduced.

L 2

76 PROFESSOR A. R, FORSYTE ON THE INTEGRATION OF

In the subsidiary equations and in the relations of identity, the derivatives with
regard to x and y are framed on the supposition that u is constant. Now the
complete solution of

0 i O i i r\

p- + q~ + 1 = °
is given, by CHARPIT'S process, in the form

p = constant, q = constant, z — px — qy •=. constant.
But these constants arise when u is constant ; hence we may take

z—px — qy- u,
where p = p (it.) and q = q (u) are arbitrary functions of u subject to the condition

p" + f + 1=0.

From the first of the three subsidiary equations, remembering that p and q are
now constant so far as concerns derivatives with regard to x and y, we have

so that some function w of x, ?/, it, exists such that

dw dw

a — pa = -- ' h — no •= -- - •

1 '' i/i/ IJ ill-

Also we have

,n <n

r = _ , h + qg = _. ,

so that

<lw dl

2<( = T + Tx '

dw dl .1
f/y + ^ I

'

From the last we have

dw dl dw dl

that is,

~ (pw - ql) + '^ (pi + qw) = 0,

so that some function 6 of x, y, u, exists such that

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.
pW-ql=f-> qw + pl=-1^,

or

''V

w = -P ;/;, +

Substituting these values above, we find

denoting an arbitrary function of a;, ?/, M.

The second of the three subsidiary equations similarly gives

77

denoting an arbitrary function of cr, y, u ; and the third of the subsidiary equations

20 = (p, <!, ~ P^~ ' ffi

gives

n = »

\l> denoting an arbitrary function.

These three arbitrary functions are not independent of one another. Comparing
the two values of 2f, 2g, 2h, respectively which have been obtained, we see that some
function <f> of x, y, u, exists such that

78

PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

X- - 9. P,

I "V '; d \

ii = (1, 0, 1 ft T ' T

\ ' .A.&; dy

\2

manifestly <£ is an arbitrary function ot x, y, u.

To deduce the value of v so far as concerns variations of x and y, we have

(Jv

fi

- = I -f- np

and

- = w -f

therefore

where thus far V is any arbitrary function of u alone.

48. As yet no account has heen taken of derivatives with regard to u ; but the
whole of the equations are subsidiary to the construction of an integral of the original
equation.

The part of the value of v given by V (u) can be dropped, as it has been considered
in the earlier stage, or it can be regarded as absorbed in the other part of the value,
all that is necessary for this purpose being that the function <£ should have a different
form.

Assuming this done, the function <£ must be such as, first, to satisfy the equation

— = n — =
du an

with the preceding notation. Now

DIFFERENTIAL EQUATIONS OP THE SECOND ORDER. 79

also

and, as
we have

that is, A behaves as a constant with regard to the operator -p ' --- \- <i - The

L il.i: J <l>j

required equation will therefore be satisfied it'

But it is necessary that the other three equations

1_ ill 1 din _ I tin fa

<J dlf, ' ' 1' du ' r. ' du du '

should also be satisfied ; that they lead to no new condition for c/> can bo seen a.s
follows. We have

so that

On account of the equation satisfied by <j>, this is equal to

80 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF

which is equal to

d d\( fp & d*\.-\

ete +9*)(*4t + ^ S*-*W*J

, d , d\ / d* P «P \

and therefore

(II

• = Act = g - -
du y du

in virtue of the equation

<•'-<£ '/20 ^ / / ';0 , '?0 \ , A'-* '^*

P s£ + </ ^ + 2 ^ ^ + '/ T,} = i ^ +

Similarly it can be verified that

din , i/z iln, dz

T~ — J ~r ' r = c ~, >

nil (in ilti ait

in virtue of this equation ; therefore it is the only condition to be imposed upon
4> iii order that all the equations may be satisfied.

49. Solutions of the equation a + I -f c = 0 have been given in §§ 31-38 ; it
is not difficult to verify that they are included in the preceding general form. Thus,
taking the solution

v=f(u)

and assuming that <£ is, as in § 48, the function through which the term V («) in v
is absorbed, it is first necessary to determine (f> so that

,_o ''"<£ I d-tf> „ d-^> i /•/ \

P~ ~j~z i *PQ ~T~i — r <r ^T = J/ (U).

This equation is easily solved ; and we have

.2

$ — i ~^f(u) + Gr (u, rf) + a;H (u, -rj),

where G and H are arbitrary functions, and 77 denotes p'x + q'y, as before.
We have

ff- • ^

^$_ _ ( ,, 1 32G ^ 92H^ .

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 81

so that

, , . 3H

X = ~ </, P, tf ' > = ~ -- ''

i , /B ,

~ -- * 2 x 2 '-

Hence

\P & + ? j X =

n =

But </> (or, what is the same thing in effect, G and H) must be such as to give

dv dz

~r — n T '

du du

so that

n = '~~T~- >

consequently we have

I m n f" (u)

i/ \ /

— p — q I A

which are the proper values of I, m, n, connected with v =f(u).
Similarly for the quantities a, b, c,f, g, It.
And if the value of H be required, it is determined by the equation

33H- =/M = /L(!i),

dr,3 A " 1 + 7/ '

the integration of which is immediate.

Application to V~v -f- K2v = 0.

50. The integrable combinations of the subsidiary system in the case of the
potential-equation seem fortuitously obtained. As one other illustration, added
chiefly to show that the subsidiary system can be used in the mode indicated in § 39
to express all the variable quantities in terms of a single quantity, we consider the
equation V2-v -f- K-V = 0, which is

a + b + c + K2v = 0,
in the notation of the present paper.

VOL. CXCI. — A. M

82 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF

We have

X = K*l, Y = *X Z = /c2n.

Thus the characteristic invariant equation is

from which we have, as before,

z = u + xp (u) + yq (u),

so that p and q are parametric for differentiations with regard to x and y. The

three subsidiary equations are

K~l + - p -', - — <l T~ — 0

fte ' (/,'.: ay ay

dli df db df

K~m + p ', + -, ? T~ == 0

// »' // /' //'J/ //?/

j
ay

and we .have the equations of definition and derivation as before.

5 L. One simple mode of proceeding is to use the six equations involving derivatives

of /, in, n, with regard to x and y — they are equivalent to five independent equations

—in order to express a, b,f, g, h, in terms of c and those derivatives. These being

obtained in the form

dl dn tin

a = ^-P^ + irc' v = ^--vc'

dm dm , , tin

b — - -- (/ - --- \- (re, / = — -- qc,

d)/ * dy J dy

dm dn dl dn

h = - -- p — + pqc = -- -- a-— -\- pqc,
dx ^ dy dij a dx

we substitute them in the subsidiary equations. The third of the latter then

becomes

d* ,\ / d d

TT + K"=2PJ- + 9'T-

dy~ / \- dx dy

the second of them, on using the first of the two values for h and also the relation
between n and c just deduced, gives

id- d* A / d? d- d- ,\

^ K" m = — ~ 2» . < ~ — K n 5

— o . ,

df j \ dx3 f dx dy

and the first of them, on using the second of the two values for h and also the
relation between n and c, gives

= (P

2

— »

df J \2 dx* ^ dx dy ^ dy

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.
Let the various operators be denoted by 8,, 82, 83, say

d2

S ° I

6, = — or — J.

z

d* d3

lx dy dij"

d*

3 =

moreover, let

83

be another operator. Tlien from the relations between in and n, I and n, it follows
that a function 0 exists such that

m = S/H),

and then, as

<lx
(lv_

dy

=

«,/; a

e

dv / <P (P ,

— = m + nu = 2 ( p — — + a -- } 0

ay \ ii.i' ihi (/-//- ,

we have

= 2 p
-1

& = 280.

From the relation between n and c, we have

or, if ® = Siv, then

28c = S3n = S32©,

We thus have, from the foregoing relations, the results

v = 2

7 O^O

I = bb^
m = 8S^
n =• S83?

c = iS

J

The values of a, &,/ g, h, con be obtained by using these values of I, m, n, c, and
we find

M 2

84

PROFESSOR A. B. FOKSYTH ON THE INTEGRATION OF

a =

7 1 S $1

7 i (\ O

g - iSjSjW

All these forms correspond to the respective expressions obtained in § 47 for the
potential-equation.

52. Account must now be taken of derivatives with regard to u, and the equations
to be satisfied are

1 di< 1 M 1 dm 1 dn dz

n dti (j dn f du c du du

Taking the first, we have
so that the equation is to be
Now, denoting

= 1 + xp' + yq' = A,

/

P

f (.1

~r

dy

and so for 8/, S3' — viz., S/ is the same as Sj except that p' and 5' replace p and g, and
so for S' — we have

— — — ^"S2 }

~
du

and therefore

28* ^

Now
so that

SA = pp' -f qq = 0,

8 (AS3w,') =
Hence the equation in w will be satisfied if

28

the condition that we may have

dv
du

dz_

du '

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 85

But now. operating on this equation with 8lf we have

8l [28 dg + 48'.] = 8l (A8.tr).
The left-hand side is

+ 4S'S'™ = 288i S + 28/V + 28/Su, + 2 (8'S, - 8/8) „
= ^{288^} + 2 (S'8t - 8/8) w.

The first term is 2 - ; the second, on reduction, gives

and therefore the left-hand side is

The right-hand side, viz., 8, (AS3w), becomes on expansion and reduction

2 (p'q - pq') — S3 «' = A2^ + 2 (//^ - ^? w.

—

j

Hence, in virtue of the above equation, we have

dl dz

-- = r/A = <7 -r-.

du " thi

Similarly we find

dm _ , dz du dz

du dii? (hi, du

and therefore the equation determining ui is

that is,

S I r» Q1' 1x0*
, + 2b w = -k&OM,
du

'>

fflu

d~w

, dw

, dw

. ' i '\ (d?'w

dtw

dxdu

" dy du

dx

dy

\ctiXs

r df

When a solution of this equation is obtained, a solution of the equation for v can
be constructed, and conversely. In particular, the identification of any solution of
the f-equation, as being included in the general solution, can be made as in
former cases.

86 ON DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

[NOTE. — Added 17th March, 1898. As indicated in the introductory remarks,
the theory given in this paper is applicable when the number of independent
variables is greater than two, and when the order of the differential equation is
higher than the first ; its application is not restricted to the case of an equation of
the second order in three independent variables.

A brief sketch of the theory for an equation of order m in n independent variables
is prefixed to a paper* which contains the integration of some differential equations
of types represented by

a_ a a \« _

• + *i + x* H h *"-' °«

when the iteration of the differential operator is purely symbolical.]

* " On some Differential Equations in the Theory of Symmetrical Algebra," Camb. Phil. Trans.,
vol. xvi. (1898), pp. 291-325.

.£

[ 87 ]

II. On the Cliange of Absorption produced by Fluorescence.

Hi/ JOHN BURKE, M.A., Berkeley Fellow of the Owens College, Manchester.

Communicated by Professor ARTHUR SCHUSTER, F.R.S.

Received June 10, — Read June 17, — Revised December 14, 1897.

Introduction.

ABOUT two years ago, in the spring of 1895, in the course of a conversation with
Professor J. H. POYNTING on the nature of the phenomenon of fluorescence, in the
study of which I was at the time beginning to engage, the suggestion was thrown
out by him that possibly fluorescent bodies absorb differently, according us they are
fluorescing or not, the rays which they give out whilst fluorescing, thus that a
body A would absorb differently, according as it is fluorescing or not, the rays from
a similar body B in a state of fluorescence. Some fluorescent bodies undoubtedly
do, others do not, absorb, except to a very small extent, the rays which they emit.
A strong solution of fluorescin or eosin, for instance, hardly permits its fluorescent
light to penetrate even a very small thickness. Glass coloured with oxide of
uranium is much more transparent, but sulphate of quinine hardly absorbs these
rays a.t all.

The question was whether during the act of fluorescing any change is produced
in the nature of the absorption itself, that is, whether during fluorescence there is
an increase or diminution of absorption in that part of the spectrum where the
emitted rays lie. For instance, with uranium glass the radiation takes place
chiefly between the D and E lines, so that the absorption power for -rays may bs
different according as the body is examined in the dark or in daylight in this part of
the spectrum. Of the five bright bands of which the radiation consists, three lie
between the D and E lines, the other two being of less refrangibility and of less
intensity in the red and orange (STOKES, 'Phil. Trans.,' 1852).* With the
spectroscope I have used I have not been able to see the band in the red, but the

* The series of experiments have been made with uranium glass, as the compounds of uranium
exhibit many peculiarities which other bodies do not appear to possess when fluorescing. For instance,
the bright bands in the fluorescing spectrum of uranium glass seem to be noticeable in uranium com-
pounds alone. It is also remarkable that crystals of nitrate of uranium have the quality as well as the
quantity of their fluorescent light altered by depriving them of part of their water of crystallization.

21.4.98

88 MR. JOHN BURKE ON THE CHANGE OF

other four were quite distinct. The spectrum seemed to me to be of the nature of
maxima and minima.

In the summer of 1895 I made some experiments at Trinity College, Dublin, on
the electrical conductivity of fluorescent solutions, and endeavoured to determine
whether the action of violet or ultra-violet light upon them gave rise to any
alteration in their conductivity.

The many difficulties which the experiment entailed prevented me from arriving
at any conclusion upon the matter, and I commenced to look for some such effect as
Professor POYNTING had suggested.

The experiments have since been carried out in Professor SCHUSTER'S laboratory
at Owens College.

1. The leading conception in the following paper may briefly be stated thus :
Suppose that a body A is transmitting light from a similar

body B, which is fluorescing. It is quite conceivable, as
Professor POYNTING remarked, that the amount of light J
from B transmitted by A should be different according as A
h fluorescing or not.

O '-t P- JW P-

2. There were some reasons which pointed to the con-
clusion that something of the sort might happen. It seemed quite possible that the
action, if any, of the fluorescent light from B might be to strengthen the fluorescence
of the body A when already fluorescing ; not that the emitted light would be
increased merely by scattering, which would be precisely the same in amount
whether the body were fluorescing or not, but that the emission of fluorescent
radiation would be increased if light of the same frequency were incident on the
radiating body.

The experiments which are described in the sequel do not confirm this supposition,
but they prove that the amount of fluorescent light apparently transmitted is very
different according as the body through which it passes is fluorescing or not, the
amount being very much less, only about one-half (0'57), when fluorescence is taking
place than when the body is unacted upon by exciting light.

3. It seemed possible that this effect might be due, indirectly at least, to an
increased absorption of the incident fluorescent light, if it possessed the property of
destroying the fluorescence of the body through which it passed. BECQUEREL has
shown that certain infra-red rays have the power of destroying phosphorescence in a
substance which is exposed to the phosphorescence-exciting rays. It was therefore
possible that the result obtained by me was an effect of this kind, but I think that
the experiment described on p. 92 sufficiently proves that, though such an expla-
nation would involve an increased absorption of the annulling rays in order that
they might be capable of destroying the fluorescence, the effect I have observed is
really not essentially of this kind, but is rather due to an increased direct absorption.

It may be useful if, before entering into an account of the experiments, I give,

ABSORPTION PRODUCED BY FLUORESCENCE.

89

as briefly as clearness will permit, a short account of the general principles on which
the coefficients of absorption may be calculated.

(fi) MODE OF DETERMINING ABSORPTION.

Let us consider two cubes of the same fluorescent substance (fig. 1 ).
We shall, by comparing the light from B when fluorescing, which is transmitted
by A according as the latter is fluorescing or not, obtain

E2 = E.,/3,

E0 being1 the intensity of the light emitted by each cube separately and independently
of any light transmitted from the other.

The quantities a. and ft are clearly the fractions of the incident light transmitted
by A according as it is or is not fluorescing. They may be called the coefficients of
transmission of the cube under these different circumstances.

Similarly, if we denote the coefficients of absorption of the cube by a and b, and
that of reflection at the two faces by >', we have

Hence

a - 1 _ (r + a) ;

&).

E1 = E0[2-(r + a)J;
Ea=E0[l-(r+&)].

It has been found more convenient to have to deal with the transmission coefficients
rather than with those of absorption, equality in the former of course implying
equality in the latter, though, as we have seen, the convei'se is not necessarily true.

Fl>S. 2.

ct -dp *$/

c, A, a,

a

d.

d,

r

p

6* \

5'

a,

J

** Ad' °*

°> A, ",
d',

Let fig. 2 represent a plan of the plane horizontal surfaces of four similar and equal
cubical blocks — Ax, Az, A'^ A'2 — of some fluorescent substance such as uranium glass,
which is singly refracting. We shall put aside for the present all minor details. Let
P be a photometer furnished with two vertical slits, the breadths of which can be
varied, and rr a screen, whose plane is perpendicular to that of the paper, serving to

VOL. CXCT. — A. N

90 MR. JOHN BURKE ON THE CHANGE OF

prevent any light from the vertical surfaces of the cubes AT and A.\ from entering
the slits s' and s respectively. We shall suppose that the cubes are illuminated by
vertical rays incident perpendicularly to the surfaces A1( A2, A'1( A's, i.e., to the plane
of the paper. By means of this arrangement we can compare the intensities of the
two beams of light coming respectively from the vertical surfaces of the cubes Aj
and A'j, opposite to the slits s, s'. None of the light from Ax and A2 is permitted to
pass sideways into A.\ and A'2, the two sets of blocks being separated by a screen of
black paper. On the other hand, the blocks At and A.\ are separated from A3 and
A'2 by a sheet of lead glass, which is very opaque to the fluorescence-exciting rays, so
that radiation scattered by A2 and A'3 cannot be the source of any part of the
fluorescence of Aj and A.\, and vice versd. The fluorescence of each block is thus
altogether caused by the rays incident perpendicularly to the plane of the paper.
If all is right the fluorescent light emitted horizontally from the surfaces «, and a\
ought to appear the same, and this is tested by the photometer.

When this condition is fulfilled three experiments are performed : firstly, to deter-
mine the coefficient of transmission of Al or A.\ when fluorescing for those rays
which the cube itself gives out by fluorescence ; secondly, to measure its coefficient
of absorption when not fluorescing ; and thirdly, to determine independently the
ratio of the coefficients, and to compare the value so obtained with that given by the
other two results. The last experiment thus serves to verify the first two
measurements.

The (a) Set of Experiments.

We may call the experiments made to determine the coefficient of transmission
during fluorescence the (a) set, and similarly the other two the (/3) and y8/(l -f- a) sets,
in all of which a proper combination of the cubes A^ A'1( A2, A'2, is chosen, and the
fluorescence-exciting light is allowed to fall upon them, whilst the remaining cube or
cubes are screened from its action.

Thus, if we wish to determine the transmission during fluorescence of the block
Aj, we screen A'2 from the incident rays as in fig. 3, these, it must be remembered,
are perpendicular to A1; A.\, A2, A'2, that is to the plane of „. „

i 1 1 i 1 1 • 1 i •..••i . * t* til i- — . ^

the paper ; hence the light emitted by the surface ar will
be that due to two cubes, and that from a\ to one only.
We can compare relative intensities of these by adjust-

j'

ing the slits s, s'. The block A'2 is not removed, as

some of the rays emerging from c} towards A2 are reflected back, and these are
compensated by a similar reflection from A'2. There is no reason to suppose that the
reflection would be different according as the reflecting surface is fluorescing or not,
because fluorescence is not confined to the surface

A portion of the light emitted by A2 is reflected by the double air layers separating
A3 from A1( but this is included in the coefficient of reflection r. The light emitted

ABSORPTION PRODUCED BY FLUORESCENCE.

91

by the surface a^ is then made up of two parts, i.e., the fluorescent light of Al and
the fluorescent light of A2 transmitted by Ax.

Hence the intensity of the light given out by a^ is E0 (1 + a). The light from A.\
consists of its fluorescent light only, the internal reflection being the same as in
the other two blocks, and is thus = E0.

Denoting the widths of the slits by s and s', and remembering that these quanti-
ties are inversely proportional to the intensities, we obtain the equation 1 + a = s'/s,
from which a may be determined.

In order to avoid errors arising from want of symmetry, the process is reversed by
screening A3 instead of A'3, arid similarly a is given by the equation 1 + a =. cr/cr',
a- and cr' being now the widths of the slits s, s'. If there are n observations with
the screen in each of these two positions,

The (ft) Set of Experiments.

To determine ft for A1( we screen Ax and A'e from the incident light, as in fig. I.
We can then compare the fraction of the light emitted by
A2 which is transmitted by Aj (when the latter is not
fluorescing) with that emitted by A.\, which is the same
as that from A3; the constant of reflection being of
course involved, as in the previous case.

The light coming from a\ is then

= Eu>
whilst that from ax

ft = ,'/«.

^8 = cr/cr',

Hence

If the process be reversed,
and after n such, observations

The removal of A'a is found to make no appreciable difference. The cube Aj and
the glass screen may be moved before the determination is made, to make sure that
A2 and A\ radiate equally.

The ft/(l + «) Set of Experiments.

The ratio ft/ (I + a) can be obtained independently by the arrangement shown in

iig. 5.

N 2

92

MR. JOHN BURKE ON THE CHANGE OF

It is easy to see, as before, that if A'x be screened, the
light coming from «j

Fig. 5.

whereas that from a\

Hence

and reversing,
so that

= E0/3.

/?/( I + a) = S/6-',
= o-'/a;

4.

This description of the general method will suffice for the present, and the account
of the general methods of carrying out an experiment will be found on p. 96.

To Determine the Alteration in the Fluorescence on the Supposition that the
Fluorescent Light is Capable of Destroying Itself.

Let us suppose that the intensity of the fluorescent light given out by each of two
cubes of uranium glass when not exposed to the radiation from each other — that is,
when there is an opaque non-reflecting screen (II, fig. 6) separating them — is equal to
E0, and that when this screen is removed the intensity is equal to E'0. The blocks
are supposed to be illuminated by rays perpendicular to the plane of the paper, and the
radiation, the intensity of which is under investigation, to be in the direction perpen-
dicular to the surfaces, ai} a',, of the cubes. The method employed for comparing
E0 and E'(J depends upon obtaining two photographs on one plate, for comparison side
by side.

We proceed thus : —

Let the slit s be completely closed. Three experiments
are made : —

1. The light coming from ai is photographed when the
cube A\ is not fluorescing, or the screen II' interposed. We
thus obtain the effect due to E0.

2. Both cubes are then made to fluoresce, and the screen is removed, in which
case we get the effect due to E'0.

It can thus be found by adjusting the slit s when the photographic effects of E0 and
E'0 are the same, from which the ratio of the two is determined. As the result of these
experiments we are not justified in saying that any material difference between E0
and E'0 exists other than that possibly due to scattering.

3. The amount of scattering from A'j into Alf or rather the amount of the A\
light scattered by At in the direction outwards from alt may be found by making A',
fluoresce alone, and noting the effect of the light then falling on the slit s. The

I

ABSORPTION PRODUCED BY FLUORESCENCE. 93

difference due to scattering did not appear to be more than 1 part in 8 at the
maximum, and was for the purpose of these experiments altogether inappreciable.

DJESCBIPTION OF APPARATUS.

1. Fluorescent Cnlcs.

2. Photometer.

1. The blocks employed are each 1 centim. cube, and are kept together by two
indiarubber bands. The two front cubes, A^ A.\ (fig. 2), are separated from the two
back ones, A,, A'2, as has already been pointed out on p. 89, by a plate of glass of
thickness (about 2 millims.) sufficient to stop the fluorescence-exciting rays scattered
by one pair of cubes from entering the other, whilst the cubes Aj and A3 are divided
by a black paper screen from A.\ and A's, so that no light could pass sideways from
Aj or A2 to A.\ or A'a.

The horizontal surfaces Ax, A2, A']( A'2, are those which are illuminated, whilst the
light studied is that emitted by the surfaces al} a\.

The surfaces 1>V, dL ; V ^ d\ ; 62, c.2, d2 ; l>'.,, c'2, d'2> as well as the lower horizontal
surfaces parallel to A1; A.\; A2, A'2, are roughened, so that there is no regular reflec-
tion from them, and thus are more or less opaque. But the surfaces ctit c] ; a\, c\ ;
a.2, a'2, are perfectly transparent, as also the upper horizontal surfaces A.lt A\ ; A.,, A';.

When Aj, A'1; A2, A'2, are all uniformly illuminated by violet or ultra-violet rays,
the fluorescence of each block ought to be the same, if we disregard the very refined
correction that the blocks A1} A.\, have each two transparent vertical surfaces, namely
«!, Cp and a\, c\, whereas the blocks A2 and A'2 have each only one, a.2 and «'.,. At
any rate, since the rays enter vertically, and therefore parallel to these surfaces, any
difference that may possibly arise in this way in the intensity of the scattered rays
would for all practical purposes be quite immaterial.

We have seen (p. 92) that the error, when sought for. was found to be quite inap-
preciable and lies within the much larger errors of observation. Apart altogether
from this consideration, the fluorescent light emitted horizontally from the surfaces
rtp ct'j, ought to appear the same, if all is right, and this is ascertained by the photo-
meter before an experiment is made.

2. Fig. 7 is a plan of the apparatus showing the position of the fluorescent
cubes, &c., in the photographic experiments. The cubes are placed Aj, &c., and
are illuminated by light incident vertically upon them either from the spark in
air between cadmium electrodes or the electric arc. The source of illumination was
almost invariably the former in the photographic determination, the poles being
about 15 millims. apart, and 2 centims. above the fluorescent surface. The spark
was obtained by the discharge of a Leyden jar charged by the terminals of the
secondary of a 10-inch Apps coil,

94 ME. JOHN BDRKE ON THE CHANGE OF

The photometer, fluorescent substance, and source of illumination in the photo-
graphic experiments were enclosed in a wooden box, shown in the figure. The
photometer was at one end, and the fluorescent substance and illuminating arrange-
ment at the other.

Fig. 7.

KK represent the electrodes, AI} &c., the fluorescent cubes, and P the photometer,
consisting of two vertical slits, s, s', and a double convex lens, L ; while tt is a card-
board vertical screen, with a horizontal slit, 3 millims. wide, placed immediately in
front of the vertical surfaces of the cubes (the slit cannot be shown in the figure).
This screen admits of being moved up and down, so that the light at various depths
below the illuminated surface may be studied. The usual depth of the upper edge
of the slit below the horizontal surface of the fluorescent cubes was about 3 millims.
Another vertical screen (rr), G centims. in length, extends along the whole length of
the box from the photometer to the other screen, and divides the horizontal slit into
two parts, the light from each part being allowed to pass only through the corre-
sponding vertical slit of the photometer.

The vertical slits s and a' were adjustable with micrometer screws.

The lens L (of focal length 4'5 centims.) served to form an image of the horizontal
slit in the focal plane of the eyepiece, or on the focussing glass of the camera, which
admitted of being raised or lowered, so that, when necessary, a number of photographs
could be obtained successively on one plate. The photometer contrivance communi-
cated with the camera by means of the conical tube T, leading up to a front slide which
remained fixed whilst the camera itself moved up and down (fig. 8). The distance
between the lens L and the focal plane of the eyepiece was 18 centims. The whole
of the interior of the apparatus was blackened, so that not the slightest trace of
reflection could be perceived.

A screen of blue cobalt glass was interposed between the source of illumination and
the fluorescent substance so as to cut off the less refrangible rays. When the appa-
ratus is in use a horizontal green band with a dark vertical line or narrow band in

ABSORPTION- PRODUCED BY FLUORESCENCE.

95

the middle is seen through the eyepiece or on the focussing plate of the camera. The
two luminous portions are images of the parts of the surfaces alf a',, cut off by the
horizontal slit immediately in front of them ; the screen rr and the paper screen
which separates Al5 A2, from A'1} A'2, divide the image of this slit into two parts, and
produce the dark vertical line in the middle.

By this arrangement, which was found to be on the whole the most convenient,
we have clearly the opportunity, not merely of comparing the two lights side by side,
which is absolutely necessary, but also of illuminating the surfaces A,, &c.,
by the same source simultaneously and under similar conditions.

ie. 8.

(C)

Any slight amount of light, chiefly blue, but partly red, which, having penetrated
the cobalt glass, may have been scattered by the fluorescent substance as false
dispersion (a term used by STOKES to designate the light scattered by irregular
reflection from small particles, which is always polarized in the plane of reflection,
and of the same wave-length as the incident light), can be eliminated, in consequence
of its being polarized, by interposing a NICOL'S prism or a pile of plates between the
horizontal and the vertical slits of the photometer. This method is open to the
objection that it would weaken the fluorescent light, and it has been thought
unnecessary to incur this disadvantage for two reasons. In the first place, the

96 MR. JOHN BURKE ON THE CHANGE OF

whole amount of light scattered has been found to be inappreciable (see p. 93), and
on this account alone the portion now under consideration may be neglected. In the
second place, if the error were appreciable, it would tend to diminish rather than
to increase the apparent difference between the absorptions before and during
fluorescence respectively. For the scattered light would add to the total light
emitted from aL, a\, and would therefore tend to reduce the effect.

The length of the horizontal slit tt being about 2 centime., and its width 2 millims.,
the image was 6 centims. by 6 millims., for the focal length was 4'5 centims., and
the focal distances 6 centims. and 18 centims. respectively. By using a lens of
shorter focal length, however, an image of tt can be formed on the horizontal slit of
a spectroscope placed so that the refraction takes place in a vertical plane, and
observations can be made of the absorption in different parts of the spectrum. This,
however, is by no means an easy matter to accomplish, and we have confined ourselves
to the light taken as a whole.

With reference to the illumination, it was thought at first that one of the
electrodes might possibly always be brighter than the other. Accordingly a plan
was adopted of continually reversing the current in the primary coil by means of a
wheel attached to the commutator of the coil, which was made to rotate by means
of a motor. Unfortunately the exposures required in the photographic work were
thereby greatly increased. The experiments, however, show that the average
illumination obtained from each electrode is, for all purposes with which we are
concerned, practically the same.

EYE OBSERVATIONS.

The accuracy of these observations has been found to depend, to a very large
extent, upon experience on the part of the observer in dealing with the relative
intensity of the lights placed side by side. It is, of course, a familiar fact in
photometry that it is very difficult to form a judgment as to the equality of two
illuminated surfaces when the illumination is either very feeble or very powerful ;
and accordingly this is a source of considerable inaccuracy in such determinations.
It is not desirable in either case to examine the relative intensity attentively for any
length of time, as it is obviously possible, by viewing two feebly illuminated surfaces
side by side for a sufficiently long time, to imagine that they are almost equal when
really they are not so, and this remark appears to be particularly applicable to such
cases in which the two colours are of slightly different tints. Instances have arisen
in which it has besn impossible to arrive at any result whatever as to the relative
intensity. In such cases, however, the difference of tint was conspicuously marked.

It is preferable, when the observations are made by eye, to make a large number
of measurements quickly, and to take the means of these first impressions, noting in
turn the maximum and minimum widths of the slit, and then reversing the system of
screens which serve to shelter the combination of the cubes from the exciting rays.

ABSORPTION PRODUCED BY FLUORESCENCE. 97

The screen is so arranged as to admit of being worked by the observer, by means of
handles, whilst he is otherwise engaged in taking observations through the eyepiece,
but the mechanism by which this is effected is not shown in the figure, which is a
sketch of the arrangement adapted chiefly to photographic work. In the eye obser-
vation an optical bench was used, being a suitable stand for the apparatus.

To test the accuracy attained and also the uniformity of the illumination, or rather
its symmetry, we may give the result of thirty observations. These observations
were made in the earlier stages of the experiments, when but little experience had
been acquired. If the illumination had been uniform, the intensity of the fluorescent
light should have appeared the same when the slits were of equal width.

Fifteen readings of the left slit, when the right was fixed at 8, give 7 '8 7 as the
mean of the readings, the probable error of a single observation being 0'54, and the
probable error from the mean 0'135. Similarly the other fifteen observations, when
the left slit was fixed at 8 and the right slit movable, gave the mean of the readings
of the right slit to be 7 '73, the probable error of a single observation being 0'53, and
that from the mean 0*133. It may be remarked that in this particular experiment
the readings of each observation were taken with an opening slit, and the apparent
equality was obtained before the real equality existed, that is that in both sets of
observations the mean of the readings was somewhat less than what it should have
been ; but the process of reversing eliminates this. Then the readings to the two
sets of observations are 7'87 ± 0'135 and 7'73 ± 0'133. The ratio of the illumination

fr.oy / A .1 OK\

of the cubes on the right to that of those on the left is - - ( 1 ± ~z~j^ ) in the first

o / / n.i QQ\

set of observations, and 77^.3 / I 1 ± » „.-, ) in the second, if we neglect the square

lid I \ {'tO /

„ /0-133

of

7-73,

The ratios of the illumination of the cubes on the right to that of those on
the left, as given by the two sets of observations, are (0'984 ± 0*017) and
(T035 ± 0'017). The mean of these two ratios is TOOO. The probable error in the
ratio is 0'017 in either case. Hence the probable error from the mean, since the
two determinations of the ratio of the two illuminations are quite independent, is

=- = 0'012. And the probable error of thirty observations is about 1 per cent.

from the mean.

In the detection of the change of absorption, an error of 20 per cent, will obliterate
the effect.

In tabulating the experiments, the following method has been adopted. The
cubes are indicated by the letters A1( A2> A'l; A'3> the arrangement being
as shown in the accompanying figure.

The intensities compared are those of the rays proceeding from the
faces ai and a\.

VOL. CXCI. — A. O

98

MR. JOHN BURKE ON THE CHANGE OF

In each of the experiments one or more of the cubes was shaded from the
fluorescence-producing light. The letter in the fifth column indicates which was
that protected.

Experiments (a).

Experiment.

No. of
observations.

s.

'•

Shaded
cube.

a.

Difference
from mean.

T.

20

5-9-2

4'

A2

0-48

+ 0-03

II.

20

5-45

8

A'3

0-4G

+ 0-01

III.

20

8

5-29

A2

0-51

+ 0-06

IV.

20

5-89

8

A,

0-36

-0-09

Mean = 0'45

Similarly taking in all eighty observations for a, ft, and ft /(I -f a), the following
are obtained, the fixed slit being at 8 :—

Mean of eighty observa-
tions of movable slit.

Probable error of a
single observation.

Probable error from
mean.

(«)

5-52

0-296

0-03

C/J)

6-20

0-514

0-05

g
(l+«)

4-17

0-50

0-05

Hence

therefore

Also we easily obtain

and

8 , 0-03

*

= 1-449 [1 ± 0-005] ;

a = 0-449 ± 0'005.

ft = 0787 ± 0-006

= 0-521 ±0-006.

It may be remarked that the probable error in a is much less than that in ft or
ft/(l + a). My attention was called to this when revising the paper. A glance at
figs. 3, 4, and 5 will make this clear. The light emanating from a^ and a\ will

ABSORPTION PRODUCED BY FLUORESCENCE. 99

be of the same colour in the determination of a ; but in the determination of ft and
ft /(I + «) we are comparing the light passing through a non-fluorescing body with
that emitted by it directly, and the result shows that there is a difference in tint
between the light given out by ax and that emanating from a\.

Substituting the values of a and ft thus obtained in ft/(l + «), we obtain 0'54,
which is somewhat larger than that independently determined value 0'52. The
difference, though only a third of the probable error of a single observation, is
nevertheless three times the probable error from the mean. But this, Professor
SCHUSTER suggests, was probably due to slight variations in the symmetry of the
illumination. The value of /3/(l + a) determined independently is, however, quite
close enough, and may for all practical purposes be considered a very good verification
of the accuracy of the values obtained for a and ft.

Taking the observed values of a, ft, and ft/(l -J- a) to be 0'450, 0787, and 0'520
respectively, we obtain by the method of least squares* the most probable values of
a and ft to be a = 0'455 and ft = 0'786, which give ft/(\ -f «) — 0'54 as the most
probable value of this ratio.

Hence it appears that the observed values of a and ft are more reliable than those
of /?/(!+ a).

*Let

«! = 1 + a, X = «j + x,

ft = ft, Y = ft + y,

ft _Y

71 -1+-*' *'-X'

where X and Y are the most probable values of 1 + a and ft. y^, ft, and 7 being the observed values
of 1 + <*, P, and pj(l + a) respectively.
Then

and differentiating, we get

'X3

Y - 7lX - X2 (ft - Y) = 0.

Substituting for X and Y their values («t + a') and (ft + y) and neglecting the powers of x and y
higher than the first, we obtain

(1) *ia! + ft2/:=0;

(2) y [1 + «!*] + ^ y - 71 «E + .ft = 0.

Substituting for »v ft, 7j, 1'45, 0'787, and 0'52 respectively, we get

x = 0-005 ;
2, =---0-01.

Hence

X = 1-455, Y = 0-786,

or the most probable values of « and ft are

a. = 0-455 ; ft = 0-786.

o 2

100 MR. JOHN BURKE ON THE CHANGE OF

The coefficients of absorption a and & are given by the equations (see p. 89)

a. = 1 — (y + a) ;
£ = 1 - (y + &).

Hence

r -f a = 0-545,

r + 6 = 0-214,
and the increase in the absorption, or the difference between a and b, = 0'331.

INFLUENCE OF TEMPERATURE.

It does not appear that temperature can have any effect on the phenomenon except
either by weakening the intensity of the fluorescence of the transmitting cube Ax or
A'j by heating effects due to the spark, or by a change of absorption due to tempera-
ture directly. But, firstly, since all the fluorescing cubes are uniformly illuminated,
and as we are merely concerned with the ratio of the intensity of the light emitted
from al and a\ respectively, the heating being the same for all the illuminated cubes
on the one hand and for all the unilluminated ones on the other, the actual intensity
E0 emitted by each cube, which may possibly be diminished by a rise of temperature,
does not enter into the results. Secondly, in the eye observations, the various com-
binations of screening were brought about with considerable rapidity, sufficient to
prevent the possibility of any such effect due to changes of temperature arising.
No doubt some of the light from A, is absorbed by A! in the determination of a and /3
for Aj ; any change of temperature, however, by this means would necessarily be
slow, arid consequently its effect, if any, small.

Photographic Method.

The following set of experiments was first carried out.

It was afterwards found not to be altogether satisfactory on account of the neces-
sity in these cases of superposing two photographs, and taking the resultant effect
to be proportional to the sum of the separate ones. The experiments, however, are
of some interest in revealing this fact : that when considerable differences in inten-
sities have to be dealt with the resultant effect of superposing two photographs is
not proportional to their sum. (It has not been considered necessary to reproduce
these particular photographs here.) Captain ABNEY has previously shown that this
is so.

The first photograph taken, which we shall call Photograph A, consisted of five
successive images, on the one plate, corresponding respectively to a gradual alteration

ABSORPTION PRODUCED BY FLUORESCENCE.

101

in the width of the slits. It was really a blank experiment to test whether the
illumination was uniform or not.

All the four cubes were exposed to the exciting illuminations.

(A.)

i

n .

s.

1

7-5

8

2

8

8

3

8

7-6

4

8

7-2

5

8

6-8

i

The table shows the width of the slits for each of the five photographs. The
nearest approach to equality was obtained in 2 and 3, and the relative intensity was
reversed. The exposures were twelve minutes each. The plate was, however, con-
siderably fogged. The next plate (B) was taken in the same way as before, but the

width of the slits was diminished.

(B.)

1

6-'.

'•

5-5

5'0

2

5-25

5-0

3

5-0

5-0

4

4-75

5-0

5

4-5

5-0

The equality was here between 2 and 3. The photographs were still too dense,
and the fogging too great, to make it possible to say with any degree of accuracy
which way the inequality, if any, went ; and it was really difficult to say if there
was any difference between 3 and 4.

(C) was taken with an experiment of six minutes.

In this case the equality was between 3 and 4.

It was not possible to obtain a greater accuracy than 5 per cent, in any single
plate, but it did not seem necessary in the determination of the absorption to attain
even to this.

In order to test whether the absorption was the same whether the substance was
fluorescing or not, the experiment was arranged in the following way : —

1. A photograph is taken with the screen, as shown in fig. 3. Let p be the ratio
of intensity of the light emitted by «/ to that emitted by at ; then

E0 (1

102 ME. JOHN BURKE ON THE CHANGE OF

2. If p2 is the ratio of the intensity of the light emitted by «/ to that emitted by
«! in the arrangement fig. 5, then

__

EO (1 + «) ~

Hence

and if a = /3,

Pi+2h= I-

The experiment is then performed by first determining plt and putting p.z = 1 — p±,
and observing whether the two lights so obtained are of equal intensity.

Again p2 is first obtained, and p} put = 1 — p2, and equality again looked for.

In the first case it is found necessary to diminish s', or widen s, to obtain equality,
and since the radiation from a} and a\ are inversely as the width of the slits, it
follows that the real value of p.2 is greater than the calculated one, and in the second
it is also found necessary to diminish ./ or widen s, which shows that pl + p2 > 1,

1 _L /Q

or that - - > 1, and therefore /3 > a. Hence the absorption is greater when the

substance is fluorescing than when it is not, a and /S being the coefficients of trans-
mission.

To test this photographically, first of all an exposure is given when the screening
of the cubes is as in (1), and then, without altering the position of the camera, the
screening is arranged as in (2). The slits are such that s' = 2s, since we want the

1 -4- R 1 -4- R

ratio - — , not ;— r --- - . Thus, if the photographic effect be proportional to the

time and the intensity, the effect in the first instance on the photographic plates may
be represented thus :

1

and in the second

Hence the resultant effects, when superposed, are

E0 (1 + 0) ; E0 (1 + «).

The camera is then slightly raised, s' is made equal to s, all the screens are
removed, and a photograph is taken on the same plate with an exposure merely
equal to that of each of the other two, the object being to obtain on the one plate
a comparison, not merely with reference to the absorption, but also with that of the
illumination. The effects on the photographic plate ought to be as follows : —

E0 a + of.)

ABSORPTION PRODUCED BY FLUORESCENCE.

103

If there is no change of absorption (a) ought to be of the same intensity as (b).
As a matter of fact, it was always found to be much denser in the negative. The
intensities of (b) and ((/) were never exactly equal, but, on the contrary, (d) was
always darker than (6). Now, the width of the slit for (d) was double that for (b),
but the exposure was half, which shows that if we double the intensity of the light
to which a photographic plate is exposed, the exposure must be less than half in
order that the same effect should be produced. In other words, the photographic
effect is not proportional to the intensity and to the time conjointly.

The actual width of the slits and the exposures in one particular instance were as
follows : —

•Superposed

(.31

8 m.

The relative intensity of (a) and (c) was not constant, which shows that the
average illumination may be different at different times, and it was consequently
necessary to make comparisons of photographs taken simultaneously, the symmetry
of the illumination having first been secured.

Fifteen plates were obtained, which are in the possession of the Royal Society and
can be inspected.

Three photographs show the degree of approximation in the uniformity of the
illuminations.

Some others exhibit the remarkable difference in the relative intensities under
conditions such that if there were no change in the absorptive power the two
intensities should be equal.

Three photographs show the superior and inferior limits of the value ot «, the A2
cube being screened, and the slits are s = 4, s' = 2'5.

The relative intensities as given appear to be the nearest approach to the limits in
the width of s' we should wish to arrive at. Thus we obtain for « the limits as

follows : —

4
1 + a < — ; therefore a < 0'48,

4
> -^

> 0-43.

The most probable value for « obtained by the eye observations »— was 0'455.
Three other photographs give the limits for /3, A't and A, being in this case
screened.

s lies between 3 and 3 '5, so that

104 ON THE CHANGE OF ABSORPTION PRODUCED BY FLUORESCENCE.

£ > -V, »>• £ > 075,

4

< — < 0-89,

the value with the eye observations p— being 0'78G.

Assuming the value of a to be 0'48, and that there is no difference between it and

f3, we obtain

+ «) = 0-32.

Consequently if we screen A.\ and adjust

s = 1-3,
s' = 4-0,

we should obtain equality of intensity on a photographic plate. This, however, does
not appear to be the case, as shown in a photograph in which two successive images
.of the horizontal slit are given, one being the result of reversing the screws and
screen, so that A} was for the second image sheltered from the illuminations, and A'j

exposed, and

s=4-0,

s' = 1-3.

It shows, moreover, that a reversal in the arrangement also reverses the effect.

Another plate exhibits the effect of superposing two such photographs, showing the
absence of want of symmetry in the illumination.

Some other photographs show the equality of illumination and change of
absorption from the calculated values, and two others the result of superposing two
photographs in the following way.

Firstly, A\ is screened, and s = s' = 4 ; and an exposure of 30 minutes gives an
effect nearly proportional to E0/3 on the left and E0(l + a) on the right. Then s is
closed completely, and the screen altered from A'j to A'^ and an additional exposure
is given without changing the position of the photographic plate, so that an increase
proportional to E0 is produced on the left. Now, since y3 does not differ so very much
from unity, we may take the resultant effect as equal to the sum, though in reality,
as shown on p. 103, it is less. The effect on the left side of the plate is then
proportional to E0 (1 + /3), and that on the right due to a single exposure to
E0 (1 + a), and if a =/J, these two should have been equal. The plates show that
they are not, but give /J > a. This is clearly the simplest method of exhibiting the
phenomenon. If the photographic effect were proportional to the intensity the
difference would clearly be more marked.

I desire to thank Professors ARTHUR SCHUSTER and G. F. FITZ&ERALD for many
useful suggestions and much valuable advice throughout this research, and also
Professor J. H. POYNTING, to whom the investigation owes its origin.

[ 105 ]

III. On the Occlusion of Hydrogen and Oxyrjen by Palladium.

By LUDWIU MOND, Ph.D., F.R.S., WILLIAM KAMSAY, Ph.D., F.R.S., and

JOHN SHIELDS, D.Sc., Ph.D.

Received December 8, — Read December 10, 1897.

CONTENTS.

PAGE

I. Introduction 105

II. Preparation of the Palladium 100

III. Density of Palladium 107

IV. Analysis of Palladium Black 107

V. Absorption of Oxygen by Palladium 108

VI. Observations on the Occlusion of Hydrogen by Palladium Black, Sponge, and Foil 111

VII. The Heat of Occlusion of Hydrogen by Palladium Black 117

VIII. Investigation of the Ratio Palladium : Hydrogen J 20

IX. On the Heat of Occlusion and the Heat of Condensation 122

X. The Influence of Increased Pressure on the Occlusion of Hydrogen by Palladium

Black 12:!

XI. The Heat of Absorption of Oxygen by Palladium Black 121

I. Introduction.

SINCE the discovery by GRAHAM ('Phil. Trans.,' I860, p. 399, also ' GRAHAM'S
Researches,' p. 235), that palladium possesses the property of occluding or absorbing
large quantities of hydrogen, chemical literature has been enriched by various con-
tributions to our knowledge of this interesting phenomenon. To mention only a few
of these, some of which will later on be considered in detail, we have the results of
investigations on the change of volume undergone by palladium on the occlusion of
hydrogen, the density of the occluded hydrogen, the heat changes which take place,
and the pressure of " palladium hydrogen " at different temperatures, &c.

The solution of hydrogen in, or the combination of hydrogen with, palladium,
which we may be permitted provisionally to term palladium hydrogen, is now often
used in laboratories as a convenient source of pure hydrogen ; and finely divided
palladium, on account of the facility with which it absorbs hydrogen, is also
extensively employed in gas analysis. Hydrogen, occluded by palladium, may con-
veniently be weighed as palladium hydrogen, and recourse was had by REISER

VOL. CXCI. — A. P 12.5.98

106 DRS. L. MOND, W. RAMSAY, AND J. SHIELDS, ON THE

('Am. Chem. Journ.,' vol. 10, p. 249) to this method of weighing hydrogen in one of
the most delicate of chemical operations, viz., that of atomic weight determination.

Any addition to our knowledge of this substance may therefore be expected to be
of interest. It will readily be admitted that the phenomenon of occlusion is one
which requires further elucidation, and in submitting the present communication to
the Society we hope that the additional work we now bring forward will not only be
of interest for its own sake, but that it will finally assist us in arriving at a definite
decision with regard to the nature of occlusion.

In previous communications ('Phil. Trans.,' A, vol. 186, p. 657, and A, vol. 190,
]>. 129) we discussed the relations of platinum black to oxygen, hydrogen, and other
gases, and on p. 152 of the latter paper we pointed out that until the relation of the
palladium to the hydrogen in the comparatively well defined substance palladium
hydrogen was ascertained, there was little hope cf solving the corresponding problem
for the less well defined substance platinum hydrogen.

We therefore turned our attention to the general behaviour of palladium to oxygen
and hydrogen, and now give an account of our work on the subject, which includes
measurements of the heat evolved on the absorption of these gases, and a compara-
tive study of the occlusion of hydrogen by palladium in different states of
aggregation.

II. P reparation oftJie Palladium.

The palladium employed in these experiments was in the form of («) black,
(6) sponge, and (c) foil.

Palladium black. — Some strips of metallic palladium, which were said to have
been prepared by WOLLASTON, were purified in the following manner : — After dis-
solving in aqua rc.gia and evaporating several times successively with water and
hydrochloric acid, the solution was treated with excess of ammonia, and filtered.
From the filtrate pure palladosamine chloride was precipitated by leading a stream
of hydrochloric acid gas through it. This was filtered off, washed with strong
hydrochloric acid, dried, and ignited. The residue, consisting of palladium sponge,
was then again brought into solution by digesting with aqueous hydrochloric acid
through which a current of chlorine gas was led. After evaporating, to get rid of
the excess of hydrochloric acid and chlorine, the solution of palladium was neutralized
with sodium carbonate, and slowly poured into a boiling, very dilute solution con-
taining excess of sodium formate. The palladium black, precipitated in this way,
was then repeatedly boiled out with distilled water until free from alkalis and
alkaline chlorides, and dried at 100° C.

Palladium sponge. — The palladium sponge employed in the course of these experi-
ments was prepared by fully charging the necessary quantity of palladium black
with hydrogen, and then igniting at a red heat in vacuo, since simple ignition of

OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM. 107

palladium black in the air or in vacua, unless conducted at an extremely high
temperature, is insufficient to remove the oxygen initially contained in it.

Palladium foil. — A portion of thin foil was procured from Messrs. JOHNSON and
MATTHEY, and was guaranteed to contain over 99 per cent, of palladium. The
special treatment to which this was subjected will be described in Section VI.

III. Density of Palladium.

It has always been customary, in stating the quantity of hydrogen occluded by
palladium, to express the result as so many volumes of hydrogen occluded by unit
volume of palladium.

This makes a knowledge of the density of palladium necessary.

The density of the foil employed was found to be 12-1, whilst a direct determina-
tion of the density of palladium black, dried at 100° 0. in a specially constructed
pyknometer, gave the value 10 '6. This refers to palladium black prepared as
described. An analysis of the black soon after it was dried showed that it contained
0'72 per cent, of water and l'G5 per cent, of oxygen, existing probably as oxide. If
we make an allowance for this amount of water contained in the black the density
rises to 11 '4.

The palladium black used for the density determination was the remainder (over
5 grams) contained in a bottle which had been frequently opened during about four
months, and which in all probability had absorbed more water. If we assume that
the percentage of water had gone up from 072 per cent, to 1 per cent, during the
four months, then the density of dry palladium black would become ITS. No
allowance, however, has been made for the fact that the palladium black contained
oxygen, which would also tend to lower the density.

Since the knowledge of the density was only essential for the purpose of translating
the actual quantity of hydrogen occluded into the number of volumes of hydrogen
occluded by unit volume of palladium, we have thought it better, in view of the
above circumstances and of the fact that the determination of the density of a fine
powder like palladium black is not at all easy, to adopt, for comparative purposes,
the value 12-0 as the density of palladium in its three states of aggregation, viz.,
black, sponge, and foil. As the actual measurements are always given in addition,
this assumption ought not to give rise to any error, and is to a certain extent justified
by the fact that, under proper conditions, the number of volumes of hydrogen
occluded by unit volume of the different varieties of palladium remains practically
the same.

IV. Analysis of Palladium Black.

Before beginning any extended experiments with palladium black a preliminary
examination was made. From the fact that platinum black prepared in the same

r 2

108 DRS. L. MOND, W. RAMSAY, AND J. SHIELDS, ON THE

way invariably contains oxygen, it was thought that palladium black would also
contain oxygen. On igniting T5650 gram palladium black in a hard glass tube in
vacuo, however, only 0'33 cub. centim. oxygen and 0'56 cub. centim. carbon dioxide
were extracted. This proves that the oxygen, if any be present, is more firmly
retained by palladium than by platinum.

Another portion of palladium black, weighing 1'9795 gram, was placed in a hard
glass bulb tube, one end of which was attached to a weighed U-tube containing
P20r), and thence to the pump, whilst the other end was connected with an apparatus
supplying pure dry hydrogen. On ignition in vacuo 0'0142 gram H20 was collected
in the U-tube. This corresponds to the presence of 072 per cent, of water. No
oxygen was given off, but on passing hydrogen over the sponge remaining in the
hard glass tube 0'032G gram H3O was obtained. The palladium black therefore
contained l'G5 per cent, of oxygen, corresponding to 23'G9 per cent. Pd2O, or 12'G9
per cent. PdO. If we assume that the oxygen exists as PdO, and if we estimate the
pure palladium by difference, we get for the analysis of palladium black : —

Palladium 8G'59 per cent. ;

.Palladium oxide (PdO) . . 12'G9 „ = TG5 per cent. Oo ;
Water 072

From the weight of water formed, it follows that the palladium black contained
22'82 cub. centims., or 138 volumes of oxygen. The oxygen initially present in
palladium black was also determined in another way.

Palladium black was fully chai'ged with hydrogen, a portion of which was really
occluded, whilst the remainder formed water with the oxygen pre-existing in the
black. The occluded hydrogen is easily determined by extracting at a red heat in
vacuo, and half the difference between the volume of hydrogen used and that
pumped off represents the volume of oxygen initially present. In this way it was
found (see table, p. 125) that the palladium black contained 140, 141, and 143
volumes of oxygen, whilst the gravimetric determination indicated the presence of
138 volumes.

Pure palladium sponge is almost white, whilst that obtained by the direct ignition
of palladium black in vacuo is much darker in colour owing to the presence of oxygen
or oxide in it.

V. Absorption of Oxygen by Palladium.

We have already shown ('Phil. Trans.,' A, 1895, vol. 18G, p. 682) that when
platinum black is heated in an atmosphere of oxygen, kept approximately at
ordinary pressure, absorption of oxygen takes place until the temperature rises to
about 360° C., when the gas is again expelled. If this is really a process of
oxidation, then, since palladium is usually regarded as a more easily oxidisable

OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM.

109

metal, it is to be inferred that palladium black will behave in the same way, although
perhaps to a greater extent.

In the first experiment 3 '5 19 grams of palladium black were heated in an atmos-
phere of oxygen in an apparatus similar to that employed for the corresponding
experiments with platinum. The following results show that oxygen was steadily
absorbed : —

Heated for

2 hours at 132"
6 „ 184°
6 „ 237°
6 280°

Total .

Oxygen absorbed.

cub. centime.
3-29

volumes.
11-2

1818

62-0

33-73

115-1

46-18

157-6

101-38

345-9

Up to 280° C. the total quantity of oxygen absorbed, viz., 101 '38 cub. centims.
= 0'1450 gram, is more than half the quantity of oxygen (0'2656 gram) necessary
to form the compound Pd20. Owing to an accident, which prevented the experi-
ment being carried further, a new portion of palladium black was taken, and the
heating in oxygen started at 280° C. The following results were obtained :—

OXYGEN Absorbed by Palladium at High Temperatures.

Palladium black used, 2'8100 grains = O234 cub. centim.

Heated for

Oxygen absorbec

1.

13

hours at 280° C

cub. centim?.
87-59

volumes.

374

gram.

0-1253

5

H

3

280° C
360° C
444° C

6-42
16-28
18-21

27
70
78

0-0092
0-0233
0-0260

3

444" C

17-79

76

0-0254

2

444° C

10-68

47

0-0153

U

600° C

9-17

39

0-0131

3

a red heat (naked flame)

60-34

258

0-0863

Total

226-48

969

0-3239

The quantity of oxygen theoretically necessary for the formation of the oxide Pd.jO
is 0-2121 gram, and it will be observed that the oxygen actually absorbed exceeds
this quantity. There is no reason to suppose that the absorption of oxygen had
ceased after the experimental tube had been heated for three hours in the naked flame.
The experiment was stopped at this stage, however, in order to see whether the

110 DRS. L. MOND, W. RAMSAY, AND .T. SHIELDS, ON THE

excess of oxygen over and above that required for the formation of Pd20 could he
removed in vacuo at any intermediate temperature.

After being placed in communication with the pump the experimental tube was
then exposed to gradually increasing temperatures, and finally to as high a tem-
perature as the hard glass tube would stand without collapsing.

Only a few bubbles of gas (about 1 cub. centim.) could be extracted, and hence
it appears that if either or both of the oxides Pd20 and PdO are formed, they must
be stable in vacuo at a dull red heat. The substance formed during the absorption
of oxygen has a dark brown colour, has lost all the appearance of palladium sponge,
and is, without doubt, an oxide or mixture of oxides.

According to WILM (' Berichte,' 1882, p. 2225), the sub-oxide of palladium, Pd20,
may be prepared by heating the sponge in a current of air until it attains a constant
weight. If this is so, it is curious that we have not been able to observe any
discontinuity in the absorption of oxygen, and we are inclined to think that palladium
sponge, if heated for a sufficiently long time in a current of air, should yield not the
sub-oxide, but palladium oxide, PdO.

We have already seen that palladium black, prepared in the way described,
contains .about 140 volumes, or l'G5 per cent, of oxygen, and the fact that the
substance was dried at 100° C. probably accounts for the presence of a certain
quantity of this oxygen. In connection with the calorimetric experiments to be
described later on, it was of interest to us to know how much oxygen palladium
black, rendered free from oxygen, would absorb directly at 0° or at the ordinary
temperature. We found it impossible to remove the oxygen without converting the
black into sponge, but the required information was obtained in the following way : —

The sample of palladium black was fully charged up with hydrogen, whereby all
the oxygen was removed as water. As will be seen presently, the bulk (over
90 per cent.) of the occluded hydrogen was next extracted by exhausting at 100° C.
To the palladium black containing only a comparatively small quantity of hydrogen,
oxygen was then admitted, in the first instance very slowly, whilst the experimental
tube was kept cold by immersion in melting ice, and in the second rapidly, whilst
the tube was simply exposed to the atmosphere. Of course a portion of the oxygen
formed water with the residual hydrogen, the rest being absorbed, in the first case
without appreciable rise of temperature, and in the second with considerable rise of
temperature. The absorbed oxygen was then determined in both specimens by
charging up fully with hydrogen and exhausting at a red heat. It was found that
about 40 volumes and 120 volumes respectively of oxygen were absorbed.

At 0° C., therefore, the absorptive power of palladium black for oxygen is con-
siderably lessened.

It will readily be admitted that the absorption of oxygen by palladium black is
really a process of oxidation, and although palladium and platinum belong to different
sub-divisions of the platinum group of metals, we think the absorption of oxygen by

OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM.

Ill

platinum black may also best be explained as a superficial oxidation, the chief
difference being that the oxide of platinum decomposes or dissociates at a lower
temperature than the corresponding oxide of palladium.

VI. Observations on the Occlusion of Hydrogen by Palladium Black, Sponge, and Foil.

A series of experiments was next undertaken with the object of comparing the
relative occlusive power of palladium in the form of (a) black, (b) sponge, and (c) foil
for hydrogen, and investigating the behaviour of the substances produced. This was
especially necessary in the case of palladium foil or wire, since the statements of
different observers are at considerable variance with each other. Whilst most are
agreed that palladium in the compact state readily occludes the maximum quantity
of hydrogen when charged electrolytically, there are many cases on record in which
the compact metal only occludes a relatively small quantity of hydrogen when it is
simply exposed or ignited in the gas.

a. Palladium Black. — 1*619 gram = 0*135 cub. centim. palladium black was
charged with pure dry hydrogen in an apparatus similar to that employed for the
corresponding experiments with platinum black ('Phil. Trans.,' A, 181)5, vol. 186,
pp. 668 and 687). Altogether 152*76 cub. centims. (Oc and 760 millims.) = 1131
volumes of hydrogen were absorbed, the greater portion of which was taken up
almost instantaneously.

From the experiments already described, it is to be expected that a certain fraction
of the total hydrogen absorbed formed water with the oxygen pre-existing in the
palladium black. In order to determine the quantity of hydrogen which was really
occluded and how much was given off in vacuo at different temperatures, the experi-
mental tube was exhausted first at the ordinary temperature, then at higher tempera-
tures, and finally at a red heat. The results are given in the following table :—

Palladium black used, T619 gram — 0135 cub. coutim.
Total H2 absorbed, 15276 cub. centims. = 1131 volume*.

Temperature.

Occluded hydrogen pumped ofl'.

°C.

'20
100
184
280
444
700 (?)

cub. centims.

105-72
2-41
2-27
3-29
1-32
0-21

volumes.
783-2
17-8
16-8
24-4
9-8
1-6

Total . £ .

115-22

853-6

The difference between the total hydrogen absorbed and that extracted at a dull

112 DRS. L. MOND, W. RAMSAY, AND J, SHIELDS, ON THE

red heat in vacuo corresponds to the presence of 1877 cub. centinis., or 139*1 volumes,
of oxygen in the palladium black. Since this volumetric estimation of the amount of
absorbed oxygen is in good agreement with the direct gravimetric determination, viz.,
138 volumes, we are justified in concluding that 115'22 cub. centims., or 853*6 volumes,
is the quantity of hydrogen really occluded.

It will be observed that practically the whole of the occluded hydrogen was extracted
at 444°, and that about 92 per cent, of it can be pumped off at the ordinary tempera-
ture. The last traces of the hydrogen given off at the ordinary temperature come off
very slowly, so that the pumping requires to be continued for a day, or longer.

It was noticed that the palladium black contracts and passes into sponge above
184° C.

This result, viz., that palladium black occludes about 854* volumes of hydrogen,
was confirmed by additional experiments (v. Table, p, 125), in which 860*, 868*, aud
868* volumes were occluded.

b. Palladium Sponge. — The pure palladium sponge remaining behind in the experi-
mental tube after the preceding experiment was completed was charged with hydrogen.
112*4 cub. centims. = 833""" volumes were occluded. Of this 111'5 cub. centims., or
826'1 volumes, representing about 99 per cent, of the whole, were pumped off at the
ordinary temperature, whilst on heating to redness 1*4 cub. centim., or 10*4 volumes,
were extracted. It would appear, therefore, from this experiment that more hydrogen
can be removed from palladium sponge than from palladium black at the ordinary
temperature.

This does not agree with GRAHAM'S observation (' Researches,' p. 287) that "in
the pulverulent spongy state palladium took up 655 volumes of hydrogen, and so
charged, it gave off no gas in vacuo at the ordinary temperature, nor till its tempera-
ture was raised to nearly 100°."

c. Palladium Foil. — The palladium foil employed weighed 1'333 gram, and was
about 0*025 millim. thick. It had been prepared by rolling out sponge, and was said
to contain over 99 per cent, of palladium.

Before being introduced into the experimental tube it was boiled in caustic potash
solution, washed, and dried between filter paper.

On ignition in vacuo a small quantity of gas, which was not examined, was given
oft. After cooling down to the ordinary temperature again, hydrogen was admitted,
but apparently none was occluded. The palladium was therefore ignited in the
hydrogen and allowed to cool down slowly. Next day it was found that only
29 volumes had been occluded. Practically none of this could be extracted in vacuo
at the ordinary temperature, but on ignition 28 volumes were pumped out.

The result of this experiment, therefore, goes to show that the occlusion of hydrogen
by new compact palladium foil is thus either (l) very small,, or (2) very slow.

* A slight correction must be applied to these numbers, the nature of which is explained on p. 120,
and the corrected values will be found in the Table on p. 121.

OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM. 113

When palladium foil is charged electrolytically with hydrogen it undergoes a
considerable increase in volume, as has been pointed out by GRAHAM, DEWAB,
THOMA, and others, and when discharged it again contracts, and it is said that its
volume becomes less than the original volume. In order to see whether the occlusive
power of the foil was altered by successive expansion and contraction, it was charged
successively and alternately with hydrogen and oxygen in a voltaic cell. After
exhaustion at a red heat in vacua, hydrogen was admitted at the ordinary tempera-
ture. The following results were obtained : —

On first admission of hydrogen . 0'3 cub. centim. = 27 volumes were occluded;
After gently heating for two hours 3'6 „ = 32'2 ,, „ „

After eighteen hours at ordinary"]^
temperature

1 0-3 „ = 27

Altogether, therefore, 4'2 cub. centims., or 37'6 volumes, of hydrogen were occluded.
The occlusive power is thus not much greater than before. If the compactness of the
metal has anything to do with the rate of occlusion, this result is what might have
been anticipated, since on charging electrolytically with hydrogen and then dis-
charging the volume of the metal becomes less, and presumably the degree of
compactness greater.

There can be little doubt that the phenomenon of occlusion consists first of the
solution or combination of the gas in the outer skin of the metal, and then of diffu-
sion inwards towards the centre.

In the case of compact foil, produced by the welding together of particles of
sponge, the hydrogen is possibly condensed, dissolved, or combined in the two chief
outer surfaces of the foil, and then diffuses inwards in both directions through half
its thickness. In the case of palladium sponge or black, however, the surface itself
is very much greater, and the diameter or radius of the individual particles probably
very much less than the thickness of the foil ; and consequently it would seem that
with palladium sponge we have to deal chiefly with the rate of solution, and that wo
are less concerned with the factor of diffusion or rate of diffusion.

Hence, if we could by any possible treatment so change new compact palladium foil
that it becomes spongy in texture, it is to be expected that it would behave more
nearly like palladium sponge as regards the amount of hydrogen occluded and the
rate at which occlusion takes place.

It is well known that when copper is oxidised and then reduced it becomes
spongy. Now, a favourite device for cleaning platinum and palladium goods is by
ignition in the blowpipe flame. Palladium, however, differs from platinum inasmuch
as it is more easily oxidised. At a still higher temperature (higher than can con-
veniently be maintained in an evacuated hard glass tube), it is again reduced to
metal. By repeatedly igniting palladium foil in this way it probably acquires a
VOL. cxci. — A. Q

114 DRS. L. MONO, W. RAMSAY, AND J. SHIELDS, ON THE

more or less spongy nature, due to the transition through the state of oxide, and
consequently we should expect it to behave more like the sponge.

Most of the palladium employed by other observers has probably been subjected
to this treatment, which possibly explains the readiness with which in most instances
it takes up hydrogen. There is also evidence, from other sources, that new palla-
dium foil possesses a lower absorption power for hydrogen. For example, a portion
of new palladium foil, examined by GRAHAM ('Researches,' p. 268), and believed to
be from fused metal, occluded only 68 volumes of hydrogen. This is ascribed by
GRAHAM to the fact that the metal had been fused. " An inferior absorbing power
for hydrogen appears to be connected in both platinum and palladium with the
fusion of the metal." GRAHAM'S explanation is quite consistent with the view here
put forward, for the fusion of the metal would simply produce (only, perhaps, in a
more perfect way) the same effect as a thorough welding or rolling.

The foil examined by us had not been fused, but it persistently refused to occlude
hydrogen in any quantity.

After it had been submitted to the preceding operations it was ignited several
times in the blowpipe flame, rolled up, reignited, and introduced hot into the
experimental tube.

On now admitting hydrogen at the ordinary temperature about 33 volumes were
immediately absorbed. When the experimental tube was warmed by a naked
BUNSEN flame an additional quantity of hydrogen, amounting to over 100 volumes,
was suddenly occluded. On heating more strongly this gas was expelled and again
reabsorbed on cooling. The experimental tube was now placed in a water bath.
When the temperature reached 88°-90° C. absorption again began and continued
slowly at 100° C., beginning at the rate of about 1 cub. centim., or 9 volumes, per
minute and gradually diminishing. In an hour and a half at 100° C. about 500
additional volumes had been occluded, and on standing overnight at the ordinary
temperature only a very little, if any, hydrogen was absorbed.

Next day an oil bath was substituted for the water bath, and the temperature was
gradually raised above 100° C. Absorption again went on until 130° was reached,
when the occluded gas began to be expelled. The temperature was therefore lowered
to 120°. Absorption continued for two hours, and then appeared to diminish. By
allowing the temperature of the bath to fall to 100° absorption again took place, and
continued for another hour. The apparatus was now allowed to remain for 44 hours
at the ordinary temperature, when we were surprised to find that a further absorp-
tion of about 100 volumes had taken place.

It would thus appear that after the occlusion of hydrogen has made a fair start, or
when the palladium has been largely converted into palladium hydrogen, a further
absorption is able to go on at the ordinary temperature. This may be due to the
change in texture of the palladium caused by ruptures between the particles of the

OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM. 115

metal produced by the expansion during the first part of the charging. GRAHAM (loc.
cit., p. 267) remarks, "The foil was much crumpled and rather friable after repeated use."

The fact that increase of temperature is necessary during the first part of the
absorption is to be ascribed not to an increase in the solubility of the hydrogen, but
to a more rapid diffusion, throughout the mass of the metal, of such hydrogen as can
be retained at that particular temperature. The meaning of this will be rendered
clearer by reference to DEVILLE and TROOST'S well-known experiment on the passage
of hydrogen through a red-hot platinum tube, or to the similar passage of hydrogen
through a palladium tube above 237° C. (cf. RAMSAY, 'Phil. Mag.,' August, 1894,
p. 206). Now platinum and palladium at this high temperature can only retain
under atmospheric pressure* a very minute quantity of hydrogen, but, minute as it is,
it can diffuse with great readiness through the hot metal, its place being immediately
taken by fresh hydrogen. At moderately high temperatures therefore, when palla-
dium can still absorb a considerable quantity of hydrogen, we should expect a more
rapid diffusion through, or permeation of, the foil, and consequently the production
of a spongy or fissured mass, caused by expansion on the occlusion of just as much
hydrogen as can be retained at this temperature, so that on cooling down to Iho
ordinary temperature the foil ought to behave more nearly like palladium sponge.

At this stage an exact measurement of the total quantity of hydrogen absorbed
was made, and it was found that 94-50 cub. centims., or 851 volumes, had been taken
up by the palladium foil.

That the absorption at 100° was complete was shown by the fact that on raising
the temperature to 100° C. about 100 volumes of gas were expelled, and of this about
three-quarters was reabsorbed on cooling down to the ordinary temperature, whilst
the remainder, along with seven additional volumes, was occluded on standing for
eighteen hours. The final measurement showed that 95'18 cub. centims., or 858
volumes, of hydrogen had been absorbed.

Just as palladium foil which has been rendered more or less spongy in texture
requires a considerable time for the absorption of its quantum of hydrogen, so also
it gives off its occluded gas extremely slowly in vacuo at the ordinary temperature.
On applying the pump a practically complete vacuum was produced in a few minutes,
when it was found that only 0'57 cub. centim., or five volumes, had been removed.
Hydrogen was, however, given off very slowly, for in another hour 2' 1 4 cub. centims.,
or nineteen volumes, were extracted. The results are contained, in the following
table, which shows that nearly the whole of the hydrogen is given off easily and
rapidly at 100° C. in vacuo. There is a difference of twelve volumes between" the
total hydrogen absorbed and the total hydrogen extracted or occluded. This may be
due to (1) experimental error, (2) all the hydrogen may not be extracted at a red

* The most recent work on the subject by Professor DKVVAR (' Proc. Chem. Soc.,' 1897, No. 183,
192) shows that a rod of palladium is still capable of occluding about 300 volumes of hydrogen, under a
pressure of 100-120 atmospheres, at a temperature of 500° C.

Q 2

116

DBS. L. MONT), W. RAMSAY, AND J. SHIELDS, ON THE

heat, or (3) the presence of a film of oxide on the foil. Of these (2) is unlikely, so
that the deficit may be regarded as due to (1) and (3), and probably chiefly to (3).

Palladium foil used, T333 gram = 0-111 cub. centim.
Total Ha absorbed, 95-18 cub. centims. = 858 volumes.

Temperature.

Occluded hydrogen pumped off.

20° (rapidly)

cub. centims.
0-57
2-14
9077
0-39

volumes.
5
19
818
4

20° (in one liour)

100°

700° (red heat)

Total .....

93-87

846

After all the hydrogen had been extracted from the foil, it was again charged with
hydrogen. A slow absorption went on, and this was promoted at the beginning by
warming. On opening the tube and bringing the foil, partially charged with hydrogen,
into contact with the air, it became distinctly warm to the touch, and after standing
i'or a short time drops of water were deposited on the walls of the tube.

Under proper conditions, therefore, as above set forth, a sample of new palladium
foil which initially would only occlude a few volumes of hydrogen may be made to
occlude 846, or roughly 850, volumes, that is approximately the same quantity of
hydrogen as palladium black or sponge.

New palladium foil, or even fused palladium, if left for a sufficiently long time
(possibly mouths or years) in an atmosphere of hydrogen at the ordinary temperature,
would probably absorb the full quantity of hydrogen, the effect of gentle warming
being to accelerate (as above) the diffusion of the hydrogen from the fully charged
outer skin of the metal towards the centre.

A few additional experiments on the behaviour of palladium to other gases may
conveniently be recorded here. Some palladium black was fully charged with
hydrogen, and from this as much hydrogen as possible was extracted at 100°. On
now admitting sulphur dioxide only about 90 volumes were taken up, in place of about
800 volumes of hydrogen removed. Palladium sponge when treated with sulphur
dioxide absorbed only two volumes. About seventeen volumes of carbon monoxide
were occluded by palladium sponge.

Palladium black and sponge, like platinum sponge, cause a jet of hydrogen to ignite.
When hydrogen was led over palladium black which had absorbed more oxygen at a
high temperature, it glowed brightly. A specimen of ordinary palladium black,
however, did not glow on passing hydrogen over it, although it became very hot.
Apparently it was not itself sufficiently rich in oxygen. Palladium black or sponge
charged with hydrogen glows like pyrophoric iron on being shaken out into the air.

OCCLUSION Of HYDROGEN AND OXYGKN BY PALLADIUM.

117

VII. The Heat of Occlusion of Hydrogen by Palladium Black.

The determination of the heat of occlusion of hydrogen by palladium black was made
in the ice calorimeter already described by us (' Phil. Trans.,' A, 1897, vol. 1 90, p. 131),
and it may be interesting to note that during the course of these investigations the
temperature of the room in which the calorimeter was placed varied between
20°-24° C. The experiments were made in a manner precisely similar to the
corresponding series with platinum black, the only essential difference being that a
constant volume manometer was attached to the experimental tube.

To pump

To experimental tube.

A constant volume manometer was constructed, as shown in the accompanying
figure. This form was chosen for two reasons. In the first place, it obviated the
necessity of calibration. The capacity of the capillary tubing between the two taps
and the mark at the point A was determined once for all and added to that of the
experimental tube. During the course of the experiments the mercury in the right
limb was always kept near the mark A, and during an actual measurement of
pressure exactly at the mark. The second and chief reason for selecting this form of
manometer, however, was that the main bulk of the gas in the experimental tube
was at 0° 0., whilst the projecting portion was at atmospheric temperature, and in
the case of an ordinary manometer it would have been very difficult to estimate the
average temperature of the projecting volume of gas, which would have been consider-
able, since capillary manometers are unreliable.

118

DRS. L. MOND, W. RAMSAY, AND J. SHIELDS, ON THE

Before being used the manometer was exhausted and found to give readings
practically identical with those of the barometer at the time.

A quantity of palladium black was introduced into the experimental tube, the
capacity of which was determined. It was then fully charged with hydrogen and
exhausted as completely as possible at 100° C. before being placed in the calorimeter,
when it was sealed on to the burette, furnishing pure hydrogen, the pump, and the
manometer.

After the experimental tube had remained several days in the calorimeter it was
found, by making connection with the manometer, that the internal pressure was
practically zero. The reduced readings of the barometer and manometer were
757'5 millims. and 7577 millims. respectively. It was estimated that the quantity
of palladium black in the calorimeter was now in a position to occlude over 110 cub.
centims. of hydrogen, which was admitted in four separate fractions.

In estimating the amount of hydrogen actually occluded in each case, account was
taken of the unabsorbed hydrogen existing in the experimental tube before and after
the admission of the gas from the burette. The quantity of unabsorbed hydrogen
was ascertained from the known capacity of the apparatus (6'30 cub. centims.), its
temperature, viz., 0° C., and the pressure indicated by the manometer.

The deflection of the mercury meniscus in the capillary tube of the calorimeter was
corrected, when necessary, for the slight errors due to the fact that the bore of the
tube was not quite uniform. The results are given below in tabular form.

HEAT of Occlusion of Hydrogen by Palladium Black.

Palladium black used, 1'6658 gram.

Heat

Pressure of

Heat

produced

Experiment.

Hydrogen occluded.

palladium Deflection.

produced,

per gram

hydrogen.

K. = 100g.eal.

of hydrogen

occluded.

cub. centims.

gram.

million-.

MlillilllS.

K.

I.

37-26

0-003353

8-1

147-5

+ 0-1553

+46-32

11.

35-16

0-003164

13-1

139-0

0-1464

46-27

III.

32-40

0-002916

106-6

129-4

0-1362

46-72

IV.

6-79

0-000611

757-6

27-6

0-0291

47-56

V.

-22-72

-0-002045

45

- 88-4

-0-0931

-45-52

Altogether 111'61 cub. centims., or 803 volumes, of hydrogen were occluded in
addition to that which was already present in the palladium, and which could not be
removed in vacua at 100°. From previous experiments (see table, p. Ill) this
probably amounted to a little over 50 volumes.

OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM. 119

It is evident from the last column of the table that the heat evolved per gram of
hydrogen occluded remains the same for the different fractions.

The mean of the four experiments is 4672 K, and if we exclude the last, in
which experimental errors have a greater influence owing to the smallness of the
quantity of hydrogen occluded, the mean of the three principal results is + 46-44 K.,
and the deviation from this mean in the case of the fourth result is only 0'6 percent.

The value -f 46'44 K. for the heat evolved per gram of hydrogen occluded may
safely be taken as correct within 1 per cent.

In Experiment V. a portion of the hydrogen which had been occluded was
exhausted by means of the pump, giving the number 45'5 K. absorbed per gram of
hydrogen removed. The hydrogen comes off so slowly at 0° C. that the experiment
could not be continued. The result does not differ very much from those already
obtained, and no doubt the conditions under which the experiment was made account
largely for the difference. In experimenting with the ice calorimeter it is essential,
in order to obtain an accurate measurement of the deflection, that the normal
creepage of the instrument should be in the same direction as the deflection to be
measured. In this case, however, the two were opposed to each other. In the
particular instrument with which we have worked, there was generally a slow
creepage inwards, due to the slow melting of the ice. In any serious attempt,
therefore, to measure the amount of heat absorbed, it would be advisable to alter
the melting point of the ice in the interior of the calorimeter by increasing or
diminishing the pressure at the open end of the capillary tube until the creepage
was either entirely suspended, or, at least, in the same sense as the deflection to be
measured.

FAVRE, by means of his mercury calorimeter (' Comptes Rend.,' vol. 78, p. 12G2),
determined the heat evolved on the occlusion of hydrogen by palladium sponge, and
obtained numbers varying from 202'7 K. to 38'9 K. The results which exceed
60 K., however, were only obtained for the first fractions of hydrogen admitted, and
were obviously due, as FAVRE himself recognised, to the simultaneous occurrence of
a second reaction, viz., the formation of water from oxide of palladium.

If we exclude four* of FAVRE'S experiments, in which the result is greater than
60 K., then the general mean of the twenty-five remaining experiments, varying
between 60 K. and 38 '9 K., is 48'6 K. per gram of hydrogen occluded, a result
which is approximately the same as that found by us, viz., 46'4 K. In another
independent series of experiments ('Comptes Rend.,' vol. 68, p. 1306) FAVRE
estimated the heat of occlusion of hydrogen at 41 '6 K. from the difference between
the quantities of heat evolved when an ordinary Smee cell was placed in the calori-
meter (i.e., when the hydrogen was allowed to escape) and when in the same cell

* Three of the four are the first members in a series of seventeen experiments, whilst the fourth is
the first of another series of eleven experiments.

120 DBS. L. MONO, W. RAMSAY, AND J. SHIELDS, ON THE

the platinum plate was replaced by palladium, so that the heat evolved on the
occlusion of the hydrogen was also included.

Independent of direct measurement in the calorimeter, the heat of a reaction can
be calculated, as was first shown by HORSTMANN (' Berichte,' 1869, vol. 2, p. 137),
by an application of the second law of thermodynamics to the dissociation pressures
of the substance at different temperatures. From measurements of the dissociation
pressures of palladium hydrogen by TROOST and HAUTEFEUILLE ('Ann. Chim. Phys.,'
(5) vol. 2, p. 279), MOUTIER (' Comptes Rend.,' vol. 79, p. 1242) calculated that the
heat of occlusion of hydrogen by palladium at 20° C. was 4T5 K. per gram, and
moro recently DEWAR (' Proc. Chem. Soc.,' 1897, No. 183, p. 197) obtained, from
similar measurements by PtOOZEBOOM (HoiTSEMA, ' Zeitschr. Physikal. Chem.,' 1895,
vol. 17, p. 1), the value (45'61 + 0'2378 T.) K. At 0° C. (273° abs.), therefore, the
heat evolved per gram of hydrogen occluded becomes 46'26 K., which agrees very
closely with the value, 4G'44, found by us. The magnitude which is directly
measured in the calorimeter, however, represents the true heat of the reaction plus
the heat corresponding to the work done by the atmosphere, viz., 27 K.,* and hence
the true heat of occlusion of hydrogen by palladium black is only 46'4 — 27 = 437 K.
per gram of hydrogen.

VIII. Investigation of the Ratio Palladium : Hydrogen.

From the analysis of palladium black (v. p. 108) it follows that the black as
prepared contained 97'63 per cent, palladium, the remainder consisting of 1'65 per
cent, oxygen and 072 per cent, water. Hence, in calculating the ratio of
the number of atoms of palladium to the number of atoms of hydrogen in palladium
fully charged with hydrogen, we ought to reduce the weighed amount of palladium
black to palladium metal per se. Strictly speaking, this ought also to be done in
expressing the number of volumes of hydrogen occluded by unit volume of palladium
existing in the substance we call palladium black.

The following table contains the atomic ratio of palladium to hydrogen for fully
charged palladium black, sponge, and foil (the atomic weight of palladium being
taken as 107), and also the corrected number of volumes of hydrogen occluded by
unit volume of palladium : —

V. p. 123.

OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM.

121

Palladium

Volumes of

Hydrogen.

Hydrogen occluded.

Palladium black

1-42

873

I *

1-41

881

II.* .

1.40

889

Ill*

1-40

889

sponge .

1-46

852

foil . . .

1-47

846

wire I. (GHAHAM)

1-37

912 (936)

„ II.

1-46

846 (867)

„ HI. „

1-44

859 (888)

. metal (DEWAK) .

1-47

847

We have also included in the table the corresponding values which we have
calculated from three experiments by GRAHAM (' Researches/ p. 291) and from an
experiment on a larger scale by DEWAR ('Phil. Mag.,' (4) vol. 47, p. 334). The
numbers given in brackets are those calculated by GRAHAM on the basis that the
specific gravity of palladium wire is 12 '38. In the last four experiments the palladium
was charged electrolytically with hydrogen, and it follows directly from a considera-
tion of the table that, no matter whether the palladium exists as black, sponge, foil,
wire, or compact metal, or whether it is charged by exposure to hydrogen gas (the
proper conditions being observed), or charged electrolytically, the amount of hydrogen
occluded in each case is approximately the same.

If we wish to express, by means of a formula, the composition of palladium
hydrogen, then, by choosing the nearest whole numbers, we arrive at the formula
Pd3H2, which was first proposed by DEWAR (loc. cit.), and which corresponds with
the ratio 1*5, instead of varying between 1'37 and 1'47, as above.

It must be observed, however, that in all cases in which the palladium has received
the maximum charge of hydrogen the amount of hydrogen taken up is always in
excess of that required for the formation of a definite chemical compound Pd3H:,.
If we admit — and this is not at all improbable — that such a compound is still capable
of occluding or condensing hydrogen in the ordinary way, then the excess of
hydrogen which is absorbed does not exclude the possibility of the formation of a
definite compound of this composition. On the other hand, evidence in favour of its
existence is of a very meagre kind, and is, we think, chiefly confined to the approxi-
mation of the ratio Pd/H to the theoretical value 1*5.

The views of chemists and physicists on the nature of palladium hydrogen have
from time to time undergone considerable change. In 1869 GRAHAM ('Researches,'
p. 290) wrote : — " The idea forces itself upon the mind that palladium, with its
occluded hydrogen, is simply an alloy of this volatile metal (hydrogenium), in which

* Details will be found on p. 125.

VOL. CXCI. — A.

122 DBS. L. MOND, W. RAMSA5T, AND J. SHIELDS, ON THE

the volatility of the one element is restrained by its union with the other, and which
owes its metallic aspect equally to both constituents."

In 1874, however, another view was suggested by TROOST and HAUTEFEUILLE
('Ann. Chim. Phys,,' (5) vol. 2, p. 279). From experiments on the vapour pressure
of palladium hydrogen at different temperatures, they conclude that the first
addition of hydrogen to palladium forms a definite chemical compound Pd2H, and
that the excess of hydrogen over and above that required to form this compound is
then occluded or absorbed in the usual way.

By subjecting the results of TROOST and HAUTEFEUILLE to a critical analysis, and
from more extended experiments by himself and ROOSEBOOM, HOITSEMA (' Zeitschr.
Physikal. Chem.,' vol. 17, p. 1, 1895) has arrived at an entirely different conclusion,
namely, that, as regards the vapour pressure experiments, there is no decisive
evidence for the formation of a definite chemical compound ; and he further suggests
that the results are in better agreement with the view that two immiscible solid
solutions are formed. This appears to resolve itself into the formation of two alloys
instead of one, as suggested by GRAHAM.

In confirmation of the arguments of HOITSEMA against the supposed formation of
a compound having the formula Pd2H, we have also the fact, first observed by
FAVRE (' Comptes Rend.,' vol. 77, p.- 649, and vol. 78, p. 1257) and confirmed by
ourselves, that the heat of occlusion of hydrogen by palladium remains constant
throughout the whole range of absorption.

If a chemical compound Pd3H were first formed, we should expect to get a certain
definite evolution of heat per gram of hydrogen combined for the hydrogen first
admitted, and then, after sufficient hydrogen had been added to form the compound
Pd2H (about 630 volumes), we should expect to find a different value for the heat
evolved per gram of hydrogen occluded or dissolved or absorbed. We have,
therefore, confidence in discrediting the supposed formation of Pd2H.

It would seem that HOITSEMA'S conclusions apply equally well to the supposed
formation of Pd3H.,. But before finally dismissing the possibility of its existence we
consider that some additional evidence is desirable. It is proposed to attack the
problem from another and independent point of view, and it is hoped that all doubts,
which are liable to affect any process in which it is uncertain whether equilibrium
has really set in or not, will be overcome.

IX. On the Heat of Occlusion and the Heat of Condensation.

It has sometimes been suggested (cf. FAVRE, 'Ann. Chim. Phys.,' (5) vol. 1,
p. 209), that the heat of occlusion of a gas represents the heat of condensation or
liquefaction of the gas in the capillary pores of the absorbing substance. Now if
this is so, and we vary the absorbing substance, at the same time maintaining it
always at the same temperature, say 0° C., so that its specific heat shall play no

OCCLUSION OP HYDROGEN AND OXYGEN BY PALLADIUM. 123

part in the reaction, then it would seem justifiable to suppose that in the liquefaction
of one gram of one and the same gas, hydrogen for example, in different absorbing
substances, the same amount of heat, viz., the heat of condensation of one gram of
hydrogen, would always be evolved.

We have found, for the heat of occlusion of one gram of hydrogen in platinum and
palladium, the following values : —

Platinum. . . . + 68'8 — 27 = 66'1 K. per gram.
Palladium . . . + 46'4 — 27 = 437 K.

In order to obtain the difference between the initial and final values of the internal
energy of the system, we require to subtract the heat equivalent of the work done
by the atmosphere. For one gram molecule of a gas this amounts to 0'02 T.K.
(2 T. g. cal.), and hence for half a gram molecule of hydrogen at 0° C. we require to
diminish the above values for the heat of occlusion by 27 K. Since the numbers
which we thus obtain are by no means approximately equal, we are of opinion that
the phenomenon of the occlusion of hydrogen by platinum and palladium black is
not simply the liquefaction or condensation of the gas in the capillary pores of the
metals.

The same arguments would; doubtless, be valid in a comparison of the heat of
occlusion with the heat of solidification or fusion.

X. The Influence of Increased Pressure on the Occlusion of Ili/drogcn by Palladium

Black.

We have already seen that the composition of palladium hydrogen, fully charged
with hydrogen at ordinary atmospheric pressure, corresponds approximately to the
formula Pd3H2. Although from the experiments of HOITSEMA and KOOSEBOOM it
would seem that increase of pressure should produce little or no effect on the
occlusion of hydrogen by palladium, an experimental demonstration was deemed
desirable. The details of the experiment were the same as in the corresponding
experiment with platinum black ('Phil. Trans.,' A, 1895, vol. 186, p. 675). A
quantity of palladium black was charged with hydrogen and shut up with an excess
of hydrogen in the shorter graduated limb of a glass U-tube. After equilibrium had
been established, the volume, temperature, and pressure of the enclosed gas were
noted. Successive quantities of mercury were then poured into the longer limb, and
the measurements repeated.

The results are contained in the following table :—

R 2

124

DRS. L. MOND, W. RAMSAY, AND J. SHIELDS, ON THE

Palladium black used, 0'878 gram.

Volume of en-
closed gas.

Temperature.

Pressure (millims.
Hg. at 0° C.).

PV.

Volume reduced
to 0° C.
and 760 millims.

cub. centime.

cub. centims.

128-1

19-14

810

103,800

127-6

74-8

19-14

1380

103,200

126-9

48-0

19-14

2151

103,300

1270

36-5

19-14

2821

103,000

126-6

29-5

19-14

3496

103,100

126-8

The fact that the volume energy (PV), or the reduced volume of the gas, except for
the first increment of pressure, may be taken as constant, satisfactorily establishes the
conclusion that increase of pressure, at least up to 4*6 atmospheres, practically has no
effect on the occlusion of hydrogen by palladium black which has already absorbed
the maximum quantity of hydrogen at the ordinary temperature. Platinum and
palladium black thus behave in precisely the same way as regards the effect of
pressure on the occlusion of hydrogen. The first increase of pressure caused the
disappearance of several volumes (about ten volumes) of hydrogen, but as this does
not continue, it may possibly be due to the presence of moisture, derived from the
palladium black, in the gas.

The influence of pressure on the occlusion of hydrogen by palladium at high
temperatures has recently been described by DEWAR ('Proc. Chem. Soc.,' 1897,
No. 183, p. 192).

XI. The Heat of Absorption of Oxygen by Palladium Slack.

It is fairly certain that, on the absorption of oxygen by palladium black, a definite
oxide, or a mixture of definite oxides, is formed. A redetermination of the heats of
formation of the different oxides of palladium is outside the scope of this inquiry, but
it seemed desirable to determine the heat of absorption of the oxygen actually
contained in palladium black, for the purpose of comparing the number so found with
the corresponding number for platinum black. Since it was impossible to make the
measurement directly, an indirect method was employed, viz., the estimation of the
amount of heat absorbed on the removal of the oxygen from palladium black.

A quantity of palladium black was introduced into the calorimeter, and fully
charged with hydrogen. A portion of the hydrogen absorbed was thus really
occluded, whilst the remainder removed the oxygen contained in the palladium black
as water. After the heat change had been measured, the occluded fraction was easily
ascertained by exhausting at a red heat, and from this the volume of oxygen removed
from the palladium black was found.

OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM.

125

Assuming, as was necessary in the case of the corresponding experiment on
platinum black (q.v., 'Phil. Trans.,' A, 1897, vol. 190, p. 149), that no appreciable
heat change is involved in the mere wetting of the palladium by the water formed,
and knowing the total heat of the reaction, the heat of formation of water, the heat
of occlusion of the hydrogen, and the quantity of oxygen removed, we can calculate
the heat absorbed on its removal.

The results obtained in three independent experiments are given below in tabular
form. The amounts of hydrogen occluded and oxygen removed are expressed in
cubic centimetres, volumes (Pd. bk. = 1), and grams ; and it will be seen that there is
a very satisfactory agreement among themselves of the values representing the number
of volumes of hydrogen occluded and the number of volumes of oxygen removed.

CALORIMETRIC Experiments on the Absorption of Oxygen by Palladium Black.

I.

II.

III.

Palladium black used . .

Grams

0-4789

1-0237

1-0570

Cub. eentims.

0-0399

0-0853

0-0881

Hydrogen occluded

Cub. ccntims.

34-32

74-08

76-51

Volumes

860

868

868

Grams

0-003089

0-006667

0-006886

Hydrogen bnrnt to water

1) )» » ...

Oxygen removed

)» M
»» )!

Heat due to formation of water

Cub. eentims.
Grams
Cub. eentims.
Volumes
Grams

K.

11-19

0-001007
5-595
140
0-008000

0-3444

24-02
0-002162
12-01
141

0-01717

0-7394

25-16
0-002264
12-58
143
0-01799

0-7743

Heat due to occlusion of hydrogen
(calculated)

K.

0-1433

0-3093

0-3195

Sum of last two

K.

0-4877

1-0487

1-0938

Heat evolved, corresponding to f
the deflection \
Difference = heat absorbed on the
removal of the oxygen . . .

K.
Millims.

K.

0-3927
373-0

0-0950

0-8424
800-0

0-2063

0-8931

848-1

0-2007

Heat absorbed per gram of oxygen
removed

K.

11-9

12-0

11-2

The three values of the heat absorbed per gram of oxygen removed, or, what
amounts to the same thing, the heat evolved per gram of oxygen absorbed, are thus
+ 11-9, + 12-0, and + 11'2 K., whilst the mean of the three is + 117 K. This
number, referred to 16 grams of oxygen, becomes + 187 K.(+ 18700 gram calories).

From the nature of these experiments it is difficult to say within what limits of
error this number may be taken as correctly representing the heat of absorption of
oxygen in palladium black. As far as such measurements go, however, the agreement
between the individual numbers may be said to be satisfactory.

126 ON THE OCCLUSION OF HYDROGEN AND OXYGEN BY PALLADIUM.

Whilst the heat of occlusion of hydrogen in palladium is less than the corresponding
number for platinum, the heat of absorption of oxygen is slightly greater. It will be
remembered that the heat of absorption of oxygen in platinum black, viz., -f 176 K.
per gram atom, is almost identical with the number given by JULIUS THOMSEN for
the heat of formation of platinous hydroxide, Pt(OH)2, viz., + 179 K., thereby
furnishing presumptive evidence that the absorption of oxygen by platinum black is
a true oxidation phenomenon, the water necessary for the formation of the hydroxide
being always present in platinum black.

It therefore remains to be seen whether there is any sort of agreement between
the numbers representing the heat of absorption of oxygen by palladium black and
the heat of formation of an oxide or hydroxide of palladium.

JOANNIS (' Comptes Rend.,' vol. 95, p. 295) determined the heat of formation of
palladous hydroxide, Pd(OH)., (oxide?), by precipitating the substance from a solu-
tion of (1) potassium palladobromide, (2) potassium palladochloride. by means of
caustic potash, and obtained the values + 193'4 K. and + 208'8 K., the mean of the
two being about + 200 K.

JULIUS THOMSEN (' Thermochem. Untersuch.,' vol. 3, p. 436) also determined the
same heat of formation by precipitating a solution of sodium palladochloride by
caustic soda, and gives as its value + 227 K., thus : —

Pd + 0 + H2O = Pd(OH)3 + 227 K.

There is thus a discrepancy of about 27 K. between the results obtained by the
two observers.

The heat of formation of palladia hydroxide was also determined by THOMSEN in a
similar way, with the following result : —

Pd + 2O + 2H2O = Pd(OH), + 304 K.

This number refers to 32 grams ot oxygen ; for 16 grams it therefore becomes
+ 152 K.

The value found by us for the heat of absorption of oxygen in palladium black lies
intermediate between the values given by THOMSEN for the heats of formation of
palladous and palladic hydroxide, and, considering the difference between the
numbers representing the heat of formation ot the lower compound as determined by
THOMSEN and JOANNIS, the heat of absorption of oxygen in presence of sufficient
moisture may be consistent with the formation of either the higher or lower
hydroxide, or possibly with a mixture of both. In any case, it is of the same order
of magnitude, and taken in conjunction with the behaviour of palladium black when
heated in an atmosphere of oxygen, is undoubtedly in harmony with the view that
the absorption of oxygen by palladium black is a true phenomenon of oxidation.

[ 127 ]

IV. Comparative Photographic Spectra of Stars to the 3£ Magnitude.
By FRANK MC(/LEAN, F.R.S.

Received March 24,— Read April 8, 1897.

[PLATES 1-17.]

THE 160 photographs which accompany the paper include, with insignificant excep-
tions, all stars equal to and brighter than the 3^ magnitude, contained in five out of
eight equal areas, into which the celestial sphere has been divided.
The diagram shows the position of the eight areas consisting of —

Two upper galactic polar regions, viz., A and A A.

zones „ B „ BB.

Two lower galactic zones ,, G „ CC.

„ „ ,. polar regions ,, D ,, DD.

The galactic zones extend to 30° from the galactic plane.

The object of the division into equal areas is to bring out roughly any differences
of distribution of the different types of spectra in relation to the galactic plane.

25.5.98

128 MR. F. McCLEAN ON THE COMPARATIVE PHOTOGRAPHIC

The areas A, B, C, D, and AA are included in the present series of photographs.
Of these the first four constitute a complete hemisphere, symmetrically divided by
the galactic plane, and the deductions as to distribution have been made from them.

The stellar spectra have been arranged in series, and classed separately for the
respective areas to which they belong. It appears at once that SECCHI'S Type I.
requires further subdivision into distinct classes. To effect this, a series of divisions
in parallel to SECCHI'S types have been adopted, in which Divisions I., II., and III.
correspond to Type I., and Divisions IV., V., and VI. to Types II., III., and IV.
respectively.

The classified tables of the stellar spectra are given at the end of the paper, and
previous classifications by SECCHI in 1866-1868, by VOGEL in 1883, by PICKERING in
1890, and by LOCKYER in 1892 are indicated in separate columns.

Division I. includes all stars whose spectra are characterised by the lines of
hydrogen and of cleveite gas.

Subdivision I. (a) shows other special lines in addition. I have made a close com-
parison with the spectra of nearly all the elements, for the purpose of identifying
these extra lines, but without definite result. The only suggestion that presents
itself to me is the possibility of their being due to oxygen.

I have placed below the scale attached to these spectra RUNGE and PASCHEN'S
spectrum of cleveite gas, which forms the characteristic spectrum of the division.
Also THALEN'S spark spectrum of oxygen. Although there is not a perfect agreement
between the spectrum of oxygen and the special lines referred to, there is a very
remarkable correspondence.

I have further placed below the scale CAMPBELL'S bright-line spectrum of the
nebula in Orion. The general coincidence with the stellar line leaves little doubt as
to the close connexion between this class of stars and the gaseous nebulae.

Evidence to the same effect is afforded by the wonderful photographs taken in
recent years, showing the physical connexion of many of these stars with the nebulae.
Such are Dr. EOBERTS'S photographs of the Pleiades and of the great nebula in
Orion. Alongside copies of these I have shown the corresponding groups of the
spectra of the involved stars ; the respective stars and their spectra are easily
identified on inspection. All six stars of the Pleiades show the cleveite gas
spectrum, but the presence of the special extra lines is doubtful. The Orion stars,
which include i and 6 Orionis, all show the cleveite spectrum, but in these the special
lines are also recognised.

We have thus evidence of both physical and chemical connexion between these stars
and the nebulse.

[Note added 8th April, 1897. — There are two other instances of the connexion of
stars of Subdivision I. (a) with nebulse.

BARNARD, in 1895, photographed an extended nebula in the vicinity of a Scorpii
(Antares). He mentions <r Scorpii as connected with the nebulosity. The photo-

SPECTRA OF STARS TO THE 3£ MAGNITUDE. 129

graphic spectra of the four stars, ft, 8, IT, and <r Scorpii, all near Antares, belong to
Division I. (a). They are in the area BB, and do not appear in the present series.

BARNARD also photographed, with a six-hour exposure, an extended nebula
contiguous to £ Persei, a star of the 4th magnitude. The spectra of e and £ Persei,
on either side of it, appear in the photographs, and belong to Subdivision I. (a).
A photograph of the spectra of f Persei was recently obtained, and it also belongs
to the same division.

Both these instances point to the same conclusion, that stars of this type are in the
first stage of stellar development from nebuke.

The star £ Persei and also t Orionis give in their spectra three lines of the second
hydrogen series, recently identified by PICKERING in £ Puppis. One of the three
lines at wave-length 4027 corresponds to the characteristic helium line. The other
two at wave-length 4201 and wave-length 4544 do not appear in any other spectra
of Subivision I. (a) in the present photographs. These lines appear to belong to the
earliest stage of stellar development.]

There is further evidence to the same effect in the similarity of the distribution of
this type of star, and of the gaseous and planetary nebulae.

The gaseous nebuke, given in the Table in FROST'S edition of SCHEINER'S ' Spectro-
scopy,' have been distributed into the same equal areas as the stars.

The following shows the relative distribution for the nebulae and for the stars : —

A. B. C. D.

Gaseous nebulre .... 3 7 1G G
Stars of Division I. ... 3 G 17 3.

Thus it appears that the helium stars of Division I. and the gaseous nebuke are
subject to a similar law of distribution in relation tq the galactic plane.

All these facts afford grounds for accepting the conclusion that the helium stars of
Subdivision I. (a) are in the first stage of stellar development from the gaseous nebuke.

Division I. has a second subdivision, viz., I. (b), which is also characterised by a few
special lines. These lines have been attributed to calcium, barium, and magnesium.
Since these special lines persist through the subsequent divisions, it may be concluded
with tolerable certainty, that in order of development Subdivision I. (b) follows after
Subdivision I. («). The K line of calcium, which first appears in this subdivision,
gradually increases in strength to such a marked extent in the subsequent divisions,
that it may be practically taken (as suggested by Dr. HUGGINS), as a criterion of the
type to which a spectrum belongs. Two bright line spectra have been treated as
belonging respectively to the divisions to which their absorption spectra belong.

The spectrum given of y Cassiopeia, clearly places it in Subdivision I. (a). It is a
helium star. The bright hydrogen lines are weak, and are placed centrally in the
difiuse absorption lines.

VOL. CXCT. — A. s

130 MR. F. McCLEAN ON THE COMPARATIVE PHOTOGRAPHIC

/3 Lyree is also a helium star. Its spectrum belongs to Subdivision I. (b). Both the
hydrogen and the helium lines appear bright. The peculiarities of this star are well
known.

A third bright line star, of another division, may be mentioned here. The spectrum
of Mira Ceti belongs to Type III. or Division V. The bright hydrogen lines appear
periodically, for a brief time, with great brilliancy. The banded absorption spectrum
brightens up simultaneously, a peculiarity which it is difficult to explain. The
exceptionally good photograph of this spectrum was taken on the 1st January last.

Division II. is especially the hydrogen type. In it the hydrogen spectrum attains
its full development both in the strength of the lines and in the extent of the
spectrum. This is shown in the ultra violet series of lines, discovered by Dr. HUGGINS.
The narrower and more sharply defined calcium line K, in itself distinguishes the
division from those which precede and follow it. The delicate absorption lines,
which also distinguish the type, are difficult to photograph, but they are well shown
in the spectra of Sirius and Vega. The spectrum of a Cygni has been sometimes
classed differently, but a comparison with the spectrum of Sirius, which -is placed
next it, clearly shows their identity, and that they only differ in strength. The fine
lines appear to be due to calcium and titanium rather than to iron, although that
spectrum is also present in an incipient form. The distribution of this division is
irregular, viz. :

A. B. C. D. AA.

10 7 0 3 3

Division III., or tl.se hydrogen-iron type, is the last of the separate divisions into which
Type I. has been divided. In its more advanced examples the iron spectrum is fully
developed. That spectrum has been plotted on the scale for comparison. This type
is more closely allied to the subsequent solar type, Division IV., than to the preceding
hydrogen and helium types. The brightness of the violet end, and the obscurity of
the red end of the spectrum, remain the same as in the preceding divisions. The
hydrogen lines remain very strong, and the calcium lines K and H are generally
subordinate to them.

This division completes the requisite subdivision of SsccHi'sType I., which has not
hitherto been fully established. The subsequent divisions remain as defined by
SECCHI in 1868. It must also be remembered that SECCHI fully recognised the special
character of the spectra of the Orion stars.

Division IV. is equivalent to Type II., or the solar type. The characteristics of this
type are well defined. They are also elucidated by our more intimate knowledge of
the solar spectrum, which forms the basis of our knowledge of astronomical physics.

Division V. is equivalent to Type III., and is the first of the banded types
investigated by Duner. The photographs of this type of spectra are difficult to
take. These stars are not numerous within the range of magnitude under con-

SPECTRA OF STARS TO THE 3£ MAGNITUDE. 131

sideration. The spectra are closely allied to those of Division IV., and have been
placed consecutively with them. The coincidences with the spectra of calcium and
manganese have been marked on the scale.

Division VI. is equivalent to Type IV. There are no stars in this division
brighter than the 5|- magnitude, but to complete the series, photographs of
the spectra of SECCHI'S Superba, and of 19 Piscium have been included. The
photographs are poor owing to the faintness of the stars, but their interest lies in
the hydrocarbon absorption bands, and further in the line spectrum, also recognizable,
and similar to that of a Tauri (Aldebaran).

There remains the question of how far the distribution of the stars into the eifht
equal areas discloses any information as to the distribution of their respective types
of spectra in space.

The dimensions of the sphere enclosing stars to the 3^ magnitude is obtained from
the light ratio. The decrease of light for each magnitude beyond the 1st is inversely
as the light ratio 2 '51. The corresponding increase of distance is as the square root
of this ratio, or as 1 to 1*58. Taking the distances of the 3f magnitude stars arrived
at in this way as the radius of the sphere containing the stars of the 3^ magnitude,
its radius will be, in terms of the mean distance of 1st magnitude stars from the sun,

= 1 X (l'58)v = 3'5 approximately.

The mean distance of 1st magnitude stars from the sun has been determined from
parallax observations to be approximately 36^ light years. Thus the diameter of the
enclosing sphere for stars of the 3^ magnitude is approximately 255 light years.

This gauge block of space, although large, might be so small, compared with the
dimensions of the galaxy, as to disclose nothing with regard to its structure. The
following table of distribution, however, shows that within this block there are
indications of differences in the distribution of the different stellar types, relatively
to the plane of the galaxy. The table shows the distribution of the gaseous nebulte,
both planetary and extended, for the same areas.

132

MR. F. McCLEAN ON THE COMPARATIVE PHOTOGRAPHIC

TABLE of Distribution of Gaseous Neblure and of Stellar Types.
Stars to the 3| Magnitude.

A.

B.

C.

D.

Total.

AA.

BB.

CC.

DD.

Total.

Planetary n elm In? .
Extended ,,

2
1

3

4

8
8

2
4

(15)
(17)

2
1

7
4

3

3

0
1

(12)
(9)

Total gaseous nebulm

o
o

7

16

G

(32)

3

11

6

1

(21)

Stellar Types.

Division I

3

G

17

3

(29)

G

„ II

10

7

0

3

(20)

3

„ III. . . .

7

8

8

4

(27)

9

„ IV. . . .

14

8

9

13

(44)

9

„ v

1

2

4

3

(10)

3

Total stellar spectra .

35

31

38

26

(130)

30

NOTF. — The gaseous nebulce are those given in the table in FROST'S edition of
SCHEIXEK'S 'Astronomical Spectroscopy.'

Wo gather from this table that as the stellar types of spectra become more
advanced they are found to be more evenly distributed in space. It suggests the
idea that stars of the solar type — Division IV. — started on their career as helium stars
of Division I., before the condensation of the galaxy.

The Procyon stars of Division III. possibly followed in the same course after the
galaxy was formed. The paucity of stars of this type in the lower polar region,
coupled with their even distribution in tbe other areas, suggests the idea that the
sun itself is situated near the lower boundary of the galaxy.

The distribution of the Sirian stars — Division II. — is irregular, and further
information is required as to their distribution in the southern areas.

The Orion stars — Division I. — are mostly confined to the galactic zones. It has
been already conjectured that they are still in the first stage of stellar development
from the gaseous nebulas.

It has been throughout assumed that the successive types or divisions are merely
the manifestations of the successive physical states, through which every star
naturally passes in the course of its career.

SPECTRA OF STARS TO THE 3i MAGNITUDE.

133

Index to the Tables.

Andromeda.

Cephens.

a. D. 3, I.

a. B. 15, III.

a.

ft C. 35, V.

ft B. 2, I.

ft

7 C. 32, IV.

7 B. 25, IV.

7

a D. 16, IV.

a B. 18, III.

a

Aquarius,
a. D. 12, IV.

Cetus.
a. I). 25, V.

n

ft D. 11, IV.

ft D. 13, IV.

i

a D. 6, II.

7 D. 7,111.

TT

£• D. 18, IV.

Aqnila.
a. C. 21, III.

i/ D. 21, IV.
0 D. 15, IV.

a.

ft C. 27, IV.

/ D. 17, IV.

7 C. 33, IV.

o D. 26, V.

a c. 18, III.

T D. 14, IV.

iT B. 7, II.
0 C. 15, I.

Corona.

a.

a A. 6, IT.

<

7

Aries.
a D. 22, IV.

(7on;Ms.
7 AA. 5, I.

r
r

ft D. 9, III.

a AA. 3, T.

£

Auric/a.

Cygnus.

0

a B. 23, IV.

x B.' 13, TI.

ft B. 9, II.

ft B: 28, IV.

P B. 20, III.

7 B. 21, III.

a

0 B. 8, II.

a B. 6, T. ft

, C. 34, IV.

c C. 29, IV.

<: c. 30, iv.

Bootes.

• a

a AA. 25, IV.

Delpliinus. ft

ft A. 28, IV.

ft C. 24, III. 7

7 A. 19, III.
c A. 23, IV.
c A. 27, IV.
,, AA. 19, IV.

Draco. .
oc A. 9, II.
ft A. 21, IV.
7 B. 29, IV.

Canes,
a. A. 4, II.

a B. 26, IV.
IT A. 3, I.
// A. 30, IV.

Canis Major,
a. B. Corapn.

, A. 32, IV. "
K A. 2, I. J

Eridanns.

Canis Minor.

ft C. 19, III. r

a B. 17, III.

7 D. 24, V. £

ft B. 4, 1.

a D. 19, IV. '

c D. 20, IV. *•'

Capricornus.

a D. 10, in.

Gemini.

x B. 12, II.

y.

Cassiopeia.

ft B. 24, IV. ft

* C. 31, IV.

7 B. 10, II. 7

ft C. 23, III.

a B. 16, nr.

7 C. 1, I.

f B. 27, IV. r

a C. 22, III. 0 AA. 16, III. »/

c B. 3, 1.

fi B. 30, V.

0

Hercules,
a. B. 31, V.

A. 24, IV.
7 A. 16, III.

A. 14, III.

A. 8, II.

A. 22, iv.

>/ A. 26, IV.

B. 22, IV.
A. 33, IV.

Hydra.

a AA. 27, IV.
AA. 21, IV.

2, I.

AA. 14, III.
7 AA. 24, IV.
AA. 15, III.
A. 25, IV.

A. 20, III.
AA. 6, I.

0 AA. 9, II.

Libra.

AA. 12, III.
AA. 4, I.

Lyra.

B. 11, II.
ft B. 1, I.

B. 5, I.

Ophinchus.

B. 14, III.
AA. 28, V.
AA. 23, IV.

Orion.

0. 38, V.
ft C. 17, T.
7 C. 9, I.

C. 2, I.

r C. 3, I.

C. 5, I.
C. 6, I.

C. 4, I.

Pegasus.

D. 4, II.

C. 37, V.
7 D. 1, I.

D. 23, IV.
D. 2, I.
C. 28, IV.

0 D. 5, II.

Perseus.

<*. C. 25, III.

ft C. 13, I.

7 C. 26, IV.

a C. 14, I.

e C. 7, I.

<r c. 8, i.

Pisces.
D. 8, III.

Serpens.

a AA. 20, IV.
ft AA. 11, III.
a AA. 17, III.
r AA. 13, III.
,/• AA. 7, II.

Taurus.
a C. 36, V.
ft 0. 12, T.
T C. 10, I.
•>, 0. 1(J, I.
A C. 11, I.

?7rsa Major.
a A. 29, IV.
y3 A. 12, II.
7 A. 7, II.
a A. 10, II.

r A. 5, II.

JT A. 13, II.

,1 A. 1, I.

0 A. 18, III.

. A. 17, III.

\ A. 11, II.

/« A. 35, V.

f A. 31, IV.

Ursa Minor.
a. B. 19, III.
ft A. 34, IV.
7 A. 15, III.

Virgo.

a AA. 1, I.

ft AA. 20, IV.

7 AA. 18, III.

a AA. 29, V.

c AA. 22, IV.

£• AA. 10, III.

,, AA. 8, II.

134

MR. F. McCLEAN ON THE COMPARATIVE PHOTOGRAPHIC

PHOTOGRAPHIC Stellar Spectra. Stars to Magnitude '6 '5.

Upper Galactic Polar Region, N.

A (Plates 1 to 3).

.g

"-E:

aj v

a c

V)

be o

.

^5

*x ^

B

vn

&1

§

a

rt •*

be

JJ

•S

|

o

I

05

?"1

^te

.

i

&

00

£

P 05*

* §S

Sj

eo to '^

s

JS

i

S

1

flj fl

00 <J

g"1^

eg

00 H +

GO eo

CO to

H

43

*

w

"o

S'S
*3 ®

a to

» b

-fa

J»c

Sg

O

o

09

°o

•s

^o

og^5

le

0*0 1

§S

w

02

£

55

13

s

d,

>

CO

o

Type I.

Division I.

1

;; Ursa; Majoris .

1-8

AW

A

1

2

k- Draconis . . .

37 A

. .

2

*>
O

r „ ... 3-3 .. ; A

••

1

Division II.

4

a. Can. Venat .

3-3 A (e)

• * • •

5

c Ursse Majoris . 1'8 A

1

6

a, Coronse . -. . ' 2'2 A (?) A

1

7

7 Ursffi Majoris . 2'3 A (e ) A

1

8

r Herculis'. . . 3'8 A

1

9

a Draconis . . . 3'6

A

1

10

F. Ursse Majoris . 3'4

. .

A

1

11

X „ ' .! 3-5

. .

A

. .

12

ft „ . 2-2

A (7)

A

1

13

?> i' *

21

#

A

1

Division III.

14

f Herculis . . .

3-2

#

A

1

15

7 Ursse Minoris

3-0

. .

A

. .

16

7 Herculis . . . 3'6

A (ft)

A

la!

1

17

< Ursse Majoris . 3'2

A

IS

0 „' .\ 3-1

F

19

7 Bootis . . . . 1 3'2

*

A

"l

20

^ Leonis ....

3-4

• •

F

• •

1

Type II.

Division IV.

21 ft Draconis .

3-0

K

. .

2

22

C Herculis ... 2'6

B(ft)

G

2

23

c Bootis .... 3-4

K

. ,

2

24

ft Herculis . . . ; 2' 7 B (ft)

K

. .

2

25

c Leonis . . . .

3-4

B (ft)

K

. .

a

26

j; Herculis . . .

3-6

B (ft)

K?

, ,

2

27

e Bootis ....

2-5

B(/3)

G?

2

28

ft „ -...••

.3-6

B(ft)

K?

. .

2

29

a. Ursse Majoris

1-9

K

...

2

30

1] Draconis .

2-8

. .

K

* •

2

31

y<- Ursse Majoris

3-2

G(ft)

K

* •

32

( Draconis .

3-3

, .

K

* •

2

83

TT Herculis . . .

3-6

BOS)

K?

. .

2

34

ft Ursse Minoris .

2-3

L?

2

Type III.

Division V.

35

ft Ursse Majoris .

3-1

• •

M

• •

••

NOTE. — For LOCKTER'S Notation, see ' Phil. Trans.,' A, 1893.

For PICKERING'S Notation, see 'Introduction to DRAPER Catalogue,' 1890.
For VOGEL'S Notation, see ' Potsdam Observatory Publications,' No. 11, 1883.
For SECCHI'S Types, see ' Mem. Soc. Italiana,' 18G7. For present order of Types 2 and 3,
see Preliminary Notice in ' Catalogo delle Stelle,' Paris, 1867,

SPECTRA OF STARS TO THE! 34 MAGNITUDE.

PHOTOGRAPHIC Stellar Spectra. Stars to Magnitude 3 "5.

Upper Galactic Zone, N.

B (Plates 4 to 7).

135

i

SECCHI'S Types.

Present classification.

,
No. of photograph.

S2
a

02

IW

o

>n

KH

Magnitude. Uranometria
Oxoniensis.

'it
?\j

^•r «
*J

73 — ' "^3

M -.

H . •,:
•^ ^ c

|ll

riCKEKISG, 1*90,

' DKAPER Catalogue.'

— i
VOGEL, 1883, 'Catalogue
of Stellar Spectra Zone
- 1D to + 20V

.

rn

o
cu
>,
H

js

"o .

£S

- 00

^ ^-<

3§
1 =

'J±

Type I.

Division I.

1

ft Lyra1 .

( MlVar

J)

G

0

2

ft Ccphei

1 I'4 /
3'4

A

i * "

2

3
4
5

c Cassiopuise ....
ft Cnnis Minoiis .
7 Lyrco ....

J'O

M

]-2

A '(-/)
\ (-i\

A

A

A

la*!

i
i
i

6

f- Cygfiii

2-8

*x i, u

1 Cy')

A

i

Division 11.

7

<C Aquilaj ...

>-l

#

A

Jd

i

8

0 Auri<*ii'

j'O

A Cf")

A

\

9

ft „

1-9

A r-/t

A

i

10
11
12
Compn.
13

7 Geminorum ....
z Lyrae (Vega) .
y- Geminoi'um (Castor)
2 Canis Majoris (Sirius)

a Cvfflli .

2-1
( + 0-9)
1-5
( + 1-9)
1-3

•"• \. U

A(«)
A (0
A(0
A (8)

A Cal

A
A
A
A?
A

la!!

i
i
i
i

2

Division III.

Compn.
14
15

a Geminornm (Uiistur)
a Opliiuclii ....
a Cephei

„

2-2
2-6

A (;)
A 03)

A Oi^

A
A

A

lo'l!

1

I
1

16
17

18

f Geminorum ....
a Canis Min. (Proeyon)

f Ceplici ...

3-5
( + 0-5)

( ^U'ar

A '(ft)

A
F

Pr

i,V:

1
1

2

19
20

i. Ursae Min. (Polaris)
f Aurigfe

I 4-9 J

2-0

( i'-Kar

B (/i)

K

2

2

21

7 Cys'ni

I 4'5 /
2-3

B (y.)

Q

o

Type 11.

Division IV.

22

/i Hcrculis

3-5

I?

1

23
24
25

a. Aurigie (Capella)
/3 Geminorum (Pollux)
7 Cephei

r+o-9)

1-4
3-5

B'(/9
B(«

F
A
K>

2

2
2

26

r, Draconis

3-0

Tr"

27

28

c Gemiuorum ....

3-3
3-0

••

K

Qv

• •

2

29

7 Draconis .

2-4

C (/5')

K

2

Type III.

Division V.

30
31

H Geminorura ....
2 Herculis

3-4
( JShar.

C(«)

C (a)

M

M

Ilia!!!

3

Compn.

j>

o Ceti (Mine)
a Orionis (Betelgeuse)

I 3'9/

136

MB. F. McCLEAN ON THE COMPARATIVE PHOTOGRAPHIC
PHOTOGRAPHIC Stellar Spectra. Stars to Magnitude 3 '5.

Lower Galactic Zone, N.
C (Plates 8 to 11).

. _j

O 0

* ••

a

1

ii

3*1

s

tfc O
ON

1

o

PI

i

H" •

if

S'3

1 2

Gd •*•* -

j

n
u
O,

td
*m

Q

PJ

|"

So

rf«*

ioe

r^>

tS

£

<•; «

~tl ~" '''•

0 «

CO IH T

cow

m

-3

o
A

£

||

"rt *r

is £

1=33

• 00
fc^H

s

49

0

Pi

«H

<M

11

l"'l

I"

£-A

I

5

I

0

6

1

|o

P ° t*

o .

|o7

1-

02

»

55

"

J

G

^

CO

o

Type I.

Division 1.

1

7 Cassiopeia; . . .

2-2

D

Q

• •

a

c, Orionis ....

| ip^Var.

A (a)

B

It!

1*

3

4

c ,. ....

K ,, ....

1-8
2-4

A (a)
A (a)

A
A

16?

1*

1*

5

r „ ....

1-8

A (a)

A

. .

1*

6

i ,, ....

3-2

A (a)

B

7

c Pei'sei

31

A (a)

A

2

8

3-1

\ /

A (a)

A?

2

9

7 Orionis (Bellatrix)

1-8

A (a)

B

id

1*

]0

t," Tauri . .

3-0

A (a.)

A

11

X

\ s

B

la!

12

1-8

A (a)

A

1

13

ft Persei (Algol) . .

^ \ /

A(v)

A

1

14

f „ . .

3-1

..

A

• *

I

15

0 Aquilffi ....

3'3

A

la!

16

>l Tauri (Alcyone)

3-1

A (7)

A

1

17

ft Orionis (Rigel)

( + 1-0)

A (a)

F

••

1*

Division II.

. .

None

Division III.

18

f Aquilffi ....

34

. .

A

la!

1

19

ft Eridani ....

2-8

. ,

A

20

ft Trianguli . . .

3-1

. ,

A

1

21

or. Aquilffi (AHair) .

1-0

A (ft)

A

la!'

1

22

f, Cassiopeia' . . .

2-9

A (ft)

A

1

23

ft ,. • • •

2-3

A (/3)

F

. .

1

24

ft Delphini ....

3-5

*

F

la!

2

25

3. Pei'sei . .

1-9

A. (ft)

F

^

1

Compn.

3-1

G

1

Type II.

Division IV.

26

7 Persei

3't

G

1

•J r

27

ft Aquilre ....

3-7

f f

I?

Ila

2

28

// Pegasi ....

2-9

B(/8)

G

2

29

2'4

C/ /J\
\l )

K

2

30

r

3'1

B (ft)

2

31

a Cassiopeia? .

B(/3)

K

2

32

7 Andromeda; . . .

2-1

ocg

K

. .

2

33

7 AquiliE ....

2-8

K

Ila!!

2

34

< Auriga; ....

2-9

..

K

••

••

Type III.

Division V.

35

ft Andromeda? .

2-2

C (a)

K?

. .

2

36

a Tauri (Aldobaran)

1-1

G(ft)

K?

Ila!!!

2

37

ft Pegasi ....

2-5

M?

• •

3

38

a Orionis ....

{ I'*} Var'

C (a)

M?

Ilia!!!

3

SPECTRA OP STAES TO THE 3| MAGNITUDE.

137

PHOTOGRAPHIC Stellar Spectra, Stars to Magnitude 3 -5.

Lower Galactic Polar Region, N.
D (Plates 12 to 14).

d

•a

»„— *

a, £

-.

Jli J?

"aS

bo o

(D

1

A

1

o

•° T*

&

o

O Cs]

J 2

H1

a

2

- ^

03 *^ .

i

o

&

bo

2

c^"T -/-

Ci "^

oo ri

"I0-

1

P»

B

o

2

•22

oo -

*•*

of10 ,

2i

H

"o

«J

o

ss
ft

55

• tn

H

'« fcr

9 n*

K a

OC' t- +

00 .£3 o

* OC

O

a

V

«*-!

o

•M
O

•| §

>H ^ e

X O S

S H

j'Sf °
H /; rH

— 1
5 »

8

CO

1

0

I

|°

O t-y r .

Q XH pH

|fi

g-s i

g=c
H

Type I.

Division I.

1

7 Pegasi (Algemb) .

2-5

AW

A

la

O

2

?t 5i

3-3

#

A

la!

1

3

a Androniedsc.

2-0

A (7)

A

1

Division ]T.

4

a Pegasi (Markab) .

2-3

A (7J

A

la!!

1

5

0

3-5

A

la!!

6

f, Aquarii ....

3-0

A

Division Til.

7

7 Ceti

3-4

*

A

la!

1

8

« Piscium ....

37

A

1

9

/3 Arietis ....

27

A (/^)

A

1

10

n Capricorni

3-0

A (7)

A

1

Type 11.

Division IV.

11

ft Aquarii ....

3-1

Kr

2

12

a ,, ....

3-0

. ,

K?

Ila

2

13

ft Ceti

2-4

B (/3)

K

2

14

•7

3-1

G?

2

15

3-4

I

2

16

S Andromeda1 .

3-2

K

2

17

i Ceti

37

K

2

18

3'5

K?

2

19

f> Eridani ....

3-0

I

2

20

6 ,, ....

3-4

, .

K

2

21

t\ Ceti ....

35

C (a)

K?

2

' r^/

22

a Arietis ....

2-1

B (y3^

K

2

23

e Pegasi ....

2-4

C (/3)

K

Ila'l!!

2

Type III.

Division V.

24

7 Eridani ....

3-0

M?

2

25

a. Ceti

2-2

C (a)

M

Ilia!!!

3

Compn.

a Orionis ....

*-* X1*/

M?

Ilia!!!

3

n

« Herculis ....

{ 3-9}Var'

O(-)

M

Ilia !!!

3

26

o Ceti (Mira) . . .

/ 3'3\Var
1 87 /V

M

3

Compn.

a Aurigse (Capella)

VOL, CXCI. — A,

138

PHOTOGRAPHIC SPECTRA OF STARS TO THE

MAGNITUDE.

PHOTOGRAPHIC Stellar Spectrum. Stars to Magnitude 3'5.

Upper Galactic Polar Region, S.
AA (Plates 15 to 17).

1

ti

..

tic 0

C

§

1

i-

1

il

HI

o5

1

|

1

? .81

g"f

« •£ * .

b

$

10

K

•

&

.

& »

*-" O

«°rj *

1

J**

i

2

i

o B

9j

0 |

00 a "**

CQ

*

o

O

.4

1

a /«f

|g

^§-8

'A

K

O

1

to

1

IM
O

6
K

IM

O

•8
13

11

JS5H

1?

fi

|?7

S ^

OQ

0

Type I.

Division I.

1

a Virginis (Spica) .

1-0

A (a)

A

1

2

a. Leonis (Regulus))

1-2

A (7)

A

la !!!

1

3

e Corvi

3-1

*

A

, f

1

4

/3 Librae

27

*

A

1

5

7 Corvi .

2-8

A

1

6

•>] Leonis

3-5

A (7)

A

la

1

Division 11.

7

ft Serpentis

3-3

..

A

. ,

1

8

rj Virginis ....

3-8

. .

A

la!

. .

9

d Leonis

3-4

A

la!!!

1

Division III.

10

f Virginis ....

3-4

#

A

la!!

11

ft Serpentis ....

3'5

. .

A

la!

12

K ! ji lira- .

3-0

#

A

1

18

e Serpentis ....

3-6

B ^

A

la

1

14

ft Leonis

2-1

#

A

la!!!

1

15

* „

2-5

A (^)

A

1

16

ft Geminorum . . .

3-6

A

. .

1

17

c Serpentis ....

3-8

. .

A

I! ?

2

18

7 Virginis ....

27

A (/»)

F

la!

1

Compn.

n Bootis .

27

B f/9)

G

la!

2

** Vr*/

Type II.

Division IV.

19

n Bootis ...

27

B r/3)

G

la!

2

^ r

20

ft Virginis ....

3-6

\"/

G

la

2

21

f Hydree ....

3-4

B (ft)

K

Ha

2

22

e Virginis ....

3-0

B(/3)

K

Ha

2

23

e Ophiuehi ....

3-3

B(/3)

K?

* *

2

24

7 Leonis

2-1

B (y3)

K

2

25

a Bootis (Arcturus)

(+07)

B(/3)

K

Ha !!

2

26

a Serpentis ....

27

Oft)

K?

Ha !!!

2

27

a Hydree ....

2-2

K

••

2

Type III.

Division V.

28
29

& Ophiuehi ....
6 Virginis ....

2-6
3-5

0(«)

M
M

Ilia!!!

3"

Compn.

a Orionis

.VcClean

1'lnl. Tram. A. 191. 1'late I.

.'i

U
LL!

Q-
(f)

CO

z
O

a
u

cr

i — I «-> DC

1 f") *f

m^

LLl 2 O

h |, ^

CD js o

<i

CL

o
o

H

o

X
CL

O

O

DC
Ul
Q.
Q.

IS ro

Median .

Phil. Trans A 191, Plate 2.

C£

H
U

Ld
CL

in

z
o

o

UJ

tr

UJ I

•3 °

CO a

o

O |

I

CL

O

o
o

X
CL

oc
u

0.

Q.

i 1

4

McClean.

Phil. Trans A 191, Plate 3.

£ I

i

Phil. Trans A 191, Ptatt 4

a:
H
U

UJ
Cu

in

HI

z
O

N

LU

Hffl

^

o -I
O § O

UJ

Q.

cr
O
O

O
CL

u-i VD

Chan.

1'lnl. Traus A. }y], Plate 5.

e

=
"=

IJ

.a ^

I

cr
0

UJ

CL

z

< ai

I z

-i :°

UJ "I 0

2 <

O I <

a.
a.

O

o

H
O
X
CL

OO Oi

McCltjn

Phil. Tram. A. 191, Plate 6

SS •

la f

I

S '~

* "i

U

LU
CL
CO

Z
O

N

<
_J

UJ 1 o

H I o PQ

rn J3 u n

o i o

CL
<

cr
O
O
H

O

oc

CL

cc

UJ
Q.
CL

t^ x> a-, o 1-1

1-1 1-1 « N CS

Phil. Tram A. 191, Plati 7.

^•J

+

_

s

— . ~

erT-

J*,

.2
|

It

Jj

*> ^?

K

.£

1

'P

±1.

JB

0

1

1

•t *»

X

|

— 5

"^s

•i 5

® ^-

5_

s

as.

?-

cc

o

5^

A
^

^

s §

I <S

~*~

-

o

s

Ul

z
O

N

U

UJ

Q-
(fi

a:

UJ

h

0!^

• — ' in

~~T ui
J-, uj

Ou °-

^^ 0-

< 3

O

O

o

s.

—

—

Median.

Phil. Trans. A . 191. Plate 8.

£

*;r-
*J*i

I

cc
o

LU
CL
CO

o:

LU

z
o

N

UJ I o

H I ji n

CO loU

2 **

o I <

— c/} (3

n: a

n , LLJ

cc
O
O

O

X
CL

. '(Clean.

Phil. Trans. A. 79?, Plate 9.

cc
H
U

UJ
CL
Ul

cr

u

z
o

N

UJ I °

CO

-" t—

60 ~ r>.

5 o 0

r ) 2 <

^-^ -2 C3

X

CL

or
u

K

O
O

O

CL

^

«

M [Clean

Phil. Tram A. 191. Plate 10

U

UJ
0.

CO

QC

UJ

z

. O
2 N

UJ

OJ ,K

•o O

I — . '5 I- r\

In I ° U

o -I

O S a

CL
<

cr
O
O
H
O
I

CL

Phtl. Trans A 191. Platt 11

CC

H
O

LU

a.

C/3

W-. QJ

<C z
I °

J - N

uui 2

o -J

: <

O

T- ^

_L LJ

5: -
<, z>

cr
O
O

O

Q,

Clean.

Phil. Trans. A . 191, Platt 12.

U

UJ

Q:
<

J

z
o

e

<

Ul E o

p I -

C/) s n

p

a
<
ct:
C
C

h
c

I
a

rt O

o

QC
LU

s
s

McClran .

Phil. Trans. A. 191, Plate 13.

1

QJ.

O

UJ

DL
c/)

<

>— I

z
o

•e j
.8 o

2 —

OS «

3 O

H -'

CL o

"^ oc

(r, uj

o S

O

I

cu

McClean.

Phil. Trans. A . 191, Plate

! •

I

H

CO
M

-

I

Phil. Trans. A. 191, Plate 15

O

LU
(X

. — I

03

z
O

O

LU

cc
ce

UJ 1 o

I— -3 Q.

I b&

C/^0<J

2 (-

r -) 8 O

v^ <j <j

0_ a

<^ ct

CC £

O o-

o 3

o
I
a.

•t ao

._

I 1

oo

S.

McClean

I-
O

LJ
CL
t/3

cc

CO

z~

o

o

UJ

cc

cc
<

H I °

CO s 0<j

2 ~

O I o

: !/3 <

CL o

cr
o
O
H
O
X
CL

cc

UJ

a.
Q.

D

Phil. Trans. A. 191 Plate 16

U-) ȣ)

Median.

Phil. Trans. A. 191 Plait 17

C£

h-

O

UJ

CO

CL

CO

z"

o

a:

a

u

<£

CE

_i

,/s cc

i ]

ro <

UJ

1 Q ,

t-

E °- ^

00

^ 0 <j

2 ~

u

H

5 °

1 — ,

i/5 <

"X

^

CL

O

<£

DC

cc

111

O

a.

a.

O

3

O

X

CL

Ol

I-

=

I

[ 139 1

V. On the Application of Harmonic Analysis to the Dynamical Theory of the Tides. —
Part II. On the General Integration of LAPLACE'S Dynamical Equations.

By S. S. HOUGH, M.A., Fellow of St. John's College and Isaac Newton Student in

the University of Cambridge.

Communicated by Professor G. H. DARWIN, F.R.S.

Received October 27,— Read December 9, 1897.

IN the former paper on this subject I have dealt with the formation of LAPLACE'S
dynamical equation for the tides, and the integration of it, subject to the limitation
that the solutions obtained should be symmetrical with respect to the axis of
rotation. In the present paper I propose to extend the method of solution so as to
free it from this restriction.

The difficulties experienced by LAPLACE in his attempts to integrate the equation
in question were so great that he abandoned all efforts to obtain a general solution,
and confined his discussion to a few of the special cases which present the greatest
interest from a practical point of view ; even in these simple cases however his
original attempts to express the solutions by means of the coefficients associated
with his name were discarded in favour of series proceeding according to powers of
a certain variable used to define the position of a point on the Earth's surface.
These power-series have been further employed by Lord KELVIN* to obtain a more
general solution of the problem, but the results obtained, though of considerable
analytical interest, do not lend themselves well to a numerical discussion. Both
AlBYt and KELVIN condemn the employment of the surface-harmonic functions
as inappropriate, but a profound conviction that the efforts of LAPLACE, though
unsuccessful, were well directed, has led me to take up the problem again from his
point of view ; with what success will be seen hereafter.

I was originally led to attack the problem by a totally different method from that
of LAPLACE based on the work of POINCABEJ and BRYAN§, and the principal
analytical results, both in this paper and in the preceding, were at first obtained by

* " On the General Integration of LAPIACE'S Differential Equation of the Tides." ' Phil. Mag.,' 1875.
t ' Encyc. Metropolitana.' Art, " Tides and Waves," Section III., § 116.

% " Snr 1'equilibre d'une masse fluide animee d'un mouvement de rotation." ' Acta Math.,' vol. 7,
p. 355, et seq.

§ " The Waves on a Rotating Liquid Spheroid of Finite Ellipticity." ' Phil. Trans.,' A, 1889.

T 2 24.5.98.

140 ME. S. S. HOUGH ON THE APPLICATION OF HARMONIC

a very lengthy analysis similar to that used by the latter writer. The comparative
simplicity of these results seemed, however, to point to the fact that they might be
more easily obtained by less pretentious means. The deduction of the formulae in
the former paper from the differential equation of LAPLACE presented no serious
difficulties, but in attempting to apply a like method to obtain the more general
formulae of the present paper, I found that formidable obstacles had to be overcome.
The method of integration now adopted seems to leave little to be desired for sim-
plicity, considering its generality, but the fact that it has been built up partly by
working forwards from the differential equation, and partly by working backwards
from the results, must account for the apparent artificiality of the procedure.

In all previous attempts at the solution of the dynamical equations for the tides,
the integration has been effected by assuming that the expression for the tide-height
could be expressed by an infinite series of terms of known form associated with
undetermined numerical coefficients. The differential equations then lead to a
difference-relation between a certain number of these coefficients from which their
numerical values are to be evaluated. The numerical determination of the coefficients
will be facilitated when this difference-relation contains as few terms as possible.
Now it is found in the present paper that, without imposing any restriction on the
period of the disturbing force, if the form we assign to the terms of the series for the
tide-height is that of the tesseral harmonics or LAPLACE'S functions, a linear relation
involving three successive coefficients only may be deduced, provided that the law of
depth is such that both the internal and external surfaces of the ocean are spheroids
of revolution about the polar axis. This however appears to be the most general law
of depth which can be employed without obtaining more than three successive
coefficients in the linear relation in question, and consequently our discussion deals
only with cases where the law of depth is subject to this limitation.

In § 1 I have collected the principal properties of the functions used in the
analysis. These properties are for the most part well known, but in consideration of
the want of agreement in the notation employed by different writers, I have thought
it best to briefly prove such of them as are required in preference to giving references
to places where they may be found. Moreover I have thus been enabled to write
the results in the exact form required for subsequent application.

§§ 2-4 deal with the integration of the differential equations and the deduction of
the linear equations (31), (40) connecting the coefficients in the expansion of the
tide-height. These equations, the analogy of which with the equations (23), (23A)
of Part I. will be at once apparent, constitute the chief analytical results of the
paper, and the remainder is occupied with the application of these formulas to
the discussion of the free and forced vibrations on lines similar to those adopted
in Part I.

§§ 5-11 treat of the free oscillations, the discussion being confined to the case
where the depth is uniform. A period-equation is obtained, and an approximate

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 141

method of determining the higher roots is given. The approximations will not
however be sufficiently close for the earlier roots, and consequently it is necessary
to evaluate these earlier roots by trial and error. The method of procedure is
indicated by numerical examples, and several of the more important roots are-
tabulated for four different depths of the ocean. The most interesting result is the
existence of a second class of free oscillations besides those whose existence may be
at once inferred by analogy from the simpler problem of the oscillations of an ocean
covering a non-rotating, globe. The characteristics of the oscillations of this class
are discussed in § 11.

In § 12 a general analytical solution of the problem of the forced vibrations due to
any disturbing force is given, but as the analytical expressions obtained are too
intricate to afford much indication of the nature of the forced tides, the various
types of oscillation which occur on the earth are afterwards treated numerically.

In certain cases, intimately associated with those actually occurring, the analytical
expressions however admit of considerable reductions. These cases are discussed
in § 14, where theorems due to LAPLACE and Professor DARWIN are obtained and
generalized.

§§ 15-18 contain numerical examples of the evaluation of the semi-diurnal and
diurnal tidal constituents. The arithmetic is considerably simplified when the
period of the disturbing force is rigorously equal to half a sidereal day or a sidereal
day, and consequently these cases are first dealt with and the results compared
with those of LAPLACE. Additional examples are however also given to illustrate
the effects of the departure of the periods from exact coincidence with half a sidereal
day and a sidereal day respectively, the cases selected for investigation corresponding
with the leading lunar constituents.

§ 1. Properties of Tesseral Harmonics.

Let P» (/A) denote the zonal harmonic of order n. Then P,, is the solution which
remains finite when p, = ± 1 of the differential equation

Let

P; (p.) = (1 — /A3)*8 ^7" (2).

Then, on differentiating the equation (I) s times, we obtain

or,

(1 _ ^v *. {( t _ ^-i. P.} - 2 (, + 1) ,* (1 - ^ ^ {(1

4. (n - s) (n + s + 1) P;, = 0,

142 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

which on reduction gives

Thus P"n is a solution of the equation (3) ; the form (2) shows that it does not
become infinite when p = ± 1> while from (3) we see that the two functions
P*, cos s<j), P' sin S(f>, or what is equivalent, the two functions P^e*'"*, are spherical
surface-harmonics of order n. In our subsequent work the latter forms involving the
imaginary exponential will be more convenient than the real trigonometrical forms.
We shall therefore describe the functions P' (/u,)e±"* as the tesseral harmonics of
order n and rank *•. In some cases it may be convenient to apply the same nomen-
clature to the "associated function" P?, (p), but whenever this is done, it must be
understood that an exponential factor is implied, though not expressed.

The tesseral harmonics of course include as special cases the zonal harmonics
obtained by putting s = 0, and the sectorial harmonics obtained by putting s = n,
while, in accordance with the definition (2), for values of * greater than n we may
suppose that P*,^) = 0.

The principal properties of the tesseral harmonics which we shall require may be
derived from those of the zonal harmonics. Thus, if we differentiate s times the well-
known relation

(n + 1 ) Pn+I - (In + ]) ,iP, + nPB_, = 0 (4),

we obtain

" d^ dp

which, on making use of the formula

gives

On multiplying by the factor (1 — /x2)=J this may be written

(n-s+ 1)P;1+1 - (2n + 1) MP; -|- (n + s)IV, = 0 . . . (6).

Again by difierentiating the equation (2) we find

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES.
in virtue of the differential equation (1) for PB ; and therefore by means of (5)

(i - ^.ffi = , p. _ OL^L± i) (« f o * *

'dp

143

2n

which with the aid of (6) may be expressed in the form

-

Let us write for brevity

d

AEE~(D)-

(8)-

Then the equation (3) may be written

, = - n(n+ 1)P:
while, if a- denote any constant quantity, we obtain from (6), (7)

(D

(9),

The relation between the operators D, A may be written in the forms

(D - o/t) (D + cr/t) - (^ ~ ^-) = (1 - M2) (A + «r) 1
(D + 07*) (D - crM) - (# - aV2) = (1 " M2) (A - a) J

which will be useful hereafter.

§ 2. Transformation of the Dynamical Equations for the Tides.

The formation of the differential equations for the tidal oscillations of the ocean
has been fully dealt with in Part I. It is there shown (§ 4) that, if U, V denote the
northward and eastward velocity-components in latitude sin"1^ and longitude </>,
when the system is executing a simple harmonic vibration in period 2ir/\, these
velocity-components will be expressible in terms of a single function i/» by means of
the equations

"

a (X» - 4» V2)

" h

v/ ( 1 -

2 - - 4« V)

_

„ (X» - 4w V)

, 1 2v

a v/ (1 - p2) (X3 - 4a>V)

144 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

Supposing that U, V, $ are each proportional to ei(M + **} , we may take 3t///3<£ = is\fi,
and therefore, if we put 2(os/\ = a- and introduce the abridged notation of the
previous section, we may write the above equations m the form —

The equation of continuity is

where h denotes the depth, and £ the height of the surface-waves. On substituting
for U, V from (13) and performing the differentiations with regard to t, <f> this
becomes

or

This equation is equivalent to the equation (17) of Part T. A second equation for
the determination of the two functions i/», £ is obtained from the pressure-condition
at the free surface. On reference to § 2 of Part I., this condition is seen to lead to

* = v'-9t + v ......... (15),

where v denotes the surface-value of the potential due to the harmonic inequalities,
and v the surface-value of the disturbing potential.

In order to effect the integration of these equations, we introduce two auxiliary
functions ^, y2, connected with i/» by the relation

*s ...... (16).

On applying the operator (D — CT/A) to the two members of this equation, we
obtain in virtue of (11)

(D - cr/t)* = (1 - ^)(A + cr)^ + (s2 -

/*s)*8 - - (17).

Now the functions 9lt ¥? have as yet been subjected only to the single condition

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES.

-145

(16). We may therefore impose on them any other arbitrary condition not
inconsistent with the former. Suppose we choose them so as to satisfy the relation

(A

(18).

The two equations (16), (18) serve for the complete definition of the two functions
<!>!, ¥., ; making use of the latter, (17) reduces to

(D -

, + (D - 07.) *,}.... (19).

Thus on replacing \fi by its value in terms of Vit V.2 in the right-hand member
of (14), we deduce

If we suppose that h is constant, the terms involving
virtue of (11) we shall obtain

will disappear, while in

or

(20).

We have now for the determination of the functions t/», £, ^^
simultaneous differential equations (15), (16), (18), (20).

the four

§ 3. Integration in Series of Tesseral Harmonics.

Let us suppose that \jj, £, v, ^"]; % are each expressible as series of tessera!
harmonics of the same rank s. Omitting the exponential factor e'^! + 5lf" , we assume
that

< ]

Then, if p denote the density of the water, and o-0 the mean density of the whole
VOL. cxci. — A. u

146 MR. S. S. HOUGH ON THE APPLICATION OF HARMONid

system inclusive of the ocean, by well-known properties of surface-harmonics it

may be shown that

„,' _ v
-2

Thus from (15), on replacing the quantities involved by means of their expansions
in terms of associated functions, we obtain

- IK P: M + V-P*. 00 ;

whence, if we equate coefficients of P"4 in the two members, we deduce

r4 = -0,d+y. ........ (22)

where we have written for brevity

From (16), (18), we Lave

\l> -- (D + cr/i) vl'i + '«sM>a - -£/* (A + <r) ¥x ..... (24).

13ut by means of (10) we iind, on replacing ^ by its expansion,

M ) 4- - ,\ M/ • v». r(rt ~ ff) (" ~ " + ]) P< (/' + ^ + 1) (M + s) p. 1

CT/i)^- ~27TT" 1"+1" 2» + i

while from (9), (6), we obtain

p (A + „) V, - S< (cr - „ (u + 1)} p^rlV, + ^ Pi-.] -
Thus

+ 2) + (r}(n + «) p.- n •

2« + l

and the right-hand member of (24) is therefore equal to

iv^.- g"-« P,

"*~a"- ~" i"+1"

Hence, on comparing the coefficients of P; in the two members of (24), we obtain

ANALYSIS TO THE DYNAMICAL THEORY OP THK TIDES. 147

Again, on -replacing ¥lf ¥3 by their expansions in (18), the two members may be
expressed as series of associated functions by means of (9), (6). Thus we find

- * I. (* * I) — J P-, = 2^* [^ P.., + i±l p. .,] ,
whence, on equating coefficients, we deduce that

Finally from (20), on expressing the two members by their expansions and
equating coefficients, we obtain

(27)

The relations (26), (27), enable us to eliminate the auxiliary constants fi',, <_,,
<+1 from (25), and thus to express F, as a linear function of C*_2, C*, C?,+2. On
substituting for <_,, <+1 from the formula (2G) in (25), we obtain

_ ., _

-s- n *

2^T **»'* ' " ST+1 K

„ - •„ - „

(M + a+ !){(•» + 2) (» + 3) + <r} {n-s+ 1 M + « + 2 1

(2n + 3) {(•« + ]) (n + 2) - a} [ 2n + I P* ' 2n + I, ^'+2 J

_ __ 2 (» - a>(» -a-l) {(» - 1) (a - 2)

(2n - 1) (2n - 3) {(n - 1) n - a]

- - _ Q" _ _ | /?;

. - i \/ii /O'ii1\/*)"iit"*\f/-ii1'WiiO\ ~\ \ " il

ft "^ J. I It ~~^ u f I _ / ' *T~ J. I ( _ // ~^~ * * / i ( /( i" i )\ ll ~} — J ^^ O" f

2 (« + 8 + 1) (» + a + 2) {(» + 2) (» + 3) + o-}
(2» + 3) (2n + 5) {(n + 1) (» + 2) - <r}

and this, by means of (27), gives

-Ml fo - s) Qt - a - 1)

< r,

4o)8a»'~ (2% - 1)(2» - 3) {(» - 1)M - <

, __ (« + * + 1) («-»-« + 2) r,

r (2n + 3) (2» + 5) {(« + 1) (» + 2) - <r}

where

__ s^ ___ (n - s) Qi + <) {(n - 1) (« - 2) + q]
" "" <r {n (n + 1) + o-} (2-n -i) (2n + 1) {(n - 1)» - <r} {m (w + 1) + <r]

(n + s + 1) (M - 8 + 1) {(?t + 2) Q + 3) + <r}
~ (2n + 1) (2» + 3) {(» + 1) (» + 2) - <r} {» (« + 1) + a}'
V 2

148 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

This expression for A* may be somewhat simplified if we separate it into its com-
ponent partial fractions; we thus find

s2 (n - s) (n + s) T (n - I)2 2n - 1 1

= " ) [n*{n(n- 1) -a] ~ n*{n(n + 1) + <r}J

(n + 2)2

3)

+ 2)2 __ __ 2n + 3 1

!)(» + 2) ~ "} (» + 1)2{« (« + 1) + *} J

1 ["^ tt2 -s2 Qi + iy - s2 1

(» + 1) + <r} to-2 """ w2 (2« + 1) (71 + I)3 (2n + 1)J

2)2 (>t - s

" 7r (2/i - 1) (27i + l){n(n-l)-<r} (n + I)3 (2n + 1) (2n + 3){(« + 1) (n + 2) - a} '
whence finally, remembering that or = 2ws/X,

X- n (/i + 1) - 2«as/X Qt - 1)- (M - s) Qi + s)

Z 4^ "~"rf~(n + ]7r " /r (27i - 1) (27», + 1) {(n - l)n - 2eos/\}

(n + 2)2 (TI -« + 1) (n + s + 1)

. ^zy;.

(71 + I)2 (271 + 1) (271 + 3) {(71 + 1) (71+ 2) - 20JS/X}

The relation (28) will hold for all values of n equal to or greater than s, provided
we suppose that C*_2 = 0 and C*_! = 0. If we put for brevity

(n —s+ l)(n- s + 2)

JL,t —

it may be written

* " (27i + 1) (2n + 3) {(n + 1) (71 + 2) - 2ws/\}>

f- • • • (30),
, __ (7^ + 5+ 1) (n + s + 2) _

= (2» + 3) (2w + 5) {(71 + 1) (n + 2) - 2a,s/\}

" - ~* p* ASP-' I nf P*

2 — Jjn-y\>n-t — •£*»v'n "T yn^n+Z-

4o;2«2
Replacing T' by its value in terms of C'n, /„ [equation (22)], we obtain

• p., T.p., I .p, W»

where x'_2, y' are defined by (30) and

T < %" /on\

L' = — T^-T -f AJ, (32),

A*i being defined by the equation (29).

On putting s = 0 it may readily be verified that the equation (31) reduces to the
equation (23) of Part I, The manner in which such an equation may be utilized for

ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 149

the determination of the free and forced vibrations has been fully discussed in that
paper. A strictly analogous procedure might be adopted in the present case, but it is
found convenient to modify the previous treatment in some respects. We shall
therefore only indicate briefly the course of procedure when our analysis corresponds
with that given in Part I., giving greater detail as regards those points where a
different method has been found desirable.

§ 4. Extension to the Case of Variable, Depth.

The formulse developed in the preceding sections apply only to the case where the
depth is uniform. We may however obtain a relation of the same nature as the
equation (31) when the depth is a function of the latitude given by the formula

h = k + l(l-p*) ......... (33),

that is, when both the internal and external surfaces of the ocean are spheroids of
revolution.

The above expression for h may be written in the form

X- -

(34)>

where

Substituting the expression (34) for h in the equation (14) we obtain
(L - ^)£= (D +

+ [(D + <rp) (D - 07.) - («2 - cr
which, with the aid of (11), becomes

= <D + ^ (D ~

The right-hand member is of the same form as that of the equation (14), except
that h is replaced by K. We may therefore introduce two auxiliary functions ¥j,
•*2, defined by (16), (18), and proceed as in § 2, and we shall obtain in place of (20)
the equation

150 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

But, if we introduce the expansions (21) for i/», £, we find

n<n + 1> + *}I"-P'- ' ' (37)'

whence the final equation which replaces (31) will be obtained by replacing C^ by

Ci + ^{n(» + l) + «r}n ....... (38)

in the right-hand member of (28), and h by K in the left-hand member.
Thus we obtain

+ tf 3;+. + s {(* + •)(» + ») + »}!:*• • . (39);

and therefore, on separating out the parts of F. due to C^, y', respectively, we find

£'-£*-.:- i:.C" +,'Q,,2 = G: ........ (40),

where

2) (» + S)+ "

and

For the special case s = 0 the equation (40) reduces to the equation (23A) of
Parti.

§ 5. The Period -equation for the Five Oscillations in an Ocean of Uniform Depth.
To determine the periods of free oscillation we may proceed exactly as in § 6 of

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 151

Part I., making use of the equation (31) instead of the simpler equation (23) of
the previous paper. On putting yj, zero, the equation (31) gives

^CU-LlCl + ^C^^O (43),

an equation which must hold for all values of n equal to or greater than .s, it being
understood that Cj_2 = 0, and C'_, = 0. The series of equations typified by (43)
may be divided into two groups, in the former of which the suffixes involved are
such that n — s is even and in the latter odd. The types of motion resulting from
these two groups may be treated independently, the former being characterised by
symmetry with respect to the equator and the latter by asymmetry. The treatment
of the two groups of equations will be exactly similar, and we shall therefore in the
main confine our discussion to the former group.
If we introduce the notation

'/ 1J I 'I I ~l II

H:< = 'if - t'lT - . . . - "L/

(i-O,

LI — J'u+s — . • . «(/ </'/.

it may be shown as in Part I. that provided L/C*,. = 0,

''J"^" . _ TT, y»-2*"'» _ . T7-. / ( r \

~^~e — in,, , <e — iv,, (4j),

and therefore the equation (43) may b« written

whence the period-equation for the free oscillations of symmetrical type is obtainable
in the form

L: - H;t_2 - K;+2 = o (46),

when n — s is an even integer.

The same equation will apply to the asymmetrical types if we suppose that n — s is
au odd integer, and that the continued fraction H" terminates with the partial

quotient -'-'-~^ - .

In particular, putting it, — n = 0, we can express the period-equation for the
symmetrical types in the form

L|+2 - !<;+. - • • • ud mj.
while, putting n — s = 1, that for the asymmetrical types may be written

L;+8 -

On the analogy of the problem dealt with in Part I., we may anticipate that, when

152

MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

X has a value in the neighbourhood of a root of the equation Lj, = 0, the continued
fractions H*_2, K'l+2 will rapidly converge to small values. Further, with large
values of n, the numerical values of H»_2, Kjl+2 tend to become equal with opposite
signs. Hence there will be roots of the equation (46) which approximate to roots of

the equation

L* = 0 (47).

Let us therefore examine the nature of the roots of this latter equation ; this
may best be done by considering the graph of the function A*. Putting y = A',
x = X/w, we have to consider the form of the curve

j o " (n + 1) — '2s/,c (ii — 1)- (n — s) (w + s)

n~(n+ I)3

n" ('2n - 1) (2>t + 1) {n (n - I) - 2s/x]
(•« + 2)2(n -s -f l)(u +s+

(n + I)2 (2n + 1) (2n + 3) {(« + 1) (» + 2) -

It is evident that this curve will have two rectilinear asymptotes parallel to the
axis of y, whose equations are

"2s

x =

n (n — 1}

(n + 1) (7i + 2)

and that on passing these critical values, with increasing x, the sign of y will change
from positive to negative. The curve will pass through the origin, while when x is
very large it will approximate to the parabola

.
J ~

»» (n + I)3

Hence it must consist of three branches as in the annexed diagram, where the
dotted lines represent the rectilinear and parabolic asymptotes.

ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 153

The roots of the equation L'n = 0, regarded as an equation for the determination
of \/a>, will be the abscissae of the points of intersection of this curve with the line

y =

Since h is essentially positive, the roots will all be real, and they will lie in the
intervals between

2s 2s

°' („+!)(„ + 2)' n («-!)'

For large values of %,,/4<u2a2 the two. extreme roots will approximate to the roots
of the equation

%„

, X3 n (n + 1) — 2(0s/\ _
""" - 3"'

~ 4w%3 """ 4^ -n? (n +TJ3
while the remaining two roots will approximate to

___ 2s__ _2s

(n + T) '(n + 2) ' n (n — 1) '

The two former roots are those which have their analogue in the special case
treated of in Part I., for which s — 0, in which case they have equal magnitudes but
opposite signs. These roots, we may expect, will approximate to roots of the period-
equation, at least when n is large.

In order to see the significance of the remaining roots, it is convenient to transform
the period-equation into a different form, which moreover is far better adapted for
the more accurate numerical determination of the earlier roots.

§ 6. Modified Form of the Period -Equation.

Eeferring back to the equations (29), (30), (32), which define xs,,, y\, I/,, we see
that (31) may be written in the form

(n — S) | v», — o— if ~8 V"- — J./ y'"-r^ p,,

oN W-2T ./o,,, , 1\ ^»

f X3 n (n + 1) — 2cos/\ hgn 1 _

. (M + 8+1) _ l"(/t + 2)3(M-S+ 1) c, (7t + S+2) c, "I _

Hence, if we introduce a new set of auxiliary constants, D], D*+1, &c., such that
for values of n equal to or greater than s

2n
VOL. CXCI. — A.

~^ «-• + !1I^TT1) c"« '= {- C + « - T} « • <49'-

154 MB. S. S. HOUGH ON THE APPLICATION OF HARMONIC

the equation (48) reduces to

(n + I)* (n - S) n* (n + s + 1) n.

2n-l •U"-1 H 2n + 3

Thus, if we write for brevity

N; = w(« + 1)-^-

and put y' = 0, the equation (43) is replaced by the two following :

(n f !)»(« - s) n, ,,, ps

2^n" u»-1"

(» + I)2 (« - *) ., S

-

1

(51),

(52).

For the determination of the symmetrical types we therefore have the series of
equations

2s + 3

(s + 2y.l

2S+1

. ,

's + i T

s+2 ~ '

2s +3

On eliminating the quantities Cj, D'+1) Cj+2, &c., by means of a continued
fraction we find the period-equation in the form

(53),

N;+1 - M;+2 - N;+s - . . . ad inf.

where we have written for brevity a* in place of

«3 (n + 2)3 (m - s + 1) (n + s + 1)

(2w 4 1) (2» + 3)
In like manner the period-equation for the asymmetrical types may be written

(54).

N* _ * * + l *+2 _ f\

* ~\/T XT Tl~]r 7 * J» "^ "

(55).

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 155

We may also write these equations in a variety of alternative forms in which
prominence is given to any one we please of the quantities M', N,. These forms
are obtained by giving n different integral values in the equations

= 0.

NS |"o.'-i «'-2 <.» "I r «; «;+i «.•+» "1 n

*~ M« - N" - M' AT" V M'

LJi«-l iNn-S ""i-a • • J LlU/( + l ^n+i — iu»+:: • • -J

In each case the former continued fraction terminates with a partial quotient
involving a", in the numerator and either M^ or N* in the denominator, while the
latter proceeds to infinity.

For the symmetrical types, if we use the form (56) we must suppose n — .t an
even integer, whereas if we employ (57) n — *• must be supposed odd. The reverse
will of course be the case for the asymmetrical types.

The continued fractions of the present section will not converge so rapidly as those
of the preceding, but in spite of this drawback they present considerable advan-
tages. In the first place the numerators of the partial quotients, which are obtained
by giving n different integral values in the expression (54), are independent of X.
These numerators, which further are in a convenient form for logarithmic computation,
may therefore be tabulated once for all, whereas the numerators of the partial
quotients in the continued fractions of the last section require to be re-determined at
each successive trial in attempting to solve the period-equation by trial and error.
Moreover, the evaluation of the denominators M',, N;\ by means of the formulae (51)
may be very quickly effected, even though a fairly large number of these denomi-
nators is required, whereas the evaluation of the quantities L* by means of (29) and
(32) is extremely laborious.

Another disadvantage resulting from the use of the preceding form is that when
X/to is near the value 2s/n(n + 1) the functions x'_lt yl^ both become large, while
Ll-b L»+i have both a zero and an infinity in the immediate proximity of this
value. Hence, in order to evaluate L^, L*l+1 for a value of X in this region it is
necessary to observe a very high degree of accuracy in the numerical work. The
singularities which occur in the left-hand member of (46) when X passes through one
of these critical values no longer appear if we write the period -equation in the form
(57) with the value of n appropriately chosen.

§ 7. Expressions for the Velocity -Components.

The auxiliary constants Dj, Dj+1, &c., introduced in the last section, may be made
use of to express the velocity -components by means of series of surface-harmonics.

X 2

156 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

Thus from the first of equations (13) we have, on replacing $ by its value in
terms of ^5 W,

On introducing the expansions for M^, ¥3 on the right we find

. 4. fl. _
. r P. |

, («• + <7 - 1) (« - s) (» - <r + 2) (n + 8+1)

~?n-l~

which, with the help of (26), (27), reduces to

//i 8\TT_?^VP« f (n-s)(n + <r-l) p,

v(L ~/*M ff/t 'r»L (2w-l){w(« + l)-(r}V h (2

or from (49),

.. . (58).

This may also be written in the form

y(i-^)u = ^s{/,a-D:(}p:! ...... (59).

Again, from (13)

~ /*$ (/.c; - D:) ?•„ + srnpn . . . (GO),

whence, by means of (6), we may express \/([ — ft2) V by a series of surface-
harmonics.

The corresponding formula? when the depth is variable may be obtained by
replacing h by K and C't by

so that we find

where the quantities D't are now defined by

ANALYSTS TO THE DYNAMICAL THEORY OF THE TIDES. 157

(n + s + 1)

The formulae (61) cease to be of use in a special case which will present itself here-
after for which K = 0. It will be seen in a later section that the expressions on the
right become indeterminate in this case, so that the determination of the velocity-
components must be effected by means of the formulae (12) or (13). These latter
formulae seem at first sight to indicate that the velocity-components become infinite
in latitude sin"1 ($/cr), but the forms (61) indicate that such cannot be the case, at
least when K is different from zero.

§ 8. Approximate. Determination of the Higher Roots of the Period- Equation.

If we take the period-equation in the form (56), and as a first approximation omit
the continued fractions from the left-hand member, it reduces to the quadratic

M:, = o,

or

For large values of n the roots of this equation will give a sufficiently accurate
approximation to the roots of the period-equation, since it may be seen that the
continued fractions tend to limits comparable with Jr/i3, and therefore small in com-

parison with na (n + I)3 " a , when n is very large and X/&> has as its value either of

~X.G) Or

the roots of this equation. We may even obtain a fair approximation by omitting
the term containing s, in which case the formula for X corresponds with that obtained
when the rotation is omitted.

A better approximation will however be obtained by representing the continued
fractions by their first convergents instead of entirely neglecting them. The
approximate form of the period-equation is then

or

_X^ fo(m+l) — 2(as/\} (n - I)2 (n - s) (n + s)

4w3 " nz (n + I)2 " n2 (2n - 1) (2n + 1) {(n - l)n - 2,

(n + 2)3 (« - s + 1) (M + s + 1)

" (n + I)2 (2ra + l)(2»t + 3) {(71 +"l) (n + 2) - 2a>s/\} 4w3ffl2 ~

We thus get back to the equation Lf, = 0. The roots of this equation may be
approximated to numerically by HORNEE'S process, the significant roots being those
which lie in the intervals between — oo and 0, and between 2s/n (n — 1) and + °° •

158 ME. S. S. HOUGH ON THE APPLICATION OF HAEMONIC

For the particular case s = 0, the biquadratic to which the equation l£ = 0 is
equivalent reduces to a quadratic, the two roots which remain finite being of equal
magnitude and opposite sign. This special case has been examined in Part L, and it
will be seen on reference to the tables there given (§§ 7-8), that the roots of the
equation L,, = 0 give a very good approximation to the roots of the period-equation
except in the case of the earlier roots when kg/Auto2 is small. In the present paper
I have examined in some detail the special cases corresponding to the values 1 and 2
for s, and the approximation is found to be equally rapid, as will be seen from the
tables given hereafter. Consequently, in these cases at least, all except the two or
three smallest roots will be obtained with adequate accuracy by finding the roots
which lie in the stated intervals of the equations L' = 0 with different integral
values of n.

The roots so found will not however form the complete series of roots of the
period-equation. We may in fact anticipate that the remaining roots of the equation
L'j = 0 will also approximate to roots of the period-equation. To obtain a better
approximation of the roots of this class, it will however be preferable to make use
of the period-equation in the form (57). As a first approximation we omit the
continued fractions and obtain

, , 2o>.s
N" = 0 or n (n + 1 ) = 0.

A,

This method of approximation will be valid if when X/w = 2s/n(n + 1) the two
continued fractions involved in (57) are small in comparison with n (n + 1). But it
may readily be verified that with large values of n these continued fractions become
comparable with a)2a?/hg, and therefore the desired condition will certainly be
satisfied when n is sufficiently large.

A better approximation may be obtained by representing the continued fractions
by their first converger) ts. We thus obtain as the approximate form of the period-
equation for the determination of the root which lies near -

* M I an .

N' -

n (n + 1)
a', a'

or

(n - I)2 (n 4- I)3 (n - s) (n + s)

]:(:: | n _

' '

(n n 2wsl
Tj

(n + 2)3 (n - s + 1) (n + s + 1)
(2n + 1) (2*1+3)

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 159

Using the first approximation in the terms on the right, we deduce

(n-lY(n+ l)°-(n _«)(» + «) n'(n + 2)*(n-s+ !)(» + 8 + 1)
nfr.il) 2(°S- (2n-l)(2n+l) (2n + 1) (2« + 3)

X *=*- + - -**- (n + !)•(» + 2

*

This formula is found to lead to-the roots of the period-equation with a surprising
degree of accuracy.

Our analysis is only applicable when /* is small in comparison with a, but subject to
this limitation the approximations of the present section will improve as hg/±<aza?
increases, that is, as the depth of the water increases or the angular velocity of rotation
diminishes. They will give good results even with small values of n when w is
sufficiently small, and they may be used to determine the limiting values assumed
by the roots when the angular velocity of rotation is indefinitely reduced.

We see then that, the roots of the period-equation are of two classes, which may
be distinguished by their limiting forms when the rotation is annulled. The roots of
the former class are such that the values of X remain finite when w = 0, their limitino-

3 ft

values being given by the formula

x =

n (n + 1) hgn

There will be an equal number of positive and negative roots of this class, but
though these approach the same limiting values their numerical values will not be
equal as in the case where s — 0, and the positive and negative roots must therefore
be determined independently.

The roots of the second class are all positive and are such that the values of X/o>
tend to finite limits when w is reduced to zero, the limiting values being given by

the formula

\ _ 2s
to n (n + 1)

whereas X. will tend to the limit zero.

The analogue of the types of motion which correspond with the former roots will
still be oscillatory when the rotation is annulled, but the types of motion corre-
sponding with the roots of the second class will cease to exist as oscillations when
the angular velocity of rotation is reduced to zero. These types of motion will
have their equivalent in steady motions, but an infinitesimal amount of rotation
would immediately convert such steady motions into oscillatory motions of very
long period.

For the particular case s = 0 the roots of the second class are all zero even
when the angular velocity of rotation is finite. Hence steady motions can exist
on a rotating globe, but these are necessarily of zonal type. We have in fact

160 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

already seen in Part I.* that the only forms of steady motion which can exist are of
this character, and have explained the fact by stating that the steady motions not
of zonal type which can exist on a globe without rotation must have their analogue
in the more general case in oscillatory motions whose period bears a finite ratio to
the rotation-period, no matter how great the latter may be. Our present work
confirms this statement and throws further light on the nature of these oscillatory
motions.

§ .9. Evaluation of the Earlier Roots.

The errors resulting from the use of the approximate formulae of the last section
may be considerable in the case of the earlier roots for which n has small values.
To obtain these earlier roots we must therefore proceed by trial and error, the
preceding method being made use of to obtain values with which to commence
the trials.

As a concrete example we will discuss in detail the computation of the positive
root of the first class corresponding to the case n = 4, s = 1, when the depth is
given by %/4w2o2 = -£$. Taking p/cr0 = 0*18093, and introducing the numerical
values of n, s, and h, the equation 14 = 0 becomes

(X/«u)4 - 0-3333 (X/w)3 - 5-5481 (X/co)2 + 1-0906 (X/w) - 0'0418 = 0.
By HORNER'S process the greatest positive root of this equation is found to be

2-43265.

Now experience shows that the numerical value thus suggested is in general too
small. t We therefore select for a first trial a value rather larger than that
indicated, say, for example,

X/w = 2-4400.

From the formula (54) we find

log a\ = 0-2553, log a\ = 1-1652, log a\ — 17289,
log a] = 2-1450, log a\ = 2'4769, log"a£ = 27537,
log a\ = 2-9915, log aj = 3-2001, log al = 3'3859,

while the values of the expression

'

for the values 2, 4, 6, 8, 10 of n are

1-6046, 18794, 84-517, 250'92, 589'4.

* §§ H, 15.

t Compare the 2nd and 3rd columns of Tables I. and II., Part I.

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 161

Thus we find from the formulae (51), with \/ca = 2-4400,

m = 6-104, Ml =9753, We = - 23'25, MJ = - 144-97, M!O = - 426'9,
Nj = 1-180, NJ = 11-180, N> = 29-180, N} = 55'180, NJ = 89'18.

It will be convenient for us now to introduce the following abridged notation : —

— M*+I — X*+2 — ... ad inf.

E;

F'

_<_
Ml

- NJU - M'n_2 -

N- - M'n_, - N'n_2 -

(64),

the last two continued fractions terminating with the partial quotient which involves
a* in the numerator, and either M* or N^ in the denominator.
From these definitions of the quantities e, f, E, F, we have :

r —

cn —

; -f.,

"n— 1

>ji — r~"
" <"+1 <•

(G5);

"NT" -- F,»
•"« r>«-i

while the period- equation may be written in the forms :

M^ - F'_, -/.'+! = 0 "1
N?, - EJ_, - e'n+l = 0 I

(GG).

Suppose that we neglect /{ ; making use of the numerical values obtained above
for the quantities M, N, a, by successive applications of the formulae (65) we obtain

loge!0 = nO-7556, log/9] = 1-2229, log el = n07829,
log/1 = 0-9666, log 4 = 5iO-9649, log/1 = 0-5606.

In like manner, if we neglect e\0, we find

log/1 = 1-2498, logcj = wO'7801, log/1 = 0'96G8,
log el6 = nO'9649, log/1 = 0'5606.

Now the two values of /! obtained by these methods are respectively the 6th and
VOL. cxci. — A. y

162 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

5th convergents of the. continued fraction fb\ Since we find that to the degree of
accuracy retained these are equal, it follows that all subsequent convergents are
sensibly equal to either of them. Hence the infinite continued fraction /j1 may be
replaced by its fifth convergent without sensible error.
Similarly we find

log F} = 0-1834, log E.l = 0-5045, logFJ = 0'8266,
and therefore

Mi — FJ — /j1 = 9753 - 6707 - 3"636 = - 0'590.

As a second trial we take

\/<o - 2-4600.

Proceeding as before, we deduce

Mi - FJ -/5' = 10-234 - 6-619 - 3'607 = O'OOS.

We conclude that there is a root of the period-equation lying between 2*4400 and
2*4600 ; by interpolation its value is found to be

2-4597.

The same method may be used for the determination ot the roots of the second
class, the initial trial values being suggested by the formula (63). As a numerical
example, if we put n = 5, s = 1, hg!±ara- = -^ in (63), we find

2&>/X = 40-974, or X/co = 0'04881.

For a first trial we take 2w/X = 41, and deduce

N£ — EJ - ej = — 11 + 8*678 + 2'602 = 0'280.

As a second trial we take 2w/X = 41*280, and obtain

N! - Ei - el = - 11-280 + 8*657 -f2'593 = - O'OSO,

and therefore, by interpolation,

NJ - E> - el = 0,
when

2o>/X = 41-253, or X/co = 0*04848.

I have selected for special investigation the asymmetrical types when s = 1, and

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES.

103

the symmetrical types when s = 2, these types presenting special interest in relation
to the diurnal and semi-diurnal forced tides. The annexed tables give the values of
the computed roots, together with the corresponding periods of free oscillation for
the four depths of 7260, 14,520, 29,040, 58,080 feet treated of in Part I., corre-
sponding to the values -f$, -£-$, -fa, and £ for

o 0

I.— DEPTH 7260 feet (%/4w2«~ = -fa, />/o-0 — 018093).

s= 1.

>\

i

A

i

^ ""

1

Approximate
root.

Corrected
root.

Period.* Approximate
root.

Corrected
root.

Period.

[

o

n = 2<

1-8331

2-0042
- 1-8472

1-6337
- 0-9834
2-0685
- 1-8234

hr.«. mins.

14 41 1-4745
24 24 - 0 6582
11 36 2-0024
13 10 - 1-7006

1-3347
- 0-6221
1-9866
- 1-6595

hrs. mins.

17 59
38 34
12 5
14 28

M

O

» = 3

« = 5

0-6932
0-05702
0-03853

0-6401
0-06251
0-03784

days hrs.
1 13
16 0 0-1532
26 10 0-08091

i

0-1671
0-08178

days hrs

6 0
12 5

II.— DEPTH 14,520 feet (%/4<u3a2 = -fa, p/tr0 = 0-18093).

s = l.

s = 2.

*

•^

f

Approximate
root.

Corrected
root.

Period.

Approximate
root.

Corrected
root.

Period.

M

0!
1

5

71=2 j

.-«{

1-9588
- 1-3021
2-4327
- 2-2949

1-8677
- 1-2450
2-4597

- 2-2907

hrs. mins.
12 51
19 16
9 45
10 29

1-6804
- 0-9128
2-4363
- 2-1706

1-6133
- 0-8922
2-4349
- 2-1547

hrs. min?.

14 52
26 54
9 51
11 8

HH
1— 1

Efi

i

5

n= 1

n = 3
n = 5

0-7502
0-08423
0-04881

0-7283
0-08673
0-04848

days hrs.

1 9
11 13

20 15

0-2054
0-10067

0-2129
0-10081

days hrs.

4" 17
9 22

* Throughout these tables the periods are expressed 111 sidereal time.

Y 2

164

MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC
III.— DEPTH 29,040 feet (%/4a>2a2 = -^, p/a-0 = 0-18093).

s = 1.

A

s = 2.

r

"

r

~\

Approximate
root.

Corrected
root.

Period.

Approximate
root.

Corrected

root.

Period.

M

05

m

ci

3

n = 2 j

n = 4 <

2-2027
- 1-6439
3-1183

- 2-9958

2-1641
- 1 6170
3-1274
- 2-9961

hr.->. rains.
11 5
14 50

7 40
8 1

2-0241
- 1-2960
3-1295
- 2-8911

1-9968
- 1-2855
3-1293

- 2-8856

hrs. nuns.
12 I
18 40
7 40
8 19

Class II.

M= 1

n= 3

n = 5

.0-8213
0-1116
0-05636

0-8149
0-1124
0-05625

days hrs.
1 5
8 21
17 19

0-2523
0-11472

0-2554
0-11472

days hrs.

3* ' 22
8 17

IV.— DEPTH 58,080 feet (Jig/tora* = |, p/<r0 = Q'18093).

s= 1.

A

A

r

i

(

i

Approximate
root.

Corrected
root.

Period.

Approximate
root.

Corrected
root.

Period.

hH
GO

1

5

n - 2 |
» = 4 {

2-6431
- 2-1726
4-1626
- 4-0501

2-6288
- 2-1611
4-1659
- 4-0506

hrs. min?.
9 8
11 6
5 46
5 56

2-5636
- 1-8623
4-1820
- 3-9610

2-5535

- 1-8575
4-1818
- 3-9592

hrs. mins.
9 24
12 55
5 41
6 4

M

I— i

M
VI
CC

5

n= 1

M = 3

» = 5

0-8886
0-13354
0-06108

0-8873
0-13375
0-06105

days hrs.
1 3
7 11
16 9

0-2865
0-12332

0-2876
0-12332

days hrs.

3" 11
8 3

The approximate roots here given have been evaluated by the method of § 7,
except in the case of the roots of the second class for n = I, s = 1, where, instead
of replacing the continued fraction involved in the period-equation by its first con-
vergent, I have made use of the second convergent, so that the approximate form of
the period-equation from which this root is determined is

or

Mi

or, what is equivalent, Li — 0.

Ni=0'

fr = °.

ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 165

By a comparison of the approximate values given in these tables with the true
values we see that for extreme cases here tabulated, the error involved in the
approximation does not amount to more than about 3 per cent., even with a depth as
small as 7260 feet. We have here a justification of the statements made in the
last section as to the approximation to the higher roots.

§ 10. Oscillations of the First Class. Determination of the Type.

We shall describe as oscillations of the first class those whose periods remain finite
when the rotation-period is indefinitely prolonged, that is, those for which the roots
of the period-equation are of the first class. The types of motion whose periods
become infinitely long with the rotation-period will be called oscillations of the
second class.

The determination of the type involves the determination of the constants Cj,
Dj+i, Q+2, &c. (supposing for convenience that we are dealing with symmetrical
types). For this purpose we may either make use of the formula} of § 6, or we may
make use of the formulae of § 5 for the determination of the C's, after which the D's
must be computed from equation (49). The latter method will be closely analogous
to that used in § 9 of Part I., but the former is the more convenient when the
numerical determination of the constants is required.

If we make use of the notation (64), the following relations may be deduced from
the equations (52) : —

C^_ In + 1 _Ci_ 2»+_l_ ~]

D " » + *) (» - Ds D',+1 " " (n - s + 1) (n + 2)' L I

But we have seen in the last section how the quantities e,f, E, F may be deter-
mined numerically. Hence the above formulas allow us to compute the ratios of the
constants C, D. One of these constants must be regarded as arbitrary, and the
ratios of the others to it can then be computed. When the type under examination
is that which corresponds to a root of the period-equation approximating to a root of
M^ = 0, we select as the arbitrary constant of integration the quantity C*, as the
continued fractions c, f, E, F required to determine the ratios of the remaining
constants to this one will then be free from singularities.

When these ratios have been determined we may substitute their values in the
formulas

166 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

//I ,,2\TT !™f H» W*0*M#> I liJ I ^ "~8 . P' I p» I _>±i p» I

V I L — A* / *~> — 7 ^nt; J* 1 • * * ' T'» "~2 T^ " ' r<» "+2 • ' ' ' f

ft L i L« u« J

r i )* iy

J i _Ll'irJ. ps i «+' p« i

_ J _i_ ^'"-i ps I -^"+1 p«

L ^« «^»

f". . . + ~ -^ 9*-* P'-2 + 9n P?, + -^T- </,,+8 Prt+
L °n °«

and we shall obtain expressions for the height of the surface-waves and the velocity-
components.

I have not thought it worth while to compute any of these series in detail, as the
general character of them may be inferred from the series computed in Part I. for
the special case where s = 0. For large values of n, and even for comparatively
small values of n when ligj^ora^ is large, the quantities C will rapidly diminish as
we pass away in either direction from Qr Hence the most important term in the
series for £ will be that involving P', and this term will in general sufficiently
predominate to decide the number and approximate position of the nodal parallels of
latitude.

If we neglect C'_i in comparison with C*.+ J and suppose that w is small in com-
parison with \, the formula (49) gives

r (r + s+ 1) CJ+1 = (r + 1) (27- + 3) Dj

and therefore Dj will be of the same order of magnitude as C*+1. Hence, when r is
less than n, D* will be of the same order of magnitude of C'+), and similarly it may
be seen that when ?• is greater than n, D'. will be of the same order of magnitude as
Q._I. Thus the predominant terms in the expressions for the velocity-components
will be those involving C*, D'(_,, D'l+1.

The exponential factors indicate that the type of motion involved will consist of
waves propagated round the sphere with uniform angular velocity X/s about the
polar axis, and that there will be s crests or troughs on each parallel. Positive
values of A. will correspond with waves propagated in the opposite direction to the
rotation, that is westwards, while negative values will correspond with easterly
waves. The paths of the fluid particles will be ellipses with their axes directed
along the meridians and parallels.

§ 11. Oscillations of the Second Class.
In dealing with the oscillations of the second class we proceed in the same manner

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 167

as before, retaining D^ as the arbitrary constant of integration when the type
under consideration is that whose period is approximately given bv the formula

n(n + 1) — 2o)S/\ = 0.

For we may anticipate that this quantity will predominate over the others, at least
when the depth is sufficiently large or the angular velocity of rotation sufficiently
small. The ratios of the remaining constants to D' may then be computed from
the formulae (67), and on substituting these ratios in the equations

,,., ,-, -- , ~ ,^ . . .

' » ' '» "x ' '» j

iv i n

JJ«+2 p> |

~Dr

[. . . + ^.P;., + ^^+IP.« + . .] .

we obtain expressions for the height of the surface-waves and the velocity-com-
ponents. The values of \ for the oscillations of this class being always positive, the
direction of the wave-propagation will always be westwards.

By way of numerical illustration 1 have computed the series for £ corresponding to
the case n = 3, s = 1, and to the values ^g-, -£$, -f$, •§• for hg/4aPa*. In these four
cases the series within the square brackets are found to be

£ - 0-1880P] + 0'3753P^ - OD916PJ + 0'01153P!0

— 0-00093P!2+ 0-000052PJ4 — 0'000002P10 + . . .

- 1-4735PJ — 0-3260P1 + 0'11690PJ — 0'01309PJ + O'OOOSOP},,

— 0-000032P1, + 0-000001 P{4 — ...

- 07296PJ - 0-2248P1 + 0-03159PJ - 0-00168PJ + 0'00005lP!0

-0'000001P!2 + . ..

- 0-3617P.1 - 0-1277P1 + 0-00814PJ - 0'0002lPJ + -000003P!0 - ...

It will be seen that as the depth increases or 01 diminishes the coefficients all
become smaller, while the convergence of the series improves. It may be inferred as
in the last section that Cj/Dj + 1 will tend towards the limit zero when r is less than

168 MR. S. S. HOUGH ON THE APPLICATION OP HARMONIC

n, while Cj+J/Dj will tend towards a finite limit. In like manner when r is greater
than n, C*/Dj_! will tend towards zero while Cv_,/D, will tend to a finite limit.

Let us examine the limiting forms assumed by £, U, V, when the rotation is
annulled. On putting n (n + 1) — 2ws/X = 0 the relations (51) give

U N« _ r (r 4. i) _ w (w _j_ j) = (r _ w) (r _|_ n _j_ !).

Thus from (52) we obtain, if r < n,

T t C' 4^ _Jr + «jtl2_ T/C'r + 1 (r - n) (r + n + 1) (2r + 3)

J «3 D'r + 1 " " ^ (r + I)2 (2r + 3) ' D'r r* (r + s + 1)

and if?' > ??,

T,_eL_ 4«2 (r - a) Ct_, (r-M)(r + » + l)(2r-l)

J to2 IV _, " ^ 7-- (2r - 1) ' I)'r (r + ]>s (r - «)

Hence if we retain only the most significant terms and put <aD'n = A*, we find

' +-^-- («-« + !)_ p. 1

h %.+, (» + I)2 (2» + 1) K+1 J

/(I _ U2\ ]J _ __ - A.g.XW + ^) p.

^ ^ ^ ' n(n+ l)/t "

+

P;

" (n "*" •*•; At»iM+its>\~*L n + s -Ds 4ffl- (n —

2a

p. , i'L3 (n - « + 1)
'"' h (n + i)'2 (2n + !) J'

If therefore we suppose that co reduces to zero while A' remains finite, £ will
reduce to zero, but the velocity-components will tend to finite limits given by

2a

in virtue of (7).

Hence the steady motions to which the oscillations of the second class reduce
when the rotation is annulled involve no deformation of the free surface. This of
course may readily be verified by a direct method. For if u, v, w denote the
velocity-components referred to fixed rectangular axes, it may be seen that all the
conditions of the problem will be satisfied by

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 169

3v 3y 3y 3y 3y 3y

II —"• tl — ^_ V XV 41 — ft ^V ^^ ~ ^V 3|* » *y» __<»i __ *t r~

(i — « ~ * -\ > V — 2 v * T~ i '" — •£ ^ '/ -\ !

y cz dy dx fa ' 3y y ox '

where x is an arbitrary function of a;, y, z independent of t. These solutions make
ux + vy + wz — 0, and therefore involve no deformation of the surface. If we refer
them to polar co-ordinates, they are equivalent to

siu 6
or

u = ^

1 tf ti<t>

2~

— fj?) V = — (1 —
This solution becomes identical with that found above if

X = - ,;

For the special case s = 0, the values of \ corresponding with the roots of the
second class will be zero, and the corresponding types of motion steady, even when «
is finite. The types of steady motion will however involve a deformation of the
free surface in all cases where the angular velocit}7 of rotation is different from zero.
These cases have been fully discussed in Part I.. § 14.

§12. Forced Oscillation*. General Analytical Solution.

The problem of the. force'd oscillations involves the determination of the quantities
C^ in terms of y". from the equation (31) or (40). Dealing first with the case of
uniform depth, and supposing that the disturbing potential involves only a single
term y*,P^ (p.) ei(M+i*\ we have to solve the simultaneous equations

= o,

with the condition that L/ C* = 0.

r=*>

From these we deduce, as in § 5, that

C> «

r-2

VOL. exci.

170 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

Thus the equation which involves y'n becomes

whence

01 =

__. - 14 + K'n+2
We may now deduce C^_2, C*,_4, . . . C',+2) C'[+4, ... by means "of the formulae

C" TT» P«

^n-2 •"•«-! ^a-t

— -- - i

'

o

and therefore

_, _2n_.

- 4«V H'a_a - L; + K;+2 [; ' ;4-2*?,-«

TJ« K"« tr« TT-. -i

^ ^ JVIV-« p, .

- ^>.+4 4- • • •
J

It should 'be noticed that the term involving P^ need not here be the predomi-
nating term of the series within the square brackets ; it will however be so when
the value of X for the disturbing force is in the neighbourhood of those roots of
the period-equation which approximate to roots of Lj, = 0. But if X have as its
value another root of the period-equation, say, for example, one which approxi-
mates to a root of L?,, = 0, the series within square brackets will differ only by a
constant multiplier from the series in the expression for the tide-height for the
corresponding- type of free oscillation, since the equations which determine the ratios
of the C's are evidently the same in both cases. Hence, for values of X in the neigh-
bourhood of this one, the predominating term will be that which involves Pf^.
Consequently, when n and m differ widely, the numerical computation of these
series will become laborious ; for as we proceed away from the term containing
'P» towards that involving P;;,, the terms will at first increase in magnitude, and
the convergence of the series will not assert itself until the term depending on
P™ has been passed. These circumstances will not however occur in any of the
cases of more practical interest.

Whatever be the nature of the disturbing potential it will be possible to expand
its surface-value in a series of surface-harmonics. Thus the most general value
of v which can occur may be expressed in the form

»=o •„=«

The deformation of the surface due to each term may be calculated independently

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 171

and the results superposed, so that the deformation at time t resulting from this dis-
turbing force will be given by

* 8=oo ,>=»r , ia<+*^> , *, -;(*«+»$) r -a* v* -in

r — ? ? v y" h °»e i n"-s ps i p« i ft-»+2 P* i

4 = 2~17» ** * * TT« T, , ir. V • • + ~. — fn-l T r,, + •— - r,, ,., + ... S .

4<»zffl- 8 = 0 » = . L "i-I ~ L» + K»+2 I »»-» 0« J J

The corresponding formulae when the depth is variable may be obtained by replacing
h, yl, 8;, H?,, K;, 14, a£, 2/L by ir, G'rt, AJ, j&, it;, £'„ £, tf, respectively, where
*r, G'n, &'n, f', >;' are defined by the equations (35), (41), (42), and

£»„» fc» „»

J^., Cn7/!! g»-2Vn-'J

<*?" TTf» 'W»

»„ JLfl-2 ' • •

m. _ . £-^"-2 .l^L
n -»« . Tirt

*n *„+:> • • •

while A^ is obtained by writing S"n in place of y* in the right-hand member of (42).

§ 13. Classification of Tide*.

In the last section we have reduced the problem of the evaluation of the forced
tides due to any disturbing force to that of the development of the disturbing poten-
tial as a series of surface-harmonics. This development for the case of the disturbance
of the ocean due to the attraction of the sun and moon has been already dealt with,
and reference may be made to Professor DARWIN'S article in the ' Encyclopaedia
Britannica' for a full account of it. We give here a short summary of the principal
results of which we propose to make use.

The principal part of the disturbing potential will consist of spherical harmonic
functions of the second order, and when expressed by means of zonal and tesseral
harmonics the terms which occur will be of three tvpes characterized bv the rank of

ti i. •>

the harmonic involved.

For the first type s = 0, and the corresponding tides will be expressible as series of
zonal harmonics. For these types the value of X will be small in comparison with
that of CD, so that the period of the disturbance is long compared with a sidereal day.
The terms will cease to be oscillatory when the orbital motion of the disturbing body
is neglected. The tides generated by these parts of the disturbing potential have
been already dealt with in Part I.

The terms of the second type are those for which s = 1 ; in certain of these terms
the value of X will be equal to w, so that the period is rigorously equal to a sidereal
day, while in the rest the period will reduce to a sidereal day when the orbital motion
of the disturbing body is neglected. If n denote the mean orbital motion of the
luminary, the " speeds" of the principal diurnal tides will be <u and a> — 2n. We
propose to neglect the sun's orbital motion, so that for each of the principal solar

z 2

172 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

diurnal constituents we shall suppose that X = ta rigorously. The same analysis will
then apply to one of the lunar diurnal constituents, while in order to illustrate the
effect of the departure of the period from exact coincidence with a sidereal day we
shall evaluate independently the lunar diurnal constituent for which

X(= (a — 2n) = 0-92700w.

The principal part of the tidal oscillations will be due to the third part of the
disturbing potential, which involves harmonics of rank 2. The period for the tides
due to these terms will differ but slightly from half a sidereal day, and will reduce to
half a day exactly when the orbital motion of the disturbing body is neglected. We
shall therefore assume that X = 2<u rigorously for the solar semi-diurnal tides, while
we shall take the value

X/2(« = 1 — w/« = 0-96350

as typical of the lunar semi-diurnal constituents. The analysis applied to the solar
constituents will be rigorously applicable to the sidereal luni-solar semi-diurnal tide
usually denoted by the symbol rL.

$ 14. Special Cases.

Instead of making use of the equation (31) as we have done in § 12, we may of
course compute the forced tides by means of the equations (49), (50) of § 6,
determining incidentally the constants D'. Thus, if we suppose that all the y's
are zero except y"+1, we have the following equations for the determination of Dj,

C]' TV .trp

**-/c 4. 1 • JL-* v 4. *>» LVU*

I 2s + 5 v

"-A ps |<r» TV I v9 + 2)2 (2s + 3) f^t

o ^.«+i — i> 8+2^8+2 T 9S + 7 '+3 —

where the terms on the right are all zero, except in the second equation. Now it is
evident that if N* = 0, or

2<M

^— r

all these equations will be satisfied if

and

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 173

It follows that if the disturbing potential be of order s + 1 and rank s, and the
period be ^ (* + 1) days, the tide will involve no rise and fall at the free surface, but
will consist merely of horizontal currents. If we put s = I the requisite period will
be rigorously equal to a sidereal day, and the circumstances will correspond with those
we have assumed to characterize the solar diurnal tides. We therefore conclude
that in an ocean of uniform depth the solar diurnal tides will involve no rise and fall.
We shall however see hereafter that for certain of the lunar diurnal tides the
difference between the period and a sidereal day may be sufficiently great to render
the rise and fall of considerable importance, unless the depth is very small.

We have seen in § 4 that the formulae applicable to the case of variable depth may

be deduced from those applicable to the case of uniform depth by replacing h by K

i P •> -~i

and C; by Q, + -—^ \n (n + 1) + - r*,.

-id) "fi" I A,

L. —1

But if y* = 0, r", = — g,,C*,, and therefore when X — 2w/(.s -f- 1) and all the y'.s
are zero except yl+i, Cj+3, Cj+6, &c., will all be zero, while

whence we obtain

c:+I=- .2^.---^-..

1 -2(s + 1)- %\,
Thus the forced tide will be similar in type to the disturbing potential which

produces it, though it will be inverted unless I < 0 or > .-

(s + l)-«/,+1

This theorem will admit of application to the solar diurnal tides on putting s = 1 ,
in which case it reduces to a theorem given by LAPLACE.^ The critical value of /,
for a system comparable with the earth, is considerably greater than such depths as
occur on the earth ; hence, for depths comparable with that of the ocean, the diurnal
tides will be inverted when the ocean is deeper at the equator than at the poles :
they will however be direct when I is negative, so that the ocean is deeper at the
poles than at the equator.

It is evident that the equations typified by (31) will all be satisfied with the Cs
all zero if h = 0, since in this case the right-hand members will reduce to zero. In
like manner the corresponding equations which apply to an ocean of variable depth
will all be satisfied when K = 0, if for all values of n

* LAIIB, 'Hydrodynamics,' § 212.

174 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

This equation leads to

/ ^ J.2flttl
(" + 1> + TJ

2ws'

There exists then a certain law of depth, depending on the period, for which the
tide will always be similar in type to the disturbing potential which produces it.
This law of depth is expressed by the formula

If we supposes that X = 2w rigorously, it reduces to

so that the depth will be a maximum at the equator, and will gradually decrease on
passing away from the equator to zero at the poles.*

For other values of X the formula for h will make the depth negative at some parts
of the surface unless / is positive and X > 2&>. The latter condition does not occur
with any of the leading tidal constituents^ but it would hold good in the case of the
semi-diurnal tides due to a satellite whose motion in its orbit was retrograde. If
however we neglect the mutual attraction of the waters, the theorem under dis-
cussion may be supposed to apply to an ocean covering that part of the surface over
which h is positive, the remaining parts of the surface being supposed to consist of
continents. When / is positive, these continents must, for the lunar semi-diurnal
tides, reduce to small circurnpolar islands, while for the same tides when I is nega-
tive they will cover the whole globe with the exception of two small seas surrounding
the poles.

For the diurnal-tides, the shores must coincide with parallels of latitude approxi-
mately 30° north and south of the equator, while for the tides of long period the
appropriate forms of sea will be bounded by two parallels nearly coincident with the
equator. A change in the sign of I in all cases involves an interchange between the
seas and the land.

It should be noticed that the formulae (12) make U, V infinite at the points where

p- — rfc £-, that is, at the shores. This indicates that the neglect of the squares of

the velocities is not allowable in the neighbourhood of the coasts no matter how small
the amplitude of vibration may be, and seems to point to the existence of "breakers"
as an essential accompaniment of the tides.

* Gf. LIMB, ' Hydrodynamics,' § 213.

t There will be small tidal constituents depending on higher powers of the moon's parallax for
which X exceeds 2w. Gf. DARWIN, " Harmonic Analysis of Tidal Observations'." ' Brit. Assoc. Report,'
1883 (Sonthport), § 3.

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 175

§ 15. Solar Semi-diurnal Tides.

In the last section we have considered some special cases in which the tide-height
is expressible by a single term. In general it will however only be expressible by a
series of terms. It may be shown, as in § 5 of Part I., that this series will be finite
when the law of depth is such that

where n is an integer, n — s being even or odd according as we are dealing with the
symmetrical or asymmetrical types of rank s. The values of I determined from this
equation will in general involve X, but for large values of n, they will approximate to
the same values of I as those required for the expression in finite terms of the lon<r-
period tides, since for such values 2«s/X will be small compared with n (n +1).

In other cases the expression for the tide-height will involve infinite series. We
deal in the present section with the case where I is zero, so that the depth is uniform.

The numerical computation of the semi-diurnal tides admits of special simplicity
when X = 2w exactly. Putting X = 2w, s = 2, in the formula? (30), (29) we obtain

i2 n — 1 ., ii + 4

~ (n + 3) (2n + 1) (2n + 3) ' = n (2n + 3) (2w + 5) '

. 8 (n - 1) (n + 2) (« - l)^ (n + 2) (n + 2)2 (ft - 1)

A,, = -

ft3 (n + I)2 n- (n + 1) (2n - 1) ('In + 1) n (n + I)2 (In + 1) ('2n + 3) '

the last of which gives on reduction

A2, =

,2 2 Qt- !)(•* + 2)

+ 3) '

This general formula for A'f, fails to hold when n = 2 ; for, in this case, n — s and
n(n — 1) — 2cos/X are both zero, and therefore the second fraction involved in the
expression for A'ft is indeterminate. To determine its limiting form we must first
suppose the period slightly different from half a day, so that X is not rigorously equal

to 2ca ; the formula (29) then gives

«

*•> _ 2 • 8 ~

=

22.33 3s. 5.7(3.4 -

which, on putting X = 2o>, reduces to

X2 _i_ !-43 JL

" 23.32 " 33.7.10 5.7 '

The formulae for xl, yl, A* are now in a convenient form for logarithmic computa-
tion, and we may readily deduce the following numerical values

* C/. LAPLACE, 'M^c. Gel.,' Part 1., Book IV., § 7.

176

MB. S. S. HOUGH ON THE APPLICATION OF HARMONIC

A?, = 0'004494
Af2 = 0'003179
Ah = 0-002367

A3 = 0-085714
A^ = 0-0233766
AJ = 0-011544

'i — 0-006823

log x\ = 3-75696
3-63639
3-45469
3-29451

log xl
log xl
log a-l

log y\
log?/;
log iji
\ogyl

2-67778
2-14569
3-81531
3-57512

Af6 = 0-001830

log x\0 = 3-1564
log y?n = 3'0360
log x\t = 4'9297

log y|0 = 3'3865
logy?2 = 3'2312
log yf4 = 3'0993

Taking hg/4(a?as = -/j-, and /5/cr0 = 0-18093, the formula (32) leads to

L'j

Ljj

Lj

+ 0-063428

- 0-0001156

- 0-012412
— 0-017379

L'r0 = — 0'019860
L?, = — 0-02128
Lf4 = — 0-02216
L?c = - 0-0228

Thus if we neglect K?8, and make use of the formula

K, ''^-ay'i-i

" — T. _ K- '

ljn -"n+i

we obtain in succession

log K?6 = n 5-671
log K^4 = n 5-922G
log K?2 = w 4-2165
logK?0 = n 4 -57 5 3

and therefore, since C"+2/Cf, = K*l+2/y',, we find

log(CJ/q) = 0-03810
log (CI/CJ) = n T-58266
= » i'22417
yCi) = /i 1-0002

log K| = n 3'03947
log Kl = w 372835

j= 2-71588

log (CyCf0)
log (C?4/Cf,)
log (C?6/C?4)

n 2-8300
n 2-6914
w 2-572.

ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 177

Butif CftP^OOe*"** denote the height of the 'equilibrium' tide resulting from
a disturbing potential ylP*,, (/u.)e;(A1+8*), we have

Therefore

Li - KJ " Li - K* l

and on introducing the numerical values for h, y2, 14 K£ we obtain

q= - 1-9476 @f.
Thus we have :

log (C;/©;) = n 0-32760 log (CV©5) = »1'9646

log(C§/€i) = T'91025 log(C?4/®i)= 5-6560

log (d/<B%) = u T-13442 log (Cie/CEl) = )'- 6-228 ;
log (C?0/@?) = 2-1346

and therefore if we suppose the exponential or trigonometrical factor to be involved
in (JCH, so that the height of the equilibrium tide is expressed by CToP? (^), the height
of the corresponding dynamical tide is given by

£ = ©5 [- 1-9476 P; - 2'12617 Pj + 0'81331 PS — 013628 Pj

+ 0-01363 P?0 — 0-000922 Pf, + 0'000045 Pf, - 0'000002 PI,; + ...]•

For points lying on the equator we have p = 0, and it may be shown that in this
case

ps .-/ v.,|3-5---(^+l)

~(~> 2.4... (2ft -2)'
whence we deduce

log P| = 0-47712 log PIO = 1 -4325

log PI = n 0-87500 log P;3 = n 1'5464

log P^ = 1-11810 log P?4 = 1-643

log PI = n 1-2941'J log P?6 = n 1-73.

Thus the values of the successive terms of the series within the square brackets
for points at the equator are

- 5-8428 + 15-9463 + 10'6746 + 2'6829 + 0'3690 + 0'0324 + 0'0020 + O'OOOl,

which, on addition, give 23'8645. But the height of the corresponding equilibrium-
VOL. cxci. — A. 2 A

178 MR. S. S. HOUGH ON THE APPLICATION OP HARMONIC

tide at the equator is 3(8%, and therefore the ratio of the height of the tide to that
of the corresponding equilibrium tide at the equator is

+ 7-9548.

In like manner the tide-height in any other latitude may be compared with the
equilibrium tide-height, but the process will be laborious in the absence of tables of
the functions P*.*

The above example has been treated in some detail as illustrative of the method to
be employed for the computation of the forced tides by infinite series ; the chief part
of the labour is involved in the determination of the quantities afM i/',, A',, but when
once these have been determined, since they do not involve the depth, it is easy
without much additional labour to multiply cases for different depths. Besides the
case already considered, which corresponds to a depth of about 7260 feet, I have
computed the series for depths of 14,520, 29,040, and 58,080 feet, corresponding
with the values -^j, -(•$, and ^ for hg/±ara~. For these depths the series within the
square brackets is replaced by

- 0-83227P; + 0-21G94P; - 0'02G15P; + O'OOlSOPj; - 0'000080P?0

+ 0 -000003 P?2 + ...,

- 191'925Pj + 15-69GP; - 0'8082Pj + 0'0256P; - 0'0005Pr0 + . . .
and

1-9610P5 - Q'06823 P; + 0'001G4P; - 0'000025P^ + . . .

respectively, giving for the ratio of the tide-height to the equilibrium tide-height at

the equator the values

- 1-5016, — 234-87, + 2-1389.

When the depth of the ocean is greater than 58,080 feet the tides are therefore
direct at the equator. They gradually increase in magnitude as the depth decreases,
and become infinite and change sign for some critical value of the depth rather in
excess of 29,040 feet after which, for further decrease of the depth, they remain inverted
until a second critical value is reached which is somewhat greater than 7260 feet,
when a second change of sign occurs. The very large coefficients which appear when
hg/ia)"a~ = YO indicate that for this depth there is a period of free oscillation of semi-
diurnal type whose period differs but slightly from half-a-day. On reference to the
tables of § 9 it will be seen that we have, in fact, evaluated this period as 12 hours
1 minute, while for the case %/4ar«3 = ^ we have found a period of 12 hours
5 minutes. We see then that though, when the period of the forced oscillation differs
from that of one of the types of free oscillation by as little as one minute, the forced
tide may be nearly 250 times as great as the corresponding equilibrium tide, a

* The zonal harmonics from P, up to P7 have been tabulated by GLAISHEU, 'Brit. Assoc. Reports,'
1879, but I do not know of the existence of any Tables of the Tesseral Harmonics, with the exception
of a few given by THOMSOX and TAIT, 'Nat. Phil.,' vol. 2, § 784.

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 179

difference of 5 minutes between these periods will be sufficient to reduce the tide to
less than ten times the corresponding equilibrium tide. It seems then that the tides
will not tend to become excessively large unless there is very close agreement with
the period of one of the free oscillations.

The critical depths for which the forced tides here treated of become infinite are
those for which a period of free oscillation coincides exactly with 12 hours. They
may be ascertained by putting A. = 2co in the period-equation for the free oscillations
and treating this equation as an equation for the determination of h. The roots may
be found by trial and error as in § 9, the approximate values with which to commence
the trials being suggested by the discussion already given. The two largest roots are
found to be given by

a3 = 0-10049, %/4w2a2 = 0'02545,

and the corresponding critical depths are about 29,182 feet and 7375 feet.

We have hitherto supposed that p/ar0 = 048093, but for purposes of comparison I
have also examined the case where /3/cr0 = 0, that is where the mutual attraction of
the waters is neglected. The series for £ in this case become

£ = Cr; [1-0927 Pj + l'91817P'i - 0-66909 PS + 0'1070lP; - 0'0103GP;0

+ 0-OOOG83Pi, - 0-000033Pf, + O'OOOOOI PI, - . . .]

£ = <£; [- 1-0733P:] + 0'24502P'i - 0'02790P; + 0'00185P; - O'OOOOSOPjo

+ 0-000002 Pj, - . . •]

£ = 1&-, [9'34370P^ — 0-70311 Pj + 0'03449P^ - O'OOlOePjj + 0-000022P'f0 - . . .]
£ = ®? [17739P; - 0-05750P; + 0'00132Ps - 0'00020Pi + . . .]

or the depths 7260, 14,520, 29,040, 58,080 feet respectively. From these series we
deduce as the ratio of the tide to the equilibrium tide at the equator the four values

' - 7-4343, - 1-8208, + H'2595, + 1-9236,

results which agree, except in the third case, with the numbers given by Professor
LAMB* deduced from the numerical formulae of LAPLACE.

It will be seen that in three cases out of the four here considered the effect of the
mutual gravitation of the waters is to increase the ratio of the tide to the equilibrium
tide. In two of the cases the sign is also reversed. This of course results from
the fact that, whereas when p/tr0 = 0-18093 one of the periods of free oscillation is

* By a carefnl re-computation of the semi-diurnal tide for the case /3=10 (notation of Professor LAMB)
I find the following series more accurate than that given for f/H"1 : —

S~ + 6-1915 -4 + 3-2447 ••« + 0-7234 ,<8 + 0-0919 1-10 + 0-0076 J* + 0-0004 *" + . . .
This series reduces to H'2595 when v = 1, thus agreeing with the result obtained above,

2 A 2

180 MR. R. S. HOUGH OX THE APPLICATION- OF HARMONIC

rather greater than 12 hours, when p/<r0 = 0 the corresponding period will be less
than 12 hours

§ 16. Lunar Semi-diurnal Tides.

A similar method to that of the last section may be used to evaluate the lunar
semi-diurnal tides for which we take X/2w = 0"96350. The arithmetical work is,
however, more severe, in consequence of the fact that the quantities a?*, y",, A* must
be evaluated from the formula? (29), (30) which do not assume the simple forms
obtained in the last section. Substituting in these formulae the value of X/2w
quoted above we deduce

Aj = 0-075603 AJO = 0-00,1845

A; = 0-019864 Af, = 0'002750

A|j — 0-009856 A?4 = 0'002028

A^ = 0-005834 Aie = 0-001968 ;

log x-. = 376026 log xic = 3'1566

log .rj = 3~-63756 log x\.2 = 3'0362

log xl = 3-45530 log x^ = 4'9299 ;
loo- yfi = 3-29488

log yr, = 2-68108 log 7/f0 = 3'3867

log y\ = 2-14687 log y\» = 3'2313

\og-yl = 3-81592 log ?/j, = 3"'0994.
log y- — 3-57549

Our procedure is now exactly similar to that of the last section. Thus, if the
height of the equilibrium tide be

W V (/*),

we find, when %/4ws«2 = ^ and p/<r0 = 0'18093,

{ = 03 [0-1039S Pi + 0-57998 Pj - 0'19273 P^ + 0'03054 Pi

- 0-002960 P?o + 0-000196 Pf, - O'OOOOIO P?4 + • • •]•

Similarly, when hg/^a* = -£$,

£=0S[- 1 -0647 P: + 0-24038 Pj- 0'02774 P^ + 0'001867 Pi

- 0-000082' Pio 4- 0-000003 Pf, -...];

when h ft/la)" a~ = -f^,

C = 05 [9-1 1 81 P^ - 0-71533 P; + 0-03621 P^

- 0-001136 PI -f 0-000024 Pjo —...];

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 181

and when A<7/4cu3a5 = j,

£ = ®| [1-7646 P; - 0-06057 P°i + 0'001447 PI - 0'000022 Pi + . . .].

From these series we find for the ratio of the tide-heights to the equilibrium tide-
heights at the equator the four values

— 2-4187, —1-8000, +11-0725, + 1'9225.

On comparison of these numbers with those obtained for the solar tides in the
preceding section, we see that for a depth of 7260 feet the solar tides will be direct
while the lunar tides will be inverted, the opposite being the case when the depth is
29,040 feet. This is, of course, due to the fact that in each of these cases there is a
period of free oscillation intermediate between twelve solar (or, more strictly, sidereal)
hours and twelve lunar hours. The critical depths for which the lunar tides become
infinite are found to be 26,044 feet and 6448 feet.

Consequently this phenomenon will occur if the depth of the ocean be between
29,182 feet and 26,044 feet, or between 7375 feet and 6448 feet. An important
consequence would be that for depths lying between these limits the usual pheno-
mena of spring and neap tides would be reversed, the higher tides occurring when
the moon is in quadrature, and the lower at new and full moon. *

There appears then to be a considerable range of depth comparable with the mean
depth of the ocean over which the reversal of the spring and neap tide phenomena
would take place, but in that the actual tides are highest in the neighbourhood of
new and full moon we conclude that the effective depth of the ocean does not lie
within this range, and that none of the periods of free oscillation of the actual ocean
lie between twelve solar hours and twelve lunar hours. The true effective depth is
almost certainly less than 26,044 feet, and therefore both solar and lunar tides will
be in the main inverted, though the configuration of the land and of the ocean bed
will probably give rise to considerable variations of phase in different places.

The shortest period of free oscillation of the second class for the case s = 2 approxi-
mates to, but is in excess of, three days. But if n denotes the moon's mean orbital
motion, the speed of the lunar semi-diurnal tide is

2 (w — n).
If we equate this to ^ <w, we obtain

•n = | to.

Hence, if the moon's orbital motion were accelerated, or the earth's rotation
retarded, until the month and day were in a ratio less than 6 : 5, it would be possible
for the period of the lunar semi-diurnal tide to confound itself with one of the periods
of the oscillations of the second class, and the tides would then tend to become very
large.

* C,f. KELVIN, ' Popular Lectures.' vol. 2, p. 22 (footnote).

182 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC

§ 17. Solar Semi-diurnal Tides in an Ocean of Variable Depth.

To evaluate the tides when the depth is a function of the latitude we must make
use of the formulae of § 4. The method will be sufficiently illustrated by the
computation of the solar semi-diurnal tide for the case where

8a8 = -2Jg- + & sin8*,
or, where

A = (14,520 + 9680 sin20) feet,

6 denoting the co-latitude.

Putting Ig/4:co2a2 = •£$, and making use of the numerical values found in § 15 for

*'?!> 2/L A«> we obtain from (41)

log £ = 3-63907 log ft = ft 3-06261

log f; = 3-12901 log f| = n 3'4369

log rj'r, = 2-17040 log 17; = n 3'957G7

loo- rj\ — n 375301 log yl = n 3'9962

while, when - - = -A-, we obtain
4o)-«~

%., = + 0-0207G5 IU = - 0-05787

Hj = - 0-039717 1-0 = - 0-0606

ILii = - 0-052592.

From these we deduce in succession, on neglecting Hjo,

log Itf0 - n 4-651 log Hi = 4-1 632

log ms- = n 4-26 1 2 log W, = 3'2089 1

and

log H£ = 3-49214.

But the first two of equations (40) give

- KQ + ,10; = [^ + 8 JMJ ,5 = [Jt + 8 ^ A;;] «. ,

which, on solution, yield

- ^« --

4«V - mi-It! " '4o,V

j^_ ., ] m| @i

4«««» A2 J ^ %| - »J

'

ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 183

On substituting the numerical values for the quantities on the right, we obtain

C; = - 2-9242 ©3, C'i = 0-28546 ©*.

The remaining constants may now be computed from the formulae Cg/C; = ftil/Tji,
g/Cg = Hg/Tjg &c., and we finally obtain

£= <££[— 2-9242P; + 0-28546P; - 0'00733P^ - 0'000147Pi - 0'000007P?0 -....].

This makes the ratio of the height of the tide to that of the equilibrium tide at
the equator

— 3-6690.

The tide will evidently be in the main inverted, the longest period of free oscilla-
tion of the first class being in excess of twelve hours.

As a further example, I have computed the series for £ when the depth is given
by the formula

-!%. -I. -i-«m-/9
4o>V~" 10

that is, when the depth is 29,040 feet at the poles and shallows to 19,360 feet at the
equator. This series is found to be

£ = ©*[— 2'366lP:i + 0-35649P; — 0'03953P^ + 0'0037GPs

- 0-000333Pi0 + 0-000029PJ, - 0'000002Pi4 + ....],

making the ratio of the tide to the equilibrium tide at the equator

- 3'4583.

4

If we put -—£-3 = 3i>) and replace X by 2<a in the period-equation, regarding this

*x(O Cv

us an equation for K, the largest root is found to be

K = 21,765.

Thus there will be a period of free oscillation coinciding exactly with twelve hours
when

h = (21,765 + 9680 shr<?) feet ;

this formula makes the polar depth 21,765 feet, the equatorial depth 31,445 feet, and
the mean depth 28,222 feet.

In like manner, when ;rf 3 = — -fo, there will also be a period of free oscillation

184 MR. S. S. HOUGH ON THE APPLICATION Of HARMONIC

agreeing exactly with twelve hours, when the polar depth is 36,970 feet, the
equatorial depth 26,290 feet, and the mean depth 29,517 feet.

§ 18. Diurnal Tides.

It has been shown in § 14 that the diurnal tidal constituents whose periods are
equal to a sidereal day will involve no rise and fall at the free surface when the
depth of the ocean is uniform. This theorem will be rigorously applicable to the
luni-solar diurnal constituent usually designated by the initial Kl9 while it may also
be supposed to apply with a fair degree of accuracy to each of the solar diurnal
constituents since the motion of the sun in his orbit is sufficiently slow. There will
however be an important lunar diurnal constituent for which the speed is 0'92700w,
in dealing with which we propose to take into account the difference between the
period and a sidereal day. The method of computation is exactly similar to that
used for the lunar semi-diurnal tides, and thus we find when hg/lurtC1 = -f0,

£ = ©i[— 0-07G38PI + 0-03543P1 - 0'00845Pi + 0'001207PJ

- 0-000114P!0 + 0'000008PJ3- . ..];

when /i<7/4ora3 = -2J0-,

£ = @.![- O'lG'JlP.l + 0-04738PJ - 0'0062SPi + 0'000480PJ

— 0-000024PJO + O'OOOOOl Pj, — . . .] ;
when hy/ihr'a" = -^,

£ = Cf.i[- 0-4 145 PI + 0-06576PJ — 0-00464PJ + 0'000184PJ - O'OOOOOSPJo + ...];

and when hy/4ara? = 5,

£ = <£![- 1-4428 PI -1- 0-1231PJ - 0'00449PJ + 0 00009 PJ - O'OOOOOlPJo +...].

It appears, then, that these tides will increase with "the depth, and that they will
be in the main inverted. For small depths the rise and fall will be small, but with
a depth as great as 58,080 feet the tide will be in excess of the equilibrium-tide.
The type will tend to approximate more and more closely to that represented by
a second order harmonic alone as the depth increases.

So long as the depth is uniform the tidal constituents whose periods are rigorously
equal to a sidereal day will never tend to become infinite, and consequently no
period of free oscillation of the type of the diurnal tides can coincide exactly with
one day. As w diminishes however the shortest period of the second class, which for
the depths under consideration is longer than the period of the lunar diurnal

ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 185

constituents, approximates to one day, and attains this as a limiting value when
a = 0. Hence, as « diminishes, or h increases, the largest root of the second class
must pass through the value 0 '92700, thus rendering one of the lunar diurnal
constituents infinite. This accounts for the rapid increase in the coefficients in the
series given above for these tides as h increases.

The roots of the first class must however all be greater than unity, no matter
how great the depth may be. Since they all decrease with the depth, they must
approach finite limiting values greater than unity as the depth diminishes to zero.

The tides of rigorously diurnal period will become infinite when the depth is
variable if / assumes the value

j tO'll'

so that with this value of I we may anticipate that there will be a period of
free oscillation exactly a sidereal day in duration. Now the above value of / will
require that the surface of the solid earth should be rigorously spherical in order
that the free surface of the ocean may be an equipotential surface under gravity and
centrifugal force. It is easy to see why in this case a free oscillation of rigorously
diurnal period must exist. For if the water be set in rotation as a solid body about
an axis not rigorously coincident with the rotation-axis of the solid earth, and the
form of the free surface be adjusted for equilibrium under centrifugal force about the
new axis of rotation, there will be no forces acting which tend to modify this state
of motion, and it will continue permanently, provided the system be free from
friction. The motion of the water will be steady in space, but it will be oscillatory
with a period of one day relatively to the solid earth.

It is easy to verify that in the forced oscillations of rigorously diurnal period the
motion of the water is of like character, involving no relative motion of the parts
and being steady in space. In this case the axis about which the rotation of the
water takes place lies in the plane containing the earth's polar axis and the dis-
turbing body.

VOL. CXCt. — A. 2 B

[ 187 ]

VI. Electrification of Air, of Vapour Oj Water, and of Other Gases.

By Lord KELVIN, G.C.V.O., F.R.S., MAGNUS MACLEAN, D.Sc., F.R.S.E., and

ALEXANDER GALT, B.Sc., F.R.S.E.

Received and Read, June 17, 1897.

§ 1. IN this paper we describe a long series or experiments on the electrification of
air and other gases, with which we have been occupied from May, 1894, up to the
present time (June, 1897). Some results of our earlier experiments, and of prelimi-
nary efforts to find convenient methods of investigation, have from time to time been
communicated to the Royal Society, the British Association, and the Glasgow
Philosophical Society.*

§ 2. The method for testing the electrification of air, which we used in our earliest
experiments, was an application of the water-dropper t (long well-known in the
ordinary observation of atmospheric electricity). Its use by MACLEAN and GOTO,J in
1890, led to an interesting discovery that air in an enclosed vessel, previously non-
electrified, becomes electrified by a jet of water falling through it. An investigation
of properties of matter concerned in this effect, related as it is to the " development
of electricity in the breaking up of a liquid into drops," which had been discovered
by HOLMGREN § as early as 1873, and to the later investigations and discoveries

* " On the Electrification of Air," ' Proc. Roy. Soc.,' May 31, 1894, and ' Phil. Mug.,' August, 1894.
" Preliminary Experiments to find if Subtraction of Water from Air Electrifies It," ' Brit. Assoc.
Report,' 1894, p. 554. " Electrification of Air and other Gases by Bubbling through Water and other
Liquids," ' Proc. Roy. Soc.,' February, 1895. " On the Diselectrification of Air," ' Proc. Roy. Soc.,'
March, 1895. " On the Electrification of Air," ' Proc. Glasgow Phil. Soc.,' March, 1895. " On the
Electrification and Diselectrification of Air and other Gases," ' Brit. Assoc. Report,' 1895, p. 630.

f KELVIN and MACLEAN, ' Proc. Roy. Soc.,' 1894, and KELVIN, MACLEAN, and GALT, ' Proc. Roy. Soc.,'
February, 1895. " Electrostatics and Magnetism," § 262 (from ' Proc. Lit. and Phil. Soc. of Manchester,'
October 18, 1859).

J "Electrification of Air by Water- jet." By MAGNUS MACLEAN and MAKITA GOTO, 'Phil. Mag.,'
August, 1890.

§ " Sur le developpement de 1'electricite a 1'occasion de la dissolution en gouttes des liquides."
' Kongl. Svensk. Vet. Ak, Handl.,' 1 c., vol. 11, No. 8, pp. 14-43 (pour 1'an 1873), -

2 B 2 25.6.98

188 LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

described by LKNARD,* in his paper on the " Electricity of Waterfalls," forms the
subject of §§ 25-37 of the present communication.

§ 3. The electrification of air by drops of water, breaking from a jet in it, or falling
through it, or striking on the ground, or on water, or on metal below it, produces
absolutely no practical disturbance of the electric potential measured by the water-
dropper in its use for the observation of open-air atmospheric electricity, but con-
stitutes a serious objection to its application for investigating atmospheric electricity
within doors, unless in a very large room or hall, and renders it altogether unsuitable
for the experimental investigations with which we are now concerned.

§ 4. We were, therefore, early led to abandon it ; and, for testing the electrifica-
tion of nil-, we have used three different methods, one or other of which we have
found convenient in different cases.

Method (l). Observation of electrification of the substance receiving the electricity
equal and opposite to that taken by air in any case of electrification of air.

Method (2). Observation of the electricity of a hollow metal vessel into which
electrified air is introduced, or from which electrified air is removed.

Method (3). Observation of the electricity taken out of air by the electric filter (§ 9).

§ 5. Method (l) was used in the experiments described in our communication to
the Royal Society of February, 1895, from which we concluded that air, and several
other gases tried, became electrified by blowing them in bubbles through water, and
through solutions of various salts, a,cids, and alkalies in water. We verified this
conclusion, for the case of common air and pure water, by collecting into a large
reservoir over water, air which had been bubbled through pure water in a U-tube.
We tested the electrification of the air thus collected by a water-dropper taking the
same potential as the air at the centre of the reservoir. We thus proved that the
electrification of the air was negative, as was to be expected from the positive
electrification which we had found on insulated vessels containing water through
which air had been bubbled.

§ 6. Method (2) was used in the first experiments described in the present paper
(§§ 16-24), which were undertaken for the purpose of determining approximately in
absolute measure the total quantity of electricity in a given mass of electrified air,
and particularly for finding the greatest electrification which we could communicate
to a large quantity of air by needle points supplied with electricity from an electric
machine. The result thus found in § 23 below, 37 X 10~* C.G.S. electrostatic, is
the gi-eatest electric density (quantity of electricity per cubic centim.) which we
have been able to communicate to air by electrified needle points. But, by an
electrified hydrogen flame a density of 22 X 10"* C.G.S. electrostatic unit was
obtained (§ 65).

§ 7. In all the experiments described in our paper after § 24, method (3) was
used ; but, probably, we must return to method (2) if, in future, we undertake

* " Ueber die Electricitat der Wasserfalle." By P. L.ENAKD, ' Annalen der Physik und Chemie,' 1892,

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 189

further experiments to find the greatest electric density which we can measure in
air or other gases,

§ 8. LENAUD'S important discovery of very strong electric effects produced by
drops of water falling on a hard surface, gave us a very convenient method for
obtaining a steady and strong negative electrification of air, which we used in
§§ 30-32 for preliminary efforts for finding a good and convenient form of electric filter
to be used in further investigations on the electrification and diselectrification of air.

§ 9. In testing the efficiencies of the electric filters used in method (3), we at first
used the filter described in our paper on "Diselectrification of Air," 'Proc. Roy
Soc.,' vol. 57, and which consisted of twelve discs of brass wire cloth, fixed in a
short metal pipe, supported in a paraffin tunnel. This filter was joined to the
insulated quadrants of a quadrant electrometer, and electrified air was sucked through
it (§ 25) till a convenient deflection was obtained. Then the filter to be tested was
connected to the sheath of the electrometer, and so placed that the electrified air
passed through the tested filter before it passed through the filter attached to the
electrometer (§ G8). In this way, by drawing equal quantities of electrified air, as
nearly as may be equally electrified, through the different tested filters, a comparison
of their relative diselectrifying powers was obtained. For example, <1 being the
deflection obtained when no tested filter was used, and dlt d* . . . d,t when the tested
filters were successively used, then the relative diselectrifying powers of the filters

would be "~v~\ " ~T~^> ... -, ", if the primary electrifications were equal.

§ 10. In other sets of experiments (§§ 32, 69) we successively joined each separate
tested filter to the insulated quadrants of the electrometer, and sucked approximately
equal quantities of electrified air through them. The diselectrifying efficiencies of
the filters were now approximately in simple proportion to the final readings on the
electrometer.

§ 11. But none of these methods gave us a means of determining the absolute dis-
electrifying power of any filter without realizing an equality of primary electrification
of the air in different experiments. We therefore, a long time later, used two
insulated filters and two electrometers, as described in § 55 and fig. 7. Let the filters
be called AB and A'B' and their diselectrifying powers n and n'. In a first experiment
the electrified air was sucked through AB and A'B' in immediate succession, and in
a second experiment the electrified air was sucked through A'B' and AB.

§ 12. On the assumption that the two filters took out the same 'proportions
(respectively n or n) of the electricity of the electrified air entering them in the two
experiments, we get the following equations :

In the first experiment

Let Q = total quantity of electricity in air entering,

ql = total quantity of electricity taken out by filter AB,
qz = total quantity of electricity taken out by filter A'B'.

190 LORD KELVIN, DR. M. MACLEAN, AND MR, A. GALT, ON THE

Then

ft = «Q ............ ghajfj • (1),

ft = n'(Q-ft) = n'Q(l-») ..... '..-(2).

In the second experiment

Let Q' = total quantity of electricity in air entering,
ft' — total quantity of electricity taken out by A'B',
q.,' = total quantity of electricity taken out by AB.
Then

7i' = «'Q' ............... (3),

gs' = «(Q'-g1') = nQ'(l-n') ....... (4).

From these four equations we find

_ rl\1\ ~ M» . > _ Mi' - gag/

"

§ 13. By taking a movable plate of a small air-condenser charged to a known
potential, and applying it to the insulated terminal of the quadrant electrometers
used as described in § 18, we could calculate ft, ft, ft', and q.2', and hence find Q and
Q', the absolute density of the electrified air or gases in C.G.S. electrostatic units.
It was thus that we found 11 X 10~* and 22 X 10~4 mentioned in §§64, 65.

§ 14. Up till the middle of December of 1895 the most efficient filter we tried had a
dist'.lectrifying power of about 0'8. During the Christmas holidays of 1895, we
succeeded in obtaining a filter of fine brass filings, as described in § 62, which
abstracted so much of the electricity from the electrified air passing through it, that
what was left was not sufficient to show on a similar filter, A'B', attached to an
electrometer, E' (see fig. 7). It was not necessary now to have two experiments,
first with electrified air in one direction through two filters to be tested, and then
with electrified air in the reverse direction through them. It was sufficient to take
a filter, the diselectrifying power of which, determined as above (§ 12), is found to be
very nearly unity, and attach it to electrometer E'. Then join the tested filter to
another electrometer, E, and allow the electrified air to pass this filter first, and
thence through the almost perfectly diselectrifying filter.

With the same notation as in 8 12, we

O '

»'

and

1

+ g2 T , 1 g2
1 + »' ft

If n' = unity, as it is for a filter of fine brass filings (§ 62),

I

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 191

§ 15. With such a filter as this it is possible to determine the quality of the natural
electricity of the atmosphere, and it may be desirable that it should be used for that
purpose in meteorological observatories.

GREATEST ELECTRIFICATION WHICH WE COULD COMMUNICATE TO A LARGE QUANTITY

OF AlR BY ONE Oil MORE ELECTRIFIED NEEDLE POINTS (§§ 16-24).

§ 1G. The first apparatus used is shown in fig. 1. It consisted of a metal can, D,
48 centims. high and 21 centims. in diameter, supported by paraffin blocks, and con-
nected to one pair of quadrants of a quadrant electrometer. It had a hole at the toj
to admit the electrifying wire, which was 531 centims. long, hanging vertically within
a metallic guard-tube, B. This guard-tube was always metallically connected to the
other pair of quadrants of the electrometer, and to its sheath and to a metallic screen
surrounding it, which is not shown in the diagram. This prevented any external
influences from sensibly affecting the electrometer, such as the working of the electric
machine, A, which stood on a shelf five metres above it.

§ 17. The experiment is conducted as follows : — One terminal of the electric machine
is connected with the guard-tube and the other with the electrifying wire, which is
tipped with needle points or tinsel, and which is let down to place the point or points
nearly in the centre of the can. The can is temporarily connected to the sheath of the
electrometer. The electric machine is then worked for some minutes, so as to electrify
the air in the can. As soon as the machine is stopped, the electrifying wire is lifted
clear out of the can. The can and the quadrants in metallic connection with it arc
disconnected from the sheath of the electrometer, and the electrified air is very rapidly
drawn away from the can by a blowpipe bellows, arranged to suck. This releases
the opposite kind of electricity from the inside of the can, and allows it to place itself
in equilibrium on the outside of the can and on the insulated quadrants of the
electrometer in metallic connection with it.

§ 18. We tried different lengths of time of electrification and different numbers of
needles and tinsel, but we found that one needle and four minutes' electrification
gave as great electrification as we could get. The greatest deflection observed was
936 scale divisions on a half millim. scale put up in the usual way, with lamp at a
distance of about a metre from the electrometer. To find from this reading the
electric density of the air in the can, we took a metallic disc of 2 centims. radius,
attached to a long varnished glass rod and placed at a distance of 1'45 centims.
from another and larger metallic disc. This small air condenser was charged from
the electric light conductors in the laboratory to a difference of potential amounting
to 100 volts, or % of an electrostatic unit. The insulated disc thus charged was
removed and laid upon the roof of the large insulated can. This addition to the
metal in connection with it does not sensibly influence its electrostatic capacity.

192

LORD KELVJN, DR. M, MACLEAN, AND MR. A. GALT, ON THE

Fig. 1.

A

The deflection obtained was 122 scale divisions. The capacity of the condenser is
approximately —

•i-n- x 1-45 "" 1-45

-" = ,• r C.G.8. electrostatic unit.

ELECTRIFICATION OF AIB, VAPOUR OF WATER, AND OTHER GASES. 193
The quantity of electricity with which it was charged was therefore

T. X ^ = v~ C.G.S. electrostatic unit.

Hence, the quantity on the can and connected metal to give 936 scale divisions was

1 936
4-3" * 129 = 1*7637 C.G.S. electrostatic units.

The capacity of the can was 16,632 cub. centims., which gives, for the quantity of
electricity per cub. centim.

= 1-06 X 10-4 of the C.G.S. electrostatic unit.

16632

§ 19. This is about four times the electric density which we roughly estimated as
about the greatest given to the air in the inside of a large vat, electrified by a needle
point and then left to itself, and tested by the potential of a water-dropper with its
nozzle in the centre of the vat, in experiments made more than three years ago and
described in a communication to the Royal Society of date May, 1894.*"

§ 20. To enable us to remove the electrified air quickly from the can, the following
modification was adopted : — The can was suspended vertically by three stout silk
threads (S, S, S, fig. 2) which had been previously soaked in melted paraffin ; it was
quite open at the bottom but closed at the top, with the exception of a central
aperture for the piston-rod of a piston, P. The piston was of wood encased in lead,
and was free to move up and down in the can by the movement of the paraffined
silk cord, C, over the pulleys, F, F. The can and the piston and piston-rod were
connected metallically by spiral springs of fine wire. The can was surrounded by a
metallic guard-screen, G, kept in connection with the sheath of the quadrant electro-
meter, and with the sheath of a vertical electrostatic voltmeter, and with one
terminal, B, of an electric machine. The other terminal of the machine was con-
nected to an insulated needle-point inside the can and to the insulated terminal of
the voltmeter.

§ 21. By working the machine the needle was kept charged positively or negatively
at 12,000 volts for four minutes. The air inside the can became charged similarly
by the brush discharge from the needle point. As soon as possible after stopping
the machine, the needle was removed and A and B were joined. The wire, W, from
the can was disconnected from the guard screen, G, and then attached to the
electrometer terminal, O, after which this terminal was insulated and the downward

* "On the Electrification of Air," by Lord KELVIN and MAGNUS MACLEAN.
VOL. CXCI. — A. 2 C

194

LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

movement of the piston begun. Usually about thirteen seconds were required to
make these changes.

Fig. 2.

§ 22. When the electrified air inside the can was expelled by dropping the piston
to the bottom, the reading of the electrometer went off the scale, and a shorter drop
had to be used to get a convenient deflection.

§ 23. A drop of 11 '5 centims., by which 3979 cub.- centims. of air were expelled,
gave a deflection of 1060 scale divisions. The quantity of electricity on the can and
connecting wire and insulated pair of quadrants of the electrometer which gives this
deflection, was (by the method of § 18) found to be 1'47 C.G.S. electrostatic. This,
therefore, was the quantity of electricity of the opposite kind in the 3979 cub.
centims. of air expelled from the open bottom of the can ; and the electric density of
this air was therefore 3'7 X 10~l C.G.S. electrostatic per cubic centimetre.

§ 24. In preliminary experiments before electrifying the air inside the can by
needle point, it was found that the dropping of the piston produced no deflection on
the electrometer.

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 195

ELECTRIFICATION OF AIR BY WATER JET (§§ 25-37).

§ 25. In previous experiments* we found air to be negatively electrified by water
falling in drops through it, and to pursue the investigation further the arrangement
shown in fig. 3 was put up. Loch Katrine water, under full pressure, issues from a

Fig. 3.

jet, J, fixed in the lid of a funnel, F, and falls down the funnel centrally into a metal
can, C, below. By means of an air-pump some of the air is withdrawn from the can
or funnel through metallic tubing, as shown in the diagram, to a metallic filter, AB,
containing 50 fine brass wire gauzes. In each experiment 200 strokes of the pump
are taken.

* " Electrification of Air by Water Jet," by MACLEAN and GOTO. ' Phil. Mag.,' August, 1890.
" Electrification of Air," by Lord KELVIN and MAGXUS MACLEAN. ' Proc. Roy. Soc.,' May, 1894.

2 C 2

196

LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

§ 26. The greatest effect, 4'5 volts negative, was obtained, as was to be expected
from LENARD'S discovery, when the air was withdrawn from a point well down in the
can, the can being close to the funnel, and the falling water rattling on the bottom of
the can. Decreasing effects were observed (l) when the air was withdrawn from the
can at increasing distances from the bottom ; (2) when several inches of water were
kept in the can ;* (3) when air was drawn from the funnel, and the distance between
the can and the funnel was gradually increased.

§ 27. A filter of 100 wire gauzes gave at the rate of 26 volts, and a filter of
2 gauzes, with a loose plug of cotton wool between them, gave 6 '3 volts, in the same
time and under the same conditions as the filter of 50 wire gauzes gave (§ 26) 4'5 volts,

§ 28. A sloping metallic plate was next fitted to the bottom of the funnel in such a
way that the falling water on striking the plate passed out by the aperture between
the funnel and the lower edge of the plate. In each experiment 120 strokes of the
pump were taken at the rate of 40 strokes per minute. Drawing the air from the
aperture near the bottom of the funnel gave, in 4 experiments, results averaging about
8 volts. These results were given by a filter containing 2 brass gauzes with cotton
wool between them.

§ 29. Two simple brass tubes of different bores with no wire gauze or cotton wool
were now tried as filters. Each was 10 ceutims. long and 1 centim. external
diameter. The following results were obtained : —

Air

drawn from side aperture of funnel : —

Internal

Near the top.

Near the bottom.

diameter
of brass

filter.
Aperture near

the bottom : —

Aperture near the top : —

Shut.

Open.

Shut.

Open.

centims. volts neg.
0-3 1-5

volts neg.
1-0

volts neg.
3-2

volts neg.
3-0

0-18 3-0

2-8

4-5

4-3

Using the brass filter of 0'18 centim. bore, drawing from the aperture near the
bottom, and varying the water pressure, we found mean results as follows : —

Full pressure . . . » . . '. ."....,. . • 3*4 volts.

Diminished pressure 2'6 ,,

.Very low pressure (200 drops per minute) . . 0'17 „

§ 30. In the long metallic tube between the funnel and the testing filter we placed,
in successive experiments, an increasing number of brass gauzes and plugs of wool.

* See a Paper by LENARD on '• Electricity of Waterfalls," ' Wied. Ann.,' 1892, vol. 4fi, pp. 584-636.

ELECTRIFICATION OP AIR, VAPOUR OF WATER, AND OTHER GASES. 197

The air h;id to pass through these and the long length of tubing before reaching the
testing filter at the electrometer. The testing filter consisted of a block-tin tube
with two wire gauzes and one plug of cotton wool. Twenty experiments were made
on air drawn from the aperture of the funnel near its lower end, and with jet from
full pressure of water. The following are mean results, for 120 strokes of the
pump :—

No gauze

between the
funnel and

2 gauzes
and 1 wool

3 gauzes
and 2 wool

4 gauzes
and 3 wool

5 gauzes 6 gauzes
and 4 wool and 5 wool

7 gauzes
and 6 wool

the testing

plug.

plugs.

plugs.

plugs. plugs.

plugs.

filter.

volts.
7

volts.

2-4

volts.
1-25

volt.
0-4

voli.
0-3

vu!t.

o-i

volt.
0-04

These results show that a large proportion of the electricity was taken, by the
7 gauzes and 6 plugs, from the air before it reached the testing filter.

§ 31. Extracting the air with no water falling gave no perceptible electrification.

§ 32. With the water again at full pressure, and falling on the sloping plate fixed
into the bottom of the funnel, the negatively electrified air was drawn from the
bottom aperture through different- filters at different speeds. Two experiments at
each speed were usually made, and whenever possible the deflection for 120 strokes
of the pump was noted. In some cases, however, the reading exceeded 8 volts, and
went off the scale with much fewer strokes ; but to preserve uniformity the tabular
results given below are a,ll calculated for 120 strokes.

Results 2 and 4 are in accordance with Results 2 and 3 of 8 68.

Duration of each stroke of the pump in seconds.

Filter used.

0-5

1

2

3

4

5

1. Platinum tube 2'4 centims. 1
long, O'l centim. bore /

volts neg.

volts neg.
4-9

volts neg.
9'9

volts neg.
14-5

volts neg.
187

volts neg.

2. Brass tube 4'1 centims. 1
long, 0'2 centim. bore J

7-8

12-5

21-8

25-8

26

25

3. Brass tube 4'1 centims. \
long, 0'34 centim. bore f

4

7-2

12

13-2

127

4. Solid brass cylinder with^

rounded ends, 8 centims. |

long and 1'8 centim. )>

20-1

34-1

527

59

62-1

diameter,insulated within j

a paraffin tunnel (fig. 3a)J

§ 33. A metallic water-dropper was now fixed into the lid of the funnel, and the
metallic plate at the bottom removed. A strong solution of common salt was placed
in the dropper, and allowed to fall down the centre of the funnel into a basin below.

198 LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

On drawing the air from the funnel by the side aperture near its lower end, and
testing it by the brass tubular filter, 0'18 centim. diameter (4 of § 32), a mean of
four experiments showed 2'5 volts positive.

•

Fig. 4.

§ 34. Arrangements were now made to test the effect of falling water upon air
only, uninfluenced by impact of drops on any hard solid. The metallic plate at the
bottom" of the funnel was removed, and the funnel, 24 6~ centlms. long, was placed
vertically in a position giving a clear fall of 640 centims. from the lower end of the
funnel to a water-trough below (see fig. 4). A new aperture was made in the funnel
near the centre. The filter used had 1 2 wire gauzes and 1 1 plugs of cotton wool.

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 199
The following results were obtained from 120 strokes of the pump :—

Water pressure.

Air drawn from aperture in funnel near the —

Remarks.

Top.

Middle.

Bottom.

Full
Reduced

volts neg.
0-5
0-16 '

volts neg.
07
0-23

volts neg.
0-5
0-33

i

Full

21-0
67

37-0

7-4

21-0

4-8

1
Sloping plate inserted in the
funnel at the bottom.
Plate removed. Funnel
tilted so that the water
struck against the side.

Fnll

§ 35. The lower half of the funnel was now removed and the upper half used. The
water now fell clear through the whole length of the funnel, and the extracted air
gave, by 120 strokes of the pump as usual, ^ to g volt negative. No electrification
in the extracted air could be detected if no water was falling.

§ 36. Putting the water-dropper (§ 33) in the top of the shortened funnel and
allowing a strong solution of salt water to fall down from the dropper, £• volt positive
was got from the extracted air. Placing pure water in the dropper and testing
again, £ volt negative was found.

ELECTRIFICATION OF AIR BY AN INSULATED WATER- DROPPER AT
DIFFERENT POTENTIALS (§ 37).

§ 37. The water-dropper was now insulated and connected with the positive
terminal of 1 or more, up to 12, cells of a secondary battery, the negative
terminal of which was connected with the funnel, and vice versa. On letting water
fall from the dropper, and testing the electrification of the air in the funnel by
drawing it through a testing filter, the results were not sensibly affected by
substituting metallic connections for the connection of the battery terminals with the
dropper and funnel. Hence, the large positive and negative electrifications thus
given to the drops as they fell from the nozzle did not sensibly diminish or increase
the negative electrification which they produced in the air through which they fell.

EFFECT OF HEAT ON ELECTRIFIED AND NON-ELECTRIFIED AIR (§§ 38, 39).

§ 38. The apparatus shown in fig. 5 was designed and used for the purpose of
trying to diselectrify.air by heat. Air is admitted into a tin plate biscuit canister, B,
near the bottom. Two metallic tubes are fixed into it at the middle opposite each
other. One of these two is plugged with paraffin through which passes a wire,

200

ORD KELVIN, DR. M, MACLEAN, AND MR. A. GALT, ON THE

ending in a needle point inside, and connected outside with the insulated terminal of
an electric machine, M. By means of an air-pump air is drawn into the canister,
where it is electrified by the needle. It passes thence through a few metres of
indiarubber pipe, to a 2-metre length of glass combustion tubing, G, 2 centims.
internal diameter, heated to a high temperature in a gas furnace, F. The hot air
passes on through a length of 3f metres of block-tin piping coiled in a large vessel of
cold water, W. The air thus cooled passes through two paraffin tunnels between
which is the insulated filter consisting of block-tin pipe with two wire gauzes and a

Fig. 5.

plug of cotton wool. There were altogether 10^ metres of tubing between the
canister and the filter. The air in the canister is kept electrified by an electrified
needle point during an experiment.

§ 39. Beginning with the glass tube cold, the air gave an electrification at the rate
of 14 volts positive for 200 strokes of the pump. On gradually increasing the
temperature of the tube the electrification correspondingly diminished to less than
3 volts. In cooling, the electrification, now become negative by an accidental reversal
in the inductive machine used, increased to 4'5 volts negative. On another occasion
5 volts negative were got with the tube cold, decreasing to 2 volts as the temperature
was raised, increasing again to 5 volts as the tube cooled. Occasionally irregular
results were noted, especially with positively charged air.

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 201

NON-ELECTRIFIED Am PASSED OVER HOT COPPER AND HOT CHARCOAL (§ 40).

§ 40. Passing air through the apparatus without first electrifying it, but keeping
the glass tube at a high temperature, we found no deflection on the electrometer.
But on repeating this experiment with copper foil in the tube an electrification of
9 volts positive for 200 strokes was observed. Replacing the copper foil by charcoal,
with temperature high enough to keep the charcoal visibly burning, we found a
negative electrification of 7 volts for 1 60 strokes.

u

Fig. G.

1

J

SD

f

\

1

7>

ELECTRIFICATION PRODUCED BY SHAKING AIR AND OTHER GASES WITH WATER
AND WITH SOLUTIONS OF DIFFERENT SUBSTANCES (§§ 41-46).

§ 41. An ordinary Winchester glass bottle, A, fig. 6, of capacity 2500 cub. centims.,
has two broad strips of tin foil cemented on its outer surface on opposite sides, from
VOL. cxci. — A. 2 D

202

LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

top to bottom. To the foil at the shoulder is attached a metallic disc, D, of 5 centims.
diameter, and having a small hole to allow a connecting wire from the quadrant
electrometer to be quickly hooked or removed. 500 cub. centims. of Loch Katrine
water having been put into the bottle, its mouth is stopped by hand, and the bottle
vigorously shaken for 5 seconds, thoroughly and violently mixing the enclosed
2000 cub. centims. of air and 500 of water. It is now immediately placed on a block
of paraffin, P, and the disc, D, is connected to the electrometer. A bent metallic tube,
T, supported by an insulating paraffin stopper, is placed in the bottle, and a foot

Curves 1, '2, 3.

0-7
0-6

0-3
0-2
0-1

o i
0-2

\

fercenf.

ren

\/

V 0-4

j:

06
07

length of rubber tube connects it to one end of a paraffin tunnel, from the other end
of which a rubber tube, 2 metres long, passes to an aspirator consisting of a large bell
jar, 13, of capacity 8500 cub. centims., filled with water, and resting on supports near
the water surface. The metallic guard-screen, S, which surrounds the bottle, is always
connected to the sheath of the electrometer. The outer surface of the bottle is always
wet. Less than half a minute is required after shaking the bottle to make the neces-
sary arrangements and connections. The electrometer terminal connected with the
bottle is now insulated, and then, by opening the stop-cock of the aspirator, " air is
drawn rapidly out of the insulated bottle, its place being taken by air flowing in

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 203

through a small vertical slit in the paraffin stopper. The electrometer shows positive
electricity, which proves the withdrawn air to have been negatively electrified.

§ 42. Very many experiments were made to test the effect of shaking up the air
in the bottle with solutions of different substances, the solutions being varied from
saturated (100 per cent.) down to practically pure water. For this purpose three
acids (sulphuric, hydrochloric, and acetic), three alkalies (sodium hydrate, lime, and
ammonia), and three salts (sodium chloride, sodium carbonate, and zinc sulphate),
were used.

Curves 4, 5, 6.
o-e

§ 43. Curves 1 to 9 show the results obtained. Generally, each test was repeated
a large number of times, and the curves are drawn for mean values. It will be
noticed that, with the exception of lime water, all the acids, alkalies, and salts, when
added to the water in the bottle in very minute quantities, showed a rapid diminution
in the negative electrification of the air produced by shaking it up with the liquid.
In some cases a single drop of a saturated solution of the substance added to the
water and shaken up with the air was almost enough to entirely neutralize the
negative electrification of the air which is obtained by shaking up with pure water.
On gradually increasing the strength of any of the solutions named, the zero line is

2 D 2

204 [LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

crossed and the electrification of the air now becomes positive, increasing to a
maximum and then diminishing till the zero line is again reached, the air becoming
negative again. The air continues to receive negative electrification, on shaking it with
the solution, for all further increase of strength of the solution up to saturation. All
the acids, all the alkalies except lime water, and all the salts tested, show two zero

points.

Curves 7, 8, 9.

§ 44. Experiments were now tried in which different liquids (500 cub. centims.)
and gases (2000 cub. centims.) were shaken up in the bottle, and the electrification
of the gas in each case was tested. The following is a tabular summary of the
results. In all the experiments, except No. 1, the gas was got into the bottle by
displacement of water. The bottle was first filled with water, and inverted in a
large trough of water and 2000 cub. centims. of the gas admitted.

ELECTRIFICATION OF ATR, VAPOUR OF WATER, AND OTHER GASES. 205

Substance

s shaken up together in the bottle.

Deflection on
electrometer.

Quantity of
electricity found
in the gas per cub.

Liquid.

Gas.

volt — 69
divisions.

centim., C.G.S.
electrostatic unit.

1

Water

Coal gas from gas mains ....

37 pos

0-17 x 10~* ne»

2
3

4
5

6

7

,,

7,

77
77

•7
77

Coal gas from pressure cylinder . . .
Oxygen from cylinder
Carbonic acid gas from cylinder . . .
Carbonic acid gas from marble and
hydrochloric acid, the gas being passed
direct from generator to bottle .
Carbonic acid gas, as in 5, but allowed
to pass into a gasholder first ....
Hydrogen from zinc and dilute sulphuric
acid, direct from o-enerator.

89 „
52 „
50 „

34 „
30 „
38

0-42 x 10-* „
0-24 x 10-' „
0-23 x 10-* „

0-16 x 10-* „
0-14 x 10-* „
0'15 x 10~*

8

77

Hydrogen, as in 7, but allowed to pass
into gasholder first

20

0-00 x 10"'

9
10

Water impreg-
nated with car-
bonic acid from
a gazogen or
siphon

Nitrogen from air, the oxygen beinsr
removed by burning phosphorus

Carbonic acid gas, as in 4 . .

71 .,
40 „

0'33 x 10-' .,
0-19 x 10-'

11

12
13

Strong solution
of common salt

77

Carbonic acid gas, as in 4
Coal gas from mains
Oxygen, as in 3

08 neq-.

71 .:

43 „

0-32 x 10-' pos.
0-33 x 10-* „
0-20 x 10-' „

§ 45. Some experiments were made to ascertain how long the air, on being
shaken up with Loch Katrine water, retains its negative electrification.

Time between stopping the
shaking of the bottle and
beginning to draw out
the air.

Deflection of the
electrometer.
1 volt = 69 divisions.

Divisions.

25 sees.

52 pos.

27 „

50

1 min.

26

2 mins.

28

5 7,

17

15 „

7

15 „

10

30 „

1

§ 46. A strong solution of ammonia (500 cub. centirns.) was placed in the bottle.
Without shaking the bottle, the mixed air and ammonia evaporating from the surface
of the liquid were aspirated. They were found i,o be negatively electrified to a
fraction of a volt.

20G

LORD KELVIN, DR. M. MACLEAN, AND MR, A. GALT, ON THE

ELECTRIFICATION or DIFFERENT GASES BV ELECTRIFIED NEEDLE POINTS AND
FLAMES. ABSOLUTE EFFICIENCIES OF FILTERS (§§ 47-66).

§ 47. The arrangement shown in fig. 7 was put up to test the electrification of
different gases by needle points and by flames. At first we only used one electro-
meter, E, and one filter, AB. The filter used was a block-tin pipe 4 centims. long,
1 centim. bore, with two brass gauzes and one plug of cotton wool between them.
A large glass cylinder, C, with a removable metal roof, R, has strips of tinfoil pasted
on its inside and outside. These strips are kept in metallic connection with R,
with one terminal, M, of an electric machine, and, with the sheath of the vertical
electrostatic voltmeter, V, and with the sheath of the quadrant electrometer, E.

Fig. 7.

A pump was used for drawing the electrified gas from the cylinder, C, through the
electric filter, AB. By this means, calculating the effective volumes of the two
cylinders of the pump, we knew the volume of electrified gas that was drawn
through the electric filter in each experiment. Placing the cylinder, C, over water, as
shown in the diagram, we found that 10 strokes of the pump raised the water
inside to a height of 8'1 centims. The cylinder was 38 centims. high and 81 centims.
in circumference. Hence the volume of gas drawn through the filter was 422'8 cub.
centims. per stroke of the pump. This agrees with the measured effective volume of
the two cylinders of the pump.

§ 48. An electrifying wire, ww, was put inside a glass tube full of paraffin and

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 207

bent as shown. A gas burner, G, was fixed in a brass tube which was connected by
sealing wax to a bent glass tube varnished with shellac leading in from the gas
supply.

§ 49. The method of experimenting was as follows : — The gas was lit at the
burner, G, and the machine started. The machine was worked for 4 minutes at
potentials from 3000 to 10,000 volts in different experiments. At the instant of
stopping the machine the gas supply was stopped, and an indiarubber cork put into
the tube, U, which was used to keep the gas in the cylinder at atmospheric
pressure during the electrification. Ten strokes of the pump were taken in every
experiment, immediately or after the lapse of certain intervals of time from stopping
the machine.

The results of a large number of experiments performed in May, 1895, are briefly
summarised in the following table : —

ELECTRIFICATION OF AIR BY ELECTRIFIED COAL-GAS FLAME.

Interval between

Average deflection

stopping

Number of

by 10 strokes of the

Remarks.

electric machine

experiments.

pump.

and starting pnrup.

(47 divs. per volt.)

minutes.

divs.

0

3

+ 210

Potential of electrifica-

0

3

-234

tion + 3000 volts.

1

•2

+ 108

I

2

-110

2

2

+ 76

2

2

- 94

3

3

+ 100

•6

3

- 82

5

2

+ 110

5

2

-107

30

I

+ 40

30

1

- 36

60

1

+ 53

0

3

+ 360

Potential of electrifica-

0

2

-395

tion + 4000 volts.

1

1

- 90

5

1

- 52

10

1

- 45

15

1

- 36

§ 50. We now electrified the air in the glass cylinder by six needle points
without flame. The method of experimenting was in other respects exactly the
same as before. A very wonderful result was noticed, namely, that short times
of electrification at 5000 volts gave greater results as indicated by electrometer and

filter than longer times of electrification.

208

LORD KELVIN, DR. M. MACLEAN, AND MK. A. GALT, ON THE

Time of electrification.

Number of
experiments.

Average deflection
for 10 strokes
of pump.

30 sees.

G

30-6

1 min. 6

2-2-1

5 „ 6

0

§ 51. A conjecture occurred to us that this very surprising result might possibly
be due to the formation of lines of conductance from the electrifying needle to the
inside of the containing cylinder, so that all the electricity from the needle might be
passing to the glass and tinfoil strips without electrifying the intervening air, and
that if means were provided for continually breaking these conjectural lines of con-
ductance the results would be diffei'ent. Accordingly a hexagonal ring of sheet tin
an inch broad was constructed to serve as stirrer. Its outer diameter was slightly
less than the diameter of the cylinder. Two rods which passed through two air-
tight holes in the metallic top of the glass cylinder were fixed at the ends of a
diameter of this stirrer. Several experiments were tried (1) with the stirrer not
moved but resting near the surface of the water at the bottom of the cylinder,
(2) with the stirrer kept moving up and down during the electrification, which was
by needle points at 5000 volts, in each experiment. The results obtained disprove
our conjecture. They are summarised as follows :—

1

Time of
electrification.

Number of
experiments.

Average deflection
per 10 strokes of the
pump.

Remarks.

30 sees.

5

Itf'O ueg.

No stirring.

30 „

(i

16'3 „ Stirring.

1 min.

6

11-6 „ No stirring.

1 „

6

13'3 pos. „

1 1

5

9'3 neg. Stin-ing.

I ,

G

10'4 pos.

)»

5 „

No deflection, stirring or no stirring !

ELECTRIFICATION OF CARBONIC ACID GAS FROM PRESSURE CYLINDER.

§ 52. One of the carbonic acid cylinders of the Scottish and Irish Oxygen Company
was taken and laid on its side near the glass cylinder, C (fig. 7). Carbonic acid gas
from it was let in by a tube into the glass cylinder, and electrified by six needle
points. To start with, the stop-cock of the carbonic acid gas cylinder was kept
shut, and ten strokes of the pump caused the water in the cylinder to rise. The
stop-cock was then opened and carbonic acid was allowed into the cylinder, the
average time of letting it in being one minute. This was performed five or six times,

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 209

and then the experiments were done exactly as already described in § 49, except that
carbonic acid gas was let in instead of air. The machine, in the two sets of experi-
ments summarised below kept the needle points at a potential of 8000 volts positive.

ELECTRIFICATION OF CARBONIC ACID GAS BY ELECTRIFIED NEEDLE POINTS.

Time of
electrification.

Interval between
stopping machine and
starting pump.

Deflection
for 10 strokes.

min.

min.

1

0

463 pos.

1

2

303 „

1

5

158 „

1

10

97 „

1

30

43 „

0

4 pos.*

1

'6

414

5

0

106

2

0

309

10

0

2

1

0

554

in 9 strokes.

0

, .

123 neg.f

1

0

554 pos.

in 7 strokes.

1

1

338 pos.

1

2

254

1

5

160

1

10

89

1

30

37

1

60

6

§ 53. The nearly perfect annulment of the electrification by 10 minutes of the
electrified needle point in carbonic acid, and by 5 minutes (§§50, 51) in common
air, is very wonderful. So far as we can see at present, its explanation seems to be
a conductive quality, such as that first discovered, we believe, by SCHUSTER, produced
throughout the carbonic acid, and throughout the air, by continued electric disruptive
action.

§ 54. It was noticed in some of the experiments with the carbonic acid cylinder
that, when it was lying on its side on the table beside the glass vessel, C, and when
the gas issued in a jerky manner, the electrification found was invariably negative,
even when the needle points were positive. This was, of course, due to the boiling

* This first experiment was made with the carbonic acid gas which had been in the glass cylinder
overnight.

f For explanation of this negative electricity in the gas with no electrification of the needle points,
see § 54.

VOL. CXCI. — A. 2 E

210 LORD KELVIN, DE. M. MACLEAN, AND MR. A. GALT, ON THE

and freezing, and ultimate evaporation from the liquid carbonic acid in the cylinder.
Various experiments were tried, with the cylinder (1) lying on its side, (2) standing,
straight up : in both cases without any electrification of the needle points. Very
slight positive electrification was found with the cylinder straight up ; and very large
negative electrification when the cylinder was lying on its side. In the latter case,
the more rapid the rush of the carbonic acid gas from its cylinder, and the shorter
the time it was left in the glass vessel before observations were taken, the greater
was the electrification observed.

TESTING EFFICIENCY OF DIFFERENT FILTERS BY USING TWO AT A TIME IN SERIES
AND TWO ELECTROMETERS (§§ 55-60, with reference to §§ 11, 12 above).

§ 55. For the purpose of testing the efficiencies of various filters, two electrometers
were fitted up side by side, as shown in fig. 7, each with a filter of block-tin pipe,
4 centims. long and 1 centim. bore, containing six wire gauzes and five plugs of
cotton wool. These filters were put in series, with a paraffin tunnel between them
to insulate the one from the other. Thus the electrified air passed the second
receiver immediately after it passed the first. Call the one filter AB and the other
filter A'B' ; then, in the first experiment, the electrified gas passed in the direction
AB A'B', and in the second experiment it passed in the direction B'A' BA. Each
filter is kept metallically connected always to the same electrometer. The sensitive-
ness of electrometer, E, with the filter, AB, was 52 '3 divisions per volt, and 100
divisions per 0'154 electrostatic unit of electricity. The sensitiveness of electrometer,
E', with the filter, A'B', was 158'3 divisions per volt, and 100 divisions per 0'071
electrostatic unit of electricity.

§ 56. Air was electrified negatively by six needle points in the glass cylinder and
the pump worked at the same time with the U-tube open, so that the water did not
rise inside. 100 strokes of the pump gave a deflection of 160 divisions on electro-
meter, E, and 32 divisions on electrometer, E'. This gives

g(1 = 0'246 electrostatic unit,
?2 = 0-023

Reversing the direction of the current of air through the filters, 100 strokes of the
pump gave a deflection of 156 divisions on electrometer, E', and 123 divisions on
electrometer, E. This gives

<?/ = O'lll electrostatic unit,
ft' = 0-189
Hence (§ 12)

n = 077 (filter AB),
and

n' = 0-31 (filter A'B').

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 211

Similar numbers were got when the air was electrified positively.

For carbonic acid gas, electrified positively and negatively, the same filters gave

n = 0-82 (filter AB),
ri = 0'42 (filter A'B').

§ 57. We now used the filter (n = 077) for determining electric density of
electrified air or gas. A known volume of the electrified gas (§ 47) was sucked
through the filter in connection with an electrometer whose constant was determined
as in § 1.8 (0'154 C.G.S. electrostatic quantity per 100 divisions deflection). We
thus found —

(1.) For air electrified, positively or negatively, by six needles at a potential of
5000 volts, an electric density of 0'92 X 10~4 C.G.S. per cub. centim.

(2.) For air electrified, positively or negatively, by electrified gas flame at a
potential of 5000 volts, an electric density of 1'98 X 10"1 C.G.S.

(3.) For carbonic acid gas, electrified negatively by gas out of a cylinder lying on
its side (§ 54), or positively by six needle points at a potential of 50CO volts, an
electric density of 2 '4 X 10 ~* C.G.S.

§ 58. We now set about to definitely determine the relative efficiencies of various
forms of filters. A standard filter of block-tin pipe, 4 centims. long and 1 centim.
bore, with 6 brass gauzes and 5 plugs of cotton wool was used, and it was per-
manently kept in metallic connection with electrometer, E'. The filter to be tested
was joined to electrometer, E. Air electrified positively or negatively was sucked
through in one direction, passing through the tested filter first, and then through
the standard filter, the diselectrifying power of which was 077 for electrified air.
Hence it is possible to determine the diselectrifying power of the tested filter by § 14.
Thus, from the numbers in the following section, we get for the diselectrifying power,
for positive electricity, of the 7 millim. brass tube — the least effective of those
mentioned — n = 1 -r- (1 + 4/077) — O'lG; and for the block-tin pipe, 90 centims.
long, and coiled into a spiral n = 1 -r (1 + 0'63/077) = 0'55.

§ 59. Let qz = quantity of electricity taken out by the standard filter of block-tin
pipe with six brass gauzes and five plugs of cotton wool ; and

(ft = the quantity of electricity taken out by the tested filter from the air before
passing the standard filter. A long series of experiments with no wire gauze or
cotton wool in the tested filters is summarised in the following tables. The potential
of the machine was in each experiment 10,000 volts.

2 E 2

212

LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

Bore of brass tube ill

i

millims.
(length = 4 centims.).

- for positive.
9fi

-— , for negative.

2-0

3-0

2-75

3-4

21

3-3

4-2

3-5

3-1

6-0

3-9

3-1

7-0

4-0

5-5

Thus the 3'4 millims. filter is the most effective and the 7'0 millims. filter is
the least effective of these five filters of equal length. The following table shows
results for different lengths and different bores : —

Length of tube.

Bore of tube in
millims.

^ positive.
1\

% negative.
1\

1. Brass, 9'9 centims.

2-0

2-9G

3-05

2. „ 10-0

4-5

2-85

4-99

3. „ 9-9

8-0

4-0

6-45

4. Block tin; 2'0

6-0

3-1

1-6

5. „ 10-0

6-0

3-26

7-5

6. „ 90-0 (coiled in

6-0

0-6.3

1-15

spiral)

7.* „ 4-0 ....

10-0

1-37

1-31

8. „ 4-0 ....

10-0

2-96

2-8

§ 60. A glass tube filter, 4 centims. long and 3 '3 millims. in bore, covered outside
with strips of tinfoil along its length, was similarly compared with the standard
filter. When newly put up, and as long as the glass was dry, it took out very little
electricity from the air ; but as the experiment proceeded, and the glass became less
dry by taking up moisture on its inner surface, the quantity of electricity taken out by
the glass tube became greater and greater. Thus in a first experiment qzlq\ = 28'5 ;
but after working the pump for one hour qzfql = 3'8.

§ 61. Up to this time (December, 1895), we had not been able to find a filter
which could take all the electricity from the air, and we now proceeded to search for
a filter which would be able to practically do so. The first filters tried with this
object were tubes filled with very small pieces of fine copper wire, and closed at each
end by a plug of cotton wool and a disc of brass gauze. The diameter of the wire
was 0'00296 centim. The containing tube was in one case block-tin pipe 10 centims.
long and 1 centira. diameter : and in another it was a glass tube of the same length
and diameter, coated both inside and outside with longitudinal strips of tin-foil.
The diselectrifying power of each was calculated from the observations by the

' Tube No. 7 had twelve wire gauzes inside it, and, as the table shows, its filtering efficiency was
more than that of the equal and similar tube No. 8, which was clear inside.

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 213

formulae in § 12. That of the block tin was thus found to be 0'93 for negative
electricity and 0'9 for positive, and that of the glass nearly the same, 0'9 for negative
and 0'84 for positive. The block-tin filter contained 18'927 grms. of wire.

§ G2. The next filter tried was a block-tin pipe, 10 centims. in length and 1 centim.
in diameter, containing 8 '3 3 grms. of clean, very fine brass filings, enclosed at each
end by a plug of cotton-wool and a piece of brass gauze. These brass filings, which
were got from a brass-finishing workshop, were poured into a glass tube about 4 feet
long and about three-quarters full of water. The filings and water were well shaken
up, and the tube was then allowed to stand for several hours, so as to give the filings
time to settle. After washing the filings three times in this manner, the top portion
was taken off and dried before a fire, and used for filling the filter. It was found that
when the electrified air passed through this filter of brass filings, before it passed
through the copper-wire filter attached to electrometer, E, no deflection was obtained
on the latter. This showed that the brass-filings filter deprived the air of practically
all its electricity. We tried also a filter with sawdust instead of brass filings, but its
efficiency was comparatively low.

FILTER OF BRASS FILINGS USED TO FIND EFFECTS OF SPIRIT FLAME, COAL GAS
FLAME, AND HYDROGEN FLAME IN ELECTRIFYING AIR (§ G3-6G).

§ 63. Having found the brass-filings filter thoroughly satisfactory, we used it to
investigate the effect of various kinds of flames in electrifying air. First of all, we
electrified the air of the laboratory by means of an insulated spirit flame, joined to
the insulated positive terminal of a Voss electric machine. The machine was worked
for 40 minutes, and then, 2 minutes after it was stopped, the electrification of the
air in the vicinity was tested by drawing some of it through a tube leading to the
brass-filings filter, joined to electrometer, E, The pump was worked at the rate of
1 stroke per 4 seconds, and after 200 strokes the electrometer read 174 divisions, or
about 3'3 volts positive. After the lapse of half-an-hour, the air of the laboratory
was found to be still strongly charged with positive electricity.

§ 64. Removing the water vessel from below the glass cylinder, C, in fig. 7, and
substituting for it a metal plate kept in metallic connection with the sheath of the
electrometer and the disinsulated terminal of the machine, we kept a coal-gas flame
burning within the glass cylinder, while the machine and pump were worked and obser-
vations taken by electrometer, E, and filter, AB. With the machine at 10,000 volts, we
found that, after two or three strokes of the pump, the deflection was about 500 scale
divisions, positive or negative, according as the machine was positive or negative.
This gives (§ 13) for the electric density, per cub. centim., 11 X 10~* C.G.S. electro-
static, which is much greater than any of our previous results (§§ 23, 57).

§ 65. To burn hydrogen, the burner, G of fig. 7, was made of rolled platinum foil
with a fine nozzle, which was kept in metallic connection with the insulated terminal
of the electric machine. The hydrogen gas was generated from zinc and dilute

214 LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

sulphuric acid in a WOULFF bottle. The electrification which we obtained in this
way, with the machine at 10,000 volts, was very large, the greatest deflection being
500 divisions in one stroke of the pump. This indicates an electric density in the air
of the glass cylinder, of 22 X 10~4 C.G.S. electrostatic, which is about six times as
great as that obtained by electrified needle points (§ 23). This electric density was
got for both positive and negative electrification.

§ 66. We next tried the effect of the insulated hydrogen flame alone, without
working the electric machine, and we found that when the height of the liquid rising
in the long, open glass tube of the WOULFF bottle was not more than about
10 centims. above the level of the liquid in the bottle, there was a small negative
electrification. When the liquid rose to a greater height than 10 centims. in the
tube (indicating that the gas was issuing at a greater pressure to feed the flame), the
electrification was positive. On one occasion, the positive electrification produced
by the name was 0'84 X 10~* C.G.S. electrostatic unit per cub. centim. of the
air which carried the electricity to the filter. This was the greatest effect obtained
from the flame without electrification by the machine, and the height of the liquid in
the tube of the WOULFF bottle was 14'5 centims.

The hydrogen gas, when not burning, gave no electrification at any pressure up to
26 centims. of water.

PLATINUM TUBE HEATED EITHER BY A GAS FLAME OR BY AN ELECTRIC

CURRENT (§ 67).

§ 67. Through the kindness of Mr. E. MATTHEY, we have been able to experiment
with a platinum tube 96 centims. long and 1 millim. bore. It was put in between
the glass cylinder, C, and the filter, AB, in the apparatus of fig. 7. The other filter,
A'B', was not used in these experiments. The platinum tube was heated either by a
gas flame or an electric current. When the tube was cold, and non-electrified air
drawn through it, we found no sign of electrification by our filter and electrometer.
But when the tube was made red or white hot, either by gas burner applied externally
or by an electric current through the metal of the tube, the previously non-electrified
air drawn through it was found to be electrified strongly positive. To get complete
command of the temperature, we passed a measured electric current through 20
centims. of the platinum tube. On increasing the current till the tube began to be
at a scarcely visible dull red heat we found butl ittle electrification of the air. When
the tube was a little warmer, so as to be quite visibly red hot, large electrification
became manifest. Thus 60 strokes of the air-pump gave 45 scale divisions on the
electrometer (0'86 of a volt) when the tube was dull red, and 395 scale divisions
(7 '5 volts) when it was a bright red (produced by a current of 36 amperes). With
stronger currents raising the tube to white-hot temperature the electrification seemed
to be considerably less. The following summary may be taken as a specimen of
several experiments. It is a copy of our notes of an experiment made on 20th July,
1895:—

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 215

Deflection after

Tim n nf

Time of
starting.

60 strokes.
52'3 divs. per
volt.

X mil, ul

60

strokes.

Current in
amperes.

Remarks.

hrs. mine. sees.
11 6 0

Divisions.
I'O pos.

miiis. sees.

5 15

0

Air left overnight in glass cylinder drawn

through platinum tube

11 17 0

3'5 neg.

4 50

197

Tube hot, but not visibly red

11 27 0

1-0 „

5 30

26-4

Tube beginning to be dull red

11 34 15

395 pos.

5 18

33-5

Tube very bright red

11 42 30

45 „

5 40

28-6

One end dull red. End next ingress of air

always duller

11 50 15

0

5 10

26-4

11 57 30

9 pos.

r> 30

27-5

Tube red before using pump

12 5 30
12 14 10

37-5 ,
190

5 30

5 47

28-6
30'5

One end perceptibly red.
One-third next ingress moderately red ;

other two-thirds (14 centims.) dull red

12 22 0

174

5 58

34-5

Nearly whole tube (20 oentims.) bright red

12 29 30

86

5 51

39'0

White hot

12 46 15

58

5 1

46-6

White hot

12 58 30

0

3 57

0

Fig. 8.

216

LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

§ 68. TJie Diselectrifying Power of Various Filters was farther tested as follows : —
Air, electrified in a metal vessel by a needle point kept electrified by a Voss
electric machine, was drawn through 340 centims. of block-tin pipe (0'91 centim.
bore), to one or other of the experimental filters which was connected to sheath
(fig. 8). After passing through A, the air was drawn through an insulated testing
filter, B, connected to the insulated terminal of the electrometer. B was a standard
filter of block-tin pipe, 5 centims. long, 0'66 centim. bore, and filled with fine brass
filings. Tests were occasionally made with A removed, to ascertain the diselectrifying
power of the standard filter (B) alone ; then A being inserted, the effect on B was
again noticed.

Deflection for 6

strokes of pump.

Percentage

F:||.,,. A

A. removed.

A. in position.

by A.

(1 ) Standard filter, similar to B

divisions.
220

divisions.
35

84

(2.) Brass tube, 5 centims. long, and O18 cen-

192

158

18

tim. bore

(3.) Solid- brass cylinder, rounded ends, 8'1 cen-

222

132

40

tims. long, and 1*8 centim. bore

(4.) Block-tin tube, 5 centims. long, and 0'91

160

158

1-25

centim. bore

§ 69. The standard filter, B, was now removed, and the various tubes used in the
last experiments as tested filters (A's), were now tried separately as testing filters
(that is, insulated filters, B), connected to the insulated terminal of the electrometer.
The following are the particulars and deflections noted for 6 strokes of the pump in
1 minute : — •

Filter used.

Divisions of deflection.

Positive.

Negative,

f 5 centims.

long and 0'91 centim. bore

none

none

BLOCK TIN 4. 10

0-91

jj

i,

1 10

0-66

4

4

r 4-1

0-18

5

none

BRASS . •< in

0-18
0-50

'5
5

5
11

t.17-5

0-90

6

8

Solid brass (rounded ends) S'l centims. long and 1'8 centiins.

45

diameter

Standard filter, block tin, 5 centims. long and 0'66 centim.

90

100

bore, filled with brass

filings

ELECTRIFICATION OP AIR, VAPOUR OF WATER, AND OTHER GASES. 217

§ 70. We then tried as filters tubes of different materials, but all of the same
length (10 centims.), and bore (0'91 centim.). Glass, brass, block tin, copper, and
zinc were used : the glass tube was covered externally with tin-foil, and also a little
way inside at each end. As usual, the air in the can was charged from the insulated
needle point at 4000 volts positive, and 3200 volts negative (§74), and drawn off through
340 centims. of block-tin pipe of 0'91 centim. bore, extending from the centre of the
can to the insulated filter, which was either glass, brass, or block tin. Before testing
the copper and zinc tubes, the can was brought nearer to the electrometer, so that
the length of the block-tin pipe conveying the electrified air to the filter was reduced
from 340 centims. to 100 centims. The mean of a large number of tests gave the
following deflections for 6 strokes of the pump in 1 minute, the mean result in every
case being very similar to the individual results.

Filter used.

Glass (covei*ed outside with
Brass

tin-foil) . .

Block tin

Copper

Zinc .

Deflection per C strokes of pump.

Positive.

I'l mean of 3 experiments
1-4 „ -2

3-3 „ 2
G-3 ,. 4

8-4 G

Negative.

7'0 mean of 3 experiments
*8 „ 2
3-1 „ 2
3-0 „ 3
2-3 I!

§ 71. The zinc filter was the only one which showed a distinctly greater deflection
for positive than for negative electricity, a result which is opposite to one obtained by
RUTHERFORD, * experimenting with air which had been electrified from an electrified
body under the influence of E.OXTGEN rays. The previous experiments having shown
that, on drawing electrified air over the insulated and non-electrified solid brass
cylinder with rounded ends, the cylinder extracted from the air a large proportion of
its charge, we now arranged to charge the cylinder and draw non-electrified air over
it to the standard insulated filter connected with the electrometer, to see if the air
would take up from the cylinder a part of its charge. The arrangements were as
shown in fig. 9. The cylinder was charged positively and negatively at potentials
varying from 1000 volts to 15,000 volts, but no trace of electrification of the air after
passing over the brass cylinder could be detected.

The Effects of the Uranium "rays," discovered by BECQUEEEL, and of a Candle

Flame, on Electrified Air.

§ 72. The air in the can (fig. 8) was charged in the usual way by a needlepoint
at 3200 volts negative for 1 minute, and the electric machine was then stopped ; the

VOL. CXCI. — A,

* See 'Phil. Mag.,' April, 1897, p. 246.
2 F

218

LORD KELVIN, DR. M. MACLEAN, AND MR, A. GALT, ON THE

electrified air was drawn from a point half-way down the can to the insulated
standard filter connected to the electrometer, and the deflection noted. The experi-
ment was repeated with a uranium plate suspended three-quarters of the way down
in the can by wires metallically connected with the can. The following results were
noted : —

Without uranium, 50 strokes of the pump were required before the electrometer
ceased to give a deflection, the total deflection being 271 divisions.

Fig. 9-

£0

The uranium was now placed in position in the can, and the air was then
charged for one minute. It was kept in position till all the electrified air was
drawn off to the filter, the total deflection being 61 divisions. When the uranium
was inserted after the electric machine was stopped, and before the air in the can was
drawn to the filter, little more than 10 strokes were required before the electrometer
ceased to give a deflection, and the deflection was now 121 divisions.

Using a very small lighted candle instead of the uranium plate, we found the
following results : —

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 219

Without the candle, 195 divisions negative for 60 strokes were noted.
With the candle inserted when about to withdraw the electrified air to the filter,
the deflection was, for 40 strokes, 81 divisions.

Best Charging Potential for Air and for Carbonic Acid Gas, in a Cylinder of
Metal, 48 centims. long and 21 centims. diameter. Greatest Electric Density
Obtained, and Loss of the Electrification.

§ 73. Numerous experiments were made to find the charging potential which
would give the greatest electric density to air drawn off from a metallic can, A,
fig. 10. An insulated needle-point, B, was fixed by a paraffin stopper in the bottom

Fig. 10.
• Ga/tzf Serfc// surrwtndtna ffl(c£roffif£er.

of the can, and was connected with the insulated terminal of a Voss electric
machine, C, and of a vertical electrostatic voltmeter, D. A pipe passed from
aperture No. 5 in the top of the can tcx a standard electric filter, E, insulated on

2 F 2

220 LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

two paraffin blocks. The filter was of block-tin tube, 5 centims. long and 0'6
centim. bore, and was filled with fine brass filings. It was connected by a short
platinum wire to the insulated terminal of a quadrant electrometer, F, and beyond
the filter were tubes passing to a mercury gauge, G, and air-pump, H. The can was
connected to the uninsulated terminal of the electric machine and to the sheaths of
the voltmeter and electrometer.

§ 74. The experiment was conducted as follows : — Apertures Nos. 1, 2, 3, 4, and 7
in the can were closed, and the electric machine started, the air in the can being
charged by a brush discharge from the needle point. The electrometer terminal
joined to the filter was insulated, and the pump worked for some time, fresh air
filtered through cotton-wool entering the can by a pipe attached to aperture No. 6.
The tests were made at potentials ranging from 2500 up to 12,000 volts, and it
was found that 3200 volts negative and 4000 volts positive gave about the best
results. The speed of the pump was kept constant, and the cubic contents of the
cylinders of the pump and the electric capacity of the insulated filter and quadrants
of the electrometer being known, the quantity of electricity in absolute measure
taken from each cub. centim. of air by the filter could be determined. In experi-
menting with carbonic acid gas, the procedure adopted was almost exactly the same
as that for air, the only difference being that instead of admitting air by the pipe
attached to aperture No. 6 the same pipe was attached to the nozzle of an upright
pressure cylinder containing carbonic acid gas. The gas was admitted to the can
under very slight pressure. For carbonic acid gas, the charging potentials which
gave the best results were found to be about 4000 volts negative and 5000 volts
positive.

In order to find out the electric density of the electrified air or carbonic acid
gas when left in the can for some time after charging had ceased, the electrification
was stopped after the machine and pump had been worked for several minutes.
The charging wire was removed from the needle and the apertures in the can
blocked. The enclosed electrified air or carbonic acid gas was left to itself for
different times in different experiments, generally just l£ hours. The gas in the
can was then drawn from No. 5 aperture through the insulated filter to the pump
(aperture No. 6 being opened to admit fresh air), and the pump was stopped when
all signs of electrification ceased.

The following results were obtained :—

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 221
ELECTRIC Densities in C.G.S. Electrostatic Units.

Gas.

Air

11
Carbonic acid

Greatest electric density,

while working electric machine

and pump.

O877 x 10 ' negative
0-370 X 10-* positive
1-17 xlO~* „
0-833 x 10-* „
0-63 x 10-* negative

Percentage loss in stated time.

73 per cent, in 90 minutes
92-7 „ ., 120 „
93-1 „ ., 14, hours
96-1 „ „ U „
98-8 14-

Diffusion of Electricity from Carbonic Acid Gas into Air.

§ 75. We next tried a scries of experiments to find if an electric charge given to
carbonic acid gas diffused from the gas into air. The method of experimenting finally
adopted was as follows : —

Carbonic acid gas was slowly passed from an upright pressure cylinder into the
metallic can, A, by aperture No. 6 ; atmospheric air was freely admitted through
aperture No. 7 ; while 12,000 cub. ceutims. of mixed carbonic acid and air were
drawn out per minute from aperture No. 2, and 6100 cub. centims. of air from the
top aperture (No. 5) ; the other openings in the can being kept closed.

In these conditions, it was found that air entered abundantly through No. 7,
showing that the volume drawn off from (2) and (5) was much greater than that of
the carbonic acid gas entering by (6). Hence there must have been nearly pure
carbonic acid gas below the level of aperture No, 7, separated by a very thin
transitional stratum from nearly pure air, above the level of No. 7. Nos. 7 and 2
were on the same level. The air drawn oft' from No. 5 passed through the iusulated
standard filter, E (already described), which was connected to the insulated terminal
of the quadrant electrometer, F. The Voss electric machine, C, was worked to give a
brush discharge from the needle point, B, the charging potential being indicated by
the vertical electrostatic voltmeter, D. Within 15 seconds after starting the machine,
a decided electrical effect was observed, the reading of the quadrant electrometer
almost immediately rising to a maximum rate of deflection of 55 divisions per minute
when the needle was charged positively, and 50 divisions per minute when it was
charged negatively. The electrification observed was not sensibly affected by
stopping the supply of carbonic acid gas. But when the working of the electric
machine was stopped and the charging wire to the needle removed, and whether the
supply of carbonic acid gas was continued or stopped, the electrical effects noticed on
the electrometer rapidly fell, and 3 minutes after the electric machine was stopped,
no further electrification was detected. The sensibility of the electrometer was
117 divisions per volt.

222

LORD KELVIN, DR. M. MACLEAN, AND MR, A. GALT, ON THE

In order to test if the carbonic acid gas in the lower half of the can still retained
any electrical charge, the connection to the filter and pump was removed from
aperture No. 5, at the top of the can, to No. 6 at the bottom, and the gas drawn
through the filter, but no electrification could be detected. We were surprised with
the results, and we do not see how to explain it : we expected that the stagnant
carbonic acid gas in the bottom of the vessel would have retained electricity as in
experiments of § 52 and § 74.

§ 76. Further experiments on diffusion of electricity were tried with a porous ball
(fig. II). The mouth of the ball was tightly closed, and through the cork passed two
glass tubes, one (B) projecting nearly to the bottom, the other being just through the

Fig. 11.

J

cork. The ball was suspended in the metallic can, which was filled with carbonic
acid gas or air. The gas in the can was electrified from the insulated needle point at
the bottom. Meantime, a strong blast of non-electrified air from a large bellows passed
into, the ball by the tube, B, and out again by the other tube to the insulated standard
filter of block-tin tubing, 5 centims. long and 0'66 centim. bore, filled with fine brass
filings, the filter being connected to the insulated terminal of the quadrant electrometer.
There was thus an air pressure from the inside of the ball towards its outside
surface, and under these conditions there was no evidence on the electrometer that
any part of the electric charge in the carbonic acid gas or air surrounding the ball
had made its way against the outward pressure of air, from outside the ball into the
interior, and thence to the filter. This experiment was varied by removing the
bellows, blocking the tube, B, partially or completely, and attaching an air-pump to the
insulated filter at C. On working the air-pump, some of the electrified carbonic acid
gas or air surrounding the ball must have been drawn inside, and thence to the

ELECTRIFICATION OF ATR, VAPOUR OF WATER/AND OTHER GASES. 223

insulated standard filter connected with the electrometer ; but, again, no evidence of
electrification on the filter could be detected on the electrometer. It thus appears as
if the porous ball itself had withdrawn the electric charge from the gas which passed
through the ball.

COMMUNICATION OF ELECTRICITY FROM ELECTRIFIED STEAM TO AIR (§§ 77-81).

§ 77. Steam was generated in a kettle, A, and electrified by brush discharge from a
needle point, B, attached to the lower end of a long copper rod, CB. The rod was
kept central and insulated in the brass tube, D, by two rubber corks. The upper
end of the rod was connected to the insulated terminal of a voltmeter, E, and of a
Voss electric machine, F (fig. 12).

§ 78. To preserve the insulating properties of the corks during the experiments it
was found necessary to keep a current of air passing in the tube between the corks,
and to surround the lower part of the tube with a jacket of oil kept at a temperature
of 235° F.

§ 79. The electrified steam from the kettle passed up into a vertical tube, G, where
it mixed with air drawn, by an air-pump shown on the right-hand side of the
drawing1, from a bottle into which the lower end of the tube was fitted. Air to take

cV

the place of that drawn from the bottle entered by a long pipe from outside a window
of the laboratory. The mixed steam and air passed from G into a LIEBIG'S condenser,
H, where the steam was condensed. The water thus formed dropped into a WOULFF'S
bottle, and the air was drawn from another neck of the bottle through a drying tube,
K, containing sulphuric pumice. From this it passed direct through an electric
filter, L, insulated by two paraffin tunnels, and thence to the air-pump. The filter was
connected to the insulated terminal of a quadrant electrometer, whose constant was
117 divisions per volt.

§ 80. When the air-pump and the electric machine were worked, with the kettle
cold, the electrometer showed no electrification. It also showed no electrification
when the kettle was boiling and the air-pump worked, but the electric machine
stopped.

§81. When the kettle was kept boiling, and the electric machine and air-pump
both worked, strong electrification, positive or negative, according as the machine was
positive or negative, was observed, 52 divisions per minute being our largest value.
This, with the known capacity of the electrometer, corresponds to O'll C.G.S.
electrostatic unit taken per minute from the air by the filter.

ELECTRIFICATION OF AIR AT DIFFERENT AIR-PRESSURES AND AT DIFFERENT
ELECTRIC POTENTIALS. MEASUREMENT OF CURRENT (§§ 82-88).

§82. In February, March, and April, 1896, we made experiments on the electri-
fication of air at different air-pressures, using for this purpose the apparatus repre-

224

LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON TTIK

r

Ifiiiiil ; i l . '

i

|i>''l ' M I'

i

ill IL "**»

1! 1 , 1 ih'fci '

i'i! .'|| !''

i

i1' 1 ' ' 1 i* V i 1

n!' 'I'f'ft1 '

I|U' i'i i t^1 i ' i

IMJ • ! • i i

••••••

, • i . .

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 225

sented in fig. 13, and electrifying by one needle point joined to the insulated terminal
of the Voss electric machine. The air was contained in a glass bell-jar, A, which
was coated inside with strips of tin-foil kept in metallic connection with one terminal,
G, of a high-resistance mirror galvanometer, the other terminal of the galvanometer
being joined to the sheath of the voltmeter, V, and to the disinsulated terminal, M', of the
electric machine. The stand of the bell-jar rested on a piece of paraffin. The pressure
of the air in the bell-jar, was measured by noting the height to which mercury rose in
the tube, B. The vessel containing the mercury was insulated by a paraffin block.

Fig. 13.

The pressure of the air after it had passed the electric filter (block-tin pipe with fine
brass filings) was also measured by means of the rise of mercury in tube, B'. A baro-
meter t\ibe not shown in the diagram gave us the atmospheric pressure. The differ-
ences of the heights of the mercury in tubes B, B', -and the height of the mercury in
the barometer, gave the pressures of the air in the bell-jar and on the exit side of the
filter respectively.

All the tubing through which the air passed was block tin.

Throughout each experiment the pressure of the air in the bell-jar was kept

VOL. cxci. — A. 2 G

226

LORD KELVIN, DR. M. MACLEAN, AND MR. A. GALT, ON THE

constant by regulating the stop-cock, C, so that the abstraction of air by the pump
was exactly compensated by the gain through C.

The galvanometer measures only that part of the electricity entering the bell-jar by
the wire, ww, which leaves it by its metal base. This part is very nearly the whole.
The observed results show that it is enormously great in comparison with the very
small part which is carried away in the current of air to the electric filter.

The galvanometer was shunted by a battery of LEYDEN jars, J, to give steady
deflections. Its sensitiveness was l/22'l of a mikro-ampere per scale division.

§ 83. First, we kept the potential of the needle constant throughout a set of
experiments made at different air-pressures, and in this way we found that the
current through the air to the metal of the jar became greater as the pressure of the
air in the bell-jar became less, down to the lowest pressure to which we went, which
was 40 millims. Curve 10 shows the relation between the current and the air-
pressure at a potential of 5000 volts. Similar curves were got for other electric
potentials.

Curve 10.

16

15

14

13

I?

10
<o

I-

t-

'•

\

•ent

Folt

'at

ls, 7 vsit

100

^oo

300

400

500

600

700

300

Pressure gfrfir in Millimetres cf Jlfercurt/,

ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES. 227

§ 84. We found also that as the air became rarer it was not so much electrified.
This was shown by the electric filter and electrometer. Thus the electrometer deflec-
tion for a pressure of 360 millims. was only about one-sixth of that for 760 millims.
with the same number of strokes of the pump, and the same potential of the electric
machine.

Curves 11 to 16.

is

J7
16
IS
14
13
12
II

10

•
•
'

6
5
4-

i

L

7

iooo aooo

3000 4000 5000- 6000

Voits

7000 -8000 9000 10.000

§ 85. We next kept the pressure of the air in the bell-jar constant and varied the
electric potential of the electrifying needle. In this manner we found how much the
current through the air in the jar was increased by increasing the potential. The
curves 11 to 16 represent the relation between the potential of the needle and the
current from the needle-point through the air to the metal of the bell-jar, for certain
definite air-pressures, and for positive and negative electrification. It will be noticed
that for the same air-pressure the current is greater for negative tnan for positive
electrification.

228 ELECTRIFICATION OF AIR, VAPOUR OF WATER, AND OTHER GASES.

§ 86. At each air-pressure the electrifying needle must be raised to a certain
potential before the galvanometer shows any current. Thus for a pressure of
342 millims. no current was shown by the galvanometer at a potential of 2000 volts
negative ; and for a pressure of 235 millims. no current was shown at a potential of
1000 volts negative or 2000 volts positive.

§ 87. We found also that air at a. given pressure was electrified to a greater extent
by a certain potential of the needle than by other potentials higher or lower. Thus
air at a pressure of 470 millims. was electrified by the needle at 10,000 volts 1'6 times
more than at any other potential we tried ; while air at a pressure of 340 millims.
received maximum electrification, whether positive or negative, when the potential of
the needle was 3000 volts.

§ 88. We are indebted for valuable help and co-operation in the carrying out of
these experiments to WALTER STEWART, M.A., B.Sc., to VINCENT J. BLYTH, to
JOHN F. HENDERSON, to WM. CRAIG HENDERSON, M.A., B.Sc., and to W. S.
TEMPLETON, M.A., B.Sc.

[ 229 ]

VII. Mathematical Contributions to tlie Theory of Evolution. — JV. On the Probable
Errors of Frequency Constants and on the Influence of Random Selection on
Variation and Correlation.

By KARL PEARSON, F.R.S., and L. N. G. FILON, B.A., University College, London.

Received October 18,— Read November 25, 1897.

CONTENTS.

PAGE
I. Introductory ......................... 229

II. On the probable errors of a system of frequency constants. — General Theorem . . 231
On the determination of the probable errors and the error correlations of the

frequency constants ..................... 236

III. On the probable errors and the coefficients of correlation between errors made in the

determination of the constants of a normal frequency distribution ..... 237

Probable error of a regression coefficient for two organs ......... 244

Probable errors of variation and. correlation for three organs ........ 245

Case of four or more organs ................... 250

IV. On the probable errors and the coefficients of correlation between errors made in the

determination of the constants in skew variation ........... 2G5

In the case of the curve

266

Numerical illustration : Incidence of enteric fever ........... 279

In the case of the curve

y = yl(l + */a,)7"' (1 — aj/aj)'"* ........ 282

Numerical illustration : Glands of the forelegs of swine ......... 289

In the case of the curve

„ — „ ___ - _ ,,-tan-i(*M) OQ7

y-y>{l + (a/a*)}"'

Numerical illustration : Stature of children .............. 303

Conclusion ...... ............ ....... 309

Note on the probable error of the criterion . . .............. 310

I. INTRODUCTORY. — GENERAL THEOREM.

(1) In earlier memoirs by one of the present authors, methods have been discussed
for the calculation of the constants (a) of variation, normal. or skew,* (6) of correla-

* "Mathematical Contributions to the Theory of Evolution.— II. On Skew Variation," 'Phil. Trans.,'
A, vol. 186, pp. 343-414.

12.7.98.

230 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

tion, when normal.* The subject of skew correlation would now naturally present
itself, but although several important conclusions with regard to skew correlation
have been worked out, there are still difficulties which impede the completion of the
memoir on that topic. Meanwhile Mr. G. U. YULE lias shown that the constants of
normal correlation are significant, if not completely descriptive, even in the case of
skew correlation.t It seems desirable to take, somewhat out of its natural order, the
subject of the present memoir, partly because the formulae involved have been once
or twice cited and several times used in memoirs by one of the present writers, and
partly because the need of such formulae seems to have been disregarded by various
authors in somewhat too readily drawing conclusions from statistical data. Differences
in the constants of variation or of correlation have been not infrequently asserted to
be significant or non-significant of class or of type, or of race differences, without a due
investigation of whether those differences are, from the standpoint of mathematical
statistics, greater or less than the probable errors of the differences. Notwithstanding
that every artificial or even random selection of a group out of a community changes
not only the amount of variation, but the amount of correlation of the organs of its
members as compared with those of the primitive group,J it has been supposed that
correlation .might be a racial constant, and the approximate constancy of coefficients
of correlation of the same organs in allied species has been used as a valid argument. In
the like manner differences in variation have been used as an argument for the activity
of natural selection without a discussion of the probable errors of those differences.

In dealing with variation and correlation we find the distribution described by
certain curves or surfaces fully determined when certain constants are known. These
are the so-called constants of variation and correlation, the number of which may
run up from two to a very considerable figure in the case of a complex of organs. If
we deal with a complex of organs in two groups containing, say, n and ri individuals,
we can only ascertain whether there is a significant or insignificant difference between
those groups by measuring the extent to which the differences of corresponding
constants exceed the probable errors of those differences. The probable error of a
difference can at once be found by taking the square root of the sum of the squares
of the probable errors of the quantities forming the difference. Hence the first step
towards determining the significance of a group difference — i.e., towards ascertaining
whether it is really a class, race, or type difference — is to calculate the probable errors
of the constants of variation and correlation of the individual groups. This will be
the object of our first general theorem.

* " Mathematical Contributions to the Theory of Evolution. — III. Regression, Heredity, and Pan-
mixia," ' Phil. Trans.,' A, vol. 187, pp. 253-318.

f " On the Significance of BRAVAIS' Formulae for Skew Correlation," ' Roy. Soc. Proc.,' vol. 60,
pp. 477-489.

J This will be sufficiently indicated in the latter part of the present memoir, but has been more fully
dealt with in a paper on the " Influence of Selection on Correlation," written, but not yet published.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 231

II. ON THE PROBABLE ERRORS OF A SYSTEM OF FREQUENCY CONSTANTS.—

GENERAL THEOREM.

(2) Let there be a group of n individuals, for each of whom a complex of m organs
is measured, and let z 8xt 8x., . . . 8xm be the frequency with which individuals having
a complex of organs lying between xlt x^ + 8xt ; x2, xz + Sx2 ; ... xm, x,a + 8xm> occur
in the total group of n. Here x shall measure the deviation of any organ from the
mean of all like organs in the group. Let hl} h.,, h.A, . . . hm, be the mean measurements
on the organs, so that hi + x,, h2 + x2, . . . hm + #/», is the system giving the actual
measurements on any individual. Then hlt h*, hA, . . . hm, are the first set of constants
of the frequency ; they determine the " origin " of the frequency surface.

Let this frequency surface be given by

Z -~ J \*^l> 3<2i X3t • • • Xm; Ci> Clt ^3> • • • Cp)t

where clt c2, c3, . . . cp, are p frequency constants, which define the form as distin-
guished from the position of the frequency surface, and which will be functions of
standard deviations, moments, skewnesses. coefficients of correlation, &c., &c., of indi-
vidual organs, and of pairs of organs in the complex.

The problem before us is to find the probable errors of the A's and the c's, which
constants fully determine the position and shape of the frequency surface. Let
o"t,» °"/v ^i,,, • • • 0X> 0V, > <rv • • • °"r-,., be the standard deviations of the quantities
7ij, h2, . . . hm, and clt c2, . . . cv. Then a knowledge of these standard deviations will
give us at once the probable errors of the frequency constants, for we have only to
multiply the former by the numerical factor '6745 to obtain the latter.

Let us now suppose that the value of the frequency constants had been Ax + AA,,
A2 + AA2, . . . hm -\- A/4, GI + Acj, c2 + Ac2, . . . cp + Acp, instead of the observed
values.

Then the frequency of any observed individual would have varied as

f(xl + AA,, x2 + AA2) . . . xm + AAM ; c, + Ac,, c2 + Ac2, . . . cp -\- Acp)

instead of as

/(x,, a;2, x3, . . . xm ; c,, c2, c3, . . . cp).

Hence on this hypothesis the probability, PA, of the set of individuals observed in
the group actually occurring is to the probability, P0, of the set occurring when the
constants are Alt h2, . . . hm ; clt c2, . . . cpt in the ratio of the product of all quantities
like / (xj + A/ij, x2 + AA2, . . . xm + AAm ; c, + Acl5 c2 + Ac2, . . . cp + Ac,) . for all
values of xlt x2, . . . xm, to the like product of all quantities like f(x\, x2, . . . xm;
Cj, c2> . . . cp), or

+ A/I,, xt

n/(xl,xt,.. .-xm; Cj.cj, . .. c,)

232 PROFESSOR K. PEARSON AND MR. L. N. G. FTLON

Taking logarithms, the products II become sums S, or

log (PA/P0) = S log/fa, + A/t,, x2 + A/i2, . . . xm + AAm ; c, + Ac,, c2 + Ac2, . . . cp + Ac,,1)
— S log/(«,, a;,, . . . xm ; c,, c2, . . . c;)).

Let the first summation now be expanded by TAYLOR'S theorem, and typical terms
up to the second order be written down. Then we have

log (PA/P0) = AA, S £ (log/) + i (A>02 S ~ (log/) + AA, A/iy S ^ (log/) .
+ Ac, S ~ (log/) + 1 (Acs)* S ~ (log/) + Ac, Ac,- S ~- (log/)
+ A/t, Ac, S -n-y- (log/) + . . . + cubic terms in A/i and Ac + &c->

where /stands for/(x,, o;2, a;3, . . . XM, ct, c2) . . . c^,).

Here r is to be given every value from I to m, and * to be given every value from
1 to p, but r' and s' in the third and sixth sums are only to be given values from 1
to m and l.to p other than r and s respectively. In the above formula we may
replace the sums by integrals, if we remember that the frequency of the system
x,, x2, . . . XM, is simply /So;, S.r2 . . . So:,,.

Writing

log (PA/P0) = A, Mi,. - IB, (A/i,.)2 + C,,, A/i, Mi, + D, Acs - P, (Ac8)2 + FM, Ac, Ac,

we will investigate the values of the constants separately.
First,

= JJ| . . . ^- dxl dx2 . . . dxm,

= III •••[/] ^1 ^2 ' • • <&r-l rfajr+l • • • dxm,

the integrals now not including one with regard to dxr and [/] denoting that / is to
be taken between the extreme limits of / for xr. Now in most cases of frequency
the frequencies for extreme values of any organ are zero.* Hence [/] equals
nothing. Thus we have A,. = 0.

* In most cases, but not invariably, as, for example, in the case of some florets and petals. In the
cases, however, in which Ar does not vanish, the conclusions finally reached will be the same, as Ar only
marks a change of origin for the constant frequency distribution.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 233

Secondly,

= III • • • ^T dxl dx, . . dxM,

d (,,
dc.g

= dnjdca,

where n is the total number of individuals measured, which is independent of c,.
Therefore Ds = 0.
Thirdly,

This will not as a rule vanish. If the frequency be normal, it will still not as a rule-
vanish. It will vanish either if the frequency be symmetrical about .<', — 0, and
xa = 0, or if there be no correlation between the x. and »•„ organs, i.e., if / be of the
form /(a:;) Xf,(x;).
Fourthly,

fl,|, ,.(!' \0" / 7 7 /•• \

G,, = IJJ • . •/ iLi.^ dx, dx, . . . (Ar,,, ...... (n.).

This will not as a rule vanish. It will vanish, however, if the frequency of x. be
symmetrical about its mean.
Fifthly,

(v.),

all of which will generally be finite, but admit, like C;,.< and G,s, of calculation when
the form of the frequency / is given. Hence

PA = Po expt. - 1 {Br(A/i,.)'- - 20,,, A/i, A/i,, - 2G,, Afc(. Ac.
+ Es (Acs)2 - 2FM, Ac3 Ac, + &c. . . .},

X expt. (terms in cubic and higher orders of the A's) . . (vi.).

This represents the probability of the observed unit, i.e., the individuals
(Xi, x2, . . . xm, for all sets), occurring, on the assumption that errors A/i,, A/(2, . . .
VOL. cxci. — A. 2 H

-'34 PROFESSOR K. PEARSON AND MR. L. N. Q. PILON

A/iw, Ac,, Ac.,, . . . Ac,,, have been made in the determination of the frequency con-
stants. In other words, we have here the frequency distribution for errors in the
values of the frequency constants.

(3) Conclusions to be drawn from the, form qf(vi.)._

(a) The distribution of the errors of frequency constants, if treated exactly, will
generally be skew, for the cubic and higher terras in the A's do not vanish. If,
however, the cubic terms are small as compared with the square terms, the frequency
distribution of errors will approximate closely to a normal correlation surface.

It would be impossible to evaluate the remainder after the second power terms in
TAYLOR'S series for any general expression f (#,, x2, . . . XM ; c,, c2, . . . cy,) for frequency.
In special cases we have found that terms of the third order amount in the most
unfavourable circumstances to 4 per cent, of the terms of the second order, generally to
a good deal less.* Probably the series in most cases converges with considerable
rapidity. The fact, however, that we are dealing with the first terms of a series should
be borne in mind. It does not seem to have been sufficiently emphasised when the
probable error of the standard deviation is taken to be G7'449/x/2>Tper cent, of the
standard deviation. The usual proof of this result, however, involves the same
assumption as to the smallness of the cubic terms.

(/3) Supposing the errors so small that we may neglect the cubic terms, we
conclude that the errors made in calculating the constants of any frequency distribu-
tion are—

(i.) Themselves distributed according to the normal law of errors,
(ii.) Correlated among themselves.

Both these conclusions are of the utmost importance. The first enables us to
obtain the probable errors of the frequency constants ; the second depends upon the
fact that C,/, G,,, and FM, are in general not zero. The standard deviations of, and
the correlations between, the frequency constant errors can now be calculated by the
ordinary theory of normal correlation.

Before, however, proceeding to these calculations, we may draw one or two other
conclusions of considerable generality and wide significance.

(y) Consider a race fully defined by the variations o-,, o-,, o-3, . . ., &c. of the organs
of its members and their correlations r12, r.a, rn, .... Now let a random small
selection be made of this race, defined by

o-! -f- So-,, o-, -f- So-,, o-;, + 80-3, . . . rr, + Sr,,, r,3 -f- Sr,3, r,3 + Sr,3, . . .,
where the magnitudes of "So-!, So-,, So-3, . . . 8rK, Br-a, Sr13, . . ., are quantities depending

* See, for the relative order of two terms of the second and third orders, " Regression, Heredity,
and Panmixia," ' Phil. Trans.,' A, vol. 187, p. 266.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY" OF EVOLUTION. 235

on the magnitude of the probable errors of the variation constants, and therefore on
the size of the selection. Then the system Scr,, So-,, 8o-3> . . . 8rK, Sr.,3, Sr,3, . . ., is not
a system of independent variations, but the changes in variation and correlation are
correlated together, since the terms FM, do not generally vanish. We therefore
conclude that if a random selection be made out of a population with regard to one
organ o-,, there will be tendencies for the variation of all other organs and the
correlation between all organs to change also in certain directions, which can be
definitely indicated so soon as the general population has been measured, and the
effect of the random selection on one organ has been ascertained. What is proved
here of random selection will be shown to be true still more intensively for artificial
selection,* i.e., every selection of one organ modifies in a correlated manner the
variation and correlation of all other organs. It is impossible to alter one organ
without altering all other organs and their relation to each other.

(§) The remarks in (y) are not only true for a random selection of variation, but
equally well apply to selection of sixe. This follows because the terms Q, and O,, do
not as a rule vanish. If a random selection of 100 or 1,000 individuals be made out
of a general population, then the mean sixe of any organ in this sub-group will
probably differ from that of the general population. The result is that the sixe of
all other organs, their amounts of variation and correlation, will probably have values
differing in definite directions from those of the general population.

(e) Take two random selections out of a general population ; the probability is that
they will have different means for any one organ, and a result will be that they will
have correlated systems of changes in the sixes, variations, and correlations of all
other organs. In other words, random selection produces a differentiation of all
characters, which differentiation will bo the more marked the smaller the random
selection. t

(£) This principle— that a random selection gives a system of correlated changes in
the deviations of all the characters of a species — seems, to some extent, capable of
explaining the small but systematic differences to be found occasionally between
closely allied species. It is not necessary to suppose them due to a long process of
natural selection acting on a variety of organs ; a small random selection, or possibly
a natural selection of one organ, might suffice to produce the systematic differences of
character in all organs.

How far a succession of random selections would give an evolution biassed by the
first random selection requires further consideration ; but it seems impossible for the
characters of a race to remain fixed under the influence of a heavy but non-selective
death-rate. They will vary from year to year, although this systematic change of

* " Memoir on the Influence of Selection on Correlation." Selection not only modifies correlation,
but the selection of one organ can create correlation between organs previously uucorrelated.

t Continuous artificial selection of an organ produces a still more marked differentiation of all other
characters, but this is treated of in another memoir.

2 H 2

236 PROFESSOR K. PEARSON AND MR. L. N. G. PILON

characters be not always in the same direction. Systematic change of characters
produced by random selection may be spoken of as random evolution. Random
evolution is -theoretically a possible cause of systematic change ; experiment only can
determine how great is its effectiveness in differentiating local races.

(77) In the case of a normal distribution of variation defined by the mean h and
the standard deviation cr, it has been usual to suppose that the error made in the
mean is independent of the error made in the variation. In other words, it has
been assumed that G,,, vanishes, although no proof has been given, or possibly it
has not been realised that a proof was necessary. In this case f is of the form

•• r 2/0 21 j i

expt. — [ar/2crj and -— — - — == — , whence

V/(27r) a- da dx CT" '

G = 2 (*"4 expt. - {^2/2o-2l - - dx = 0.
J-oo a ' v/(27r)o-

Thus there is no correlation between error in the mean and error in the standard
deviation. This assumes that we stop at the square terms in (vi.). If, however, we
include the cubic terms, &c., product terms in A/i and ACT do arise, and we cannot
state straight off that no correlation exists, although it may be very small. In the
case of all skew variation, such as is so frequent among plants and animals, a corre-
lation will always be found between deviation or error in the mean and the like in
the standard deviation. In other words, to alter the mean by selection (artificial or
random) is to alter the variation of an organ.

With the exception of the statements in this paragraph (17), the whole of our
general conclusions in this section are independent of any particular law of frequency.

(4) On the Determination of the Probable Errors and the Error Correlations of the

Frequency Constants.

Let iji, 7)2, f]3, . . ., be the frequency constants, whether they be the means, standard
deviations, or correlations of a complex of organs. Then if we neglect cubic and higher
terms in the deviations A^1} A^, A^3, . . ., the frequency surface giving the distribu-
tion of the variations in the deviations is

PA = Po expt. - l [S {«,,(A>?,)2} - 28 K A^A^}],
where

It is required to find 2,., the standard deviation of AT;,., and R,s, the coefficient of
correlation between A??, and A^.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 237
Let A be the discriminant :

where ars = «.„.. Further, let A,, be the minor of the rtlx row and sth column ; then

22 = A,.,/ A and Rr, = A^AS/E,) (v"'-)

give the required values of S,. and R,.8.*

Further, the standard deviation of AT;, for selected values of all the other AT/S
is 1/v/a,,..

This value can often be of service. Thus suppose a considerable number of
skeletons found, but that only in a comparatively few cases is it possible to pair
together the femur and tibia of the same individuals. Then the variations of femur
and tibia in the race will be known with great exactness, but the probable error of
the correlation between femur and tibia will be given with close approximation by a

form like '67449 —. — . In fact, whenever we have obtained from a large number of

V"rr

observations the values of the frequency constants of a race with great exactness,
then, using these values to obtain an additional variation or correlation constant from
a few observations, the probable error will be of the form just indicated, and not of
the form '67449 A,,./A. It is needless, perhaps, to remark that the former is far easier
to calculate than the latter.

III. ON THE PROBABLE ERRORS AND THE COEFFICIENTS OF CORRELATION BETWEEN
ERRORS MADE IN THE DETERMINATION OF THE CONSTANTS OF A NORMAL
FREQUENCY DISTRIBUTION.

(5) In order to exhibit more clearly the method of investigation, it is desirable
that a simple case be first taken, Accordingly we will start with the following
problem :—

To find the Probable Errors and Error Correlations of the Constants of a Normal
frequency Distribution for Two Organs.

Let hi, h», be the means, o-,, ar.>, the standard deviations, r,2 the coefficient of correla-
tion, n the total number of pairs of observations, xlt x2, any pair of corresponding
deviations of the organs. Then the frequency z dxL dx2 is given by the surface

1s

o *•

z =

/• ,

* See " Regression, Panmixia, and Heredity," ' Phil. Trans.,' A, vol. 187, p. 301 (e).

238 PROFESSOR K. PEARSON AND MR. L N. G. FILON

"We require to find

£„,, the standard deviation of errors in o-lt

— (Tj) )! » )> °~2)

^VJ2> J) )! 5! '12)

2,,, ., „ ,, pi = r^a-i/a-.i,

which is the coefficient of regression of a:,,*
R,1<ra, the coefficient of correlation between errors in crl and or.,,

<ri>'n> '••• '•' " ""l " *">2'

^ cr2rjj» » )) 5* )» ^2 )> '12)

S/,,, the standard deviation of errors in h},

Sft,) j) ;) ,) "2)

H/,,;,.,, the coefficient of correlation between errors in h, and h2.

It follows that RVl, R,,^, R,,i<Ta, R,,^, R^.,12, R,,a,.2, are all zero, since, by (ii.) of p. 233,
Grt will vanish when the distributions of Xi and x., are symmetrical. These correla-
tions would not, however, vanish for the skew frequency distributions, which are of
most frequent occurrence in problems of heredity and fertility in man, &c.

The first -stage in the investigation is to write down the second differentials of the
logarithm of z for all quantities occurring in it.

We find

7 » <> 1 A 9/1 9 \ *T 9 \ [ J

rtcrf erj [ 07 (1 — r,-2) cr^cr., (1 — »'J2) J

? — — ~ ~ T 1 — ~ ~~ -f" "~ [" a ('

d*qogz) _ rwlt

2i' t& •' ' •> • / 1 t *,2 \ T

1*1 iX'jiicg ( J- ~r * j9 / I

, ,, " , v, -r, rr, \ >

[(1 - »B)! 0-10-2 (1 - »nrJ

— l ±1** 1 + 3?j8 /M_ i ^_\ i 6?'ia

of (1 — ?•?,)

* See the memoir on " Heredity, Regression, and Panmixia," p. 268.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 239

Now these must be multiplied by 2 and integrated for xl and x., from — oo to + co.
These integrations follow at once, if we remember that

na''i = llzx* dxt dx.2, ncr\ = \\zxi dxi dx.,, ncr^-.r^, = \\zxlx., dx: dx.it

by definition of o^, cr.,, and rr,.

Thus we obtain the following system :

n 2 - rl, n 2 - 7-f.,

0 — if fl _— .

11 — 01 o ' W22 — o-i " ., '

o"j 1 — rjo cr; 1 — ?'jo

nr\->

ao \
— /72)

where the a's are those of Eqn. (vii.), p. 236, obtained by taking the above differentials
of the logarithms of z in order.

We can now write down the correlation surface, giving the frequencies of errors in
the constants :

n

, L j

" ^ IT? a-4) " ^(i - "/-fD "h ^a

f it, (2 — rr,) . /; 2 — rr.

x expt. _ i L J - .

—

| i ,,

cr, 1 — / 12 a.,, i — i }., — i r,)

2m»-,. 2/M-,., 2/n-;, 1 ...

-- Ao-., A?',., — ----; ACT, Ar,., — — - - Ao"! Ao-., ^ . (XL).

o-2 (1 - rl,) ff, (1 - r|,) cr^^l - rla)

Now several important conclusions follow at once from this result :
(a) For the case of normal frequency (but in general only for this case) the errors
in the means are uncorrelated with the errors in the variations and correlations.
The error correlation surface breaks up into two parts, of which the first part we
have written down involves only the means, and would coincide exactly with the cor-
relation surface (ix.), with which we started, if we write in (ix.) A/i, for xlt A/;, for
x.,, and o-i/^/n, cr.>/</n, for o-j and cr, respectively.

It follows accordingly that the standard deviations of the errors in the means are

....... (xii.),

and the correlation of the errors made in the means is

240 PROFESSOR K. PEARSON AND MR. L. N. G. PILON

Further, the standard deviations for errors in ht when the error in h2 is known, or
for errors in h» when the error in A, is known, are respectively

The results (xii.) are, of course, well known ; the results (xiii.) and (xiv.) are, we
believe, novel and important. An illustration may be of service.

Suppose the stature and arm-length of a population to be under consideration.
Let us suppose the mean stature of the population known from a great number of
observations. Now let the arm-lengths be determined for a random selection of the
population ; then, if the stature of these individuals so selected differs A^ in excess
from the mean stature of the whole population, the arm-length of the random
selection will most probably differ from that of the whole population by A/;..,, a
quantity fixed by (xiii.), or rather by the coefficient of regression

Ro^X/ij/S/ij) i.e., ?'i2cr2/crj.
Thus

A/io = r^o-o/o-^A/i,,

with a probable error of '67449 X ~ (1 — r",.,).

In other words, if the arm-length is to be found from a selection of the general
population only, and the stature of this selection differs from that of the general
population, it is most reasonable to take the arm-length of the general population to
be the mean arm-length of the selected population less the quantity ?-,,cro/cr1.A/i1.

Or, again, if a selection from a general population show a mean organ &Ji1 in excess
of that of the general population, the whole system of correlated organs will exhibit

changes of which the magnitudes are most probably given by the type - -- A/;,.

The bearing of this on what we have termed random evolution will be obvious.

(/3) Turning to the second part of the error correlation surface, we note at once
that, if two organs be correlated, random selection will give a system of correlated
deviations in their variations and their correlation.

Random selection (and a fortiori it may be added,, artificial or natural selection),
which alters an organ's variability, alters the variability and correlation of all other
organs. In fact, when it is once realised how two random selections from a general
population will as a rule have organs of different means, of different variabilities, and
of different correlations, the means among themselves and the variabilities and corre-
lations among themselves forming systematic groups, it becomes obvious how any
assumption of the coefficient of correlation as a constant for local races runs wide of
the mark ; and this, whether natural or random evolution, is to be looked upon as
the source of the observed differences in character. What chiefly concerns the
biologist in this matter at present is this, that even a random selection of one organ

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 241

will produce changes in all other organic characters, which, if small, are still sensible
and capable of quantitative expression.*

(6) Returning now to the algebra of our investigation, \ve have to discuss the
second part of (xi.) by aid of the formulae given in (viii.), p. 237.

We require, in the first place; to evaluate the determinant A.

Now

A =

n 2 — .)•?.,

»»

«»•,,

<r; 1 — /•;.,

" 0-^(1 -»14) '

<r,(t-4)

M>'l2

/< 2 — j'js

7M'1S

"Wl-^i)'

« 1 - '7, '

T, (1 - 4)

nrn

nrt.

//.(! + »'i.,)

" <r, (1 - 4) '

o-.,(l - 4)'

(1- - w

Divide the first row by - ~T\> ^ne second row by
• j- , the first column by cr,, and the second by a-.,. Hence

, the third row by

Subtract the second column from the first, and then add the first row to the
second ; we have

A =

2

r,

1 + •>*,

The minors are now easily found to be

2n-

Au =

4n

2»*»'.

Ao, =

Aoj

* Take (xvii.) below, for example; it expresses for the first time quantitatively the important
biological principle that, if a group be selected at random from the general population, and it has more
variability in one character, it will be more variable than the general population in all other characters.
VOL. CXCI. — A. 2 I

242 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

From these expressions the values of the standard deviations and error correlations
are at once found by (viii.).
We have

v ~~ffl * "*-§: (xv-).

\/2n '
1 - rf»

E' I- T> ' 1* / ••• \

'-.'•., = 7£. K"-'.. = ^ ....... (XVIII.).

The result (xv.) is, of course, old ; the results (xvi.) to (xviii.) are novel, and lead
to interesting conclusions, which are considered in the following paragraphs.

(a) The probable error of a coefficient of correlation ?-,, is '67449(1 — r^)/^/n. If,
therefore, the correlation between two organs is less than once to twice '67449/^/w,
they cannot be safely assumed to be correlated at all.

(ft) If we know definitely the errors made in o-, and a-., — if, for example, we know
the variations accurately — then the probable error of 1\., is that of an array of rt.2's
for definite ACT, and ACT,. It is given at once by the coefficient of (Ar,2)2 in (xi.) as

•G7449(l --<'?,)/x/Ml +/•?,)} ....... (six.).

This is the value given for the probable error of rn by one of the present authors
in a former paper.* At that time he had not fully realised the importance of the
principle of tbe correlation of errors made in determining the magnitude of
frequency constants.

The following table will enable the reader to appreciate the difference in magnitude
between the •' absolute " and " partial " probable errors of r,2 :

0 1 1

•1 -99 -985

•2 -96 -94

•3 -91 -87

•4 -84 -78

•5 -75 -67

•G -64 -55

7 -51 -42

•8 -36 -28

•9 -19 -14

1 0 0

"' Heredity, Regression, and Panmixia," ' Phil. Trans.,' A, vol. 187, p. 266.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 243

It will thus be seen that when r,2 is small the absolute and partial probable errors
are both large and nearly equal ; that when rl2 is large the absolute and partial
probable errors differ more widely, but, as both are small in this case, their difference
is not of much importance. Bearing these facts in mind, it will be found that the
reasoning based on the partial error in previous memoirs remains valid, even if the
partial error be replaced, as it generally should be, by the somewhat larger absolute

error.*

(•/) If we know definitely the variability of any organ, and we take a definite
group of the general population to find the variability of a second correlated organ,
then there will be correlation between the deviation of the variation of this group
with regard to the first organ from the variation of the first organ in the general
population and the deviation of the variation of the group with regard to its second
organ from that of the general population's variation for the second organ. This
correlation is measured by r?2, and thus, if the organs be slightly correlated, it is
small ; but if the organs be closely correlated, it is large. Suppose, for example, we
know the variability of the tibia, and require to find that of the ulna from a com-
paratively few specimens. Let cr, -}- Ao^ be the variability of the tibia in the
specimens for which the ulna can be measured, and »*,» the correlation observed
between the ulna and tibia in these specimens ; then, the variability of the ulna being
observed as cr/ = cr2 -f- ACT,, the most probable variability of the ulna in the general
population is

a-.,' — /''f^Ao-i/X,
or

/ 4°"i .

cr, — - Ao-,,
<TI

or, since the second term is small, we may write cr./ for cr2, and the above expression

equals

,/', ., A

= cr, 1 — r]., -

'

For the long bones, r,., = '9 roughly, and therefore we have the ratio of variability of
the ulna in the general population to the variability observed in the group
= 1 — -8 Ao-j/cr,.

It is clear that this expression also measures the change in the variability of the
ulna due to a random selection of tibia.

(8) Although the correlation between deviations in the variability of two organs from
their mean variabilities only varies as the square of their correlation, the correlation
between the deviations in the variability of an organ and in its correlation with a second
organ varies as the first power of the correlation of the two organs. In other words,

* See, for example, the reasoning as to the non-constancy of the correlation coefficient for local races
in ' Phil. Trans.,' A, vol. 187, pp. 267 and 278.

2 I 2

244 PROFESSOR K. PEARSON AND MR. L. N. G. FILQN

while a selection of variability may produce only a small or moderate change on the
variability of correlated organs, a selection of correlation or a selection of variability
is likely to produce considerable changes on variability and correlation respectively.
Let tr1; cr2, rn, be the mean values of the standard deviations and the coefficient of
correlation for any three organs ; let cr, -f- ACT,, o-2 -f* Ao-2, r,2 + A>'J2, be the like
quantities for a group selected at random. Then the principle of regression tells us
that most probably

Substituting the values given by (xv.) to (xviii.), we find

''a i

Ao-, .= £0-, 1 ---- ; Ar,,

.

'

Now these equations lend us to some important conclusions. In the first place, if
the correlation be very small or very large, then a random selection of variability (Ao-,)
makes only a small change in correlation (Ar,2). The change in correlation for a
selection of variability is greatest when rt., — l/\/3, and then is approximately
•385 Ao-i/cr,, or over 6 per cent., if Ao-,/0-, were as high as -^j. On the other hand, the
change in variability (Ao-,) due to a selection (A'/1,.,) of correlation is small if the
correlation be small, but increases rapidly if the correlation become nearly perfect.
Of course, for perfect correlation the probable error of rl2 is zero, and accordingly it
is infinitely improbable that a selection can be made with Ar,2 differing from zero.
But if rv, be not unity, then a selection in which Ar12 is large, however improbable,
will give very large changes in the variability, if r,2 be very large. Our conclusion is
accordingly that considerable changes in variability are likely to be produced whenever
there is a correlation selection among highly correlated organs.

(7) To find the Probable Error of the Regression Coefficient for Two Organs.
The regression coefficient />, is given by

»

Pi = rna-i/crz.
Its standard deviation SP| is given by the summation equation

MATHEMATICAL CONTRIBUTIONS TO THE T&EORY OF EVOLUTION. 245

To find its value we adopt a method, which we give on this first occasion at
length, as it will be frequently used in the sequel.
Take logarithmic differentials

Ap, A/-,, AO-, A<TO

7T! ' ^' '~<rT ~^'

Square and divide by n after summing

S (Ap,y _ SjA^ S (Ao-,)2 S (Ao-,)- 28 (A/-u Ao-,) 2S(ArliAo-s) -'S (Ao-, Ao-.,)
w.s 7M,2^ 7l£r2 1ta-i nr.cr nr

Now, remembering the definitions of standard deviations and coefficients of
correlation, this may be written

- -

^iL 4. i^iL _ £f*i*2i£5D?4 a. £±rs!±!sc:

" °1 O'lO'a ''l2°"l

Now all the quantities on the right have already been found in equations
(xv.) to (xviii.). Hence, substituting, we have

pi

I fence

Thus the probable error of a regression coefficient

- '67449 *

This is of fundamental importance for testing the significance of results obtained
by applying the theory of regression to problems in heredity, panmixia, &c.

The probable percentage error in a regression coefficient = — ~ — , and

Vn M2

hence is small if the correlation be close, and increases rapidly if the correlation be
small. This again illustrates the point to which reference has been, made in another
memoir,* namely, that when only a few individuals can be measured, the most
reliable results for the purposes of the quantitative theory of evolution are to be
found from the measurements of the most highly correlated organs.

* PEARSON and LEE : " Con-elation in Civilised and Uncivilised Races," ' Roy. Soc. Proc.,' vol. 61,
p. 345.

246 PROFESSOR K. PEARSON AND MR. L. N. G. F1LON

Attention should be drawn to the fact that we have replaced errors by differentials.
This is only legitimate so long as product terms in the errors are negligible as com-
pared with linear terms. This is the assumption almost universally made by writers
on the theory of errors.* It will not lead us astray, so long as we take care in any
practical applications to verify the smallness of Ar,,, ACT,, ACT,, as compared with
r,2, tr,, and <r., respectively.

(8) To find the Probable Errors and Error Correlations of the Constants of a
Normal Frequency Distribution for Three Organs or more.

It will scarcely have escaped the attentive reader that our investigation hitherto,
only involving two organs, has left several important problems untouched. For
example, it has dealt only with the direct effect of random selection. But we may
ask such a question as this : Wliat is the change in the correlation of two organs
when the variability of a third is randomly selected ? Or again : What is the
change in the correlation of two organs when the correlation between one of these
and a third, or between a third and a fourth, is randomly selected ? All these are
important problems in the theory of evolution.

The general equation to a normal frequency-surface for m organs is : —

where H is the determinant

1 ?Y> 'Y; •

rai 1 rn . . .

r,, r,, 1 ...

and Rsj, is the minor of the term in the sih row and s'th column.
We require first to find the quantities like (vii.) of our Art. (4).

log * = log (nj(^r -) k log II - S, (log „.) - $8, - S,,,

._. s

Bo? (ll w)\

* GAUSS, admittedly, ' Theoria CombinatioDis Observatiounm ' .... p. 53, Problema; LAPLAOK and
POISSON, actually but obscurely ; see ' Theorie analytique des Probabilites, Liv. II., chap. IV., and
' Recherches sur la Probabilite des Jugements,' chap. IV. ; more cleai-ly in TODHUNTER'S account,
'History of Theory of Probability,' Art. 1,002 et seq. Further, CROFTOS, Article 'Probability,' § 48,
for a like assumption.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 247
d(logz) _ (logR) d /E^\ d

~

Differentiating the first of these again with regard to o-,, and summing for all
possible values of x's, we find

«ll =

O", \ It ]'

But
Hence

•:1 /I! -L I> V

(xxiv.).

Differentiating (xxii.) with regard to cr_, and summing, we have at once

log 2) , /' r,.,!!,.,

< ix, doc., . . . dx = «•,, = — - . . . (xxv.).

r, ((&., <7,o-., 1 1

Differentiating (xxii.) with regard to i\., and summing, we have

rrr rf*(logs) , 7 , // [ // /R \ '/ /i{,/\

• • • z T^fc" (/Xl f/;''- • ' ' t/a!»' = ]Cl- = 7" P + k"''1- JT PF /

JJJ na'lnrt., &l ["/V> \, K / "* ii \ " '

A

o-, l^A K

or

,cr_, = — nl^o/o-JI (xxvi.).

Differentiating (xxii.) with regard to r.fi and summing, we find

n d E,, n

__
^ dra H ' <r, rfr. i \ E

»i rf /E.. + S. («,.»•,.)'

... \ E

or

,ca = 0 (xxvii.).

Our next step is to differentiate (xxiii.) with regard to r,2 and sum. We have

248 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

n d*(logR)

~

/R\ rf /E12N

" E + * " •"'

Now, since R = Rn + Ss (Risr,,),

mR = S

and therefore

Sa,(R.,

Substituting, we find

<i ,, i '/i; '/];

But — (log R) = : - -- and 2R12 == ^ ; hence

Thus finally

where R12, rj is the second minor, found by striking out from R the first and second
rows and columns.

In the next place let us differentiate (xxiii.) with regard to r,3 and sum. We find
in precisely similar manner

fff .. <P lpg s

j . . . ,<, ^

«•.,._. </i\

or

/i)ff T> T> T> \

7 (Jxv10ivi3 — ltl\12 i.) / • \

1.,b13= —n- - (xxix.).

Similarly, we deduce

2E12E3t -

7, _ „ ' _ _ „ i

2t>34 _ n — I —

^T \ E I ~ R2

This completes all the possible types.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 249

We are now in a position to write down from (vii.) of Art. (4) the complete
frequency surface for errors in the constants of a normal frequency surface for
•m organs. It will suffice to write a type of each term. We have

n fl! + RufAo-,)* 2R,«r,, A«r, A<r4
PA = P0 X exponent - c \ --=- -f- H -- £-

°~\ ffi<rt

(A,B)S + . . . + ? AS 4,,s + . . .

+ 2 =***,.„„ + . . . + 2 s-. a,.*., + . . .(xxxi.).

Now this result again seems at once to give conclusions of considerable importance.
Thus :—

(a) Since there are no terms of the type ACT, A/V,, we infer (i.) that the random
selection of variation in one organ will most likely only vary the correlation between
two other organs by terms of the second order ; and that (ii.) the random selection of
correlation between two organs will in all probability only change the variability of
a third organ by terms of the second order.

(/3) The selection of correlation- between any two organs will most probable vary
the correlation between a second pair, i.e., terms exist in Ar,., A;;t, &c.

(y) The selection of the variation for any organ varies the correlation between
that organ and a third organ, and vice versd the selection of correlation between two
organs changes the variability of both organs. And lastly

(8) The selection of correlation between two organs varies the correlation between
either organ and any other organs.

We may exhibit these results more clearly by taking four special organs, say,
femur, tibia, humerus, and radius. Then a group having the variability of its femur
different from that of the general population, will also have, in all probability, the
variability in its tibia, humerus, and radius different ; the correlations femur-tibia,
femur-humerus, and femur-radius different; but those of tibia-humerus, humerus-
radius, and radius-tibia only slightly different. Further, a group having the
correlation of its femur-tibia different from that of the general population, will also
have all the other correlations, humerus-radius, femur-humerus, femur-radius, tibia-
humerus, tibia-radius, different from the values for the general population. Further,
the variability in femur and tibia will be changed ; but in all likelihood the variability
in humerus and radius only slightly changed.

These general conclusions, which seem to cast considerable light on the manner
in which selection influences the variability and correlation of organs, must now be
reduced to quantitative expression.

VOL. CXCI. — A. 2 K

250

PROFESSOR K. PEARSON AND MR. L. N. G. PILON

3

S

5>

O

O

o o

Pf

P/ft

-e

I I I

"^ pffpT JjpT

i i i

s pfipT pfR

*

i i i i i

o
or I

„ 5^

4^

J J

! I

0_

8
"g

s

§

t-i w

O cl

a a°

>-i O

e*- 5

< — i O

cs «*5

e
«

^iW I'lp^

fl?

S I b

^j i — ( ^j

"•53

: j.
re

i i i i i

p

•

o tf»J o

I I I

ift iiS- ^_2

I I !

i pfjpq pf|w

1 -S

^i

I

pf!«'

sf «

•S

1

S

I I

Frt

P

o

§ P

-8 -2

•« -£i

D p
O i— i
O C^
*- >
" <K

"

w?1

W PH"

P?

W pf

b" p=l
¥

w

a

b

PH

P

* b

I I

p

W

fft

**

iW a
: b* PJ

: b §

JF

PH

\\ PI w" PI o"

! b § b

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 251

Divide each row by n, the first, second, third, and fourth rows by cr,, cr,, 0%, <r4
respectively, and the first, second, third, and fourth columns by cr,, cr,, cr:!, er4
respectively. Multiply the fifth column by rvi and subtract it from the first and
second ; the sixth column by r,:, and subtract it from the first and third ; the seventh
column by r,4 and subtract it from the first and fourth ; the eighth column by r.,3
and subtract it from the second and third ; the ninth column by r24 and subtract it
from the second and fourth ; the tenth column by r:J4 and subtract it from the third
and fourth. Remembering that

rx. = -tin T" ^'12^12 ~T ^*uj"ia ~T **i4-"Hi

R = R22 + r.,iR12 + r23R23 -f r24R24, &c.,
and that

R12 d /R13\ d /R,.;\ d

12 I;i ,,

' r»" I3 U

r12Rls + '/'13R13 + ?'HRU\ d /Ra

dr,, \~ ~^~

A » , ,

7l3 "- "

we find, writing cZ,2 for d/drK, &c., for brevity,

2Rn/R, 0, 0, 0, R)2/R, RI3/R, RH/R, 0, 0, 0,

0, 2R22/R, 0, 0, R12/R, 0, 0, R,3/R, R24/R, o,

0, 0, 2R33/R, 0, 0, R13/R, 0, R23/R, 0, R34/R,

0, 0, 0, 2RM/R, 0, 0, R14/R, 0, R21/R, Rs4/R,

/R.A , /R^\ , /R»\ , /R«\ , /R«\ , /Rw\ , /R«\ ., /E«\ r/ /Ry\ r/ /»«\

ME"/' 12U ) 12\"R ) 12U/' 12U/' I2U/' 12U/' 12U/' 12l R /' 12v « /'

E.A /R^\ /R^A /Eu\ /Rw\ /Rw\ , /Ru\ , /R^X , /E2A /E^X

' I3"' I3 '

/Rn\ /EaX /RsA /Ru\ /E12\ /Rw\ ; /Ru\ , /R«\ , /R«

"B ) Kv HU) MU / 14(^/' • -w j w 14U/' K B

/Rn\ /R^N /R^X /R44\ /R,A , /R,A /E,_A /R«\ , /R«\ , /B,,\

/' 23U} 23U/' 23U /' ^U/' 23U/ 23l R /' 23\ R /' \ K /' \ R /'

Rn\ /KaX , /R^X , /RM\ , /Bu\ , /R13\ , /B.A , /Bg\ ," /B«\ , /B»\

' 24' " ; " ' 24 ' ^4' M ' M ' E '

Bw\ , /E.A 7 /B«\ , /B^X , /B.A

AW^^

2 K 2

252 PROFESSOR K. PEARSON AND MB. L. N. G. FILON

Here the signs of the terms in the last six rows have been changed from minus to
plus.* Now multiply the first four rows by R12/R and add them to the fifth row, the
•first four rows by Ri3/R and add them to the sixth row, the first four rows bv RU/R
and add them to the seventh row, and so on. Then, remembering that

/H.A 2RUR12 1 <jBy

13 \ B / " Rs R dra '

R- ' R dra'

2R3tR,, _1_ dEn

and, further, that lijv does not contain r,,, so that <7(Rw,)/dr,,, = 0, we have, taking
a factor 1/R out of each row,

A=(_=4)^l 2RM, 0, 0, 0, R,,, R1:), RI4, 0, 0, 0,

0, 2R,,, 0, 0, R,,, 0, 0, RM, R,,, 0,

0, 0, 2R38, 0, 0, J113, 0, R,,, 0, R34,

0, 0, 0, 2R44, 0, 0, R14, 0, R24; R34,

0, 0, — > - -' 1 - — ! — » - — > - — ) - — >

arn arK ara «»•,. dr}, ara drv drn

'/RSS dEH (lEu dE13 dElt dEn dE

u> :7T~ ' u> TET ' ^7~ ' ~TT ' j ' ' ~7~:

drls dru dr13 dra
dR.,, dE.a dEK dRa

r/K.,., rllty r/K)2 dRa dEu dE.,..

0, — > ^— --- ' (), •— — » -- — - > > -

aru aru arlt dru drlt drlt

— > 0, 0, ~ > > : > I - — » - — J - — j;

o, P> o,

J J

lr^ dr.,t dr.^ dr*t
-— — > > 0, 0, ~ — " > ~ — '•" > ~ — > — -'- > > — — > (xxxin.).

The form of A is now clear for the case of any number of correlated organs."!"
(10) Case (i). Let us evaluate A and its minors for the case of three correlated
organs.

* The factor ( — l)to(»-i) must be introduced if we deal with p organs.

f We may reduce this determinant as follows to one of the 6th order. Divide the 1st, 2nd, 3rd, 4th,
5th, 6th 7th, 8tb, 9th, 10th columns by 2Bn, 2R2!, -21^, 2R«, BIS, B,,, BJ4, RU, BM, BM respectively.
After this division, subtract the first column from the 5th, 6th, and 7th. The determinant reduces to
one of the 9th order ; subtract the first column of this new determinant from its 4th, 7th, and 8th

MATHEMATICAL CONTRIBUTIONS TO THE THEORY" OF EVOLUTION. 253
In this case R = 1 — r|, — rj, — r?., + 2r.ar3lrK, and we have

1 - r?3, R33 = 1 — r\,,

whence we have

A_(rJ^A; 2 (!-<•»), 0. 0, »V»-»;«, rl4ra-»-lfc 0,

ttl RK 0> 2(l-ri,), 0, rwru-rM, 0, ,-31r,2 - ,-23,

0, 0, -(1 — >•!•>) 0, /Yz'-'s — r,3) r,,-;-,, — )•,.-,,

°. °> — -'V' — 1, ''••a, v-31,

°. — 2rig, 0, r,3, — 1, r,,,

-2r.,3) 0, 0, r.fl) r,2, - 1.

Divide the first three columns by 2 ; multiply the last row by ryi and subtract
from the first row ; the fifth row by r,3 and subtract from the second; the fourth row
by ra and subtract from the third ; we find

column, it again reduces by one degree. Repeat the process twice more, and after a slight rearrangement,
the fact that d (R>pp)idrfg = 0 being remembered, \vc have, if £„. = R«/V'RM/.Rr>l

A.IO *?*R Ti T? 1-?

d •* d i cl> i /"/ ,

57" °8'^" gjrlog£m JT ^ *'" ,77~ sf:" "

<i i '? , d ,

-r- log ft. , j- log ft, , log f,, ,

w™a «7 1S d) j-j

^— log f,., , _ log f,3 ,
aru rt?-]S

^lofffi., 4; log ft,,

rf ,
^logf!3>

rf ,

J^ log ft:. ,

the general run of terms being obvious.

In precisely similar manner the value of A for p organs can be written down, its degree being p Jess
than the form given in (xxxiii.). We have not succeeded in reducing A for the general case [since
writing this, Mr. ARTHUR BERRY, of King's College, Cambridge, has succeeded in reducing the deter-
minant for p = 4, and also in showing its relation to elliptic space], but we feel fairly confident that its

value will be found to be -

—

0(0} . . . <rf

254 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

871" (-1)' 1

1.

0,

0, - r)2, - r,3

A 'r^ """

ffiajo? R

o,

1,

Q r ^ y.

o,

o,

1 r r

o,

o,

— r}.2, —I, r,.,,

o,

- »-,„

0, ra, - 1,

— r.a,

o,

0, »•„, r,2,

r23»

— r.

'-.I!

F3I>

Add the first column multiplied by r,2 to the fourth ; add the first multiplied by
?*13 to the fifth ; and subtract the first multiplied by r.,3 from the sixth, the determinant
will reduce to the minor of the first row and column. Continuing this process twice
more, we ultimately deduce

o-?o|o-f K6

J*1IJ ~~" -"12» -"13

-tvl2) -t*-22j ~~

-*-M3» -"23> ~"

in"

We will now proceed to calculate such of the minors of A as will give us results
beyond those obtained for two correlated organs.*

We require the correlation of cr, and r.,3, and of rn and r,:j.

Taking a-l and r2s we must strike out in (xxxii.), as we have taken only three
organs, the 4th, 7th, 9th, and 10th rows and columns straight oflf, and for the required
minor the 1st row and the 8th column. We have then for the minor M (or,, r23),

ra"

r]2R12

R + R22

»«ESS

Eu

o,

Txdyr\

R

R

R

R '

7'lSRIS

j-jgEa

R + EB

R1S

R

R

R

,

R '

K]2

R12

rf /R,,\

d /KB\

R

R

(

^7'12 \ R /

rfrMU/'

R"S)

o,

EM
¥'

^ /EW\

AfKjfV

rfrls V R / '

0,

E,,

^ /Rl8\

d /E1S\

R

R

rfj-^ \ R / '

rf^ \ R /

To reduce this expression take the third row multiplied by r,2 from the first, and
-the fifth row multiplied by r,3 from the first. Then take the fourth row multiplied
by 7*13 from the second, and the fifth row multiplied by r23 from the second, then
remembering that

As a matter of fact all the minors were worked out and the results of (xv.) to (xviii.) thus verified.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 255

and that generally,

We find :

M (o-1; r*) = -

R =

R =

<* (R,

?l5

o,
o,

-It JO

2K22

o,

d /Ra\ ^ /JW\

R '

o,

R '

o,
¥ '

rf»-is \R/ ' " rfrls \R/ '

tt / -tV;j<{ V tv / 1 l-j;;; \

R '

o,
IT'

rZ?* \ I"L / ^t?* V Ti /

R '

rf /RB\ rf /R,,\

R '

o,

• drn U / ' " rfru I It ,' '

d /R12\ fZ /R13\

R '

• dr,U/' ^«U/'

Multiply the first two columns by Er>/R and subtract their sum from the fourth ;
multiply by the second two columns by R13/R and subtract their sum from the fifth ;
divide out by the factor 1/R;\ We obtain

mr< 1

0, 2R22, 0,

0, 0, 2R33!

Rl2, Rl2, 0) ~T|

0, R23, R.VJ, -y;

Now remembering that

R = 1 — rit — iii — r'f, + 2?-23r31r,,,

substitute the various terms, and we find :

0, 1 - r?3,

dE*.

dra ' dra '

dE,,
dr,.

M (o-!, r23) = —

Rr>

0, 0,

~T **2S^*13i '*12 +

_j_ r&rl2, 0,

0, — ra +

0,
0,

0, - r,;
- »'„, 0,

256

PROFESSOR K. PEARSON AND MR. L. X. G. FILOX

Subtract r,3 times the fifth from the second, and rtt times the fourth from the third
column :

M (o-,, r,:i) = -

0,
0,

— »*12 +

— ris +
o,

i, o,

0, 1,

— n«, n-j,

o, - m

— r,2, 0

— 1, ra

n,, — 1

Add r13 times the second column to the fifth and r,, times the third to the fourth,

we have

1,

0,

0,
0,

0,

K»,

R1:i) R,,,

0,
1,

0,
0,

0
0

-R22

<r,o-.>ff;

(xxxv.).

Now we have seen that

^ = ^^ by (viii.) of Art. (4),

and further 2,, and S^ are given by (xv.) and (xvi.) of Art. (6). Hence

/ ;\

In the next place we will determine R,.liriJ- The minor M (r12, ?-,3) is given by

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 257

' /r r \ — —

\? !.'> nji —

n"

R+B,,
E '

'•|*EIS

'•I»EW

Bu

E '

o,

*?«&

E '

E '

T) »

E '

E '

,

E '

E '
By

E '
Eg

It '

It- '
d /E,,\

E '
(1 /Ea,\

It '

E '

'

••^AB"/1

" <77'~ (^ E / '

0,

E*

EM

d /Els\

rf /EjA

It '

K '

''>:,,. ( E ) '

(/>'... \ 1! / '

trjcrfffpl5

R+RU, »-,2R,...,

r,2Ria> R+R22,

r T? i- T?

' 13*->'13> '28-"28l

0, R,3,

o,

o, ^~

2E1.,!lt,
E

0,

E

2E|
E

Add the first three rows together, multiply by R,2/R and subtract from the fourth,
and by R..,.,/R and subtract from the fifth, we find

M (r,,, r,,) = —

R + Ru,

R + Roo,

— R12,

o,

0,

TV,,,

RM,

Multiply the fourth column by ?',3, and the fifth by r.,3, and subtract from the third
column

M (r12) ria) = -

+ RH. ^3i2,

,.Rls , R + R,2 ,

0,

0,

Rl3

o
R«

-2R.,3, -R,3, 0,

Multiply the third row by r]2, and the fourth by 1 — r;2 or R33, and subtract the
latter ; the determinant now reduces to one of the fourth order, and we find : —
VOL. cxci. — A, 2 L

258

M (r,2, r,3) = —

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

R + R,,, r12R12, R13> 0

r12R]2 , R + R22 , 0, R23

R , — r i

Add the third column, multiplied by r,3 to, and subtract the fourth multiplied by
•.,;, from, the second, and then subtract the first

n v __ __ _
<ri«rl<r«R5

R + Rn , - 2Tln ,

igBja + ^'is

— 2l\,o3 ,

0,

R,,3 , R,:,

Divide out the 2 in the second column, add it to the first, and subtract the third
column multiplied by r13

4jj5 j> ?-R. — R R 0 I

M (?>, r13) = — -- , „-,. !

: R - ?',3R,:i , R,2 , 0, R23

?-12R;i:! , 0, R2:i , R,3

— rv>:j , ?Vj . — ?*|2 ,

Add r.,;, times the last row to the first

M (r,,, ?•„) =

i> o .•) -r > -

crjcrjcrjli-'

133>

R -

&aa> R22> 0, R2;

r ^R 0, Ro. R ,

- "R • r - r 1

-"23? ' 23» ' 12i

Subtract r23 times the first row from the last, and remember that — R23 — ?'23R33=r,2R

1s,

R ?*23R23» R22, 0, R23

»*»Riii o» R23) RIS

r,2R13, 0, R12, Ru

Multiply the first row by R22 and add it to the second ; the determinant reduces
to the third order,

M (>',!>, ra) = —

R — ?'23R

2323

13R12

R23J

R,2,

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 259
Expanding this determinant from its first column, remembering that R,3Rn— R13R12

M

«
ri,, r13) = -- , 8 ,T>5 { — r.an (R — r23R23 -f R22Ra3) + »Via (R,,Rn — RfzJ

WJWJWJ-MP

Or, since R22Rn — R'^ = R, R12RJ3 — R^R^ = r,3R, and R — r,3R,3 +
= 2R,2R33 — i\.,r^ R,3,

M (rla> ria) = -7^7, {2A3R,,R33 - rBru (RM + v,all2a + r^U,,)],

o jtr.)CTyit

__ §!^L- {,. li T? _ £i^j? -
- fffa|ailll I " M 2

Hence, since

T> _ ^-....V

Lt"*"> ~ AS,2S,3 '
we have by (xvi.) of Art. (6) and (xxxiv.) of Art. ((J)

TJ - ,, . 1 ,. ,,

«,= 'i

/V'/'i.., (1 — '4; — ''Is — ''i- + 2/wr,.,)

- -- -

To complete the theory for the errors made in an investigation of the constants
for a system of three correlated organs, we require to determine the probable error of
a regression coefficient for a partial regression of a first organ on a second, the third
organ being constant. This coefficient is given by

Take logarithmic differentials

S«pig _ ____ 8»js_ raSra ^ L^~Jjs __ • J^

sPw " ~ ''is — Vis ra - r*ri3 !3 Vis - '^ri3 1 - rl

Let this be squared and divided by n, and then the values found above for the
standard deviations of the errors in rn, rn, r.,3, o-j and <r,, and for the correlations of
errors in these quantities be substituted. After some lengthy algebraic reductions,
which it seems unnecessary to reproduce, there results

2 L 2

2GO
or

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

1:! .

- .... (xxxvm.).

The percentage probable error in a partial coefficient of regression is accordingly

67-449

Before discussing the significance of these quantitative results for three organs, it
seems desirable to complete the general case by investigating the con-elation between
the errors made in the correlation coefficient* of a first pair of organs and a second
different pair of organs.

(11.) Case (ii.). Case of Four or more Organs. — In the case of four or more organs
the only new probable error will be that of a partial regression coefficient, but this
can theoretically always be found by the method of the preceding paragraph, provided
we know all the error correlations. The only novel correlation among the errors
will be that of r^r^ and this we shall now proceed to investigate. The discovery of
an error correlation coefficient of this type completes the theory of the errors of
normal frequency constants.

Instead of evaluating A of (xxxiii.), which in the case of four organs appears to be
very laborious, we may proceed as follows :—

If A be written in the form

and M denote the minor of the corresponding a in A, we have by a well known
property of the determinant,

Divide by A,

a M('-.a. °-i) 4. a Miriam 4. a MiDiLD.) + . . + a ,. M(r'2' rsi) = 0.

"^•"i A ' "<f-i A ' "•?* A

MATHEMATICAL CONTRIBUTIONS TO- THE THEORY OP EVOLUTION. 261

Now — b*sA t —p^- , M&e^ , . . . —<!!*'•<»> , are the correlations between the errors
AS^ AS^S,, A2JS, A2.,U2,J4

made in the various quantities o-j, cr2, o-3, c7-4, rJ2, r,3, ru, r23, r24, r3,, every one of which
is known by the previous investigations except that of r,2 and rM or M(,.ui ...^/AS,.^,^.
Hence the above equation will suffice to find the latter quantity, since 2,.u and S,.M are
known. We have

1

°~.=if('+!).j

n RI^'H 7/. Ii2,n,

T> > "'<T4<7. ~~ T> J

rj(7j J.X CTjiT.) It I

by (xxiv.) and (xxv.),

Further,

M(,v, „.,) _
-~ —

^i^> = ji r,, (1 - rr,, by (xvi.) and (xviii.),

by (xxxvi.),

- ^ {*•« (»'„ - r12rS3) + r. (ra - rBru)}

a j-

+ rM (»•« - r^u,

J

r;4) - ir^K^},

). by (xxxvii.),

/^,) - £*V«1U, J
- r'34), by (xvi.).

— J Iv

Now substitute and divide out by common factors, and we find

(v^Kii + 'VJU rlz(l - ?•?,) + rullu {rw(ra - r,,r,,) + ra(i:a - /•„»',.
+ (R + E44) {r,4(r,4 - rMr,2) + ru('/.>4 - r,,ru)}
+ 2K14 {»'M(1 - r?2)(l - 7-i4) - J rI2ruR33}
4- 2K24 {»•„(! - r?2) (1 — ri) — -^r^r^R^}

262 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

Now

R = R44 4- ^i4Ri4 + »'24Rz4 4- ruRu,

0 — <i« T? I T? I M T? I M "R
' I4"4» "I -"14 T ' 12"24 T^ ' al-t^Sl)

0 = r24R44 + r12R14 4- R24 4- r32R34.

Multiply the third of these by r,4 and the second by r24 and add, we have
0 = 2r,4rS4R44 4- r,4R14 4- rl4R,t 4- r,8 (r,4R14 4- ?-,4R24) 4- (»•„»•« 4- r,3r,4) R,4.

Hence, by the first,

• T i , T> .1) /,. o,, „, \ TJ,, i 1) />f ,, ,, ., „, ,, \

' 24"14 "T" 'H-"M ^I44\'l^ ~~ "'IVyttj ~~ •"'1J T ^134 \' 12' 34 ~~ ' 13' 24 ~~ ' 23' 14/J

while

^'l4J-XI4 ~T ''34"24 -— ^v "4* ?'34il34-

By means of these relations, let us get rid of the terms in R14 and R24 in (xxxix.)
above.

Re-arranging we have, after some reductions,

— 2r,o(l — 4,) R + 2r,,R33R44 4- R34 (r34r,3 (r13 — ?',,7'23) 4- v34r13(?-23 — 7'12?'13)

4- 91} 1} /"I - . r" Wl - . r2 ^ — 0

-(- ^ Lt34lV,.u,.3t ^ I '12/V1 '3t) ~ ' V.

But

Hence we can divide out by R34, and accordingly,

= — {wM(rn — rur.a) + r34r13(/-23 — >-,,r13) + 2(1 — '/•;,)( ri,»-al — ?-I

+ 2» w« - »'i2» M (» M + H2) +- 2r,,R34] .
Noting that

Rat = — v34 (1 — 'T>) + r31»-41 + rs,rK — )',, (r31r4, + r3,r41),

we have, after substitution and rearranging,

2R,.13,31(1 — r'i,) (1 — ?-3-4) = (ra — r,,rM) (r,4 — r,3r34) + (r,4 — »V»)(r

+ (^13 — rltrM) (r,t — r,2r14) + (^"u — J'la*^) (^3 — rMrM).
Or,

| (''is - »'is»'ss) (»2i - »'2srsi) + (''u - rsi»'is) (''23 ~ rn'\s) 1
13 . I + fas - yi^'st) Q'it ~ ^I^H) + (>'n - VgQOa - nt^t)J

- ~ ~-~"

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 263

If we put 4=1 in this result and remember that r,, = 1, we find, after some
reductions,

T> i 1 ~ ?"Ti ~ 9*18 ~ ?'li + 2?V/B)?'a1

'''""" (l-ry(l-^3)

which 'agrees with (xxxvii.), and may be taken as a verification of this result.*

(12.) We may draw several conclusions from the results (xxxvi.), (xxxvii.), and (xl.).
(a.) While errors in the correlations of a first organ with a second and a third
have a correlation themselves of the first order, errors in the variation of a first
organ and the correlation of two others, or in the correlation of two organs and in
the correlation of a second two, have only correlation of the second order. Thus a
selection of the correlation between two organs modifies the variation of all organs
correlated with one or other or both of the first, but only in the second degree.
Again, a selection of the correlation between two organs modifies the correlation of
every other pair of organs, one or both of which are correlated with one or both of
the first pair ; but this is only in the second degree.

(j8.) If two organs be entirely uncorrelated a random selection of the variation of
a third organ correlated with both of them will tend to generate correlation between
the hitherto uncorrelated organs, i.e., put ?v, = 0 in (xxxvi.), and we have

If a variation Ao^ be made in crl the probable value of r,:; is

A<7,

An, -- * -

which may clearly be of sensible magnitude. Thus correlation may be generated
by selection of variation, and vice versa.

(y.) If two organs be each entirely uncorrelated with a third, yet a random
selection, which produces a correlation between one of these organs and the third, will
produce a correlation of the first order between the other of these organs and the
third, i.e., put r,2 = r13 = 0 in (xxxvii.), we have

a correlation of the first order between the probable changes.

(8.) Consider four organs of which the first is alone correlated with the third and
the second with the fourth, the third and fourth being themselves uncorrelated.
Then any random selection which produces a correlation between the first and

* The probable error of a partial regression coefficient for p organs has not been worked owing
to the labour involved, but judging by the cases on pp. 245 and 260, it may safely be taken as
67-449 v/H/^w (— R12), where R is now the determinant of they"1 degree.

264 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

second will tend to produce a correlation between the third and fourth, i.e., if
r,2 = r,4 = ?%, = r.,4 = 0, we still have from (xl.)

(e.) We may further illustrate these principles by one or two hypothetical examples
drawn from actual organs.

Let the actual organs be (1) physique of father, (V) artistic sense of mother,
(3) physique of offspring, (4) artistic sense of offspring. Suppose in the general
population there is no correlation between physique of father and artistic sense of
mother, or between physique or artistic sense of parent, and artistic sense and
physique respectively of offspring. Then rn = 0, rH = 0, r.,% = 0, and, presumably,
r34 = 0. Hence

Bv« = »V«

is the product of the two coefficients for inheritance of physique from father to
child, and for inheritance of artistic sense from mother to child.

Now let a random selection be made out of the general population in which assor-
tative mating between physique in the male and artistic sense in the female presents
itself, i.e., let Ar,._> be sensible ; then we have, most probably,

Ar:u — Pi,[3,3i -i Ar]2 = rnr,4 Ar,,,
*""u

or, a correlation between physique and artistic sense in the offspring will tend to be
developed. Generally, when r;!4 and rn do not start from zero, we have,

A ,- 'Vil + 2?W,t01.1 + 9'il) . „

(1 - rtf

or, any increase of sexual selection in a group tends to emphasise the correlation of
the selected qualities in the offspring.

Let the three characteristics be artistic sense (1) in a man, (2) in his mother, (3)

in his wife. Then

i »s _ ,3

*D 1 ~ ' I4* ' 13

if we suppose n3 to be zero.

Hence any selected group with a higher coefficient of maternal inheritance of
heredity will have a less coefficient of sexual selection than the general population,
and vice versd. The tendency is, of course, independent of the magnitude of r, and
really of the particular character. Supposing likeness of faculty or character to be
a rough measure of " sympathy," we might conclude for any population with inheri-
tance and sexual selection, that on the average a selected sub-group of men having
greater sympathy with mothers than the general population will have less sympathy
with wives, and vice versd.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 265

(13.) Many like propositions may be stated with regard to the action of selection
on the correlation of characters. They require but little modification to state them
for artificial or natural selection, as they are here stated for what we have termed
random selection. The above will, however, suffice to indicate how every form of
selection of variability or correlation influences in a manner capable of quantitative
expression the variability and correlation of all other directly and indirectly corre-
lated organs. Selection cannot be of service in altering one organ only, it alters at
the same time the whole inter-relationship of a complex of organs. Evolution by
natural selection can never be the change of one organ to suit a particular environ-
ment ; it is the balance of advantage and disadvantage produced by the change of
all organs involved in the attempt to select one of them. The moment the intimate
correlation of organs in animal or plant life has been fully realised — and this i-ealisation
owing to recent statistical investigations has become fairly easy — then the conception
of natural selection as moulding any single organ to what may be fittest to its sur-
roundings must be discarded. The selection of the " fittest " in one organ would
probably mean the selection of the unfit in other organs, and a general balance of
fitness in the complex of organ is all that is possible.*

IV. ON THE PROBABLE ERRORS AND THE COEFFICIENT.-; OF CORRELATION BETWEEN
ERRORS MADE IN THE DETERMINATION OF THE CONSTANTS IN THE CASE OF
SKEW VARIATION.

(14.) The case of Skew Variation has been dealt with at length by one of the
present authors in the second paper of this series. He has shown that in a great
variety of cases it can be dealt with by a series of curves having three principal
algebraical types, each defined by a certain number of constants. The probable
errors of the determination of these constants were not then investigated, but it is
clearly of great importance for the practical use of these curves to know how far
these constants can, for any given number of observations, be depended upon to give
an accurate measure of the skevvness and its special features. At the same time an
investigation of the probable errors of these constants leads us to a number of novel
properties which are connected with the theory of evolution in the frequent case of
skew variation.

* Take, for example, result (xxxvi.) ; as far as terms of the second order are concerned Rffli'.,s = \/2 . rl2rl3.
Hence, with positive correlation between three organs, the effect of trying to get a group very stable in
oue organ, i.e., with a negative Ao-j, is to reduce the correlation between every other pair of organs ! In
other words, we have to reduce variation at the expense of correlation, increased stability of one organ
is gained at the expense of decreased stability in the inter-relationship of other organs. This may
possibly be illustrated by the long bones of the French, where the lesser variability of the male relative
to the female connotes also a lesser correlation. See LEK and PEARSON : " On the Relative Variation
and Correlation in Civilised and Uncivilised Races," ' R. S. Proc.,' vol. 61, pp. 354-350,
VOL. CXCI. — A. 2 M

266 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

We shall deal first with the skew curve of Type III. (''Phil. Trans.' A, vol. 186,
p. 373), because its treatment is less complex and leads at once to some general
principles which must be borne in mind, whenever natural selection acts upon an
organ exhibiting skew variation.

(15.) Probable Errors and Error Correlations of the Constants of the Generalized
Probability Curve of Type y = yl (l -\- - 1— 1 e~yx .

This is the equation of the curve referred to its mean as origin, where

7C-OHl)(jj + jy>

Further, the moments about the centroid vertical are given by,

p_±l 2(p + l) 3(p + l)(p+ 3)

p.,= -^ , &= •-,-- --, /*4= —r-

or,

y = 2p,/^, p - W/W - l ....... (xliii.).

The criterion for the application of this curve to any frequency distribution is

2fu (3/v - p4) + 3fi,2 = 0,
or, if we write & = ju,4/ju./, ft = pf/pS,

6 - 2ft + 3/8, = 0 ........ (xliv.).

Lastly,

Sk. = the skewness = |^3/ (/A2)3/2 =

and the modal frequency,

We require to know the probable errors of p, y, yu y0, a- = \/p.>, p2, p.3, p4, and the
skewness. We must discover the best physical constants to describe such skew
frequencies and we shall at the same time succeed in deducing certain — we believe —
novel properties of normal frequency distributions as limiting cases of this skew type
of distribution.

* See ' Phi], Trans.,' A, vol. 186, pp. 373-4.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 267

(16.) The first stage in the investigation is to apply the general proposition of our
Art. 2, to

logy = log n+ logy- (p + l) + plog(p + 1) — logr(p -f 1)

We find :

if- (log//) i> 1

-- • ''-

... ,

^ <•? +

1

(1 + '''A')2/'

civ dp a \1 + ;>•/" 2' + 1

c]-(loS>,) __ P _L . t
da; (fry p + 1 (I 4- <•/")-

^y)=-^lo£r^+ ') + ,,+ 11^

fr-(iog//)_
" •

P. -\

i vi -•- / ^ __ -r_ ' ~ .__; i

Let I, = Vi ( 1 4- --Ve'^daj, then we easily find I , — — I , and

J J J-«\ «/ i"

(P +

^

By aid of these we can at once write down the integrals of the above expressions
multiplied by y, since n = I,,. We find with the notation of p. 243,

r, , .. r^ (los y) ^ _ "T'J

'ti

f
_,

tfxdp PCP-I)'

C (P (log y) j 2n

y ~-~ dx= — : ,

J _„ * rfa; f?7 p — 1

(V (log y) . Iff-, . . -2

-

a33 = —

-— r_»'

2 M 2

268 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

Before we consider the determinant and its minors, we may note that

log

i) logp -

where the B's are the BERNOULLI numbers. Hence

and we have the convenient form,

,., = n log r (p

'

p(p-

JP —

S

where S is the semi-convergent series B,/2> — B.(/p:t + B.7j»>r' — , &c.
Now we have at once,

, c
i"

+ 1 2 (/> + 1)

Divide all three columns by \j(p — 1) ; the first row and first column by y, and
then the last row and last column by 1/y ; we find,

A =

n''

I,
1

;•

P

-2,

0(0 , "T" ^ I i *

• P- \*p-i •/ p
p + i o f i v

P "^1J ''

Divide the second row and column by l/p ; add half the last row to the second,
and we find,

1, 1, -2

0, (p - 1) S, 0

- 2, - (p + 1), 2 (p + 1)

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 269

Subtract the first column from the second, and add the double of it to the last,
we deduce

A __ »* 1, 0, 0

P ^ " 0, (p — 1) S, 0

or,

A = ., --' - . (xlvii.).

The minors can now be written down at once.

M_J t // | -L / Ifj O
M = ^/,-^iy '

whence

, _ _ p±_} ...

or

2,, = cr/^/n . (xlviii.).

• M., = — '- ,
p — 1

whence

2;, •=• Mo., /A = p'/nB,
or

V _L . / _ ^vllY ^

2,t, — —^ A / Q ^AIIA.^.

v n v o

P (.P — ^ )
whence

2; = MM/A = ~ ( 1 + ,
tn \ ^o/

or

+ i) (i.).

\ *"/

M» = -^y ,

whence

Wtt + S)} ("•)•

M12 = 0,
whence

B*, = 0 (Hi.)-

270 PROFESSOR K. PEARSON AND MR, L. N. G. FILON

Lastly,

M1:i = ",~^T) >
whence

This completes the direct series of probable errors and error correlations. By aid
of the above correlations and standard deviations we can now find a further series.

From (xlii.) we have for the standard deviation cr (about the mean), &' = {*., =

/> + 1
, ' - — —

v
or cr = • Hence

A5 , A,, A7

o- "~r+ i 7 '

Square both sides of this, divide by n and sum, we have at once from the definition
of a coefficient of correlation

('v c- *?- v- P v v

-°- \ — i _ ^ _ 4. -v _ £*si±z*a .
*+lS 'J ' +1

70' +1)
Hence, using (xlix.), (1.), and (li.), we find, after reductions,

Multiply (liv.) by A/;, sum and divide by n, we have

^

o- " ! p + I

Whence, by (Hi.),

RA.T = ---

7 s»

or, reducing by (1.), (liii.), and (lv.),

Next, if S<; be the skewness, we have from (xlv.)

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 271
or

Sfc-*j+iv'»;7{(jTT)S) ...... <lvii-)-*

Similarly

S(ASA)
J*w = v v — = 0, since Rhj, = 0,

~St^»

or

RAS,, = 0 ........ . (Iviii.).

We easily obtain, by multiplying (liv.) by A/>, the result

We can now obtain R,ffS,, for every ASt is negatively proportional to the corre-
sponding Ap. Hence

= {1 + SQrTiys"}1 ........ (lx>)'

We next pass to the mean and modal frequencies as given by (xli.) and (xlvi.).
We have, by taking logarithmic differentials,

where

* If S' = -A^ - B3 + - B5 - - &c., then it is easy to show that i + / S' = (p + 1)' S.
Remembering that, if Si = a, q — — - , we easily deduce

J = + n«g r (jp + i)} - log (p + i).

' + f j« + (B5 + 2

+ 3(Sk.)'+j

as far the 7th power of the skewness inclusive.

Very generally the probable error of the skewness may be taken as equal to

\'2n! v/{l +3(Sk.)2}'
and it is always less than ^(3/2%), its value in tlte case of a normal frequency.

272 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

Squaring, introducing the standard deviations, and rearranging, we find

^i — -* ci — w ) 4- / J — ?*- ?s?Y s2 = — 1 1 4- — cj» — iv

yf ' " T2^ \ 2* 7 / ij) 2n [ S W t;

We must now evaluate Jp — ^. This is easily shown from the BERNOULLI
number expansion for log F (p + 1) to be given by

p

where

Thus we determine

Expanding the expi-ession in brackets in inverse powers of p we find

y, 49 28 248 1*

Besult (Ixi.), however, with S and T calculated to 1/p3, gives a better value than
(Ixii.).

To find the modal frequency error we must take the logarithmic differential
of (xlvi.) and proceed in the same way. We find almost at once

2/(l "7 P
Whence on squaring and completing the square of the factor of %>2, we find

v- if 2TS]
±2, _ J_ J ! j_ ±L I ,

tf 2n\ § J

and

Expanding as far as powers of 1/p8 exclusive we obtain

a very simple expression for the probable error of the modal frequency, y0.

MATHEMATICAL" CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 273

In like manner we shall now determine the probable errors of the moments and
their error correlations.

Take the logarithmic differentials of (Ixii.)

p + I 7

u,; A/J 3A7

~

Pa ~ P + 1 7

/i, _ 2 (p + 2) Aji _ 4A_7 .
~ y) 7

Squaring eacli of these in succession and using the known values of -.„ -y, H;,y
we find

4 I 4-

-"• " ^ 7(2.7) 1 h i

Now multiply A//,2//x.. by A/x,3//A3 and we find, after some reductions,

1 + j> + 3

iy (Ixviii.).

2

18S

Next multiply A^,//*., by A/^//^, and we ultimately have

1 +

_

4" "7T77 i _Vi

V \( f2S^TiyV\

Lastly, multiplying A^3//A3 by A/i4//A4, w^ deduce, after some reductions,

u . bEKjM */ ___ Qxx)

-tv.,.u. = ~ ~7^r^7 — '/„ , ".J\2 \~T /r/ /'9n 4- 3V \1 " " "

VOL. CXCI. — A.

274 PROFESSOR E. PEARSON AND MR. L. N. G. FILON

We may add to these results the values of 2^ and 2ft> where & and ft, are given
by (xliv.) ; we find

4 p ^ 6 p

' ' ' '

The distances from the mode to the mean, d, and from the mean to the end of the
range, a, are given by

d = 1/y and a = - •

7

Hence

is

and further

(Ixxiv.).

The results (xlvii.) to (Ixxiv.) must be now considered at length.

(17.) («.) The frequency curve of the type considered is fully described by the
three constants, the mean, y, and p. But, since any three constants would do
equally well — for example, what may be termed the three physical constants : mean,
standard deviation (or variation), and skewness — it becomes of some importance to
inquire which constants have the least percentage of probable error.

Now (xlviii.) shows us that the probable error in the mean is precisely the same
as in the case of the normal curve and

= >67449o-/v/n.
Thus, the percentage error in the mean

A i/n

•67449 .

= '—r— X coemcient ot variation,
V n

and will certainly be small whenever the coefficient of variation is small. Its value
is quite independent of the order of p.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 275

On the other hand, the percentage probable errors of p and y are from (xlix.)

and (1.)

1 67-449 // 1\ L. .

» and -J&) v (l + is) resPectively-

67449 1

Here S is equal to the series Bi/p — B3/p3 + B0/ps — ... which tends to zero as
p increases.

The errors in p and y thus tend to increase indefinitely as p increases. It may then
be asked how the form of the curve can be determined with any degree of accuracy.
The answer is simple : Equation (li.) shows us that the correlation between errors in
p and y tends, asp increases, to become " perfect," i.e., unity. But as p increases
indefinitely, it has been shown that the frequency curve of this type passes over into
the normal form.* It is the high correlation between errors in p and y which
renders the curve, when plotted to observations, such an excellent fit. If the errors
injp and y were independent, this would not be so. At the same time it renders
j) and y unsuitable for tabulation as physical or biological constants of the frequency.

Turning to (Iv.) and (Ivii.) we see that the standard deviation, cr, and the skew-
ness, Sk., are suitable constants for tabulation. Their probable errors do not tend to
increase indefinitely with p, and will always be small, if n be large.

Hence a frequency distribution of this type is best defined by its mean h, its
standard-deviation cr, and its skewness Sk. These are constants characteristic of
the group, for they are given with small probable errors. If it be desired to draw
the form of the frequency-curve, then its algebraic constants, p and y, may be found
from

and the possibly considerable errors in p and y will not vary largely its actual shape.

(/3.) The nature of the probable errors of the other allied constants may now be
considered. The mean and modal frequencies per unit variation of organ, or y^ and
y0 are seen by (Ixii.) and (Ixiv.) to have small percentage probable errors, and are,
therefore, good for use as characteristic physical or biological constants. But it
should be noted that the modal frequency is considerably more exact for moderate
values of p than the mean frequency. For example it would be somewhat better to
tabulate the modal than the mean frequency of the barometer as a physical charac-
teristic of climate.

The probable errors of the distances from the mean to the mode arid from the
mean to the terminal of the range are given by (Ixxii.) and (Ixxiii.). Since
d = \ly = cr/ */ (1 -f p), we may write the first

oV(d

+ p • 2 (1 + p)S

* ' Phil. Trans.,' A, vol. 186, p. 374.
2x2

276 PROFESSOR K. PEARSON AND MR, L. N. G. FILON

This remains finite, even if p be indefinitely great. On the other hand, the
probable error of a, and even its percentage probable error, becomes indefinitely great
with p. It is to be noted that a in this case becomes infinite.

(y.) Results (Ixv.) to (Ixvii.) give the probable errors of the second, third, and
fourth moments. It will be seen that roughly, for a large p, the percentage error of
the fourth moment is about double that of the second. It might thus appear, at
first sight, safer to work with the second than with the fourth, but this is by no
means necessarily the case, for to deduce any quantity from one or the other they
must be reduced to the same order. For example, the square root of /u,2 must be
compared with the fourth root of /*4. and the probable errors of ^/^ ar>d
be sensibly of the same order.

Remembering that fi3 = ~~T~ = ~77~ ~~T)> we may wr't-e

6g» // 1

*M,= 77IT> VU hl +
v(*11) » \p + j-

(P +

18$(p + I

This tends to a finite limit as p increases indefinitely, and we conclude that the
probable error of /i3 is always finite, and will in general be a small fraction of the
cube of the standard deviation. The above remarks are a justification for the use of
higher moments in frequency calculations.

Equations (Ixviii.) to (Ixx.) give the error-correlations between the first three
moments. They show that an error in the value of one of these moments will most
probably lead to an error in the other two. We see that for p fairly large R^4 is a
large correlation, while R^ and RMjMt are small. In other words a random selection
of an even moment makes a far larger correlated change in another even moment
than in an odd moment. If p increase indefinitely we find the ratio RM-1J / RWi
approaches the value 2/3 ; in other words, fj.3 is more closely correlated to the higher
moment /*4 than to the lower moment /i2.

Formulae (Ixxi.) give the probable errors of the useful constants j8, — /il//n* and
$, = p,t/p%. We see that they are small and approach the value zero as p is
indefinitely increased.

(8.) Let us restate the formulae for p indefinitely great, i.e., for the normal curve of
frequency

-'' S

In this case we have //,> = or, //3 = 0, //,4 = So-4, /?, = 0, /?2 = 3, skewness = 0,
mean and mode coincide. Several of these zero quantities, however, tend to have
definite probable errors.

We have

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 277

2, = <r/v/(2n),

// ?, \

= =

The first, second, fourth and fifth of these results are old ; the rost appear to bo
novel and of some importance.

In the first place we notice that given a population which is really normal, we
should not expect a random selection to exhibit all the signs of normality. Its

skevvness will differ from zero with a probable error of '67449 A/ (;j- ). For example,

in a random selection of 600 from a normal population, the skewness will be as likely
to exceed as to fall short of '034. Hence an exhibition of skewness of less than once

VI '•* \
( — ) must not in itself merely be taken to indicate an absence of
\2w/

normality in a general population.

Again, in a random selection from a general population, the mode will differ
from the mean, even if the population be normal, with a probable error of

•G7449 A/ (—-} o: Thus, in a population of 600, a difference between the mean and

the mode of '0340- should not be taken to indicate want of normality. Generally,
the divergence between mean and mode in a population must at least exceed once to

//3\

twice '67449 A/i^)0"* f°r us to be able to argue on this ground alone that the

population has not a normal distribution.

Again, the third moment not being zero, but having a value of once or twice '67449

X 2 A/ ( — j cr3, is not in itself an argument for skew frequency.

The above statements are an important addition to the second memoir of this
series ; they give us the criterion, there wanting, to distinguish between a skewness
which is characteristic of a population and one which might arise by the random
selection of a population of the given size out of a larger, but really normal, popula-
tion.

(e.) We may now note the exceedingly interesting conclusions which these results
have for the theory of evolution.

Suppose an organ to have, as so many do, skew variation, then we notice

278 PROFESSOR K. PEARSON AND MR. L. N. G. PILON

(i.) Any selection of the organ by size tends to alter its variability, but not its
skewness; this follows from (Ivi.) and (Iviii.). Further, if, as we have supposed, the
range be limited on the side of dwarf organs, then any increase of size means a
decrease of variability, and vice versd.

(ii.) Any selection of variability is a selection of skewness; this follows from (lx.).
If a selection be made from a general population, which has less variability, then it
will tend to greater normality. In other words, it would appear that stringent
selection tends to generate normal distribution. Thus, if out of a skewly distributed
population we make a number of random selections, that with the least variability
will be most normal. Select at random again out of this latter selection, and the
least variable group will again be the most normal, and so on.

Now take a problem of this kind involving group, and not individual, selection.
Let a large general population break itself up at random into groups, and let us
suppose these groups, not individuals among them, to carry on a struggle for
existence — an inter-group, not an intra -group, struggle. Then, if it be an advantage
to a group that its members shall be among themselves close to a type, i.e., less
variable, then the more normal groups will survive, for variability is positively
correlated with skewness. Now suppose each group to be periodically subdivided at
random into new groups — the mathematical description of some process of group
reproduction — then we see how normal distribution may be a result of a stringent
inter-group selection of groups whose individuals have the closest resemblance to
each other — intra-group resemblance.

(iii.) Any selection of the size of an organ produces by (Ixxiv.) an alteration in
the distances between the mean and the mode, and between the mean and the end
of the range.

A random selection which has its mean larger than that of the general population,
will, if the mode be on the dwarf side of the mean, tend to have its mode and mean
nearer together than are the mode and mean of the general population, while on
the other hand, to raise the mean is to raise the dwarf limit to the range.

A considerable number of like results might be stated, but the above will be
sufficient to emphasize the general principle that a random and d fortiori an
artificial selection of the size of an organ, does, whenever its distribution is skew,
influence in a definite manner the variability of the organ. It is quite safe to assert
that it will also influence the correlation of organs. When we notice how wide-spread
is skew variation in nature, we may assert that the general rule is that no modifica-
tion can be made in any of the features — mean sizes, variabilities and correlations —
of a group of organs without at the same time modifying all the others.*

* A paper has recently been published by Messrs. DAVENPORT and BULLARD in the ' Proceedings of the
American Academy of Science' (see Illustration II. below) on "The Variation and Correlation of the
Glands in the Legs of Swine." Unfortunately the authors have overlooked the markedly skew character

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 279

As a result of Articles (15) and (16), it is possible to use the frequency curve of
type y = y\ (1 + a5/ct)pfi~wl>with as much certainty as to the nature and magnitude
of the errors made in the constants as has hitherto been possible in the case of the

normal distribution y = - -.- e-"7'-*"'. The method has been exemplified numerically

in twenty-three cases in a memoir on the " Variation of Barometric Frequency " (see
'Phil. Trans./ A, vol. 190, p. 423). It may not, however, be amiss to illustrate it
further in a special case having closer bearings on the theory of evolution.

(18.) Numerical Illustration — Incidence of Enteric Fever.

In a memoir in the 'Phil. Trans.,' A, vol. 186, p. 391. it is shown that the curve
y = 1894-57 (l + 3.^0 J'^V--

closely represents the distribution with age of 8,689 cases of enteric fever received
into the Metropolitan Asylums Board Fever Hospitals. The unit of x is five years,
and the origin is the mode at 14 '3025 years. The criterion is not very nearly zero,
although small, but the curve is graphically a good fit (see Plate 12, fig. 9).
The following are the numerical values of all the constants : —

Mean = 18'9691 years. d = mean-mode.
Mode = 14-3025 years. = '933,313 unit.

Sk = skewness — '462,594.

7=1-071,453. /> = 3-673,042.

« = 3-428,094.

y0 = modal frequency = 1894*57.

yl = mean frequency = 1687'80.

o- = standard deviation = 2*01756 units = 10'087« years.

From these the numerical values of the probable errors and of the correlations
between the errors of the constants were found by the processes indicated and the
formulae given above.

We found

T = -022,525, S = -044,735,

whence

of the distribution. It is, however, clear from their tables and plate that no selection could be made of
the absolute number of glands without altering tire variability of the gland distribution and the
correlation between different systems of glands.

280 PROFESSOR K. PEARSON AND MR. L. N. G. PILON

Probable Error. Percentage Probable Error.

p '125659 3-4211

y -019130 1-7854

Correlation of errors in i> and y = '9581
a -061202 1-7853

.y,, 'J-8029 -5174

yl 23-5465 1-3951

mean = -014600 = '073 year,

mode = -024126 = '121 year.

These are the constants which determine the position and algebraical equation to the
frequency curve, and we see at once that they are all determined with a close degree
of accuracy. The largest percentage probable error is in p, but this is under
3 -5 per cent., and, owing to the high correlation between p and y a much larger
error would produce no sensible change in the shape of the curve.

Two important facts may also be drawn from these results, which indeed follow
from the general formulae, namely :

(i.) The position of the mean is sensibly more exactly determined than the position
of the mode. Here about 1'7 times as accurately.

k/

(ii.) The modal frequency, on the other hand, is sensibly more accurate than the
mean frequency. Here about 2-8 times as accurate.

Hence the advantage of using the mean as origin of measurement for the curve is
accompanied by the counterbalancing, and here relatively greater, disadvantage of
the increased inaccuracy of determination of the mean frequency.

Passing to the "physical" constants of the curve, we have

Probable Error. Percentage Probable Error.
a- -012693 "6291

Sk. -022815 1-3445

d -016663 17854

These fully determine the non-symmetrical nature and spread of the curve, and
since the errors in the skewuess and in the distance between the mean and mode are
less than 1-4 and 1*8 per cent, of the respective values of these quantities, we
conclude that skewness and divergence between mode and mean are characteristic
features of enteric lever distribution, and not mere anomalies due to a random
selection of cases. They are significant constants peculiar to each type of fever
distribution and no description of such a distribution is sufficient unless their values
are stated.

Before giving a table of the correlations between what we have termed the
" physical " constants, it may be well to write down some of the correlations between
the errors in the physical and algebraical constants, which arise in the course of their
calculation. We find

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 281

R,ilv = '1875, B,,, = - 1.

By aid of these we find the following table of error correlations : —

Mean.

2/,,.

a.

d.

Sk.

Mean.

1

•6469

-•5321

-•1875

0

?/„•

•6469

1

-•8908

-•4260

-•1489

a.

-•5321

-•8908

1

•7905

•1584

d.

-•1875

-•4260

7905

1 -9592

Sk.

0

-•1489

•1584

•9592 1

Now this table enables us to draw some remarkable conclusions with regard to

O

enteric fever. We see at once that no random selection of a group of individuals,
which has any single characteristic differing from that of the general population will,
except in the case of mean age of incidence and skewness, leave the other charac-
teristics unmodified. Thus the most probable result of any selection which alters the
nature of the distribution of enteric fever can be predicted. The reader will possibly
appreciate this better, if we replace the above table by another giving the absolute
progressions in years, number of cases per thousand, &c.

PROGRESSION TABLE.

Unit change of

Corresponds to a
probable change in the
same units of

One year
in mean
age of
incidence.

One case per
cent, in
modal year of
frequency.*

One year in
standard
deviation
or " spread."

One year in
number of
years between
modal and
mean incidence.

A unit of
1/10 in the

skewness.

Mean age of incidence

1

•0909

- -6120

- -1643

0

Modal frequency

4-5852

1

- 7-2626

-2-6456

- -3372

"Spread" . . . .

- -4626

- -1093

1

•6022

•0440

Interval between mode
and mean
Skewness

- -2140

o

- -0686
— -0657

1-0377
•5702

1
2-6301

•3498

1

* The frequency of incidence in the modal year = y0 X ^ since the unit is five years = 1894-57 X i

To make this 1000 we must multiply by

1000 x 5

1894-57

year. Thus we have to replace Ay0 by i5 ^? ( l^S 1000 ] = -^7^7- (error per thousand in modal

1000 \ *'-
year of incidence).

VOL, CXOI. — A. 2 O

Similarly A?/0 X | = error in incidence of modal
1000"

282 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

We see at once from the above table that if the mean age of incidence of enteric
fever in any group were raised, the disease would be concentrated in fewer years, the
modal and mean incidence would be brought closer together, and the incidence in the
modal year of frequency would be heavier. The changes here are very sensible.
Thus, if we raised the mean age of attack to that of phthisis, or about nine years, the
modal frequency would be increased about 41 per cent., the concentration of the
incidence of the fever increased about 40 per cent., while the distance between mode
and mean would be reduced to nearly 2/5 of its original value. The skewness would
not be changed. Much less marked effects would arise from a selection of modal
frequency. Any increase of modal frequency tends to slightly raise the mean age of
attack, to increase slightly the concentration, to draw the mode towards the mean
and reduce the skewness.

The changes produced by closer concentration of the attacks of the disease, i.e., the
limitation of its incidence to fewer years, would be of a more marked character, they
would raise the mean age of attack and the modal frequency, they would decrease the
interval between mode and mean, and reduce the skewness. Concentration of the
disease would thus tend to render its distribution more normal.

To increase the interval between mean and mode lowers the mean age of attack,
reduces the modal frequency, increases the period of liability to incidence, and much
increases the skewness.

Finally, increase of skewness decreases the modal frequency, increases the period
of liability and the interval between mean and mode.

These statements with regard to the manner in which enteric fever would affect
different groups selected at random from the general population seem of considerable
interest, for there is reason to believe that what is thus stated for enteric fever in
different groups may be applied to different fevers in one and the same group. For
example, the lower the mean age of attack of any fever, the greater its concentration ;
the less the concentration, the more nearly normal is its distribution, &c., &c.

(19.) Probable Errors and Error Correlations of the Constants of the Generalised

X \"'i / 3C V'1*

1 + -- ) 1 1 — -- ) .

Transfer the origin to one end of the range, and the equation to the curve becomes

n T (m. + mt + 2) / x V'" / a \»«

T r (Wl + 1) r (WlTT) (T) (l ' ' T) • • • • <]xxv->'

where n is the number of observations and b is the range.

The following values are given in 'Phil. Trans./ A, vol. 186, pp. 368-9, where
r = m, + w, -j- 2 :

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 283

TIT * ("(l + 1) • \

Mean x = - ....... (Ixxvi.),

mi + m* + 2

•mr i '"i^ •• \

Mode or, = ......... (Ixxvii.).

w, + m,

. J> (in, — w.) . ... .

rt = a: — a:, = . - ...... (Ixxvin. ),

(»/, + •«„) (?n, + w, + 2)
J> (•/«, + 1)(wis + D

-- -

' . . . . (lxxx>).

/(, + ])(m, + 1)J

For the moments we have

//'(//(, + l)(w., + 1)

^= - Hffr + i)— ...... r ...... (IXXXL)'

L'6:;(m, + 1) (ws + !)('".. - »'i) /,

Ma= "r:i(,+TK7T^r

36' (w i + 1) (w, ^ 1_)JO«iJ- J ) (/"..^1 )Q- - <J) +_-£} /i ••• \

^= H(r + l)(r+~2)(r+3)

Lastly, for the mean and modal frequencies of ^ Sx and T/, 8.T, we have

S (,„, + ,»,) -S (,«,)-S (,«,)! . . (1.X.XXV.).

where

Let

J6 / ,. \nii / (). \M»Z

,(f) ^-T) ^

where

^ r(»<i + M., + 2)
Tr7i«r+ i)r(»n, + i)'

then I (mu m.2) = ?/, and we easily find, by the fundamental property of T functions,

that

2 o 2

284

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

T , , wi, + in., + 1

I (m, — 1, m,) = - n,

m.

m, + m, + 1
•i,, mz — 1) = - - n,

. (»il + m.,) (in, + m* + ])

1 *7 <115 1 J _ 11

'i — ^j wijj — - - n,

MI, (m, — 1)

I (»i,( m.j — 2) —
From (i.) we have

m,, («i4 — 1 )

n,

. (Ixxxvii.).

wliui

r o<,

•n i 1.- i , i ^ (loir v)

It will be needful to find - , - z ,

«//;;

oL;1 v) , d- (\u<> y)

^- , and - v*

//(.] am, dm.,

•'•"" 7 ) (Jxxxviii.),

• dtii\

(hn\ dm'(

(//i, + m., + 2) >t- T (;/i, + 1 )

= e. — e, . . . . (Ixxxix.).

Similarly

where

d (M, + ?/(.» + 1)- dmi

cl~ (log x)/(/'»:^ — e» - e- (xc-)'

c/J (log x)/dmi dm, =; e3 (xci.),

Cl = d1 T (m, + l)/t/m'i, e, = d1 Y (m, + l)/t?»4

£, =^ (?- r (TO, + ;», + '2)/d (m, + m, + 1)-

. . . (xcii.).

e,, e.j, and e3 can now be readily expressed in semi-convergent series admitting of
calculation.

e _ '« - . i i

'2iit'i mi

2in., -J. S (in.,)
'* "

. . . (xciii.),

in,

where
and

2(ml + ms + !)•'
S ( «) = B,/» - B,/j

"

S (/<t,

+~His"+"l?" J

J

(xciv.).

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 285

It is clear that if w, and m., be at all large, which they frequently will be, we may
omit the series S, or even reduce e,, e,, e3 to l/m^ 1/m.f, and l/(m, + m.t + 1)>
respectively.

Making use of (Ixxxvii.) and (Ixxxviii.) we easily find

... = n6,., = -

a,4 = H.U — —

a,, = w,, = -

The next stage is to calculate the determinant

6,,, 6,,,, 6,3, 6,4

6,3, 6,3, 633, 634

6,,, 6,4, 631, 644
and the minors B,,,, &c., of ba-, &c. We shall then determine

S* = V A7 ' S': = T A7 ' 2'"' = T A' ' ""•

fy g^cfa= -f (M' + ""X"" t "^ 1> ...... (xcvl).

Jo"7 tfoao &- ?/;., — 1

f <• «P (log ;/) 7 w )M, 4- m, + 1 / • • \

•?/ —,-f— ax = ----- - .......... (xcvn.).

Jo dxdml I) tiii

6 if (logy) . n (w, + at., + 1)
/ -- -v--- cto =-- - ......... (xcvni.).

o dxdm., I M,,

rf2(logw) , ra / ,- /", + 1 / • \

— (^ ^ ='js ("»i + '"^ + 1) -; — ! • • • • (x"x-)-

(^(log.//) , //• ,

^bd^dx'-

d- (log v) , n MI + 1 , . .

^^!dx = -T~^~ ............ ^-

d*-(logy) , x y .. K

'fa° J> dx = n (e. - e,) ............. (cii.).

280 PROFESSOR K. PEARSON AND MR. L. N. G. F1LON

and the correlations

P B" P _A» _ P B»«

KM v/(BuB.) ' """ " v/(BnBS!) ' - vABuB,,) .

T r IL. f ' ' ' (cvi')-

-p _ PM -p _ __ £31 _ p __ 5*

Lt("'" - v/(BKBSi) ' - y(BwBu) ' *•"*"• - v/CBsA.) J

The algebraic expressions for the expanded determinant and its minors are very
lengthy, and it will be found easiest in any numerical case to calculate the numerical
values of the bn> l>\-i, ^ia, blt> b-n •••••, and then find the values of the determinant and
its minors numerically. So soon as the above four standard deviations and six
error correlations have been calculated, the determination of the probable errors in
the fundamental constants of the frequency distribution becomes easy.

We have for the mean organ M(, if Ji now defines the origin of coordinates,

AMt = A/i, + Ax

. ///,+ ! I (//*,+ 1) Aw, li(nt, + 1) AM. - .. ,

Whence

«, + 1 r v., 4s (HI, + I)'-' , ir (w, + 1)-

H- -j- a^-h -

.,

1 )(>, + I) ^(/w, +_!).-- p

^ -26S(lilKSwi - -jj- SftS^Ki,,,,,

!)(M, + 1)

, - - S^B ....... , ........... (cvm.).

Similarly, the modal value of the organ can be found from*

•in. l)n „ 6w, . ,

AM0 = A/i + - A?> + . AJH, — ,., A?H, . (cix.).

»//., + m., (Ill, + M.,)* (7/lj + ?HS)-

* The easiest numerical method in this, as in the previous case, -is to proceed as follows : — Write

v _ v . 111, v , &TOJ „ bm{ v

'Mh + 7/1, + in, "^^ (m, + m,)2 "l'Xn" ~ (in, -f- ^ i«^»i1

where the x'« *"'e umbral symbols, and let N,, N.;. N3, N4 be the numerical values of the coefficients on
the right. Then put

-M0XMo = NlX/l + N2,V/J + N,X-WI - NlXw/

Now square this equation, and, whenever a product, x7X,/» occurs, multiply it by the corresponding error
correlation, R,/v', already calculated, putting R / = 1 if 5 = 5'. Then, actually,

SMO = N? + NJ + NJ + Nj + aN.N.Rift + 2N1N8R/,,,ll - '2N.N.R/,,,,,, + 2NsNsR6mi

BARLOW'S Tables rapidly give the squares and CRELLE'S Tables, or a Brunsviga, the products.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 287
In like manner from (Ixxviii.)

M A6.fl 1 1

- = — + Aw, < - - — ---

d 1'iHi — ni* w, -f- m., ?/?, + in* + 2J

* 1

+ m* 4- -J

4- PI Am, 4- C'.«

say, where the numerical values of c, and c, can easily be found in any actual case.
Hence

Again, from (Ixxix.)

A«r A6 /I

7 : : T

Or AO-/O- = A&//> + f , Aw, +

§- = f- + eiS;,, + ^S:,, + 2^2S,,,^,,,2-R,MiM2 + f ' S^,,,^,,,,,, 4- ^ S^K,,. (cxi.).

Further from (Ixxx.)

AS,__ J __!__ _l_

' " '- • ' J

4

= /, Am, 4-/, Am,, say.
Hence

;/aSWlS)I1,EtolW,} .... (cxii.).

From the results given above we can deduce the effects on size, range, variability,
or skewness of a selection at random of any one of these four.

Writing (cvii.)

AMe = A/i 4- yl A& 4- g.. Am, — g* Am,
we find

R<r \ ^b^t^-bh • f/i <2 i '•<•!• v V T? 8* V ^ P _l_ /- •S1 V T{

MH7 = g— ^r — ^— 4- "5" *» + y A6iWl±ttB!l --- ^ *s2«,J*ta,, -| t^Zfci,,,, n,;,,(tl

4- e^.^^Ri-, + e^,S;, - e^^^^, + ^S^B*™, 4- ^/iS4S,%RAfflJ
4- ^S^SA^ - ft^S2., ......... (cxiii.).

288 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

-r/'-'' + #-S""R""" ~ ffi2"F'»>j} • • '•''•"• • • • (cxiv.).
J^ -f SV/X.SA,,

(cxv.).

These results show us (hat it is, in the general case of skew variation, impossible to
select any one of the quantities — mean size, range, variability, or skewness of an on/an,
inthout at the same time in all probability modifying all the others.

For example, the frequency of the incidence of certain types of diseases at different
ages follows a distribution of this character. Hence, if any special class of the
community had a mean age of incidence differing from that of the general population,
we should expect correlated changes in such other characteristics of the disease as
(i.) its first appearance ; (ii.) its last appearance ; (iii.) its tendence to heavier
incidence above or below the mean age of incidence ; (iv.) the concentration of its
incidence about the mean age of incidence for this selected class.

Precisely similar series of changes would arise in the case of a random selection of
individuals having the variability of a certain organ greater or less than that of the
general population, there would be correlated changes in the size, range, and
skewness of the distribution of this organ.

Turning to the mean and modal frequencies, we have

Ay, _ Ai , [ 3 _ _1 i I _ , dti(m, + m.t + 2) __ r£S(?«1+ 1

~2 w, + 7?J + 2 " 2 ?», + 1 " ' r?Oi + '"'2 + 2) " " <l(mi + 1) riim'

_ JL • <'S(»'i + *»s+ 2)

" "

1 a 7/f, + m., + 2 2 w, + 1 (f (»?, + m, + 2) d (m, + 1

/i,A?», -J- //._> Aw.,, say (cxvi.).

Similarly,

A?/., At f 1 1 1 rfl3(m, + 7/O rfS(m,)l

-' =•• ; — r i~ 7 + ;^ ~; — ;, ~ + ' r — " r A?n,

i/j o [?«! + ?«» +1 2 (m^ + wi») 2ml d(ml + m*) dml \

' |m, +m, + 1 "^ 2 (m. + ws) ~ 2v?72 "" ' ~d(m^~+ in,) ' ~ dm. JAWl'-
= — A6/6 + ki Am, H- h Amo, say (cxvii.).

Here /i,, /i2, 7c,, k.2 can be easily calculated, if we note that

. . (cxviii.).

= — T/p, where T is the same as on p. 272.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 289

2y, and Sy, can be found in the usual manner by squaring after the insertion of
the numerical values.

From (Ixxxi.)-(lxxiii.) the probable errors and correlations of the moments can be
found if required, the calculation being numerically somewhat laborious, but presenting
nothing of novelty.

The probable error of the criterion

K = 3ft - 2/3, + 6,

where ft = pi/fd and ft = /u4//4 may be found as follows : Put c = (w, + 1) (in, + I),
r = nil + m-i, + 2 ; then we find

and accordingly

A« / 2 2 .1 1 \ / 1

/• + 1 + e e

r + I

A

r + 2 ,- + 8r + l + « (/•+! + e) (/«t + I)

./A __2_ _1 J_ J' + L

V /• ' " r + 2 " /• '+ 3 1r r + I' + e ~~ (/• + 1 + e) (m, + I} L

= ii Am, + i> Awa ............... (cxx.),

where z\ and t: admit of easy calculation. Hence

S*/K2 = tiS;,, + ^S;la + 2MoS,1,1SM,R(lll,11J ..... (cxxi.).

The value of S« can thus be found, and the steadiness of the curve to its type
ascertained.

Illustration. — Glands of the Fore-legs of Swine.

In the ' Proceedings of the American Academy of Arts and Sciences,' vol. 32,
p. 87, 1896, is a memoir by C. B. DAVENPORT and C. BULLARD, on the variation in
number of the Mtillerian glands in the fore-legs of 4,000 swine. The paper especially
attracted our attention, because the authors are content to describe the frequency
distribution of these glands by means of a normal curve. They write, after
discussing the plotting of the normal curve on their diagram (pp. 90-91) :—

"These and other characters of the ' probability' curve are indicated in that shown
in dotted line* in the accompanying diagram. The diagram also shows the curve of

* The authors actually represent the normal curve by an 18-sided polygon.
VOL. CXCI. — A. 2 P

290

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

distribution of the various numbers of glands occurring on a leg from 1 to 10""".
This curve is drawn from the right female leg only ; the curve for the other legs
would be very similar. We shall speak in a moment of the method of construction
of these curves ; but we want now to call attention to the fairly close similarity of
the two curves — that gained by observation and the theoretical one — a similarity
so close that we are justified in concluding that the law of distribution of the
variants in the leg glands of swine is the same as that of accidental errors."

+.TC

Now, in our opinion, the curve was markedly skew, and it seemed to us that most
interesting properties bearing on the action of selection on the Milllerian glands in
swine actually depended on this skewness. We have taken the distribution of
glands for 2,000 ? swine.

To illustrate the difficulty of applying the normal curve we may remark that it

gives about 6 swine per mille with — 1 gland, and about 1 '5 with — 2 glands, while

»

* The aatliors have forgotten that there is a sensible percentage of zero-glands.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 21J1

it gives 30 per mille instead of 10 per mille with no glands. These difficulties are
entirely met by the skew curve, which gives no frequency whatever of negative
glands (see figure).

Taking the number of glands in the right fore-leg of female swine, we have the
frequency series :

No. of glands .

0

1

2

3

4

5

6

7

8 | 9

10

Frequency . .

15

209

365

482

414

277

134

72

22 8

o

Per mille . . .

7-5

104-5 182-5 241 207

138-5

67

36 11 4

1

We have worked with the frequency per mille for convenience of reduction,
although the actual number of observed cases, 2,000, is used, of course, in the
determination of the probable errors.

Using the method of the paper in the ' Phil. Trans.,' A, vol. 186, p. 367, we found

mean = 3'501 glands.
Hz = 2-824,999
/*,= 2-417,278
j*4 = 24-826,297

6-f

0-= 1-680,774
/8, = 0-259,1825
&= 3-110,8211
?, — 2/32 = 0-555,905

Thus the criterion is greater than zero, or the frequency distribution is of Type I.,
or has a limited range.
Proceeding we found

r= 19-985119
ml = 3783718
a, = 3-79623
I = 18-0446
d = 0-522996
Mode = 2-978 •

e= 72-71918
m, = 14-201402
a, = 14-24837
y0 = 237-263
Sk. = 0-311164
Start of curve = — 0*818 gland.

Thus it would appear that both the distance (d) from mean to mode and the
skewness are very sensible, and that, unless their probable errors be very large, it is
quite impossible to represent the results by a normal curve.

We may note that the range starts from - 0'818 gland and runs to 17'227 glands,
so since it gives zero at — 1 gland, we see that it sensibly confines the possible
number of glands between 0 and 17, but we should have to examine considerably
more than 2,000 swine to have a probability of more than 10 glands occurring. The

2p2

292

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

total range given is thus both in magnitude and position extremely satisfactory, and
supposing only the frequency, not the actually measured quantity, i.e., number of
glands, to be known, the theory would have given a very accurate determination of the
limits of possibility, especially the start of possibility with the whole number of glands.

In order to work out easily the determinant, A, and its minors, we found it
desirable to bring out certain factors and reduce the formulae given above to slightly
different forme, which, as they are likely to be of general service, are here repeated.

Let a = ?n,e, — (TO, + ^2)^3. ft = "Ms — (mi + ^2) es> where e,, c2, e3 have the
values given on p. 284, then we found

A =

«*(»!, +m.,

?,-i)(«h-i)

nil + in-> 0,

— (TO, — 1), m,a + m

TO,(TO, - 1), m2&

m, + 1, - 1,

1, 0,

— TO,a, — (TO, + w- — 2)

Zr///;»*| (•»?.> — 1 )

TO, a + ??i,p, — m,a, — 1

— (TO,— 1), — TO, (TO, — 1), (TO, + 1) (TO, -f TO, + 1)

A.» =

n*(mi + ffl3 + 1)
ilw5(w, - I)(w2— 1)

w,

1),

TO, + 1 ,

0,

TO, — 2TO2 -|- 1

2 (TO, + TO, + ] )

AM =

, + w(.a +1)

wll + m.,, _ i, o

— (TO, + 1) (TO, — 1 ), TO? (e, — e3), m, — <2ml + 1
«^ + 1 , 0, 2 (TO, + w, + 1)

_ M3(mt + m2 + 1)

;,! + ?»2 +1)

i, — 2), 0, (m, — I)(TO,— 1)

0,

^ — 1)

— m,m,e3,

0

0, TOoyS, - 1

(TO, + m,), — (TO, + 1) (TO, — 1), m, H

+ ???., + 1)

0,
-* >
(wi, -{- m,),

«

(TO, — 1),

— 1

0

'!+ '

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 293

ml (m, — 1)

»s(w,

o,

m., —I,
— (TO, + TO.,),

TO,« + TOo/3,

m&,

— m,

4. m.,t

A24 =

?l3 (7?l,

lsmtmr, (m1 — 1)(»?2 — 1)

- i, o,

t,m.,e3, in, — 2»io -4- 1

0, 2(m,+ TOO + 1)

1 \ / 1 \

1,
0.

TOO — 2m, + 1, — ('»i + m-2 — 2)

TOO (TO, — 1)

W,«, — TO,TO2C;,

— (TO, + in., — 2), w, — 2»j,, + 1)

w, + TOO, — (TO, — 1),

— 1,

0.

In our particular case we found

e, = -232,4012, €., = "067,9945, e:! = '051,3099,

m,a = — -164,4934, m2^8 = '607,8513.

With the aid of the values for TO, and ni.2 given above, the determinantal parts of
the A's were then calculated. If these be 8, «,,, «•«, «:i,, a,4, a,2, a,:!, «14, a.,.,, a.,4, a;J4,
we have

8 = '153,7969,
«„ = '348,3713,
«22= 13-018,7332,
«33 = 47-671,9443,
«44 = 13-357,1309,

«,., =r '274,9969,
a,3= -111,8280,
a,4 = '211,9650,
a.,3 = 22706,5156,
a.,4= 11-507,0153,
OL^= 25-088,6121.

From these the standard deviations and correlations of errors in the algebraical
constants are at once found. We have

S7l = -2325,
2ffll= 7784,
tat = 5-5908,
2,, = 3-7602,

Rimi = -9387,

R,jM2 = -7548,
•Ru = 7143,

RMi,, = -8726,
='9942,

294

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

Then as a step towards the determination of other probable errors, the standard
deviation and utnbral equation* for y = ml + nh were found

This led to

2, = 6-3085,

Xv = Antl. l'091,2932Xwi + Antl. 1'947,5528X,V

Ryra, = -9312,

y, = '9888,

Ry,, = 7848.

By aid of these auxiliary results the probable errors of all the algebraical and
" physical " constants were determined.

PROBABLE Error Table.

Constant.

Probable error.

Percentage probable
error.

•§ji

[m,

0-5250

138762

g-S

m.,

3-7709

26-5533

|f

-; o

Cla .

0-1748
2-4351

4-6056
17-0903

<1 0

fRange

2-5362

14-0554

Start of range

0-1568

•a .§

o a
'$ £<(

Mode
Mean

0-0398
0-0253

—

>* 01

f3 <=

PM 0

Standard deviation ....
Mean to mode

0-0183
0-0294

1-0911

5-6308

Skewness
Modal frequency.

0-0158
3-2455

5-0655
1-3679

Now it will be clear from an examination of these results that all the " physical "
constants are determined with great accuracy, t The mean is subject to less probable
error than the mode, the modal frequency has a slightly less probable error than the
mean, and as it is less than 1*4 per cent, in the former case, either are closely
known. The skewness and distance from mean ta mode are known respectively
with less than 5'1 and with 5 '6 probable errors. Thus they are both significant
constants. In other words, the curve differs significantly from a normal curve, and
it is erroneous to represent the frequency by such a normal curve. The range which
ought to be such that there is no frequency at — 1 gland, gives no frequency at

* See footnote, p. 286, and later, p. 305. It may be as well to remind the reader that here, as in the
other illustrations, logarithms of the full, not the cited values, were used in the calculations.

t The probable percentage errors in^, m}, aL, «3 are high, but this, as we have several times pointed
out, is of small importance, as, owing to their high correlation, the actual shape of the curve is not
changed sensibly by large changes in i»i and mt.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 295

— '818 gland with a probable error of ± '157. It is, therefore, clear that our
method gives the start of the range with very considerable accuracy. The whole
length of the range runs to 17*227 glands, with a probable error of 2'536. We may,
accordingly, conclude that the maximum possible number of glands is hardly likely
to be less than 16 or more than 20. We consider that this example is a good
illustration of the accuracy with which the principal "physical" characteristics of a
distribution may be obtained by aid of skew curves, and how they provide much
information which is not given by the use of the normal curve.

The next point is the determination of the umbral equations giving the error
correlations of the " physical " constants. They are, if Antl. stands for antilogaritlun :

X mean = Antl. 792,415Gx/1 — Antl. 1-380,2040x6 — Antl. 1-153,9020
+ Antl. l-508,1033x«,,,

X range = X&>

X,jo = —Antl. l-011,7885x4 — Antl. •248,2856x,,li + Antl. I'097,6534x,,l3,
X, = Antl. 1-109,9660x6 + Antl. -168,8507x,,,, - Antl. 1-151 ,0582xW3,
Xd = Antl. '397,2701x6 - Antl. •274,1702x,,il - Antl. I'S 10,31 80 x,,,,,
x,4. = — Antl. -381,5919x,,, + Antl. >367,7012x,,li.

Multiplying these out pair and pair, we found

ERROR Correlation Table.

Mean.

Range.

Modal
frequency.

Standard
deviation .

Mean to
mode.

Skewness.

Mean

1

•0232

•3400

-•3493

•0500

•1309

Rane'c .

•0232

1

•6284

•0906

•2132

•2175

Modal frequency .

•3400

•6284

1

-•6944

-•1473

-•0141

Standard deviation

-•3493

•0906

-•6944

1

•5891

•4394

Mean to mode ....

•0500

•2132

-•1473

•5891

1

•9847

Skewness

•1309

•2175

-•0141

•4394

•9847

1

^

Hence, proceeding to multiply rows and divide columns by the corresponding
standard deviations, we have, after altering the units, the following

296

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

PROGRESSION Table.

Unit change of

Corresponds to probable
change iu same units of

One gland
in the
mean.

One gland
in the
range.

One
per cent.
in the
modal
frequency.

One gland
in the
standard
deviation.

One gland
in the
interval
from mean
to mode.

•fjj in the
skewness.

Mean

1

•0002

•0063

— •4818

•0430

•0210

Range

•2-3231

1-1651

12-5279

18-3604

3-4992

Modal frequency ....

18-3876

•3389

1

-51-7973

-6-8411

-•1225

Standard deviation ... — '2533

•0007

-•0093

1

•3668

•0511

Interval from mean to mode

•0583

•0025

-•0032

•9459

1

•1840

Skewness

•8156

•0135

-•0016

3-7761

5-2706

1

An examination of this table brings out several interesting features of the
frequency distribution of Mlillerian glands in the fore-legs of swine. If a group of
swine were isolated, and found to have a higher mean number of glands, then this
group would most probably have an increased possible range, but at the same tirae a
decreased variability and a marked increase of skewness. This increase of the
possible range with a decreased variability is especially notable, since the rough-and-
ready class of statistician is very apt to treat the range observed as a measure of
variability ; we have here a case in which the same cause, raising of the mean,
produces opposite effects on range and variability. Increase of range, it will next be
observed, produces very little effect on any of the physical constants, but such effect
as there is, is an increase of them all. To increase the modal frequency is to increase
the range and to reduce both the variability and the skewness. Thus the more
mediocre swine there exist in any group, the more nearly their distribution will be
normal. Change in the variability is the cause which on the whole produces most
effect. Increased variability means lowered mean and less mediocrity, but much
increased skewness. Finally increased skewness denotes probable increase of range,
variability, and mean.

As we have suggested in a previous illustration the principles of multiple corre-
lation easily enable us to predict the probable change in a random selection in which
two or more of the characters differ from those of the general population.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 297

(20.) Probable Errors and Error- Correlations of the Constants of the Generalised

Probability Curve of Type

II — -)/„ ,,->> tan -'(*/«) ^pvvi! ^

" — y° n j. /»/«vi» \^x.x.u.f.

i i i v*'/'*/ j

This curve is discussed at length, 'Phil. Trans.,' A, vol. 186, pp. 376-80. The
chief constants are given as follows, if m —«('>' + 2), z = v1 -f- r~, and h denote the
origin : —

Moments —

«* , , , ,v , ... v

ju.2 = ~^~~t r: ('" + v) (cxxm.),

4ft:li/ (/•- + !/'-) . . ,

/".•i = — <i:, /,._]) (r _ 9) (CXXIV.),

Ll.j ^^ 7 1~T7 O\ ' ^\ ..... ( CXXV. J.

Distance of centroid from origin = — av/i- .... (cxxvi.),
Size of mean organ = /i (cxxvii.),

Size of modal organ — h — — -} (cxxviii.),

Distance from mean to mode = d = —-- . . . (cxxix.),
Skewness = - ^ /y/hj™A . • . (cxxx.),

Standard deviation = o- = —r.- \ x/(/1- + v') . . (cxxxi.),

>' vv '

y0 = -'- e-v"/G (r,v) (cxxxii.),

where

and

G(r,v)=

G (r, v) = r

is the formula of reduction.

VOL. CXCI. — A.

. (cxxxiii.),

2 Q

298 PROFESSOR K. PEARSON AND MR. L. N. G. FILON

Further, we have the following BERNOULLI number series for G (r, v), where

tan <f> = v/r :—

log {e~^ G (r, v)} = log y(27r/r) + (r + 1) log cos <f> + v<f>

(2s

COS

To find ?/, Sx, the mean frequency, we have only to put x = — avjr in (i.), and we
have

log y, = log y0 + (r + 2) log cos <j> + v<f>

= log n — log a — log [e~1"" G (r, v)} + (>• + 2) log cos <f> -\- v<f>
= log n — log a — log v/(27r/?1) + log cos <b — x,

where x stands for the summation in (cxxxiv.). Hence

^ = ~v i^e~Xco8<^ • • (cxxxv-)>

or,

* ..... (cxxxvi.)-

As typical constants we require the probable errors of the mean, the standard
deviation, the skewness, and the mean frequency. It is clear that these will require
us first to find 2,,, 2,., £„, 2,,, and R,,((, R,,,, R,,,, R,m, R,,,, Rm, R_a.

We shall only indicate briefly the steps towards finding the integrals of the second
differentials of log y.

log y = log i/o — in log U

^ (log?/) v 1

' T T+7-eay " "

^-(log?/) _ 2 f v.v/a m 2m "I

rf*2 " "2 I ~ {1 + («/«.)2}2 ~" 1 + (a/a)* + (1 + ^/«2)2 J

= — (2/a-) { — v sin 0 cos3 6 — m cos'2 0 + 2m cos4 0

-- e-v

" —ir/2

whence, remembering that 2m = r + 2, and integrating the first integral by parts,
we find

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 299

, ., J.IW O V * L / •> / A v

'7a- «' x ' v- + (/• + 4)

« i/* + (>• + 2)-

_ a,

n (r + 4) {v* + (r + 2)-} G (r + 2, v)
" «2 " »» + (r + 4)3 G (r, i»)

or,

»i (r + 1) (r + 2) (r + 4) ..

- «" = ' -# • ^ + <y + 4)2

Precisely similar reductions lead us to

f+» <P(logy) , nv (r + 1) (r + 2) ... .

— «,.,= V — 7— ~dx= rri — r^ TST • • • (cxxxvin.),

J -XJ dxda a- {v1 + (/• + 4)2}

- dx = — r-; — „ .... (cxxxix.),

gy) 7 « (/• + 1) (/• + 2) / i \

55-S£-i ft T» — — — — — / r»v I i

rZv a j/2 + (r + 2)- '

\ *> i t"l / i^ j \

7? - - Mi.),

(cxlii.),

f '/) 1 '» I> (/' + 1) / !••• \

;t/;^ = — — ---~— — 9 , (cxlni.),

[ + * ''MlOg'/) , « ' M f~i , M / V \

Ms= M- ^dx=— n — {logG(r,v)} . . . (cxliv.),
]__ (/?•- "'"

- rfe = — n — — {log G (r, v)} . . . (cxlv.),

= — n y-s (logG(/', i')} . . . (cxlvi.).

It will now be needful to find easily calculable series for the second differentials of
log G (r, v). These can be obtained from (cxxxiv.). We find

^ (log G (r, v)} = - — + log cos e£ + '^r-

_ o Ba.+i(~ !)' < l _ 22"+2 cos2s+2<f> cos ("2s + 2) <j>] . . (cxlvii.),
«(2s+2)^°- {

d ., ~ , ,,, , sin <f> cos <f> , ±

-r {logG(r5^)}=A7r-

r

22s+2 cos"*^ sin (2s + 2) ^ . ' . . (cxlviii.).
2 Q 2

300 PROFESSOR K. PEARSON AND MB. L. N. G. FILON

Hence

l {^g G (r, „)} = -5 + siity (r -1-2 cos2<£)

+ S ~ "*' (1 -
o *

{log G (r, „)} - ~ cos^ (2-1+2 si

cos

(2s + 3) <f>] l(cxlix.),
J

. . . (cl.),

ilr dv

— {log G (r, „)} = siu <£ cos <^ (2 cos2^. - r)

- S - -

22s+2 cos&+3<^ sin (2* + 3) A 1 . . . (cli.).

J

These allow of the fairly rapid calculation of «33) ait, a.H. The values of the
standard deviations of the errors, and of the error correlations, can then all be
calculated from the determinant

A =: a,,, «,,, «13. ((,4,

and its minors.""'

Let bpp, = - apjl; and let A' = — A, then if BA/ be the minor corresponding to

'/& fv

l>n,, in A', we must work out for any special numerical case

n A

~ n A' :
R B«

ail /fO. TJ V

V (Bii B22)

I ^S.
n A"

»7 A7'

BIS

T? — — ^

^"- v/(BMB1(y

v/(Bs3B44y

* As in the former case, these were all developed, but the extreme length of the resulting formulae
gives them no advantage over working in any special case with the numerical determinants.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 301

In general none of these correlations vanish, and their values must all be found
before the errors and correlations of the chief characteristics of the frequency can be
found.

The following results, easily obtained by aid of the relation tan <f> = v/r, will be of
service

$1 =C^{S;+tair^-2tan^SA,) .... (clii.),

B^-^^S, -tan^R,,} ......... (cliii.),

' •*<£

B*, = ^{2,11,,- tan f£,} ......... (cliv.),

E,,,,, = -^{SA,, -tan^Ee,} ........ (civ.),

T2.$

R,i = ^{SA,-tan^,.R,,} ........ (clvi.).

i — ,f

By (cxxvii.) and (cxxxi.) if x be the mean size of organ,

A,x = A/i — tan d>Aa — - - Ar/>.

cos- <j>

Ao- A'< , . A/-

= - + tan (f>k(p — jf - 7 •

(7 ft i' — I

Hence

^ = ^ + tan^ ffi + c~—Z,-2 tan <^2KR,., - ^ Sfc2^t

X^Ra, . (clvii.),

= + tan^ ^ + S; + tan

T^S.R*, .......... ". . . . (clviii.),

y -~~ -L

and

ET \ 2aS A/, tan0 v^ ^a^Rn^ I fnl, J.V V "R - tan'2 AS? ~$ T?

-„ = =r^r < - --- - X — ;dhr T tan (P-*2'*it** t'an <PAl^«f.^a«

2taz,^. L fl cob 9*

tan < , ft V ^ Ti Mix ^

^^ '

From (cxxx.) we have, if S* = skewness,

302 PROFESSOR K. PEARSON AND MR, L. N. G. FILON

Or, taking logarithmic; differentials,

AS JS, = cot <£ A<£ - (^ - i^) Ar,
whence,

SI/SI = C0ts 0$ + ' '

In a similar manner EjSl and RaSli can be found if desired. None of these quantities
will, as a rule, vanish, and as very many measurements on animals give curves of
the tangent type, we conclude that in general all selection of the size of an organ
alters its variability and the sketvness of its distribution, and again all selection of
variability connotes alteration of the size and skewncss of the selected organ.

The probable errors of p.2, p.A, and p.t, as well as the error-correlations of these
quantities, can all be found from the differentials of (cxxiii.), (cxxiv.), and (cxxv.) ;
the calculation is laborious, but presents no novelty.

Lastly, the probable errors of the mean and modal frequencies may be deduced.

For the mean frequency we start from (cxxxv.) and use (cxxxiv.).

This requires us to know AX, where

' ' C°S 2S

We have, as in (cxlvii.) and (cxlviii.),

cos

s^ sin 2^

where c, and c-, admit of fairly easy calculation.
Hence, by (cxxxv.), we find,

+ (i + siir <^> + c,) Ar/r — (c2 + cos <f> sin <£)

a* + (i + si"2 <^> + o,)2 S;/7-2 + (c2 + cos <j> sin ^»)2

2 / 1 • 2 j \V-C-D . 2 (c2 + cos ^> sin </>)

-- (| + sm2 <j> + c,) 2aS,.Ra, + ~ - S

i fL tt/

sina + <g) (c2 + cos < sin <>)

If the problem be to find the modal frequency y-, 8x, we easily deduce y-, by putting

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 303

x = — a . 2- in the equation to the curve. Writing tan <f>' = v/(r + 2) and y/ the
same function of r + 2 that y is of r, we have

r + 1
Since - . •-- is greater than ^/r, and $ and Y are less than <£ and x respectively,

it follows that y., is greater than yl} as it should be. Further we find

/ sin (£') - . (clxiv.).

Here c,' and c2' are the same functions of r + 2 and <£' that c, and c2 are of r and <£.
The usual process of squaring and introducing the standard deviations into the square
terms and the product of standard deviations and correlations into the product terms
will give us 2?,,.

(21.) Illustration. — Stature of Children.

In order to illustrate the difficulties which may arise in determining the probable
errors of the constants and the error correlations, we have selected for this illustration
not a curve markedly skew, but one which is extremely nearly normal. The problem
in this case is accordingly the following one : Are the values of the constants obtained
for the distribution and distinguishing it from a normal distribution really significant ?
The difficulties which arise in the course of the arithmetical work depend upon the
fact that, as the distribution is nearly normal, its constants approach the values at
which the type of the skew curve passes over into the normal curve, and conse-
quently not only will their probable errors be large, but, as in all cases of approach
to limits, they will depend upon expressions tending to become indeterminate. Thus
in the evaluation of the determinant A and its minors, we at once found our results
depended on the ratio of the differences of very small quantities. We were accord-
ingly in this case obliged to calculate our constituents to a degree of accuracy which
will, in general, be quite unnecessary, and which was only possible and straight-
forward owing to the ready help of a large sized Brunsviga. That the method, even
in a critical case of this kind, gives correct results is evidenced by the agreement of
our values of the constants with those (probable errors of mean and standard
deviation) which can be readily calculated by other processes.

The example we have selected is that given for the stature of 2192 St. Louis
school girls of 8 years of age in ' Phil. Trans.,' A, vol. 186, p. 386.

The equation to the frequency curve is

304 PROFESSOE K. PEARSON AND MR. L. N. G. FILON

x= 14-9917 tan 0,

y - 235-323 cos82'8023 tfe-4'56967',

the axis of x being positive towards divarfs and the origin 2'224l on the positive side
of the mean. The unit of x is 2 cms. of height, and all the constants except the
mean height are given in two-centimetre units.
We have the following values of the constants : —

*
p.2= 770739, Mean height = 118'27l centims.,

p, = — 2-38064, a- = 277622,

^ = 192-17419, yi = modal frequency = 324'18;

y., = mean frequency = 3237G,

d — -135,606, Sk. = skewness = '04885,

r= 30-8023, m= 16'4011,

v— 4-56967, ft-— 14-9917.

It will be seen at once that the skewness is small, that the mean and mode are
close together, and the mean and modal frequencies are almost identical. Our
problem is : Are these differences significant or not ?

Let n = 2192, the total population ; then the values of a, r, v given above were
assumed to be absolutely exact, and A calculated with its constituents to 9 places
of figures, as it depends on the differences of vei-y small quantities. We shall
indicate one or two stages in the arithmetical work

A _ | -131,108,064 -017,214,971 —'008,837,638 '063,438,906

•017,214,971 -010,392,042 - '003,264,669 '008,837,638

- '008,837,638 - '003,264,669 '001,144,775 - '004,422,657 j

•063.438,906 '008,837,638 — '004,422,657 '030,799,043 I

The evaluation of this determinant and its minors was then carried out by means
of the Brunsviga, and we found

A_ "104,824,.472 A^ _ 1200-842,528

•/7 : 101S 1* ~~ 1012

A,, 670-195,695,496 A1S 5059-387,378

n* 101- n"

AH _ 4606-123,523 Ay _ 1762-570,609

w:1 = 1012 n3 ' 101S

AS., _ 76025-131,845 A^ _ 18675-261,289

w3 = 1012 ws = 101S

AM _ 4828-382,384 Ay _ 3833-460,555

?is " 1012 nx ~ 10l

Ay 15979-332,581
ns ~ 101

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 305
We can now, remembering that

write down the standard deviations and error correlations of the algebraical constants

2/i = 17078, R,re = - -6835,

S«= 4-4773, R,,, = - 7088,

S, = 18-1898, II,,,, = — '9798,

?.„ = 4-5840, R,,, = -9980,

R«,= '8129,
ft,, = -8340.

Here h marks the position of the origin of the curve, and the numerical values are
only retained to four places of figures, although, of course, in the further calculations
the logarithms of the full values of the 2's and R's have been used.

It will be noticed at once that though a, r, and v have very considerable probable
errors, the correlation between them is very high. In other words, as the curve
approaches its limiting shape, a, r, and v may vary very considerably, but owing to
their close correlation this will not sensibly affect the geometrical shape of the
curve.

The next stage was to determine the standard deviations and error correlations
of certain subsidiary constants. Here, as in the determination later of the like
quantities for the " physical " characters, we found the umbral notation of great
service. It consists, as we have seen, in writing down a .difference equation between
any constants, and then replacing the differences Bu by 2,,x,,> $v by 2^, &c., whore
X"-> X»> &c>> are quantities which obey the relations ^; = 1, ^; = 1, X,,XB = R,,,. Thus,
if tan<£ = v/r, we find for the umbral equation

<? v - -" v -
-^A* ~~ , A."

Whence, putting in the numerical values, we have

2^X4 = Autl. 1-163,2J 15Xll - Antl. 2-933,070Gx,

where Antl. stands for antilogarithm, in which form we found it easiest to keep the
umbral coefficients. The square of this result gave at once

$t= -087,926

and dividing out by its logarithm, we have the pure umbral equation
x^ = Antl. -219,0937x, - Antl. r988,9728x,..

VOL. CXCI. — A '2 R

306

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

Our object was then to find such pure umbral equations connecting all the
" physical " constants with the algebraic constants. Their products will then give
the error correlations of all the " physical " constants in terms of the correlations
already known between the algebraical constants.

For example, multiplying the above equation for x$ by x/,, X«> X" X- we nave> since
X*X« = B»0> x«X.- = R«,-, &c., are already known,

= T'969,2668,
=1-572,0115,
= T'608,8746,
= 1-925,8379.

R/10 = — '9317, actually log (— ]

E,,£ = -3733, ,, log R(,^

R,^, = -4063, „ log R,^,

R* = -8430, „ log R^

It was these logarithms, of course, which were used in the further calculations.
Since h is measured negatively (i.e., towards dwarfs, x is positive), we must write
for transferring origin to the mean

x' = x + a tan <f>,

where a tan ^> is the distance between the old origin and the mean, or if m be used

to represent the mean we have

m = h + a tan </>.

Hence we find the umbral equation

S,,,Xlrt = Antl. '232,4493x4 + Antl. I-822,3l79x« + Antl. -I29,4233x*.

Hence we determine

2,,, = -0549,
and the pure umbral equation

X,,,= Antl. 1-492,5897x4+ Antl. T082,4583x« + Antl. r389,5637x*.

In precisely the same way all the other " physical " constants, i.e., the standard
deviation, <r, the mean frequency, ?/.:, the distance between mean and mode, d, and
the skewness, Sk., were found, and the uuibral equations investigated. It is only
necessary here to give the results.

Quantity.

Probable error.

Percentage
probable error.

.a *

2 a

[A, position of origin ....
a

1-1519
3-0199

20-1438

jz a J

JB-S

c .

12-2688

39-8309

tlC 12

^8
.

v
"Position of mi .1 H

3-0919
0-03705

67-6612

"3 £

Position of modo .

0-05950

§ a

Mean frequency, y., . •

4-4362

1-3703

Jl"

£§

Standard deviation, a .
Mean to mode, d

0-02984
0-0497

1-0750
36-6420

Skewness, Sk. . . . .' .

0-02661

54-4690

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 307

Now it will be seen at once that the probable errors in the algebraic constants are
large, but that the probable errors in the position of the mean, of the mode, and in
the magnitudes of the mean frequency and standard deviation are small. The position
of the mean is sensibly more correct than that of the mode. On the other hand, the
distance of the mean from the mode and the skewness have large probable errors, not,
however, so large but what these quantities are probably significant. The frequency
distribution probably differs significantly from the normal distribution, but the
difference is small and would require a very large number of observations to deter-
mine it with extreme accuracy. That there is a significant divergence from normality
is also indicated by the sensible difference between the percentage errors in y., and cr,
which would be equal for a normal distribution. Had we taken a normal distribution,
the pi-obable error of the mean would have been '0400, and of the standard deviation,
•02831. In fact, the standard deviation of the standard deviation, if calculated for
the normal curve = '04197, if calculated by our present method = '044246, and if
calculated by a modified form of the fourth moment formula given by CZUBER*
= '044240. This shows that the arithmetic of our process has been substantially
correct.

We now place together the umbral equations for the correlations of the errors in
the " physical " constants. They are

Xm — Antl. l'492,5897x/, + And. r082,4588x« + Antl.
X, = Antl. l-272,7484x« - Antl. 1-282,1306X, + Antl. r913,0028x*
x,2 = Antl. T-816,696Gx^ — Antl. l']G7,3480x, + Antl. 1-155,7688X>.
X,i = Antl. -266,3616x, + Antl. T'740,1625x,, — Antl. -323,825Gx,
X*.= Antl. 1-865,6420x4, - Antl. 'Ol7,0466xi.

From these results any correlation between pairs of errors, " physical " or algebraic,
can be found at once. The following table gives the chief results : —

CORRELATION Coefficients between Errors in Constants.

m.

(Tt

y>-

d.

sL

m

I

•0772

-•0584

•0826

•0126

a

•0772

1

-•7062

•1177

•1431

2/2

-•0584

-•7062

1

•1779

•4086

d

•0826

•1177

•1779

1

•6843

8k.

•0426

•1431

•4086

•6843

1

* ' Theorie der Beobachtnngsfehler,' p. 133.
2 R 2

308

PROFESSOR K. PEARSON AND MR. L. N. G. FILON

Now it is clear that although the curve is nearly normal, there is still sensible
correlation between quantities — e.g., mean and cr or d — which would have no cor-
relation between them if the curve were absolutely normal. This will be clearer if,
as in the previous illustration, we replace this table by a table of regression
coefficients.

m.

2/2-

a.

d,

ik.

m

1

-•0005

•0959

•061(5

•0-5935

yt

- (3-9883

1

-104-9745

15-8875

68-1271

(J

•0622

-•0047:.

1

•0707

•16045

d

•1108

•0020

•1960

1

1-2780

tk.

•0306

•00245

•12755

•3665

1

1

This table has now finally to be thrown into more suitable units and attention
paid to the fact that TO increases towards divarfs. We have, after the proper changes,

the following results :-

PROGRESSION Table.

Unit change o

f

Corresponds to probable
changes in the same units of

One centim.
in mean
stature.

One child
pei- hundred
in frequency
of mean
stature.

One centim.
in standard
deviation.

One centim.
in interval
from mean
to mode.

1/10 in the
skewness.

Mean stature

1

•0032

-•0959

-•0616

-•0119

Mean frequency

1-0798

1

-16-1745

2-4536

2-1042

Standard deviation ....

-•0622

-•0308

1

•0707

•0321

Interval from mean to mode

-•1108

•0129

•1960

1

•2556

Skewness .

— 1530

•0794

•6378

1-8333

1

This table is extremely suggestive. It shows us that a random selection of girls
of eight which had an increase of stature would have a less standard deviation, less
distance between the mode and mean and less skewness. In other words, a selection
giving taller children would be less variable and more nearly normal. Now as
children grow older their stature increases, is less variable, and is more normal in its
distribution. Thus, a selection of taller children from among children of eight
would broadly tend to reproduce the characters of the stature distribution of older

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 309

children. In the same manner a selection of shorter children is more variable and
less normal than the distribution of the general population of eight years of age, i.e.,
tends to reproduce the characteristics of a younger population. Generally, a random
selection, which increases variability, very sensibly increases skevvness and decreases
stature. What, perhaps, would hardly be expected, is that increase of skewness as
well as increase of interval from mean to mode, i.e., greater divergence from normality,
increases the frequency of the mean stature.

It will be clear that by aid of this table we are able to predict the probable
changes in all the other physical characters of the distribution when any sub-class
has been selected at random from the general population with a difference of one
character. If two or more characters differ in the sub -class, the probable changes
in the other characters can be found by the principles of multiple correlation from
the correlation table on page 307.

(22.) Conclusion.

This study of the probable errors and error correlations shows us that these
quantities can be determined for the most complex system of organs in the case
of normal correlation, and in the case of either normal or skew variation with con-
siderable ease. It is only in the case of skew variation that the arithmetic becomes
at all laborious. But numerical examples suffice to show that the errors here
made are of the same order as in the case of normal variation, if we confine our
attention to the characteristic features of the frequency, e.y., the mean or modal
frequency, the standard deviation, the skewness, &c. Certain constants of the
algebraic form of the frequency curves have large probable errors, but these errors
are so highly correlated, that their existence does not suffice to substantially modify
either the form of the curve, or the " physical " characteristics of the distribution
calculated from such values.

For the theory of evolution certain very important principles flow, beyond the
mere advantage of knowing the probable errors made in the measurement of racial
or organic characters. Above all we note the importance of a random selection in
altering in a systematic manner all racial constants. In most cases even size cannot
be altered without alteration of the size, variation and correlation of all correlated
organs. This principle is developed more at length in a memoir, nearly completed,
on the influence of directed selection, which covers as a special case that of random
selection.

Later, we hope to apply the general theorem from which our memoir starts to
determine the probable errors in the constants of the components into which
a heterogeneous frequency distribution may be resolved by the method of the first
memoir of this series.* It applies equally to such an investigation.

* The importance of such a determination was emphasized by Professor GEORGE DARVCIN in the
discussion which took place at the reading of that memoir.

310 PROFESSOR K. PEARSON AND MR, L. N. G. FILON

[NOTE. — Added May 25, 1898. One point ought to have been more fully dealt with
in the above memoir, namely, the probable error of the criterion K = 6 -f 3j8, — 2fi.,,
upon which the selection of the type of the frequency depends. Clearly, if the
probable error of this criterion is as large as the criterion itself, there can be no
stability of type, or the frequency may change over from one type to another.

On page 289 we have found the standard deviation of the criterion in terms of
known quantities for the curve

It is in fact given by the umbral equation

X.A/K = tXjfe, + i.&.jG.. ....... (clxv.)

where i} and I, are functions of ml and m., given in (cxx.) and x»»,X/». = ^>»,™, '8 known
from (cvi.).

The standard deviation of the criterion for the curve of type

y = y0e-™-*™l{\ +(aj/o)s}M
may be found by taking differentials of

K = 6 + 3/3, - 2& = - ^~ j 4 ^n^~~ + 1 J ,

a value readily obtainable from 'Phil. Trans.,' A, vol. 186, p. 377. We thus find
the umbral equation

^ J 0 , . , , i* - 3 ;• + 1 12 1 96 sin 0 cos <f> (r - 1) ^

XA - {96 sin- ^ j—j—— + (7_^J 2,Xr -- (, _ 3) (r _ 2)3 - S,X, (clxv,.)

where 2,, S^, and x/X* — ^ are given by (clii.) and (cliv.).
Applying these results to the numerical examples, we find : —

(a.) For the glands of swine

^ X. = - -070,5459 2WiXMi - -038,4629 2^XlBi,

K

whence the probable error of K = '67449 2* = '1012 ; or,

K— -5559 ± '1012.
(b.) For the stature of children

S.x« = -01 5,6206 S7.Xl. - -018,0067 2,x«»
whence the probable error of K = '67449 2K = '1919 ; or,

K= - -4330 ± '1919.

MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 311

In both cases, therefore, we may consider that the sign of K is heyond question, or
that the type selected is really a significant character of the frequency.

With regard to the probable error made in estimating a criterion to be zero, and
using a curve of type

we must remark that, the criterion being assumed zero is equivalent to assuming
that its probable error is zero. Accordingly the only satisfactory method of testing
whether a curve really falls under this type is to work out the probable error of its
criterion on the hypothesis that it belongs to one or other of the two types, with
positive or negative criterion as the case may be. If the probable error of the
criterion thus calculated is sensibly as large as the criterion itself, then we may
assume that the frequency distribution is of the type

[ 313 ]

VIII. A Compensated Interference Dilatometcr.

By A. E. TUTTON, Assoc. R.C.S.
Communicated by Capt. ABNEY, G.B., F.It.S.

Received March 8,— Read April 28, 1808.

THE instrument now described is a form of Fizeau interference dilatometer, in which
the sensitiveness of the interference method is so largely increased as to render it
unnecessary to employ a plate or block of the solid, whose expansion is to be
measured, of greater thickness than 5 millimetres. The author has been led to
adopt it for the accurate determination of the thermal expansion of the crystals
of artificial chemical preparations, which it is frequently impossible to obtain
sufficiently large, and at the same time homogeneous, to furnish plates a centimetre
thick, as demanded by the forms of apparatus hitherto described.

The exquisite method devised by FIZEAU (' Compt. Rend.,' vol. 58, p. 923, and
vol. 62, p. 1133; 'Ann. Chim. Phys.,' [4], vol. 2, p. 143, and [4], vol. 8, p. 335)
depends essentially upon the determination of the difference of expansion, which
accompanies rise of temperature, between the screws of a small metallic tripod
and the object under investigation which is supported by it. In the form of
apparatus now described the expansion of these screws is compensated and elimi-
nated, thus rendering the total expansion of the object available for measurement.
Hence, it is possible to obtain a result with a small crystal of as satisfactorily
accurate a character as was formerly only to be obtained with a much larger crystal.
Besides the introduction of this compensating principle, the new instrument com-
bines, in the author's opinion, the best features of the several forms of Fizeau
dilatometer previously described ; at the same time it is essentially different from
any one of these previous forms, and includes many details of a novel character.
The author is largely indebted to the Memoirs of BENOIT (' Trav. et Memoires du
Bureau Int. des Poids et Mesures,' vol. 1, 1881, p. 1, and vol. 6, 1888, p. 106),
concerning the Fizeau apparatus belonging to the Bureau International des Poids et
Mesures, Paris, and the classical work which he has accomplished by the use of it ;
and to the later one of PULFJRICH (' Zeitschrift fvir Instrurnentenkunde,' 1893, p. 365)
concerning an improved form of apparatus embodying the important modifications
introduced into the method by ABBE in the year 1884 and described by WEIDMANN
in 1889 (' WIEDEMANN'S Annalen,' vol. 38, p. 453).

VOL. CXCT.— A. 2 S 21.7.98

314 MB. A. E. TCJTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

DESCRIPTION OF THE APPARATUS.

The Interference Tripod.

The first essential of an interference dilatometer is the tripod support for the
object and for the glass lens or plate to be suspended, without touching, above the
latter. The upper surface of the object and the lower surface of the glass are
the two surfaces, by reflection of pure monochromatic light from which interference
curves or bands are to be produced ; and it is the movement of these bands upon
alteration of the temperature which affords the measurement of the change in
thickness of the air space which separates the two surfaces. For the passage of two
successive curves or bands past any given spot determines precisely that a change
of thickness equal to half a wave-length, of the particular light employed, has
occurred during the interval.

The tripod employed by BENOIT was constructed of platinum-indium ; that of
ABBE and PDLFRICH was of steel. The tripod of the apparatus now described is
constructed of platinum-iridium, which, in addition to its obvious advantage on
the score of unalterability, and its exceptionally small coefficient of expansion,
is specially suitable for a reason which will be apparent when the method of compen-
sation is described. It is represented in fig. 1.

Fig. 1.

It is entirely constructed out of a single homogeneous casting of the alloy of
platinum with 10 per cent, of iridium, which was supplied by Messrs. JOHNSON and
MATTHEY for the purpose, in order that the thermal expansion should be uniform
throughout its whole extent. It consists of a solid horizontal table 7 millims.
thick, circular in shape, but with three short equi-distant projecting arms, through
which are threaded the three screws, of very fine pitch, each 35 millims. long,
upon the bluntly-pointed upper ends of which the glass cover-wedge is to be
supported. The distance between the centres of any two adjacent screws is
30 millims. The screws are provided, near their lower rounded ends, with milled

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 315

heads for convenience of rotation ; and each of the arms is slit from one side, the
slit passing through the centre of the screw hole and slightly beyond the latter, and
provided with a small clamping screw, which draws the two sides of the slit together,
binding the main vertical screw and thus enabling it to be firmly fixed in any
position. Both sides of the table are polished, and on one side three concentric
series of raised points are provided, three equi-distant ones in each series, and each
series increasing in height as the centre is receded from. These serve as rests foi
objects of different sizes.

The Compensation.

It is an interesting fact that pure aluminium expands 2'G times as much as
platinum-iridium for the same increment of temperature, and it is this fact which the
author has utilised for the purpose of effecting compensation for the expansion of the
platinum-iridium screws. A circular disc of aluminium, whose thickness is about ten
twenty-sixths of the length of the parts of the platinum-iridium screws projecting
through and above the table, is laid upon the latter. Provided adequate care has
been taken to adjust the screws to the length calculated for complete compensation,
from a knowledge of the two coefficients of expansion, it will be evident that the
expansion of the apparatus is entirely eliminated. The length of screw provided
affords, after compensation, an available space of some millims. above the aluminium
block in which to place the crystal or other substance to be investigated, and as this
space remains unaltered by change of temperature, any alteration of the thickness of
the air layer between the upper surface of the object and the lower surface of the
glass disc laid upon the screws must be entirely due to the expansion of the crystal.
Moreover, the relative positions of the aluminium block and the crystal may be inter-
changed, it being immaterial which is uppermost. For the case in which the crystal
rests on the block, a series of circular aluminium blocks of 25 millims. diameter
and of respectively 12, 10, 8, G and 4 millims. thickness are provided, suitable for
use with all the various sizes of crystals which are likely to be met with. The
thinnest one is shown in position in fig. 1, resting upon the raised points. They
were constructed out of a homogeneous casting of the purest aluminium obtainable.
It is only necessary to select the block which is most suitable for use in connection
with the particular size of crystal to be employed, and to set the screws to the proper
corresponding length, which has been calculated, verified experimentally, and
recorded in a table once for all.

The polish which many crystalline solids take upon their ground plane surfaces is
rarely of a-character equal to that of glass, and it is particularly desirable that the
two reflecting surfaces relevant to the interference should reflect as equally as
possible, in order that the interspaces between the bands shall be as dark as possible.
Moreover, it is likewise desirable that the lower reflecting surface should be fairly
extended, in order to afford an ample field of interference bands ; and that a crystal

2 s 2

316 MB. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

should not have to be rejected as unsuitable because either one or both of the
parallel horizontal faces, perpendicular to which the expansion is to be measured, is,
relatively to the other dimensions of the crystal, small. For these reasons the
author usually lays upon the crystal a circular plate of glass, sufficiently thick to be
adequately rigid, the lower surface of which is ground dull and the upper polished
and worked perfectly plane. The glass employed is black astronomical glass, in
order that no disturbance may arise from an illuminated lower surface, as would be
more or less the case with colourless glass. The expansion of this glass, which has
been accurately determined, is very nearly the same as that of platinum -iridium, so
that the very slightly coirected measure of the thickness of the plate has simply to
be added to the length of the screws calculated for compensation by the aluminium,
and the screws sot to this total length. Three series of such discs, of respectively
25, 15, and 10 millims. diameter, have been provided for use with crystals of greater
or iess lateral extension ; for it is unwise to employ the 25-millim. disc with
crystals of small diameter, a columnar type placed on end for instance, and the
smaller ones still provide an adequate field of bands. One of medium size is shown
resting on a crystal in fig. 1. Each series consists of three plates, varying from
2 to 4 millims. in thickness, so that there is ample choice afforded. Such a choice is
preferably made as will leave, when the crystal and compensator are in position, an
air space, between the upper surface of the glass plate and the lower surface of the
large disc resting on the screws, of less than half a millimetre, in order that brilliant
interference bands may be obtained. The size mostly employed by the author with
crystals of artificial salts is 10 millims. diameter and 2 millims. in thickness.

[The second method of using the compensator, namely, above the crystal, is the
one which the author prefers to employ whenever possible. Three series of
aluminium discs are provided for use with this method, of respectively 15, 10, and
6 millims. diameter. Each disc is polished as truly plane as possible on one surface,
intended to be the upper surface when in position on the crystal, and carries on
the other side three equi-distant points, the ends of miniature screws of the same
aluminium firmly screwed into the disc. Each series consists of two discs, the
thicknesses of which, including the points, are respectively 4 and 5 millims. One
of these compensators is shown resting on the ground in fig. 1. By this method
the lower surface of the crystal rests directly on the three points of the
platinum-iridium table, the particular three varying with the size of the crystal but
being usually the innermost three, and the aluminium compensator rests by its
three points upon the upper surface of the crystal. Hence any movement due to
very slight convexity of the surfaces, or possible rolling upon a speck of dust
included between the surfaces, is entirely avoided by this arrangement of three-point
contact only. Moreover, the calculations are simplified and sources of error reduced
by avoiding the use of a glass crystal-covering disc. In this connection it is par-
ticularly fortunate that aluminium does not take a very high polish, but one which

MR. A. E. TCJTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 317

happens to reflect light about equally with the lower surface of the glass cover-
wedge laid on the screws, and thus affords most excellent bands. The smallest
compensators, of 6 millims. diameter, still exhibit a field of about eight bands of
normal width, ample for the purpose of the observations ; the points of these
compensators are only 5 millims. apart, so that the method is applicable down to
crystals of only slightly greater extent of surface than is adequate for these points
to rest upon. — May, 1898.]

The Cover- Weaye.

The thick glass disc of 40 millims. diameter, which is placed over the screws of
the tripod, and whose lower plane surface is to form the upper of the two surfaces
relevant to the production of interference, is not lenticular, as employed by FIZKAU
and by BENOIT, but is possessed also of an upper plane surface, as used by ABBE and
by PULVRICH, in order that parallel rectilinear bands and not curved ones may
be produced. The surfaces of this plate are not precisely parallel, but are inclined
at an angle of thirty-five minutes, forming in reality a wedge of extremely small
angle. By this device tho reflection of the illuminating light from the upper surface,
which would otherwise illuminate the interspaces between the bands, is deflected out
of the field of vision.

General Arrangement of the Dilatometer.

The interference apparatus which has now been described is supported during
the observations in a manner similar in principle to that employed by AISHK, but
differing in the constructive details; and the heating apparatus is an air bath of
special construction instead of an oil bath. Moreover, the interference tripod rests
on non-conducting material in direct contact with the heated air, whose temperature
is measured by the bath thermometers, and its actual temperature is determined by
a third thermometer, so bent that the bulb is in direct contact with it. Further,
this portion of the dilatometer, which may be termed the expansion apparatus, is,
unlike the ABBE arrangement, separated from the illuminating and observing
apparatus, which latter is removed to a considerable distance in order to be well out
of the range of the heated atmosphere in the neighbourhood of the air bath.

A general view of the dilatometer and its accessories is given in fig. 2. The two
parts, the expansion apparatus on the left, and the illuminating and observing
apparatus on the right, are mounted at the two ends of a rigid slate table six feet
long.

The Expansion Apparatus.

The details of the expansion apparatus will be more readily understood with the
aid of the section, fig. 3. The tripod, a, stands upon a thick circular glass plate, I, of
50 millims. diameter, which forms the floor of the interference chamber. This

318 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE D1LATOMETER.

bo

£

MR. A. E. TCTTTON ON A COMPENSATED INTERFERENCE DILATOMETER, 319

chamber consists of a wide gun-metal tube, c, very little longer than its diameter,
and the cylindrical wall of which is so considerably cut away in three equi-distant
parts as to form three relatively large windows separated by three equi-distant

Fig. 3.

pillars adequate to maintain rigidity. It is closed at the top by a stout diaphragm,
d, whose aperture is filled by a circular glass plate, e, 27 millims. in diameter, which
serves the double object of acting as a non-conducting roof to the interference
chamber, to prevent upward radiation and consequent loss of heat into the supporting

320 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

tube, and of correcting the slight prismatic deviation of the light rays introduced by
making the cover-wedge slightly wedge-shaped. The correction is achieved by
making the disc, e, equally wedge-shaped. In fact, the two discs are cut from the
same piece of plate glass, which has been ground down so that the two perfectly
plane surfaces are 35 minutes out of parallelism. The direction of the wedge is
marked by a straight line engraved parallel to the long edge of the slab before
cutting the discs from it. The latter are so cut that this engraved line comes near
the margin of each, forming a chord of the circle. It is only necessary to place the
two discs so that the engraved chords are parallel and on the same side of the centre,
but so that the marked surface of the larger disc of the interference apparatus lies
uppermost while that of the smaller diaphragm disc lies underneath, when the
refractive effect is fully counteracted. In order that no reflections shall be visible
due to the two surfaces of the countei-acting diaphragm wedge, the latter is slightly
tilted by means of two very small screws driven through the rabbet of the aperture
from the under side in such positions that the line joining them is parallel to the
engraved chord of the wedge ; when the disc is laid in its rabbeted aperture it
rests upon the two screw points and a third equi-distant point on the surface of the
rabbet, the lines joining the three points of contact thus forming an equilateral
triangle. The slight tilt given to the disc when the screws are properly adjusted is
sufficient to throw the reflected rays so far out of the optical axis of the system as
to be no longer visible through the micrometer eyepiece.

The chamber terminates above the diaphragm roof in a screw thread, f, which
engages with one on the lower end of the supporting apparatus. Below, the
chamber ends in an adjustable table, g, upon which rests a disc of asbestos millboard,
which, in turn, supports the glass floor disc. The adjustment is effected by three
milled-headed screws, h, worked from underneath, which raise or lower the table
above the rigid gun -metal base, &, of the chamber through which the screws pass.
This rigid base is also perforated by a central hole, through which passes a short
cylindrical rod fixed to the under side of the adjustable table ; the rod terminates in
a boss, I, between which and the under side of the fixed base a spring is confined,
which causes the adjustable table to be always pulled firmly down upon the ends of
the three screws. The chamber can be closed when desirable, in a light-tight manner,
by a concentric outer tube, m, capable of sliding over it from above, with sufficient
friction to maintain it up when the chamber is required to be open during adjustment
of the interference apparatus. The author removes it altogether when the interfer-
ence chamber is immersed in the air bath for the purpose of observations at the
higher temperatures. The rigid base of the chamber and its adjustable table may
be readily detached from the chamber if required, but the windows are sufficiently
large to admit the tripod without this necessity arising merely for that purpose. To
provide for its occasional desirability, however, the lower end of the chamber wall
terminates in an annulus and flange, n, which rest upon a corresponding rabbet cut

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 321

in the rigid base ; rotation is prevented by a slot and pin. The rigid base and the
flange are bound together by means of a large milled nut, o, which has above an
inner flange projecting over the chamber flange just referred to, and whose thread
engages with one turned in the periphery of the base.

The interference chamber is supported at the lower end of a non-conducting tube
of Berlin porcelain,^?, supported from an arm, q, carried by a stout gun-metal columnar
pedestal with thiee-legged base, provided with levelling screws i-esting on toe-plates.
The column is provided with a strongly-made vertical adjustment by rack and pinion,
worked by a milled ebonite disc, and provided with an arresting clamping screw,
manipulated by a lever to securely fix it at the required height. Sufficient rack is
given to enable the interference chamber to be 1'aised above the height of the air
bath, and conversely to be readily lowered into the latter. Particular cave has been
taken to render the working of the rack smooth and rigid, in order that no movement of
the adjusted interference apparatus shall occur during the operation. The arm is rotat-
able about the upper part of the column, and can be fixed when desirable by a clamping
screw. The outward end of the arm carries, as part of the same casting, a short thick-
walled tube, r, of the same diameter as the interference chamber ; at the lower end of
this tube the porcelain tube is supported. In order to be prepared for the possible occa-
sional fracture of a porcelain tube during heating, half-a-dozen such tubes were specially
prepared, from the author's pattern, at the Berlin porcelain works. They are glazed
outside and biscuit within, and have each a flange at either end ; by subsequent
trimming in the lathe the flanges have all been reduced exactly to the same size, so
that a fractured tube can be readily replaced by another. This is also facilitated by
the method of suspension, whieh is likewise one that renders fracture very improbable,
as it admits of freedom of expansion. The flange, s, passes easily up the metal
supporting tube carried by the arm, and after its insertion the two halves of a collar,
t, are passed up the metal tube under the flange until they are flush with the lower
end of the tube, when they are secured by screws passing through the tube and
collar from the outside. In order that the porcelain tube, thus easily suspended, may
be sufficiently rigid for the purpose in view, slight pressure is brought to bear on the
flange from above by a pair of circular bent springs, u, confined between the flange
and another internal collar, v, in the upper part of the metal tube. The lower flange
of the porcelain tube is similarly attached to a short metal tube, which carries inside,
at its lower end, an adequately long screw thread, which engages with the one at the
upper end of the interference chamber, by which means the latter is attached.

The incident light rays are directed into the expansion apparatus from the illumi-
nating apparatus, and the reflected rays back into the latter (which, being arranged
for autocollimation, serves also as observing apparatus) by means of one of two inter-
changeable pieces of deflecting apparatus. The first is a single total reflecting prism ;
this is employed during the preliminary adjustment of the interference apparatus,
with the aid of ordinary white light, and also when it is only desired to produce the
VOL. cxci. — A. 2 T

322 MR, A. E. TDTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

interference bauds by monochromatic flames. The second is a train of two flint glass
refracting prisms, which is employed for the production and observation of the bands
produced, with the aid of a Geissler tube containing rarefied hydrogen and mercury,
by the monochromatic light corresponding to the red and greenish-blue hydrogen
lines, C and F of the spectrum, and to the green mercury line found so suitable by
PULFRICH. The interchangeability of the two is very simply attained. In the top
of the tubular termination of the arm, above the internal collar, there fits a very
much shorter tube, w, terminating above in a stout flange ; it is capable of rotating,
for the purpose of azimuth adjustment, in the main tube, without the possibility
of vertical motion, by means of an annular groove, into which project a couple of
screws driven through the outer tube from opposite sides. The flange, whose central
aperture is of slightly greater diameter than that of the porcelain tube, carries at
two opposite sides raised and grooved guides, in which are capable of sliding the
correspondingly rabbeted basal annular supports, x, of the single or double prism
arrangements.

The single reflecting prism is supported in suitable bearings by central axles
rigidly attached to the metallic case, and provided with milled bosses at the ends for
convenience of rotating the prism ; this enables the latter to be adjusted for altitude,
and as the axles fit fairly tightly in the bearings, the setting remains unaltered after
adjustment.

The pair of refracting prisms are mounted in a light framework, whose shape is
calculated to ensure rigidity. Each prism is cased in metal, and the case carries
centrally at its two sides short but thick axles, narrowing somewhat after a distance
equal to the thickness of the mount, and terminating in a longer screw thread. The
axles gear at their thickest part in corresponding bearings in the supporting
framework. Over the two projecting axial screws on one side of the mount are
passed in each case first an arm carrying at its end a silver arc as if for a vernier,
and which slips over the screw and narrower portion of the axle only and is
arrested at the thickening, and then a milled nut engaging with the thread, by which
the arm may be clamped firmly to the axle and hence to the prism. As the thick
part of the axle projects a little beyond its bearing, however, the arm is not clamped
to the mount. The silver arc carries a central indicating mark, and by rotation of
the milled head and the prism travels over a divided silver quadrant carried by the
mount, and thus the prism may be set to any angular reading which may at any
subsequent time be reproduced. The axial screws at the other side of the mount
receive in each case first a washer and then a milled nut similar to that on the other
side ; the thick part of the axle being in this case slightly shorter than the thickness
of the frame, the prism is clamped firmly to the mount on screwing home the nut,
and so the prism may be rigidly fixed to the indicated circle reading. The refracting
angle of each of these prisms is approximately 57°, which in the case of the particular
flint glass employed gives a total minimum deviation due to the two of 90° for the

MR, A. E. TUTTON ON A COMPENSATE) D INTERFERENCE DILATOMETER. 323

middle part of the spectrum, and of about 88° for the red hydrogen line C, and 92°
for the greenish blue hydrogen line F.

The heating apparatus consists of a double-chambered cylindrical air bath,

mounted on a stout annular support resting by means of three levelling screws upon

toe-plates also earned by the movable slab. During adjusting operations, and

observations at the temperature of the room, the bath is removed from its stand.

It is constructed throughout of brazed copper, and is covered outside with a thick

coat of asbestos cloth. The inner chamber, y, is similar in shape to the outer, z,

from which it is separated all round by the air space of about 4 centims. Within

this inner bath the interference chamber of the dilatometer is to be lowered, by

means of the rack and pinion, for observations at higher temperatures. The opening

in the top of the bath is necessarily a little larger than the widest part of the

interference chamber, the flange of the rigid base, k (fig. 3). The inner bath is

supported within the outer one by the tubular wall of the opening, 3^ centims. deep,

which connects the top of the outer with the top of the inner bath. The top of this

tubular opening is closed, when the interference chamber has been lowered into

position, with the two overlapping halves of an asbestos-lined and closely-fitting lid,

which encircles the porcelain tube of the dilatometer, and is provided with suitable

handles above and a deep rim beneath, which latter passes down into the tubular

neck sufficiently far to obviate any appreciable diffusion of cold air from outside.

Through a wide tubular opening at the top and near the side of the bath there is

inserted into the outer bath, for the purpose of enabling the temperature within

the bath to be regulated with constancy, a Muenke thermostat, provided with a

regulator to control the size of the flame of the gas-burner when the mercury in the

thermostat closes the orifice in the steel gas-supply tube. There are also two short

tubes, on opposite sides of the centre, which pass air-tight through the tops of both

baths into the inner one, for the passage of two thermometers. Both the thermostat

and the thermometers are suspended free of the metal by non- conducting stoppers,

and the thermometers are arranged so that the bulbs are on either side of and in

close proximity to the interference tripod. A third thermometer, of such special

construction that its bulb lies in actual contact with the tripod, passes through a

hole in the front half of the lid of the bath. Details concerning the thermometers

will be given at a later stage. A twenty-jet Fletcher ring gas-burner is employed

as source of heat, the size of the jets being -controlled by the thermostat, which has

been adjusted so as to obtain a constant upper temperature, as near the particular

desired limit as possible. Of very great utility in attaining constant temperatures

has been found a gas-tap provided with a long pointed lever arm, travelling over a

graduated quadrant, divided to read directly to degrees. By its aid the temperature

can be very nicely regulated, leaving only such little further regulation to the

thermostat as is rendered necessary by the very slight variation in the pressure of

the gas supplied from a meter furnished with a governor.

2x2

324 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

Illuminating and Observing Apparatus.

The illuminating and observing apparatus consists of an autocollimating telescope,
mounted separately upon a stout pedestal similar to that which supports the
expansion apparatus, provided likewise with a very strong rack and pinion vertical
adjustment, in order to enable the telescope to be adjusted to the height required by
the particular deflecting arrangement mounted on the expansion apparatus. The

Fig. 4.

u —

— w

construction of the optical system will be more clearly evident from the section given
in fig. 4. Another view of the apparatus is given in fig. 5, so as to exhibit more
clearly the illuminating arrangement.

The main optical tube, a, carries at the end furthest from the observer an achromatic
objective, 6, whose focus is at c, which is likewise the focus of the rays from the
illuminating lens, d, carried at the end of the side tube, e. At c is an iris diaphragm, /,
immediately behind which are a fixed diaphragm, </, pierced by a relatively large

MR. A. E. TCTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 325

aperture, and a totally-reflecting prism, h, covering one-half of the aperture and so
arranged that the rays from the illuminating lens, d, are reflected through this half
of the aperture towards the objective, b, which latter they are capable of filling with
light. The surface of the prism nearest the objective is intended to be masked by
one or other of a series of interchangeable stops, k, having larger or smaller
rectangular apertures. A short distance after the prism comes a stout milled
flange, I, to the thread within the aperture of which either of two interchangeable
observing optical arrangements is capable of being attached. One, m, is an ordinary
eyepiece, for use in adjusting the interference apparatus by observation of the images
of the rectangular signal aperture, k, in front of the reflecting prism, reflected from
the various reflecting surfaces of that apparatus. The other, n, is an optical com-
bination of a micrometer two-lens eyepiece, o, with a third movable lens, p. The
system is such that when the lens p is at that end of its path nearest the prism, the
reflecting surfaces of the interference apparatus and the interference bands which they
exhibit are clearly focussed to the observer looking through the eyepiece o, when the
expansion apparatus is approached very near to the observing apparatus ; and when p
is at the end of its range nearest to the eyepiece o, the same result is achieved for
the separation of the two parts of the apparatus by about five feet. There is no
necessity for a separate adjusting apparatus, as used by PULFRIOH, for when the first
of the two conditions just referred to obtains, the two parts of the apparatus are so
close together that it is quite easy to manipulate the screws of the interference tripod
while actually observing. The relatively large windows of the interference chamber,
as distinguished from PULFRICH'S closed chamber, greatly facilitate this.

For convenience in moving the expansion apparatus between the two positions, the
toe-plates of its pedestal do not, like those of the telescope, rest directly upon the
long slate table upon which -the instrument is mounted, but on a smaller rigid slab of
hard mahogany, lined underneath with thick baize, and furnished with a couple of
stout brass handles.

In addition to the vertical rack and pinion motion provided in the pedestal of the
telescope, the latter can be adjusted for altitude by a tilting arrangement. It is
mounted about trunnions and the axis of rotation is made coincident with that of the
illuminating side tube, which latter forms a prolongation of the hollow axle on one
side. This is convenient when employing a source of light which does not move
with the apparatus as does the Geissler tube, the illuminating centre remaining in
the prolongation of the axis of the side tube whatever the tilt of the telescope. The
latter may be laid in the bearings with the illuminating tube on either side according
to convenience ; it is only necessary to remove the two screws which in each case
secure the caps of the bearings in order to effect a change of side. The telescope is
suspended sufficiently high above the transverse part of the bearing casting to admit
of considerable inclination from the horizontal position, and the depression or eleva-
tion is effected by a milled-headed screw which passes through a rigid arm radiating

326 MR. A. E. TDTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

from and forming part of the bearing casting, and which presses upwards on rotation
against a V-piece attached to the optical tube. Two such V-pieces are provided, on
the under and upper sides of the tube. It is convenient to have a means of recording
the proper tilt of the telescope corresponding to the adjustment of each of the
spectral lines (and therefore of the corresponding signal images and interference

Fig. 5.

bands), whose light is to be employed in the observations. This is delicately
achieved by placing a sharp edged flange above the milled head of the tilting screw
and employing it as the indicator of a vertical scale, divided on silver to half milli-
metres, and suspended from the horizontal arm which carries the screw.

The method of suspension is by a slider whose rectangular aperture is slightly larger

MB. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 327

than the dimensions of the arm at this point, affording means of adjustment by four
small adjusting screws driven through the top of the slider, whose points are pulled
tightly down upon the arm by two larger milled-headed binding screws passing
through the under side of the slider, one on each side of the scale. By means of
these screws the scale is readily adjusted to its proper position with respect to the
indicating flange edge. The scale is 60 millims. long, this distance corresponding to
the traverse by the telescope of rather more than the whole length of the spectrum.
Hence a delicate means is afforded of setting the telescope to the proper tilt for the
use of either C or F hydrogen 1'ght, or that corresponding to the green mercury line,
without actually observing the signal image for that colour at every observation. It
is only necessary to ascertain once for all, with occasional verifications, the scale
readings corresponding to these colours.

The bearing casting of the telescope carries below a vertical conical axis, which
rotates within a corresponding hollow cone or vertical bearing, thus admitting of
rotation of the telescope in the horizontal plane ; the greater portion of this vertical
conical bearing passes down into the wide internal boring of the pedestal, and it is
rigidly attached to the latter by a screw thread turned on the thickened upper part
of its cylindrical exterior and engaging with one in the upper part of the boring of
the pedestal. The telescope is prevented from rising in the vertical bearing by a
screw driven into the lower end of the solid cone, and a washer broad enough to cover
also the end of the bearing. This vertical bearing carries at the top as part of the
same casting, immediately above the thread by which it is attached to the pedestal,
a broad collar, passing on one side into an arm, intended for use in connection with a
fine adjustment fitting placed immediately above it. This latter takes the form of
another collar passing round the thickened upper end of the conical axis and con-
tinued on two opposite sides into arms. That on one side is short, and through it
passes a milled-headed clamping screw to attach the fitting rigidly to the cone, and
hence to the telescope ; the opposite arm is of the same length as the fixed one
carried by the vertical bearing, and its end carries below two elbow pieces, forming
two vertical claws, between which the narrower end of the fixed arm passes, with the
necessary amount of free play within which the fine adjustment can be effected. The
end of the fixed arm is pressed between the fine adjustment screw passing through
one claw, and a spring piston carried by the other. Hence, in order to effect fine
adjustment for azimuth, it is only needful to tighten the clamping screw, and then to
manipulate the fine adjusting screw.

The telescope is provided with a rack and pinion adjustment for the objective,
which is of 4 centims. aperture, and is placed at the end of an inner tube, q, which
carries the rack, r. This enables the objective to be brought exactly to the focus of
the signal aperture of the reflecting prism. There are also provided the means of
slightly tilting the objective, in order to throw the troublesome reflection from the
first concave lens surface just behind the diaphragm of the micrometer, and hence

328 MR, A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

out of the field of view. This is achieved partly in the mount of the achromatic
combination itself, and partly as a final adjustment, by screwing the mount, not to
the draw tube itself, but to a cap or short tube, s, slipping so easily over the latter
that when a pair of screws are passed through both at opposite ends of a horizontal
diameter, the cap is free to tilt slightly about them as an axis. By means of another
pair of screws at the ends of a vertical diameter, passing through the cap tube only,
the latter may be fixed at the slight tilt required. The objective cap is further
provided, for use when desirable, with an outer lengthening tube not shown in the
figures, projecting outwards 4 centims. beyond the objective, in order to shield it
from any extraneous side light.

The reflecting prism, I, is cased in metal on the hypotenuse side towards the eye-
piece. Between the glass surface facing the objective and the fixed diaphragm there
is just room to insert, through a niche cut in the outer tube, any one of the series of
interchangeable stops, k, suitable guides being provided to prevent actual contact
with the glass during the insertion or withdrawal. The stop is in its proper position
when it has been pushed home, as far as a handle at the top will permit. The
rectangular aperture is placed as near one edge as possible, and when the stop is in
position this edge is almost identical with the vertical diameter of the large circular
diaphragm aperture, and consequently with the common edge of the semicircular free
aperture and the prism ; also the centre of the stop lies on the horizontal diameter of
the circular aperture. Hence the axis of the bundle of rays incident from the rect-
angular signal aperture is as nearly as possible identical with the optical axis, and if
the rays reflected from the interference apparatus are brought to a focus in the free
aperture, so that the image of the signal is almost in contact with this same vertical
diameter, the two paths are practically identical, and the best conditions for inter-
ference attained. The apertures of the stops vary from 2 millims. by 1 millim. to
3 by 5 millims., and two series are provided, one with the longest sides of the
apertures horizontal, and the others with them vertical ; the former are naturally
preferable for use with the refracting deflection apparatus of the dilatometer. The
smallest size is, of course, preferable, provided the source of light is sufficiently
powerful. With the author's Geissler tube the illumination of the bands is many
times more powerful for C or F light than it is when using a sodium flame. An
additional stop is also provided, which is furnished with a means of varying the
vertical height of the aperture, and which is intended for use as a fine slit. At that
position in the metallic strip, which will be irfthe horizontal diameter of the field of
the telescope when the stop is in position, a rectangular aperture of 4 millims. width
is commenced, and continued upwards as far as and right through the handle. This
long aperture runs, of course, parallel to the edges of the strip, and very close to the
edge on the side which is to be nearest to the optical axis when in position. Through
the handle a movable strip, the same width as the aperture, but longer, so as to
project through the handle, passes, adequately guided by miniature guides in the

MB. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 329

edges of the aperture and corresponding bevels on the slider, the lower horizontal end
edge of which forms the upper jaw of the slit. Both this edge and that of the main
strip which forms the lower end of the aperture and acts as the lower jaw of the slit,
are bevelled. The top of the movable slider projecting above the handle is furnished
with a smaller handle, for convenience in setting it so as to adjust the fineness of the
slit. This slit enables the spectrum of the illuminating source of light to be studied.
As the light traverses the two refracting prisms twice, the apparatus aftbrds a disper-
sion equal to a spectroscope of four such prisms, and hence the study of the spectrum
is highly interesting.

The iris diaphragm, f, is placed almost in contact with the fixed diaphragm, r/ ;
it is manipulated from outside by means of a lever handle traversing a slot, upon the
edge of which is a graduated silver arc, the indications of which, corresponding to
definite apertures, are recorded by an indicator attached to the handle.

The lens of the illuminating tube is carried in a short sliding tube, t, so that the
most favourable position for the illumination of the rectangular signal aperture can
be attained. When white light is being employed for adjusting purposes, or when
sodium light is being used for observations of the bands, this is all that is necessary.
When a capillary Geissler tube for longitudinal vision is being employed as source
of light, a further arrangement is needful for adjusting the capillary so as to attain
the maximum illumination of the signal. The Geissler tube employed is of the
H type recommended by PULFRICH, which consists of two wide terminal tubes, >i,
arranged vertically and fitted with aluminium spiral terminals, and a horizontal
capillary connecting tube, v, the brilliant glow in which, regarded end on, is the
source of light. The tubes, after Dr. EIEDEL'S pattern, were supplied by ZEISS.
They contain a hydrogen vacuum, and a globule of mercury in one of the wide
limbs, which is placed furthest from the lens, and on gently warming which the green
light due to mercury vapour, corresponding to the green line of the mercury spectrum,
makes its appearance. This particular radiation, when separated from the hydrogen
radiations by the train of prisms of the expansion apparatus, forms the most
effective monochromatic source of illumination for the production of interference
bands when the two reflecting surfaces are at a considerable distance.

The supporting and adjusting arrangement for the Geissler tube affords the means
of both centering and adjusting, so as to bring the capillary exactly into the pro-
longation of the optical axis of the illumination tube, and also for moving it
longitudinally in this axis to the most suitable distance from the lens. It consists
of a tube, iv, which, fits fairly tightly over the broad flange which forms the lens
mount of the sliding tube ; this flange is of slightly greater diameter than the
illuminating tube itself, hence the latter is not injured by the placing in position and
removal of this attachment. On the outside of this attached tube slides another, x,
movable over it by means of a pinion gear carried at the end of the tube nearest
the observer and travelling with it, which gears with a rack carried by the inner
VOL. oxci. — A. 2 u

330 ME. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

attached tube. At the outer end of the movable tube is carried a three-disc center-
ing apparatus, y ; the second and third discs are complementarity pivoted about
points near their periphery, the second about the first, and the third about the second ;
they are maintained rigidly in position by similar pins on the other side of the centre,
working in slots, and their movement is effected by a pair of screws and spring
pistons, one carried by the first and another by the third disc, working against pro-
jections carried on each side of the middle disc and which are movable in larger slots,
cut likewise in the first and third discs. The three discs are pierced by central aper-
tures sufficiently large to admit the rays from the capillary to the lens. Attached to
the third disc is a vertical tubular holder, 2, for the Geissler tube, one limb of which
slides easily in it, a slit being provided in its outer side for the passage of the capillary
as far as the centre. Owing to the presence on this limb of some of the author's
Geissler tubes of a little sealed side tube, which had been used for the purpose of the
exhaustion, the sliding into position of the Geissler tube is brought about from
below. When the capillary is raised as far as the slit permits, it is approximately
central ; a loop support is then attached below it to maintain it in position, by means
of a pair of milled-heacled screws, which secure the loop to the stout bracket which
attaches the tube holder to the third centering disc. The inner side of the tubular
holder has a central aperture corresponding to those of the three discs. The bracket
is attached in such a manner to the third disc as to provide a means of adjustment
for altitude, the claws, one on either side of the holder, not being secured by the
pair of screws directly in contact with the disc, but being able to swivel more or less
about a horizontal axis formed by a pair of short pins, one carried by each claw
between the two screw holes and on the side which would otherwise be in contact
with the disc. The slit in which the capillary slides being adequately large, adjust-
ment for azimuth is readily attained. Hence every required adjustment of the
capillary is provided for. In order that the Geissler tube may be firmly held, and
also that the glass may not be in direct contact with the metal tube, a pair of
caoutchouc rings of suitable thickness are introduced as packing, one near the top
and one near the lower end of the holder. As the Geissler tube fitting requires to
be frequently removed, and it cannot be laid down without the risk of mercury
getting into the capillary, an inclined cloth-covered supporting pillar, on a suitable
stand, is provided for its reception when not in use. The pillar is of slightly less
diameter than the inner tube of the fitting, so as to pass inside it until arrested by
the centering discs. The Geissler tube can thus be always left in position in the
fitting. It is shown on the side table in fig. 2.

The milled flange, I (fig. 4), which completes the main optical tube, and to the
aperture of which the observing eyepiece or micrometer combination is attached, has
been constructed in duplicate, one of metal, for use whenever the Geissler tube is not
necessary, and another of ebonite, for use in connection with the Geissler tube.
For it is found that when the latter is employed, the relatively powerful electrical

MR. A. K. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 331

induction discharge, corresponding to a 5-centim. spark in air, leaks slightly along
the metal of the observing apparatus, and if no insulator is interposed, small sparks
are discharged from the eyepiece to the observer's eyebrows. The thick ebonite
disc acts as a perfect insulator, and prevents this disconcerting occurrence.

The ordinary eyepiece, m, slides in a short tube which is directly screwed to the
flange, I, the amount of slide being adequate to enable the observer to sharply focus
the vertical edge of the reflecting prism, h, which divides the iris diaphragm into
two halves, a semicircle of light (clear aperture) and a dark half covered by the
prism.

The micrometer optical combination, n, slides for about 3 centims. in a similar
but slightly longer tube attachment to the flange, it is then prevented from sliding
further by the thickening of the tube. This thickened part eventually passes into
a still thicker one, which is directly attached to the micrometer box, a ; in front
of the latter is attached a short tube within which slides the Ramsden double
eyepiece, o, which focusses the micrometer spider-lines. Three such eyepieces, of
graduated magnifying power, are provided, suitable for all the widths of interference
bands likely to be employed. As the movable lens -p is only required in two positions,
one corresponding to the close approximation of the expansion apparatus and the
telescope, and the other to their removal to the two ends of the slate table, two
inner sliding tubes are provided, of respectively 15 and 57 millims. length, each
carrying at one end an inner thread corresponding with one on the lens mount.
When the lens is attached to the shorter tube, and the latter is pushed in the long
tube of the combination until the ends of the two tubes are flush with each other,
the combination, when in position, focusses the bands for close quarters ; when the
longer tube, p ' , is employed, and similarly pushed in until flush, the lens being then
in the beginning of the thickest part of the long tube of the combination, the bands
are focussed when the expansion apparatus and the telescope are separated at
opposite ends of the table. If angle marks are placed on the table as indicators for
the position of the corners of the movable slab upon which the expansion apparatus
is mounted, corresponding to the sharpest focus under the two conditions, it is easy
to bring the slab exactly to one or other of these positions.

The micrometer employed by the author is one in which the spider-lines are
moved, not the eyepiece as in the Abbe apparatus. Moreover, it is a double one,
carrying a divided drum on each side of the box. Each drum is divided into
100 parts, every 10 being figured. It may be set so that the zero corresponds
exactly to any desired position of the spider-line or lines by means of a milled
clamping boss carried at the end of the axle, which is here of square section ; the
aluminium drum itself, being on a round part of the axle, is free to move independently
of the spider-lines until the boss is pushed firmly against it and fixed there by a
screw driven into the axle, which does not extend so far as to be quite flush with
the boss ; the drum is then held firmly between the boss on the outside and

2 u 2

332 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

the thickening of the axle on the inner side. There are two movable vertical
spider -lines, and a fixed horizontal one; below and parallel with the latter is the
finely-serrated edge of a diaphragm, forming the lower boundary of the field. Every
fifth niche of this edge is more deeply indented, and every tenth more still. One of
the drums, b', moves one only of the vertical spider-lines, and thus enables it to be
set at any recorded distance from the other, or to measure in drum units the distance
of any object in the field from the fixed line. The distance between any two teeth
or niches of the diaphragm edge is equal to the distance traversed by the spider-
line for one complete revolution of the drum. The other, c, moves both vertical
spider-lines simultaneously, the movement from one tooth or niche to the next
corresponding to one complete revolution of the drum. Hence, eacli niche corre-
sponds to 100 drum units, and each of the deepest niches to 1000. The micrometer
thus affords every convenience for the measurement of the thickness of the bands
and the precise location of any one with respect to any point on the horizontal
spider-line, and will be equally suitable for any proposed method of procedure. It is
particularly convenient for determining the position of a band by bringing it between
the two parallel vertical spider-lines, which can be arranged at such a distance apart
as is suitable for the purpose.

The fixed point with reference to which the position of the bands at the two
limiting temperatures is to be measured, and the number of bands passing which
during the change of temperature is to be ascertained, is of the kind adopted by
PULFEICH, namely, the centre of a silvered circular spot in the middle of the lower
surface of the glass wedge of the interference apparatus. The spot is about 1 millim.
in diameter, and the author has removed the central portion so as to leave a silver
ring, the inner circular edge of which is employed as the reference circle. Such a
ring is readily made by silvering the whole surface by means of a milk-sugar ammo-
niacal silver solution, and removing the greater portion by ordinary means, and the
portion about and within the ring with the aid of a rigidly supported needle and a
microscopist's turn-table. The spider-lines are to be placed at such a distance apart
as is just slightly smaller than the diameter of the inner silver circle. This enables
the centre of the spot to be accurately located by setting the pair of spider-lines so
that an equally small arc is cut off from the silver "circle on each side. The inner
portion is found by the author to be very advantageously left clear, as it enables a
band passing the centre to be actually observed at the centre.

To illuminate the Geissler tube, the author employs a 15-centim. spark induction
coil, actuated to the extent of a 5-centim. spark. A 100-volt direct supply current
is used, filtered through five 32-candle-power lamps arranged in parallel circuit. As
each lamp allows 1*2 amperes to pass, the current supplied to the primary coil is
6 amperes, and this current affords a very satisfactory and constant illumination of
the tube.

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DTLATOMKTRR. 333

The Thermometers.

The two thermometers, whose lower halves were immersed in the inner bath on
each side of the interference chamber, were the two excellent instruments supplied
by FUESS, with the most recent form of heating apparatus in connection with his
largest gonio-spectrometer No. IA. They had previously been tested several times,
and twice during the course of this expansion work their zero and 1 00° points were
again determined with great care. The zero of both remained exact throughout, and
the correction for boiling-point never exceeded 0°'18. Experience has shown, how-
ever, that the temperature of the interior of the interference chamber never quite
attains the temperature of the inner bath at the higher limits. The difference for
the neighbourhood of 70° is about 2°, and for the higher limit of 120° it is rarely less
than 4°. These bath thermometers were merely used, therefore, for the purpose of
attaining and maintaining a constant temperature in the inner bath. For it is found
that if the temperature of the inner bath can be maintained constant to 0°'2, which
can readily be attained by the combined use of the graduated gas-tap and thermostat,
the actual temperature of the tripod remains constant to well within 0°'l, and as it is
capable of exact measurement, and the bands move precisely with the temperature of
the tripod, all error of temperature disappears.

The measurement of the actual temperature of the tripod is attained by a third
thermometer, specially constructed for the author by Messrs. NEGRETTI and ZAMBRA.
This thermometer is bent at right angles just above the bulb, and it is so suspended
alongside the interference tube that the small elbow carrying the bulb passes right
into the interior of the interference chamber itself, without contact with any part of
the walls of the chamber, and the cylindrical bulb rests in actual contact with the
upper surface of the platinum-iridium table and one of the screws. Special care has
been taken to determine its fixed points from time to time ; its capillary was a
specially selected one, and its zero and boiling points have required only the minutest
corrections. It was so constructed that the 70° mark came just outside the bath, so
that no correction for exposed stem is required for the lower limit. For the higher
limit of 120° this correction is necessary, and to enable it to be made another much
smaller thermometer was attached to the stem, with the bulb opposite to the 90°
mark. That the indications of this thermometer actually express the temperature of
the tripod and whatever it supports, is proved by the fact that the bands follow it
exactly, their motion being arrested simultaneously with that of the mercury column.
The importance of actual contact of the bulb of the thermometer with the interference
tripod has been very exhaustively proved. Mere hanging of a straight thermometer
alongside the chamber, as in the apparatus employed by PULFRICH, affords no
guarantee, especially with a closed interference chamber, that the tripod and its
contents actually attain the temperature indicated thereby. In the author's expe-
rience, even with an open chamber, it never does. The use of oil in the bath cannot

334 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

do more than slightly reduce the source of error, besides introducing an error due to
the impossibility of stirring the oil, on account of the disturbance which a stirrer
would cause to such highly delicate observations as are here in question.

i • < *

Recording Apparatus.

ABBE has shown that it is not necessary to actually follow and count the number
of bands which pass the silver spot during the interval between the two limiting
temperatures, it being possible to deduce the number by calculation from the data
afforded by observations of the initial and final positions of the bands adjacent to the
silver spot for light of two different wave-lengths. It is necessary, however, that
these initial and final positions should be determined with the utmost precision. In
addition to employing this method, which has been elaborately worked out by
PULFRICH (loc. cit.}, the author prefers to remove the slightest possibility of doubt as
to the number of bands which pass the reference spot by actually following and
counting them, as a mistake of a single band is highly important when the observa-
tions relate to the very slight differences of expansion between the members of a series
of isomorphous salts. Moreover, it is far more satisfactory to observe the transit of the
bands throughout, in order to be assured that it has been unbrokenly uniform, and
that there has consequently been no derangement of the adjustment, or cracking, or
irregular expansion of the crystal. In order that no error of counting may occur, a
method has been adopted by means of which a permanent record of the passage of
each band is obtained. A small recording apparatus has been constructed, of such a
nature that, by pressing down a key with the finger the moment each band passes
the silver spot, a puncture is made in a paper tape, and on releasing the key the
tape is moved on a short space ready for the next puncture. When the bands
become stationary again, on the attainment of the higher limit of temperature, the
tape is cut off behind the last puncture, and thus a permanent record of the number
of bands which have made their transit is obtained, which can be counted at leisure,
and verified as frequently as may be desired.

The recorder is shown to the right on the accessory table in fig. 2. A drum,
carrying a roll of Morse tape, is suspended on an axle inside a rectangular box,
the front side of which is hinged below in order that the box may be thrown
open, for the purpose of replenishing the drum. For this latter purpose the axle
is fixed, and the drum provided with a corresponding central bore ; the drum can
consequently be drawn forwards right off the axle, and replenished, and is again
exactly in position when pushed on the axle as far as it will go, a stout boss
attached to a disc screwed to the back of the box arresting its further progress. In
order to prevent the drum coming forwards more or less off the axle during working,
which would pull the tape out of position, a spring is fixed to the inside of the front
lid, and when the latter is closed and is fastened by the two hooks and pins provided

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 335

for the purpose, this spring presses lightly against the drum and maintains it in
position. The tape comes up through a slot in the top of the box, and passes over a
small roller raised on supporting bearings about half an inch above the surface.
From this roller the tape passes horizontally through the punch and thence between
a pair of caoutchouc rollers, which compress it sufficiently to be able to determine its
movement when the lower driving roller is rotated. The upper roller is free to rotate
on its axle simultaneously with the lower one, but in the opposite direction, by simple
contact with the latter ; its axle is adjustable for height in order to obtain the
necessary amount of pressure between the two rollers, by means of two little milled-
headed screws carried on the top of these supports ; the screws press down upon
rectangular sliders carrying the axle and sliding in slots cut in the upright supports.
The axle of the lower roller is continued backwards just beyond the back of the box,
where it carries, rigidly attached to it, a ratchet wheel. The movement of the latter
is determined by a click, maintained in position against the teeth by a spring ; the
click is pivoted to and in front of a short lever, the fulcrum of which is the end of
the roller axle, upon which it freely turns. The teeth of the ratchet are so arranged
that upward movement of the lever and click causes the latter merely to pass round
the tooth, while downward movement causes the click to force the ratchet round
by one tooth. The lever is connected with a longer one by a short vertical link ; the
longer one is pivoted at the end of a prolongation of the drum axle, and its move-
ment is directed by another vertical link, which is hinged to the back end of the
punch lever. .This latter is supported by two fulcrum upright supports, which form
the bearings of the horizontal axle about which it is capable of movement. The
longer front end terminates in a knob above and a cylindrical rod beneath ; the latter
passes slightly through a hole in the top of the box when the lever end is up at its
normal height. The lever is kept up by a strong spiral spring wound round the rod
and slightly compressed between the lever end and the top of the box. Between
the knob and the fulcrum the lever carries below a thickening, directly over the
punch head, not in contact with the latter, but separated from it by about 4 millims.
The small cylindrical punch is provided at the top with a circular head carrying an
annular groove, in which gear the two prongs of the forked end of a strong strip of
steel spring. The spring is fastened at its other end to a suitable raised support, and
the punch passes down, but not quite so far as the tape, through a guiding block
provided with a corresponding vertical bore and fixed to the upper plate of the punch
bed. Underneath this, leaving a space sufficiently thick for the passage of the tape,
is the steel plate of the punch bed, whose sharp-edged circular aperture corresponds
to the punch, which fits it sufficiently tightly. It will be at once evident that on
pressing down the knob of the lever and thus compressing the spiral spring round
the rod, after traversing the 4 millims. space the punch will be pressed through the
tape, and the click of the ratchet will be slipped upwards round one tooth ; on
releasing the knob, the spring pushes it upwards again, but the click now drives the

336 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

ratchet round for the space of the tooth, thus causing the caoutchouc rollers to draw
the tape onwards for a corresponding space. The size of the ratchet teeth is so
arranged, relatively to the amount of movement given to the click, that the move-
ment of the ratchet and the tape does not occur until the punch has been extracted
from the tape.

Readiny Telescope.

A further accessory is a small reading telescope, by means of which the ther-
mometers can be watched by the observer without leaving his seat. This is mounted
upon a standard provided with the usual means of adjustment for height, altitude,
and azimuth. An inner sliding rod, capable of being arrested at any height by
means of a clamping screw passing through the upper boss of the fixed standard,
affords the means of adjusting the height of the telescope. Azimuth is arranged
for by attaching the telescope to a short tube sliding over the lengthening rod, and
fixable at any position by means of a slit and tightening collar carried at one end of
the tube. Altitude is provided for by the means of attachment to the sliding tube just
referred to, which carries a short projecting horizontal rod over which slips tightly a
corresponding short, narrow, horizontal tube, carried by the wider collared tube
through which the telescope is inserted ; rotation of the narrow tube over the rod
enables any desired tilt to be given to the telescope, or the latter to be arranged
precisely horizontally with the aid of a spirit level laid along the optical tube. A
further adjustment is attainable by means of a joint in the lengthening rod, which
enables the telescope to be brought to the most convenient position for the eye
when it is impossible to do so by moving the standard bodily. A strong clamping
screw is provided with the joint. At the summit of the lengthening rod is carried a
clamp which supports a large circular screen of dark-green silk, which serves the
double purpose of protecting the observer's eyes from the glare of the vacuum tube,
and interposing a broad non-conducting surface between him and the electrical
connections of the tube. The whole arrangement is seen in position in fig. 2 (p. 318).

Accessory Interference Apparatus for Observing the Fizcau Phenomenon.

The " Fizeau phenomenon " of periodical secondary interference, produced when
light not strictly homogeneous is employed for the generation of the bands, may be
very conveniently observed with the aid of the little accessory to the interference
apparatus which is shown resting on the movable slab of the expansion apparatus
in fig. 2, and which enables the separation of the two reflecting surfaces to be varied
at pleasure.

It consists of a special base similar to the ordinary one (k of fig. 3), and adjustable
table (g of fig. 3) ; the latter carries three adjustable screws to act as an interference
tripod for the support of the large glass cover-wedge, and a platform, adjustable for

ME. A. E. TUTTON ON A COMPENSATED INTERFERENCE D1LATOMETER. 337

height, for the support of the object whose upper horizontal surface is to act as the
lower reflecting surface. The base has a much larger central aperture than the
ordinary one (k of fig. 3) in order to permit the passage of a stout, hollow cylinder,
rigidly carried below by the table ; otherwise it is similar to the ordinary one,
having three screws passing through it from beneath, pressing upwards against the
table for the purpose of adjusting the latter, which is pulled firmly down upon them
by a stout spiral spring, confined between the under side of the base, and a collar
carried by the hollow cylinder. Within this hollow cylinder slides another, also
very stout walled, and closed at the upper end, which is truly plane perpendicular to
the axis and forms the vertically movable platform. The fit of the two cylinders is
a very close one, and the movement is effected by a pinion, whose manipulating
milled head gears in a rack sunk in the movable internal cylinder. It is imperative
that there should be absolutely no alteration of the parallelism of the vertical axis
during movement, as the minutest amount of such alteration is sufficient to entirely
alter the position and width of the bands ; hence the closeness of the fit and the
relatively considerable length of the cylinders. In order that the amount of
separation of the two surfaces may be accurately determined, the milled head of the
pinion carries at its inner side a divided drum, reading directly to quarter millimetres
with the aid of an indicator carried by the outer cylinder. The three tripod screws
are adjustable for height by manipulation of the milled flanges carried some little
distance below the pointed upper ends, which enable the screws to be more or
less screwed into the fixed nuts carried on the surface of the table. The base carries
a screw thread on its periphery, similar to that on the ordinary one, for attachment
by means of the tapped flange to the lower end of the interference chamber.

Thickness Measurer.

The measurements of the thickness of the objects investigated, and of the various
accessory aluminium blocks and glass discs employed in the work, are carried out by
the author with the aid of an admirably accurate thickness measurer of the same type
as that furnished by ZEISS, for use in connection with the Abbe dilatometer, and
described by PULFRICH in the ' Zeitschrift fur Instrumentenkunde,' 1892, p. 307.
An improvement is introduced, however, into the method of suspending the
counterpoised vertical silver scale, which does not form the continuation of the
agate-pointed contact rod, but is suspended in front of the latter, in such a manner
as to allow of a delicate means of adjusting the zero of the scale exactly to the centre
of the pair of parallel horizontal spider-lines of the micrometer eyepiece of the
microscope, when the agate point is resting upon the glass disc fixed in the base.

The instrument is shown in fig. 6, from which the nature of the improved method of
suspension will be apparent. Eound the rod, at the suitable height, a collar is fixed
by a clamping screw ; the collar is continued forwards and sideways into a bracket,

VOL. cxci. — A. 2 x

338 MB. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMBTER.

between the two arms of which the scale is indirectly suspended. The upper end of
the scale slides tightly in a short jacket which m directly suspended from the arms of
the bracket by two screw-pins. The jacket is pierced in front by a square aperture,
through which passes a projecting piece, slightly less in size, screwed firmly to the scale ;
above and below the aperture the jacket carries two adjusting screws between which the
projecting piece is firmly gripped, and which afford the desired means of adjustment.

Fig. 6.

A sliding piece, fixed to the scale half way down, but sliding round the rod, maintains
the scale strictly parallel to the rod. The mode of counterpoising the latter so that
the agate point may not exert undue pressure upon the object, is the same as in the
Zeiss instrument. The scale is divided directly to fifths of millimetres, each division
being consequently 0'2 millim. The drum of the micrometer is divided into 100
parts, the tens being figured as units. For one complete revolution the pair of hori-
zontal spider-lines travel vertically through half a scale division, that is, O'l millim

ME. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETKR. 339

The tenths of millimetres are also indicated by the teeth of a serrated diaphragm,
which forms the left edge of the field of the micrometer eyepiece, and whose centre,
corresponding to the zero of the scale, is specially indented. The figures on the
drum thus represent the second place of decimals of millimetres ; and the small
divisions of the drum represent the third place. Hence the instrument reads directly
to O'OOl millim.

ADJUSTMENT AND USE OF THE APPARATUS, AND DETERMINATION OF ITS

CONSTANTS.

Before the apparatus can be employed for the determination of the coefficient of
the absolute expansion of any substance, it is necessary to determine the expansion
of the platinum-iridium alloy of which the interference tripod is constructed, in order
that the expansion of the three screws of the tripod may be known. Having
ascertained this, the expansion of any substance may be determined by FIZEAU'S
method, provided a homogeneous specimen of the substance can be obtained in the
f-rm of a stout block, furnished with two parallel surfaces at least a centimetre
apart. For the author's purpose of being able to determine the expansion of a
substance with a block only half a centimetre thick, it is essential to proceed imme-
diately with the determination of the expansion of the specimen of aluminium from
which the series of compensators are cut. A block between 12 and 13 millims. thick
was employed for this purpose. It' the method is adopted of employing the com-
pensator above the crystal, these are all the data required before being able to
employ the apparatus directly for the determination of the absolute expansion of a
crystal or other small object. If the compensator is employed below the crystal, a
determination of the expansion of the glass of the small plates, to be used as covering
plates with crystals whose surfaces take a duller polish than glass, also requires to
be carried out. A block of the glass, 13 millims. thick, was similarly employed for this
purpose. The operations of making these preliminary determinations of the con-
stants of the interference apparatus will so fully include all the necessary manipu-
lative and adjusting operations in connection with the use of the apparatus, that
the description of the mode of conducting them, and the discussion of the mode of
calculating the coefficients of expansion from the experimental results obtained, will
at once be proceeded with.

Determination of the Expansion of the Platinum-iridium Interference Tripod.

This is the most difficult of all the operations in connection with the work, on
account of the fact that it ia necessarily performed with the two reflecting surfaces,
the under surface of the large glass cover-wedge laid upon the three screws and the
upper surface of the platinum-indium table, separated at the relatively long distance

2x2

340 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

of somewhat over a centimetre. At this distance it is most difficult to generate
satisfactory bands with C or F hydrogen light, and it is only at all possible with
sodium light provided the exact distance of a FIZEAU maximum is attained. The use
of the light corresponding to the green mercury line, as recommended by PULFRICH,
affords, fortunately however, excellent bands in the author's apparatus, even at
distances considerably beyond 2 centims. At 12 millims., between glass
surfaces, the bands are really admirable. A further difficulty, however, arose,
owing to the greater volume of reflection from the polished plane surface of the
platinum-iridium table than from the glass cover-wedge, the excess of light from the
former illuminating the dark spaces and causing the bands to appear very faint.
This difficulty was overcome by depositing upon the relevant glass surface, by means
of a milk-sugar ammoniacal silver solution, a thin film of silver of the right density
to afford the necessary increased reflection from the glass and diminished reflection
from the platinum-iridium. After numerous attempts a film was obtained winch
caused the reflections to be almost exactly equal, and consequently enabled excellent
bands to be obtained. The reference mark of the disc was a minute central ring,
where the silver had been scraped off by a needle point whilst rotating the disc on a
microscopist's turn-table. It should be remarked that this cover-wedge, employed
for the purpose of the determination of the expansion of the tripod, was a duplicate
of the ordinary one carrying the silver ring previously referred to. A corresponding
duplicate smaller disc was employed in connection with it to close the upper aperture
of the interference chamber, both being cut from the same slightly wedge-shaped
slab of glass, as in the case of the ordinary pair. The two pairs, being of slightly
different angle, were provided with distinguishing marks on their edges to prevent
bhe use of a wrong combination.

The screws of the tripod were adjusted so that about 12 millims. length projected
oeyond that side of the table which carried no points, and the tripod was placed in
the interference chamber with this side uppermost. The glass floor of the chamber had
previously been made exactly horizontal with the aid of a circular spirit level and the
adjusting screws. In order to prevent reflection of light from the marginal portion
of the platinum-iridium surface, a lens atop of 15 millims. aperture and convenient
diameter was laid upon it as a mask. The cover-wedge was then laid upon the
screws with the silvered side downwards, and the engraved chord, marking the
direction of the wedge, arranged perpendicular to an imaginary line at this height
ioining the pedestals of the expansion and the observing apparatus, that is, running
right and left of the observer in front of the latter. The counteracting upper wedge
was arranged with its chord in a parallel direction and on the same side of the centre
to that of the cover-wedge, but below, while that of the cover-wedge was uppermost,
on the non-silvered side.

The expansion apparatus was arranged close up to the observing apparatus for the
adjustment, and carried the large single reflecting prism in position at its summit.

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 341

The heating bath had been temporarily removed. The telescope was fitted with the
simple eyepiece and was arranged for parallel rays, the eyepiece focussing the vertical
line of division of the iris aperture formed by the edge of the little reflecting prism.
An incandescent electric lamp with ground glass globe, and provided with an opaque
shade pierced by a circular aperture the same size as the illuminating lens, was
brought close to the end of the side illuminating tube, the Geissler tube fitting
having been removed. The telescope was then directed at the reflecting prism of
the expansion apparatus, and the prism of the latter adjusted so that the images of
the rectangular signal stop in front of the small reflecting prism of the telescope,
reflected from the four glass surfaces of the two wedges and from the platinum-
iridium table, could be viewed and any one brought into the centre of the clear
half of the iris aperture by slight movement of the telescope. The iris aperture
should be open to the full during this operation. By use of the rack and pinion of
the telescope any slight want of sharpness of the images can immediately be
corrected.

There are, in general, five such images from the interference chamber, two due to
the two surfaces of the upper wedge, two to the cover- wedge, and one from the object,
in this case the platinum-indium table. They are easily recognisable. By slightly
lifting and replacing the cover-wedge, the two due to that are distinguished by their
temporary movement or disappearance. The angle of the wedge is such that the two
images are so far apart in a horizontal line as to be just incapable of being simul-
taneously visible in the semicircular field of view. The pair of images due to the
upper correcting wedge ave equally distant from each other, but they should be
about the same distance below the other pair as they are apart from each other, and
a little to one side, owing to the suitable tilt given to the wedge by the two small
tilting screws. These four images are thus arranged at the four corners of a
rhombus.

The first thing is to distinguish which of the two cover-wedge images is afforded
by the lower surface. The author always works with the thickest side of the wedge
to the right, and this causes the image from the lower surface to appear to the right
of the other. Hence, the image in question forms the right top corner of the rhombus.
Moreover, if the lengths of the platinum-iridium screws had been adjusted to
approximate equality, the image due to the object or tripod table would lie very near
to this. The next thing is to adjust the image from the lower surface of the cover-
wedge so as to lie symmetrically to the horizontal diameter of the field, and almost in
contact with the vertical edge which divides the light half from the dark half of the
field. This is achieved by movement of the telescope by the fine azimuth adjustment
screw and the tilting screw. The next operation is to bring the object image exactly
to the same height as the one just adjusted, that is, also symmetrical to the horizontal
diameter. If this is not done exactly, the interference bands will be oblique to the
vertical diameter of the field instead of parallel thereto, as is desired. To achieve it,

342 MR. A. E. TUTTON OX A COMPENSATED INTERFERENCE DILATOMETER.

one of the screws of the tripod is manipulated, and if the tripod has been conveniently
placed so that one screw is to the left side, in the diameter of the chamber floor at
right angles to the imaginary line at this height joining the pedestals of the
expansion apparatus and the telescope, the screw to rotate may be either the front or
back one, at pleasure. It then only remains to manipulate the screw to the left, in
order to bring the object image so as to more or less cover the adjusted image,
according to the size of signal stop employed and the width of interference band
desired. If the two images were absolutely coincident, the conditions would be those
for the production of alternate fields of light and darkness, rather than bands, during
alteration of temperature and the resultant alteration of the thickness of the air space
between the two reflecting surfaces, which, under these conditions, would be precisely
parallel. As parallelism is more and more deviated from, by manipulating the left
screw so as to bring the object image to cover less and less of the cover-wedge image,
the conditions are those for the production of bands, at first few and very wide, but
becoming narrower and narrower, and proportionately more numerous, as the process
of uncovering the first image proceeds. If an object were supported on the tripod,
and the air wedge consequently thin, the bands would be immediately visible if the
iris diaphragm were so far closed as to exclude all images but the double one, the
simple eyepiece replaced by the micrometer combination, and the white source of light
exchanged for a sodium flame.

In order to employ mercury light for the determination of the expansion of the
screws of the tripod, or as in ordinary determinations, red or greenish blue hydrogen
light, the single reflecting prism of the expansion apparatus is replaced by the pair of
refracting prisms. These have previously been set for minimum deviation for the
green mercury line, which occupies a convenient mean position in the spectrum, with
the aid of the special slit signal stop. The source of white light is removed, and the
Geissler tube and its fitting attached, and connected to the induction coil. The
telescope is raised to the height of the upper refracting prism, and the induction coil
set in action. Instead of the four simple images, the right top one a double one,
there are now visible four corresponding vertical spectra, each showing sharp
brilliant images of the signal stop in the red and greenish blue, corresponding to
C and F hydrogen light, and a fainter one, so long as the tube is at the ordinary
temperature, in the bright green, corresponding to the green mercury line. The
reason for arranging the white light images as a rhombus, rather than as a square, is
now apparent ; for if the double image in white light had another image vertically
below it in the field, in the Geissler tube light the spectrum of the lower image would
partially overlap the double one due to the surfaces whose reflections are to produce
bands, and the interference bands would be blurred by the illumination due to the
undesired radiations. The angle of the rhombus is such that when the iris is arranged
no as just to permit of the complete passage of the desired double image in any of the
three colours, the other spectrum is entirely excluded.

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILA.TOMETER. 343

The telescope is now to be so arranged for height, tilt, and azimuth that the
double image in the desired colour is symmetrical to the horizontal diameter of the
field, and almost in contact with the vertical line of demarcation between the light
and dark halves of the field. The paths of the incident and reflected light are then
almost identical, and the condition for maximum brilliancy of bands is attained. The
micrometer combination, fitted with the movable lens carried in its shorter tube, is
now attached in place of the simple eyepiece, and its eyepiece is withdrawn, to
enable the observer to inspect the double image after the change and ascertain that
it is still in position. For it conveniently happens that when the eyepiece of the
micrometer is withdrawn, the movable lens left in position acts as a simple eyepiece,
although not quite so well as the simple eyepiece supplied for the purpose, being
further from the eye and not quite at the focus of the vertical edge of the semi-
circular field ; it affords, however, images which are sufficiently good for the purpose
of adjustment of their proper position with respect to the iris aperture, or of the
relative position of the two overlapping images required to produce the bands.

On replacing the eyepiece in front of the micrometer, the interference bands are at
once seen in the colour adjusted for. If an object is being used, and the air wedge is
thin, the bands are exceedingly brilliant in red hydrogen C light, and only slightly
less so in greenish blue F hydrogen light. . In green mercury light they are faint at
the ordinary temperature of the Geissler tube, but most brilliant if the capillary and
the limb containing the mercury globule are gently warmed. The observer no
longer sees a semicircular field, as with the simple eyepiece, but a full circular field,
completely covered with bands if the object is as large as the diaphragm containing
the upper glass wedge of the interference chamber, The semicircular opening,
which is really very small when the iris is arranged as has been described, now
simply acts as a diaphragm to cut off undesired light. The disturbing reflection from
the inner surface of the objective, which would otherwise be very serious, is also
thrown behind either the iris diaphragm, or the micrometer serrated diaphragm, by
the device for slight tilting of the objective which has been described in the earlier
portion of the memoir (p. 328).

The bands can be adjusted to the desired width by manipulation of the left-hand
screw of the tripod, and any inclination from the vertical, as observed by reference
to the pair of vertical spider -lines, can be corrected by slight rotation of the tripod,
or, if preferred, of the micrometer itself within its outer carrying tube. The most
suitable width is dependent upon the nature of the object. If its reflecting surface
is truly plane, as proved by the severe test imposed by its use for the purpose under
discussion, namely, by its generation of rectilinear bands at regular distances, the
width of band chosen may be more considerable than when the bands are slightly
curved or the distance is not quite regular, as the irregularities largely disappear
when the bands are closer together, produced by an air wedge of greater angle.
Usually a breadth of about 100 drum divisions, one complete rotation, is a con-

344 MB. A. E. TUTTON ON A COMPENSATED INTERFERENCE UILATOMETER.

venient one. The two vertical spider-lines are then to be arranged at such a
mutual distance as best facilitates the setting of a band to their centre, and if the
width mentioned is chosen, the spider-lines are then also at the most suitable
distance apart to enable the centre of the reference mark — the inner edge of the
silver ring of the ordinary cover-wedge, or of the clear ring of the silvered cover-
wedge used in the observations of the tripod expansion — to be located ; in either
case the desirable small segment of the circle is visible outside each spider-line when
the ring is set for a drum-reading of its position. The horizontal spider-line is to be
arranged as a tangent to the ring at the lower end of its vertical diameter. This is
achieved by use of the rack and pinion of the pedestal.

The surface of the author's platinum-iridium tripod, truly plane to all ordinary
tests, such as reflection of goniometer signals, proves to be very slightly concave, the
bands in green mercury light being slightly curved, that is, arcs of large circles.
They are perfectly regular, however, and move parallel (concentrically) to themselves
when the temperature is altered, demonstrating the fortunate perfect equality
of expansion of the whole of the parts of the tripod, and particularly of the screws,
and enabling excellent determinations of the expansion to be made by observations
of the number of bands passing the centre of the reference circle. A width of band
of about GO drum divisions was found most suitable in these determinations of the
expansion of the tripod itself'.

The preliminary adjustment having been carried out, the screws of the tripod are
rigidly fixed by tightening the little clamping screws. The bent thermometer is
then to be carefully attached. It is held near its upper end by a pair of loops at
the ends of a circular brass spring collar, fitting just under the flange at the base
of the prisms. This mode of support admits of sufficient play to enable the half of
the lid, through a hole in which the thermometer is to pass, to be slid over the bent
lower end and up the stem into position, and the half lid is supported by the upper
end of the interference chamber while the bulb of the thermometer is carefully
arranged in proper contact with the upper surface of the tripod table and one of the
screws ; the bulb may then be secured in this position by means of silk thread.
After ascertaining that this operation has not impaired the adjustment, or correcting
any slight derangement, the expansion apparatus is smoothly removed, by means
of the handles of the cloth-lined slab upon which it is mounted, to its marked
place at the further end of the slate table. The micrometer is detached, and
its movable lens extracted, unscrewed from its short tube, and fitted to its longer
tube, which is then pushed home in the tube of the micrometer until the ends
are flush ; the micrometer is then placed in position, being now in a condition to
sharply focus the bands and reference mark for the new position of the expansion
apparatus. The height of the telescope will require slight alteration, and a little
further adjustment for azimuth may be necessary ; removal of the front eyepiece
will, as before, enable the double image to be properly adjusted. The height

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 345

adjustment is best achieved while the eyepiece is in position and the bands and
reference circle in view, the rack and pinion of the pedestal being manipulated
so as to again bring the lower margin of the little circle into apparent contact with
the horizontal spider-line.

The heating bath may now be placed in position upon its annular support, which
latter is always left in situ. To enable this to be done, the interference tube is
gently raised, by the rigidly moving rack and pinion, to the necessary height, indi-
cated by a mark on the sliding column. A large circular sheet of asbestos millboard,
pierced by the necessary apertures, is intended to be laid on the top, and a large
semi-cylindrical screen of the same material is placed between the bath and the pedestal
for the full height of the latter. It is most convenient that the circular sheet should
be arranged while fixing the bent thermometer, above the two halves of the lid, which
are readily supported on the upper end of the interference chamber. Several circular
sheets of asbestos are already laid beneath the Fletcher ring gas-burner, alternated
with porcelain tiles. The interference tube is then lowered into the bath, another
mark on the sliding column indicating the proper position, which, as adjustment had
designedly been made with the column at this same height as indicated by the mark,
will be the same as before. The two halves of the lid and the asbestos circle are left
in position by the sinking of the interference tube, and the wide aperture necessarily
left in the asbestos for the passage of the interference chamber is covered with two
additional overlapping semicircles of asbestos. The thermostat and the bath ther-
mometers may next be placed in position and the gas connections made. In order to
protect the dispersing apparatus as much as possible during the heating a further
circular screen of mica, arranged in two halves, attached by studs at each end and
perforated by suitable apertures at the overlapping inner edges, is fixed about the
height of the top of the porcelain tube. It is conveniently supported by the top of
the semi-cylindrical screen and a couple of asbestos supports laid on each side of the
front part of the asbestos circle, leaving ample space between for a clear view of the
thermometers. By this means the temperature at the dispersion apparatus is only
appreciably higher than that of the room when the thermometers are indicating the
highest limit of 120°.

It should be here stated that the author was much troubled at first by the
absorption or condensation of moisture in or on the copper walls of the inner bath.
It is quite unnoticeable at the ordinary temperature, but when the temperature is
raised and the transit of bands commences, the latter become suddenly obscured and
the observation is lost, owing to the condensation of a film of distilled moisture on
the glass wedges in the upper part of the interference chamber. This awkward
circumstance necessitates the abandonment of the work for the day, as the apparatus
requires to stand at least twelve hours to re-take the atmospheric temperature. It
can be completely avoided, however, by placing on the floor of the bath, after each
determination, so as to desiccate it before the next, a shallow dish of vitriol, and

VOL. CXCI. — A. 2 Y

346 MR. A. E. TTJTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

leaving the interference tube immersed in the bath, so as also to share in the
desiccation. It is a great advantage if the adjustment can be carried out the evening
before a determination, as not only is there then ample time for the temperature of
the room and apparatus to attain equilibrium, but also vitriol can then be left in the
bath during the night, and removed just before the observations commence in the
morning. For if the bath and interference apparatus are long exposed to moist air
during adjusting operations, and the tripod subjected to much manipulation with the
fingers, condensation again occurs to a greater or less extent which it is desirable to
remove. The fixing of the mica screen should be left till this is carried out. The
momentary raising of the interference apparatus from the bath in order to remove
the vitriol dish serves, moreover, to enable the observer to note the reading of the
inner bent thermometer, whose bulb is in contact with the tripod. This temperature
is, of course, taken as the starting temperature in preference to that indicated by the
bath thermometers, in case any slight difference is apparent.

Finally, before commencing actual observations, the bands are closely examined,
and any slight alteration of the height of the reference ring above the horizontal
spider-line corrected ; also the eyepiece should be momentarily removed to satisfy
oneself that the double- image is still correctly placed. The observation of the
position of the bands for the starting temperature can then be proceeded with.

The method adopted for determining the position of the bands, with reference to
the centre of the silver ring, is that recommended by PULFIUCH. Two things require
to be ascertained. First, the distance between the centres of two adjacent dark
bands, which may be termed the width of the bands ; and second, the distance
between the centre of the nearest dark band and the centre of the reference ring.
The quotient of the second by the first is evidently the fraction of a band by which
the centre of reference is distant from the nearest band. When the temperature is
raised, and the bands move past the vertical spider-lines, the first dark band which
comes between the pair of spider-lines is to be counted, not as one band, but as the
fraction of a band thus determined, in case the movement of the nearest band occurs
in the direction of the centre ; and in the contrary case as the complementary
fraction, that is, unity minus the determined fraction. The whole observation is
completed by counting or otherwise determining the number of succeeding dark
bands which pass during the interval until the temperature again becomes constant
at the higher limit, and by then determining in a similar manner as for the lower
limit the fraction of a band to be added, beyond the last which has passed the centre.
Hence, the essentials of the determination are the whole number of bands which pass,
and the initial and final fractions of a band to be added to this whole number ; and
the determination of each of these fractions involves the two operations previously
referred to.

To determine the width of the bands the author measures the five whose middle
one is immediately to the left of the centre of the little reference ring. Duplicate

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DII^ATOMETER. 347

readings of the micrometer drum, the one which moves both spider-Hues simul-
taneously, are taken, corresponding to the setting of each of these five bands in turn,
precisely midway between the two spider-lines. It is convenient to call these values
a, b, c, d, e, letting a be the one most to the left ; the values will then be in ascending
order of drum divisions, each complete revolution of the drum being 100. In addition
to the four direct values for the width of a band obtained by taking the differences
between successive readings, a considerable number of other combinations can be
obtained for the width from the five readings, and PULFBICH has shown that the
whole of the values reduce eventually, converting his symbols into the author's, to

the simple formula :

w = i (d + e) - (a + b).

To determine the distance of the middle band, c, from the reference point, it is only
necessaiy to determine the true position of c by taking the arithmetical mean of the
five readings, and to determine the drum reading of the reference point by taking
the mean of two or three independent readings corresponding to the adjustment of the
silver ring so that equal small arcs are visible outside each of the vertical spider-lines.
The difference between these mean readings for the band c and the centre of the
reference ring is evidently the distance S required. The fraction of a band corres-

^s

ponding to this temperature is, then, either S/iv or - — , according to the direction

in which the bands are found to move on raising the temperature.

The Fletcher ring gas-burner is then ignited, the gas supply to it being regulated
by the graduated lever tap, so that the temperature of the bath may not rise too
rapidly. As has already been explained, the author prefers to actually observe the
passage of the bands, and to count their number with the aid of the recording
apparatus, rather than to rely exclusively upon the ABBE method of calculation from
observation with two wave-lengths. Moreover, in the case of the determination of
the expansion of the tripod, the bands are so very much clearer with green mercury
light than with either C or F hydrogen light, or sodium light, at the necessarily long
distance apart of the two reflecting surfaces, that the author prefers to base his
determinations entirely upon observations in green mercury light. The recorder is
placed upon the little cloth-covered accessory table to the observer's right, and as
each dark band (after the first, which has the determined fractional value) advances
to the middle of the two vertical spider-lines, the knob is pressed and the puncture
made in the tape.

For a reason which will be fully discussed in considering the mode of calculation of
the experimental results, the author makes two series of observations, the upper
temperature limits of which are in the neighbourhood respectively of 70° and 120°.

The rise of temperature is so regulated, that fully an hour is occupied in attaining
the neighbourhood of the first higher limit, about 70°, the transit of the bands
occurring with more or less regularity during this interval. A further valuable

2 Y 2

348 MR. A. E. TTJTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

control of the number of bands passing the spider-lines is afforded if the observer
notes the passage of each quarter of a band in his note-book. The author calls the
quarters respectively 1, 2, 3, 4, and as the four successively make their transit, puts
down their number on the same line ; after 4 is so put down, the puncture is
immediately made on the recorder, and a " p " added after the 4 in the note-book, to
signify that the puncture has been made. If, also, the number of this particular
band is added, an additional check is obtained. It may be mentioned that, in the
case of the tripod determination, and all other cases where the bands are not so
numerous as to pass with great rapidity, it is not necessary to keep the induction
coil in action during the whole time ; it is sufficient to run it intermittently in such
a manner as to observe the passage of the four quarters of each band. This prevents
overheating of the Geissler tube, and considerably prolongs its efficiency. The
hydrogen spectrum rapidly deteriorates if the coil is worked continuously. A further
period of two to three hours is then allowed for the attainment of equilibrium at this
temperature, during the last hour and a half, at least, of which the temperature is
maintained constant with the aid of the thermostat and the graduated gas-tap. It
is found that if constancy to within 0°'2 is attained in the inner bath, as indicated
by the outer two thermometers, a degree of constancy fairly easy to maintain,
the actual temperature of the interference apparatus, as indicated by the inner bent
thermometer whose bulb is in direct contact with the platinutn-iridium tripod,
remains absolutely constant. For the last ten degrees or so of the rise of tempera-
ture, the bands move precisely with the mercury column of this thermometer, and
their motion ceases exactly with the attainment of constancy by it, a circumstance
which is eminently satisfactory. When the last band to pass the vertical spider-
lines for this temperature interval has been recorded, and no further movement of
the bands is observable, the tape is drawn forward some distance, in order to inter-
pose an interval between the last puncture for this temperature interval and the first
of the next, and a series of measurements similar to those at the starting temperature
are made, in order to determine the fraction of a band which has passed since the
last transit.

When these measurements are completed, the gas supply is again increased so as
to gradually raise the temperature to the highest limit, about 120°. The transit of
bands is again observed and recorded, the same interval as before is allowed for the
attainment of equilibrium and constancy, and finally, the determination of the last
fraction of a band is made in the same manner as the two former determinations.
The gas supply can then be turned off, and the apparatus allowed to cool. The tape
is again drawn forward, and as the determination is finished, cut off. It is con-
venient to leave ample tape on each side of the punctures for a brief description of
the determination to which it refers ; also to add at each end of the row of punctures
corresponding to each of the two temperature intervals the values of the fractions to
be added. In the case of the first puncture of the second interval, this only

MB. A. E. TTJTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 349

represents the complementary fraction to that to be added to the punctures of the
first interval, so this fraction is set, by exception, over the top of this particular
puncture rather than alongside it. It is convenient to actually make this puncture,
because the total number of punctures made for the two sets of observations then
represents the actual number of dark bands which have made their transit between
the starting temperature and the highest limit.

Three quantities are the result of these observations :

(a.) The number of whole bands and fraction of a band which have passed the
point of reference between the starting temperature and the first higher limit
(about 70C). This is given by the number of punctures for the first interval plus the
initial and final fractions for that interval.

(b.) The number of whole bands and fraction of a band which have passed the
reference point between the first higher limit and the highest limit (about 120°).
This is equal to the number of punctures for the second interval minus one, plus the
initial and final fractions ; or, which is the same thing, the number of punctures plus
the final fraction for this interval minus the final fraction for the first interval.

(c.) The number of bands and fraction of a band which have made their transit
past the reference point between the starting temperature and the highest limit.
This is given by the sum of the total number of punctures and the initial and last
fractions. Naturally, c = a -\- b.

If we retain FIZEAU'S symbol, f, for the number of bands and fraction of a band
passing the reference point during a given interval of temperature, the alteration of
thickness of the air film denoted by the observations for that interval is approxi-
mately equal to/^X, where X is the wave-length of the particular light employed to
generate the bands. The wave-length of the green mercury line is 0'00054GO millim.
The word approximate is introduced because there is a correction to apply for the
alteration of the refractive index of air, and consequent variation of the wave-length
involved, due to change of temperature and pressure. The result of this alteration
in the wave-length is that the number of bands actually observed to pass the fixed
point is slightly different to the number which would have been observed if no
change in the wave-length had occurred. This correction is very small, only a small
fraction of a band, when the air film is thin, as is usually the case ; but it is very
considerable, amounting to a tenth of the whole value, when the thickness is
relatively great, as it is during the determination of the expansion of the tripod.
The nature and extent of this correction have been fully discussed by both BENOIT
and PULFKICH (loc. cit.), and the formulae arrived at by each of these observers
lead to the same result. After employing both, the author finds the form of
expression given by PULFBICH (' Zeits. fur Instrumentenkunde,' 1893, p. 456), rather
more convenient, as the values of several of the factors can be taken directly from
LANDOLT'S, ' Physikalisch-Chemische Tabellen.' The formula is as follows : —

350 ME. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

where/' is the corrected number of bands, f the observed number, d the thickness
of the air layer, 1 1 and t., the limiting temperatures, and 6, and 6, the corresponding
barometric pressures ; a is the coefficient of expansion of air 0'00367, and n the
refractive index of air for the wave-length A. of the light employed. The logarithmic

values of 2 - -" and 2 ~^r can be found once for all ; they are respectively

3-59901 and 3'15353. The logarithms of the factors —:, , and can

7bU 1 -f- atj 1 + a.tt

be extracted directly from LANDOLT'S tables. Hence the expression, although
apparently long and troublesome, lends itself to very easy computation. For the
purpose of the calculation of this correction it is necessary to take the reading of the
barometer at the time of making the observation of the exact position of the bands
at the initial temperature, and again when they have attained their condition of rest
at each of the two higher limits. These barometric observations are of particular
importance in the case of the determination of the expansion of the tripod, as the
share of the variation of pressure in the correction is then an appreciable one
The author employed a standard barometer suspended in the same room.

With regard to the sign of the correction, the signs given in the above formula for
the temperature and pressure portions are the correct ones for use in all cases where
the result of the increase of temperature is to effect an increase in the separation of
the two reflecting surfaces, as in the case in question of the determination of the
expansion of the tripod. In such cases the effect of change of temperature is to
cause the number of observed bands to be less than it would be if such a change did
not occur. The contrary is the case where the thickness of the air-layer diminishes,
as in the cases of the determination of the expansion of an object supported on the
tripod table, whose expansion is greater than that of the screws, and in all
determinations of expansion involving the use of the compensator. In such cases
the signs of the temperature and pressure portions of the above formula should
be respectively — and -)-.

The temperature portion of the correction is usually much the larger, and
its sign governs that of the total correction. Indeed, in all cases when the
air-layer is very thin, the share of pressure can be neglected. Moreover, as
will be evident from the formula, the sign of the pressure portion varies according
as frj is less or greater than 62. With regard to the sign of the more important
temperature portion, the fact that it follows the rule just indicated will be evident
from the following considerations. Rise of temperature causes a diminution in the
refractive index of the air, and a corresponding increase in the wave-length. The
same amount of separation of the surfaces will, therefore, contain fewer wave-lengths,
and if no expansion of the tripod and object (if one is being used) occurred, the

MR, A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 351

observer would perceive the transit of the bands to the extent indicated by the
correction, and in the same direction as the bands move when the two surfaces
approach each other ; so that the effect of rise of temperature on the refraction of
the air is similar to that produced by approach of the two surfaces. Hence the
correction requires to be added when the effect of rise of temperature is to cause
recession of the surfaces, and to be subtracted when the surfaces approach each other.

After the conclusion of the observations of the number of bands which effect their
transit for the two intervals of temperature, it is the author's practice to leave the
apparatus untouched for about sixteen hours, during which time it regains the
ordinary temperature of the air, and subsequently to repeat the whole series of
operations and observations for two similar temperature intervals on the following
day, and on as many succeeding days as it is desired to make independent deter-
minations. After the completion of the series, and the final slow cooling of the
apparatus during the night, the measurement of the exact length of the screws of
the tripod above the reflecting surface of the table is made, with the aid of the thick-
ness measurer. In the case of the determination of the expansion of the tripod, this
is the only additional quantity required before being able to proceed with the
calculation of the results.

The full data now available for this determination are as follows : —

Ltlt length of platinum-iridium screws at the initial temperature, tl ; in this case

= d thickness of air-layer.

ti} initial temperature ; t2, first higher limit, about 70° ; tt, highest limit, about 120°.
6,, &2, &a> corresponding barometric readings.
fz, observed number of bands for interval £2 — <:.
fs, observed bands for interval t3 — tt.

By means of the correction formula and these data we first calculate f2' and/,', the
corrected number of bands for the two intervals. These quantities, when multiplied
by ^X, in this case, for green mercury light, 0'000273 millitn., afford the amounts of
expansion of the screws, or alteration of thickness of the air-layer for the two intervals
of temperature ; and therefrom, by addition to the measured initial length, Ltlt we
obtain the lengths Lt2 and Lt3, at the first and second higher limits (near 70° and
120°) respectively. That is to say, the length of the platinum-iridium screws is now
known at three temperatures in the neighbourhood of 10°, 70°, and 120°.

The author's object in making observations for two different temperature intervals
is to be able to determine not only the mean coefficient of expansion between
two limits of temperature, but also the absolute coefficient of expansion at any
given temperature, and the variation of the coefficient with change of temperature.
For it is well known that, in general, the linear coefficient of expansion is not a
constant quantity but varies slightly with the temperature ; the increment of the
coefficient per degree, however, remains practically constant. If, therefore, the

352 MR. A. E. TTJTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

length of the solid substance at 0° is represented by L0) and the length at f by ~Lt,
the nature of the change of length is adequately represented by the formula

Lt = L0(l + at + It2).

If « be the absolute coefficient of linear expansion, then the constant increment
per degree is A«/A£, and their relations to the constants a and b of the above
formula can be ascertained at once by successive differentiation with respect to the

temperature.

« = a + 2bt, Aa/A£ = 26.

The mean coefficient of expansion between 0° and t° is therefore a -\- bt; but the
true coefficient at any particular temperature t, and also the mean coefficient between
any two temperatures whose mean is t, is a + 2bt.

Hence in order to be able to determine the true and mean coefficients, and the
increment per degree, which together afford full information as to the nature of the
expansion, it is only necessary to ascertain the constants a and b in the general
expression above quoted for L£.

In this formula ~Lt and t are known, and there are three unknown quantities,
L0, a, and b. To determine them three equations, for three different temperatures,
are required. The data derived from the observations at the ordinary temperature, ^
(about 10°), at the first higher limit, tz (about 70°), and at the highest limit, t
(about 120°), enable the required three equations to be compiled. For L^ is the
length measured at the ordinary temperature by means of the thickness measurer ;
L£> is L£[ -\-f-i \ X ; and L^3 is L^ -f-,/7 \ X.. We have then the three equations

3

Lt, = L0(l + at2+ bfi),
Lt3 = L0(l +a«3+ bfy.

By subtracting respectively the first from the second, and first from the third
equations, we obtain a pair of equations which, by complementary multiplication
with the coefficients of a and b respectively, enable each of these constants to be
eliminated in turn and the other to be obtained in terms only of L0 and the known
quantities. On substituting these values of a and b in any of the three fundamental
equations L0 is at once obtained, and its substitution in the expressions for a and b,
just referred to, affords the desired numerical values of these constants.

The three expressions for a, b, and LO, in the form actually employed by the
author in the reductions, are as follows :

MR. A. E. TTJTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 353

e

1 1* L ft - <i> ft - Q ft - <•) ft - y

^0 L ft I/ ft 2/ ft — 'l) ft — "!.) . *A>

L_ T j /Jj JL*2

0 — Jjtj — t/ti — <pf].

Although the expressions 6 and </> appear long they are really very readily
computed from the experimental data, as there are only three sets of quantities
involved, the differences of the temperatures, the sums of the temperatures, and
the differences of the lengths at these temperatures ; also the two pairs of denomi-
nators are inversely identical.

The results of the observations connected with the determination of the expansion
of the platinum-iridium tripod will now be given. Five independent series were
carried out, in order that this fundamental constant might be ascertained with the
utmost precision. With regard to the length of the screws of the tripod, given in
the tables of results, it should be remarked that the actual lengths of the three
screws are necessarily very slightly different, in order to produce the slightly wedge-
shaped air-layer required for the generation of the bands. The length given is
of course the mean length, and is the thickness of the air-layer at the centre, where
the passage of bands is observed. In order to determine this mean length of the
screws, besides direct measurement of each screw, which is somewhat difficult on
account of the more or less pointed character of the screw-ends and of the agate end
of the rod of the thickness measurer, recourse was had to the use of a plane parallel
disc of thick glass, whose thickness was accurately known, and which was laid
on the screws.

Expansion of Platimim-iridium Tripod.

SERIES 1.

Length of screws above table, L£, = 12*369 millims.

Thickness of air-layer, d = 12*369 millims.

Temperatures of observations, t, = 11°*1 C., t, = 68°*5, «, = 1170<8.

ti — tl= 57*4. £3 — «! = 106*7.

Barometric pressures, b^ = 763*9, fc2 = 765*2, &3 = 766*4 millims.
Number of transited interference bands observed during the two intervals,

/2 = 20*63,/3 = 39*40.

Correction for alteration of refraction of air, + 2*17 and + 3*52.
Corrected number of bands, // = 22'80, /3' = 42*92.
Wave-length of light employed (green mercury line), X = 0*0005460 millim.

£X == 0*000273 millim.
VOL. cxci. — A. 2 z

354 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER,

Elongation of screws, L«, - L^ = //|-X = 0'0062243 raillim.

Lt3 — Lti = /3'|X = 0-011717 millim.
0 = 0-00010622. <£ = 0-00000002791S.
L0 = 12-3678 millims.
a = 0-000 008 588 2. b = O'OOO 000 002 257 3.

SERIES 2.

Length of screws above table, L^ = 12'369 = thickness of oir-layer, (I.
Temperatures., tl = 12°'0, t, = 70°'0, tt — 118°'5.

t, — tl = 58-0. t3 —tl= IOG'5.
Pressures, bl — 7(>3, b., = 762'1, 1>3 = 76 T6 millims.
Number of transited bands, f, = 20'91, f3 — 39'44.
Correction, + 2-19, + 3'5'2.

Corrected number of bands, /,' = 23' 10, /,' = 42'96.
Wave-length of light employed, X = 0'00054(J. 1 X = 0'000273.
Elongation of screws, L<, — Ltt —f.!^\ = 0'00630G2.

Lts — Ltt =f3'±\ = 0-011728.
f) — O'OOO LOG 37. </> = O'OOO 000 028 739.
L0 = 12-3677.
« = O'OOO 008 GOO 6. b = O'OOO 000 002 323 7.

SEKIES 3.

Length of screws above table, L^ — 12'369 = thickness of air-layer, d.
Temperatures, t, = 12°'0, t, = 70°'3, t3 = 122°'2.

tz — tl= 58-3. t3 — tl= 110-2.
Pressures, ^ = 761'0, b., = 761'2, b3 = 761'9 millims.
Number of transited bands, /^ = 21'04, /, = 40'97.
Correction, + 218, + 3' 57.

Corrected number of bands, /2' = 23'22, // = 44'54.
Wave-length of light employed, X = 0'000546. |X = 0'000273.
Elongation of screws, ~Lt2 — Lt1 = 0'006339. L«3 — L<, = 0'012159.
0=0-00010618. ^ = 0-000000031011.
L0 = 12-3677.
a = 0-000 008 585 2. b = O'OOO 000 002 507 4.

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 355

SERIES 4.

Length of screws above table, L^ = 12'369 = thickness of air-layer, d.
Temperatures, «x = 10°'8, t, = 70°7, *, = 121°'l.

t2 — «, = 59-9. ta — ti = 110-3.
Pressures, &, = 761 '5, Z>2 = 76 TO, &3 = 7597 millims.
Number of transited bands, fz = 21-61, fs = 4078.
Correction, -f 2'26, + 3'65.

Corrected number of bands, /2' = 23'87, f3' — 44'43.
Wave-length of light employed, X = 0'000546. £X = 0'000273.
Elongation of screws, L«2 — L^ = 0'0065164. Lta — L^ = 0'012129.
^=0-00010688. (j> = O'OOO 000 023 394.
L0= 12-3678.
a = O'OOO 008 642 1. & = O'OOO 000 001 891 5.

SERIES 5.

Length of screws above table, L<i — 12'369 — thickness of air-layer, d.
Temperatures, *, = 10°'3, t, — 680'1, t3 = 1180>8.

«, — <! = 57-8. t3 — ti = 108-5.
Pressures, ^ = 765'3, 62 = 766'0, b3 = 766'8 millims.
Number of transited bands, /2 = 20'78, /3 = 40'14.
Correction, + 2'19, -f 3'58.

Corrected number of bands, // = 22*97, f3' = 4372.
Wave-length of light employed, X = 0 "000546. |X = 0 "000273.
Elongation of screws, Lt2 — L«, = Q-0062708. ~Lt3 — Ltr = 0"011935.
0 = 0-000 106 15. <jt = O'OOO 000 029 846.
L0= 12-3678.
a = O'OOO 008 582 8. b = O'OOO 000 002 413 2.

The results dei'ived from the five series are compared in the following table, and
the mean values for a and b extracted : —

a.

&.

Series 1 ....
„ 2 ....
„ 3 ....
„ 4 ....
,5

0-0000085882
86006
85852
86421
85828

0-000 000 002 257 3
•2 323 7
25074
18915
24132

356 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

Mean values, a = O'OOO 008 599 8, b = O'OOO 000 002 278 6.

The final result, therefore, for the mean coefficient of expansion a + bt of the
platinum-iridium of the tripod, between 0° and t°, is

O'OOO 008 600 + O'OOO 000 002 28£, or 10-9(8600 + 2'28«).

The true coefficient of expansion at t°, or mean coefficient between any two
temperatures whose mean is t, a. = a + 2&£, is

a = O'OOO 008 600 + O'OOO 000 004 56t, or = 10~9(8600 + 4-56«).

It will be interesting to compare the value for the expansion of the specimen of
platinum-iridium composing the author's tripod with the values found by BENOIT
for various specimens of JOHNSON and MATTHEY 10 per cent, alloys of iridium with
platinum. BENOIT'S values are as follows ; they refer to the mean coefficient between

0° and t° :—

Specimen 1, 10-9(8615 + 2'21«).

2, 10~9(8593 + 2'40«).

3, 10-9(859S + 2'22«).

4, 10-° (8575 + 2-38<).

5, 10~9(8575 + 2-54«).

It will be observed that the author's value agrees most satisfactorily with these
values of BENOIT. As the coefficient of iridium is much lower, namely, C'0000065 ,
than that of platinum, 0'0000089, a slight difference in the proportions of the two
metals present in the alloy might be expected to produce just such slight variations
in the coefficient as are observed.

DETERMINATION or THE EXPANSION OF THE ALUMINIUM COMPENSATOR.

The aluminium compensating blocks were cut from the same casting of the purest
obtainable aluminium. The thickest block of the 25 millim. series, slightly over
12 millims. thick, was. used for the purpose of determining the expansion by the
differential Fizeau method. The platinum-iridium tripod screws were preliminarily
set to about 12'5 millims., on that side of the table which was furnished with
raised points. The aluminium block was placed upon the table, resting on
the three outer highest points. The tripod and block were then placed in the
interference chamber, and the large glass cover-wedge provided with the minute
central silver ring (not the cover-wedge silvered all over, which is used only for the
tripod determinations) was placed over the screws. The corresponding counter-
balancing wedge had previously been exchanged for the former one in the upper
aperture of the chamber, and the engraved direction chords were, as before, arranged

MR. A. B. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 357

parallel, but that of the cover-wedge above and that of the counteracting wedge
below.

The adjustment of the upper surface of the aluminium, and of the lower surface of
the cover-wedge, was carried out precisely as has been described for the tripod table
and cover-wedge ; the two parts of the apparatus were arranged at close quarters,
and the images of the signal-stop obtained in white light with the aid of the single
reflecting prism were brought to the desirable partially overlapping position, and
isolated by the iris diaphragm from all other radiations. The single prism was then
exchanged for the train of refracting prisms, the white light was replaced by red
hydrogen light from the Geissler tube, and the simple eyepiece was replaced by the
micrometer combination, when very brilliant bands were at once observed. After
adjustment of the width of the bands and their parallelism to the vertical spider-
lines, the platinum-indium screws were fixed, the bent thermometer was arranged
in position, and the expansion apparatus removed to its proper distant position. The
interference tube was then raised, while the bath, containing its dish of vitriol, was
arranged in position; the interference apparatus was then lowered into the bath,
the thermometers fixed, and the whole left overnight to attain equilibrium of
temperature.

Next morning the vitriol was removed, and the inner thermometer read while the
interference apparatus was raised for this purpose ; the barometric pressure was also
taken, the adjustment of the bands for height effected, the silver ring being brought
into contact at its lower limb with the horizontal spider-line, and the measurements
of the positions of the centre of the silver ring and of the five adjacent bands were
carried out in red hydrogen light. This light is particularly suitable, on account of
the bands being separated at a greater distance than with the other available wave-
lengths, and also by reason of the particular brilliancy of this radiation afforded by
the Biedel- Geissler tube at the ordinary temperature of the tube. The better of the
two parallel surfaces of the aluminium block, which was placed uppermost for use,
afforded bands which were strictly regular and almost perfectly rectilinear, yielding
excellent measurements. The temperature was then raised to the neighbourhood of
70°, recording the transit of bands on the tape, and after a couple of hours' constancy
measurements were made at this temperature. After the attainment of another
interval, the passage of bands being recorded as before, a final set of measurements
were made for a constant temperature in the neighbourhood of 120°.

After cooling overnight, a duplicate series of determinations for three similar
temperatures were made next day.

Lastly, after cooling during another night, the exact lengths of the platinum-
iridium screws projecting above the height of the points were measured by the aid of
the thickness measurer ; similarly, the known thickness of the aluminium block was
verified.

358 MR, A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

After these duplicate determinations with the same screw-length, an independent
series was taken on another day with a different setting of the screws.
Following are the results obtained :—

Expansion of Aluminium.
SERIES 1.

Thickness of aluminium block, L^ = 12'188.
Length of screws above table points, I = 12'400.
Thickness of air-layer, d = 0'212.
Temperatures, t} = 10°'l, t2 = 68°'9, t3 = 123°'4.

t3 — t! = 58'8. t3 — ti = 113-3.
Pressures, 6, = 754, &2 = 754'8, 63 = 756 millims.
Number of transited bands, /, = 30'35, f3 = 60'32.
Correction for air refraction, — 0"04, — O'OG'.
Corrected number of bands, /2' = 30'31, /,' = GO'26.

Wave-length of light employed (C hydrogen), X = 0'0006562. |X = 0'0003281.
Diminution of thickness of air-layer, /„' £ X = 0'0099449. /,' i X = 0'019772.
Elongation of tripod screws = measured length of screws X (a + 2fo for tripod
alloy) X temperature interval,

I

10-9(8600 + 4-56 -^y
10-° (8600 -1- 4'56 ~

(t2 — «,) = 0-0064074.
(«, - <,) = 0-012510.

\

Expansion of aluminium block = diminution of thickness of air-layer + elongation
of screws,

Lf2 — Lti = 0-0099449 + 0'0064074 = 0'0163523.
L«, - L«! = 0-019772 +0-012510 = 0'032282.
8 = O'OOO 268 23. <j> = O'OOO 000 125 31.
L0= 12-1853.
a = 0-000022013. b = O'OOO 000 010 284.

SERIES 2.

Thickness of aluminium block, L£, = 12'188.
Length of screws above table points, I = 12'400.
Thickness of air-layer, d = 0 212.
Temperatures, «, = 80'0, tz = 70°'0, <3 = 1200>6.

t^ — tl = 62-0. «3 — ti = 112-6.

Pressures, 6, = 755'1, &2 = 7537, 63 = 752'7 millims.
Number of transited bands, /2 = 32'20, ft = 60 '22.

MR. A. E. TTTTTON ON A COMPENSATED INTERFERENCE D1LATOMETER. 359

Correction, — 0'04, — 0'06.

Corrected number of bands, /2' = 32-16, /3' = 60'16.

Wave-length of light employed, X = 0*0006562. |X = 0'0003281.

Diminution of thickness of air-layer, /2' £ X = O'Ol 05520. /,' $ X = 0'019739.

Elongation of tripod screws,

I

I

10-9(8600 + 4-56 -l-

10-9 8600 -f 4-56 '

(t.z — £,) = 0-0067481.
(t, — ti) = 0-012417.

\ 2

Expansion of aluminium block,

Lt2 — L«, = 0-010552 + 0-0067481 = 0'0173001.

U3 — LtL = 0-019739 + 0-012417 = 0-0321 56.
0=0-00026894. <£=0-OOOC0012937.
L0= 12-1858.
a = 0-000 022 070. b = O'OOO 000 010 617.

SERIES 3.

Thickness of aluminium block, Li, = 12"188.
Length of screws above table points, I = 12'812.
Thickness of air-layer, d = 0'624.
Temperatures, «, = 9°'8, tz — 67°'6, t3 = 118°7.

tz — ti= 57-8. t3 — ti = 108-9,
Pressures, bt = 772'6, 62 = 771 '7, b3 ~ 771 '9 millims.
Number of transited bands, /2 = 29'43, f3 = 57'] 8.
Correction, — O'll, — 0'18.

Corrected number of bands, /2' = 29'32, /3' = 57'00.
Wave-length of light employed, X = 0-0006562. J X = 0'0003281.
Diminution of thickness of air-layer, // 1 X = 0'0096199. f3 % X = O'Ol 8702.
Elongation of tripod screws,

4-56 '

(«2 — «,) = 0-0064992.
(«, — ^) = 0-012407.

\~ 2

Expansion of aluminium block,

L«2 — L^ = 0-0096199 + 0'0064992 = 0-0161191.

Lt3 — Lti = 0-018702 + 0-012407 = 0'031109.
0=0-00026858. <£ = O'OOO 000 132 75.
LO= 12-1854.
a = O'OOO 022 041. b = O'OOO 000 010 895.

360 MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.
The three results for aluminium are collated below :—

a.

6.

Series 1 ....
„ 2 ....
„ 3 ....

0-000 022 013
22070
22041

0-000 000 010 284
10617
10895

Mean values, a = O'OOO 022 041. b = O'OOO 000 010 599.

The agreement of the results is very gratifying, and leaves no doubt as to the
accuracy of the observations. The final result may be expressed as follows :

The mean coefficient of expansion a + bt of the pure aluminium of the compen-
sators, between 0° and t°, is

O'OOO 022 04 + O'OOO 000 010 Qt, or 10"8 (2204 + 1'06«).

The true coefficient of expansion at t°, or the mean coefficient between any two
temperatures whose mean is t, a. = a + 26^, is

a = O'OOO 022 04 + O'OOO 000 021 2t, or = 10~8 (2204 + 2'12«).

The value given by FIZEAU for aluminium for 40° (' Compt. Kend.,' vol. 68, p. 1125 ;
and ' Ann. Chim. Phys.,' [4], vol. 8, p. 335), is O'OOO 023 13 ; he further gives for the
increment, Aa/Ai = 2fc, the value 2'29. This increment agrees fairly well with the
author's value of 2 '12 ; and if the value for 0° is calculated by diminishing
O'OOO 023 13 by 40 times the increment 2'29, the number O'OOO 022 21 is obtained, a
figure which also agrees satisfactorily with the author's constant a. The coefficient
for 40°, calculated from the author's formula, is O'OOO 022 89.

DETERMINATION OF THE EXPANSION OF THE CRYSTAL-COVEKING-GLASSES.

For the purpose of determining the expansion of the black glass of the covering-
glasses employed with crystals whose surfaces cannot be made to take a polish equal
to that of glass, a block of the same glass, 13 millims. thick, was procured. The
mode of adjustment and observation was precisely similar to that for the aluminium
block. Two series of determinations were made, employing red hydrogen light. In
each case the platinum-indium screws were adjusted so as to leave an air-layer about
a quarter of a millimetre thick between the upper black glass surface and the lower
surface of the large cover -wedge.

This glass proved to have an expansion coefficient slightly less than that of
platinum-iridium, so that the converse was observed of what happens in the case of
aluminium ; that is to say, instead of a reduction of the thickness of the air-layer by

MB. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 361

the approach of the two reflecting surfaces, a slight recession was found to occur.
Moreover, as the metallic tripod was affected more rapidly by the rising temperature
of the air bath, a recession for several bands was first observed, due to the expansion
of the screws ; the movement of the bands subsequently became arrested for a con-
siderable interval of time, until, when the badly conducting glass eventually
approached the same temperature as the tripod, one or two of the transited bands
retraced their steps past the spider-lines, so that the actual number of bands
eventually found to have effected their permanent transit was very small, corres-
ponding to close similarity of expansion between the glass and the tripod.
The results obtained were as follows :

Expansion of Black Glass.
SERIES 1.

Thickness of black glass block, L^ = 12'956.
Length of screws above table points, / — 13'120.
Thickness of air-layer, d = 0*164.
Temperatures, t, = 8°'9, t, = 68°'9, t, - 122°'6.

t., — ti= GO'O. ts — «, = 1137.

Pressures, 6, = 739% &2 = 739'2, ba = 739'4.

Number of transited bands, f, = 2'83,fa = 4'66.

Correction for air refraction, + 0'03, + 0'05.

Corrected number of bands, f> =• 2'86,f3' = 471.

Wave-length of light employed, X = 0-0006562, £X = 0-0003281.

Increase of thickness of air-layer, //£ X = 0 "00093838. /,'^X - 0'0015454.

Elongation of tripod screws,

10-9 8600 + 4-56

10-9f8600 + 4-56

ti +

(tz - «,) = 0-0069092,
(ta — t^ = 0-013275.

Expansion of black glass block,

L*2 - L«, = 0-0069092 — G'00093838 = 0'00597082.

L<, — IA = 0-013275 —0-0015454 = 0'0117296.
^ = 0-000094226. (f> = O'OOO 000 067 939.
L0= 12-95514.
a = O'OOO 007 273 3. b = O'OOO 000 005 244 2.

VOL. CXCI. — A.

362 MB. A. E. TUTTON ON A COMPENSATED INTERFERENCE D1LATOMETER.

SERIES 2.

Thickness of black glass block, Lt, = 12'956.
Length of screws above table points, I = 13'221.
Thickness of air-layer, d — 0'265.
Temperatures, t, = 6°'0, t.2 = 69°'0, t, = 1190'i.

«2 — «, = 63. tA — tl= 113-1.
Pressures, I, = 743, b, = 745, b3 = 746'5.
Number of transited bands, f, = 3'24,/3 = 5 '19.
Correction, + 0'05, + O'OS.

Corrected number of bands,// = 3'29,/3' = 5'27.
Wave-length of light employed, X = 0'0006562, £ X = 0'0003281.
Increase of thickness of air-layer, // 1 X = 0'0010795. /,'i\ = 0-0017291.
Elongation of screws,

I

10-9(8600 + 4-56^-

10-9 8600 + 4-56

(t, — f,) = 0-0073052.
(ta — t,) = 0-0132850.

Expansion of black glass block.

LL - Lrf! = 0-0073052 — 0'0010795 = 0'0062257.

L<3 — L^ = 0-0132850 — 0'0017291 = 0'0115559.
0 = O'OOO 093 804. <j> = O'OOO 000 066 894.
L0 = 12-95514.

a = O'OOO 007 240 7. I) = O'OOO 000 005 163 6.
The two results are compared below, and their mean extracted : —

a.

b.

Series 1 ....
„ 2 ....

0 000 007 273 3

72407

0-000 000 005 244 2
5 1636

Mean values, a = O'OOO 007 257 0. b — O'OOO 000 005 203 9.

Hence the mean coefficient of expansion a + bt of the black glass of the crystal-
covering-discs, between 0° and t° is

O'OOO 007 257 + O'OOO 000 005 2Qt,

or 10-9(7257 + 5'2«).

The true coefficient of expansion at t°, or the mean coefficient between any two
temperatures whose mean is t, a = a + 26i, is

a = O'OOO 007 257 + O'OOO 000 010 4«, or
= 10-° (7257 + 10-4*).

MR. A. E. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 363

THE DETERMINATION OF THE EXPANSION OP A CRYSTAL.

It only remains now to briefly indicate the mode of employing the compensator
for the determination of the expansion of a parallel-faced block of a crystal or any
other substance, whose thickness between the parallel plane surfaces need not exceed
5 millims.

In case the method is followed of employing the compensator below the crystal, one
of the two thinnest aluminium compensating blocks, respectively 4 and 6 millims.
thick, is employed, together with one of the 2 millims. thick black glass discs
of 10 millims. diameter. The exact length of platinum-indium screw which expands
to the same amount as each of these, is readily found from the knowledge now
available of the coefficient of expansion of each. Knowing, therefore, the length
of screw whose expansion would be exactly counterbalanced by the added expansion
of the particular aluminium compensator and crystal-covering-glass, the screws are
to be set to this length above the raised points by the aid of the thickness
measurer. As regards choice of compensator, that one is selected whose thickness,
added to that of the covering-glass, will be less than the corresponding screw-length
by an amount which is just slightly greater than the thickness of the crystal block,
slightly greater in order to leave room for the necessary air film between the two
reflecting surfaces.

The compensator is placed on the raised points of the tripod. A masking disc
(lens stop), similar to that referred to in connection with the determinations of the
tripod expansion, but of smaller aperture, is next laid on the compensator. Its
aperture is slightly less than the diameter of the black covering-glass, so that it
arrests all light other than that from the covering-glass. The crystal block is then
laid with one of its two plane parallel surfaces resting on the centre of the aluminium
block within the aperture of the mask, and the black crystal-covering-glass, polished
face upwards, is placed on the top of the crystal. On then laying the large cover-
wedge over the screws, its lower surface should be found to be separated by the
desired air-layer from the polished surface of the black glass. The whole is then
transferred to the interference chamber and the adjustment of the images from the
reflecting surfaces proceeded with.

If the method of employing the compensator above the crystal is chosen, the latter
is laid first on the three points of the tripod table, and on it is then laid one of the
smaller compensators furnished with three points, these latter resting directly on the
crystal. No covering-glass is required, the aluminium surface reflecting light
adequately. The screw-length should be that calculated for complete compensation
by the compensator chosen.

In effecting the adjustment of the lower reflecting surface by slight variation of
the screws, it is preferable to do so in such a manner that any difference of screw-
length introduced, from that which is compensated, should be an excess. This can

3 A 2

364 MB. A. B. TUTTON ON A COMPENSATED INTERFERENCE DILATOMETER.

readily be done by making the adjustment by screwing further out for the required
minute distance one, or, if necessary, two of the screws, rather than by slightly
withdrawing the opposite screw. The final measurement of the actual screw-length,
after the completion of the observation, affords the information required for the
calculation of the exact amount uncompensated. This minute amount is most
conveniently taken into account in the calculations in the form of a correction to
the change of thickness of the air-layer, or, which is the same thing, amount of
expansion or contraction of the crystal. If, as usual, the crystal expands with rise of
temperature, the correction is positive ; the effect of the slight apparent expansion of
the screws being to reduce the observed amount of approach of the two reflecting
surfaces ; if the crystal is a contracting one along this particular direction, the
correction requires to be subtracted, as the effect is to enhance the apparent contraction.
The amount of the correction is the difference between the calculated amounts of
expansion of the screws and the compensator ; these are obtained by multiplying the
measured screw-length and the thickness of the aluminium block respectively, by the
now-ascertained mean coefficient of expansion of the tripod alloy and aluminium, for
each of the two intervals of temperature, and by the number of degrees of rise of
temperature, in each case.

The mode of carrying out the determinations of the number of bands which make
their transit during the two intervals of temperature is precisely similar to the
procedure which has been described for the cases of the determination of the
expansion of the aluminium compensator and of the glass of the covering-discs.

In a subsequent memoir, the author hopes to present to the Royal Society the
results of a series of -determinations, carried out in this manner, of the thermal
expansion of the crystals of the sulphates and selenates of potassium, rubidium, and
caesium, an investigation which is now engaging his attention. Further details,
therefore, as to the mode of carrying out determinations with crystals, and con-
cerning the preparation of suitable crystal blocks, will be left for consideration in
that memoir.

The dilatometer and its accessories have been constructed for the author in an
altogether admirable manner by Messrs. TROUGHTON "and SIMMS.

INDEX SLIP.

VELKY, V. H., and MANLEY, J. J.— The Electric Conductivity of Nitric
Acid.

Phil. Trans., A, vol. 191, 1X98, pp. 365-398.

d.

Curator, Maydalen

MANLI.Y, .1. J, and VELEY, V. H. — The Electric Conductivity of Nitric
Acid.

Phil. Trans., A, vol. 191, 1898, pp. 365-398.

Electrolytic Cells, construction and calibration of.

VBLEY, V. H., and MANLEY, ,1. J.

Phil. Trans., A, vol. 191, 1898, pp. 365-398.

Nitric Acid — preparation and purification, properties of anhydrous acid,
hydrates of, specific resistances, conductivities and temperature
coefficients. VKLEY, V. H., and MANLEY, J. J.

Phil. Trans., A, vol. 191, 1*98, pp. 365-398.

3id, from their evident
lie tiiose of sulphuric
glected ; on the other
ed with considerable
rabtless on account of
:uid preserving it in a
.is, the alterations in
icy nitrogen peroxide
ttle attention, though
of a Grove's or other
iming acid, originally
than probable that of
tance are continually

of purification

ssage of a few bubbles
iuced an alteration of
jcted to a less degree
to artificial illumina-
of conductivity pro-
i of acid purified with
on the subject have
lent as to any method
with a substance so

our results from the

standpoint 01 me nyurate tiieory, ana to point OUt ally aLU'Upl alterations of physical
properties of nitric acid corresponding to any particular concentration, adopting in

28.7.98.

366 MESSES. V. H. VELET AND J. J. MANLEY ON

outline the methods and arguments used by PICKERING* in his investigations on
sulphuric acid.

Pure acids and clean vessels are glib phrases, which represent unattainable ideals ;
we only trust that we have adopted reasonable precautions of purification, and that
even if our determinations are not of absolute exactitude, yet the arguments deducible
from them are not seriously affected.

Lastly, we desire to express our obligations to the Rev. F. JERVIS-SMITH, of
Trinity College, Oxford ; Mr. E. H. GRIFFITHS, of Sydney Sussex College, Cambridge,
for assistance in the experimental portion ; to Mr. R. T. GUNTHER. for photographs ;
to the Chemical Department of the University of Oxford, for loan of apparatus ;
to Mrs. V. IT. V.ELEY, for certain illustrations ; and to the Government Grant Com-
mittee of the Royal Society for moneys to defray a moiety of the expenses. The
work was conducted partly in the Chemical Department of the University Museum,
and partly in the laboratory of Magdalen College.

FORMER INVESTIGATIONS ON ELECTRIC CONDUCTIVITIES.

KOHLRAUSCH and GROTRiANt determined the conductivity of dilute nitric acid
of concentrations varying from 6'32 to 62-07 per cent., seven points in all, and at
three different temperatures for each concentration, in order to deduce the values of
the constants a and /3 in the equation

These writers adopted their well-known method of alternating currents, but allowed
that the acid used was impure, as they wrote : — " Die concentrirte Salpetersaure,
ursprunglich farblos, hatte nach langerem Stehen eine hellgelbe Farbung angenommen,
offenbar durch Bildung eines Spurs von salpetriger Siiure unter dem Einfluss des
Lichtes." Their investigations, however, brought out an interesting detail, namely,
that the conductivity increases with the percentage concentration up to 30 to 33 per
cent., and subsequently decreases ; in the sequel this point will be more fully
discussed.

OSTWALD,! adopting KOHLRAUSCH'S telephone method with slight modifications,
made a number of determinations of the molecular conductivity of nitric acid, diluted
so as to contain a gram-molecular weight in volumes of water varying from 2 to
8,192 litres.

Finally, BOUTY§ studied the changes in conductivity caused by the solution in

* ' Journal of the Chemical Society,' 1890, Trans., 64-184.
t ' Ann. Phys. Chem.,' vol. 154, pp. 215-239.

J 'Journ. Prakt. Chem.,' vol. 30, p. 228, and vol. 31, p. 437; also ' Zeits. Physickal Chem.,'
vol. 1, p. 75.

§ ' Compt. Bend.,' vol. 106, p. 595.

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID. 367

nitric acid of small quantities of metallic nitrates. This work has, however, only an
indirect bearing upon the present investigation.

PKELIMINABY EXPERIMENTS.

In consequence of certain initial difficulties, experiments were made upon the
effects produced by (l) an added impurity of nitrous acid ; (2) sunlight; and (3) im-
perfect insulation of the electrolytic cell. The results obtained may serve to explain
certain discrepancies between our values and those of previous workers.

(1.) Added Impurity of Nitrous Acid. — The following experiment serves to illus-
trate this point, to which reference was made in the introductory section : — The
resistance of a mixture of 1 vol. of acid (I '48 sp. gr.) and 4 vols. of water was found
to be 6'97 ohms ; a few bubbles of nitric oxide gas passed in so as to form a trace of
nitrous acid caused an increase of -07 ohm, or 1 per cent, in round figures.

(2.) Sunlight. — It is a common matter of observation that the space above the
liquid in a partially-filled bottle of concentrated nitric acid contains coloured fumes,
though the liquid itself may be almost colourless. This would show that the vapour
rather than the liquid was decomposed by the action of light. In order to decide
this point a comparative experiment was made by taking two stoppered flasks of the
same capacity, and presumably of the same glass. One of these was completely, and
the other partially filled with acid of 85 per cent, concentration, which contained one
part per million of nitrous acid. Both flasks were exposed to the direct sunlight of
an April day for two hours thirty minutes. Though the amount of nitrous acid in the
former sample was unaltered, that in the latter was increased to 21 parts per million,
and it had become bright yellow in colour.

As a further illustration, the following experiment may also be quoted : — The
resistance of a certain length of acid (1 : 4 as above) was found to be 7'18 ohms, but
after exposure for six hours to the diffuse daylight of a dull and cloudy day in
October, the resistance had altered to 7'31 ohms, or an increase of 2 per cent, in
round numbers, both experiments being made under the same conditions of
temperature.

(3.) Imperfect Insulation of Electrolytic Cell. — As it appears from the diagrams
given in certain standard text-books on physical chemistry that the electrolytic cells
may be kept rigid by metallic clamps, it was thought desirable to ascertain if such an
arrangement might lead to incorrect determinations. Our experiment showed that
(l) winding the ends of a copper wire around the cell in the neighbourhood of the
electrodes, (2) wrapping tin-foil round the whole cell, produced a diminution of
resistance amounting to 0'3 and 1'G per cent, respectively.

368 MESSRS. V. H. VELEY AND J. J. MANLET ON

PUEIFICATION OF WATER.

Ordinary distilled water was purified by redistillation with a few crystals of
potassium permanganate and a few cubic centimetres of concentrated sulphuric acid
instead of potassium hydrogen sulphate, as recommended by STAS. A slow current of
purified hydrogen was passed through the water during the operation, which was
conducted in a laboratory used for no other purpose. The condensation was purposely
kept somewhat imperfect to prevent any ammonia gas from being redissolved in the
cool water ; the first and last fifths of the distillate were rejected, and the inter-
mediate portion again distilled over barium hydrate to remove any sulphuric acid.
On the day before any conductivity experiments were made the water was distilled
a third time in a platinum still fitted with a condensing tube and receiver of the
same material, and again also with imperfect condensation. It was observed that
after several litres of the water had been distilled a deposit of silica, distinctly
crystalline, was formed on the still. Finally, a few hours previous to the commence-
ment of a series of experiments, the water contained in small bottles was partially
frozen (by immersion in a freezing mixture) so as to obtain a hollow tube of ice ; the
central core of water was poured off, the ice quickly melted and mixed with a
quantity of nitric acid necessary for the concentration required. It is not, of course,
presumed that, by the method adopted, water was obtained of the degree of purity
arrived at by KoHLRAUSGH and HEYDWEILER ; * indeed, it was not considered
necessary to use such precautions, when the solutions to be measured would possess
,i conductivity many million times greater than that of the purified water.

PURIFICATION OF NITRIC ACID.

Two different methods were adopted for this purpose according as the concentration
required was less or greater than 68 per cent. In the former case, acid of sp. gr. = 1*4,
sold as pure, and nearly colourless, was distilled continuously under a pressure of a
few millimetres in an apparatus constructed entirely of glass, as to the essential parts,
namely, the still, condenser and receiver. The acid could be drawn up into the still,
and syphoned off from the receiver by glass tubes . provided with stop-cocks; it was
stored in tubes of hard glass (a precaution adopted by STAS in his investigations upon
atomic weights) the ends of which were drawn off but not sealed, as it was found
that the vapour of the acid given off at ordinary temperatures decomposed to produce
nitrous fumes in the operation of sealing.

When acids of concentration greater than 68 per cent, were required, a proportion
of the water was removed by distillation of such an acid with an equal bulk of
sulphuric acid ; the pale yellow acid thus obtained was then redistilled with barium

* ' Zeits. Physikal Chem.,' vol. 14, p. 317.

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.

369

and silver nitrates dissolved in it, the former to remove any sulphuric acid which
might have been mechanically carried over, the latter to remove haloid acids
remaining dissolved.* The acid thus obtained was subjected to a third process of
fractional distillation t in an apparatus constructed all of glass as follows.

A (fig. 1) is a distillation flask of about 400 cub. centims. capacity, the neck of
which, considerably lengthened out, was sealed at its upper end to the top of a spiral
condenser, B, kept cool by a current of ice-cold water ; to the lower end of the latter
the receiver was attached, which was provided with a stop-cock for drawing off the

Fig. 1.

condensed acid from time to time. A side trap, C, was sealed on to the upper end of
the receiver, in order to retain any accidental spirting from the sulphuric acid towers,
D and E ; these latter were about half a metre high, and filled for three-quarters of
their length with glass beads moistened with sulphuric acid, which could be renewed
by means of stop-cocks sealed at their lower extremities. This process of cleansing
and renewal was repeated not only previous to, but also during each distillation. It
was found by experience that these towers were sufficient to prevent any moisture
from passing from the water pump to the anterior portion of the apparatus. The

* This process of purification was previously adopted by KOLB, ' Aim. Chim. Phys.' [4] vol. 10, p. 140.
f Attempts were made to remove the water present by means of phosphorus pentoxide, but were
unsuccessful owing to the rapid decomposition of the 99 per cent, acid by this substance.
VOL. CXCI. — A. 3 B

370 MESSRS. V. H. VELEY AND J. J. MANLET ON

gauge, F, introduced between the two towers served to indicate the pressure,
generally a few millimetres, under which the distillation was conducted, while the
glass springs, G and G', decreased the rigidity of the apparatus. The process of
distillation consisted in drawing up the acid into the flask by means of the stop-
cock ; the whole apparatus was then exhausted, and the water-bath surrounding
the distillation flask gently warmed. The nitrous fumes present in the acid were for
the most part immediately swept through the apparatus, and the remainder collected
in the first portions of the nitric acid condensed in the receiver. This was drawn off
and the whole process repeated until a small test sample of the acid produced no
decoloration of a drop of standard potassium permanganate solution after the lapse
of thirty minutes. The most effective method, according to our experience, for
removing the last trace of nitrous fumes without either diluting the acid or adding
any other impurity, consists in passing a stream of dried ozonised oxygen through
the acid, and then subsequently distilling it in the above vacuum apparatus to
eliminate any dissolved gas.

DETECTION AND ESTIMATION OF IMPURITIES.

The principal impurities likely to be present in any sample are (i.), nitrous acid,
formed from the decomposition of the liquid itself, or its superincumbent vapour,
(ii.), sulphuric acid, (iii.), the haloid acids, namely, hydrochloric and hydriodic,
derived from the previous manufacture of the acid from " caliche," and, (iv.), alkaline
silicates or silica from the glass of the containing vessels. The methods used to
estimate the three first-mentioned impurities are described in detail.

(i.) Nitrous Acid.

For this purpose the metaphenylene-diamine method was employed, by means
of which 1 part in 20 millions (according to GRIESS, its author) can be detected,
and according to our own observation a quantity less than 1 in 15, but greater than
1 in 20 millions is revealed. The standard test adopted was that of 1 part nitrous
acid in 1 million parts of water, produced either by the liberation of that amount of
nitrous acid by the acidification of sodium nitrite (obtained from silver nitrite) or by
an imitation tint formed from Bismarck-brown dissolved in dilute alcohol ; the
comparisons were made by the tintometer described by one of us in a previous
communication.

(ii.) Halogen Acids.

The solubility of silver chloride was fully investigated by STAS* in the course of
his investigations upon the atomic weights. According to this writer, the solubility

* ' Ann. Cbim. Phys.' [4], vol. 25, p. 22, and [5], vol. 3, p. 145.

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID. 371

varies according as the chloride is in the pulverulent or caseous condition. More
recently the same subject has been investigated by J. P. COOKE,* who states the
solubility is not affected by nitric acid. The method of estimating the traces of
halogen acids in the samples was as follows : —

10 cub. centims. of the sample to be examined were placed in a miniature flask
and 1 cub. centim. of a silver nitrate solution containing '05 gram of the salt was
added ; the turbidity, if any, was compared with that produced by the same
quantities of purified water, the silver nitrate solution and a fractional part of
1 cub. centim. of a solution of hydrochloric acid of known strength. It was found
in the course of experiment that a sample of nitric acid of 20 per cent, concentration
gave no turbidity with the silver nitrate, but on addition of '5 per million of
hydrochloric acid a turbidity was produced which appeared to be equal in amount
to a blank experiment, made at the same time, with water only ; but after standing
for some hours the turbidity in the former case was less than that in the latter, and
remained nearly white, instead of changing to a violet tint on exposure to diffuse
daylight. It was probable, therefore, that the chloride was being slowly decomposed
by the nitric acid. As a result of this observation, acids of concentration greater
than 20 per cent, were diluted down to this strength, and the comparison made with
the blank water experiment after the interval of one hour. It is not presumed
that the estimations are of absolute accuracy, especially as the greatest quantity of
haloid acids found in any sample was 3'8 parts per million, the average being less
than 2'0 parts. The correction to be applied for these amounts of impurity is of
the second order, and may reasonably be neglected.

(iii.) Sulphuric Acid.

The observations upon the solubility of barium sulphate in water and in dilute
nitric acid appear to be very discordant. According to FEESENIUS and HiNTzt one
part of barium sulphate in 400,000 water can be detected provided that a slight
excess of barium chloride is added, but only 1 in 100,000 parts if no such excess is
present ; under the same conditions, in presence of nitric acid of 7'8 per cent,
concentration, only 1 part of barium sulphate in 33,000 can be detected.

These estimations were made with a view of examining the accuracy of the
quantitative determination of sulphuric acid by barium chloride, as the precipitant,
in the presence of various salts and acids which might be present in the course of
analytical experience ; otherwise the use of barium chloride rather than the nitrate
in the presence of nitric acid might be open to objection. Furthermore, it is evident
that the delicacy of the reaction between a barium salt and sulphuric acid will
depend cceteris paribus on the relative masses of the reacting substances. The work

* 'American Journal of Science,' vol. 21, p. 220.
t ' Zeits. Analyt. Chem.,' 1896, vol. 33, p. 170.
3 B 2

372 MESSRS. V. H. VELEY AND J. J. MANLEY ON

of the above .writers has been criticised by KUSLER,* who found by an electric
resistance method that a saturated solution of barium sulphate in water contains
1 part of salt in 425,000 water (equals 2'35 parts per million), a result in accordance
with the former observations of ROSE, KOHLRA.USCH, and HOLLEMANN.

In the estimation of the amount of impurity of sulphuric acid in the samples used,
1 cub. centim. of a 10 per cent, solution of barium nitrate was added to 10 cub. centhns.
of the nitric acid, which had been neutralised previously by a solution of ammonia ;
the quantity of the barium niti'ate was more than a hundredfold in excess of that
required to precipitate the greatest amount of sulphuric acid likely to be present. In
a blank experiment with water which had been distilled over baryta it was found that
2-3 parts of sulphuric acid per million water produced a turbidity with the barium
nitrate ; but in the case of a sample of nitric acid of 50 per cent, concentration, after
neutralisation with ammonia, 2'8 parts of sulphuric acid per million produced a
turbidity. The mean of these two numbers, namely 2'5, may reasonably be taken as
the limit of delicacy of the reaction within the limit of '0650 per cent. ; from the
results calculated upon this basis the greatest quantity of sulphuric acid found was
4'3 parts per million. A selection is given below (p. 385) of various acids used in order
to show the degree of purity attained.

DETERMINATION OF THE CONCENTRATION OF THE SOLUTIONS.

For this purpose standard solutions of sodium hydrate were made from metallic
sodium, purified by melting it in vacuo and filtering the molten metal through iron
gauze ; it was then placed in a silver dish and allowed to hydrate slowly in the vapour
of water from a dilute soda solution ; the method adopted has previously been fully
described.

The solution thus obtained was made up to a definite volume and standardised as
against (i.) sulphuric and (ii.) hydrochloric acid ; the acidity of the former was deter-
mined volumetrically by sodium carbonate, and gravimetrically as barium sulphate,
that of the latter (i.) volumetrically by pure metallic silver, by the method of STAS,
and (ii.) volumetrically by sodium carbonate. Three different methods and two
different indicators, namely, litmus and methyl-orange, were used by two different
individuals. The degree of concordance obtained for one such standard solution of soda
is shown by the following table : —

' Zeits. Anorgan. Cbem.,' vol. 112, p. 261.

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.

TABLE I.

373

Acid.

Method.

Sodium hydrate in
1 cub. centim.

Sulphuric
ii

»»

Barium sulphate
Sodium carbonate
ii

gram.
0-03912
0-03912
0-03914

Hydrochloric
i»

>!
li

Silver

)!

Sodium carbonate
»

0-03914
0-03912
0-03912
0-03912

For the determination of the nitric acid solutions a weighed quantity was expelled
from a pyknometer, neutralised by a slight excess of the standard sodium hydrate,
and the amount of such excess added determined by a solution of hydrochloric acid of
Too equivalent strength of the soda.

STANDARDISATION OF THERMOMETERS.

The thermometers employed were graduated to '1° C., and were compared with
(i.) a standard Kew thermometer which could be read to '05° F., and (ii.), within
certain limits, with another standard graduated to '05° C., constructed by BAUDIN
and formerly in the collection of the 8th Duke of MARLBOROUGH.

The zero displacements of all the instruments were determined by immersing them
in a cylinder filled with washed ice, and surrounded by an outer jacket packed with
the same material. The readings were made with a telescope, furnished with cross
wires and mounted on glass plates, after the manner of QUINCKE'S reading microscope.
For the estimation of the errors at higher temperatures, the thermometers were
suspended in a deep tank (containing several gallons of water), so that the thread of
mercury at the point to be observed was just visible above the surface ; the instru-
ments were arranged in a line with the standard in the middle, so that by a slight
rotatory movement of the telescope stand each of them could be brought into the
field of view in rapid succession. The water was warmed about 2° above the point
required, allowed to cool slowly, while thoroughly stirred, and a first set of readings
taken ; the process of stirring was repeated, and a second set taken ; in case of any
variation, the mean value was accepted. At the commencement and conclusion of
each set of observations, the standards were read, and .their constancy throughout
was regarded as a safe criterion that the temperature of the large mass of water had
not appreciably altered. The whole series of observations was repeated for every 5°
within the range required, and in the case of two thermometers found to be less
uniform in bore than the others, readings were taken at an interval of each degree.

374

MESSRS. V. H. VELEY AND J. J. MANLET ON

This method of standardisation, though tedious, probably eliminates the errors due
to a lag of the mercury thread to a greater degree than a more rapid process. It
was not thought necessary to apply the corrections for variations of atmospheric
pressure, or others adopted by the Bureau International, as the methods of deter-
mining the electric conductivity of a highly decomposable electrolyte are not
susceptible of such a degree of accuracy that such corrections would materially affect
the result.

CALIBRATIONS OF ELECTROLYTIC CELL.

In order to determine the mean cross-sectional area of this apparatus, two methods
were adopted : (i.) that described by BUNSEN (Gasometrische Methode), namely, by
pouring in equal volumes of mercury and reading the different levels .by a telescope,
the absolute volume of the measuring tube being subsequently ascertained by two
independent weighings of mercury at the temperature of the observation ; (ii.) by
pouring in from a funnel such a weight of mercury as would fill the burette for every
tenth graduation in cubic centimetres, the temperature being noted at the time.
The following results are given to illustrate the concordance obtained by the
two methods : —

TABLE II.

Method I.

Method II.

Difference.

cub. centime.

cub. ccntims.

cub centiras.

10' 12

10-04

+ •08

10-02

10-02

Nil

10-02

9-99

+ •03

10-33

10-36

-•03

10-02

10-04

-•02

9-82

9-82

Nil

10-02

9-96

+ •06

9-82

9-88

-•06

10-02

10-06

-•04

10-02

10-05

-•03

10-22

10-25

-•03

The mean values were accepted as the true volumes.

In order to measure the different lengths so as to deduce the cross sectional
areas, a pair of slide callipers was used, reading to '01 centim. and compared with a
standard metre. The highest and lowest value for the area within the limits of
observation were 4'4732 and 4'3839 sq. centims. respectively, so that the burette
was of a fairly uniform bore.

KESISTANCE COILS.

The resistance coils, supplied by Messrs. ELLIOT BROS., and marked as true ohms
at 16° were standardised by Mr. E. H. GRIFFITHS as against his coils, which had been

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.

375

compared with the standards in the possession of the Cavendish Laboratory,
Cambridge. We desire to express our appreciation of Mr. GKIFFITHS' courtesy and
kindness in this matter. Two sets of determinations were made at different times
and at slightly different temperatures. The results are given as true ohms, when the
thermometer used always for the box registered a temperature of 16°.

TABLE III.

Coils as marked.

Series I.

Scries II.

Moan.

ohm-.

50

ohms.

49-9994

ohms.
49-9927

ohms.

49-9961

20

19-9999

20-0003

20-0001

10 A

10-0026

10-0026

10-0026

10 B

10-0031

10-0028

10-0029

5

4-9995

4-9996

4-9996

2

2-0022

2-0034

2-0028

] A

1-0026

1-0018

1-0022

1 B

1-0022

1-0018

1 002

•5

•5027

•5017

•5022

•2 A

•2005

•2007

•2000

•2B

•2001

•2007

•2004

•1

•1005

•1007

•1006

With the exception of the 50-ohtn coil the results are concordant to within
'001 ohm, and in the majority of cases to '0005 ohm, which is within the limit of
experimental error.

The values of the higher coils were referred to the 50-ohm coil by reducing the
bridge arm to y^y ; they are not, therefore, of such a high degree of accuracy, but
these coils were seldom required.

Coils as marked.

Values.

ohms.

100 A

ohm*.

99-9895

100 B

99-9956

200

200-0117

500

500-1618

TEMPERATURE COEFFICIENT OF BRIDGE WIRE.

The bridge wire, supplied by Messrs. JOHXSON and MATTHEY, was of platinum-
(90 per cent.) iridium (10 per cent.) alloy ; its mean diameter was 1*5 millim.

In order to determine its temperature coefficient, a small portion was drawn out to
•005 millim. diameter, and the wire introduced into a Call endar- Griffiths pyrometer.
Its resistance at the three temperatures of steam, ice, and 20' 15 was determined
by one of their platinum resistance boxes, in which the compensating leads are

376

MESSES. V. H. VELEY AND J. J. MANLET ON

thrown into one arm of the bridge, while the coil and its leads are in another arm
thus, changes in resistance of the leads produce no effect on the readings. The
values for R in steam and ice were ascertained by different arrangements of plugs,
and hence different bridge wire readings. The coils used are indicated by letters.

STEAM.

Coils.

Readings.

Temp, of resistance
box.

Bar.

R. con.

C.F.H.
C.F.H.
C.F.G'.
C.F.G.H.

+ 6-04
+ 6-04
+ Ml
- 4-05

19-95

20-03
20-01
20-12

millims.
748-8
748-76
748-76

748-78

191-351
191-355
191-359
191-353

Mean ....

748-78

191-353

The boiling point of water for -p = 748*78 is 99*59 ; hence the value for R under
this condition is 19T355.

ICE.

Coils.

Readings.

Temp, of resistance
box.

R. con.

C.G.
C.G.H.
C.G'.

+ 204
- 3'095
+ 2-04

2038
20-4
20-4

172-288
172-286
172-289

Mean ....

172-288

Hence II = 172-288, if t = 0 ; similarly, for t = 20'15 R was found in the same
manner to be 176-149. Taking R at 0 as 1, R at 20'15 = 1*02241, and at
99-59 = 1-11067, the curvature is very slight, and is expressible by the equation

R,= l + -0011123 t;
but if R15 := 1, then R = -0010911, and the equation for the temperature coefficient is

SB, = R15 (l + (t- 15) X -001094),
with a probable error of 1 in 1000.

CONSTRUCTION OF THE BRIDGE.

In preliminary experiments with the ordinary form of metre bridge used with a
telephone and small induction coil as described by KOHLRAUSCH and others, difficulty

THE ELECTRIC CONDUCTIVITY OP NITRIC ACID.

377

was experienced in obtaining silence in the telephone within the limits desired. This
was eventually traced to imperfect insulation of the wooden board, even though well
seasoned and previously soaked with paraffin wax (figs. 2 and 3). It was there-

Fig. 2.

Fig. 3.

•.

fore thought desirable to construct a bridge in which the wire was an airline, and all
the strappings were mounted upon ebonite pillars. The wire was stretched between
two massive brass castings, aa, and insulated from them by ebonite plates, bb ; at one
end it was fixed to a brass pin, and at the other to a stretching screw, s', so arranged
VOL. cxci.— A. 3 c

378 MESSES. V. H. VELEY AND J. J. MANLEY ON

as to tighten but not twist the wire ; its effective length was 2 metres. As this
arrangement obviously required a special form of slider to tap, without sagging the
wire, the following plan was adopted : two weighted opposed cones, cc, moved upon
two uniform glass cylinders, cy, placed side by side and set parallel to the wire,
according to the principle of Lord KELVIN, that a body provided with four points will
always move parallel to itself along a cylinder. A brass shoe, s, carrying a glass
plate projected from the centre of the cones at a right angle and rested upon the
millimetre scale of the bridge ; the position of the slider was indicated by a line
etched upon the lower surface of the glass plate, and observed through a little
window, iv, situated vertically above it. A platinum contact piece, f, mounted upon
an ebonite rod, g, was adjusted to within a fraction of a millimetre of the wire.
When it was desired to make contact, an insulated arm, H, was brought down upon
the wire from above by tilting the lever, /. In order to move the slider a wheel, wh,
Avas attached to one of two pulleys, over which passed a cord kept stretched by a
spiral spring. For the purpose of fine adjustment, the wheel, wh, was furnished
with a rubber band, cemented at its edge, on which a second and smaller wheel, ivh',
could be brought to bear by releasing the lever, J ; from the latter wheel projected
an arm, A, which rested upon the end of the milled head screw, S. By this arrange-
ment the' slider could be moved by the observer from the extremity of the bridge,
and all thenno-currents due to his proximity and direct handling were avoided.

CALIBRATION OP BRIDGE WIRE.

The wire was placed in the bridge, stretched to the requisite extent, and allowed
to rest undisturbed for nine weeks ; then it was calibrated by CAREY FOSTER'S
method, using as a gauge a short length of the same wire soldered on to massive
copper terminals and a copper connector of the same length, both of which were
joined to the bridge by mercury cups furnished with stout copper tags. The process
of calibration was conducted on two different dates. The wire was found to be very
uniform throughout, the highest and lowest values for the resistance of 1 millim.
being '0001516 and '0001510 ohm respectively.

The total resistance of the 851 millims. in the centre of the bridge, as also of the
ends of the bridge, were determined by a fall of potential method, using one Daniell
cell working through a resistance of 200 ohms, a standard coil of '1 ohm value, and
a high resistance galvanometer ; the current was uniform within limits of time
greatly in excess of those required for the observation ; the necessary corrections for
the temperature coefficient were subsequently applied. Ten independent obser-
vations were made of the total resistance, one set of two on one date, and another
set of eight after an interval of a week. For the interchange of the standard coils
and electrolyte, a switch board, constructed upon S. P. THOMPSON'S pattern, was
used. The mercury cups were mounted upon ebonite stems to secure better insulation,

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID. 379

and for the purpose of better contact between the copper connectors and tags the
former were furnished with leaden weights instead of the more troublesome elastic
bands generally used.

THE COMMUTATOR.

In our earlier experiments a small induction coil was used according to the well-
known KOHLRAUSCH method, but owing to the susceptibility of the nitric acid to
polarisation, this apparatus was found to be unsatisfactory. A commutator of the
disc pattern was therefore substituted, being driven by a small water-motor served
through a Kelvin tap to secure a regular flow of water ; the necessary electric
current was obtained from a one-quart Daniell cell.* The speed of the commutator
disc was determined by a kymograph, a lever arrangement being fitted up so as to
give one vibration for each revolution. A number of experiments showed that the
revolutions varied between the extreme limits of 16 and 20 per second; as the
disc carried nine commutators, there were for each revolution 18 makes and breaks
and 9 reversals; this gives per second 288-360 of the former and 144-180 of the
latter. In order to test the efficacy of this method, the alternating currents were
led directly into nitric acid of various concentrations for one hour. No evidence of
polarisation could be detected when the commutating arrangement was switched oft'
and the electrodes simultaneously connected up with a very sensitive galvanometer,
and, further, no trace of nitrous acid could be detected by the metaphenylene-
diamine test.

The commutator, working-cell, and water-motor were placed in a different wing
of the building, the current being brought into the experimental room by leads
twisted together.

GENERAL ARRANGEMENT OF APPARATUS (Diagram I).

C represents the commutator, L its leads, K a key for either short-circuiting the
current (position 1), or connecting up with the bridge and electrolytic cell E
(position 3) or breaking connection altogether (position 2). E3 is a small bifilar coil
to the extremities of which the leads of a call telephone, t, were attached, the latter-
serving to detect any alteration or failure of the commutator. R^ is a resistance box,
the coils of which were adjusted so as to approximately compensate any alteration in
the standard resistance, R, and therefore to maintain the current uniform through
the whole series of experiments ; B is a switchboard, E2 a box of auxiliary cells used
in connection with CAREY FOSTER'S method, and tl a balancing telephone. The leads
Lj and L2 connecting the switch-board with the bridge consisted of ten strands of

* The current of this cell, measured by a standard tangent galvanometer, was found to be '22 ampere
•when flowing through the commutator at rest, but it would probably be rather less when the
commutator was rapidly running, owing to a consequent increase of resistance in the contacts.

3 C 2

380

MESSRS. V. H. VBLEY AND J. J. MANLEY ON
Diagram I.

Showing general aiTangement of the apparatus.

No. 18 copper wire soldered on to massive copper tags ; their resistances were deter-
mined in position and found to be as nearly as possible equal ; L3 the leads (of
similar construction) to the standard coils were of resistance '005 ohm., a constant
factor added to all values of II.

THE ELECTROLYTIC CELLS.
(i.) Standard Cell.

Several forms of cell were adopted in the earlier ex-periments, but as all those of
the U or bottle pattern, as described by KOHLRATJSCH, involve their calibration by a
solution of an electrolyte of known conductivity, it was resolved to use a cell of a
straight burette form, with one electrode fixed, and the other movable, so as to
throw different lengths of the acid into circuit, and thus obtain independent
measurements. For this purpose the movable electrode was welded on to a
platinum wire passing through a glass tube to the binding-screw, S (fig. 4) ;
above this a similar tube was fastened, upon which was etched a centimetre scale
copied directly from the standard. In order to ensure that the whole arrangement
should slide parallel to its axis, the graduated tube was allowed to run in V grooves

THE ELECTEIC CONDUCTIVITY OF NITRIC ACID.

381

mounted upon a block of ebonite, and provided with a small spring to avoid
accidental displacement ; their rotation was prevented by a small arm, A, working in
glass tube guides, y. A cord passing over the pulleys, P, P! and P.,, was attached to
the upper end of the graduated tube, and the other end to an arm ; this cord could

Fig. 4.

be shortened by suitable loops, and the final adjustments effected by means of the
kymograph screw, K. The divisions of the tube were observed by a telescope
mounted on a rigid stand.

To secure perfect insulation the cell rested upon and against blocks of ebonite, a, b
and c, the two latter being V grooves, against which it was kept pressed by elastic
bands. Preliminary experiments already described showed that all fastenings of a

382

MESSRS. V. H. VELEY AND J. J. MANLET ON

conducting nature should be avoided, especially if high tension currents are used. So
far as can be judged by the description and diagrams of certain previous writers,
sufficient attention does not seem to have been given to this essential point. Finally,
to avoid contamination of the acid by dust, the cell was closed by a platinum disc.

(ii.) Temperature Coefficient Cell (fig. 5).

This was a U tube of 177 sq. centims., cross sectional area, rigidly attached to an
insulating stand containing the electrodes, both movable in this case and welded
on to platinum wires passing through capillary tubing. The upper ends of these were
tied and cemented on to an ebonite block which could be made to rest upon either of
two brackets furnished with a hole, slot and plane arrangement similar to that used
for a KELVIN galvanometer. When the block rested on the lower bracket (position

Pig. 5.

Temperature coefficient cell.

Electrolytic cell No. 2.

A), measurements were made of the resistance of the leads and the acid in the bend
of the U ; when, on the upper bracket (position B), the total resistance was made
up of these factors, together with the introduced length of acid, being double that
of the height through which the electrodes had been lifted. To obtain, so far as
possible, independent measurements the process of raising and lowering the electrodes
was repeated during any one series of experiments. The top of each limb of the
U tube was closed with platinum discs.

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID. 383

(iii.) Cell for Acids from 70 to 99'8 per cent. Concentration.

As the acid above 70 per cent, is very hygroscopic and its fumes are liable to
attack the electric connections, a cell for use not only at a particular but also at any
required temperature was devised, as follows : a cylinder of glass, with an etched
centimetre scale, was drawn out at its lower end and a vacuum tap sealed on ; the
upper end was closed by an accurately ground glass cap carrying a tube on which
were etched two lines at a distance of 10 '24 centims. apart. Within this tube moved
a piece of thermometer tubing so as to form a stuffing-box arrangement containing
within it the platinum lead from the movable electrode ; one line was etched upon
it, which could be made to coincide with either line upon the outer tube. The
method of measurement was similar to that described in (i.) (standard cell), and the
whole was, as in previous cases, thoroughly insulated. It was found that acid could
be preserved in this cell for a considerable time without any appreciable change of
concentration.

The electrodes for all the above cells were coated with the velvet-like deposit
of platinum, which, according to the experience of KOHLRAUSCH and successive
writers, gives the most satisfactory results. After preparation they were kept
immersed for some weeks in the prepared distilled water, which was frequently
changed.

TANKS FOR ELECTROLYTIC CELLS.

Two zinc tanks, of many gallons capacity, and furnished with covers, were used in
the course of the investigation ; they were both similarly constructed, one of two, the
other of three concentric chambers ; the innermost, of small dimensions, containing the
electrolytic cell with its accessories, stood on three legs, so as to raise its level about
six inches ; the other, of large dimensions, furnished with ring stirrers, was filled with
water to a level above that of the top of the electrolytic cell. It was found that a
difference of 20° between the tank and room temperatures caused only a fall in the
former of 1° after two hours ; there was, therefore, no appreciable alteration of
temperature in the course of any set of experiments lasting only a few minutes.
Standardised thermometers were placed both in the outer and inner chambers ;
when their corrected readings differed only by O'l it was assumed that equilibrium of
temperature in the outer and inner chambers had been established.

When a temperature of 0° was required, the outer chamber of the triple-chambered
tank contained air only, and the middle chamber was filled with finely-powdered ice,
the water melting from which was continuously drained off. For other temperatures,
water was run into the outer chamber, and steam blown in, if necessary.

384 MESSRS. V. H. VELEY AND J. J. MANLEY ON

THE METHOD OP PERFORMING AN EXPERIMENT.

The electrolytic cell was charged with the required mixture of nitric acid and
recently-melted ice as previously described, then placed in the inner chamber of the
tank and its electrodes connected up with the bridge leads. When an hour had
elapsed after equilibrium of temperature had been first observed, preliminary
measurements were made in order to ascertain approximately the conductivity of the
acid, and thus not only to facilitate the final measurement, but also to eliminate the
risk of decomposition. The cell was then emptied, recharged with a second portion
of the acid, and connections made as before ; after an interval of some hours the final
measurements were made as follows : —

The graduated glass tube was adjusted so that division 5 of its scale coincided
with the point of intersection of the cross threads of the telescope ; the standard
coils were placed in the left gap (L) and the electrolytic cell in the right gap (R) of
the bridge ; if then the current from the commutator was found satisfactory,
connection was made with the bridge and its accessories, and a first balance
obtained.

The current was then switched off, the position of the coils and electrolytic cell
interchanged, the current switched on, and a second balance obtained. The movable
electrode was then raised to division 15 on the scale, and the value of the standard
coils altered so that their combined resistances were approximately equal to those of
the introduced length of acid and leads (as determined by the preliminary experiment),
while simultaneously the resistances in the control box were also changed so as to
maintain the current intensity uniform. Then balancings were obtained with the
electrolytic cell successively in the II and L gaps as before. The whole series of
operations were repeated for the introduction of the second length of acid caused by
raising the graduated glass tube through another five divisions.

The thermometers in the standard resistance box, bridge and tanks were read
before and after each set of observations, and any necessary corrections for alteration
of temperature subsequently applied.

As an illustration, all the measurements and observations for two samples of acid,
namely of 1'299 and 30-52 per cent, concentration are given ; these particular values
have been selected as those of maximum and minimum conductivity for each form of
electrolytic cell.

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.
TABLE IV.

385

Series.

Percentage of acid.

Impurities in parts per million.

I.

1-299

f Halogen acids = 0'5
< Sulphuric acid = TO
(_ Nitrous acid = 0'07

II.

30-52

f Halogen acids = 3'7
< Sulphuric acid = 2'3
Nitrous acid = nil

45-01

{Halogen acids = 0'29
Sulphuric acid = 2'3
Nitrous acid = 0'16

58-2

f Halogen acids = 2'7
< Sulphuric acid = 3'6
1 Nitrons acid = 0'75

Other samples used
in the course of
the investigation

73-82

f Halogen acids =1-1
< Sulphuric acid = 3'16
Nitrous acid — nil.

90-37

f Halogen acids = nil
< Sulphuric acid = nil
[ Nitrous acid = 0'83

98-87

Nitrous acid = I'l

TABLE V.

Temperature (corr.) before

Posi-

and after each observation.

Resist-

Mean

tion OI

ance

Resistance

cross

mov-

flfll A

Tank.

(corr.)

of acid

sectional

Resistance of

«l I ' 1O

Resist-

of coils

+ leads.

area of

acid.

.

elec-
trodes.

Outer

Inner

ance

+ leads.

tube.

chamber.

chamber.

coils.

ohms.

ohms.

sq. centims.

ohms.

c

5

15-7

15-6

16-5

15-0106

14-9658 =R,

4-4591

33-0002 =R.2-R1

Series I.<

15

15-7

15-6

16.5

48-0260

48-2660=R2

1 • |

4-4560

50-0521=R8-R,

L

20

15-7

15-6

16-5

65-0179

65-0179=R,

r

5

13-7

13-6

14-2

22-9031

22-9031 = R,

4-4591

3-0474=Rs-Rl

Series II. <

15

13-7

13-6

14-2

25-8988

25-9505=R,

I

4-4560

4-5779=R8-R1

L

20

13-7

13-6

14-2

27-3989

27-4810=R,

VOL. CXCI. — A.

3 D

386

MESSRS. V. H. VELEY AND J. J. MANLEY ON

The following table serves to illustrate the method of determining the temperature
coefficients, the acid used in Series II. being selected as an example : —

TABLE VI.

Temperature.

Resistance (corr.) of acid
and leads.

Difference (B— A).

0-2

ohms.

17-2093 (A)
29-7130 (B)

ohms.
12-5037

14-5

15-9305 (A)
25-7937 (B)

9-8592

27-9

15-1748 (A)
23-4070 (B)

8-2322

Hence, from these data, the values for the constants a and
( = HO (1 ± at ± fit") can be deduced.

in the equation

PREPARATION AND PROPERTIES or ANHYDROUS ACID (HN03).

By the vacuum apparatus described in the previous part of the paper it was not
found possible to obtain acid of concentration greater than 99 '88 per cent. With a
view of eliminating the remainder of the water, another apparatus was constructed,
all of glass, but without any glass taps, through which, however perfect, moist air
may find a passage.

TT (Diagram II.) are drying towers similar to those in the first form of apparatus.
c, c', c" are thick- walled capillary tubes, which can be sealed off in succession, F is the
distillation flask of about 200 cub. centims. capacity, "and E the electrolytic cell fur-
nished with another capillary, t, sealed before distillation, but which subsequently served
to withdraw a sample of acid for analysis after the determination of conductivity. The
apparatus anterior to the towers was dried by exhausting through the capillary, t,
and subsequently admitting dry air, the whole process being repeated a number of
times, while the glass was kept continuously warmed by a flame ; t is then sealed,
acid of 99 -8 per cent, drawn up into the distillation flask through the suction tube,
which was then sealed off. The whole apparatus was then exhausted by the pump,
and the flask gently warmed until the acid began to distil over into the towers ; when
this was observed the source of heat was removed, and the whole of the tube c"

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.

387

continuously heated to free it from any condensed acid ; then it was sealed off at the
lowest possible temperature while the pump was kept in action. By immersing E in
a freezing mixture acid passed over from F, and this process was allowed to continue
until the upper electrode was well covered ; the electrolytic cell was then detached
by cutting off at c', plugging the opening immediately by a sealed capillary, and
covering the whole with a glass cap. Though it is true that a trace of moisture
might be introduced here, yet the error due to this would be less than that caused

Diagram II.

ro water
supply.

Apparatus employed for the preparation of the anhydrous acid (HN03).

by the decomposition of the vapour of acid if c had been sealed in a flame ; thus, of
two evils, the less was chosen.

The acid thus ob tained was perfectly colourless. Analyses made after measurement
of conductivity gave the following values : —

Acid, per cent.

(1) 99-97

(2) 99-98

99-975 mean.

Determinations were made of its density at 4°, 14'2°, and 24'2° (vide infra), and a
further pair of analyses made, with the result :

Acid, per cent.

(1) 99-97

(2) 99'97
3 D 2

388 .' MESSRS. V. H. VELET AND J. J. MANLEY ON

The acid may therefore be regarded as anhydrous (HN03) within the limits of
experimental error. Hitherto the highest recorded values appear to be 99'45 per
cent. (PERKIN),* 99'67 per cent. (LuNGE and REv),t and 99'84 per cent. (ROLE),}
but doubts have been expressed as to the correctness of the determinations of the
last observer.

CALIBRATION OF CELL AND WIRES USED FOR ANHYDROUS ACID.

As the method originally proposed by KOHLRAUSCH and adopted by subsequent
workers for determining the " resistance capacity " of their cells appeared in this
particular case to involve manipulative difficulties and errors consequent therefrom,
the following method was used. The distance between the inner surface of the
electrodes was determined along nine different lengths ; the mean value found
was 9'77 centims. ; the volume of the portion of the cell between the electrodes was
measured by water delivered from a calibrated burette connected with the cell by
pressure-tubing, readings being taken for the levels of the upper surface of the
lower electrode and lower surface of the upper electrode. Five or six series of
experiments were conducted, different burette levels being taken in each such series;
the mean of all the values was 35 '2 cub. centims., with an error of ± '1 Per cent.
From these data the mean cross sectional area was calculated to be 3'603 sq. centims.
In order to determine the resistance of the platinum leads sealed into the cell,
their lengths were measured and the resistance of a known length of the same wire
determined by CAREY FOSTER'S method •, the value thus calculated was added to
that of the copper wire leads which joined the cells to the bridge, due allowance
being made in each case for its coefficient at the temperature of the experiment.
Such indirect measurements cannot, of course, lay claim to absolute accuracy, but.
appeared to be the best attainable after the cell had been fitted up and used for the
experimental determinations.

(i.) Chemical Properties of Anhydrous Acid.

As a considerable quantity of the anhydrous acid was prepared in the course of
the enquiry some of its properties were examined. It has no action upon the
following rnetals (i.) copper, (ii.) silver, (iii.) cadmium, (iv.) mercury, all of a high
degree of purity, and (v.) commercial magnesium at ordinary temperatures ; purified
iron and commercial granulated tin were unaffected by the acid even when boiling.
Zinc purified by frequent distillation in vacuo was slightly acted upon at ordinary
temperatures, but sodium immediately caught fire. The acid has no action whatever

* ' Journal of the Chemical Society,' 1893 (Trans.), p. 65.

t ' Journal of the Society of Chemical Industry,' 1891, p. 543.

J 'Ann. Chim. Phys.' [4], vol. 10, p. 140.

THE ELECTRIC CONDUCTIVITY OP NITRIC ACID.

389

on finely -powdered calcium carbonate (Iceland spar and marble) either at ordinary
temperatures or the boiling point. Vitreous phosphorus at first melts in the acid
and then catches fire, while a considerable quantity of a dark-red sublimate (pro-
bably phosphorus suboxide P40) condenses in the walls of the tube above the surface
of the liquid ; flowers of sulphur, as also iron pyrites, dissolve quickly and completely
in the gently-warmed acid. Nitration of aromatic hydrocarbons is readily effected
by the acid, whilst such a substance as chloral hydrate is decomposed, with complete
conversion of the chlorine into hydrochloric acid.

(ii.) Determination of Density.

For this purpose the form of Sprengel pyknometer, as modified by DITTMAR, was
used, and of capacity about 54 cub. centims. ; the ends of both limbs were closed by
caps, and on account of the high coefficient of expansion a series of bulbs were blown
on one limb of the U tube.

The necessary corrections were made for reducing the observations to weighings
in vacuo.

The following results were obtained :—

Density 4/4 = 1-54216 (i.)
- 1-54209 (ii.)

Density 14'2/4 = 1-52236 (i.)
= 1-52231 (ii.)

Density 24'2/4 = 1-50390 (i.)
= 1-50398 (ii.)

Mean

1-54212

Mean

1-52234

Mean

1-50394

The values obtained by former observers are given for the sake of comparison.

Observer.

Percentage of acid.

Values.

KOLB
LUNGE and NEF
PEBKIN

99-84
99-65
99-45

/Do = 1-559
\ D15 = 1-530
DI54 = 1-520
( 0,5/15=1-5191
\ D25M = 1-5043

The results of LUNGE and NEF, as also of PERKIN, do not differ materially from
those obtained by us after allowance is made for the slightly higher percentage
strength and lower temperature, while those of KOLB are considerably higher ; the
method adopted by the last for the determination of acidity was, however, open to
considerable objection, though his preliminary, methods of purification were similar
to those described above. It is proposed to publish shortly an account of the
determination of densities and of other physical properties of the acid to confirm the
conclusions drawn from -the present work.

390

MESSRS. V. H. VELEY AND J. J. MANLET ON

SUPPLEMENTARY NOTE ON THE APPLICATION OF CAREY FOSTER'S METHOD FOR

COMPARISON OF RESISTANCES.

It was thought advisable to obtain certain measurements by CAREY FOSTER'S
method for the comparison of resistances, not only to serve as a check upon the rest
of the work, but also to ascertain if this method could readily be applied in the
case of a decomposable electrolyte.

For this purpose the auxiliary coils were wound upon glass tube frames and soaked
in paraffin wax ; the frame was then enclosed in a box which was placed in a larger
one, the space of two inches between them being packed with cotton wool. Two
thermometers were introduced through the lids, so that their bulbs were in the
centre. The maximum change registered during an hour was never more than one-
tenth of a degree.

The process of working was as follows : — The ordinary method was first employed
to obtain the value for K X 10", then the auxiliary coils introduced into circuit and
the determinations repeated. It will be seen from the following table that the results
obtained by the two methods are concordant when due allowance is made for the
slight variation of temperature.

TABLE VII.

Strength of
acid per cent.

K x 10s by
C. FOSTER'S
method.

Temperature.

KxlO'by
KOHLUAUSCH'S
method.

Temperature.

Difference.

25-23

7094

0

17-6

7079

17-3

-15

45-34

5984

12-6

5980

12-6

- 4

51-77

5294

12-0

5277

11-8

-17

56-07

4402

6-6

4398

6-6

- 4

61-60

3829

5-5

3841

5-6

+ 12

64-87

3521

4-7

3514

47

- 7

As a matter of experience it was found that for CAREY FOSTER'S method the
electrodes should be continually recoated with the platinum deposit, otherwise it was
difficult to ascertain exactly the point of minimum sound in the telephone.

EXPERIMENTAL RESULTS.

The approximate temperatures selected for observation were 0°, 15°, and 30° for
acids of percentage concentration from 1'3 to 50, but 0°, 10°, and 20° above this
strength, as the more concentrated acids are not only extremely volatile, but also
liable to decomposition.

In Table VIII. the values are given for the specific resistances in true ohms, corrected

THE ELECTRIC CONDUCTIVITY- OF NITRIC ACID.

391

at 0°, 15°, and 30° for temperature coefficients ; these of the first and second order,
multiplied by 10* and 10" respectively, are shown in the fifth and sixth columns.

TABLE VIII.

Percentage
concentration.

Specific resistance in true ohms.

Temperature coefficients.

AtO.

At 15.

At 30.

alO*.

/310«.

1-30

19-924

14-883

11-974

-204

+ 238

3-12

8-739

6-53

5-279 -205

+ 244

5-99

4-767

3593

2-927 -200

+ 237

10-13

3-022

2-306

1-895 -192

+ 224

15-32

2-236

1-723

1-416 -184

+ 206

20-11

1-910

1-478

1-224 -182

+ 206

25-96

1-746

1-368

1-120 -169

+ 165

3042

1-722

1-342

1-105 -174

+ 183

3381

1-740

1-351

1-109

-177

+ 188

35-90

1-771

1-372

1-195

-178

+ 184

39-48

1-837

1-415

1-146

-181

+ 186

45-01

1-969

1-511

1-222

-183

+ 190

51-78

2-248

1-694

1-36C

-198

+ 225

53-03

2-316

1-726

1-427

-211

+ 275

58-20

2-571

1-916

1-576

-210

+ 277

61-20

2-755

2-016

1-375

-190

+ 77

65-77

3-074

2-317

1-963

-208

+ 292

69-53

3-227

2-523

2-150

-177

+ 228

73-82

4-153

3-270

2-831

-174

+ 237

76-59

4-7584

3-599

2-302

-153

- 6

78-90

6-239

4-836

4-493

-206

+ 377

84-08

10-188

8-164

7-177

-166

+ 226

86-18

12-535

10-106

8-874

-161

+ 212

87-72

16-624

13-320

12-220

-177

+ 294

89-92

24-012

19-929

15-847

-120

+ 46

91-87

37-618

31-077

27-967

-146

+ 202

94-32

53-847

45-270

41-108

-133

+ 182

96-12

73-663

66-069

63-517

-192

+ 152

98-50

54-733

56-057

60-730

- 4

+ 136

98-85

50-182

51-586

53-966

+ 122

+ 43

99-27

26-554

25-075

26-164

- 69

+ 215

99-97

25-688

24-286

23-713

- 47

+ 71

In Table IX., the values for K15X108 for the different concentrations obtained in
Cell No. I. represent for each length of acid the mean of two, and in some cases three,
four, and six, independent observations ; the values are concordant in the majority of
cases to '05 per cent., which is probably within the limit of experimental error ; the
values for Cells II. and III. also represent the mean of several experiments.

In Table X., the values for K0X108 and ICjoXlO8 are given for the sake of com-
parison of our results with those of other observers. Throughout the conductivity of
mercury at 0° is taken as unity, and its specific resistance as 94'07 microhms per
1 cub. centim.

392

MESSRS. V. H. VELEY" AND J. J. MANLET ON

TABLE IX.
Measurements with Cell No. I.

Values of K16 x 108.

Percentage
concentration.

I.

II.

III.

IV.

Length of acid
= 15 centims.

Length of acid
= 10 centims.

Mean.

1-30

63186

632-25

632-06

3-12

1440-2

1441

1440-6

5-99

2617-3

2618-9

2618-1

10-13

4077-9

4080-4

4079-2

15-32

5458-9

5462-2

5460-5

20-11

6361-8

6365-7

6363-8

25-96

6873-6

6877-8

6875-7

30-4-2

7006-1

7010-4

7008-2

33-81

6960-7

6964-9

6962-8

35-90

6855-5

6859-7

6857-6

39-48

6646-5

6650-5

6648-5

45-01

6224-1

6227-8

6225-9

51-78

5552-5

5555-9

5554-2

5303

5448-2

5451-5

5449-8

58-20

4907-3

4910-3

4908-8

61-20

4663-7

4666-6

4665-1

65-77

4059

4061-5

4060-3

Measurements with Cell No. II.

69-53

3728-7

73-82 2877

76-59 2613-5

78-96 1945-1

84-08 1152-2

86-18 930-86

87-72 706-23

89-92 472-02

91-97 302-7

94-32 207-8

96-12 142-38

.

98-50 167-81

98-85 182-36

99-27

375-15

Measurement with Cell No. III.

99-97

387-34

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.

393

TABLE X.

Percentage
concentration .

Values of K0 x 108.

Values of K30 x 108.

1-30

492-15

785-63

3-12

1076-4

1781-9

5-99

1973-3

3213-8

10-13

3112-7

4963-6

15-32

420G-5

6641-0

20-11

4925-4

7686-1

25-96

5386-2

8399-1

30-42

5464-4

8515-4

33-81

5405-7

8480-1

35-9

5312-3

8402-9

39-48

5119-5

8205-7

45-01

4777-1

7701-2

51-78

4183-9

6884-5

53-03

4061-4

6594-5

58-20

3C58-6

5967-8

61-20

3414-9

6842-5

65-77

3059-8

4792-1

69-53

2915-3

4375-9

73-82

2265-2

3323-2

76-59

1976-9

4085-8

78-90

1507-75

2093-5

84-08

923-3

1310-6

86-18

750-5

1060

87-72

555-9

769-8

89-92

391-8

593-6

91-87

250-1

336-4

94-32

174-7

228-8

96-12

127-7

148-1

98-5

171-9

154-9

98-85

187-5

174-3

99-27

354-3

359-5

99-97

366-2

396-7

From the data given in Table VIII., it is evident that the specific resistance of
nitric acid decreases for percentage concentrations from 1'3 to 30, at first more, then
less rapidly ; from this point the resistance increases slowly up to 76 per cent.,
thence more rapidly, until a maximum is reached at 9G'12 per cent., when a sudden
reversal takes place.

KOHLRAUSCH (vide supra), by interpolation of his results, gives the points of
maximum conductivity at 29 per cent, at 0°, 297 per cent, at 18°, and 30'2 per cent,
at 40°, data which are in general accordance with our results, though it is evident
that the exact minimum point could only be ascertained to within one-tenth per cent.,
by a number of experimental determinations made on either side of the point in
question, whereas KOHLRAUSCH gives data only for 23, 30'95, and 37'36 per cent.,
respectively.

As the reversal between 96'12 and 98'5 was wholly unexpected, and at first sight
appeared anomalous, the determinations were continued within small percentage
limits, and the results thereby confirmed.

VOL. cxci. — A. 3 E

394 MESSRS. V. H. VBLET AND J. J. MANLEY ON

It will also be observed that whereas nitric acid behaves as other electrolytes,
namely, in possessing a positive temperature coefficient of conductivity for percentage
concentrations from 1'3 up to 96'12, yet from this point up to 99'27 per cent, it
behaves as a metallic conductor in possessing a negative temperature coefficient.

Formerly, negative and positive coefficients were regarded as the essential point of
differentiation between metallic and electrolytic conductors respectively, until
ARRHENIUS,* led by theoretical considerations, based upon the ionic dissociation
hypothesis, observed negative coefficients in the cases of hypo-phosphorous acid (of
concentration = I '014 normal) and phosphoric acid (of concentration = 3 X '95 normal).
Upon this matter ARRHENIUS expressed himself as follows : — " Es ist wohl kaum
denkbar dass man nach den bisherigen Ansichten ohne Zuhilfenahme der Dissociations-
theorie die Existenz von negativen Temperaturkoeffizienten erklaren konnte."
OSTWALD, and other writers of the same school, when quoting the results of
ARRHENIUS, have adopted the same view, with greater or less modification. But in
the case under discussion, nitric acid of concentration 96-99'97 per cent, would,
ex hypothesi, contain few, if any, free ions, and, therefore, the theory would lead to a
totally opposite conclusion. For the present it is not proposed to offer any expla-
nation of the phenomenon observed, but merely to call attention to the somewhat
analogous example of the density of sulphuric acid, which attains its maximum at
9878 per cent. (PICKERING, LUNGE, NEF, and ISLEE) and thence decreases up to 99 '8 f>
per cent. (PiCKERiNG).t

In the paper quoted above, ARRHENIUS calculates, by interpolation, the temperature
of maximum conductivity of nitric acid as at 668°, but to our minds it is idle to
speculate upon its properties at a temperature far above that at which it could have
any existence.

HYDRATES OF NITRIC ACID.

The existence of definite hydrates of nitric acid has been discussed by various
writers^ from the standpoints of (i.) direct experimental observations ; (ii.) analogy
of the composition of the metallic nitrates and of the phosphoric acids ; (iii.) differen-
tiations of observed values ; cumulative evidence from the first of the methods
appears the most cogent. The following resume gives briefly the conclusions of the
several writers : —

* ' Zeits. f. Physikal. Chem.,' vol. 4, p. 96.

t Note added June 17, 1898. — Since the above was written, the suggestion has been put forward that
in the case of nitric acid of 96-99'7 per cent., there is an initial decomposition into water and the
anhydride N306, precisely as sulphuric acid to a greater degree at a higher temperature, and to a less
degree at a lower temperature, decomposes into water and its anhydride SO8.

J MBNDELEEFF, " Principles of Chemistry " ; W. H. PEEKIN, Sen., ' Journal of Chemical Society,'
Trans., 1889, p. 724, and 1893, p. 65 ; KOLB (vide supra) ; BEKTHELOT, ' Mecanique Chimique,' pp. 1, 397 ;
PICKERING, ' Journal of Chemical Society,' Trans., 1893, p. 436 ; CEOMPTON, ibid., 1888, p. 121 ; VELEY,
' Ber. Deutsch. Chem. Ges,' vol. 28, p. 928.

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.

395

HNO3 10H2O 25 '5 percent. HNO3 Differentiation of density curve (PICKERING).

I Specific magnetic rotation (PERKIN).
„ •< Freezing of solution (PICKERING).

I Chemical activity towards metals (VELEY).

„ Density curve (MENDELEEFF).

/Isolation in crystals (PICKERING).
I Metallic nitrates (MENDELEEFF).

("Vapour pressure (MENDELEEFF).
I Heat of dilution (BERTHELOT).

[Density determinations (KOLB).
J Isolation in crystals (PICKERING).

HNOS 7H,O 33-3

HNO3 5H20 41-2 „
HNO3 3H2O 53-85 „

HNO3 2H20 63-63 „

HN03 H20 78-02 „

•s

I Specific magnetic rotatory power, metallic
[_ nitrates and analogy with H3P04 (PERKIN).

70

Curve 1.

6C

'40

10

10 20 SO 40 50 60 70 80 SO 100

flsr cent; acid

Specific resistance /> of nitric acid at 15° C.
3 E 2

396

MESSRS. V. H. VBLEY AND J. J. MANLEY ON

60

Curve 2.

70

60

so

so

10

0 20 JO 40 SO 60 70 6O 3O IOO

Per cent Add.
Specific resistance /> of nitric acid at 0° to 30° C.

In Table XI. the values for a X 104 and /3 X 106 are given in relation to the
molecular proportions of water from 3 '07 to '07, as it would seem that the method
adopted by comparatively few writers on such u subject as the present is the more
rational, as nature deals with molecules, but art with percentages.

THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.

397

8,000

7000

6000

5.0OO

<0

1

I"

3,000

ZftOO

1,000

Curve 3.

Per cent HN03

92 93 34 95 56 97 96 99 (00

\

\

\

\

\

\

\

30C

ZOC

100

io so .50 •fo so 60 TO so 9o ioo Per cent HN03
Conductivity in mercury units of acid at 15° C.

398

ON THE ELECTRIC CONDUCTIVITY OF NITRIC ACID.

TABLE XI.

Molecular proportions
of water.

Value of
a x 10*(-).

Value of
/3 X 10»( + ).

3-07

211

275

2-43

210

277

2-25

190

77

1-84

208

292

1-59

177

228

1-30

174

236

1-09

153

6

0-99

206

377

078

166

226

0-65

161

212

0-58

177

294

0'47

120

46

0-39

146

202

0-27

133

182

0-18

192

152

0-07

4

136

From these figures it will be observed that the minima values for alO4 and
maxima for /SlO6 occur in the cases of 3*07, 1'84, '99, and '55 molecular proportions
of water, or very approximately HNO3.3H20, HNO3.2H20, HN03.H20, and
2HN03.H2O, and the curve representing the conductivity units in terms of per-
centages is markedly discontinuous at points corresponding to the three last hydrates,
the existence of which has also been confirmed by a series of experimental determi-
nations of the densities and contractions of samples of acid of various concentrations.
Further evidence is thus added by an independent method to that already accumu-
lated as to the existence of definite combinations of nitric acid with water.

In conclusion, we trust that these observations may serve as an addition to our
knowledge of the electric and chemical properties of dilute, and especially of the
most concentrated nitric acid, purified, as we hope, with such methods as are
available in the present state of chemical art. It appears that for this and similar
investigations a material, which should at once be perfect in its transparency, its
non-conductivity, and its unalterability by the strongest reagents, remains still an
ideal.

[ 399 ]

X. On the Thermal Conductivities of Single and Mixed Solids and Liquids
and their Variation with Temperature.

By CHARLES H. LEES, D.Sc., Assistant Lecturer in Physics in the Oivens College.
Communicated by Professor ARTHUR SCHUSTER, F.R.S.

Received November 30, — Read December 16, 1897.

PART I. CONDUCTIVITIES AND TEMPERATURE COEFFICIENTS OF SOLIDS.

Sketch of Method,

IN determining thermal conductivities* of solids not very good conductors of heat, the
method least open to objection on theoretical grounds is the one in which a spherical
shell of the substance to be tested is filled with, and the exterior surrounded by, some
good conductor of heat, the temperatures of the conductors, inside and out, being
observed by means of thermometers or thermo-junctions, and maintained constant by
heat supplied, e.g., electrically, to the inner conductor at a measured rate (fig. 1).
Difficulties, mainly of a mechanical kind, present themselves, however, in the carrying
out of this method, which render it advisable to sacrifice some of the theoretical

simplicity, in order to make the method more practicable.

Fig. 3.

Fig. 1.

Fig. 2.

DA.CBE

These difficulties are overcome most easily by having the material to be tested, the
conductor to which heat is supplied, and the outside conductors, in the form of flat
circular discs of the same diameter (fig. 2), the good-conducting disc, C, to which heat

* The extremely good account of previous methods and results given by GBAETZ in WINKELMANN'S
' Handbuch der Physik,' vol. 2, pp. 273-314, renders an account of such work unnecessary here.

5.9.98.

400 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OP SOLIDS

is supplied, being placed between two discs of the material to be tested, A and B, and
two other good- conducting discs, D and E, placed in contact with the outer surfaces.
By placing thermo-j unctions in each good-conducting disc we can determine the
relation between the heat supplied to the inner disc and the differences of temperature
between the inner and outer discs.

If all the heat supplied to the inner disc flowed through the substance to be tested
•to the outer discs, the thermal conductivity of the material would be thus deter-
mined, but some heat is lost from the curved surfaces of the discs by conduction and
radiation through the air to the enclosure in which the discs are placed. The amount
thus lost can be determined by observing the temperature of the enclosure, if the loss
per square centimetre per degree excess of temperature of the edges of the discs over
the inside of the enclosure were known.

To enable this quantity to be determined directly by experiment, the whole of the
hea,t supplied to the discs must be lost in the same way, i.e., by conduction and
radiation through air, and to satisfy this condition, the collection of discs should be
placed in the centre of an air bath kept at constant temperature (fig. 3). The heat
lost from the curved surfaces of the discs then follows with sufficient closeness the
same law as that lost from the flat surfaces of the outer discs, and the relative
amounts'of the two can be determined from a knowledge of the areas and temperatures
of the various surfaces, if the surfaces have the same "outer conductivity" or
" emissivity," an equality which is easily secured by varnishing them.

Once the method of surrounding the discs by an enclosure at constant temperature
has been adopted, a further simplification of the arrangement of discs is possible. If
one of the discs of the substance under test and one of the outer discs were removed,
the relative amounts of heat lost by the heating disc, C (fig. 4), by conduction and
radiation directly to the air from its exposed, surfaces, and by conduction through the
disc, B, of material experimented on, partly to the air directly by conduction and
radiation from the surface of B, and partly by conduction to the outer good-conducting
disc, E, arid radiation from its surfaces, could still be calculated, and, the total heat
supplied being known, the conductivity of the material determined. On account of
this method requiring only one disc of a substance the conductivity of which is to be
determined, it has been adopted in the experiments to be described.

Description of Apparatus.

The materials to be tested were cut into discs of 4 centims. diameter, and of
various thicknesses, in order to make the magnitude of the difference of temperature
of the two flat surfaces, suitable for observation with the thermo-j unctions employed.
The discs between which the material was placed were of copper, and had the same
diameter. On one side of the substance to be tested was placed a single copper
disc, M (fig. 5), -320 centims. thick, on the other side a compound disc, consisting of

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE.

401

two discs of copper, C, "103 centim., and U, '312 centim. thick, with a flat heating
coil of double silk-covered No. 25 platinoid wire, P, between them, well insulated by
means of mica and shellac, the total thickness of the coil and mica being '110 centim.
The contacts between the surfaces of the copper discs U and M, and the substances,
were improved by smearing them with glycerine and sliding them together. The
glycerine was generally sufficient to keep the discs together during an experi-
ment, but to ensure the surfaces remaining properly in contact, a few turns of silk
thread were wound round the discs.

At opposite extremities of a diameter of each disc, two holes, '7 millim. diameter
and 3 millim. deep, were bored towards the centre of the disc, and into one hole of

Fig. 4.

Fig. 5.

Fig. 7.

each disc a platinoid, and into the other hole a copper, wire was soldered. The
wires were double silk covered, about 30 centims. long, and were all cut from the
same two coils.

The surfaces of the pile of discs were varnished to give them the same emissivity,
and the pile placed in the centre of a double-walled air-bath (fig. 6), of 17 centiras.
height, 15 centims. length, and 7 centims. breadth, supported on legs about 4 centims.
above the table. The top of this bath consisted of a double wooden lid, cut down
the centre to allow the wires from the discs to pass out. Th« edges of the lid were
covered with green baize, in which the wires embedded themselves, and which
closed the bath sufficiently to prevent convection currents between the inside
and outside. On the outer walls of the air-bath a coil of insulated platinoid wire
was wound, and an electric current could be sent through the wire to keep the
bath at any required temperature. The inner walls were about 1 centim. within
the outer, and between the two water could be circulated, or the space could be filled
with air simply. A thermo-j unction was soldered to the centre of an inner surface,
and enabled the temperature of the enclosure to be determined. The wires from the
heating coil, P, after passing out through the lid, dipped into mercury cups, from
which wires passed to the cells supplying the heating current, the power absorbed
being regulated by a rheostat and measured by a wattmeter.

VOL. cxci. — A, 3 F

402 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

Each of the thermo-junction wires was brought to a mercury cup, made by cutting
4 centim. lengths from a piece of thin glass tubing with a bore of about 3 millims.,
sealing one end of each length, and fastening it, with the sealed end down, in a brass
tube 5 centims. long and 3 centims. in diameter, mounted in a block of wood and
containing mercury. Ten of these glass tabes nearly full of mercury were arranged
in a circle round the bulb of a thermometer graduated in -^ degrees, which was
placed in the centre of the brass tube. To diminish loss or gain of heat, the brass
tube was wrapped with green baize (fig. 7). The ends of the wires leading to the
potentiometer arrangement for measuring the electromotive force in the thermo-
electric circuits, could be placed in any two of these mercury cups, and by means of
an additional short wire of platinoid, any two thermo-circuits could be placed in
series, and the difference of the electromotive forces in the two determined.

TJieory of Flow of Heat in Discs.

Let H = heat generated per second in the heating coil.

h = heat lost per second, per square centimetre per degree excess of tempera-
ture of the discs over that of the enclosure.

v = excess of temperature over that of enclosure.

t = actual thickness of a disc, plus a small correction, p. (409) to allow for
heat lost along thermo-wires.

r = radius of discs.

Subscripts C, U, M, S refer to " cover " " upper " " middle " discs,* and disc of
substance respectively, the heating coil coming between C and U, and the substance
to be tested between U and M (fig. 5).

When the " steady state " has been attained, the heat received per second by the
copper disc M from the disc of substance S, and given up to the air

Similarly-, the heat received by the disc S, of material tested and given up to the
air and to the disc M

-9 .
r

If I is the thermal conductivity of the material, the heat flowing through the disc
tested

= 7n^^L=l^

The reason for the choice of these terms will be understood from Part II.

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 403

Hence we have, to a very close degree of approximation,*

The total heat imparted to the discs and lost from their surfaces
= H = ^h % -f t>c -f ft** + ts v-^

Eliminating & between these equations we have

and from the second equation we have

H

l + - *M + M + l + - *C + - + •

which will enable us to determine & and h from observations of H and the tempera-
tures of the discs.

In the foregoing theory it has been assumed that the steady state of temperature
distribution had been attained. As, however, an infinite time is required before this
condition is satisfied, it remains to determine at what previous period observations
may be taken without the results deduced from these observations being in error by
say £ per cent.

If the temperature of the " middle disc " is increasing at a rate dv^/dt, the expres-
sion for the heat it receives becomes

i M
-f- - - VM

T I

(It

where mx is the mass, and CM the specific heat of the material, of the disc.

Similarly, the expression for the heat received by the disc of substance, since we
may assume that the discs increase in temperature at the same rate dv/dt, becomes

+ msCs) .

* It will be noticed that the heat flowing through the substance has been taken equal to the mean of
the heats flowing into and out of it respectively. The closeness of this approximation may be tested by
using the values of v in terms of x, obtained on the usual assumption of plane-isothermal surfaces, i.e.,

v = A cosh ax + B sinh cue.
3 F 2

404 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

Hence,

and the total heat imparted to the discs

= H = irrVi jvM + *>c + 7- (MM + *s *M + % + *uVu + ^U + —
Eliminating & between these equations, we have

r -4

For the discs used wiMcM = 3 '2, mscs = 1'8, Imc — 11 '5. and during the experi-
ments II was about '22, and the value of the fraction on the right-hand side of the
equation, '4. Hence, for the error introduced by neglecting dv/dt to be less than
5 per cent., we must have

or

dv/dt < '00045,

so the change of temperature in 1 minute ought not to exceed '03° 0.

No observations in which this rate was exceeded have been used in what follows.

Loss of Heat along Thermo-Wires.

In the above theory, the heat conducted away from the discs by the wires of the
thermu-j unctions, has been taken into account by adding a small correction to the
thickness of the discs. To determine the amount, of this correction for any wire,
we write

-•v x

v = v0e v «*

where v is the temperature excess at a point of the wire distant x from the disc,

p •= perimeter of section of the wire.

q = area

k = conductivity of the wire.

li = emissivity „

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 405

The heat conducted away by the wire per second

= — qk —-, for x = 0,

= \/qk . ph . VQ.

For the copper wire used (No. 28), p = '15, allowing for silk covering ; h = "0003,
q — -0011, and k = I'D, therefore

Vqk.ph = 2'2 X 10~4.

For the platinoid wire, which had the same diameter, we have k = 'OS, and

\/qk.ph = '6 X 1CT4,

therefore total loss of heat along thermo-wires from one disc per second

= 2-8 X 10~*. v0.

If w is the thickness of a disc which would lose the same amount of heat from its
edges, we have, since the loss of heat from such a disc = 2-irrwhvQ,

2-8 x 10-* 2-8 x 10-'

W = —^T = 12-6 x 3 x 10- = >075 C6ntimS '

and the loss along the thermo-wires may be taken into account in the theory of flow
of heat in the discs, by taking as the thickness in the formula, p. 403, the measured
thickness of the disc plus this quantity.

Constants of the Discs.

r = 2 centims., therefore irr3 = 12'5 sq. centims.
10 = '075 centims.

ZM = '32 + -075 = '395 centims., therefore 1 + — <M + *?- = 1-395 + -T.

r 4 4

tc = -103 + '075 + '055* = -233 centims., therefore 1 + — tc= 1-233.

r

tv = -312 + -075 + -055* = -442 centims., therefore — «n + 4r = '442 + %-.

T & i

* Half tbe thickness of the heating coil and mica insulation is added to the upper disc and half to
the cover.

406 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

Hence the formulse, p. 403, become

1-395 + VK+VV
4 4

1-395 + -:§-•

"442 + --.%

'5"D "

H

1-395 + 4-

l-233t;0 + '442 + .

1 *
12-5

Measurement of Power.

The power given to the heating coil of the discs was measured by means of a
wattmeter, with its series coils in the main circuit, and the terminals of its shunt
coil dipping into the same mercury cups as the terminals of the heating coil. The
shunt coil had a resistance of 12 ohms, and was suspended binlarly by means of the
two leading wires. The instrument was standardised by sending a current through
a 2-ohm resistance coil of thick platinoid wire, and regulating the strength of this
current till the electromotive force at the terminals of the resistance was equal to
that of a Clark cell. The wattmeter readings were taken by reversing the con-
nections of the suspended coil to the heating coil in which power was to be measured.
The series coils were placed parallel to the magnetic meridian, so that the effect of
the earth field on the deflection would be small, and there would be no necessity for
reversing the current in the heating circuit. For convenience in working out results
approximately, the resistance in series with the suspended coil (320 ohms) was
adjusted till the wattmeter deflection was nearly 1000 divisions per gram degreet of
heat, generated per second in the heating coil.

If H is the heat generated per second in a coil of resistance li, and d is the
deflection on the wattmeter, the shunt coil of which has a resistance of r ohms, we
have

f Tho value of h is only required in this work to make certain small corrections in the course of the
calculation, and the above equation will not be further referred to. The mean values of h found during
the course of the work, are : —

Mean temperature of
discs and enclosure.

li.

24 C.
54
72

•000276
316
339

t Water gram degree at 15° C. = 4'20 joules.

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE, 407

H = K-'l-BT-r '

where K is a constant.

In one of the standardisation experiments the Clark cell had a temperature of
19° C. and an E.M.F. of 1-428 volts. The deflection on the wattmeter was found
to be 245'5. Hence, since R = 2'018, r = 332 ohms, we have

1 334

When, therefore, the wattmeter was connected to the heating coil of the discs, which
had a resistance of 3 '3 ohms,

990

H = -000986 -£— . d = -000976 . d,

ooo'o

from which the heat generated in the coil could be found from the wattmeter
deflection.

The power thus measured is that spent in the heating coil and the connections
leading to it. A portion only of this power is imparted to the discs themselves, the
rest is spent in the short lengths (1'6 centim.) of the platinoid wire projecting
outside the discs, and in the No. 22 copper wires leading to the mercury cups outside
the air-bath. The total length of the platinoid wire was 150 centims., and
3 '2 centims. of it, and 50 centims. of No, 22 copper wire, were outside the discs.
The resistance of the coil and connection was 3'3 ohms, "02 ohms being due to the
copper, and '07 ohms to the projecting platinoid wire. Of the total power generated,
therefore, only 321/330 = '973 is imparted directly to the discs.

There is, however, an indirect method by which some of the remaining power may
be communicated to the discs, i.e., by thermal conduction along the coil wire, and to
determine the amount of this, we require to know the distribution of temperature
along the wire.

Assuming that the isothermal surfaces in the wire are planes perpendicular to the
axis, we have for the " steady state "

7 (Pv -,

qk d& + w = Ph • v>
where

2 = cross section of wire,
p = perimeter of wire,
k = conductivity of wire,
h = emissivity of wire,

w = heat developed in 1 centim. length of wire,

v = temperature excess over air, at a point distant x from the junction of the
platinoid and copper wires.

408 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

The solution of this equation is

/M/I . /)*'*

10 'v • x I-* "v — • x

7?/i

where A and B are constants of integration.

Fig. 8.

For the platinoid wire, w = '21/150 = '0014, p = '027, h = '0003, g = "002,
k = '08, therefore

4 = 170 and A/4='22'
vh v gk

Hence in the platinoid wire

Vl- 170 = A,e~ + B,e-"'.

For the copper wire, u< = '0014/40, p = '029, h = '0003, q = '004, k = 1,
therefore,

= 4 and - -047.

ph v j*

Since the temperature of the copper wire to the right is nowhere infinite, we have
in the copper wire

v, - 4 = B,e-0471

where B2 is a constant of integration.

When x = — 1'6 in the expression for the temperature of the platinoid wire, the
temperature excess v{ should be equal to that of the discs, which, on the average
was 17°C.

Hence,

— 153 = A,e-35 + B^'35 = 705A, + 1'42B1(

Also at x = 0 the temperatures and fluxes in the two wires are alike.
Therefore,

170 + A, + B, = 4 + B2

and 35 (A, — B,) = — 190B2.

These three equations determine Au B,, B2, and give for the temperature in the
platinoid wire,

«, - 170 = - 100e22x - 57'8e-22*.

The heat flowing per second from the wire to the discs at x = — 1'6

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 409

= qk-22(— lOOe-36 + 57'8e+'35)

= '00016 X '22 (— 100 X '705 + 57'8 X 1'42)

= -000035 (- 70-5 + 82-1)

= '0004 gram degrees.*

Hence, '0008 gram degrees of heat flow per second along the two wires to the
discs.

Of the '21 gram degrees of heat imparted in the coil '21 X '027 = "0057 is
expended outside the discs, and of this '0008 is conveyed to the discs by conduction,
leaving '0049, or about 2'4 per cent., which does not reach the discs. The wattmeter
readings must therefore be diminished by this amount to give the power actually
imparted to the discs.

Combining this result with the one previously found, we have for the heat, H,

imparted to the discs,

H = -000953 d,

when d is the deflection on the wattmeter.

The Thermo-electric Measurements.

The principle adopted in measuring the electromotive forces in the various
thermo-electric circuits, was that of balancing them against the fall of potential
down a high resistance in series with a cell of known electromotive force.

The arrangements of the circuits will be understood from the accompanying

diagram.

Fig. 9.

RI, R2, and Rz are resistances in series with the Leclanche cell, L. R2 and R3 are
resistance boxes of platinoid, of 30,000 ohms and 10,000 ohms respectively. R3 is
adjustable, and it was varied until the fall of potential down the 30,000 ohms was
equal to that of a Clark cell, C. This adjustment was not absolutely necessary, but
was carried out for convenience by the arrangement shown in the figure. The thermo-
couple, T, is balanced against the fall of potential down a part of Rlt which represents
three dial resistances of copper wire, only one of which is shown in the figure. The
movable arms of the dials consisted of short thick copper wires dipping into mercury
cups. The dials read respectively, ohms, '1 ohms, and '01 ohms. They were made

* More generally the heat conducted along one wire to the discs = '0077H — -000071t>, where H is
the heat generated in the coil, and v the temperature excess of the discs.
VOL. CXCI. — A. 3 G

410 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

of copper wire to get rid of thermo-electric forces in the circuit, T, other than those to
be measured.

The galvanometer enabled the determinations to be made to about '003 ohm. By
means of the mercury cups (fig. 7) any one of the thermo-electric circuits could
be connected through the galvanometer' to RI, and the electromotive force- in it
determined, or by arranging two of the circuits in opposition, the difference of electro-
motive forces in any pair could be found.

Standardisation of Thermo-couples.

To determine the constants of each of the thermo-couples, the various copper
discs were bound together by tape, a thin sheet of mica being placed between
each disc for insulation, and the bulb of a thermometer graduated in ^ degrees was
placed in contact with, and bound to, the discs by a few turns of the same tape. The
combination was then placed in its usual position in the air bath, and the bath heated
by sending a current through the coil encircling it. When the temperature of the
bath had been raised to about 50° C., the current round it was diminished till the
temperature ceased to rise, and further adjusted till the temperature remained
constant. After about half-an-hour the indications of the thermometer in the bath,
and of a similar thermometer in the mercury cups in which the other junctions were
placed, were read, the electromotive forces in each therm o-circuit found, and the
thermometers again read. After a few minutes' interval the process was repeated.

Since iu the test of conductivity it is the difference of the temperatures of the discs
which is required, observations of differences of the electromotive forces in the various
circuits were in each case taken. As these differences were small, and could be
observed with the same degree of accuracy as the actual electromotive forces, which
were comparatively large, the calculations of conductivity have all been based on
them, one of the observations of actual electromotive force having been used to fix
the mean temperature of the disc experimented on.

The following table gives the results of the observations : —

6 = the temperature of the discs in the bath.

00 = that of the mercury cups.

U = the number of ohms of the potentiometer necessary to give a balance in the

circuit from the "upper disc." 1 ohm corresponds to 1 '434/30, 000

= '000044 volt.
U — M = the ohms giving a balance when " upper " and " middle " circuits were

opposed.

U — L = ditto . . . ditto, when " upper " and " lower " opposed.
B = ditto . . . ditto, when thermo-junction from inside surface of air bath was in

circuit.

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE.

411

For convenience of interpolation the numbers headed " ohms per degree " are taken
as ordinates, and 6 + #o as abscissae, of the curves which follow the table.

CONSTANTS of Thermo-j unctions.

Ohms.

Divisors.
Ohms per degree.

U.

U-M.

U- L.

B.

U.

B.

47°-68

20-80

10-31

o-io

0-05

9-72

0-384

0-3G2

68-5

47-57

20-91

10-21

o-io

0-04

9-67

0-383

0-362

68-5

82-63

21-75

24-46

0-26

012

t ^

0-402

. .

104-9

83-13

21-87

24-67

0-24

0-10

23-73

0-402

0-387

104-9

97-63

21-87 31-34

0-31 0-16

B f

0-414

. .

119-5

97-63

21-88 31-34

0-32 0-17

29-70

0-414

0-392

119-5

72-60

19-92 20-91

0-18

20-05

0-397

0-381

92-5

72-66

19-41

21-15

0-19

20-26

0-397

0-381

92-1

••*/

•40
•S3

Ohms -se

Per -S7

degree. ,M

30 40

so eo ro so

90 IOO IIO ISO OO I4O ISO 160

It will be noticed from the table that although the wires used for the thermo-
junctions were cut from the same bobbins and treated alike subsequently, the
thermo-circuits had different constants, the difference in the case of the upper and
middle circuits amounting to 1 per cent., and in the case of the upper and heater
circuits to 6 per cent.

This prevents the difference of the temperatures of two junctions being determined
directly by dividing the electromotive force in circuit when the junctions are placed
in series, by a constant. The method to be used is indicated subsequently.

The "cover" and "upper" circuits were found to give identical results, and the
upper junction was therefore adopted as the standard, and the differences between its
electromotive force and those of the other junctions determined in each case.

To determine the temperature of the junction in the upper disc, given the
temperature of the junction in the mercury cups, and the electromotive force in the
circuit, we require to divide this electromotive force by a quantity which itself
varies to a small extent with the temperature required. If, however, this tempe~

3 G 2

412

DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

rature is known approximately, the value of the divisor may be found from the upper
of the preceding curves, and the temperature found more accurately. If necessary
this new value of the temperature may be used to 6nd the divisor more accurately,
and from it a second approximation to the temperature.

To obviate the necessity of determining the temperature of the disc in this way by
two stages, a table of divisors was constructed having a double entry, one, the
temperature of the junction at the mercury cups, the other the resistance of the
potentiometer for a balance. The form of the table will be seen from the part of it
given below : —

Temperature of mercury cups.

Resistance.

16°.

17°.

18°.

19°.

Ohms.

1

•365

•366

•367

•368

2

•367

•368

•369

•370

3

•368

•369

•370

•371

16

•386

•387

•388

•390

17

•387

•388

•390

•391

' 18

•389

•390

•391

•392

To determine the temperature of the upper junction when the junction in the
mercury cups had a temperature of say 18° C., and the resistance for a balance was
say 3 ohms, we find from the table that the divisor is '388, and the temperature
difference between the junctions is 3/'388 = 773, and the temperature required
therefore = 2573° C.

The same plan was adopted for the junction in the air bath.

Tlieory of Observations with Thermo-junctions in Series.

If two homogeneous wires of materials a and b have their ends soldered together,
and the two junctions are kept at temperatures #1 and 60, the electromotive force Ej
in the circuit is given by an equation of the form

where e and /are constants.

If the junctions of a second pair of materials, a and b, where a' differs slightly
from a in thermo-electric properties, are kept at temperatures 6t and 60, we have

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 413

E2 = (ft - ft) e' (\ + /' ftT~ft)

where e and/' are constants.

If at the junctions at temperature ft, the two circuits are joined in series, the
electromotive force in the circuit is the difference of the above, and if we write it
E;j10 we have

E210 = (ft - 6.} e' (1 +/' 6, + ft) - (6, - ft) e (1 + /ft + ft).

Let ft become identical with ft, and let E1IO now be the electromotive force in circuit.
Then

E110 = (ft - ft) e (1 +/'.ft + ft) - (ft - ft) e (I + /ft + ft).
Therefore

E210 - E110 = (ft - ft) e' (1 +/' ft + ft),
or

ft - ft = (E210 - Euo)/e' (1 +/' ft + ft),

which furnishes a means of determining ft — ft from observations of the electromotive
force when the circuits are in series, and of the electromotive force iu the same
circuits when the temperatures ft and ft are identical. In the tables which follow
the quantity E210 is indicated by the term " Observed," and E210 — Euo by the term
"Reduced." -

In the experiments, ft is the temperature of the system of mercury cups (fig. 7,
p. 401), indicated in what follows by J. The junctions in the discs are indicated by
the letters U, M, C, B, for upper, middle, cover, and junction inside the air bath
respectively.

Method of Experimenting.

In carrying out the test of a disc of any substance, the copper discs, between
which the disc to be tested was to be placed, were first brought together, and the
total thickness of the combination measured by means of a micrometer wire gauge.
The thickness of the disc to be tested was then measured, the disc placed in the
proper position between the copper discs, glycerine being used to improve the
contacts, and the total thickness measured. The differences of the observed thick-
nesses is the thickness of the disc of the material used, plus that of the two layers
of glycerine, and we know, therefore, the thickness of the glycerine layers, and can
calculate their effect on the flow of heat.

The combination was then suspended in the centre of the air bath, with the plane
surfaces of the discs vertical. An electric current was then sent through the heating
coil of the discs, which increased in temperature and began to lose heat to the bath.

414 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

If the test was to be carried out at a moderate temperature, current was also sent
through the coil encircling the air bath for a short time, then stopped. Owing to
the heat supplied by the coil to the discs, and given up by them to the bath, the
temperature of the bath remains higher than that of the air, and would eventually
become constant. The currents through the coil of the discs, and through the coil
round the bath, were, however, adjusted from time to time, in order to attain the
steady state of temperature as quickly as possible. When the temperatures of the
discs and of the air bath had remained constant for 10 or 15 minutes, observations of
the wattmetei-, and of the electromotive forces in the various thermo-circuits, were
made, an interval of 5 or 10 minutes allowed to elapse, and the observations repeated.
If the second set did not show that the steady state had been reached, the tempera-
ture of the bath was varied suitably, and observations repeated. In this way it was
insured that the observations were made during a state which approximated with
sufficient closeness (p. 404) to the steady state.

If the test was to be carried out at a lower temperature, cold water was allowed
to circulate between the inner and outer walls of the air bath, the observations being
taken as before.

If the test was to be carried out at a higher temperature, the current through
the heating coil was increased for a time, and current sent through the coil on the air
bath till its temperature had been raised to the required amount. The current
through the heating coil of the discs was then reduced to its normal amount, and the
temperatures throughout allowed to become steady. Observations were taken as
before. In cases where there was any suspicion of a change of the material under
test having been brought about by the increase of temperature, for additional security
the temperature of the bath was again lowered and a second test made at the original
temperature.

The following tables and curves show the substances tested and the results
obtained :

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE.

415

i

r» 10 co co oo os CM

O "^ O rH rH O rH

°«b cb CM cb do rH 4fi

rH i— 1 1C rH CO 1C CO

^^

n^j< ic oo ic >c r^ cs

°CO CO CO CO iC CO t-

q?

1C CN iO OO iO iO E*-

CO 00 >C CO I> CO O

CO rH OS CM CO rH O
•^ ^? iC ^ iC CD C^

rH CO -H rH OS CM 10
| | CO | rH CO •*

'^lAfJOUpUOQ

888 8888

„ i>» crj r*~ co m

•J081ATQ

T? •*? CO -^ iO <SO l>-

CO CO CO CO CO CO CO

•ado[s

00 CM OS OS •* t^ iC
0rH CM 00 rH O OS 00

•* CO 00 00 1C * rH

ajti^'B.iadiuajJ

CO CO CM CO CO CM CM

•srai{o g

O rH rH O t- rH CD

• oii^ueo

CO CO 0 -* CO CD O
OO £>- 1C !>• t^- !>• t^-

iC 00 t>. CO CO "SO

§Q O O O O O
OO O O O O

s

oip -* rH CO 1C O OS

* * H

<&

CO CO CO CO iC CO !>•

•pe^tiu

OS rH rH CM
GO l^ ^ CO t^ t^- CO

•

-anna; ^^a^j

§Os Os OJ Os OS O»
O O O O O O

i

oOS O rH OS >O CO O

1

COOS 00 00000000

5?

^* "-JI *^l vjl •.]' "^ rf

MOSIAIQ

CO 00 rH CO O rH CO'

•paonpa?i

CM 00 <O iC C"» "O C^

•rang

co >o co ic ic ic

o

CO O CO 1— 1 rH OS CM

§

So o ic r~ o o
O CM O OO -^ OS

n»(8?!+m.)

O rH OS rH O OS OS
rH rH rH rH

t>

-^ rH CO "^ OS -HH rH

T* OS 00 CO 00 00 O
•* CO OS CM CO 00 c~

'Oft ' 886-1

CO ^ O ^ rH rH O
CM CM CM CM CM CM !M

^ T| j,— ».,_ j^-j 1* |^ .»,

CO CO CO CO iC CO t^

OS CO t— 1 O O CO !>•

|

1C rH -^ ^ OS rH CO

•^(^f+sefi-T)

CD l>~ ^^ 00 iO "^ CM
CM CM CM CM CM N <M

CO iO t^ 1^ CD OS O

rH rH -^ 1 — 1 CO ^ CD

OS !>• CM O rH to CO

1C O rH CM IO "^ O

O O CO 00 OS -* -*

•rang

•M CM CM CM CM CM CM

Cb

°ib cb do ib i'c do o

CO CO CO CO >C CO CO

•afl • *l 1

CO CO ~H CO CM CM rH

^

CO CO O i — I CM O iO

QS
1

rH t^ . — 1 O 1C 1>- CO

rH l— 1 -* rH CO •* CO

•W«(^T + S68-I)

i^ CO Ol CO CO CO r^
CM CM CM CM CM CM CM

•JosiAirr

I> r~ cr. t- 00 OS O

CO CO CO CO CO CO •*

•saaagapra^Sui

O O CO O O O O
CM CM rH CM CM CM CM

pai'BjanaS TVSTT
"

CM CM CN CM CM 'N CM

•srauo n

CM I>- I> iO CO CO CO

CO »C OO CO ^ OS -^

rH rH rH CM

CO CO rH O 1C CD CO
-* O rH CM CO OS t-

5a

i>» co *b do cb 10 ^f

•* CO O r^ CM CO O

rH rH rH CM

t- •* CM o o os oo

CO OS OS rH CM t^ 1C

oo do >b os i>> cb ib

^

0O5 CM 1C CO •* C~ O

!>. O O I> CO OO OS
rH CN CM rH i— 1 i— 1 rH

CO 1C iC OS !>• iC OO

•sranraao s;

rH rH
OO 00

»

oo 6s cb 6^ i> i^ <b

o

1 .

CO

i

w "*

>> bb

"^ •§

00 CQ

cs S

oo o co §

CO Q °2 «

416 DR. 0. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

•uqu-dnKX

CO <M CO CM CO

°CO CO CO CO CO

CM rH CO

co co co

CM rH CO

CO CO CO

CO I-H CO

co co co

CM rH CM
CO CO CO

CO rH COCO

co m co co

CO CO CO

cSScS

•.fyiArpnouoQ

•# X •* CM rH
X CO CO ~>f •*
OS X OS X OS

OO iC IN

CO COCO

rH 1— 1 O

OO CO X

co co co

OS CM OS

r- in IN

l-H rH l-H

<O O 00 rH

CM I-H os OQ

O O O 3

CN i— i in

IN O X

IN tN IN

•adojs

! aancfiuaduiajj

CO CM r-H CM OS

°IN do IN, do t-

CM !>• X

CO CO CM
OO CM OS

os os

X rH 00

rH (M l-H
O rH O

iC CO 1C
OS iC OS

CO ^f CO

p CM ic in

cb cb cb ic

OO CO IN

rH CO iH

in CM m

00 -* OO

dc os ob

' * ' ' PH

UUISUU.I) ^ajj

iC OJ CM CO CM

OS X OS CO OS
CO CO X X 00

p p p p p

iC iC O:
CO •* CO

00 X 00

o p p

CO IN CO
CO ^ 1C

OS l-H tN
iC iC iC

X CO OO

rjj 'V' ~f

pop

rH O O CO

1C CO CM r-

OOOO X X

poop

rH *^ 1C

ic co in

X OO OO

pop

X rH O

IN tN x

9?p

•HmnBjoo.TO

GO "M IN CM t-
CM Cl CM <M CM

<M O CM
CM rH CM

OS rH •*
rH rH i-H

1 — 1 O rH

IN l-H IN

I-H m -* CM

^^^^

l-H l-H CO

1— 1 O rH

^Jf rH in

CM CM -SI

•*•*-*

• n^ ( **j -

+ ?f f ) +

X ^ ^ CD CO

iC CM O

CO CO !>•

O OS CM

CM O -*

IN 00 CO X

IN. ,-H OS

ic m O

"'il pp7, T -4- ^*,'

(s?f + yee-l)

— -^ rH CO O

CO iC CD iC CD

OS rH O
>C 1C -O

00 rH O

1C 1C CO

8S§

O CO rH

••O 1C CO

00 CM Oi IN
1C 1C ^P iC

CO OS X
iO "^? iO

X i-H O

ic ic co

O OS CM CD OS

rH O CO

•* CM r-H

CO CM O

IN. CO CM

rH •<}< OS OO

r-l> CO

CO IN iC

A^J-H

a(8?l + -6S'T)

CD CM CD CM iC
CM CM C 1 <M CM

1C rH 1C

CM CM CM

•H^ r-H 1C
CM CM OQ

••Jl l-H 1C
CM CM <M

iC CM CO
CM CM CM

-* rH OS CO
CM CM i— i CM

-*< os •*

CM i-H CM

CM O) CM

^OUOS^H

os os os o os

O O O i— 1 O
"M CM CM ^1 CM

iC CO CD
0 0 O
CM CM CM

CM CM CM

CM CM CM

I>- IN, tN

o o o

CM CM CM

IN 1C CO OO
O O O OS
CM CM CM i-H

IN. 00 IN.

o o o

CM CM CM

f- IN IN,

OO O
CM <M CM

iC *C OS 00 CO

I-H IN O "-O rH

OS OO X
CO OS 00

O CO i-H

os -Ji os

CO 00 -*
O 1C i-H

iC CM •<?

O CO r-H

X CO i-H OO
CO iC 1C '•fl

iC CO -H

1C rH IN

CM m os

CM >C CM

<a

iO iC iC CO »C

-rfl 1C -^
rH -^fl rH

•*J! CO "^fl
i— 1 -^ i-H

1C 1C 1C
rH -^1 i — 1

1C CO 1C
l-H -* I-H

iC iC CO 1C
rH CO -* r-H

1C t^ ic
i— 1 Tfl rH

ic co in

rH -^t l-H

SCO O CM OS
X d iC OS

0 0 rH
OS CO OS

1C CO CD
IN. 1C X

CM O O
CO CM X

co I-H ov

OO CO OS

r-H X CO -<Cf<
000 X rH

X ^ O

CO OS •*
OS CM O

•sau[o g

i-H OS i-H O O

1 1^1

0 O O

O 0 O

O O 0

0 0 O

O CO O i-H

O Or-
1 ^ 1

?^7

CM CM i—i iC OS

IN fM X OS iC

i-H rH CO
IN fM O

CD 1C rH
•*f IN OS

O 1C rH
IN CO CO

CM 1C rH
>C iC OS

CS 1C rH OS
00 'M rH IN

OS CD O
X Tfl CM

•^1 rH CO
CM •* CO

*

CM rH CM rH (M

CO CO TO CD 77

rH O •M

co co co

rH O rH

CO CD CO

I-H OS i-H
CO 1C CO

r-H O 1-H
CO CO CO

1— 1 O <M r-H

CO 1C CO CO

rH O CM

'.^ -0 CC

CM i-H CM

CO CO CO

T

iC CO IN. CO O

Qr— ' rn rH CO CM

l-H X r-H
CO X •*

l-H (M rH

'Jl 00 'Jl

m IN, -31

OS i-H OS

£38

00 i-H 1C CM

O rH rH O

OS Tfl CO
OS rH X

O rH O

cr?

CM 75 CM CM

rH CM i-H

l-H l-H rH

^5 02

•poonpa'jj

CO ^D t"^ (O IN

•^l iC TJ( ic ^P

O TO O
1C IN 1C

iC O O
ic IN, iC

iC

^H OS -*
IN X IN

X O 00
•* CO -*

IN

OS 1C OS IN
IN. oo X t-

CO CO CO
IN. X IN.

l-H I> l-H

t> 0

.

IN 1C X 00 OS
QX 1C OS CM IN.

CM OS •*

0 O ^

IN IN. CM

x m co

in CM ic

CO CO IN

X l-H IN
INO i-H

)>. CO CO r-H
OS CO CM X

oooco

X CO O

l-H 1C O
CO 1C OS

*

CO •» CO CO CO

co cb co co co

CO d CO
CO CD CO

CM CM CO

CO CO CO

co — i co

CO CD CO

CM CM CO

CO CO CO

CO CM •* CO

CO 0 CO CO

CO CM •<?

CO CO CO

CO CM CO
CO CO CO

O, CO O OS O
0-* rH CD X •*

1C rH IN,

1C CD OS

CO CD iC

-* l-H X

Os cc co

rH CX CM

CM •* 00

CO iC CO

CO O OS i-H
1C OS IN CO

rH ^ft rH
•* I-H CO

coo^

CC,

"^ CO ^fi CO "*?

CO CO CO CD CO

CO CM CO
CO CO CO

CO CO CO

•rfl CM •*

CO CO CO

CO CM CO
CO CO CO

-* (M •* -*

co m co co

•rjl CO -*

CO CO CO

CO CO •*
CO CO CO

•saiqo £}

•* X X IN CM

00 i— IN IN OS

X 00 IN

X CO OS

OS CO OS

X CO rH

(M CO I-H

O OO tN

IN rH CO

r-H CM x m

r-H O X CO

CO IN <M
O CM X

IN CO 00

1C IN. ic C-* 1C

rH I-H

1C IN, 1C

rH

1C IN 1C

1 — 1

CO tN CO
I-H

m r» in

i-H

co co IN. in

r-H rH

CO IN, iC

rH

ic IN. m

rH

•siuiio o

IN rH I— 1 O 1C
O Tjl O O rH

X X IN
O 1C l-H

CO O IN.

1— 1 OS -H

OO co O
"^1 1C CO

O OS CO
OS CO CO

CM CO OS -Ji

CO CM O X

CO X CM
CM •* O

OS 1C O

CO I> CO X CD
i — 1 rH

CO IN CO

l-H

CO IN CO
rH

co IN co

rH

1C tN 1C

rH

CO CO CO iC
i-H i-H

CO tN CO

i-H

m IN co

l-H

^

O i— 1 IN. o O
CO iC 1C C*- O

O CO CM
CO "C 1C

t~ CO O

O CM ^

CO CM CO

X CO -^

iC X O

CM iC CO X

IN Tp IN IN

O CO O
CO •* CO

8S8

do cjo do IN x

rH'J-H'rH'

IN IN tN

CO IN IN.

IN. C~ OO

IN X 00 X

IN. X CO

CO 00 CO

'STHT1U9O 82

X
iC

l-H

CM

l-H

CO
CO

rH

CO

os

rH .

m

OS

l-H
CO

I

CO l-H
CO CO

i-H
(M

rH

-

Substance.

Naphthaline

a. Naphthol . .

ft Naphthol . .

3
,3
OH

"3

QQ

rrj £

11
O2

Ebonite . . .

1

QQ

1

£

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE.

•0028 •
' 26

417

•oozo

16

k-

16
14
12

•0010
S
6
4
2.

•^Sajphitr&fretich

50°
Temp.

fbre

70°

f,ha)ine

so"

Remarks on preceding Tables.

In the first table the whole of the observations and redactions are shown, but in
the second the observations and only the important steps in the reduction.

The thickness given in the tables is that of the material under test plus the two
layers of glycerine. The correction for the glycerine is too small to be taken into
account, except in the case of glass and of the mixture of sulphur and French chalk.
The corrected values for glass are : at 35°, '00248, '00243, '00244 ; at 55°. '00258 ; at
68°, '00261, '00263 ; at 79°, '00272, and for the cement at 32°, '00182 ; at 61°, '00155-

The glass was ordinary window glass, known as 22 oz. In the second series of
experiments with it, the glycerine contact layers had been replaced by shellac (see
p. 420).

The naphthaline disc was cast between glass plates and ground down to the required
thickness.

The two naphthol discs were obtained by sawing slices from two large blocks of the
material and grinding the surfaces.

The sulphur disc was made by pouring melted sulphur on to a glass plate and
grinding the upper surface down till it was parallel to the lower, and the disc of the

VOL. cxcr. — A. 3 H

418 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OP SOLIDS

requisite thickness. The surface which had been in contact with the glass had a
crystalline appearance, the other not.

The disc of sulphur and French chalk was made in the same way. It contained
about equal quantities of the two constituents.

The shellac disc was cast between glass plates, and contained a few small air holes.

The fibre was the ordinary " white fibre " of electrical instrument makers.

As the result of these experiments, it may be stated that the tendency at about
40° C. of the thermal conductivities of solid substances not very good conductors of
heat, is to diminish with increase of temperature, at a percentage rate which appears
to vary with the nature of the substance. Amongst the materials tested, glass is the
only exception to this rule.

Substance.

^35° c.

Mean percentage change
per degree
between 35° and 60° C.

Glass (22-oz.' window) ....

•00245
•00095

+ -0025
- -0052

•00076

- -0105

H

•00080

- -0085

•00067

- -0036

Sulphur and French chalk .

•00180
•00042

- -0052
- -0019

Shellac

•00058

- -0055

White Fibre ....

•00079

- -0020

The above values of the conductivities agree with those found by a different method
(LEES, ' Phil. Trans.,' A, 1892, p. 506) for temperatures between 25° and 35°:— glass,
•00243; sulphur, -00045 ; ebonite, "00040 ; shellac, '00060, except in the case of
sulphur, where the large difference is probably due to the disc tested in the present
experiment not being wholly crystalline.

PART II. CONDUCTIVITIES AND TEMPERATURE COEFFICIENTS OF LIQUIDS.

Preliminary Experiments, .

The apparatus used originally in the experiments on the conductivities of liquids,
consisted of two circular copper discs, U and M, cemented by rubber cement to the
opposite sides of a circular sheet of rubber, R, fig. 10. To the free surface of the
upper disc, a flat coil, C, of insulated platinoid wire was cemented, and served to
supply the heat required in the experiment.

These discs were supported by means of a three-legged support, T, screwed to the
upper disc, above the third copper disc, L, which was provided with a raised edge.
The legs of the support were long enough to raise the lower surface of the disc, M,
about '13 centim. above the upper surface of the disc, L. The liquid, the thermal

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 419

conductivity of which was to be determined, was poured on to the lower disc, where
it was retained by the raised edge. The three discs were then placed on the
horizontal top of a closed vessel through which a stream of cold water was kept
flowing, and they could, if necessary, have a cover placed over them. On sending a
current through the coil, (7, part of the heat generated flowed through the discs
to the top of the cold water vessel, the rest was lost by conduction and convection
in the air above the coil. After the distribution of temperature has become steady,
the heat flowing through the rubber from U to M proceeds either through the layer
of liquid to L, or is lost from the surfaces of M and of the liquid. As the tempera-
ture of the water flowing through the vessel underneath the discs was always a few
degrees less than that of the air of the room, it was possible, by regulating the rate
of flow of the water, to arrange that the temperature of the surface from which this
heat was lost was nearly identical with that of the air, and the loss was thus
reduced to so small an amount that it could be neglected. By means of tbermo-
junctions of copper and platinoid wires, soldered into holes about 3 millims. deep in
the edges the discs, U and M, and in holes 2 centims, deep in the edge of the disc, L,
the temperature of the upper disc, and the differences of temperature between the
upper and middle, and between the upper and lower discs, were found by balancing
the thermo-electric E.M.F. in each case against the requisite fractional part of the
E.M.F. of a Leclanche cell, which had been compared with a standard Clark cell. It
was found that for small temperature differences the E.M.F.s so determined, could be
taken as proportional to the differences of temperature.

Fig. 12.

v c

U

From the difference of temperature on the two sides of the rubber sheet, and the
thickness area and conductivity of the rubber, the amount of heat entering the
liquid could be calculated, and if the thickness of the liquid, the area of flow, and the
difference of temperature are known, the thermal conductivity of the liquid can be
found.

The curvature of the lines of flow in the liquid near the edge of the disc, M, will
render the area of flow in the liquid greater than in the rubber, and a small correction
would have to be applied if the thermal conductivity were made to depend on that
of the rubber. It is better, however, to base the determination on the conductivity
of water, which can be substituted for the liquid, and tested under the same con-
ditions. In this case the areas of flow in the two liquids may be assumed to be equal,
and the calculation simplified.

8 H 3

420

DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

With the apparatus in this form, determinations of the thermal conductivities
of water, ethyl alcohol, acetic acid, and glycerine were made.

The values found in different experiments for the relative conductivities of water
and glycerine, and mixtures of them, were concordant, and are given in the following
table along with H. F. WEBER'S* values for water and glycerine between 9° and 15°,
but experiments on ethyl alcohol and acetic acid showed discrepancies which were,
after some time, traced to the evaporation of the liquid at points not covered by the
middle disc. As most of the liquids to be tested were more volatile than water and
glycerine, it was decided to get rid of this evaporation by entirely enclosing the
liquid, and the apparatus used throughout the greater part of the investigation was
then constructed.

Liquid.

U - M.

M-L.

&.

Temp.

H. F. WEBEK'S
values.

ohms.

ohms.

Water

•97

•28

•00140J

20 C

•00136

+25 per cent. "1 , .
20-« ? glycerine in water

•93

•32

•00119

)»

50 1

44-2 J

•953

•387

•00101

»

75 1

70-4 /

•91

•46

•00081

»»

Glycerine ...

•89

•52

•00070

•00067

>»

Description of Apparatus.

Between two nickel-plated copper discs, U and M , fig. 1 1 , p. 4 1 9, 4 centims. in diameter
and '3 centim. thick, a disc of glass, C, '28 centim. thick, was cemented by means of
thin layers of shellac. In order to prevent bubbles of air occurring in the layer of
shellac, the surfaces of the copper and glass discs to be cemented together were each
covered with a thin layer of shellac, which was allowed to dry. Both surfaces were
then smeared with glycerine, and slid carefully together, air being prevented from
getting between the surfaces by the presence of abundance of glycerine. Pressure
and heat were then applied, the pressure forcing out the glycerine and bringing the
layers of shellac into contact, and the heat joining them. Both contacts were made
in this way, and withstood a large amount of usage without coming apart.

A flat spiral coil of platinoid wire, P, was placed on the top of the upper disc, U,
and was held down by a thin copper disc, C, MO centim. thick, screwed to the disc, U.
The coil was insulated from both discs by means of mica, and the surfaces of the
discs exposed to the air were varnished to give them the same emissivity. This
combination rested on the inner edge of a ring of ebonite, E, of 7 centims. external,

* H. F. WEBEK, ' Berliner Ber.,' 1885, p. 809.

t The upper numbers are percentages by weight ; the lower percentages by volume.

t Taken from the table, p, 425.

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 421

and 371 ceutims. internal diameter, which in turn was supported by a disc of nickel-
plated copper, L, 7 centims. diameter and '3 centim. thick, to which it was cemented
by rubber cement.

The copper discs will be called in order downwards, the " cover," " upper," " middle,"
and " lower " copper discs, respectively.

The discs were placed with the flat surfaces horizontal,"' in a flat cylindrical air
bath, fig. 12, p. 419, consisting of an upper and a lower half, each capable of being
heated by means of an insulated coil of platinoid wire wound round it, and the
lower half provided with a compartment through which a stream of cold water
could be sent to cool the bath rapidly. During a test of thermal conductivity, this
compartment contained air only.

A thermo-j unction was soldered inside the top of the upper part of the enclosure,
and it is assumed that the air to which the top and side surfaces of the discs lose
heat, has the temperature given by this thermo-junction. The thermo-junctions in
the upper discs were those described (p. 419), and were arranged vertically under
each other. In order to bring the junction in the lower disc vertically under those in
the upper, the holes in the sides of the lower disc were made correspondingly deeper,
and for a depth of 1'5 centims. wider than those in the upper discs, and were lined
with thin glass tubes to prevent the wires making contact with the disc at any
other points than the junctions.

The liquid to be tested was placed in the ebonite ring, and had therefore the same

thickness as the ebonite.

*

The heat flowing through the under surface of the middle copper disc was con-
ducted away partly by the liquid under test, partly by the ebonite. The amount
conducted away by the ebonite could be calculated if the shape of the stream lines
and the thermal conductivity of the ebonite were known. It is, however, much more
accurate to determine this quantity by a separate experiment, substituting air
for the liquid and so arranging the amount of heat supplied, that the temperature
difference between the middle and lower discs is the same as in the experiment on
the liquid. The heat conducted through the air is then small, and can be calculated
from the known thermal conductivity of air.t The remainder of the heat flowing-
through the under surface of the middle disc is conducted through the ebonite to
the lower disc, and is the quantity required.

Theory of Ebonite Ramg Apparatus.

If r = radius of copper and glass discs,
ta = thickness of glass disc,

* The disc of liquid being thin, convection currents only come in to a slight extent when the discs
are not qnite horizontal. In the case of a disc of water, an inclination of 15° to the horizontal only
increased the apparent conductivity 1 per cent.

t WINKELMANN'S value has been nsed, ' Wied. Ann.,' vol. 48, p. 186, 1895.

422 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OP SOLIDS

tm = thickness of copper disc below glass, plus correction for thermo-wires,

TL = radius of disc of liquid,

t, = thickness of disc of liquid,

vv = excess of temperature of upper disc over that of the air in the enclosure.

•VM = excess of temperature of middle disc „ „ ,,

Vj, = excess of temperature of lower disc ., „ ,,

then the heat flowing through the glass disc per second

= TTT~KQ - ,
the heat entering the middle disc

and the heat leaving the middle disc by its under surface

tG 4

The amount passing through the disc of liquid

the amount entering the ebonite ring surrounding the liquid may be put *

where A is an unknown constant depending on the breadth of the ring under the
middle copper disc, and on the distribution of the lines of flow in it. Between these
quantities we have the relation t

from which we can determine kt if A, kQ and h are known, by observations of vv, VK,
and VL.

* Strictly, a term BvL ought to be introduced here, since some of the lines of flow from the middle
disc may pass through the ebonite to its upper surface. On account of the smallness of B, this
term has been neglected.

t See note, p. (403).

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 423

The value of ka and h are known from the observations made in Part I.

The value of A is determined most accurately, by carrying out an experiment with
air substituted for the liquid, so that the term involving kt on the left-hand side of
the equation has a known small value. From the observations of temperature the
value of the right-hand side of the equation is calculated, and hence the value of A.

Constants of Apparatus.

r = 2 centims., therefore irr* = 12 '5 sq. centims.
rt — i'855 centims., therefore -nr\ = 10'8 sq. centims.
tt= -127

tQ = -281 „ therefore -- = '070 centims.

tu= '320 + '075 = '395 centims.
efore ZTrrh -|- =

and 2-vrh tx = '00148

h = '00030, therefore ZTrrh -- = '00026

Sum = '00174

Method of Experimenting with Ebonite Ring Apparatus.

In carrying out a determination of the thermal conductivity of a liquid, the surfaces
of the discs with which the liquid was to come into contact were cleaned, and the
lower nickel-plated surface of the middle disc wet with the liquid. Liquid was then
poured into the space inside the ebonite ring, till the upper surface of the liquid
was higher than the upper surface of the ring. The middle disc was then placed on
the ebonite ring with one edge touching the liquid, which was drawn into the space
between disc and ring by capillarity. The disc was then slid slowly over the ring
till the liquid was entirely enclosed, and any excess which flowed over the ring was
removed by filter paper.

To prevent the upper discs moving during an experiment, they were secured to the
lower, either by a silk thread passing from one side of the lower disc over the upper
discs to the other side of the lower disc, or by a wire frame which could be attached
to the lower disc, and was provided with a screw which pressed on a small disc of
cork on the copper disc covering the heating coil.

The combination was then placed in the air bath and the process described (p. 413)
carried out till the steady temperature state was attained, when the observations of
power supplied, temperatures, and temperature differences were made.

Experiments were made on four liquids, with the results given in the following
tables and curves.

424

DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

•3Jn^lBJ9draajl

«3SSS8

CO

us co

CNT?

»O i-H
CM IO

tN. IO

•* CM

•.-y = 'rat}
-two g-reribs jg'd

p9^lUI8U'&IJ (JTJ9JJ

t^ CM CD ts.

CO CM CO rH

I-H rH , — I rH
8OOO
O O O

i

(M CO

CO rH
CD CO

coos

CM CO

•* CO
00
CO CO

iO IO

-* OO

o o

'pinbij qoUOjq}

CO CM IN. CO
•<^ CO "ft CM

r-l rH 1— 1 rJ

OO O 0

CD

8

IN. CM

CO CO

IN. CO

O 00
CO OS

•* co

0 0

OO •*
OS CM

-P iO

o o

pinbi[ •qSuojq'}
odojs ;iun jgd

IN. i-H CO IO

IO ^ iO CO
r— I i-H i-H rH
O 0 O O

CM

rH
O

iO O
CM iO

00 IN.

pp

Op CO

•*? oo
o o

O5 CM

CD rH

PP

•pmbij ui
odojs ojn!).Ba9dui9T

r-1 rH O CO
6 rH 6 rH
i — 1 rH rH i — 1

O

OS CM

CO CM

Oi (M

CO OS
i-H i— i

co co

CM CM

co -^

<M G<I

•o^moq9 puu
pinbij qgno.iq'j

OS CO CO CO

CM

00

CO ^?

r- IN-

••# co

CO '00

cc ^

.... . . . .

•"a 4100-

O O 0 O

o o o o

CM

i— 1
O

CO IN-

O OO
i-H O

o o

t*» <o

O rH
00

•-1* 9SOOO-

CM i-H CM O

6
o

CM CM

O 0

O 0

o'S

O 0

CM CM

O O

p9}}irusuu.u ^TJOJJ

CO CD CD O

CD

OS

CD CO
CD iO

Oi £-*

iO O

^ CO

CM

o* ^o

CO CD

•ssiqS m
odo[S 3jn^i3J3duiojJ

OS OO OS -H
CD rH 10 O

OS l^

10 OO

CO O
CD-*

iO iO >0 iO

CO

10^

iO -^

rp US

cq

CTS rH CO OS
CCM O I>- 0

rH

CO O
O CM

iO OS
CD i-H

CM CO
-* CM

6 •* 6 cb

CM -^ CM -*

O

CM

(M iO
CM ^

ooo

CM •*

•* CM

03

^ CO CM OO CO

iO
O

IN- lO

OS t-

IO rH

OS rH

,3 6 rH r^H

i— 1

rH 0

1 — 1

O i-H
i-H

OS rH

<r

•^ CO OS O
IO Oi CO iO

OS

00

O CO

CO O

IO IN.

CO CO

CD O

-*l CO 10 00
CM •* O1 "*

CO
CM

co os

CD CM
CM iO

CO IN-
^P CM

••>«-»a='i0-"0

CO i— 1 !>• CO
oCM "^Jl O') ^^

CO
CO

CO "^

O CO
-* 10

•^ CO

o o

I — i rH i — I i-H

CD

CM CM

CO CO

CO CO

-d 1*

O —
0 f^

„• 0 O iO 0

p co o i-^ t-
b ^? 10 ^ 10

CD

0 0
o; r-

00 OS

CO -*1

rH CO
CM rH

06666

CM

OO

I-H rH

rH rH

"2 S
1

« o o o co

a O CO Gi CO

CO

0 0

OS ^
iO iO

OO
1^ CO

10 10

O -0
CM IN.

IO iO

<£

*3< CC'^i —t
0rH CO OS C5

00

rH^

i-H OS

IN. CD

OS 10

CO CO 0 OS

CO
CM

CO O
CM 0

CO CO
CM iO

OS CO
^ CM

t>

,/ OO CM rH 00

£ ** 0 co co

01
CM

•^ OJ

O CO

OS OS

I>- <~°l

CO •<?
rH OO

,3 CO CM CO CM
O i-H pH

^

-* CM
rH

CO •*?
rH

CM CO

r— 1

•-5

-* i-H O O
OIN. OJ CM 00

CM

I> O

0) iO

10 CO

OS CO

O -<
CO CM

CO CO IN- CO

IN- !>.

IN. r»

CO CO
rH rH

1

53
1

]

Glycerine . .

'o

o
o

•3

"o

r«
O

13
^

1

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE.

125

•0014
•OOIZ
•0010

k.

•0008
•0006
•0004
•0002

"-.

^

*NN

—

***

*"->

>

rt'dl

V.

—

___

'

—

• -

•1/yi

enr

-

— —

H

t-,/

<f(h
-'£!,

•/-A

y-

•xh

m

it.

'«/

£0° 30° a 40° 50° 60° 70°

o.

Liquid.

K.&.

Mean percentage
change per degree between
25° and 45°.

7,;a_iy, H.F.W.

Water

•00136

-•0055

•001:30

Glycerine
Methyl alcohol ....
Ethyl alcohol ....

•00068
•00048
•00043

-•0044
-•0031
-•0058

•00067
•000405
•000423

On account of the rapid way in which the methyl alcohol evaporated, it was
necessary to work quickly, and the numbers given for it are not quite as reliable as
those given for the other liquids.

The last column contains the values obtained by H. F. WEBER,* for the mean
thermal conductivities between 9° and 15°.

As a result of the experiments, we may state that the thermal conductivities of
liquids decrease with increase of temperature in the neighbourhood of 30° C., at a
percentage rate which appears to be roughly the same for a number of liquids.

PART III. CHANGE OF CONDUCTIVITY ON MELTING.

Combining the last result with the one obtained for solids, Part I., and with the fact
that solids in general conduct heat better than liquids, and liquids better than gases,
we are led to conjecture that a given substance will, as its temperature is increased,
decrease in conductivity, and that the decrease will continue during any change of
state which the substance may undergo, owing to the increase in temperature. The

VOL. CXCI. — A.

* H. F. WEBER, ' Berliner Ber.,' 1885, p. 809.
3 I

426 DR. 0. H, LEES OST THE THERMAL CONDUCTIVITIES OF SOLIDS

question then arises : Is there any sudden change in the conductivity as the substance
passes from one physical state to another, or is this change continuous ?

In order to test what change occurs in the thermal conductivity of a substance as
it passes from the solid to the liquid state, the apparatus used for liquids was
modified slightly, so as to allow space between the lower and middle discs for change
of volume of the substance tested on melting.

The ebonite ring was replaced by one of "white fibre," pierced by a hole
4M 6 centims. in diameter, i.e., greater in diameter than the middle copper disc (Kg. 13),

Fig. 13.

with three strips, S, of fibre projecting inwards, in order to provide support for the
middle disc, and keep the layer of substance tested of constant thickness. By means
of a wire frame attached to the lower disc, and a screw, the upper discs were pressed
down on to the strips of fibre, and the lifting of the upper discs, owing to expansion
of the substance tested on melting, prevented.

Some of the substance to be heated was placed within the ring, and the lower disc
heated till the substance melted. The upper discs were then placed in position and
the apparatus allowed to cool. The experiment was then conducted as in a test of
a liquid.

The area of the projecting lugs of the fibre disc, through which heat could flow,
was estimated as approximately '4 centim., hence the heat transmitted per second per
unit slope of temperature through the fibre, and not through the substance under test,
would be = "0003 gram degrees approximately.* The nett area of flow through the
substance has been taken — 13'3 sq. centims., but as this number is, to some extent,
uncertain, owing to the difficulty of obtaining discs which, in the solid state, com-
pletely filled the space within the fibre ring, the results given in the following table
for different substances are not comparable with each other to within 3 or 4 per cent.,
although those for the same substance, at different temperatures, are.

* The conductivity of the fibre = '0008 (pp. 417, 418).

AND LIQUIDS AND THEIR, VARIATION WITH TEMPERATURE.

427

tmuedrau

•3 §3838

•^ O rH

CN CO •*

CM

CM

US rH CM <M

C-l •* US CO

CD CO rH -* OO
CM CO US CO <M

'AllAWOnpUO'^)

^H CO ^^ ^^ ^f
CO CM CM rH CM

O •* CO

US rH O

05
US
O

O5 US 00 00

CO CO CM CM

OO CO O5 t^ CO
CO CO CM CM •*

88888

SO 0
O O

0

O O O O
O 0 OO

§iS88

"OuOIS

"^ CO ^D O5 *O
l^ CO CO us CD

O CM rH

O US •*

oo

o

liil

rH -rfl O5 CO t~
IO 'rjl CO CO US

/aonB^squs q u uq-} ^Bajj

O O 0 O O

O O O

0

o o o o

o o o o o

•edoIS/^q wu

!>• CO CO us 00
i>. CD CO U7 CO

CO US ^*
O us ^fl

rH

i — 1

^ 00 O 0

iO Tf^ ^* CO

O CO *r?l TP O
iO ^* -^ CO CD

0 OO 0 0

O OO

O

o o o o

o o o o o

*aouij}sc| ns

CM <M CM CO

rH IO CO CM CO

CM CO O

C5

^ •* CM OS

US O CD US CM

ui odo[s aan^jadoiajj

G5 C5 O5 O O5

— H

CO O rH
rH i — 1

CO

t~ O •* CO
CM CO CO CO

CO C5 rH IM O

CM CM CO CO CO

•ojqg pui3 aoLHji)sqns

rH CO t^ 00 CO
CO us iO us CD

J^ •* CO

CO CD iO

CO

o

O CO OO us

us -^ co co

CO CO -? CO i— I
-fl CO CO CM GO

H 1 " L " . ±i

al[Qf)-

co r^ co -!fi oo

O O O O O

o o o o o

00 CO 00

CO

o
o

O 00 00 00

rH O O O

o o o o

O t- O5 CO rH

rH rH O O O
O O O O O

>n<* 95000-

CM CM CM rH CM

O O O O O
0 O O O O

CM CM CM

000
000

CM

o
o

CM CM CM CM

o o o o

0 O O 0

CM CO CM CM rH

O 0 O O O
O O O O O

•SS13[3

qSnruqq po^iuismjjc) ^t!3H

rH t^. US CO CO
!>. CO CO CO t^

I> -f CO
!>. i> CO

CO
iO

o

CM 00 00 iO

CD us ^ "rJH

US CO us CO CO
iO US -fl CO OO

•SS,IS

rH US I>- O O

OO 10 CO CM OO

CO O "r?

CS 00 -*

1 — 1
CO

CO CM O CO '
Tjl r-l I> -)H

CO CO C5 CO ' — i

CM — 1 .0 CM rH

iL

US US us US us

us US us

r— 1

US US ^fl ^

10 US Tfl Tfl CO

t^ -ft iO CM CO
CO •* 1^ C5 O

o

O5 O 10

CO

CO CO rH CM
!>. IN. IO i— 1

CO US CM CD CM

rH CO C-J C5 rH

Q?

O5 CO >O CO CM

rH CM CO -* CM

O5 CO 0
rH CM CO

rH

O t^ t^ C75

CM CO "^ iO

rH Gi t^ rH Ci
CM CM -r? 0 CM

K O CM t~ US CO

a CO IO rH "^ O5

CM -31 CO

00

23

CO i> US CO

O CO rH t^

US ^ — ^rH O5 lO

rH CO CO CO 05

,C rH Tp 1^ rH rH

O rH

rH CO !>•

o

rH rH

r-H CD O CD CO
^— 'rH i — 1

O5 CO O CM rH
CM !>• •* -H I>

-^ CO CM
us CO CM

rH

US 00 •* C-

10 CO CM CO

p — 1 O !>• IO O
CM rH CM in CO

<ss

•* CM 05 OJ CO
CM CO CO -* C-J

-? O rH

CM CO -r)<

CM

CD CM CM CO
CM Tti iO CD

t- O5 CM CO O5
CM CO us CO CM

•<JI C5 i— 1 1^- I—I
"rH rH CM CM CM

co CM r^

O CO CO

o>

CO us CO CO
C3 CM CO CO

CO
-* O CO CO CO
OO rH CO ^f CM

rH rH r-l rH rH

i — 1 rH rH

•*

CM CO CO CO

CM CO CO CO CO

US CO US UI
C 1 IO t^ O O

US US

co o co

CO us iO

iO
CO

I>> US

O !>• CO C5

CD o us 10 CM

O CM CO -rft CM

S o

0 0 OO 0

o o o

rH

rH rH rH i— '

° 1 'TO-fl

!>. CO 1C US

rH Oi 00 00 !-•

CO us us 10 CD

CO CM O
CO CD CO

as
1 — 1

US CO O
OO CO CO CM

10 us 10 us

iO CO

10 co CM o in

us us us us CO

CM O5 rH CO -*<

O5 CM C35 IO CO

-* O5 US
CM CM t^

CM
CM

O5 CM CD CO
O OO US C5

CO iO CO ^ l^
CO us us Jr~— CO

*

"us •* o o do

CM CO •* us CM

CO CM OJ
CM CO -*

US
CM

CO CO CO "*
CM -* UJ CD

co o co r^ rH

CM -* us CO CO

a CM CO CO rH CO
g I>»»I>. rH O5 CO

O O5 rH
CO £"* C*~

s

rH US O O

CO 1^ CO CM

"*? ^ CM O O

O CO CO CO CO

13

£1 CO CD O5 CM Tfl
0 rH

CO US O5

CM

CO C5 CO 00
rH rH

•rfl 00 CO O5 -^fl
rH i — 1

CM CO 00 f~ t>.

CO CM CO CO CO

CD rH US
-* CO rH

O
CM

CM CO •* O

OO rH CM US

CM US US CD O

CO US CO I>- US

«?

°ib cb cb cb cb

CD CO I>

OO

rH

r^ co co oo

r^ co co co co

,un,uao '8SaUWu

10
CM

rH

>o

CM

rH

;

o

o

rH

i

1

02

NasHP04+12HsO
m.p. = 35° C.

0,05

co ||

+ A

Q

tj

Naphthylamine .
m.p. = 50° C.

Para-Tolnidine
m.p. = 45° C.

!

i 2

428

DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

•0016
•00/4
•0012
•OOIO

k

•O008
•0006
•0004
•0002

—too,

#,0"

30°

Temp.

6O°

The experiments show no apparent break in the regularity of the change of
thermal conductivity at the melting point in the case of three of the solids, and a
decrease of about 20 per cent, in the case of CaCL + GILO.*

This salt was extremely difficult to work with, on account of the rate at which it
absorbed moisture, and the results obtained cannot be relied on to the same extent
as those for the other substances.

It seems, then, reasonable to conclude that, for salts at least, change of state on
melting is not invariably accompanied by an abrupt change of thermal conductivity.

PART IV. THERMAL CONDUCTIVITIES or MIXTURES.

After coming to the above conclusions with respect to pure, or approximately pure,
substances, one is led naturally to experiment on mixtures of substances, and, since
mixtures of liquids are most easily made, they were the first on which observations
were carried out.

It is obviously advantageous to investigate mixtures of pairs of liquids, the con-
ductivities of which differ as widely as possible. The number of miscible liquids which
satisfy this condition is small, owing to the great number of organic liquids which
have conductivities nearly alike. Water and glycerine are much better conductors
than most liquids, and have, therefore, been used in several mixtures.

The following mixtures of water, ethyl alcohol, methyl alcohol, acetic acid, glycerine,
and sugar were made, and their thermal conductivities investigated : —

* BARUS (' Sill. Journal ' (3), vol. 44, p. 1, 1892), has found a decrease of about 15 per cent, in the case
of thymol, which molts at about 12'5° C.

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 429

f 25, 50, 75 per cent., by weight.

Glycerine 111 water •{

I 20-8, 44'2, 70-4 per cent, by volume.

f25, 50, 75 per cent., by weight.
Ethyl alcohol in water- . . . . J

|_30, 08, 81 per cent., by volume.

T23-9, 50, 74'7 per cent , by weight.
Methyl alcohol in water . . . .4 t

I 28'8, 58, 807 per cent,, by volume.

T25, 50, 75 per cent , bv weight.

Acetic acid in water <

L24, 48 '7, 74 per cent., by volume.

25, 49-2, 74-8 per cent., by weight.

C41ycerine in ethyl alcohol . . . ,

I 17'4, 387, 65'3 per cent., by volume.

Methyl alcohol in ethyl alcohol . 25, 48'5, 74 per cent., by weight and by volume.

25, 50, 67 per cent., by weight.

•U7-2,38-8, 62-6 per cent., by volume.

Some of the experiments were carried out with the ebonite ring apparatus, fig. 1 1 ,
p. 419, some with the earlier form of apparatus, fig. 10.

The following table gives the experiments made with the former apparatus :—

Constants of Apparatus.

T = 2 centime. irr2 = 12'5 sq. centims.

rt = 1-855 „ 7T/7 = 10-8

tt =0-127 „

tQ = 0-281 ,. (i = -070 centims.

£M = -320 + "075 = "395 centims.

h = -00030, 2irrh -^ — '00026

4

2TrrhtN = '00148

Sum — -00174

* HENNEBEEG, 'Wied. Ann.,' vol. 3G, p. 146, 1889, obtained results for mixtures of water and ethyl
alcohol which ngree closely with those that follow.

430

DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

•a RiadrnaT

orH

0

l-H

CM

0

_

rH

rH

i-H OS

'

1-1

.il!W™0

O

p

rH
O

p

Oi

O

o
p

1

00
I-H
rH
O
O

§8

o
o
p

1

CO I>

•*? o

O rH

•ado[s/pmb;i qSno.!!^ ^Bajj

rH
O

CM

rH
rH
O

CD
p

p

rH

i

t~

rH
p

1

o

ss

pp

•adojs/^aq r^J,

Oi

CD
rH
O

rH
CM

O

iO

o
p

CO
t—

8

o

CO

co

CO
rH
O

-o
o

rH
O

Oi

co 10

iO CM
OP

•ado[s a-m^eaadraajj

lp

rH
rH

00
rH

Oi

o

CM

CO
CM

CM

CO

O
rH

CM

Oi

rH

CM
CM

Oi •*

cb cb

CO rH

•a^nioqa pun pinbij

3

rH

O

rH

CM

Oi

rH

CO
Oi

0

1

l—<

Oi

rH

O iO

Oi O

rH CM

•Mo,

Oi
O
O
O

'l

Oi
rH
O
O

'l

CM

8
i

O

1

p

1

O
CM

I

Oi -*
O CO

PP

OJ

"«ooo.

I

co

P

g
O

p

p

r— 1

O

8

CM

O

8

r*» ^^

§§

oo

>

rl

qSno.it[i paq^irastrB.!} ^'OJJ

Oi

1 — t

1

oo

Ci

rH

OS

i — i

CO
Oi
rH

CM
O
CM

Oi
Oi
rH

rH

Oi

rH

O CM
Oi O
rH CNI

|

ut adojs a.ra^G.i8duiojJ

CD

cb

p

Oi

CO

cb

CO

oo
cb

O

OI

cb

Oi
CO

cb

CM iO

CO O

cb t~

•5

4

j

O

°cb
i — i

rH

CO
1 — I

cb

i-H

Oi
l-H

Oi
CM

rH

CM

cb

rH

CO
CM

rH

r^ op

CM CM

l-H rH

s

>•>

ri

oiCM

39
1 +

CM
j

CM
'l

rH
CM

'l

CM

oo
'l

CO

CO

'l

CM

CO

'l

O
'l

O Tjl

CM CO
'l 'l

05
HP

<£

— H

i— 1

rH

CO

rH

CM

rH

CO
rH
-H

o

CM

rH

r— 1

cb

rH

CM 00

cb 6

rH i-H

(L,

-0 - "o

•3

rH
CM

10
CO

CM

CO

CO

cb

CO
rH

l — 1
Oi

CM

00

o

cb

rf< CM

1
E

T3 -rr T<T
I J\L

0

is

"oo

CO
0

Oi
O

CM

rH

CM

Oi

rH

rH
O5
CO

O

Oi

00
0

o

I-H
rH
l-H

l-H CO

iO iO

rH 6

1

o

PH | -re - n

z oo
'o •

IO
1 — 1
t-

CM
g

i — 1

00

CO

CM

Oi
CO

rH

0

O

CO
CO

O iO

r>» rH

co t»

=-

. £

rH

•*
cb

•*

rH

CO
rH

I-H

CO
CO
rH

Oi

cb

rH

O
l-H

rH 00

iO CM

rH rH

0

g

t;

J«

p

CO

CM

?

CM
CM

co

CM

'l

Oi
0

'l

CO

•* CO
CO CO

*

,£>

ocp

CM

CM

cb

co
cb

co

Oi

cb

OO

cb

CM

Oi
CO

<M

rH

CD
GO
CM

GO C^

o^ oo

coco

1

! .

<D

1

'1° j Ethyl alcohol in H20

o

•*!—*•

-<f—^

!

-T3

JS

O

rO"

'i

'I

1

,
o

6
T3

v

6

v

• O

.W

'-3
• ja

.8

it

, o <a

10 O
CM CO

OO
IO u

s r>. 3

iO •*?
CM (M

O CC

iO •*

•^ CM

AND LIQUIDS AND THEIR VARIATION WITH T 431

The conductivities given in the preceding table depend on that of the disc of glass
between the upper and middle copper discs, which was determined in Part I.

Previous to the apparatus with the glass disc being constructed, experiments had
been carried out with the apparatus in which rubber replaced the glass. As the
conductivity of this rubber was unknown, the results obtained were all expressed
in terms of the conductivity of water. By means of the value for water found in
Part IT., these relative values have been converted into absolute values, and are
given in the table which follows.

The discs were not in this case surrounded by a vessel of known temperature, but
were exposed to the air, the temperature of which was observed. The lower disc was
placed in the top of a vessel through which cold water circulated, and the observa-
tions were taken in the manner previously described.

The theory of the method is identical with that given on p. 422, except that the
constant A is zero.

Constants of Apparatus.

r = 2 centims. TTT~ = 12 '5 sq. centims.

rt = 1-855 ,, irri = 10'8

tt = '132.

tR =• '110, therefore fu/4 — '027 centims.
«M = '320 + '075 = -395

h = -00030, therefore 2irrh £E/4 = '00010

. = '00148

Sum . . = -0016

432

DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

'0 oll/y

I-H rH O O O
£""} L~5 (^5 CO C^

CM

iO

00

IO

SS 3

' rH rH

'8 8

o

CO

o

8

1

CM O CD CO

lO 10 ^H -*

o o o o

8888

1

a.

2
a

I

<o
cc

3

.-

1

i

a
EH

3
O

60

§

_a

"o

r4

CO

1
1

2

rQ

a

o

a.

•pinbij jo oju^u-iodraaj,

- CO CO OS OS OS

os

OS

Oi O Gi

05

OS

OS O COCO

tN.

rH

i-H

00 00

1-H rH

*»-«

I-H OS "^ CO CO
rH rH. O O O

So o o o
o o o o

O

CD

55

O

8

O rH J^
IN. ^4 ^)

O rH rH

O O O

CO
1—

I-H

co

O 00 iO ^

8C O O
O 0 O

o o o o

CM

rH
0
9

CM

O
i-H

8

IN. OS

OO O
O rH

1 8

•ado[8 ^iun
jod pjnbi[ qSno.u^ ^B8jj

CM rH CO CD CO
rH rH O O O

i
P

o

CD

8

CM CD

. lO i-H
. rH I-H
0 0

."*

1

-* CM O- IN.
iO iO •<? •£<

rH
CO

O

o

rH
0

^ GO
O> rH
O rH

•pmbi[
ut ado[s Mn^BJcbdxnoj]

IN. 1>

CO CO CO CO O

cb cb cb do do

rH i — 1 rH

CO
i-H

9°

rH

. *r}< CO

• -b cb

IN.

cb

i-H

0

cb

rH

CO CO CO r-

»o cb ib cb

rH I-H I-H rH

t-

do

fcO

GO 05
Oi I>-

•a^iuoqa pue pjnbi[
i{i5 no-it^ pe^iuisuur} ^8 jj

CO IN. •* rH OS

O O O O o;

rH f— i rH i-H O

s

rH

8

rH

• o o

• I-H i-H

1

1

O CM rH CO

3S§S

rH
O

rH

8

rH

1 §

-.9,00.

•M CM rH O O

o o o o o

O 0 O O O

'l 'l 'l '

i-H

8
l

F-H

'l

C-1 CM

SO
O

' 'l "l

CM

8
i

CM

o
o

'l

CM i-H CM CM

'l 'l + +

I-H
'l

rH

8
l

rH i— '

8 8
l l

•^1000-

CM CM CO CO CO

o o o o o

O O 0 O O

p p p p p

CO

o
o
p

CM

p

CM CM
, o O

. o o
p p

1

CM

CM CM •<}( r}i

CM

CM

CO CO

o o
o o
p p

•aaqqtia qSnoaq^ ^t'3H

rH IO CO rH OS
O O O O OS

I — 1 rH rH rH O

rH

O

rH

rH
O
rH

-M -M

• o o

• rH l— 1

IN.

OS

p

iO

p

00 rH CO iO
OS o c: OS

p rH p p

8

rH

OS

os
p

SOS
os

CO -* O CM O

3 rH rH O O O

111 +

0

1

OO
O
1

CM K5

* rH rH
' 1 1

I-H
rH
1

1 — 1
1 — 1

1

P CO rH CM
rH O rH rH
1 1 + +

OS

0

1

IN.
O
1

CO -*
0 0

1 1

' -I.-,.

•* CO rH rH CO

- "SJ OO ^< -* CO
CD O CM CM CM

rH

10

gs

rH

•*}< >O
.IO OS

. 00 O
O rH

rH
rH

rH
IN.

rH

OS Ol CO rH

O CM O CM
CM CM CM CM

IO
OS

6

rH
rH

rH

o -^

CO p

r-H rH

•*rt - 0.»

IN. CS CO IN- CM

o i — I CM CM i — l i — I

cb cb cb cb cb

1 — 1
CO

00
1 — I

cb

O rH

; CM CM

" cb cb

p

CO

CO

OS

CM

CD t^ CO IN.
O rH OS OS
CO CO CM CM

CM

CO

OS
0
(XI

^ OS
I-H 0

cb cb

•flfl

OS OS IN. -^ r-l
rH rH CM CO CO

IN.

CM

• CM rH

rH

OS
rH

rH •* O CM

CM
CM

CM

lO IN.

CM CM

•a-H

2 O IN.

^ rH CM OS OS IN.

— CO CO 00 CO CO
o

OS

IN.

00

—H
IN.

IO ^JH
" rH CD
" CO -^

CM
US

CO
CO

IN. O-l CD rH
IN- CO IN- 00

CO

i — l

IO

TO »O CO
•jrr p» p I>- f— ' O 1>- O

O rH Al i— 1 rH r-1

iO
rH
rH

10
IN.

rH

p — 1

CO

. co au

• T1 T1

rH rH

Cl

1 — 1

rH

0

\ — 1

rH

>0

CO IN. CO O5
rH rH O O
rH rH rH rH

rH
r-i

rH
I-H

CD TJI
rH rH
rH ^-,

•n

2 O O O CO CO

E IN. IN. O CM ,_,

"o O O rH rH i — 1

p

rH

O
OS

o

CO rH

• rN. co

*O O

os

CO

o

OS
CO

6

IN. O CD •*}<

IN- OS •Tfl O
O CD rH rH

CM

CO

o

OS
0

CQ O

os O

O rH

•'*

o 0 0 0 0 0
CM CM CM CM CM

O

CM

1 — 1
CM

rH CO

•o o

• CM CM

IN.

O
Cl

rH

CM

OS OS CO IO

o o do do

CM CM rH rH

00
rH

OS
rH

OS OS

I-H rH

I "

b-.

... "So

• • • c

o

* *

. . . -* IN.

ift IN- o do

. CM i-H 1O CO

' ~o

' ' O

o

Li » £

0) T"

-4J . -^.

<-^-

co

iO IO
IN. CO

'-^-

' ' "o

' ' |

' 'I

. . g

^ 2

O

* " ?-l

' ' 'r-2-

CO

. 10 do
o ^

.9 -^~^~

3^

O CO
10 iO

•— V—1

, — * —

IN.

iO O
t- 00

>-^-

• "3 • • • '3

.•§ . rl*

_O <« c8

r- " " -ti 0

. >•> a a
~ f-> o c
73 os o a

• * % ' ' r. §

.-S 3 &

. -t ^~~^^^
0 CD CO

^ s s R CM co

• rV ^ * " ||

IO O iO ^ r~

• CM Jrt IN. &• o

,§ CM CO CO '||
iO IN. CM CO IN. CM 4

,S CMrHlOCOCOCO J"

j * " !T^TT j

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE.

•0015
14

433

0010

000,

O /0% 20
100% SO SO

SO 40 M 60 70
70 60 SO 40 JO

Percentage by volume.

BO so 100

£0 10 0

When the results given in these two tables are expressed by curves, with com-
position by weight as abscissas and conductivities as ordinates, the curves for
mixtures of methyl and ethyl alcohol and glycerine with water differ widely from
each other, and no general law connecting the conductivities of mixtures directly
with the conductivities and relative masses of the constituents can be traced. If,
however, proportions by volume are taken as abscissae, instead of proportions by
weight, the curves for mixtures of methyl and ethyl alcohol and acetic acid with
water become nearly identical, and we are led to the conclusion that in further
work proportion by volume should be made the basis of comparison.

A further conclusion from an inspection of these curves is that the conductivity of
a mixture of two liquids is less than the value calculated by the linear formula

, vl

Vj + V,

when l~i and L are the conductivities, i\ and v., the volumes, of the constituents
present, and that the difference is greater the greater the difference in the con-
ductivities of the constituents.

VOL. CXCI. — A. 3 K

434 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES 01* SOLIDS

It is evident, therefore, that to investigate the law of variation of conductivity
of a mixture with its constitution more closely, it is advisable to work with mixtures
of constituents which differ in conductivity to a greater extent. If we confine our-
selves to liquids, we are limited to ratios of conductivities of about one to three, and
we must turn to mixtures of solids, or of solids and liquids to get greater ratios.
In the case of mixture of solids with liquids which dissolve them, a difficulty arises
as to which conductivity of the material entering into solution ought to be used in
making the comparison — that of the material when in the solid, or when in the
liquid state.

[The results given in the previous table for solutions of sugar in water are almost
identical with those of mixtures of glycerine and water, so that it may be said that
a solution of sugar behaves as if it were a mixture of water and a liquid having the
same density as sugar and a thermal conductivity of '0007. I have, however, found,
by the method described by me in ' Manchester Memoirs,' vol. 42, No. 5 (1898), that
the thermal conductivity of the solid sugar was about '0012, so that from a know-
ledge of the thermal conductivity of a solid we can infer little or nothing as to how
it will behave when it enters into solution. Thus, JAGER, ' Wien. Ber.,' vol. 99,
p. 245 (1890), found that solutions of sodium chloride, potassium chloride, and zinc
sulphate of about the same strengths had conductivities nearly alike and slightly
less than that of water, whereas I have found (' Manchester Memoirs' as above) that
the first two salts in the solid state conduct eight or nine times as well as water, and
the latter a little better than water. The result for ammonia solution given in
the above table points in the same direction, so that it seems advisable, for the
present at least, to confine our attention to mixtures in which neither constituent
has changed its physical state. The further question, whether such mixtures should
be treated as physical mixtures or as chemical compounds, must be held over till
more information has accumulated. — 13th June, 1898.]

If. on the other hand, solids are mixed with liquids in which they are not soluble,
there is great difficulty in keeping the mixture experimented on uniform, and in
determining its exact constitution.

These considerations point to the conclusion that mixtures of solids would be
most useful in leading to the discovery of any law. They have, however, the dis-
advantage of not being as readily made as mixtures of liquids, a disadvantage which
may be removed by using as one constituent a semi-solid like lard or vaseline.

A series of experiments have, therefore, been made with the apparatus, fig. 11,
p. 419, on mixtures of one of these substances with various amounts of reduced iron,
marble, zinc sulphate, and sugar, in the form of powder, and the results are given in
the following table and curves : —

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE.

435

•»

HI

Is

OO

lO C- CO

CM 00

-p

8

o

-* CO CO OS

•*f< CO O 1N.
§O rH rH
0 O O
OO O

*

rH

O

o

•ptnbi[ t[Snoaq}

in

I> rH

OO

o o o

CO >O

10 co

OCD
OO

rH
IO
0
O

r- oo t^ co

-* CO rH OS
O O i-1 rH
O O O O

!

aad powmlT^H

CM 00 •*

O rH CO

coo

O CO

i— 1 rH
00

O CO
CO OO "S
r- CO O

88S

t^- OS

•* CO
CO t^

0 O

O

o

CO
0

o

00 rH
»O CN- CO CM

10 r- CM O

SO rH CM
o o o

:

t~ !>• 'O

CO rH

OSiO-JI

co co

10

rH IO CS lO

•adojs aan^uiodraajj

CO CO CM

co co

rH i — 1

lO CM CS
CM CM rH

O CO
CO CM

r-l
CO

iO >O CO O
CO CM i— 1 rH

:

. •ojiuoqo

i — 1 O CO

os os cs

rH rH i— 1

CS O
CS rH
rH CM

00 OS OS
rH i— 1 rH

00 !>•

OS CS
rH rH

OS
CO
1 — \

CO 1^ 00 CO
OS CS rH T— 1
rH rH CM CM

:

•K«uoo-

CO CM CO
O rH O
O O O
O O O

»O

§ci
0

O O O
O OO

00 O
O O
0 O
O 0

CM

O

o

0

00 O •* 00
O O TJH H<
0 O C O
O 0 O 0

.

1

1 1

1

1

1

+ 1 1

•n* 92000-

CO 00 O

o o o

0 O O

o o o

CO CM

O O

iO "*^ C1*

-^ CO

§o
o
o

o

I-~ CO <M CM

O C O 0

So o o
o o o

1

1 1

•«K[8 qSno.^

CM CM CO

os cs cs

r— 1 i — 1 rH

00 CO
CS O

rH CM

OS CO 10

co os cs

rH rH i— 1

00 00
OS CS
rH I — 1

OS

os

rH

CO CO OS 00
CS C. O O

rH rH CM CM

;

'

CM I>- CM

OS US

O t^ CO

iO O

rH

iO O CO 00

•*a

oO O O
+ 1

O CM

1 1

O O O
1 +

o o

1

O
1

O O CM CM
+ 1 1

•

•i-i %

CO O CM
O CS rH

CO CO
CO CO

os co co

CM 00 H4

CS 00
CO CO

O

o

CO -F 10 CO

-* CM rH CO

°"^ CO CO

CM rH

CO CM CM

CO CO

-*

-T? CO CM rH

•MI %

1O IO CO
00 00 CO

rH CO

os os

CO — 1 OO
CO OS CO

— I rH

os cs

CO
CO

rH rH Ol rH

CS CS O O

OrHrHrH

rH rH

rHrHrH

rH rH

f—J

rH rH CM CM

O »O £*»

<O lO

00 CM CM

•^H cs

^

^ OS CO 00

oCM CM rH

rH O
1

i — ( i — 1 CM

rH —1

rH

CM rH O O

! 1

*

'1— H

CO CO O
"O rH CO

p-< O
10 O
CO CO

CO CO O
OS CO CS
i— O CO

O CM

HH CM

CO

t>- CM CS CO
rH t^ t~ 00

CO rH !>• -*

•

rH i— 1 rH

0 0

1— 1 rH O

rH rH

rH

rH rH O O

•re-n

i — 1 Cj •<?
I>> CO t-~

CO CO CO

CO t>
OS rH
CO £>•

rH -- i— 1
CO OS CO
CO CO CO

CO i— 1

cs os

CO CO

CO
CO
CO

O rH rH f.

CS 'Cs CC CM
CO CD C~ 1^.

:

CO CM —1

r- os co

r- co

CO rH

lO CO CS
CO 'tf !>•

0 O

i — 1
CO

OO O CM 00
CO 1^ CM CM

.

1

+

1 1

• r/,

r~ co

.

rH i— 1

'

§ .
^ *
&J

oT o

. . O

. . ffl
O-,

' 'CO

i

o . ~

rj - -
13

-* CO

O

CO £ -

a - -

Q) - -

o

o

PH
OO iO lO

So p

g

"S

0 "

P-I

00

•

5 per cent, marble

JJ J!

o

•

'S ^
1

. .0

1— 1

OS CM
rHCO

co t^ o

CM CO lO

CO CO
rH CO

f*

CM •* CO
g + + +

!

A

uj

>

CM

cs

CO

o

CS

S3
g

H

3 K 2

436

DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OF SOLIDS

•006
•007
•006

•005

k.

•004
•003
•002
•001

O 10 £0 £O 4O SO 60 7O 80 90 100

PercenCage by volume.

These results, taken in conjunction with those found for liquids and solutions,
show conclusively that the thermal conductivity of a mixture is not a linear function
of its composition. The observed conductivity is always less than that calculated
from the linear law, and, on the other hand, greater than that calculated on the
assumption that the resistivity is a linear function of the composition. The
second assumption is, however, a closer approximation to the observed facts than
the former.

Since neither of these simple assumptions seems capable of representing the facts,
which, by their uniformity, appear, notwithstanding, to point to some general law of
mixtures, it seemed advisable to calculate the conductivity of a model of a mixture
built up in some simple way so as to lend itself readily to the process.

Suppose a cubic centimetre of some substance, having a conductivity p0, to be
divided by equidistant planes parallel to its faces, into 1000 small cubes of 1 millim.
edge ; and let n small cubes of the same size but of a material of conductivity p be
substituted for n of the cubes of the cubic centimetre chosen at random. The large
cube is then a mixture of two materials, and its conductivity may be readily
calculated if the lines of flow are assumed to be parallel to one edge of the cube.

If p0 is the thermal resistivity of the original, p that of substituted small cubes,
the probability that, when n are substituted, p of them will be found in any column
chosen at random,

«-! (1000-?i)! / 1000!

~ pi (n—p)\ (10 —p)l (1000 - n - lQ~-Jj)l / 10! 990!

«!(1000-n)! 10! 990!

p\ (n - p)! (10 - p)\ (990 —n+ p)\ 1000!

and the heat conducted through such a column, when its ends differ in temperature 1° C.,

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE. 437

n\ (1000 -n)! 101990! 10

= p\ (n - p)\ (\fT- p)\ (990 -n+p)\ 1000! ' (10 - p) pn + pp '

Hence the conductivity of the mixture on the assumption that cross-transmission
from column to column may be neglected,

i'=10

M!(1000 -n)\ 10! 990!

! (990- n + p)\ 1000! (10 - p) pu

The values of the factorial expressions have been calculated for n = 200, 400, GOO,
and 800 with the aid of DE MORGAN'S tables of n !* and are as follows : —

P-

n = 200

« = 400

0

•1062

•0054

I

•2684

•0397

2

•3036

•1204

3

•2021

•2156

4

•0878

•2521

5

•0260

•2013

6

•0053

•1112

7

•0007

•0420

8

•0001

•0103

9

•0

•0015

10

•o

•0001

n— GOO

n = 800

•0001

•o

•0015

•o

•0103

•0001

•0420

•0007

•1112

•0053

•2013

•02GO

•2521

•0878

•2156

•2021

•1204

•3036

•0397

•2684

•0054

•1062

The results of calculations of the conductivities of mixtures for which pn = 1, and
p = gij- and yg- respectively, are given in the following curves, along with the
observed curves for mixtures of water and alcohol, and of vaseline and marble,
and the curves for conductivity and resistivity following the linear law, for com-
parison. The calculation gives conductivities always below the observed values,
a result which is due in part, and it may be entirety, to the neglect of the trans-
verse transmission of heat from column to column of the model. The agreement
is, however, sufficiently near to justify the statements that the thermal conductivity
of a substance is not greatly modified when it enters as one constituent in a physical
mixture, and that the thermal conductivity of a mixture depends directly on the
amounts and conductivities of its constituents.

* ' Encyclopedia Metropolitana,' vol. 2, p. 486.

438 DR. C. H. LEES ON THE THERMAL CONDUCTIVITIES OP SOLIDS

Xx

*s

<«K

20%

#t

M

40% 60%

Constitution.

MrP .

60%

5O

2-O

1-0

A reference to the values obtained for the thermal conductivities of solutions of a
gas or a solid in water, will show that the above law does not apply to cases in which
one constituent of the mixture changes its physical state on mixing. Further work
is necessary before any law can be stated, but it seems probable that if the thermal
conductivity of the gas or solid when liquefied were known and used as that of the
material entering into solution, the above law would be found to cover solution as
well as physical mixture.

PAET V. CHANGE OF CONDUCTIVITIES OF MIXTURES WITH TEMPERATUEE.

In order to determine the effect of temperature on the thermal conductivities ot
mixtures, experiments were made on several of the 50 per cent, mixtures of liquids
mentioned in the preceding section. They were carried out with the enclosed
apparatus, fig. 11, p. 419. The observations and results are given in the following
tables and curves : —

AND LIQUIDS AND THEIR VARIATION WITH TEMPERATURE.

439

•oan^ednzax

rj •<? oo co

o CM ^J* CM

10 co 10 i— i

<M "f< CM rH

•*$< C~ •*}< rH
CM •<}< <M rH

co r- co

CM -* CM

-

O OO O
rH O rH

888

^ -^ iO CO
00 t^ CO CO

rH CO rH OS
CO CO CO t~
O O O O

S3S

§o o
o o
o o

•edojs ^jun .tod
pmbif nSnoaij^ ^uofj

O 10 CM
i— 1 OS i— 1
rH O rH
OOO

rH O CM
OS CO OS .
O 0 O .

o o o

r^ -*jt co
CO to. CO .

888 •

iO "^ ^

ooo
pop

**-

— 1 O CM
i — 1 1 — 1 ^H

pop

CO CO rH

oco o •

rH O rH

pop

iO CO tN.
OS 00 OS •

CM CO O
CO t- CO
CO IO CO

CM CM 00

•* CO CO

-* iO CO

rH CO IO

•edojs aati^aaatuaj,

CO ••*< CO
rH i — 1 rH

iO IO iO .
rH rH rH

CO !>. CO .
1— 1 rH rH

CO CO CO
CM CM CM

•a^iuoqo puB
pinbi[ qSnaiq!}

>0 •* CO

•>Jt CO IO
IO CO IO '

!>• CO CO
iO -f CD '

CD CO CO
-* CO •*

pOllITUSU'GJQ. lljOTI

.

•Kkj / Ton.

i>- iO CO

888

r- iO CO

888 :

r- co 10

o o o •
o o o •

O r~ co

rH O O

ooo

•a''95000-

CM rH rH

888

CM r- CM

o o o •

O O O '

CM CM rH

ooo •
o o o •

CM CM -M

ooo
ooo

paWTmsLafn°^lH

CD iO t^

1— 1 rH 1— 1

CO •* CO

co •* co •

i — 1 rH rH '

CD 10 co ;

oo >o co

iO -^ iO
rH 1 — ! I — 1

•gsiJ[3 ut

O) CO ^JH

iO OS CO

rH CO ' — 1
IO CD IO '

OS CO O
iO OS C- •

O >ra o

CO CD CO

ado[s o.ui^B.ioduioj,

iO ^ iO

10 -* iO

10 -* iO

iO •* 10

•** os r^

os co i^
o »o o .

I>» iO *O
rH rH CM .

OS CO IN.

-* 00<M

•»

0 10 CO
CM •* CM

rH CO CM '

CM TP CM

O •* CM •
CM ^? CM

O CO CM
CM -* CM

| rH rH r^

CD I>- CO CM

O "S CM CO

CO OS O CM
OS O CO CM

iO CD CO
rH OO CD

PQ

SS rH O rH
0 rH

r- 1 OS rH I

O rH |
rH I

rH OS rH

!>• CO IO

^H O CM

rH 1^ CO
CO OS CM .

CO rH CM
CM O IO .

CO I— 1 rH
•* CO OS

>0 OS t^
CM •* CM

IO CO CO '
CM -* CM

1(5 CO >O '
CM -^ CM

CM -* CM

•'*v-Ko

CO O iO
°CD 00 I>

10
iO 00 •<? .

OS OS OS .

IO
CO CM -* .

O CM rH .

•CO O OS
OS O OS

rH rH rH

p— 1 ^H rH

CM CM (M

CM CO CM

|
ID TT — -p^

i cog eg

CO t> OS
CO OS CM t»
L^-. !>• t- CO

CO 1>
CO CO O iO
l>- CO CO OS

iO c^1* r^~
O OS CM

i — 1 i — 1 rH

o
a

o

i — 1 i — 1 i — 1

Tl

2 <M

9 OS CO fM
f, iO iO CO

CM CM Tfi

CO CM CO O

o 10 10 i^

rH CM CM

OS CO O O
iO 1Q CD 1>»

lO CO CO
CO CM CO
iO iO iO

•^ 00 Os

Or» "^t* co

CO t^ CO
CO CM CO .

CO rH CM
CO ^ rH .

iO CM O

OS CO ^

*

CO O CO
CM iO CM

t~ 00 t- •
CM -* (M

CO OS I>- '
CM •* CN

00 OS OS
CM •* CM

^: 10 CM t-

O -* CO OS
•^ -^ CO O

CD rH CO CO
•* CM CO CM

CO rH O

o; rH co

t3

.= CO CM CO
0 rH

CO rH CO 1
rH 1

CO CM CO
rH

CO CM -*

rH

•3 CM CO
Tfl OS D-

O O •* CM

CM ^1 t- rH

O CO "* CO
iO iO CO C~~

CO O "^

CO O CO

Cb°

0 OO CO

CO 30 CO -*

t^ t^ t^ so

r~ co r^

Mixture.

50 per cent.* glycerine
and water

'3
5

'•Si

8

CS

•g -2

"SnS
o g

|^
*1

0

us o

. "*
^ ^

o

bD
0

o

To

440 ON THE THERMAL CONDUCTIVITIES OF SOLIDS AND LIQUIDS.

•OOI6
•00/4
001S
•0010

k.

•0008
0006
•0004
•0002

•«.

^

"•»

t,

"^

~-~

~^

'•— — .

— -^,

•^^

aijft

•erl

IS (

m

i,or

--

~~ — -

^

— -

>A

•eh

:t

f/Ai

or

~*t

"hyl

ci/C

&

Wdl

r.r.

i .

—(

lyc

?rir

n&

M

K*

l/C.

Iff £0°

4ffL SO" 60° TCP 30°
G

Mixture
50 per cent, by weight of

Glycerine and water . . .
Acetic acid and water .
Ethyl alcohol and water .
Glycerine and ethyl alcohol .

•00103
•00085
•00080
•00050

Mean percentage

change per degree

between 25° and 45° C.

- -0063

- -0058

- -0068

- -0050

It will be seen from this that the thermal conductivities of mixtures decrease with
increase of temperature at roughly the same rate per cent, as their constituents.

STATEMENT or EESULTS.

The preceding results may be summarised as follows : —

1. Solids which are not very good conductors of heat, in general decrease in
conductivity with increase of temperature in the neighbourhood of 40° C. Glass is
an exception to this rule.

2. Liquids in the neighbourhood of 30° C. follow the same law.

3. The conductivity of a substance does not invariably change abruptly at the
melting point.

4. The thermal conductivity of a mixture lies between the conductivities of its
constituents, and seems connected with the constitution and conductivities by a
simple law.

5. Mixtures of liquids decrease in conductivity with increase of temperature in
the neighbourhood of 30° C. at about the same rate as their constituents.

[ 441 ]

XI. Experiments on Aneroid Barometers at Kew Observatory, and their Discussion.

By C. CHREE, Sc.D., LL.D., F.R.S., Superintendent.
Communicated by the Author at the Request of the Kew Observatory Committee.

Received May 5,— Bead June 9, 1898.

TABLE OP CONTENTS.

SECTIONS. PAGE

1. Preliminary 441

2, 3, General character of phenomena ; apparatus 442

4-6. Differences of descending and ascending readings, from Kew verifications .... 444

7, 8. Differences of descending and ascending readings, from special experiments . . . 448
9. Formulae for variation of the differences of descending and ascending readings

throughout the range 451

10. Position of maximum difference, descending less ascending reading 452

11-15. Relations between phenomena over different ranges 453

16-19. Pall of reading at the lowest pressure 457

20, 21. Recovery after pressure cycle 461

22-25. Effects of temperature 463

26-29. Secular change in aneroids 468

30-33. Influence of rate of change of pressure 473

34-41. Effects of stoppage 480

42-46. Theoretical deductions 487

47-54. Discussion of previous work 492

55, Conclusion, acknowledgement of assistance 499

Preliminary.

§ 1. THE ordinary aneroid barometer is an instrument whose chief recommendations
are its portability and the ease with which it can be used by travellers. It is essen-
tially a field instrument, and it would be largely waste of time to treat it as if
intended for the laboratory.

The first object of the present investigation is to acquire knowledge likely to
increase the usefulness of the aneroid under the conditions in which it is actually
employed.

Aneroids have for many years been tested at Kew Observatory, and records exist
of the performance of many hundreds of various sizes. The test has hitherto been
applied as follows : The aneroid has been put in the receiver of an air pump, and,

VOL. CXCI. — A. 3 L 11.10.98

442 DE. C. CHRBB, EXPERIMENTS ON ANEROID BAROMETERS

pressure being reduced at the rate of 1 inch in 3 or 4 minutes, readings have been
taken of the aneroid and a mercury gauge attached to the pump. The readings are
taken with the pressure temporarily stationary, usually for each inch, but sometimes
for each half-inch of pressure. After the reading has been taken at the lowest point
for which verification is desired the pressure is reduced a very little further, and then
maintained constant for some minutes. It is then allowed to rise by the readmission
of air, stoppages being made and readings taken during the ascent of pressure
precisely as during the descent.

From the observed differences between the aneroid and gauge — the readings of
the latter being corrected of course for temperature — corrections are calculated
separately for the descending and ascending readings of the aneroid, and these are
given on the certificate issued.

If the aneroid possesses, as is most usual, a scale of altitudes in feet or metres,
that is read as well as the pressure scale, and corrections are given to it based on
Airy's table connecting altitude and barometric pressure. Whether the unrestricted
application of Airy's table is the best means of interpreting observations taken during
mountain ascents is an interesting question, but I do not propose to discuss it here,
confining my attention to the aneroid as a measurer of pressure.

Aneroids intended for the Meteorological Office are usually constructed only for
the narrow range, 31 to 26 inches. In their case there are exact regulations as to
the size of error permissible and other points, and the issue of a certificate implies
the attainment of a certain standard of excellence. With this exception, however,
there exist no rules for rejection, and the certificate merely represents the report of
an expert on an instrument submitted to him. The owner is supposed to draw his
own conclusions from the figures submitted. This view of a certificate was apparently
that originally dominant at Kew Observatory, but of late years there has been an
increasing tendency on the part of the public to substitute the idea of a " certificate
of excellence" for that of a " certificate of examination." What is sought is protec-
tion against inferior instruments rather than the means of applying corrections to
observed readings. It has thus become desirable to ascertain how the excellence of
an aneroid may be judged of, and one of the chief objects of the present investigation
was to obtain data suitable for the purpose.

General Character of Phenomena.

§ 2. An aneroid is usually graduated by direct reference to a mercury gauge during
a reduction of pressure. An examination, made under the same conditions as the
graduation, tests the accuracy of the workman, but not necessarily the quality of the
instrument. That something more is desirable becomes obvious, when, after reducing
the pressure several inches, one keeps it constant and continues comparative readings
of the gauge and aneroid for some time. A gradual fall in the readings of the latter

AT KEW OBSERVATORY AND THEIR DISCUSSION. 443

soon becomes apparent. Again, on allowing the pressure to rise to its original value,
one finds the aneroid read lower than at the start, this depression gradually
disappearing. The instrument, in fact, behaves as if an imperfectly elastic body.
The strain, under a uniform stress, tends to increase, and as a natural concomitant,
there is elastic after-effect (elastische Nachwirkung). That the whole of the difference
between readings with pressure descending and ascending (or, as we shall call them
for brevity, descending and ascending readings) represents, in all cases, a true after-
effect, I am not prepared to say. If we regard a rise of temperature as analogous to
a fall of pressure, a thermometer presents a somewhat parallel case ; and we know
that there the difference between ascending and descending readings is due to at
least two causes, viz., change in the mercury meniscus, with consequent alteration of
internal pressure, and temporary change of zero following exposure to the higher
temperature. Of these two causes, the latter only is a true after-effect ; but the
former may also conceivably have its counterpart in the mechanism of the aneroid.

An aneroid showing large after-effect is not a suitable instrument for travellers.
Readings taken with it, for instance, during a mountain descent, are largely influenced
by the elevation of the summit and the time spent there. Thus, from the outset. I
have regarded the after-effect phenomena as specially requiring investigation.

§ 3. As a first step, I examined the records of about 300 aneroids, tested over the
ranges 30-15, 30-18, 30-21, 30-23, 30-24, and 30-2G inches. The differences
between the descending and ascending readings were noted for the several ranges,
and the general character of the phenomena ascertained. There existed, however, no
data for connecting the results from the several ranges, it being customary to test
each aneroid over one range only. Special experiments were thus requisite, in which
the same aneroids should be taken over a variety of ranges. For such a purpose it
was impossible to employ the ordinary working apparatus, with a due regard to the
regular work of the Observatory. Application was accordingly made by the Kew
Observatory Committee to the Government Grant Fund, and £30 was obtained for a
new air-pump and receiver, and for a set of aneroids to be experimented on.

The only special feature of the apparatus* is a second or auxiliary receiver, between
the pump and the main receiver which contains the aneroids. On the tube connecting
the two receivers are two stop-cocks, a third cock being placed between the auxiliary
receiver and the pump. One can thus exhaust the auxiliary receiver separately, and
then by manipulating the cocks lower the pressure in the main receiver at any
desired rate. This arrangement avoids the sudden changes of pressure to which
aneroids are exposed when in a receiver connected directly to the pump. The pump,
the receivers, and the mercury gauge — which shared the pressure of the main
receiver — were rigidly attached to a stout board which rested on a table. The

* The apparatus was obtained from Mr. J. J. HICKS, 8, Hatton Garden, London. It was solidly
constructed and proved very satisfactory. The experimental aneroids were likewise purchased from
Mr. HICKS.

3 L 2

444

DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

board was tapped before each reading, this being the invariable practice in the
ordinary Observatory test. Though not free from objections, this seems to reproduce
most closely natural conditions.

Differences of Descending and Ascending Readings, from Kew Verifications.

§ 4. It being most convenient to extract data from books not in current use at the
Observatory, I have employed, in the following discussion, data from aneroids tested
between 1885 and 1891. I have, however, examined a sufficient number of the more
recent data to assure myself that the ordinary aneroid has since then undergone no
important modification. On the whole, there is, perhaps, a slight reduction in the
average size of the after-effect phenomena.

Table I. shows the mean excess, in inches, of the descending over the ascending
reading at each inch of pressure for the several ranges specified, also the sum of the
differences between the descending and ascending readings, and the mean difference
for each range. It also mentions the number of instruments on which each set of
results is based. The results, with the exception of those from the second group of
30 aneroids taken over the range 30-24 inches, are shown graphically in the several
curves of 'fig. 1. Abscissae represent air pressures, ordinates the excess of descending
over ascending readings.

Fi£. 1.

!

7

7

15

16

18

ZO Zl ZB 23 24 25 Z6

Pressures in inches of mercury.

£7

^9 JO

AT KEW OBSERVATORY AND THEIR DISCUSSION.

445

O

03

fcJO

g

^3

03
<O

PH

fcJO
G

^3
i

05
02

a>

i— H

&0

c

^3

G
CD
O

03

o
ft

rC
Hi
-
^

H

1

I
1

I*

11 8

•i S3 B

rH 00

co as
0 O

rH iO

O r-

CO
rH

CM
CM

00

8

If

? rrj <U

\

b

3 !§ °°

CO •*

s s

CM O
O •*
rH- <M

CM

CO

1

r— 1

o

*o

rH

i-H

CO

117

o

CO

1 1

CO CO
OS CO
O rH

rH
rH

e»
i— i

CM

CM

CM

1s- iO

CO O

0 rH

-<}< CM
0 rH

rH

CM

os
os

CM

CO
CM

0 -*

O rH
O rH

CO rH
CM OS

r- 1 rH

O
CM

CO

CO
-*<

CO

CM

CO CM

O rH

O OS
CO OS

1— 1 rH

i— '
CM

CO
CM

CO

CO

CO

CO
CM

•^ *-O

rH rH
O rH

CO CO

co os

rH rH

CO
CM
Cl

rH
CO

CO

o

co

CO

o

CO CO
CO CO

tO ^H

CO
1 — 1

n

o
>o

CO

32

0>

CM

CO

>o

rH O
>O CM

0 rH

OS

i

CO

co
?

o
s

•r-i

a

%

: :

CO
0

rH

CO
CM

CO

CO

CO

T

2

I

CM
CM

; ;

: :

0
CM

rH

o

r—*

CO

3>
CM

•jf

£

PH

rH

CM

: :

' '•

CO
O

CM

"f

CO

»0

CM

• •

•

*

CM

CO

OS

rH

g

1-1

rH

cp

00

CO
OS

rH

I-H

0

cp

^

i

rH

CM

CO

CO

I-H

1— 1

Ui

CO
CO

rH

1

111

i-H CO
O rH
rH

O CO
CO CO

S

US

CO

o

1

M CO ^
^CM CM

li =

•^ CO
CM CM

i-H
CM

CO
i-H

I-H

446 DR. C. CHREB, EXPERIMENTS ON ANEROID BAROMETERS

The second group of aneroids tested over the range 30-24 inches were all large
and of a special make, and so best treated apart. They showed such exceptionally
small after-effect that I multiplied the actually-observed mean differences by seven,
except in the last two columns, to make the figures similar in size to those given by
the first group of 13 aneroids tested over the same range. In this way one sees
more clearly that the general laws obeyed by the after-effect are to all appearance
independent of its absolute amount.

In no case were the aneroids read to nearer than O'Ol of an inch, the third
significant figure in Table I. being introduced merely in taking arithmetical means.
Thus the law of variation throughout the range of the quantity tabulated is
difficult to ascertain with great exactness over the shortest ranges, especially when
the aneroids are of such exceptional quality as the second group tested between 30
and 24 inches.

When results are derived from a number of aneroids, such disturbing factors as
local irregularities of graduation lose their influence ; whereas with one or a small
number of aneroids they may affect the smoothness of the results, however numerous
be the observations.

§ 5. A glance at fig. 1 shows a similarity of type in the curves ; but this is some-
what obscured by the variety in the pressure ranges and in the lengths of the
maximum ordinates. The method of bn'nging this similarity into clearer relief will
be most easily followed by taking a particular case, say, that of the range 30-21
inches. Here there are 10 differences of descending and ascending readings. Call
these, starting from 21 inches and proceeding upwards, dl} d2 . . . dw. The mean
difference is

and dj/d, d.,/d, &c., are the ratios of the several differences to the mean. Now draw
a curve whose ordinates represent the size of these ratios on any convenient scale,
while the abscissa represent the corresponding fractions, 0, 'i, % &c., of the range
measured from the lowest pressure. The curves obtained in this way for the four
longest ranges of Table I. are shown in fig. 2. In a short range the number of
points on the curve determined by the data is so small as to leave the shape a little
uncertain. Thus to have given together curves for all the data in Table I. would
have sacrificed the clearness of detail without securing adequate compensation. It
will be observed that the curves in fig. 2 are almost coincident near the central part
of the range. At the lower end of the range there is a slight but appreciable
difference, the ordinate tending to decrease as the range becomes longer. The
closeness of the curves, striking as it is, would, I think, be improved by altering the
cycle and the method of treating the results in the following three respects :—

(i.) Making the duration of the stoppage at the lowest point proportional to
the length of the range.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

447

(ii.) Making the number of points of observation the same in all ranges by
suitably varying the size of the pressure steps.

(iii.) Allowing only half-weight to the two terminal differences ; for instance
in the range 30-21 inches taking c? as the mean of nine quantities, of which
(c£i + c?io)/2 is one.

The last of these changes was, of course, within my power, but as its advantages
after all could not be large, and might be nil, it did not seem worth while to under-
take the large amount of arithmetic that would have been entailed by its substitution
for the method originally adopted.

Fig. 2.

•/ -i -3 -4 -5 -6 7 S -9

Fractions of pressure r&rtge from lowest point

10

§ 6. In actual practice, the change (ii.) specified above would be inconvenient,
entailing readings at fractions of inches difficult to determine exactly on an ordinary
gauge. In ranges where the points of observation are sufficiently numerous the
same object can be obtained pretty satisfactorily by interpolation. For instance, in
the range 30-21 inches, the middle of the range is 25'5 inches, and (d5 + d6)/2 may
be regarded as a tolerably good first approximation to the corresponding difference.

448

DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

I have calculated in this way from Table I., in the case of all but the shortest range,
the differences of the descending and ascending readings answering to the fractions
0, '1, '2, -3, &c., of the range, measured from the lowest point. For the range 30-24
inches the first group of aneroids was used. The ratios borne by these calculated
differences to the mean differences in Table I. are given in Table II.

It should be noticed that the mean of the 11 differences found for the

fractions 0, '1, &c., of the range is not generally identical with the difference cl given
in Table I. ; consequently the sum of the 1 1 entries in each row of Table II. is
not identically 11, though nearly so. .In deciding to take the d of Table I. as the
consequent of the ratios, I avoided the confusion entailed by having two slightly
different mean differences for each range. There are, however, disadvantages,
which seemed to me to prevail in the case of the mean results for the 5 ranges
of Table II. In their case, accordingly, I made the small alterations required — an
increment of less than \ per cent. — to make the sum of the eleven ratios exactly 11.

TABLE II. (Old Observations). — Ratios of Differences Descending less Ascending

Readings to Mean Difference.

Fraction of range —

Range.

0

•1

•2

•3

•4

•5

•6

.7

•8

•9

1-0

30 inches to 24

•541

•743

•936

1-115

1-210

1-265

T204

1-145

1-089

•998

•888

€>Q

>i » «J

•413

•666

•898

1-100

1-213

1-265

1-274

1-238

1-159

1-034

•858

,, 21

•370

•661

•943

1-106

1-215

1-274

1-270

1-223

1-143

•986

•814

18

•342

•670

•900

1-081

1-192

1-271

1-278

1-226

1-149

•971

•804

15

•245

•628

•931

Till

1-232

1-281

1-318

1-235

1-137

•953

•724

Mean for 5 ranges

•382

•674

•923

1-104

1-214

1-273

1-270

1-215

1-137

•989

•819

Before treating further the results shown in Tables I. and II. it is convenient
to notice some of the special experiments.

Differences of Descending and Ascending Readings. Special Experiments.

§ 7. The first 24 special experiments were intended to serve for comparison of
the phenomena over different pressure ranges. They took place as follows, the
numbers showing the chronological order : —

Range.

Order of experiments.

Number over range.

30 inches to 26

Nos. 1, 2, 12, 13, 22, 23, 24

7

» 24

3, 4, 5, 20, 21

5

21

6, 7, 18, 19

4

,, 18

8, 9, 16, 17

4

» 15

10, 11, 14, 15

4

AT KEW OBSERVATORY AND THEIR DISCUSSION.

449

Two aneroids, No. 1 and No. 4, were subjected to all these experiments ; two
others, No. 2 and No. 3, divided only to 20 inches, were not taken below 21 inches
of pressure.

The order of the experiments was adopted with a view to the possible permanent
influence of the lower pressures on the aneroids.

The procedure in these experiments was uniform. The change of pressure during
both the descent and the ascent was at the rate of 1 inch in 5 minutes. At the
lowest point there was a stoppage of 10 minutes, preceded however by no slight
lowering of pressure below the point at which the last descending reading was taken.

After each experiment the aneroids were left for some days at atmospheric
pressure, so as to be fully rested at the beginning of each pressure cycle. The
experiments were made in 1895, and extended over a period of five mouths. As will
be seen later, there would not appear to have been any serious change in the after-
effect phenomena during this interval.

§ 8 The first thing investigated was the law of variation throughout each range of
the differences of the descending and ascending readings, it being advisable to make
sure that the aneroids were fair specimens, and gave results sufficiently similar in
type to those derived from the old verification books. I have thought it unnecessary
to reproduce the results analogous to Table I., but pass at once to those analogous to
Table II. The procedure followed was exactly the same as that already described, so
that the results of Tables III. and II. are immediately comparable.

TABLE III. (Special Experiments). — Ratios of Differences Descending less
Ascending Readings to Mean Difference.

Fraction of range.

Range.

0

•1

•2

•3

•4

•5

•6

•7

•8

•9

1-0

30 inches to 26

•430

•707

0-984

1-168

1-260

1-352

1-292

1-231

1-140

1-017

•895

» !) 11 ^4

•437

•771

1-029

1-137

1-195

1-228

1-293

1-309

1-230

0-974

•630

„ „ „ 21

•'294

•708

0-992

1-163

1-306

1-334

1-250

1-169

1-116

0-975

•693

1 ft

11 11 11 AO

•262

•715

0-940

1-142

1-276

1-416

1-425

1-227

1-049 0-887

•495

» >i » 15

•197

•650

1-084

1-228

1-255

1-323

1-442

1-274

0-989

0-823

•433

Mean for 5 ranges

•323

•707

1-002

1-163

1-252

1-325

1-334

1-237

1-100

0-931

•626

The alteration required to make the sum of the ratios in the last line exactly 11, was
slightly greater than in the case of Table II., but still was less than one half per cent.

There is a tendency in the values at the extremities of all the ranges to be lower
in Table III. than in Table II., entailing of course a slight difference in the opposite
direction towards the centre of the range. This is probably due mainly to the
difference between the procedure followed in the old and new observations, The

VOL. cxci. — A. 3 M

450

DR. C. CHEBE, EXPERIMENTS ON ANEROID BAROMETERS

difference in the results, though worthy of notice, is for practical purposes of little
consequence. As we shall see later in Table VI., unity in Table III. represents at the
very utmost 0'25 of an inch, so that a difference of 4 or 5 per cent, in one of the
ratios answers to a quantity which is in general considerably less than the probable
error of an observation.

Fig. 3.

0 •/ -Z -3 , -4 -5 -6 -7 -a 3 I'O

Fractions of pressure range from lowest point.

To facilitate comparison of the old and new observations, I show side by side in
fig. 3, curves of the same type as in fig. 2, the one representing the mean results of
Table II., the other the mean results of Table III. The former curve is exceptionally
smooth, and the latter even is very fairly regular ; but curves for the individual
ranges of Table III. — which I do not reproduce — show very appreciable irregularities.
This of course is only what one would expect from the limited number of observations.
Whilst the data in Table III. thus possess defects which might have been much
reduced by an increase in the number of experiments, they appeared quite sufficiently
consistent for the purpose I had in view.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

451

Formula for the Variation of the Differences of the Descending and Ascending

Headings throughout the Range.

§ 9. The absolute size of the differences between the descending and ascending
readings varies largely from aneroid to aneroid ; thus formulae reproducing the data
of Table I. would not be immediately useful. On the other hand, the law of
variation, as shown by the ratios of these differences, at different points of the range,
to the mean difference, appears to be nearly the same in all ordinary aneroids. I
have thus tried to represent the variations shown in Tables II. and III. by simple
algebraic formulae of the type

(1),

y —

3x2 -f-

where y is the ratio of the difference of the descending and ascending readings to
the mean difference, x the fraction of the range measured from the lowest point.

One can secure absolute agreement between the observed and calculated values in
such a case by taking enough terms ; but, for practical purposes, the question is
whether with a comparatively small number of terms one can secure a sufficiently
close agreement. In the present instance, four terms are enough for practical
purposes. To get the best general agreement we ought, of course, to determine «0,
«i, «2, a3 by the method of least squares, taking account of the whole eleven values
in each range. In dealing, however, with material such as that here available, the
slightly increased accuracy thus attainable would not, in my opinion, be an adequate
return for the labour expended. I thus simply determined the constants so that the
observed and calculated values were identical for the values '1, '3, '7, and '9 of x.
The values thus found for the constants are given in Table IV., the first six sets of
values relating to the old observations of Table II., the last set to the mean data of
Table III.

TABLE IV. — Values of Constants in y = «„ + a\x + ff-a*2 +

Range.

a0.

«!•

Qj.

Oy

30 inches to 24
90
»> )» » ***

>! !! •:> "1

» >) >i 18

» )! 1> 15

+ •425
+ •332
+ •322
+ •377
+ •264

+ 3-680
+ 3-677
+ 3-808
+ 3-219
+ 4-067

-5-209
-4-079
-4-264
-2-947
-4-391

+ 2-031
+ 0-958
+ 0-948
+ 0-115
+ 0-802

Mean old observations
„ new „

+ •345

+ •358

+3-699
+3-922

-4-186
-4-362

+ 0-969
+ 0-792

The value of a, shows little variation. In the shortest range the values of «2
a3 are undoubtedly unduly large numerically, and in the range 30-1 8 inches there is

3 M 2

452

DE. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

a similar defect in the opposite direction. In these cases the method of least squares
would no doubt have given values closer to the mean.

The closeness with which the data of Tables II. and III. are reproduced by the
formulae based on the above values of the constants is shown by Table V., which
gives the algebraical excess of the observed over the calculated values. There is, it
will be remembered, absolute agreement when x = 'I, '3, '7, and '9.

TABLE V. — Observed less Calculated Values. (Ratios of Differences to Mean

Difference.)

x =

Range.

Mean of
differences.

0

•2

•4

•5

•6

•8

1-0

80 indies to 24

+ •116

-•033

+ •016

+ •048

+ •007

+ •014

-•039

± -039

9q

» '1 !) —5

+ •081

- -014

+ •002

-•006

-•003

+ •006

-•020 ±'019

., ., 21

+ •048

+ -021

-•OOP

-•004

-•007 +-01 9 , -000

±•015

„ „ 18 | --035

- -004

-•009

+ •007

+ -005

+ •023 +-040 i ±'018

>, „ „ 15

-•019

+ •024

— •006

-•018

+ -022

+ •018 , --018 ; ±'018

Meau old observations

+ •037

-•003

-•003

+ -003

-•004 +-015 --008

±•010

,, new „

-•035 +-028

-•024

-•002

+ •022 --010

-•083

±•029

There is not a single instance here where the difference between the observed and
calculated values represents as much as O'Ol of an inch in absolute pressure.

For ordinary purposes — unless special importance is attached to the phenomena at
the two ends of the range — the formula

y = -35

(2)

is quite good enough for experiments made on the old plan over any range not
shorter than 6 nor longer than 15 inches.

Position of Maximum Difference, Descending less Ascending Reading.

§ 10. As illustrating the practical use of (2), let us find where it makes the
difference between the descending and ascending readings largest. For this, putting

we find
whence

2-91cc2 — 8'4z + 37 = 0

x = '54.

(3),

AT KEW OBSERVATORY AND THEIR DISCUSSION.

453

Here x, it will be remembered, represents the fraction of the range measured from
the lowest point. To test the accuracy of this result, I have calculated the position
of the maxima for the several ranges from Table I., which contains, it should be
noted, the direct results of observation. The results are as follows : —

I

iange 30 i

uches to-

26

24

23

21

18

15

Position of maximum, x =

•56

•50

•54

•56

•58

•58

•55

Relations between Phenomena for different Ranges.

§11. Our previous work may be regarded as determining the differences between
the descending and ascending readings at every point of any range in terms of the
mean difference for that range. The next step is to express the mean difference for
a range of n inches as a product of / (n) — a function whose sole variable is n — and a
factor which is constant for a given aneroid.

The mean differences between the descending and ascending readings (defined as
in § 5) given by the first 24 experiments with the 4 aneroids are recorded in
Table VI.

TABLE VI. — Mean Differences (in inches) from the first 24 Special Experiments.

Range 30 inches to —

26

24

21

18

15

Aneroid No. 1 . . . '043

•084

•146

•217

•315

j! » * • • i

•033

•063

•123

. .

„ 3 . . .

•046

•089

•179

. .

?» H T" • • •

•027

•043

•077

•122

•190

The most convenient way of dealing with these results is to find the ratios borne
by the several mean differences to one of their number for each aneroid separately.
For this purpose, I have taken the differences in the range 30-21 inches as standards.
The results are given in Table VII.

454 DR. C. CHRBE, EXPERIMENTS ON ANEROID BAROMETERS

TABLE VII. — Katios of Mean Differences to Mean Difference in Range 30-21,

Ra

nge 30 inches

to—

26.

24.

21.

18.

15.

•295

•575

1

1-486

2-158

o

•268

•512

1

!> )! «

4

•257
•351

•497

•558

1
1

1-584

2-468

Mean for all 4 aneroids .
„ „ Nos. 1 and 4 ...

•293
•323

•535

•566

1
1

1-535

2-313

Assume now that the ratio of the mean difference dn for a range of n inches to the
mean difference d9 for a range of 9 inches is given by

dn/da = 0 (n) = b0-\- bt

(4).

Determining b0, blt b2 from the three ranges in Table VII. common to all four
aneroids, we find

<f> (n) =— -028 + '053/1 + '0068n2 ....... (5),

with of course

0(9) =1.

In arriving at this expression, we utilised only the data from the three shortest ranges
in Table VII. Thus a check on the general suitability of the formula is supplied by
the good agreement between the mean results for the two longest ranges in the table
and the corresponding values

= 1-587, 0(15) = 2-297

calculated from (5).

§ 12. As illustrating the practical application of (4) and (5), suppose we find that
in a particular aneroid the mean difference between ' the descending and ascending
readings for the range 30-15 inches is 0'338 inch. Then we should conclude
that the mean difference for a range of m inches, m being any specified number, is
given by

3,= '338 X <f>(m) -r- 0(15).
For instance, we should find for this aneroid

J4 = '043, J6 = -078, J7 = -099, 39 = '147, 312 = '234.

AT KEW OBSERVATORY AND THEIR DISCUSSION. 455

The value assumed in the above illustration is that found in Table I. for the range

30-15 inches. Hence the values just deduced for <F4, &c., are what the mean
differences for the other ranges would have been in the case of the average aneroid
sent for trial over the range 30-15 inches. Comparing these calculated values with
the mean differences actually recorded in Table I., we should draw the conclusion
that the average aneroid sent for testing over the longest range is less subject to
after-effect than the average aneroid sent for testing over any other range except the
shortest. This is, I believe, really the case. The exception in favour of the shortest
range is due to the exceptional size of a number of the aneroids intended to cover the
range 30-26 inches.

§ 13. The sum of the differences of the descending and ascending readings is
really the source of our knowledge of the mean differences, and for some purposes it
may well be the more convenient quantity of the two. The formula for it is
deducible at once from (5), for

Sum of differences when range n inches __ «, + !,, ,
Sum of differences when range 9 inches 10

= — -0028 + -0025n + -0060rr + -00068n3 . . .(6),

= \ji (n), say.
The following results are easily verified : —

= -147, ^(6) = '375, t/»(9)=l, ^(12) = 2-066, ^(15)=3'680.

Using S,t to denote the sum of the differences for a range of n inches, then, according
to the formula,

Sn-5-tJr(n) = 8w-5-iJr(m)
for all values of n and m.

To illustrate the degree of accuracy attained by the use of such formulae, I have
taken the case of the first 24 special experiments, and determined the arbitrary
constant for each aneroid so as to make the total sum of the differences over all the
ranges for which the instrument was tried the same for the calculated as the
observed values. As it may be well to show how this was done, take the case of
aneroid No. 2. Here the sums of the differences observed in the three shortest
ranges were respectively '163, '443 and 1'228 inches, amounting in all to 1'834 ;
while by the formula,

*(9)= 1-522.

Thus, for a range of n inches with this aneroid, we may assume
Sum of differences = </» (n) X 1'834 -=- 1'522.

456 DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

In Table VIII. C denotes the values calculated in this way ; O, those actually
observed.

TABLE VIII. — Sums of Differences Descending less Ascending Readings.

Range 30 inches to —

26

24

21

18

15

O.

C.

O.

C.

0.

C.

0.

C. 0.

C.

Aneroid No. 1

•214

•204

•589

•523

1-463

1-393

2820

2-878

5-040

5-126

o

•163

•177 | '443

•452

1-228

1-205

. .

. .

. .

,, 3

•231

•254

•622

•651

1-790

1-737

. .

. .

. .

„ 4

•134

•118

•304

•302

0-770 0-804

I

1-588

1-661

3-048

2-959

The differences between the observed and calculated values in Table VIII. are of
the same order as those met with in different experiments over the same range under
the most favourable conditions.

§ 14. The preceding results go a long way to secure one of the principal objects
of the investigation — the more complete utilisation of the ordinary Kew test.
Suppose, for instance, an aneroid has been taken over any convenient range in the
usual way, and a summation made of the differences of the observed descending and
ascending readings. This sum, being based on a considerable number of independent
readings, is but little affected by errors of observation, if ordinary care is used. It
is also but little affected by any ordinary irregularities in the sub-divisions, and thus
affords a satisfactory basis for a general diagnosis of the quality of the instrument so
far as after-effect is concerned.

By means of §§ 11, 12, and 13 we can deduce from the observed sum, or from the
mean difference, the values of the corresponding quantities for any other range. In
this way a common standard of excellence could be utilised for all aneroids of given
type, irrespective of the range covered by their graduation.

§ 15. In the laboratory one can change the pressure at a uniform rate and keep
the temperature within narrow limits. In the field, however, the conditions are very
variable. It was thus obviously desirable to aim at a fuller knowledge of the
properties of aneroids, which might be of service to a wider circle than that concerned
with the mere testing of these instruments.

Some valuable work in this direction was done many years ago by Dr. BALFOUR
STEWART, and more recently by Mr. EDWARD WHYMPER. Neither writer, however,
made any very serious attempt to ascertain the exact connection between the several
phenomena. I thus postpone consideration of their work to a later stage, when its
significance will be more easily understood.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

457

Whilst I regard the present experiments as throwing much fresh light on the
subject, it will, I hope, be clearly understood that I do not profess to have established
a complete physical theory, by the aid of which one can foretell the exact behaviour
of an aneroid under any arbitrary set of conditions.

Fall of Reading at the Lowest Point.

§ 16. In the 24 experiments described in § 7, readings were taken at 2-minute
intervals during the 10 minutes' stoppage at the lowest pressure. It was soon
obvious that the fall in a given time was at least approximately proportional to the
range, i.e., to n, where 30-w represents the lowest pressure in inches.

To show this in a simple way, I give in Tables IX. and X. the mean results obtained
by multiplying the observed falls by (18,000 -f- range). The constant was chosen so
as to avoid decimals. The actual fall in inches is of course obtainable by multiplying
the figures in the table by (range -f- 18,000).

TABLE IX.— Fall at Lowest Point X (18,000 -4- range).

Aneroid No. 1.

Aneroid No. 4.

Range.

Time.

Time.

2

4

6

8

10

2

4

6

8

10

30 inches to 26

39

45

71

77

90

26

39

45

45

51

24

24

54 54

78

102

6

30

36

42

60

21

30

65 80

85

95

10

30

30

30

40

18

34

53

68

83

98

11

15

23

30

41

„ 15

27

45

63

78

87

]2

15

21

30

36

Means

31

52

67

80

94

13

26

31

35

46

TABLE X. — Same quantity as in Table IX.

Range.

Aneroid No. 2.

Aneroid No. 3.

Time.

Time.

2

4

6

8 10

2

4

6

8

10

30 inches to 26
24

,, 21

32

18
20

39
54
55

58
54
60

64 71
66 90
65 80

32
24
30

45
54
60

64
66
65

71

84
80

77
108
95

Means

23

49

57

65 ; 80

29

53

65

78

93

VOL. CXCJ. — A.

3 N

458

DR. C. CHBEE, EXPERIMENTS ON ANEROID BAROMETERS

Considering the limited number of experiments, the irregularities in the tables are
by no means surprising, when we reflect that even at 15 inches pressure the average
fall of reading in 10 minutes was only about '07 of an inch for aneroid No. 1,
and '03 of an inch for aneroid No. 4. In the case of the readings after 2 or 4 minutes'
exposure to pressures such as 26 or 24 inches, the uncertainty is of course much
greater. It will be noticed that the figures for aneroids Nos. 1 and 3 are very
similar and are approximately double the figures for No. 4.

§ 17. Subsequently three experiments, Nos. 33, 34, and 35, were devoted to
elucidating the law connecting the fall of reading with the time elapsed since the
stationary pressure was reached. Pressure was reduced at the normal rate, 1 inch
in 5 minutes, but kept for 2|- hours at the lowest point of the range, readings
being taken at intervals. The lowest pressures in these three experiments were
respectively 24, 21, and 18 inches. It would occupy too much space to give full
details, but Tables XL and XII. show the falls observed at 30-minute intervals.
To avoid decimals the observed falls are multiplied by 104/range ; so, to deduce the
falls in inches, the figures in the table must be multiplied by n X 10"4, where n is
the number of inches in the range.

TABLE XI. — Fall at Lowest Pressure X (I04/range).

Range.

Aneroid No. 1.

Aneroid No. 4.

Time, in minutes.

30

Time, in minutes.

30

60

90

120

150

60

90

120

150

30 inches to 24
» 21
18

83
89
92

133
111

108

150
133
142

155
156

158

172
156
175

50
44
50

78
44
67

83
66
83

83
78
83

100
89
92

TABLE XII. — Same quantity as in Table XL

Range.

Aneroid No. 2,

Aneroid No. 3.

Time, in minutes.

Time, in minutes.

30

60

90

120

150

30

60

90

120

150

30 inches to 24

» 21

100
111

128
122

155
144

167
156

172
178

117
111

161
133

172
156

183

178

205
189

AT KEW OBSERVATORY AND THEIR DISCUSSION.

459

In these tables each entry depends on only one experiment, so the agreement
with the law, " fall proportional to range," could hardly be better.

In the interval between these experiments and those dealt with in Tables IX. and X.,
aneroid No. 1 had broken down and been altered. The figures obtained from the
other three aneroids stand to one another in almost exactly the same proportion
as before.

§ 18. To show more clearly that the law of variation of the fall of reading with the
time was nearly if not exactly the same for all four aneroids, I have in the following
Table XIII. multiplied the figures actually found for Nos. 1, 2, and 4 by 6/5, 8/7
and 2 respectively. The results are the means of the several ranges included in
Tables XI. and XII., but the observations at all the time intervals are included.

TABLE XIII. — Quantity for Aneroid No. 3, same as in Table XII. ; for the others,

multipliers are applied, as explained above.

Time, in minutes.

2

4

6

8

10

15

20

25

30

45

60

75

90

100 105

120

135

150

No. 1 . . .

7

34

63

8G

83

101

89

96

IOC

128

140

158

170

173 174

187

196

202

2

10

31

51

70

Tl

91

102

109

120

134

143

155

170

183 i 184

185

191

200

„ 3 . . .

9

33

47

69

75

89

98

103

114

133

147

156

164

173 ! 175

180

189

197

„ 4 . . .

16

40

46

58

58

78

94

100

96

124

126

142

154

160 :.162

162

192

198

After trying some logarithmic functions, I found that the results in Table XIII.
were more in accordance with a formula of the type

fall, under stationary pressure = Ctv (7),

where t denotes time elapsed since the pressure became stationary, while C and q are
constants for a given previous rate of fall. C varies, of course, from aneroid to
aneroid, but to all appearance there was at least a clcse approach to equality in the
values of q for the above four aneroids. Accordingly, I took the mean of the results
from all the aneroids in Table XIII., and determined q by trial. Table XIV. com-
pares the mean thus found for the observed values with results calculated from the
formula

constant X fall = (31'3) t'm (8).

The corresponding mean results from the first 24 experiments, as given in
Tables IX. and X., are added.

3 N 2

460 DR. C. CHREB, EXPERIMENTS ON ANEROID BAROMETERS

TABLE XIV. — Law of Fall of Reading under Constant Pressure.

Time, in minutes.

2

i

6

8

10

15

20

25

30

45

60

75

90

100

105

120

135

150

Calculated. . . .

40

52

61

67

73

85

95

103

110

127

142

154

165

171

174

183

191

199

Observed from —

ExDts. 33 to 35 .

11

34

S3

71

72

90

96

10

109

130

139

IB3

164

172

174

179

192

199

1 to 24 .

22

42

51

60

73

After the first 6 minutes the agreement between the calculated and observed
values could hardly be better. The observed values at 2 and 4 minutes are con-
spicuously lower than the calculated, but this discrepancy is least in the case of the
24 experiments, which, on account of their much greater number, should give a more
reliable mean than the 3 special experiments Taking either set of figures, however,
one would most naturally draw the conclusion that the formula (8) does not apply
exactly when t is small. Another explanation is, however, possible. On an average,
an aneroid's reading at two successive intervals would be taken as unchanged if the
true alteration were less than -005 of an inch, and this unquestionably was likely to
happen frequently in the case of the readings at t = 0 and t = 2. That the falls
observed, during at least this interval, suffered from this or some similar cause, can
hardly be doubted, for it is most improbable, as would appear from the 3 special
experiments, that the fall in the first 2 minutes should be less than the fall in the
second 2. Again, as all the operations, such as the reading of the aneroids and
gauge, occupied some time, it is possible that the method of procedure adopted really
tended to curtail the first 2-minute interval. It is certainly a little suggestive that
the values calculated for t = 2, 4, 6, are so nearly identical with those observed in
the first 24 experiments at t = 4, 6, 8, respectively.

§ 19. In connection with §§ 16 to 18 several considerations should be borne in
mind. The change of pressure prior to the stationary stage is gradual. The
circumstances are not analogous to that of a wire suddenly loaded. Thus, the fact
that the fall of reading in a given time is proportional to the pressure range can
hardly be regarded as merely a special case of the ordinary after-effect law. As we
shall see very clearly later, the phenomenon is largely dependent on the way in
which the pressure is reduced. A preliminary stoppage on the way down, or a
change in the rate of reduction of pressure, each exert an important influence ; the
one affecting the magnitude of the fall, the other its law of variation with the time.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

461

Recovery after Pressure Cycle.

§ 20. As already stated, it is customary for an aneroid to read lower on completion
of a pressure cycle than at its commencement. Let the departure from the original
reading be called the deficiency, and let D< represent the deficiency when t minutes
have elapsed after return to the original pressure. Thus D0 — which we may call
the original deficiency — is the difference between the descending and ascending
readings, answering to the pressure of 30 inches.

According to the first 24 experiments, D0 is, at least approximately, proportional
to the pressure range. The evidence for this conclusion is summarised in Table XV. ;
the figures are deduced from the mean values calculated from the several experiments
over the same range.

TABLE XV. — Value of D0 X (1800/range) in inches.

Aneroid No.

Range.

1.

2.

3.

4.

30 inches to 26

15

13

16

13

24

19

12

17

9

„ 21

19

15

29

11

18

17

, .

9

„ 15

16

..

10

;

The abnormal entry 29 for aneroid No. 3 is really due to the occurrence of
appreciable permanent changes of zero in the course of 2 experiments over the
range 30-21 inches. On one of these occasions, after a full day's rest, the aneroid
read no less than '08 inch lower than before the experiment.

The irregularity in the other results is due partly to the comparative fewness of
the experiments, and partly to the fact that the pressure at the end of the experi-
ment was in reality to some extent different from that at the start. It was usual, in
fact, to treat the atmospheric pressure at the time being as 30 inches, unless it
exceeded this value by several tenths of an inch.

§ 21. In the 24 experiments the aneroids were read at intervals of 5, 10, 15, 20,
60, 120, and 1440 minutes after the return to atmospheric pressure. The similarity
of the phenomena for the several aneroids and the several ranges is brought out
more clearly by tabulating D,/D0 than D( itself. This is done in Table XVI.
Initially, of course, i.e., when t =. 0, D(/D0 is unity.

462

DR. .0. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

TABLE XVI. — Recovery after Conclusion of Pressure Cycle, Values of D,/D0.

Range.

Aneroid
No.

t =

5.

10.

15.

20.

60.

120.

1440.

r

1

•74

•65

•52

•39

•22

•13

•17

2

•65

•65

•50

•40

•25

•20

•25

30 inches to 26 <

3

7-2

•64

•48

•44

•32

•16

•20

|

4

•62

•57

•57

•52

•43

•52

•19

L

Mean

•68

•63

•52

•44

•31

•25

•20

r

1

•84

•76

•60

•56

•44

•48

•16

2

•81

•69

•62

•44

•25

•31

•25

„ 24 J

3

•83

•70

•57

•48

•39

•39

•13

1

4

•92

•75

•67

•50

•58

•75

•17

L

Mean

•85

•73

•62

•50

•42

•48

•18

r

1

•74

•68

•66

•61

•42

•32

•03

2

•77

•77

•68

•58

•45

•35

•16

„ 21 ^

3

•86

•81

•74

•69

•53

•48

•29

]

4

•81

•71

•67

•57

•43

•48

•14

I

Mean

•80

•74

•69

•61

•46

•41

16

r

1

•84

•73

•76

•67

•49

•40

•24

18 <^

4

•87

•74

•61

•57

•26

•22

-•09

I

Mean

•86

•74

•69

•62

•38

•31

•08

r

1

•79

•70

•66

•58

•32

•23

•04

15 4

4

•82

•79

•70

•64

•39

•36

•33

I

Mean

•81

•75

•68

•61

•36

•30

•19

In dealing with Table XVI. one must bear in mind the smallness of D0 in the
shorter ranges, especially in the case of aneroid No. 4. The law of variation of a
quantity of the size '03 or '04 of an inch is difficult to ascertain with great exactitude
unless one can read to much less than '01. The figures, as they stand, certainly
suggest that in its early stage the recovery was distinctly more rapid in the case of
the shortest range than in any other. Since, however, there is no trace of a like
phenomenon in the next shortest range, its reality is at least doubtful, and I should
be disposed to attribute it to experimental errors.

Over the other ranges the law of recovery is unquestionably almost, if not exactly,
identical ; and if, instead of grouping the results under pressure ranges, one had
grouped them under the several aneroids, one would have found that the mean
results were almost identical for the four. Thus, in deducing the law of variation, I
combined the results from all the aneroids, utilising all the ranges except the shortest.
The mean values thus obtained are compared in Table XVII. with values calculated
from the formula

D,/D0 = (1 + t X -1357)-

(9),

AT EEW OBSERVATORY AND THEIR DISCUSSION.

463

where t represents as before the time elapsed in minutes since the return to atmos-
pheric pressure.

TABLE XVII. — Law of liecovery, Observed and Calculated Values of D/D0.

t

=

0.

5.

10.

15.

20.

60.

120.

1440.

D//D0 observed . .
„ calculated .

1
1

•83

•825

•74
•729

•67
•664

•59
•616

•41

•442

•37
•349

•15
•143

The agreement is certainly closer than I anticipated from the use of any formula
with only two arbitrary constants. Of course I was led to try — '369 in the index
of (9) from having already found + '369 in the index of (8) ; but this does not affect
the significance of the fact that formulae with the same numerical index should so
closely reproduce the depression and recovery phenomena.

The expectation that (9) would apply to all aneroids treated as the 4 were, could,
of course, only be justified by very wide experiment. As will be shown later, the law
of recovery certainly depends on the rate and type of the pressure changes.

Effects of Temperature.

§ 22. Aneroids are usually " corrected for temperature," i.e., there is a compensating
arrangement to prevent the reading altering when the instrument is exposed to
varying temperature at ordinary atmospheric pressure. The experimental aneroids,
tried as usual at three temperatures at atmospheric pressure, appeared correctly
compensated. It would appear, however, from the special experiments, that this is
not incompatible with imperfect compensation at lower pressures.

Temperature is likely to influence several parts of the mechanism, and that
possibly in more than one way. It presumably alters the elasticity of the vacuum
box and iron spring, and causes slight changes in the dimensions of the apparatus.
The compensation works by producing slight curvature in a lever, but as the position
of the lever varies with the pressure, a curvature that suffices at one pressure may
be insufficient at another. Again, temperature might influence the after-effect
phenomena, and so modify the reading in a variety of ways.

I thus investigated the influence of temperature on the readings with pressure
descending, on the fall at the lowest pressure, and on the differences of the descending
and ascending readings. The experiments bearing most directly on the question
consisted of a group, Nos. 51 to 55, made on five consecutive days, May 10 to 14,
1897. No. 51 was preliminary, and was not utilised in the calculations. Of the

464

DR. C. CHREB, EXPERIMENTS ON ANEROID BAROMETERS

others, Nos. 52 and 55 were made at a temperature of 50° F., Nos. 53 and 54 at a
temperature of 81° F. Use was made of the room employed for testing chrono-
meters, the temperature of which is readily controlled. Information as to tempe-
rature effects is also obtainable from the other special experiments, because, being
taken at all seasons of the year in an ordinary room, they answered to considerably
varied temperatures.

The investigation of the influence of temperature on the descending readings is
complicated by the fact that the index error of an aneroid is a somewhat variable
quantity. To get rid, so far as possible, of this uncertainty, I subtracted from all the
readings a constant equal to the error observed at 30 inches. This is eqxiivalent to
making the instrument correct at the start of each experiment. The excesses of the
readings so modified over the true pressures at lower points of the range I shall call
the corrected errors.

Table XVIII. shows the algebraic excess of the corrected errors from the mean of
experiments 52 and 55 over the corrected errors from the mean of experiments
53 and 54.

TABLE XVIII. — Corrected Errors temperature 50° F., less Corrected Errors

temperature 81° F.

Pressure in inches.

A 'A

29

28

27

26

25

24

23

22

21

No. 2 . .

•005

•00

•04

•05

•055

•095

•09

•11

•13

,,3

•01

•02

•055

•045

•065

•085

•085

•105

•12

,,4 . .

•015

•02

•03

•045

•005

•095

•08-5

•09

•11

,,8 . .

•02

•01

•035

•055

•055

•085

•075

•105

•115

Mean for 4 aneroids

•01

•01

•04

•05

•06

•09

•08

•10

•12

Mean corrected error

•012

•006

•013

•012

•012

•01 *>

•012

•01 ^

•niQ

30 — pressure

There is here a very decided fall of reading accompanying rise of temperature, and
the fall appears to increase directly as the pressure interval measured from 30 inches.

§ 23. Similar conclusions follow from the experiments as a whole. For the two
aneroids Nos. 2 and 3 there were 22 experiments over the range 30-21 inches at the
normal rate. These I divided into two groups, the 11 colder in the one, the 11
hotter in the other, and found the mean corrected errors for the two groups separately.
Aneroid No. 4, for some reason, changed its behaviour during a long rest between
experiments Nos. 46 and 47. Prior to the change, however, there were 25 experi-
ments in which the pressure was lowered to or below 21 inches. Twelve of these
with temperatures from 77° to 71° formed the hotter group. The colder group

AT KEW OBSERVATORY AND THEIR DISCUSSION.

465

included the rest, but half-weight only was attached to each of two experiments in
which the temperature was 70°. The treatment applied to these groups was the
same as for aneroids Nos. 2 and 3. Aneroid No. 1 was somewhat erratic almost from
the start, and the results from it are not reproduced : they showed clearly enough
the same general phenomena as in the others. The results for aneroids Nos. 2, 3,
and 4 are given in Table XIX. The temperatures quoted are the means for the
colder and hotter groups.

TABLE XIX. — Corrected Errors, influence of Temperature.

Aneroid No. 2.

Aneroid No. 3.

Aneroid No. 4.

Pressure

<B - •

O hH

o 1 £

% -

g .

g'2

e .

§

t

in inches.

0> °

H , w

£00

c 3

SOQ

B 0

c

'•r

56" F.

75° F.

a <-'

o ' 2

56° F.

75° F.

o 2

; 01

iS o

61° F.

74° F.

£ 2

£

i

8g t,

^ ' — '

££ t-

rS

IS t.

&£

PH

SU

S-8

5 ,.!,

(5

1

0

Ico

•CO

•n

29

-•005

-•021

+ •016

+ •016

-•012

-•028

+ •016 +-016

+ •040

+ •041

-•001

-•001

28

-•019

-•037

+ •018

+ •009

-•029

-•062 +-033 +-01 7

-•020

-•029

+ •009

+ •004

27

-•049

-•087

+ -038

+ •013

-•029

-•083 +-054

+ •018

-•037

-•047

+ •010

+ •003

26

-•026

-•069

+ •043

+ •011

-•047

-•092 +-045

+ •011

+ •004

-•on

+ •015

+ •004

25

-•045

-•096

+ •051

+ •010

-•059

-•111 +-052

+ •010

+ •022

-•008

+ •030

+ •006

24

-•053

-•127

+ •074

+ •012

-•089

-•159 +-070

+ •012

-•030

-•060

+ -030

+ •005

23

-•077

-•163

+ •086

+ •012

-•130

-•205 + -075

+ •011

-•004

-•038

+ •034

+ •005

22

-•136

-•232

+ •096

+ •012

-•186

-•276

+ •090

+ •011

-•053

-•095

+ •042

+ -005

21

-•198

-•305

+ •107

+ •012

-•214

-•304

+ •090

+ •010

-•074

-•117

+ •043

+ -005

The experiments being numbered in chronological order, the arithmetic means of
the experiment numbers in the several groups were, for aneroids Nos. 2 and 3,
colder 38, hotter 37 ; for aneroid No. 4, colder 24, hotter 27. This alone would
suffice to show that the phenomena cannot be ascribed to any gradual change in the
aneroids.

§ 24. I next consider the possible influence of temperature on the fall of reading
at the lowest pressure. Taking the 16 earliest experiments in which pressure was
reduced at the normal rate to 2 1 inches, and then maintained steady for 1 0 minutes,
I divided them into four groups, as follows : —

Group I., Experiments Nos. 6, 7, 18, 19.

II., „ „ 29, 30, 31, 32.

„ III., „ „ 34, 36, 39, 40.

„ IV., „ „ 41, 42, 43, 44.

Temperature did not vary much between individual experiments of the same
group.

VOL. CXCI. — A. 3 O

466

DE. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

Table XX. gives the mean temperature for each group, and the mean falls of
reading at the intervals 2, 4, 6, 8, and 10 minutes after the lowest pressure,
21 inches, was reached. To avoid decimals, the unit chosen is the ^-Q of an inch. •

TABLE XX. — Fall at Lowest Pressure (21 inches), unity = '0025 inch.

Group.

l|
||

Aneroid No. 1.

Aneroid No. 2.

Aneroid No. 3.

Aneroid No. 4.

Time.

Time.

Time.

Time.

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

H

I.

o

6

13

16

17

19

4

11

12

13

16

6

12

13

16

19

•2

6

6

6

8

IT.

5()|

3

7

11

18

18

4

7

10

16

17

4

9

10

15

18

0

2

4

6

6

III.

75

. r

. .

2

6

12

13

15

4

12

16

21

24

1

2

4

5

6

IV.

72|

3

8

12

16

18

4

8

11

16

20

1

3

7

11

12

The results in the last two groups for aneroid No. 1 are omitted as not comparable
—owing to repair of the instrument — with the earlier ones.

In Table XX. there is no certain indication of any temperature effect.

In the special temperature experiments, Nos. 52 to 55, the observed mean falls
in the colder and hotter groups were as follows : —

TABLE XXI.— Fall in 10 minutes at Lowest Pressure (21 inches) in inches

(Experiments 52 to 55).

Experiments.

Temperature.

Aneroid.

No. 2.

No. 3.

No. 4.

No. 8.

.

52 and 55
53 and 54

50" P.

81° F.

•035
•035

•065

•05

•03
•025

•04
•025

There being only two experiments in each group, it would be unsafe to base on
this any more definite conclusion than that if there is any temperature effect it is
certainly not large.

§ 25. The influence of temperature on the differences of the descending and
ascending readings next calls for consideration. The experiments bearing most
directly on this point are the special ones, Nos. 52 to 55. The results derived from
them are given in Table XXII., unity representing 1 inch.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

467

TABLE XXII. — Mean Sums of Differences Descending less Ascending Headings

(from Experiments 52, 53, 54, 55).

Aneroid.

Experiments.

Temperature.

Sum for
4 aneroids.

No. 2.

No. 3.

No. 4.

No. 8.

52 and 55

50° P.

0-96

1-35

1-19

0-98

4-48

53 and 54

81° F.

0-98

1-54

108

0-88

4-48

Here there is certainly no suggestion of any temperature influence.

The great majority of the later experiments had some peculiarity, such as a
stoppage, which prevents their use for investigating the influence of temperature on
the sum of the differences. I thus confine my attention to the earliest 24 experi-
ments, subdividing those over each range into a colder and a hotter half.* The
mean sums of the differences, in inches, are given in Table XXIII.

TABLE XXIII. — Mean Sums of Differences Descending less Ascending Readings

(Experiments 1 to 24).

Aneroid.

Range
30 inches to 26.

Range
30 inches to 24.

Range
30 inches to 21.

Range
30 inches to 18.

Range
30 inches to 15.

Temperature.

Temperature.

Temperature.

Temperature.

Temperature.

57° F.

67° F.

66°i F.

72° F.

62°i F.

70°£ F.

68°i F.

72° F.

69° F.

74°i F.

No. 1

•207

•217

•530

•625

1-375

1-550

2-750

2-890

„ 2
„ 3

„ 4

•170
•217
•143

•177
•210
•143

•400
•500
•290

•480
•740
•315

1-230
1-745

•785

1-225
1-825
•755

1-555

1-620

2-920

3-175

Some of the data, more especially those for the range 30 to 24 inches, suggest an
increase in the sum of the differences as temperature rises. In the range 30 to 26
inches, however, where the temperature difference was greatest, there is no trace of
temperature effect. In the two longest ranges the temperature difference was small
and the experiments very few.

In the four experiments dealt with in Table XXII. the deficiency in the reading

* In the case of the range 30 to 26 inches the two groups consist of the three hotter and three colder
experiments, the odd experiment being left out of account. The hotter were also the earlier experi-
ments in this instance.

3 o 2

468 DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

on return to atmospheric pressure was exceptionally small in the two high tempera-
ture experiments ; there was, however, no corresponding phenomenon in the case ol"
the experiments dealt with in Table XXIII.

Secular Change in Aneroids.

§ 26. Variations in the reading of an aneroid at a standard pressure, say 30 inches,
may arise from two causes — true secular change of zero, and elastic after-effect. In
practice it is by no means easy to separate the two causes with absolute accuracy ; it
is desirable, however, that the theoretical distinction should be clearly grasped.
When a low pressure is maintained for a long time, the interval required for the
after-effect to disappear on return to the standard pressure is, as Mr. WHYMPER has
found, correspondingly great. This increases, of course, the difficulty of tracing the
true secular change. In all but the latest experiments at Kew Observatory there
was little trace of after-effect when 24 hours had elapsed since the return to
atmospheric pressure. It is thus comparatively easy to trace the true secular change
in the experimental aneroids, with at least a close approach tp accuracy. This is
done in Table XXIV. The quantity tabulated for each aneroid is the error or
algebraic excess of its reading over the true pressure, as registered by the mercury
gauge, at the commencement of experiments preceded by at least 40 hours' exposure
to atmospheric pressure. Experiments during the same month were grouped
together, and the mean of the errors found.

All the aneroids except No. 4 had a zero error to commence with.

AT KEW OBSERVATORY AND THEIR DISCUSSION,

469

TABLE XXIV. — True Secular Change of Zero, error in inches.

Date.

Aneroid.

No. 1.

No. 2.

No. 3.

No. 4.

No. 7.

No. 8.

1895.

April . . . .

-•09

-•05

-•23

May . . . .

-•05

+ '13

-•23

•00

June . . . .

-•02

+ •16

-•39

+ •05

July . .

-•02

+ •16

-•54

+ •09

August . . .

-•11

+ •17

-•54

+ •10

September

-•19

+ •19

-•51

+ '11

October . . .

-•25

+ '13

-•59

+ •11

November . .

-•01

• •

..

+ •07

December . .

-•14

+ •12

-•59

+ •05

1896.

January ....

-•34

+ •10

-•62

+ •04

February . . .

-•39

+ •09

-•61

+ •06

March ....

-•34

+ •09

-•62

+ •06

April

-•29

+ •12

-•59

+ •08

May

— •42

+ '13

-•57

+ •08

June

-•56

+ •15

-•56

+ •10

July . .

— •67

+ •16

— •57

+ •13

August ....

-•71

+ •14

-•58

+ •14

December . . .

-•69

+ •14

-•56

+ •16

1897.

April ....

, .

. .

+ •09

-•05

-•04

May

+ •15

-•55

+ •12

+ •06

+ •01

August ....

••

+ '15

+ •07

+ •03

October ....

..

+ •02

-•08

-•03

November .

, t

•00

-•21

-•06

December .

••

-•04

-•23

-•08

1898.

March ....

• •

+ •08

-•60

-•04

-•30

-•11

A horizontal line indicates the break down and repair of the instrument ; readings above and below

it are not comparable.

The changes were certainly in a good many instances discontinuous. Thus aneroid
No. 1 experienced at least 9 sudden changes of reading of "1 inch or more, 3 of
these being rises and 6 falls. No. 2 had only one sudden change so large as this.
It occurred in May, 1895, and amounted to + '18 inch. No. 3 had two sudden falls
of appreciable magnitude, both in June, 1895. No. 7 had two sudden changes
exceeding '1 inch, one a rise, the other a fall. In the case of Nos. 4 and 8, except
on the occasion of the former breaking down, there was no sudden change as large as
•1 inch. The constancy of zero in the former aneroid was especially notable, because
it experienced more variations of pressure than any of the others. The cause of the

470 DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

discontinuities is not known. In some instances they may have been brought
about by a somewhat too vigorous tapping preparatory to taking a reading. Some-
times they occurred during exposure of the aneroids to atmospheric pressure. Unless
a discontinuous change were of a size considerably larger than '01 of an inch it
would be difficult to demonstrate its existence. It is quite possible, in fact, that all
the changes in the experimental aneroids were discontinuous. There is certainly
no clear indication of a general tendency in the zeros of these aneroids to alter in a
definite direction with age. The fact that discontinuous changes may not unlikely
arise from time to time in the zero of. an aneroid, shows the absolute necessity of
periodic comparisons with a mercury barometer.

§ 27. Apart from shift of zero, it is conceivable that secular change might occur in
at least two ways. There might exist something equivalent to gradual alteration
of elastic moduli, whereby changes of reading answering to definite pressure changes
would either increase or diminish. Quite independently, there might be change in
the qualities whereon depend the elastic after-effect, manifesting itself in the
alteration of such quantities as the sum of the differences of the descending and
ascending readings in a given pressure cycle.

To examine into the first possibility I took note of the size of the corrected errors
(see §22) in the descending readings of a number of the experiments. The earliest
24 experiments were dealt with first. Aneroids Nos. 2 and 3 were used in 16 of
these experiments. In the earlier 8 the pressure 24 inches was reached 5 times,
the pressure 21 inches only twice ; the later 8 experiments differed only in that the
pressure 24 inches was reached no more than 4 times. Subtracting the mean
corrected error at each inch of pressure in the earlier group from the corresponding
mean error in the later group, we obtain the algebraical increase in the error (or
rise in the reading) during the interval.

Aneroid No. 4 was exposed to the whole 24 experiments. In the earlier 12 the
pressure 24 inches was reached 9 times, 21 inches 6 times, 18 inches 4 times, and
15 inches twice ; the later 12 experiments differed only in that the pressure 24
inches was reached but 8 times. The mean corrected errors in the two groups
of experiments were dealt with exactly like the corresponding errors in aneroids
Nos. 2 and 3. Aneroid No. 1, being so much more erratic than the others, is not
discussed. The data obtained from the other three aneroids are recorded in
Table XXV.

AT KBW OBSERVATORY AND THEIR DISCUSSION.

471

TABLE XXV. — Corrected Errors from later less corrected Errors from earlier of

first 24 Experiments.

Pressure in inches.

29

28

27

2G

25

24

23

22

21

20

19

18

17

16 15

No. 2

-•001

-•018

-•Oil

-•010

-028

-•074

-•095

-•085

-•07

„ 3

-•002

-•017

-•018

-•024

-•022

- -029

-•025

-•05

-•04

„ 4

-•on

-•021

-•020

-•014

-•010

-•012

-•on

-•01

-•008

-•02

-•01

-•012

+ •02

+ -03 + -035

The table shows pretty decisive evidence of a slight fall in the readings of Nos. 2
and 3 as time advanced. In their case the differences in the table increase on the
whole fairly regularly as pressure falls, a phenomenon at least consistent with a
diminution of true elastic moduli. With No. 4 the figures suggest a slight fall in
reading down to the pressure of 18 inches, but a rise at lower pressures. Both con-
clusions, however, are very doubtful. In only 4 of the experiments was the pressure
carried below 18 inches, and in the two earlier temperature averaged 5° F. higher
than in the two later. As we have seen in §ij 22 and 23, the difference in tempera-
ture would tend to relatively depress the readings in the earlier experiments. Again,
while the entries in the last line of the table are persistently negative between 29
and 18 inches, they show no tendency to increase numerically as pressure falls. In
aneroid No. 4, as might be conjectured from Table XIX., the scale was not very
regular near 30 inches, so that the apparent index error, answering nominally to
30 inches, but in reality to a pressure which might be slightly greater, or slightly
Jess, was exposed to a fictitious variation, liable to influence the corrected errors at
lower points of the range.

§ 28. To throw more light on the general question, I calculated the mean corrected
errors shown by aneroids Nos. 2 and 3 in the years 1895, 1896, and 1897 separately,
confining my attention to the part of the scale most used, and including only those
experiments in which pressure was lowered at the normal rate. Particulars are
given in Table XXVI.

TABLE XXVI. — Corrected Errors, Yearly Means.

Aneroid No. 2.

Aneroid No. 3.

Year.

Number of
Experiments.

Mean
Temperature.

29

28

27

26

29

28

27

26

inches.

inches.

inches.

inches.

inches.

inches.

inches.

inches.

1895

18

G4°F.

-•008

-•022

-•060

-•038

-•012

-•031

-•037

-•045

1896

13

68° F.

-•016

-•031

-•072

-•049

-•020

-•048

-•063

-•074

1897

7

64° F.

-•009

-•027

-•067

-•054

-•020

-•044

-•051

-•074

472

DR. C. CHREB, EXPERIMENTS ON ANEROID BAROMETERS

In accordance with § 22 a trifling + correction ought to be applied to the figures
for 1896 to allow for its 4° excess of temperature over 1895 and 1897. On the whole,
there appears to be satisfactory evidence of a slight secular lowering of reading in the
two aneroids, such as would accompany alteration of a true elastic modulus ; but it
certainly did not increase to any sensible extent after the end of the first year.

In pushing the inquiry further in the case of aneroid No. 4, I took the 25
experiments already dealt with in Table XIX., but divided them into an earlier and
a later half, treating experiments Nos. 16 and 17 as but one. The results appear in
Table XXVII.

TABLE XXVII. — Mean Corrected Errors in Aneroid No. 4.

1

Pressure in inches.

Tempera-

ture.

29.

28. 27.

|

26.

25.

24.

23.

22.

•21.

Earlier half . .

68° F.

+ •042

-•023 i — -043

-•006

+ •001

-•052

-•031

-•079

— •103

Later half

07° F.

+ •039

— •025 . — -041

-•001

+ •012

-•038

-•012

-•068

-•088

Later less earlier

••

-•003

-•002 : +-002

+ •005

+ •011

+ •014

+ 019

+ •009

+ •015

At the higher pressures the apparent difference between the earlier and later
series is microscopic ; at the lower pressures the figures would point to a secular rise
in the corrected errors ; but, as there is no visible tendency in this to increase at the
very lowest pressures, its true character is somewhat doubtful. Probably the only
• safe conclusion to be drawn is that if any real change occurred in aneroid No. 4 it
certainly was very small.

§ 29. For investigating possible secular change in the differences of the descending
and ascending readings, the data are somewhat limited, because few of the later
experiments were altogether of the normal type. The most suitable data are
utilised in Table XXVIII. , which gives the sums of the differences of the descending
and ascending readings in experiments of the normal type separated by appreciable
intervals of time. When two experiments are combined, the mean sum is taken to
the nearest '01 of an inch.

AT KEW OBSERVATORY AND THEIR DISCUSSION. 473

TABLE XXVIII. — Sums of Differences, Descending less Ascending Readings.

Range 30 inches to 26.

Range 30 inches to 24.

Range 30 inches to 21.

Range 30 inches to 18.

Aneroid No.

Aneroid No.

Aneroid No.

Aneroid No.

Experi-

Experi-

Experi-

Experi-

ments.

ments.

ments.

ments.

1.

2.

3.

4.

1.

2.

3.

4.

1.

2.

3.

4.

1.

4

1,2

•22

•14

•13

3,4

•63

•47

•68

•33

6,7

1-48

1-18

1-83

•74

8,9

2-85

1-52

12, 13

•21

•18

•24

•14

20,21

•61

•44

•63

•30

18, 19

1-45

1-28

1-70

•81

16, 17

2-80

1-65

23, 24

•21

•16

•21

•14

36, 39

. ,

1-15

1-67

•72

56

1-09

1-67

•73

The blanks are due to some uncertainty in the readings of No. 3 on one occasion,
and to the later experiments with No. 1 not being comparable with the earlier.
Between experiments No. 6 and No. 36 there intervened fully a year, and between
No. 36 and No. 56 nearly a second year.

The table does not, I think, afford any clear proof of a change in the sum of the
differences. If any change occurred, the evidence is in favour of its being a
decrease.

In all the experiments utilised in Table XXVIII., the aneroids had been exposed
to atmospheric pressure for at least two days previously. This introduces a difference
between these experiments and five of the ordinary type, Nos. 52, 53, 54, 55. and 59,
which were preceded by experiments over the same range, 30-21 inches, on the pre-
vious day. The difference in this respect is probably the chief reason for the relative
smallness of the mean sums of the differences of the descending and ascending readings
given by these five experiments, viz., '96 for No. 2, 1'43 for No. 3 and '66 for No. 4.

Influence, of Rate of Change of Pressure.

§ 30. In ordinary mountain ascents the rate of change of pressure is much slower
than 1 inch in 5 minutes ; it is thus important to know how the various phenomena
are modified when the rate of change of pressure is reduced. A variety of experi-
ments were devoted to this investigation. In eight of these, Nos. 25, 26, 27, 28,

37, 38, 57, and 58, pressure was altered at the rate of 1 inch in 10 minutes, or
half the normal rate, with a stoppage of the normal length, 10 minutes, at the lowest
point. In experiments 27 and 28 the pressure range was 30-15 inches, in the other
six instances it was 30-21 inches.

The two pairs of experiments, 37, 38 and 57, 58, were each preceded and followed
by an experiment of the normal type over the same range. The experiments 36, 37,

38, 39 were taken on consecutive days, July 6-9, 1896 ; the experiments 56, 57,
VOL. CXCI. — A. 3 P

474

DR. C. CHRBE, EXPERIMENTS ON ANEROID BAROMETERS

58, 59 were taken on May 19, 21, 25, and 26, 1897. During either of these two
groups of experiments the condition of the aneroids may reasonably be regarded as
constant, except in so far as earlier experiments of a group may have affected the
later. The temperature was practically the same for all the four experiments of
either group ; so that the corresponding observations are peculiarly well fitted for
investigating the influence of the rate of reduction of pressure on the descending
readings. The general nature of the phenomena will be found, I think, sufficiently
illustrated by Tables XXIX. and XXX., which give the corrected errors at the lowest
point of the range.

TABLE XXIX. — Corrected Error at Lowest Point.

Experi-
ment.

Tempera-
ture,
Fahren-
heit.

Aneroid No. 1.

Aneroid No. 2.

Aneroid No. 3.

Aneroid No. 4.

Time to fall an.
inch.

Time to fall an
inch.

Time to fall an
inch.

Time to fall an
inch.

5 mins.

10 mins.

5 mins.

10 mins.

5 mins.

10 mins.

5 mins.

10 mins.

-•16
-•16

36
37

38
39

Mea

O

73

74
75
76

ns . . .

-•23

• •

-•22

-•27
-•19

-•29
-•34

-•32

-•32

-•30
-•29

-•32
— •30

"

-•11
-•14

-•225

-•23

-•315

-•32

-•295

- 31

-•125

-•16

TABLE XXX. — Corrected Error at Lowest Point.

Experiment.

Derature,
hrenheit.

Aneroid No. 2.

Aneroid No. 3. Aneroid No. 4. Aneroid No. 7. Aneroid No. 8.

Time to fall an
inch.

Time to fall an Time to fall an
inch. inch.

Time to fall an Time to fall an
inch. inch.

<n
EH

5 mins.

10 mins.

5 mins.

10 mins. 5 minw.

10 mins.

5 mins.

10 mins.

5 mins.

10 mins.

56
57
58
59

Mef

O

66
66
64
63

ins.

-•27
_-22

-:29
-•27

-•30
-:20

+ -06
-•31

-•28
+ •07

+ :04
+ •05

-'.07
-:03

-:05
-•06

-•03
•'60

-:04 -
-•02

-•245

-•28

-•25

-•295 +-065

+ •045

-•05

-•055

-•015

-•03

In both tables the mean corrected error is more negative, or the aneroid reads
lower, when the rate of change of pressure is reduced. The difference between the

AT KEW OBSERVATORY AND THEIR DISCUSSION.

475

two rates appears larger in Table XXX. than in Table XXIX. ; this is mainly due,
however, to experiment No. 59, and very probably arises from the fact that the time
interval between this and the previous experiment was shorter than the average.

Taking the figures as they stand, we should conclude that the mean reading at
21 inches was lowered by only '02 of an inch when the time of reducing the pressure
was increased from 45 to 90 minutes. A.t first sight, the smallness of this difference
seems truly remarkable, because after lowering the pressure to 21 inches at the
normal rate we should, in accordance with Tables IX. and X., have observed a mean
fall of '02 of an inch in some 4 minutes.

The influence of a still slower rate of pressure was tried in a very recent group of
experiments, Nos. 71, 72, 73, 74, and 75, carried out on March 16, 18, 22-23, 29-30,
and April 1, 1898. Of these, Nos. 71, 72, and 75 may be regarded as of the normal
type, except that the stoppage at the lowest pressure, 21 inches, lasted 2 hours.
Nos. 73 and 74 differed from these only in that all the pressure intervals were nine
times longer. These last experiments lasted of course nearly two days, the length of
stoppage at the lowest point being chosen so as to make this feasible without night
work. In reality, it was found convenient to reduce the stoppage in experiment
No. 74 by 45 minutes, and the corresponding curtailment, 5 minutes, was made in
the case of No. 75. I have disregarded this trifling difference, treating Nos. 71, 72,
and 75 as identical, and Nos. 73 and 74 as identical. From these experiments I have
calculated, and give in Table XXXL, the differences between the mean corrected
errors throughout the range for the slower rate (1 inch in 45 minutes) and the faster
rate (1 inch in 5 minutes). The results are given to the nearest '01 inch.

TABLE XXXI. — Mean Corrected Error Slower Rate, less Mean Corrected Error

Faster Rate.

Pressure in inches.

A 'J

29.

28.

27.

26.

25. 24.

23.

22.

21.

No. 2 . .

+ •01

•00

-•01 -'02

-•02 -'03

-•04

-•04

-•06

„ 3 . .

+ •01

-•01

-02 -'03

— •02 —-04

-•05

-•07

-•08

„ 4 . .

+ •02

+ •01

+ •01

•00

+ •01 -00

-•01

-•01

-•02

„ 7 . .

•00

-•01

•00

+ •01

-•01 -'01

-•03

-•04

-•04

„ 8 . .

•00

-•01

•00

-•01

-•01 -'02

-•03

-•03

-•04

Means

+ •01

•00

•00

-•01

-•01 --02

-•03

-•04

-•05

Here, as in Tables XXIX. and XXX., there is, at least at the lower parts of the
range, a lowering of the reading accompanying reduction in the rate of fall of pressure.
Also the mean lowering of the reading at the lowest point, '05 of an inch, is notably

3 P 2

476

DR. C. CHEBE, EXPERIMENTS ON ANEROID BAROMETERS

greater than in the two previous tables. As temperature was about 4° lower in the
experiments at the slower rate, its influence must have tended if anything to reduce
the apparent difference.

As we should have anticipated a priori, the differences in Table XXXI. are greatest
in the case of the two aneroids Nos. 2 and 3, which showed the largest after-effect
phenomena.

Taking both the Tables XIX. and XXXI. into consideration, we see that the fall of
temperature naturally experienced in mountain ascents, and the reduction in the rate
of change of pressure relative to that of the ordinary test, must occasionally neutralise
one another's effects ; but the consequences, however satisfactory, can hardly, I think,
be justly ascribed to design. Though the differences exhibited in Table XXXI. are
certainly quite appreciable, they are very much less than would naturally be antici-
pated from experiments in which pressure is reduced rapidly and then maintained
constant for a lengthened period.

§ 31. The influence of the previous rate of reduction of the pressure on the fall of
reading at the lowest point is hardly capable of being satisfactorily ascertained from
the earlier groups of experiments, because with a stoppage of only 1 0 minutes we
have to do with changes of reading so small relative to the probable error of observation
that a large number of experiments are required to give results of much value. In
the last group of experiments, Nos. 71 to 75, however, the stoppages were sufficiently
long to produce considerable falls of reading. To avoid decimals I have applied a
common large multiplier, and give the results in Table XXXII.

TABLE XXXII. — Fall of Heading at Lowest Pressure (21 inches).

Rate 1 inch in 5 minutes.

Rate 1 inch in 45 minutes.

Aneroid.

Time in minutes.

Time in minutes.

2.

5.

10.

15.

20.

30. ; 60.

120.

18.

45.

90.

1080.

No. 2

6

12

24

30

30

44 54

68

9

12

18

69

„ 3 .

6

16

28

34

42

48 68

86

6

6

12

75

„ 4

0 4

8

12

15

24 32

36

3

3

6

33

„ 7 .

2 8

14

18

18

28 36

40

6

6

12

39

» 8 •

0

0

4

10

12

20 32

36

6

6

12

42

Sums .

14

40

78

104

117

164 222

266

30

33

60

258

The agreement between the falls in 120 minutes in the one set of experiments and
in 1080 minutes (120 X 9) in the other is remarkably close. A comparison of the
falls at the three shorter intervals, 2, 5, and 10 minutes in the one case, and 18, 45,

AT KEW OBSERVATORY AND THEIR DISCUSSION.

477

90 minutes in the other, though naturally showing conspicuous irregularities, points
to the same general conclusion. It is clear that the creep, or fall of reading, at the
lowest point is largely dependent on the rate of the previous change of pressure. A
slow rate of change affords opportunity for simultaneous creep, but at the same time
it reduces the tendency to creep at lower pressures.

If we may judge from Table XXXII., supposing the time taken to produce a given
fall of pressure to be increased in a given proportion, then the time required for the
subsequent creep to attain a certain value is likewise increased in the same or nearly
the same proportion. If the proportion were exactly the same in the two cases one
would hardly, I think, expect to meet with the differences in the corrected errors
shown by Tables XXIX., XXX... and XXXI.

§ 32. In treating of the influence of the rate of change of pressure on the differ-
ences between the descending and ascending readings, I shall consider first the final
group of experiments Nos. 71 to 75. The following Table XXXIII. shows the ratio
of the mean difference of the descending and ascending readings at each point of
the range from the two experiments at the slower rate (1 inch in 45 minutes) to
the corresponding mean from the three experiments at the faster rate (1 inch in
5 minutes).

TABLE XXXIII. — Ratios Differences Descending less Ascending Headings,

slower : faster.

Pressure in inches.

, . ,

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

No. 2 .

1-02

1-05

1-03

1-08

1-14 1-08

1-05

1-08

1-19

•94

„ 3 .

•87

1-01

1-04

•97

I- 11 -98

1-00

•99

1-04

•87

„ 4 .

•92

•99

•99

1-03

1-14

•99

1-07

Ml

1-24

•81

,, 7 .

•98

•90

1-01

1-15

1-09

1-18

1-03

•94

rn

1-33

„ 8 .

1-17

•92

•94

•96

1-08

1-10

1'14

1-07

1-06

•98

Means .

•99

•97

1-00

1-04

I'll

1-07

1-06

1-04

1-13

•99

The mean of the means is I "04.

We can hardly avoid drawing the conclusion that, provided all the time intervals
in the cycle be altered in the same proportion, the sum of the differences between the
descending and ascending readings, and the law of variation of the differences
throughout the range, are either absolutely unaffected or very nearly so. The
departure of the mean ratio in Table XXXIII. from unity is mainly due to experiment
No. 72. The interval allowed for recovery after the immediately preceding
experiment was possibly insufficient. The agreement between experiments 71 and
73 and between experiments 74 and 75 could hardly have been improved. In the

478

DR. C. CHREB, EXPERIMENTS ON ANEROID BAROMETERS

latter pair, for instance, the sums of the differences of the descending and ascending
readings were as follows : —

Aneroid.

No. 2.

No. 3.

No. 4.

No. 7.

No. 8.

Experiment No. 74 (rate 1 inch in 45 minutes)

f.' K

„ >, fo „ „ „

1-36
1-42

1-93

1-88

•84
•85

1-14
1-12

1-15
1-24

In devising the earlier groups of experiments, for comparing the two rates 1 inch
in 10 and 1 inch in 5 minutes, I did not at the time foresee the laws of the
phenomena, and so allowed equal lengths to the stoppages at the lowest point in the
two cases. The results are notwithstanding of considerable interest. Taking the
experiments over the range 30-21 inches, in order to allow for alterations in the
aneroids, I grouped them as follows :—

Group.

I.

II.

III.

Experiments Xos.

6, 7, 18, 19
36, 39
56, 59

nutes.

1 inch in 10 minutes.

9

25, 26
37, 38

57, 58

Taking Group III. for example, the differences of the descending and ascending
readings at each inch were summed for the aneroids Nos. 2, 3, 4, 7, 8 in experiments
57, 58, and the ratios taken to the corresponding sums in experiments 56, 59. The
preponderance thus allowed to aneroids Nos. 2 and 3, in which the after-effect was
largest, is of little consequence, as all the aneroids showed the same general
phenomena. The results appear in Table XXXIV.

TABLE XXXIV.-

-Ratios Differences Descending less Ascending Readings
(l inch in 10 : 1 inch in 5 minutes).

Pressure in inches.

Group.

Mean for

21.

22.

23.

24.

25.

26.

27. 28.

29.

30.

group.

I.

•94

•84

•86

•90

•84

•87

•90

•91

•84

•81

•87

11.

•62

•79

•87

•90

•83

•84

•88

1-04

•77

•75

•83

III.

i-oo

•91

•81

•96

•98

•98

1-07

•86

1-01

•97

•95

Means .

•85

•85

•85

•92

•88

•90

•95

•94

•87

•84

•88

AT KEW OBSERVATORY AND THEIR DISCUSSION.

479

The ratios would seem to be somewhat above their average near the centre of the
range, and below it at both extremities ; but there are hardly sufficient experiments
to justify any too positive conclusion. That the mean ratio is less than unity is
absolutely clear ; and there can be but little doubt that this is due to the fact that
the stoppage at the lowest point in the case of the slower experiments was not
doubled along with the other time intervals.

I likewise compared experiments Nos. 27 and 28 over the longest range, 30-15
inches, at the slower rate, 1 inch in 10 minutes, with earlier experiments, Nos. 10,
11, 14, 15, over the same range at the normal rate. Only one aneroid, No. 4, was
available for calculation, owing to erratic behaviour in No. 1. It will thus suffice to
mention that the results obtained were analogous to those in Table XXXIV., and
that for the ratio of the sums of the differences of the descending and ascending
readings at the two different rates I found

case 1 inch in 10 minutes
case 1 inch in 5 minutes

In both cases, as already stated, only 10 minutes stoppage was allowed at the
lowest pressure.

§ 33. The influence of the rate of change of pressure on the rate of recovery at the
end of a pressure cycle was less exactly ascertained. After experiments in which
the pressure changes are slow, the variations observed in the reading during the first
hour or two of the recovery are so small as to be largely affected by small errors of
reading. After 12 or 24 hours recovery, readings are affected by various sources of
uncertainty, e.g., change of temperature or atmospheric pressure, and possible
permanent shift of zero.

Table XXXV. compares the recovery following pressure cycles a.t the two rates
1 inch in 10 and 1 inch in 5 minutes.

The data for the normal rate are taken from Table XVII. , and D,/D0 has the same
meaning as there. The data for the slower rate were obtained from the experiments
25, 26, 37, 38 over the range 30-2 L inches.

TABLE XXXV. — Recovery after Pressure Cycle, Values of D(/D0.

Interval in minutes.

Rate in cycle.

0.

5.

10.

15.

20.

60.

1 inch in 5 minutes

1

•83

•74

•G7

•59

•41

„ 10 „

1

•87

•83

•72

•74*

•49

* If No. 4 omitted, only '69. For some reason, reading recorded for No. 4 showed on two occasions
apparent fall of '01 inch in interval 15 to 20 minutes.

480

DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

The number of experiments at the slower rate was insufficient to give very smooth
results, and it would be waste of time to fit a formula to them. We should not be
justified in regarding the coincidence between the data under 10 and 20 minutes in
the second line with those under 5 and 10 minutes in the first as more than
accidental. At the same time there seems conclusive evidence that the rate of
recovery becomes slower when the rate of the previous pressure changes is reduced.

This is further illustrated in Table XXXVI. , which compares the recovery in
experiments Nos. 73 and 74 with that exhibited in experiments N'os. 71, 72, and 75.

TABLE XXXVI. — Recovery after Pressure Cycle, values of D</D0.

Pressure rate 1 inch in 5 minutes

Pressure rate 1 inch in 45 minutes

(stoppage at lowest point 2 hours).

(stoppage at lowest point 18 hours).

Aneroid.

Time in minutes.

Time in minutes.

0.

5.

10.

15.

20.

120.

210.

0.

15.

30.

45.

1020.

2460.

No 2 .

1

•89

•85

•78

•70

•29

•20

1

•94

•88

•83

•65

•47

,, 3 . .

1

•91

•84

•81

•74

•38

•27

1

•96

•88

•80

•(30

•44

„ 4.

1

1-00

•92

•85

•85

•67

•20

1

1-14

1-14

1-00

•14

•00

!! 7 .

1

•89

•83 ; -89

•87

•67

•13

1

•87

•87

•86

•50

•38

„ 8.

1

•80

•65

•65

•60

•50

-•06

1

•85

•77

*-80

•38

•31

Means .

1

•90

•82

•80

•75

•50

•15

'

•95

•91

•86

•45

•32

The uncertainties referred to above as unavoidable in a small number of experi-
ments of this kind are conspicuous in Table XXXVI. ; but as to the recovery being
very much slower in the experiments with the slow pressure change there is no
room for doubt. That it is exactly or almost exactly nine times as slow in the one
case as in the other is a conclusion for which the evidence is insufficient ; but there
is at least nothing to demonstrate the contrary. The observations in the case of
the experiments at the slower rate would naturally have been taken at the
intervals 45, 90, 135, &c., minutes after return to atmospheric pressure, but daylight
did not last long enough to admit of this. The extreme slowness of recovery after
experiments such as Nos. 73 and 74 is a somewhat serious obstacle to research.

Effects of Stoppage.

§ 34. The influence of a stoppage on the readings at lower pressures is shown by
three groups, each of three experiments, in all of which pressure was reduced at the
rate of 1 inch in 5 minutes— in most cases with a subsidiary stoppage— to 15 inches,
and maintained at that point for an hour.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

481

In the first two groups there was in two out of the three experiments a subsidiary
stoppage of 1 hour at a pressure above the lowest. In the third group there was a
subsidiary stoppage of about 26 hours' duration in all three experiments. As the
order of the experiments is of significance, I give particulars : —

Experi-
ment.

Date.

Stoppage
at—

Experi-
ment.

Date.

Stoppage
at —

Experi-
ment.

Date.

Stoppage
at —

1897.

inches.

1897.

laches.

1897.

inches.

62

Oct. 11

18 and 15

65

Nov. 15

21 and 15

68

Dec. 21 j 16 and 15

1898.

63 „ 13

15

66

,, 17

15

69

Jan. 5 18 and 15

64, „ 15

21 and 15

67

„ 19

18 and 15

70 „ 20

21 and 15

As experiments Nos. 63 and 66 were of the normal type, so far as descending
readings are concerned, I have used them as a standard to which to refer the others.
To see, for instance, how an hour's stoppage at 18 inches influences the reading when
the pressure 15 inches is reached, I compare the mean error at 15 less the mean
error at 18, prior to the stoppage, in experiments 62 and 67, with the mean
difference between the errors at 15 and 18 in experiments 63 and 66. For instance,
in aneroid No. 4, I found : —

from experiments 62 and 67, mean error (before stoppage) at 18 = -|- '01,
62 and 67, „ .. .. at 15 = -f '015,

and so error at 15 less error at 18 = + '005 ;

from experiments 63 and 66, mean error . . . . at 18 — + '01,

„ 63 and 66, „ .. .. at 15 = -f '04,

and so error at 15 less error at 1 8 ~ + '03.

Thus an hour's stoppage at 18 has lowered the reading at 15 by

•03 — '005, or '025 inch.

Again, to determine the influence of the above stoppage on the total creep which
takes place as pressure is being lowered from 18 to 15, I compare the mean error at
15 less the mean error at 18, subsequent to the stoppage, in experiments 62 and 67
with the difference between the mean errors at 15 and 18 in experiments 63 and 66.
For instance, in aneroid No. 4,

from experiments 62 and 67, mean error (after stoppage) at 18 = — '06,
62 and 67, „ .. .. at 15 = + '015,

and so error at 15 less error at 18 = + '075 ;
YOL. cxci, — A. 3 Q

482 DR. C. CHRBB, EXPERIMENTS ON ANEROID BAROMETERS

and this is greater than in experiments 63 and 66 by

•075 — '03, or '045 inch.

Thus an hour's stoppage at 18 diminished the total creep between 18 and 15 by
•045 inches.

These examples should suffice to explain Table XXXVII.

TABLE XXXVII.

Stoppage.

Consequent lowering of reading, in
inches, at 15 inches.

Consequent diminution of
total creep, in inches, between stop-
page and 15 inches.

At

For

No. 4.

No. 7.

No. 8.

Mean.

No. 4.

No. 7.

No. 8.

Mean.

inches.

hoii rs.

21

1 '015 '025

•005

•02

•04

•05

•05

•05

18

1 . -025 -015

•05

•03

•045

•06

•04

•05

21

2C

•04 '04

•07

•05

•08

•10

•06

•08

18

26

•12

•135

•14

•13

•06

•065

•06

•06

16

26

•185

•185

•19

•19

•035

•045

•04

•04

§ 35. In arguing from a limited number of experiments allowance must be made for
accidental coincidences. The general conclusions to be drawn from Table XXXVII.
seem however prett}r obvious.'5" A stoppage has two effects, tending somewhat to
neutralise one another. It lowers the reading, but while doing so it diminishes the
tendency to further creep as pressure is further lowered. For instance, stoppage for
2G hours at 18 inches lowered the reading at 18 inches in aneroid No. 4 by '18 inch,
but lessened the subsequent creep between 18 and 15 inches by '06 inch. Conse-
quently on arrival at 15 inches the aneroid read only '12 inch lower than if there
had been no stoppage at 18 inches. When the stoppage occurred at 21 inches, and
lasted only an hour, the diminution in the creep between 21 and 15 inches nearly
obliterated all influence on the reading at 15 inches. With increased duration of
stoppage the direct lowering of reading tended more distinctly to predominate.

The lower the pressure at which stoppage occurs, the greater influence does it
exert in both directions. To realise this one should notice that stoppage at 21 inches
influenced the creep over the 6 inches 21 to 15, whereas stoppage at 16 inches
influenced the creep over a single inch only.

§ 36. The way in which the influence of a stoppage tends to disappear the further
the pressure is lowered below the stationary point, is brought out most clearly by a
consideration of the phenomena observed after the lowest pressure is attained.

In treating of this I have utilised, in addition to experiments Nps. 62 to 70, two

* At the same time, so far as concerns phenomena occurring daring actual change of pressure, it
would be difficult to distinguish between a temporary rise in true elastic moduli and a diminished
tendency to creep.

AT KEW OBSERVATORY AND THEIR DISCUSSION..

483

earlier experiments, Nos. 60 and 61, made respectively on August 17 and 20, 1897.
Of these 60 agreed in type with 63 and 66, while 61 was identical with 62 and 67.
Taking the means of the observed falls in reading during stated intervals of exposure
to 15 inches pressure in experiments 60, 63 and 66 as standards, I have found the
ratios borne to these by the corresponding mean falls in the other groups of experi-
ments during both the subsidiary stoppage and the stoppage at 15 inches. The
results appear in Table XXXVIII. The other data in the table are of the following
character : —

"Mean ratio" signifies the mean deduced from the values at the seven intervals
5, 10 ... 60 minutes. " Theoretical ratio " is what the value should have been accord-
ing to the law that the creep in a given interval of time is proportional to the
pressure interval measured from 30 inches. For instance, in experiments 64 and 65
the reading during the subsidiary stoppage at 21 inches should according to the law
have fallen (30-21) -~ (30-15), or '60, times as much in any given interval as it
did in the same interval during exposure to the pressure of 15 inches in experiments
60, 63 and 66. In reality the fall in the former case was on the average only '58 of
that in the latter. Accordingly the presumption is that the tendency to creep in
experiments 64 and 65 was less than in the three standard experiments in the
ratio 58 : 60. This result is utilised in calculating the true influence of the stoppage
at 21 inches on the fall of reading during the subsequent stoppage at 15 inches.
According to the actual observation figures the mean ratio of the falls at 1 5 inches in
experiments 64 and 65 to the falls in the standard experiments was '93, but on the
assumption that this suffered from the same cause as the creep at 21 inches, it should
be multiplied by 60/5B, in order to eliminate the difference between the aneroids'
mean state during the two groups of experiments. The result of this multiplication
is termed the " Corrected ratio."

TABLE XXXVIIL-

-Eatios to Falls of Reading (in same intervals) at 15 inches
in Standard Normal Experiments.

»

Time in minutes.

Experiments.

I7all of

Mean
ratio.

Theoretical Corrected
ratio. ratio.

j

5.

10.

15.

20.

30.

45.

60.

64, 65

inches.
21

•57

•79

•55

•47

•55

•55

•CO

•58

•60

64, 65

15

1-07

•96

•88

•97

•87 -85

•91

•93

. .

•96

61, 62, 67

18

•62

•71

•65

•76

•70

•76

•77

•71

•80

. .

61, 62, 67

15

•71

•85

•77

•85

•81

•79

•83

•80 |

•90

68

16

not ob

serve

d

68

15

•29

•00

•20

•17

•08

•07

•10

•13

• •

69

18

•86

•75

•85

•78

•76

•71

•65

•77

•80

. .

69

15

•43

•33

•33

•33 -42

•39

•39

•37

. .

•38

70

21

•57

•50

•65

•6ti

•55 i -60

•65

•60

•60

70

15

•71

•66

•85

•78 -72

•71

•72

•74

. ,

•74

; t .

3 Q 2

484 DR. C. CHEEE, EXPERIMENTS ON ANEROID BAROMETERS

We should conclude from the table that an hour's stoppage at 21 inches lowered
the creep during the subsequent stoppage at 15 inches by only 4 per cent. An hour's
stoppage at 18 inches was distinctly more effective, but still not very serious.
Stoppage for 26 hours had a much larger influence, reducing the creep at 15 inches
from 100 to 74 when it occurred at 21 inches, from 100 to 38 when it occurred at
18 inches, and from 100 to 13 when it occurred at 16 inches.

According to the table :

Influence of an hour's stoppage at 21 _ _ jt

Influence of an hour's stoppage at 18 10

Influence of 26 hours' stoppage at 21 26

• • — • "T^T T~ 4^.

Influence of 26 hours' stoppage at 18 62

This suggests that the influence of a stoppage on the creep at a lower stationary
pressure may be expressible as a product of two factors, one a function of pressure
only, the other having for sole variable the duration of the subsidiary pressure.

Again, taking the three experiments when the subsidiary stoppage lasted 26 hours,
it will be noticed that the mean ratios, "13, '38, and 74, are to one another nearly in
the proportion 1:3:6, which corresponds to the pressure intervals 16 to 15, 18 to 15,
and 21 to. 15 inches.

Whether we have to do here with chance coincidences or with physical laws it
would be impossible to say without further somewhat elaborate experiments.

§ 37. Before leaving Table XXXVIII. we may note its bearing on the laws
established in §§ 16 to 19 for change of reading under steady pressure. The data
deduced from the subsidiary stoppages at 21 and 18 inches, and the stoppages at
15 inches in the three standard experiments, are in good agreement with the laws
that the fall of reading in a given time during a stoppage, preceded by a reduction
of pressure at a uniform rate, is proportional to the pressure range, and that the
mode of variation of the reading with the time is independent of the range. The
first of these laws is supported by the closeness of the " mean " and " theoretical "
ratios, the second by the absence of any clear tendency in the ratios of the falls at
the specified intervals to either increase or decease as the intervals become longer.

Further support of the first law is supplied by the fact that the mean observed
falls during 26 hours' exposure to the pressures of 21, 18, and 16 inches were respec-
tively '13, '193, and '226 inch, and we have

•13/9 = 'Ol'l, -193/12 = '0161, -226/14 = '0162,

values nearly equal.

A very interesting point is that the ratios borne by the falls at 15 inches pressure,
in the several experiments where a subsidiary stoppage existed, to the corresponding
falls in the standard experiments are, to all appearance, constant throughout the
hour's stoppage at 15. There are, of course, irregularities in the figures, but there

AT KEW OBSERVATORY AND THEIR DISCUSSION.

485

is no clear tendency in the ratios to increase or decrease as the time increases from
5 to 60 minutes. There would thus appear to be a very vital distinction between
the influence of a stoppage and that of a reduction in the rate of fall of pressure. In
both cases there is a reduction in the creep observed in a given time while the
pressure is maintained steady at its lowest point, but the stoppage has apparently
no appreciable influence on the law of variation of the creep with the time, while the
reduction of the rate of fall of pressure has a notable influence in this direction.

§ 38. The influence of a stoppage is also brought out in an instructive way by
comparing the differences of the descending and ascending readings at points below
and at points above the stationary pressure with the corresponding data from experi-
ments of the normal type. Taking us standards the normal experiments 63 and 66,
I have instituted this comparison for experiments 62 and 67, 64 and 65, 68, 69, 70,
and exhibit the results in Table XXXIX. The following example will show how
the results were arrived at and what they signify.

The mean of the experiments 63 and 66 gave, in the case of aneroid No. 4, for the
sum of the differences, descending less ascending readings, from 15 to 18 inches
inclusive '605 inch, and from 18 to 30 inches inclusive 2'635 inches. Now in
experiment 69, with a prolonged subsidiary stoppage at 18 inches, the sum of the
differences at 15, 16, 17 and 18 inches (after stoppage in descent) was "26 inch, while
the sum of the differences at 18 inches (before stoppage in descent) and 19 to 30 inches
inclusive was 3'19 inches. Thus stoppage for 20 hours at 18 inches diminished the
sum of the differences below this point by 100 ('605 — '26) -f- '605, or 57 per cent.,
and raised the sum of the differences above this point by 100 (3'19 — 2'635) -f-2'035,
or 21 per cent.

TABLE XXXIX. — Percentage Change in Sum of Differences, Descending less
Ascending Readings, Below and Above Pressure of Stoppage (+ gain, — loss).

Aneroid.

Stoppage at 21 inches.

Stoppage at 18 inches.

Stoppage

at 16 inches.

For 1 hour.

For 26 hours.

For 1 hour.

For 26

hours.

For 26 hours.

Below.

Above.

Below.

Above.

Below.

Above.

Below.

Above.

Below.

Above.

No. 4
,, 7

,, 8

-7
-6
-6

+ 3
+ 8
+ 4

-24

-22
-20

+ 12
+ 9
+ 13

-19
-23
-19

+ 10
+ 4
+ 11

-57
-48
-43

+ 21
+ 19
+ 17

-83
-68
-83

+ 57
+ 49
+ 51

Means .

-6

+ 5

-22

+ 11

-20

+ 8

-49

+ 19

-78

+ 52

The percentage loss in the sum of the differences below the stationary pressure is,

486

DR. C. CHREE, EXPKRIMENTS ON ANEROID BAROMETERS

as a rule, considerably larger than the percentage gain throughout the higher part of
the range. The latter, however, is a substantial quantity except in the case of the
stoppage at 2 1 inches for an hour only.

§ 39. As pressure was raised above the point where it was stationary in the
descent, the influence on the difference of the descending and ascending readings
gradually diminished, and by the time atmospheric pressure was reached there was
in every case a close approach to the normal deficiency in the reading. In some
cases the deficiency was even below the normal. There is, however, ground for
believing that even after return to the atmospheric pressure the stoppage exercised
some influence. The evidence for this conclusion is contained in Table XL., which
shows the rate of recovery in cases in which there was, and in other cases in which
there was not, a subsidiary stoppage, in addition to the stoppage of an hour at the
lowest point, 15 inches. The results are means for the three aneroids Nos. 4, 7, and 8.

TABLE XL. — Ratio of Deficiency to Original Deficiency, D,/D0.

C4

62

erimcnts

Additional stoppage.

Time interval in minutes —

Nos.

At

For

0.

5.

10.

15.

20.

and GO

inches,
none

hours,
none

1

•88

•81

•61

•58

„ 65

21

l

1

•81

•75

•71

•67

„ 07

18

1

1

•92

•88

•86

•84

68

16

26

1

•89

•90

•87

•83

1

The experiments are, of course, not numerous enough to justify exact numerical
deductions, but it seems quite clear that the recovery in the two last cases in the
table was decidedly slower than in the first case.

§ 40. In addition to the experiments already referred to, there were two of earlier
date, Nos. 45 and 46, in which there was a subsidiary stoppage, lasting 24 hours,
during the descent. In the one the stoppage occurred at 26, in the other at 24 inches.
In both, pressure was reduced to 21 inches, the stoppage there lasting only 10 minutes,
and the ascent was of the normal type. At points below the stoppage the differences
between the descending and ascending readings were very decidedly less than in the
case of the normal experiments Nos. 36 and 39 treated as standards. The effect was
considerably larger when the stoppage occurred at 24 than when it occurred at
26 inches. In neither case, however, was the recovery at the end of the experiment
noticeably slow.

§ 41. After our discussion of the effects of a stoppage during the reduction of
pressure, a brief reference will suffice to a variety of experiments, Nos. 29, 30, 31, 32,
40, 41, 42, 43, and 44, in which there was a prolonged subsidiary stoppage during
the ascent of pressure. In all, the lowest pressure was 21 inches, and the stoppage
there lasted 10 minutes.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

487

Speaking generally, the reading of the aneroids tended to fall or to rise during the
subsidiary stoppage according as the stationary pressure was below or above
26 inches ; but unless it was below 25 or above 27 inches such change of reading
was extremely small.

The influence of the stoppage in these experiments on the readings during the
subsequent rise of pressure, can be traced with nicety, because the part of the cycle
preceding the subsidiary stoppage was strictly of the normal type, and so could be
utilised at once in conjunction with complete normal experiments to fix the standard.
In this way I calculated the percentage increment produced by the subsidiary
stoppage in the sum of the differences of the descending and ascending readings at
points further up the scale. The results appearing in Table XLI. are the means for
aneroids Nos. 1, 2, 3, and 4 ; + signifies an increase, — a diminution. The subsidiaiy
stoppage lasted 24 hours when it occurred at 24 or 26 inches ; in the other cases it
lasted only one hour.

TABLE XLT. — Influence of Stoppage during Ascent of Pressure. Percentage Change
of Sum of Differences (Descending less Ascending Readings) at Higher Pressures.

Percentage change .

Stoppage at

22.

23.

24.

26.

27.

28.

29.

+ 10

+ 17

+ 27

+ 15

-11

-38

-40

A slight rise of reading during a prolonged stoppage is not incompatible with an
increase in the sum of the differences of the descending and ascending readings at
higher pressures. For, as we have already seen, there is reason to believe that a
stoppage tends to make subsequent recovery slower.

Theoretical Deductions.

§ 42. Reasoning from the experimental data I have built up a theory, of a some-
what empirical kind it is true, which reproduces satisfactorily the phenomena
presented by the normal type of test at Kew Observatory.

A lowering of pressure is supposed to produce two independent lowerings of an
aneroid's reading. One is a perfectly reversible or wholly elastic phenomenon, the
other is the source of the after-effect and is termed the creep. It is supposed that
during the constant interval— called 5 minutes for brevity — occupied in the fall of
pressure from 30 — n to 30 — n + 1 inches the magnitude of the creep is
k (n + n + l)/2. Here k is constant, for a particular aneroid, so long at least as the
temperature is unchanged. During the fall of pressure to. say, 30 — p inches, the
accumulated creep thus amounts to

488 DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

During the stoppage for a constant interval— termed 10 minutes for brevity — at
the lowest pressure, the creep augments by kmp, where m is independent of the
aneroid.

During the rise of pressure, matters are more complicated. As pressure rises from
30 — p -j- r — 1 to 30 — p + r inches, it is assumed that in virtue of the rise of
pressure above the lowest point the creep will diminish to the extent U (r — I + r)/2,
or Td (2r — l)/2, where I is constant in the same sense that m is. On the other hand,
the creep tends to increase in virtue of the pressure being still below 30 inches.
During the 5 minutes ascribe to the pressure its mean value 30 — p + r — \. The
length of time elapsed from the beginning of this 5 minutes to the beginning of the
corresponding 5 minutes during the descent is 5r + 10 + 5 (r — 1), or 5 (2r -j- 1)
minutes. Then what I assume is that the creep downwards in this 5 minutes bears
to the creep in the corresponding 5 minutes in the descent the same ratio that the
creep in the interval from 5 (2r + 1) to 5 (2r 4- 2) minutes after pressure becomes
steady bears to the creep in the first 5 minutes after pressure becomes steady. This
does not absolutely assume the creep to be the same in magnitude when pressure is
changing as when it is steady ; but it is, of course, considerably speculative.

In the ordinary Kew test, the time occupied by the experiment is not in excess of
the duration of the steady pressure during the special experiments Nos. 33, 34, 35,
used in constructing Table XIV. In the following calculations I have employed, not
the direct results of these experiments, but the values supplied by the formula based
on them.

§ 43. An example of the calculations will serve, I hope, to illustrate the process.
Take the cycle 30-24-30 inches, supposed of the normal type.

As pressure falls the creep augments by '5k between 30 and 29 inches, by
(1 + 2) k/2 or l~5k between 29 and 28 inches, and so on. Thus the accumulated
creep amounts to '5k at 29, 2k at 28, 4'5& at 27, and so on.

During the stoppage at 24 inches, the creep augments by Gmk. As pressure rises
from 24 to 25 inches we regard it as 24'5, or 5 '5 inches below 30. Also the 5 minutes
occupied by the rise represents the interval 15 to 20 minutes elapsed since pressure
in its descent entered the stage 25 to 24 inches.

Now, by Table XIV.,

fall in interval 15-20 minutes _ 10 _
fall in interval 0-5 minutes 57 ~

Thus, for the increase of creep during this 5 minutes, we take '18 X 5'5k, or I'Ok,
as against a decrease of '5^.

The following scheme shows the magnitude of the creep as the pressure attains
the values specified in the first column.

The common factor k is to be understood.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

489

Accumulated creep.

Pressure

Descending less ascending

in inches.

reading.

Descent.

Ascent.

30

0

6m + 20-3 - 18-01

6m + 20-3 - 18-01

29

0'5

6m + 20-3 — 12-5Z

6m + 19-8 - 12-5Z

28

2-0

6m + 20-2 — 8-01

6m + 18-2 - 8-OZ

27

4-5

6m + 20'0 — 4'5Z

6m + 15-5 — 4-5Z

26

8-0

6m + 19-6 — 2-OZ

6m + 11-0 - 2-OZ

25

12-5

6m + 19-0 — 0-5Z

6m + 6-0 - 0-5Z

24

18-0

6m + 18-0

6m

This gives sum of differences descending less ascending = k(4:'2m + 91 '9 — 45'5Z).

§ 44. As A; is a common multiplier; all quantities such as ratios of differences of
ascending and descending readings to the mean difference, or ratios of sums of such
differences for different ranges, depend on only two constants, I and m. When these
are known, k may be found for any given aneroid from the observed sum of the
differences of its descending and ascending readings over any given range.

For the old Kew observations, summarised in Table I., I found by trial

I = 0-83,
m — 1-3,

employing these values for all the ranges.

To test the theory severely, it seemed expedient to compare with it the observed
results for the law of variation of the ratios of the differences of the descending and
ascending readings to the mean difference over the several ranges. For this com-
parison I have utilised the figures resulting directly from the observations at each
inch of pressure, and not the data in Table II., as the latter arise from a combination
of actual observations at adjacent points of the scale. This should obviate the
suspicion that naturally arises when data compared with theory have been subjected
to any kind of manipulation.

The observed, "0," and the calculated, " C," values appear side by side in
Table XLII. To adequately appreciate the agreement, the reader should remember
that unity in the table answers to the mean difference between the descending
and ascending readings. It thus answers to only about '34 inch even in the range
30-15 inches. In the range 30-26 inches it represents only a tenth of this.

VOL. cxci. — A

490

DR. C. CHEEE, EXPERIMENTS ON ANEROID BAROMETERS

TABLE XLTL— Ratios of Differences of Descending and Ascending Readings to the

r 7 __ Q-g'3

Mean Difference. "0," observed results (1885-91); "C," calculated from |m _ 1<3' '

Range.

Pressure in inches.

30-26
inches.

30-24
inches.

30-23
inches.

30-21
inches.

30-18
inches.

30-15
inches.

0.

C.

0.

C.

0.

C.

0.

C.

O.

C.

O.

C.

i K

0-25

0-28

16

0-52

0-53

17

0-74

073

18

0-34

0-33

0-93

0-90

19

0-63

0-61

1-05

1-04

20

0-84

0-83

1-17

1-15

21

0-37

0-40

1-00

1-01

1-23

1-23

22

0-69

0-72

1-14

1-15

1-27

1-29

23

0-41

0-46

1-00

0-97

1-20

r-24

1-29

1-31

24

f t

0-54

0-50

0-77

0-83

1-15

114

1-27

1-29

1-32

1-31

25

. .

0-88

0-90

1-08

1-08

1-26

1-25

1-29

1'30

1-26

1-28

26

0-35

0-G1

1-17

1-14

1-24

1-23

1-29

1-29

1-25

1-27

1-21

1-22

'27

1-14

1-05

1-26

1-26

1-28

1-28

1-24

1-26

1-19

1-20

1-14

1-13

28

1-51

1-23

1-16

1-25

1-23

1-22

1-18

1-17

1-12

1-09

1-02

1-02

29

1-19

1-19

1-07

1-11

1-11

ros

1-01

1-01

0-93

0-94

0-88

0-88

30

0-82

0-91

0'89

0-85

0-86

0-83

0-81

0-79

0-80

0-74

0-72

0-71

Mean difference

observed -»- calcu-

lated

± -14

± '037

±•024

±•016

±•020

±•013

Representing in

inches ....

±•004

± -004

±•004

±•003

±•005

±•004

By varying I and m one could of course have slightly improved the agreement in
individual ranges.

For instance, the assumption

I = 078,

m- 1-07,

would reduce the mean difference of the observed and calculated values in the
range 30-23 to ± 'Oil, representing ± '002 inch.

§ 45. As an example of the calculation of k, we have for the sum of the differences
of the descending and ascending readings in the cycle 30-24-30 : —

from mean of aneroids in Table I. (first group) '684 inch,

from formula k (42m + 91'9 — 45'5Z).

AT KEW OBSERVATORY AND THEIR DISCUSSION. 491

Putting I = '83, and m = T3, as in Table XLIL, we find

k = '684 -r- (54-6 + 91-9 — 37'8) = '0063.

This is a mean value for the 13 aneroids dealt with as a group in Table I. ;
individuals of the group differed, of course, amongst themselves.

§ 46. In the case of the experimental aneroids, it might appear that m and I are
at once determined by the experiments.

For instance, it might be supposed that

fall of reading in 10 minutes at lowest point
tn * — — • >

fall of reading in 5 minutes at lowest point

recovery in first 5 minutes at end of cycle
fall in first 5 minutes at lowest pressure

If this were so, we should have from the first 24 experiments for the mean of
aneroids Nos. 1, 2, 3, and 4,

m = 73/47= 1-55,
I = '65, roughly.

In the case of tn, if we preferred to use the formula (8), we should have

m = 73/57 = 1-28.

These deductions, however, assume the creep phenomena the same whether
pressure is steady or changing, which may not be strictly true. Further, even if
there were no theoretical objection, the method would not, in practice, be very
satisfactory, owing to the large probable error involved in determining such small
quantities as an increment or decrement of creep in 5 minutes.

Another method that naturally suggests itself is a comparison of the observed
sums of the differences of the descending and ascending readings in three ranges.
This supplies two simple equations involving I and m as unknowns. This method,
however, is in practice no better than the other, as will be recognised on inspection
of Table XLIII. The quantities tabulated there are the ratios borne by the sums of
the differences of the descending and ascending readings over several ranges to the
sum for the range 30-21 inches. The observed values are taken from the first
24 experiments.

3 K 2

492 DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

TABLE XLIII. — Ratios of Sums of Differences to Sum for Range 30-21 inches.

Range.

30-26

30-24

30-21

30-18

30-15

inches.

inches.

inches.

inches.

inches.

•146

•375

1

2-00

3-65

Ratios calculated from 1 = 0*65, m = 1'5 . . .

•145

•370

1

2-08

372

0-83, „ 1-5 . . .

•148

•374

1

2-06

3-67

0-83, „ 1-3 . . .

•144

•369

1

2-08

3-73

i-o, „ i-o. . .

•138

•363

1

2-10

3-79

The difficulty here is that large variations in the values of I and m have but little
influence on the calculated ratios. The differences between the observed and
calculated values in Table XLIII. are in no case in excess of reasonable limits of
experimental error. The agreement is certainly best for I = '83, m = 1'5 and
worst for I = m = 1.

The variation of the ratio of the difference of the descending and ascending
readings- to the mean difference over a range affords a much more sensitive method
of determining I and m. The application of this method shows that, for the special
experiments, I = m = I is in reality a better choice than I = '83, m =• 1'3. The
evidence for this conclusion is supplied by Table XLIV.

TABLE XLIV. — Mean Differences between Observed and Calculated Values of Ratios
of Differences of Descending and Ascending Readings to the Mean Difference
(Experiments 1-24).

Range.

30-26

30-24

30-21

30-18

30-15

inches.

inches.

inches.

inches.

inches.

Mean difference, using 1 -~ 1, 7;;, = 1 .

±•066

±•070

±•057

±•043

±•048

= 0-83, =1-3. .

±•080

±•094

±•063

±•086

±•095

Previous Work Bearing on the Subject.

§ 47. Of investigations into elastic after-effect, more especially in metal wires,
there is an excellent summary in vol. 1 of WINKELM ANN'S ' Handbuch der Physik.'
This I studied, having previously read a number of the papers it refers to, notably
some by KOHLUAUSCH. Of quite recent papers on the subject I have not seen many,

AT KEW OBSERVATORY AND THEIR DISCUSSION. 493

and with one or two exceptions — e.g., a paper by Mr. L. AUSTIN, in ' Wied. Ann.,'
vol. 50, 1893 — the theories they dealt with seemed of a hopelessly involved character
for practical application. I did indeed try one or two" of the formula? proposed, but
conditions so complicated as those of the Kew aneroid test soon led to expressions
of a formidable character. The numerical results I actually reached did not show
a promising accord with experiment, and having regard to the many claims on my
time, 1 did not push the calculations further.

8 48. The following are the only earlier investigations on aneroid barometers,
having any direct bearing on the results of the present paper, that have come under
my notice : —

1. "On the Errors of Aneroids at Various Pressures" (Dr. BALFOUR
STEWABT), ' B.A. Keport,' 1867, pp. 26, 27, of 'Transactions of the Sections.'

2. " An Account of Certain Experiments on Aneroid Barometers made at
the Kew Observatory at the Expense of the Meteorological Committee " (Dr.
BALFOUR STEWART), ' Hoy. Soc. Proc.,' vol. 16, 1869, pp. 472-480; 'Phil.
Mag.,' vol. 37, 1869, pp. 65-74.

3. " How to Use the Aneroid Barometer," by EDWARD WHYMPER ;
London : JOHN MURRAY, 1891.

Of these, No. 1 is a preliminary report apparently of the work more fully dealt
with in No. 2. It mentions the tendency in the reading to fall under continued
exposure to low pressure, and also the general nature of the recovery after return to
atmospheric pressure.

No. 2 commences with a slight reference to the secular variation of zero. It then
describes some tests showing that at atmospheric pressure changes of temperature
have little effect on "well-made compensated" instruments. The writer adds, "I
am unable to say what effect a change of temperature would have at a diminished
pressure."

The main part of the paper describes the phenomena observed in pressure cycles,
30-19-30 inches of the following kind. Pressure was reduced an inch and then
kept steady for 10 minutes, when a reading was taken; it was then lowered a
second inch, and so on. At the lowest pressure there was a stoppage of l£ hours, and
thereafter pressure was raised at the same rate as it had been lowered. The
aneroids were always tapped before reading. The general superiority of large to
small aneroids, and the increase in the tendency to creep when the range is lengthened
are duly noted. The effects of stoppage on the top of a mountain are also
remarked on.

A considerable number of aneroids were examined. The object was apparently,
however, rather to ascertain the conditions under which they behaved fairly con-
sistently than to investigate whether the variation in the readings obeyed any
ascertainable laws.

494 DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

One experiment, at least, was, however, made to ascertain whether the rate of
reduction of pressure was of any consequence. Two aneroids were tried, a comparison
being made of their readings when pressure was reduced at the two different rates,
1 inch in 1.0 and 1 inch in 30 minutes. In one of the aneroids the readings were
sensibly lower all through the range at the slower rate ; in the other the change of
rate had no certain influence.

§ 49. No. 3, which has been already referred to, is divided into parts. Part 1,
treating of " Comparisons in the Field," enumerates changes of zero and of reading
observed in a number of aneroids used at lofty stations, but hardly bears on the
present paper. Part 2, " Experiments in the Workshop," gives numerous details as
to the creep during several weeks' exposure to low pressures. It also records a
considerable number of observations showing the recovery of aneroids on return to
atmospheric pressure after a week's exposure to a lower pressure.

Mr. WHYMPER was apparently unaware of Dr. BALFOUR STEWART'S work, and
regarded the creep at low pressures as a new phenomenon. He noticed that the
rate of creep diminished as the time increased since the low pressure was reached, and
that the recovery on return to atmospheric pressure was most rapid at first. He
seems, however, to have made no serious attempt to study more exactly the laws of
the phenomena. Judging from some remarks on his p. 25, he was discouraged from
doing so by the belief that different aneroids varied very much in the laws they
followed. The fall of reading during the first day's exposure to a low pressure might
vary, he says, from about one-third"" to more than three-fourths of the fall in the first
week.

For evidence of this he refers to a table on his p. 26, giving particulars of the creep
in 29 aneroids during the first day and the first week. The evidence seems to me
somewhat inconclusive. In the first place, there seems no information as to whether
the original lowering of pressure was carried out at an invariable rate. Thus it is
not clear whether an exact comparison is possible between the aneroids which were
exposed to different pressures. Those exposed to the same pressure were presumably
exposed to the same conditions, and in their case there are no variations at all
approaching those mentioned by Mr. WHYMPER. In fact, I notice only two instances
in the whole table in which the creep in the first day was less than the half of that
in the first week ; and in both these cases there was no companion aneroid tried at
the same pressure. In the first instance a 3-inch Watkin aneroid, exposed to a
pressure of 24 inches, is credited with a creep of '111 inch in a day and '273 inch in
a week ; in the second instance, that of a 4|--inch Watkin aneroid, the creeps were
•052 inch in a day and '144 inch in a week. The presumption is that the aneroids
were only read to '01, or at most to '005, of an inch, the third decimal coming from
the reading of the mercury barometer ; in the second instance mentioned above,
a trifling error in the reading would make a large difference in the result.
* A footnote says " less than a fifth. " in some exceptional cases not quoted.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

495

In the other 27 aneroids the creep in the first day bore to that in the first week a
ratio lying between '6 and '9. It is also not unreasonable to suppose that some at
least of the extreme values were appreciably influenced by permanent changes of zero.

§ 50. Notwithstanding the absence of information as to the exact nature of the
operations, Mr. WHYMPER'S experiments are of much interest, inasmuch as they
supply information of an almost unique kind relative to the effects of very prolonged
exposure to low pressures. I have thus examined the results with considerable care,
with a view to seeing what light they throw on the phenomena already described.

There are, in the first place, several tables whose contents afford the opportunity
of determining how far it is possible to represent the creep over a series of weeks
at a low pressure by a single algebraic term Cf, as in (7), § 18.

Thus, on his p. 17 Mr. WHYMPER gives the errors, relative to a mercury
barometer, in six aneroids exposed together for six weeks to a pressure of 22| inches.
Readings were taken at the end of each week. How exactly equal the " weeks "
were is not stated ; but a few hours' variability in such a case would hardly matter.
At the end of four weeks the readings had become, if not absolutely stationary, so
nearly so that the probable error of reading began to be too serious.

In Table XLV. I give the ratios I find borne to the first week's creep by the total
creeps up to the end of the second, third, and fourth weeks; also the values found
for q in the algebraic term by taking separately the mean values of the above
three ratios.

TABLE XLV. — Creep Eatios, calculated from data on Mr. WHYMPER'S p. 17.

[

Aneroid

Mean.

Corre-
sponding q.

No. 1.

No. 2.

No. 3.

No. 4.

No. 5.

No. 6.

2 week's creep/1 week's

1-06

1-10

1-15

1-05

1-12

1-19

1-11

•151

1-13

1-20

1-26

1-21

1-15

1-33

1-20

•166

A,
» )» »

119

1-22

1-23

1-29

1-20

1-31

1-24

•155

The irregularities apparent in the data for some of the aneroids are not surprising,
in view of the fact that the average creep recorded in the fourth week was only
"013 of an inch.

The mean results agree well with the single term formula (7).

The next set of data I have utilised are given on Mr. WHYMPER'S p. 19. They
refer to three aneroids — distinct from those of Table XLV. — exposed for eight weeks
to the pressure of 16 inches. In these the increase of creep can be fairly traced up
to the end of the sixth week. The individual aneroids show somewhat irregular
behaviour. I give the mean results, obtained in the same way as the corresponding
results in our last table. " Iw " stands for " one week's creep, ' and so on.

496 DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

TABLE XL VI. — Creep Ratios, calculated from data on Mr. WHYMPER'S p. 19.

2iv/liv.

3w/lw.

4wjlw.

bio/lie.

Qwj\w.

Mean ratio

1-21

1-29

1-38

1-41

1-48

Corresponding q .

0275

0-232

0-232

0-213

0-219

The mean value for q, viz., '234, is considerably higher than that given in
Table XLV. There is also at least a suggestion that the first two weeks' creep
was, relatively speaking, somewhat greater than according to the single term law.

§51. The last and much the most varied set of data which I have utilised are
taken from tables on Mr. WHYMPER'S pp. 26, 27, and 28. The table on p. 26 has
been already described. That on p. 27 gives with some blanks the errors at the end
of the 1st, 2nd, 4th, and 7th days' exposure in 21 of the 29 aneroids included on
p. 26. The table on p. 28 gives the errors at the end of the first hour, and the end
of the first day, in all the 29 aneroids. Though not apparently mentioned, it is
clear, from the size of the errors specified at the end of the first day, that the three
tables refer to the same experiments. I have thus dealt with them together. The
results of my calculations are given in Table XLVII. for all the aneroids for which
complete sets of readings are given, with the exception of one, of which a reading is
queried by Mr. WHYMPER. In the case of the 5 lowest pressures in the table, the
results are means for the 2 or 3 aneroids exposed to the same pressure.

TABLE XLVII.— Creep Ratios, calculated from data on Mr. WHYMPER'S

pp. 26, 27, 28.

Constant
pressure in
inches.

Number of
aneroids.

1 Hour's

2 days'

4 days'

7 days'

1 Day's

1 day's

1 day's '

1 day's '

14

2

•62

Ml

1-27

1-50

15

2

•58

1-17

1-27

1-42

21

2

•42

1-14

1-32

1-41

22

3

•47

1-17

1-35

1-50

23

2

•63

1-13

1-41

1-64

24

1

•44

1-50

1-92

2-46

26

1

•62

1-14

1-23

1-33

Mean from 13 aneroids "1
Corresponding q . . J
Mean from 12 aneroids 1
Corresponding q . . J

•53
•200
•54
•194

1-17
0-227
1-14
0-189

1-36
0-222
1-32
0-200

1-55
0-225

1-48
0-201

The aneroid exposed to the pressure of 24 inches is one of the two exceptional
ones already referred to. The mean values in the last two lines of the table are
obtained by omitting it.

AT KEW OBSERVATORY AND THEIR DISCUSSION.

497

The agreement between the mean results from the 12 aneroids and those deduced
from the single term

a-196

is extremely close.

There are naturally very considerable irregularities amongst the individual results
in the table, but, with the solitary exception already alluded to, the phenomena
observed at the different pressures are very similar. The aneroids were made mainly
by CASELLA and HICKS, but they varied in diameter from 2 to 4^ inches.

§ 52. Mr. WHYMPER'S data as to recovery, though less varied, are also of interest.
On his p. 30 he gives the original errors, the errors on return to the original
pressure, and those observed 1 day and 1 week later for 36 aneroids, exposed for one
week to pressures varying from 14 to 26 inches. In 12 instances the reading was
higher after a week's recovery than it was prior to the experiment.

On his p. 31 Mr. WHYMPER repeats some of these data for 20 of the aneroids,
adding also the errors observed after only 1 hour's recovery. I have omitted one of
the 20 aneroids, No. 40, because after a week's recovery it read '29 inch higher than
prior to the experiment, the figures suggesting the occurrence of a large permanent
change of zero before the end of the first day's recovery. According to the figures on
p. 31 there was also a permanent rise of zero in three other aneroids, Nos. 42, 22,
and 23. In the last-mentioned case, however, this seems due to a misprint of
•236 for '206 in one of the data, the latter figure appearing on p. 30. In the
case of Nos. 22 and 42 the final reading exceeded the original by so small an amount
it seemed best to retain the aneroids. This retention accounts for the minus sign in
the second entry of the last column of Table XLVIII. When more than one aneroid
was exposed to a given pressure, I give only the mean values of the ratios for the
group. In many cases the aneroids are the same as were dealt with in Table XL VII.

TABLE XLVIII. — Ratios of Deficiencies to Original Deficiency (data from

Mr. WHYMPER'S pp. 30, 31).

Constant

Number

Time.

in inches.

of aneroids.

0.

1 hour.

1 day.

1 week.

14

2

1

•52

•17

•04

15

2

1

70

•11

-•06

18

2

1

•62

•30

•22

20

6

1

•63

•37

•09

22

3

1

•63

•35

•19

23

2

1

•57

•28

•04

24

1

1

•61

•56

•35

26

1

1

•70

•50

•17

Mean from

19

1

•62

•32

•11

VOL. CXCT. — A.

3 S

498 DR. C. CHREE, EXPERIMENTS ON ANEROID BAROMETERS

Evidently, little weight attaches to individual results after a week's recovery, A
small permanent change or a slight initial fatigue in an aneroid would have a large
effect on the final data. Even after a day's rest the results are somewhat erratic.

So far as one can judge from the recovery during the first hour, the lowness of
the pressure to which the aneroids were exposed was immaterial. There is certainly
no indication of the recovery becoming slower as the stationary pressure is more
remote from 30 inches. It must be remembered, of course, that the aneroids
exposed to the different pressures were different, so that individual peculiarities
were not eliminated.

Comparing Table XL VIII. with Tables XVII. and XXXVI. , it will be seen that
the rate of recovery in Mr. WHYMPER'S experiments was slower than in the case of
the normal Kew experiments, but faster than in those experiments where pressure
was reduced at the rate of an inch in 45 minutes, and where the lowest pressure was
maintained stationary for 18 hours.

§ 53. There are other data on Mr. WHYMPER'S p. 33 relating to the recovery of
22 aneroids, which had been exposed for a week to a pressure of 21 '692 inches.
These data I have not considered, for the reason that the reading during recovery
proved higher than the reading prior to the experiment in

1 aneroid on return to atmospheric pressure,
5 aneroids 14 hours after return to atmospheric pressure,
9 „ 12 days
1" » 30 ,, ,, ,, „ ,,

Such a wholesale tendency to a rise of zero, whatever its cause, would have
introduced great uncertainty into any conclusions drawn, even as to the recovery
during the first 14 hours, the shortest interval for which results are given.

§ 54. Part 3 of Mr. WHYMPER'S pamphlet, dealing with the "Determination of
Altitudes," discusses the best method of utilising the readings of aneroid barometers.
I hardly think that Mr. WHYMPER makes due allowance for the fact that in
mountain ascents the reduction of pressure is very gradual, even at the lower stages.
This allows accommodation to take place, so that on arrival at the summit the
instrument will not behave as it would have done if rapidly transported there.
Again the difference between descending and ascending readings is very different in
aneroids whose readings as pressure falls at a given rate may be equally correct, and
the difference increases ceteris paribus with the length of time spent on the mountain
summit. Thus the advantage to be derived from taking the mean of ascending and
descending readings — a course which Mr. WHYMPER seems to suggest in some
instances — is very problematical.

On his p. 51 Mr. WHYMPER gives an example of a Kew aneroid certificate and
then devotes several pages to a criticism of it. On his p. 54 he says, " In the
absence of directions to the contrary, it may be assumed by persons into whose

AT KEW OBSERVATORY AND THEIR DISCUSSION. . 499

hands similar certificates come that the corrections stated in them are good under
all conditions and for all time." In view of the footnote to the certificate, calling
attention to the probable recovery from the residual error, exhibited on return to
the original pressure, this standpoint could hardly, T think, be justified ; and judging
by a footnote to his p. 52 Mr. WHYMPEB would presumably allow this himself. At
the same time, he apparently considers it a point of view likely to present itself to
travellers, a class as to whose scientific knowledge and general intelligence he is
better qualified to speak than I am. If he is right in his conclusion, a change in the
certificate is certainly desirable.

Before concluding my reference to Mr. WHYMPER'S interesting pamphlet, I would
take the opportunity of explaining that the rate of change of pressure in the
ordinary Kew test is not, as he states on several occasions, about t inch in 2 minutes ;
the actual rate is only about half this.

§ 55. The 75 special experiments — some of a very tedious and exacting character —
on which this paper is mainly based, were carried out with great care and discretion
by Mr. W. HUGO, Senior Assistant at Kew Observatory. In addition to the obser-
vational work, Mr. HUGO reduced all the barometer readings and carried out some
of the subsequent arithmetical operations. The bulk of these and the checking of
the reductions were undertaken by myself.

It is, I allow, anomalous, and from various points of view undesirable, that a
scientific man should have himself performed none of the experiments which he
discusses. When, however, as in the present case, observation is being pushed
to the utmost capabilities of the instruments employed, the absence of preconceived
ideas in the actual observer is a valuable compensation.

3 s 2

[ 501 ]

XII. On the Heal Dissipated by a Platinum Surface at High Temperatures.

By J. E. PETAVEL, 1851 Exhibition Scholar.

Communicated by Lord RAYLEIGH, F. R.S.

Received May 19,— Bead June 9, 1898.

[PLATES 18-23.]

FROM the beginning of the century, and more especially of late years, an almost
uninterrupted series of papers have appeared on the above subject. Unfortunately
few of the authors have reduced their measurements to absolute units, thus rendering
comparison of the results obtained very difficult.

The first part of this paper is to some degree an extension to higher temperatures
and pressures of the work so ably done by Dr. BOTTOMLEY,* whilst the names of
LANGLEY, PASCHEN,t WEBER, J and many others will occur to anyone glancing
through the second and third parts.

With regard to temperature measurements, the present work is mainly founded on
the researches of CALLENDAR and GRIFFITHS, § which have been amply confirmed by
the later investigations of HEYCOCK and NEVILLE. ||

Full references to the above papers are given here once for all, as we shall have
continually occasion to refer to one or other of them.

On the Apparatus Employed.

The preliminary experiments soon showed that to obtain reliable results a very
thick wire must be used for the double purpose of radiator and thermometer, even
if it were at the cost of a slight loss of sensitiveness, the reasons being the
following : —

* ' Phil. Trans.,' A, vol. 178, 1887, p. 429, and 'Phil. Trans.,' A, vol. 184, 1893, p. 593.

f ' Wied. Ann.,' vol. 49, p. 50, 1893 ; vol. 58, p. 455, 1896, and vol. 60, p. 663, 1897.

| ' Physical Review,' vol. 2, p. 112; ' Berichte der Preussischen Akademie,' 1888, p. 933, &c.

§ ' Phil. Trans.,' A, vol. 182, p. 119, 1891.

|| ' Journ. of the Chem. Soc.,' vol. 67, p. 160, 1895 ; same vol., p. 1024; ' Phil. Trans.,' A, vol. 189,
p. 25, 1897. On this subject see also C. M. CLAKKE, ' The Electrician,' vol. 38, pp. 175, 241, 372, and
J. D. H. DICKSON, ' Phil. Mag.,' December, 1897, p. 445.

25.10.98.

502

MR. J. E. PETAVEL ON THE HEAT DISSIPATED BY

1. A thin wire maintained at any temperature above 1200° is subject to a rapid

increase of resistance.

2. It is impossible to obtain consistent results with a thin wire at atmospheric
pressure, owing to the constant fluctuation in temperature due to convection

currents.

3. As pointed out by Professors AYBTON and KILGOUR, with fine wires the
emissivity, referred to unit of surface, depends largely on the diameter, but the
rate of change decreases rapidly as the diameter increases.

The diameter of the wire used was '112 centim., or nearly 1 sq. millim. in cross-
section. The resistance of the thermometers was about '01 ohm at 0° C., or about
one five-hundredth of the resistance usually employed.

Fig. 1.

ST

(Standard Resistance)

To Che Potentiometer

Jo the Potentiometer

The electrical connections can be briefly described as follows : — At C, in fig. 1, is
a set of forty cells, which by the aid of a mercury commutator can be placed in any
desired combination, either series or parallel. This commutator was used to make
the first rough adjustment of the current. R is a variable resistance consisting of
thirteen German silver tubes, which are kept at a constant temperature by a water
circulation. The resistance is designed to carry up to 150 amperes. By two sliding
contacts, one of which serves as a fine adjustment, the current can be regulated to
within '01 of its value. The current through the radiator is measured by aid of the
standard resistance ST, the exact value of which, according to the Keichsanstalt
certificate, is '0099965 ohm at 16'55 C., its temperature coefficient being '0003 per
cent. A small motor was used to maintain a circulation of oil round the wires of
this resistance, the oil being in its turn cooled by a water circulation. By this
means the temperature of the standard was prevented from rising above 25° C.

A PLATINUM SURFACE AT HIGH TEMPERATURES. 503

All electrical measurements have been referred to this standard and to a Clarke's
cell (N 5217 A), its value being taken as 1'4331 volts at 16° C.

Both the current through, and the electromotive force at the terminals of the
platinum wire, were measured on a Wolff potentiometer. In none of the coils did
the error exceed '02 per cent. The temperature coefficient was less than '001 per
cent.

The radiators used were calibrated at frequent intervals, a slight increase in
absolute resistance and in temperature coefficient being noticeable.

One of these sets of readings is given below : —

»

Radiator R^.

R100 = resistance at 100° C ..... '012252
R0 = resistance at 0° C ...... '009907

c = ^"^ ........ '00002345

8= 1-304.

The value of S is deduced from the resistance of the wire in sulphur vapour by
CALLENDAR'S formula,

where t is the temperature in degrees Centigrade,

P _ p

pt = temperature in platinum degrees =• - - » arid

c

S = a constant.

The platinum thermometer has rarely been used to determine temperatures above
1100°. It is therefore necessary to obtain some confirmation of the above formula for
temperatures up to 1500°. For this the melting point of palladium was chosen, and
the value given by VIOLLE,* 1500° C., taken as correct.

A length of about 3 millims. of palladium wire '01 centim. in diameter was
held against the platinum radiator. The temperature being gradually increased, the
palladium soon became viscous and adhered to the platinum, finally fusing at a very
sharply-defined temperature.

To this method there is an obvious objection. With each fusion there is a certain
weight ('0004 gram, or '03 per cent, in this case) of palladium added to the platinum.
This would produce an appreciable change in the electrical constants. To obviate
this difficulty it is necessary to fuse the palladium on a part of the platinum wire, not
between the potential leads, but sufficiently near them to be substantially at the same
temperature.

* ' Comptes Rendus,' vol. 87, p. 983, 1878.

504 MR. J. B. PETAVEL ON THE HEAT DISSIPATED BY

The melting point of palladium calculated in this way from the formulae of
CALLENDAR and GRIFFITHS was for the thermometer B, 1489°, as against 1500° as
determined by VIOLLE.

A second confirmation of the temperature determinations will be found in the fact
that in alt the curves given below, the calculated temperatures agree with those
determined directly by the fusion of platinum or palladium.

Finally, it may be well to state here that although every precaution was taken to
ensure the accuracy of the results, the absolute resistance of the wires used was too
small for me to lay claim to as great a degree of precision as that obtained by
HEYCOCK and NEVILLE in their valuable researches on the melting point of metals.
Up to about 500° I believe the results given to be correct to a fraction of a degree,
but above 1200° it is difficult to prove that they are absolutely exact.

In Table I. will be found a determination of the volume specific resistance of
platinum and palladium. The values obtained for platinum show that the tempera-
tures above 1500° cannot be calculated by the ordinary equations ; these, for
instance, in the case of R2 would give for the melting point of platinum a
temperature more than 100° too low.

Part I. — ON THE EMISSIVITY OF A BRIGHT PLATINUM SURFACE IN AIR

AND OTHER GASES.

Throughout the experiments, the results of which are given in Tables II. to XIII.,
a cylindrical glass enclosure, 5 '8 centims. in diameter and 24 ceutims. in height, was
used. The straight platinum wire, which served at the same time as radiator and
thermometer, was placed in the axis of the cylinder. A platinum wire, 2 millims. in
diameter, bent into the form of a " J " and fused at its lower extremity on to the
thermometer, formed the return lead. This wire, as well as the potential leads, was
kept well to the side of the enclosure. The enclosure was always used with its axis
vertical. This arrangement forms the simplest geometrical disposition of the surfaces
of the radiator and enclosure that can be used with the method adopted, as for
obvious reasons a sphere enclosed in a sphere is out of the question.

A sufficient length was left between the point of contact of the lower potential
lead and the terminal of the radiating wire, for the wire to acquire a constant
temperature.

As pointed out by Dr. J. T. BOTTOMLEY, the low thermal conductivity of glass is
a very serious objection to its use as an enclosure in experiments on emissivity, it
being possible for the inner surface to rise to a temperature very appreciably above
that of the water circulation. Owing to this fact, it was not thought advisable to
take any observations with the radiator at much less than 100° above the enclosure.
At temperatures higher than this a slight error in the estimation of the temperature
of the enclosure would not alter the results.

A PLATINUM SURFACE AT HIGH TEMPERATURES. 5CT5

A large number of experiments were made, but only the most typical results have
been recorded here. The observations are given in the tables according to the order
in which they were taken. In Table HA will be found an illustration of the way in
which all the results were worked out.

The curves relating to air, hydrogen, carbon dioxide, oxygen, and steam are given
in figs. 2, 3, 4, and 5. The pressure in centimetres of mercury is indicated on each
curve. Observations for each of the first three gases are given at three distinct
pressures, for steam and oxygen the emissivity is only determined at a pressure of
76 and 228 centims. respectively.

The degree of purity of the hydrogen used was 97 per cent., the remaining
3 per cent, being, in all probability, air. The carbon dioxide, which was prepared
from calcium carbonate, contained less than 1 per cent, of impurities.

During the course of each experiment the exact height of the barometer and the
temperature of the room, the standard resistance, and the potentiometer, were
recorded. The temperature of the enclosure was also read at short intervals of time
on a thermometer divided in tenths of a degree. These factors have too little effect
on the main point at issue for each of the readings to be recorded here.

In the course of the preliminary experiments it was found that the emissivity, as
far as these results are concerned, is practically independent of the condition of the
glass surface (whether perfectly clean and freshly polished or covered with a thin
layer of lamp-black), and of the exact position of the radiator in the enclosure.

The emissivity does not seem to bear any simple relation to the specific gravity
of the gas. In hydrogen, and to a lesser degree both in carbon dioxide and in
oxygen, the emissivity is greater than in air.

In fig. 2 the rise in emissivity caused by saturating the air with moisture will be
seen. The difference in emissivity is nearly constant throughout the entire range of
temperature, averaging '0001 therm, per second per square centimetre of surface, per
degree above the enclosure. This is all the more remarkable as the average tempe-
rature of the air, and therefore the quantity of water vapour present, naturally
increases with the temperature of the radiating wire.

In fig. 6 some curves will be found giving the relation between the temperature of
the wire and the mean temperature of the gases in the enclosure. These curves were
obtained by using the enclosure itself as a rough form of air thermometer, the
measurements being made at atmospheric pressure. The three gases are here in the
same order as when classed with regard to the experiments on emissivity. For all
the gases included in this study, the rise of emissivity due to any given increase of
pressure is not proportional to the initial value of the emissivity, but is more nearly
constant at all temperatures.

For facility of reference some parts of the hydrogen curves given both in fig. 2 and
fig. 3 (Plate 18), and the curve representing the emissivity in air at three atmospheres,
will be found again in fig. 4 (Plate 19).

VOL. CXCI. — A. 3 T •

506 MR. J. B. PBTAVBL ON THE HEAT DISSIPATED BY

The emissivity in steam at atmospheric pressure is given in fig. 5. The slope of
the curve is much sharper than for any of the gases previously studied. This fact, as
we shall see from the results given below, cannot be entirely accounted for by the
higher temperature of the enclosure.

I hope shortly to undertake a research on the emissivity of platinum in gases at a
much higher pressure. For the present the results are too incomplete to allow any
general theory to be formulated.

The values obtained for the emissivity are to some extent dependent on experi-
mental conditions, such as the diameter of the radiating wire, its position in sp;ice
(whether horizontal or vertical), the dimensions of the enclosure, its absolute tempera-
ture, and the material of which it is made. For the results to be of any general
application, it is necessary to form an idea of the extent to which they are affected
by these divers factors.

Curve I. in fig. 7 (Plate 20) is taken with the radiating wire in the square gun-metal
box described in Part III. of this work. For Curves II. and III. the wire was placed
in the axis of an iron cylinder 2'6 centims. in diameter, 27'5 centims. long. In all
three cases the wire was horizontal.

Curve IV. has already been given in fig. 2 ; it refers, a.s do all the results given
above, to a wire '112 centim. in diameter placed vertically in the axis of a glass
cylinder 5 '8 centims. in diameter.

The difference between the Curves I. and IV. is mainly due to the change of posi-
tion. When the wire is vertical, the part on which the observations are taken is
surrounded by a layer of gas which rises from below after being heated to sub-
stantially the same temperature as the wire itself. When the wire is horizontal, the
hot gas rising from it is replaced by a fresh supply at nearly the same temperature as
the enclosure. The variation between Curves I. and II. is about 10 per cent. It is
a difference of this oi'der that we may expect when the shape, size, and material of
the enclosure are radically changed. In cases, however, where the ratio of the surface
of the radiator to the surface of the enclosure is not kept very small, the total loss of
heat depends very largely on the dimensions of the enclosure.

Curve III. is taken in the same iron cylinder as used for Curve II. In this case
the iron was kept at a high temperature by surrounding it with a jacket containing
boiling sulphur. The temperature of the enclosure, as measured by the platinum
wire subsequently used as the radiator, was somewhat high. This temperature,
namely, 455° C., or ten degrees above the usual boiling point of sulphur, can easily
be accounted for by the direct heating effect of the flame, for the prevention of which
no precautions were taken.

The slope of the curve of emissivity obtained at this temperature is considerably
steeper than that relating to experiments with the enclosure at a lower temperature.

The above-mentioned curves, which are given in fig. 7, all refer to the emissivity
in dry air at atmospheric pressure.

A PLATINUM SURFACE AT HIGH TEMPERATURES. 507

The usual definition of the emissivity of a body, and the one I have adhered to
throughout, may be expressed as follows : — The emissivity is numerically equal to the
number of gramme-degrees, or therms, dissipated per second, per square centimetre of
surface, per degree above the enclosure. The " true emissivity " at any temperature,
t, might be defined as the slope at the temperature t of the tangent to the curve
representing the total heat dissipated plotted in terms of the temperature. In other
words, if h is the loss of heat per square centimetre per second at any temperature, t ,
the " true emissivity " at this temperature is dh/dt.

The Curves II. and III. (fig. 7), if plotted in terms of the true emissivity, would almost
coincide, showing that in this case, at any rate, the value obtained depends to a very
limited extent on the temperature of the enclosure.

A comparison of Curves I. and V. will show the effect of using a wire '6 millim.
instead of 1*12 millims. in diameter.

Finally, in fig. 7, the points marked AA' refer to a platinum wire, 2'005 millims.
in diameter, that is to say, nearly twice the diameter used in all previous experi-
ments. The wire was fused in the gun-metal enclosure, and the results must therefore
be compared with the point B, obtained under identical conditions, but with a wire
1'12 millims. in diameter.

It will be seen that the decrease in emissivity due to the use of the larger wire is,
at this temperature, less than 7 per cent.

The emissivity at the points of fusion of palladium and platinum were deduced
from the readings of instruments of the Weston type, which were afterwards cali-
brated by aid of the potentiometer. The wire was raised to within about 100° or
200° of the melting point. The current was then increased by equal amounts of
\ or £ per cent., and current and electromotive force readings were taken at each
successive rise. The current was noted at the instant the wire fused, the correspond-
ing electromotive force being easily found by extrapolating the last few readings.
0-ver 100 amperes were needed to fuse the 2 millims. wire, and for accurate results to
be obtained it was necessary for this large current to remain constant to within a
tenth of a per cent. This experiment, as many of those previously recorded, was only
rendered possible by the exceptional facilities available at the Davy-Faraday Labora-
tory.

A large proportion of the work done on the subject of emissivity refers to the
cooling of comparatively large bodies. This method was used by DULONG and PETIT,
J. C. NICHOL,* McFARLANE.t C. F. BRUSH,! Dr. J. T. BOTTOMLEY, and many others.
These results cannot serve as points of comparison. Added to the effect of the abso-
lute size of the bodies used, the question of the relative surfaces of the enclosure and
radiator comes here into play. Whenever the ratio of these two surfaces is not very

* ' Proc. Roy. Soc.,' Edin., 1869, vol. 7, p. 206.
f ' Proc. Roy. Soc.,' 1872, vol. 15, p. 90.
J ' Phil. Mag.,' vol. 45, p. 31, Jan., 1898.
3 T 2

508 MR. J. E. PETAVEL ON THE HEAT DISSIPATED BY.

small, its absolute value has a considerable influence on the rate of cooling of the

enclosed body.

SCHLECEKM ACKER'S* results only apply to radiation in a high vacuum, but BOTTOMLEY,
in his researches on the same subject, incidentally gives one value at atmospheric
pressure ; this is for a temperature of 408° C. It works out at nearly one and a-half
times the emissivity for the same temperature as shown in Curve I., fig. 7.

From the resistance given in Dr. BOTTOMLEY'S paper, it is probable that the wire
used was about '3 millim. in diameter. If this be the case, the difference is what
might have been predicted from the results obtained by Professors AYETON and

KlLGOUK.t

The formulae given by them for the connection between the emissivity of the wire
and its diameter are

At 200° C. e= -00111 + -014303d-1.
At 300° C. e = '001135 + •016084C?-1.

The wires were heated in a horizontal position and the diameters measured in
millimetres.

The diameter of the wire used was 44 millims. Inserting this value for d in the
above formulae gives for the value of the emissivity at 200° and 300° C. "00144
and -00150 C.G.S. units- The values given in Curve 1., fig. 7, are '00107 and '00121.
The largest wires used in establishing the above formulae were 14 millims. It is
therefore not surprising if, after extrapolating for a wire over three times this
diameter, the calculated and experimental values differ considerably. This diver-
gence is also in some degree accounted for by the fact that the enclosures used
in the two cases were of an entirely different shape and size.

In a paper recently published, Mr. C. F. BRUSH gives a number of observations
" on the transmission of heat by gases," at temperatures of from 0° to 15° C.
Unfortunately it is impossible to reduce these to absolute units. The relative
value of the emissivity in air and in hydrogen is much the same as that which may
be deduced at some hundred degrees higher from fig. 2. My measurements show
that the emissivity in carbon dioxide is greater than in air at high temperatures,
converging to the same value at 150° C. At about 10° C., according to Mr. C. F
BRUSH, the relative value of the emissivity in these two gases is reversed.

Many formula have been given, principally with regard to radiation in vacuo ;
but I am aware of none that will apply at atmospheric pressure and high tem-
peratures.

Until more is known as to the physical constants of gases at temperatures
ranging from the melting point of silver to the melting point of platinum, it is
doubtful whether any general law can be obtained.

* ' Wied. Ann.,' vol. 26, p. 287, 1885.

t ' Phil. Trans.,' A, vol. 183, p. 371, 1892.

A PLATINUM SURFACE AT HIGH TEMPERATURES. 509

Part II. — A BOLOMETRIC STUDY OF THE LAW OF THERMAL RADIATIQN.

The relative merits of the laws of DULONG and PETIT, and STEFAN, have formed
the subject of a number of researches. It is sufficient to recall the experimental
work of SCHLEIERMACHER * and ScHNEBELi,t which appeared in 1884, and the
theoretical discussion of the subject by FERREL^ some five years later. In 1888
Professor WEBER § proposed a new expression connecting the thermal radiation with
the absolute temperature of the radiator and the enclosure. This expression, as
he showed, accounted satisfactorily for the greater part of the experimental results
then available. It was with the hope that some additional data with regard to the
total radiation at higher temperatures might help the elucidation of the problem that
the present experiments were carried out.

Most of the apparatus employed has been already described in Part I. For the
experiment we are now about to describe the radiating wire was placed in a metal
enclosure kept at a constant temperature by a water circulation. A number of
diaphragms, also provided with water circulations, shielded the bolometer from all
external radiation.

The bolometer used was of a somewhat special construction. Many weeks were
spent in preliminary experiments before the design of the instrument was finally
fixed. It is with a desire to spare this loss of time to others that a somewhat full
description of the bolometer is given here.||

The instrument is symmetrical throughout, one side containing the active platinum
film which is exposed to the radiation, the other side an exactly similar film forming
the second arm of the bridge. For the platinum silver foil from which these films
were made I am indebted to the kindness of Mr. J. S. SELLON, of Messrs. JOHNSON,
MATTHEY and Co.

A thin sheet of platinum was welded to a thick sheet of silver, the two sheets,
protected on each side by a copper plate, were then rolled together and a leaf was
thus obtained, of sufficient thickness to be quite easily handled. This I cut on the
dividing engine into the shape of a grid (as shown in fig. 8, F). It was then placed
in the holder (fig. 8, A), and the silver dissolved in dilute nitric acid. The film of
platinum left was found to have a thickness of '00011 centim., or about twice the
wave-length of sodium light. The active surface has the appearance of a grating,
consisting of fifteen bars, 3 millims. wide, with a space of 1 millim. between each
bar. The span from the upper to the lower edge of the carrier is 56 millims. The

* ' Wied. Ann.,' vol. 26, p. 287.
f 'Wied. Ann.,' vol. 22, p. 430, 1884.
J ' American Journal of Science,' vol. 38, p. 3, 1889.
§ 'Berichte der Preussischen Akademie,' 1888, p. 933.

|| Those interested in this subject will do well to refer to an article by LUMMER -and KOELBAUM
(' Wied. Ann.,' vol. 46, p. 204, 18D2 ; or, ' The Electrician,' vol. 34, pp. 168, 192, 1894).

510 MR. J. B. PBTAVEL ON THE HEAT DISSIPATED BY

practical difficulties in the construction of so thin a film are considerable, and in
designing a new instrument it would be advisable either to reduce the size of the
filuTor to increase its thickness. The very thin platinum, obtained in the way we
have just described, though capable of withstanding a large amount of vibration,
cannot resist the slightest mechanical strain, a touch with a single hair being sufficient
to break it. Great care is needed in the manipulation of the film from the moment
the silver is dissolved off' it until it is screwed up in position in the instrument.
During this time it has to receive an electrolytically deposited coating of platinum
black, be washed several times, and carefully dried. When once in position in the
instrument there is, however, comparatively little chance of an accident.

Fig. 8.

Section a.b.

The desiderata of a good bolometer may be enumerated as follows : —

1. Great sensitiveness.

2. Constancy of zero.

3. Rapidity of action.

The first and third conditions are fulfilled, thanks to the extreme thinness of the
platinum and to its high temperature coefficient. The sensitiveness was found to be
increased by coating the films on one side only with platinum black, and by placing
behind them a nickel mirror. With regard to rapidity of action it was found in the
course of some subsequent experiments that the platinum rises in half a second to
over 90 per cent, of its maximum temperature.

The second condition, namely the constancy of the zero point, is the most difficult
to fulfil. It has already been pointed out that the four arms of the bridge must be
in pairs as nearly as possible identical, not only with regard to resistance, but also
with regard to shape, size, position, and temperature coefficient. But these pre-

A PLATINUM SURFACE AT HIGH TEMPERATURES.

511

cautions are not sufficient. To avoid thermo-electric disturbances it is necessary to
keep all the working parts of the bolometer at the same temperature. To attain this
object the entire instrument was made of metal. The insulation of the films at their
upper and lower extremities where they are secured to the carriers is provided for by
very thin strips of mica. The working parts of the bolometer were protected by two
massive metal covers, D, and D2, fig. 8, fitting one over the other, and between which
room is left for a water circulation.

The variations due to convection currents formed one of the most serious
difficulties. Following the suggestion of Professor DEWAR, the instrument was
constructed to allow of the films being used in a vacuum. As shown in fig. 8, the
double cover is bolted down on to the bedplate, E. The joint, is kept covered with
oil which is poured into the rim, R.

Pig. 9.

BaCtery

A vacuum can only be used for experiments in which the absorption of the quartz
plate, Q, is not a serious objection. The screens, B! and S2, are intended to regulate
the path of the convection currents when the instrument is used under atmospheric
pressure ; they also screen the edges of the film from direct radiation. The two
other arms of the bridge are made of manganin wire, the coils being placed in an
earthenware vessel filled with oil.

The resistance of each of these two coils is 20 ohms, and the resistance of each of
the bolometer films 23 ohms. The connections of the instrument are diagram-
matically shown in fig. 9. The bar, A, is of German silver, and serves to adjust the
zero of the galvanometer. The bar, B, between the potential contacts, c and d, is
divided into 1000 parts of equal resistance, the length between the potential contacts
is 80 centims., the resistance '007730. This bar serves to calibrate the galvano-

512 MB. J. E. PETAVEL ON THE HEAT DISSIPATED BY

meter. By the switch, S, one terminal of the galvanometer can be thrown over from
the zero point of the bar to the sliding contact, E, which has been previously set at
any desired division.

If n be the number of divisions at which the sliding contact is set, n X 2 X '00000773
is the change in resistance of the bolometer film,* which would produce an equal
galvanometer deflection. Where the total radiation received is large, a zero method
can be used, and the change in resistance measured directly by the position of the
slider, E, on the bar, B.

As regards sensitiveness, the deflection obtained with a candle at 1 metre from
the instrument was 360 millims. The total electromotive force on the terminals of
the bolometer was 2 volts, the time period of the galvanometer 10 seconds, and
distance of the scale 1 '3 metres. Each film before being mounted was tested at a
pressure of 30 volts, so that any electromotive force up to 60 volts could be used
without danger, but no advantage would be gained by increasing the pressure above
10 or 15 volts. The main object in view has not been to obtain excessive sensitive-
ness, but to construct an instrument which would give thoroughly reliable results.
In almost all cases a much smaller degree of sensitiveness than that given above has
been found amply sufficient.

The law of thermal radiation may be studied by two entirely distinct methods.
We can either measure the heat lost by the radiating surface when in a nearly perfect
vacuum, or the heat received by the sensitive surface of some type of radiometer.

The determinations by the first method will exceed the true value by some quantity,
w.R, representing the heat carried away by any gas or vapour remaining in the
enclosure.

The numbers obtained by the second method will fall below the actual value of the
radiation by some quantity n.R, where n.R represents the part of the radiation not
absorbed by the irradiated surface.

If R be, at any given temperature, the true value of the radiation, and O, and O2
the observed values obtained by the two methods described above, we have :

I. O, = ct (R + m.R),
II. 02 = c2 (R - nB),

where c, and c.2 are constants depending merely on the experimental conditions and
on system of units chosen.

In Equation I., m represents the ratio between the heat dissipated by convection
and conduction and the heat radiated. Now it is well known that this ratio decreases
as the temperature increases. The slope of the curve of radiation obtained by this
method will therefore always be somewhat less steep than the true curve of radiation.

* The change in the resistance of the bolometer film produced by the radiation being thus known, it
is an easy matter to express the heat received in gramme- degrees. In the present case, however, as we
are only dealing with relative values, little advantage would be gained by the use of absolute units.

A PLATINUM SURFACE AT HIGH TEMPERATURES. 513

Considering now Equation II., we see that here also the second term decreases as
the temperature rises ; for it has been shown that whereas a surface coated with
platinum-black or lamp-black almost totally absorbs all radiation of short wave-length,
it reflects a comparatively large proportion of the infra-red rays. The curve obtained
by any form of radiometer will therefore be steeper than that representing the actual
relation between total radiation and temperature.

The foregoing conclusions may be summed up as follows : —

The rate of change of the total radiation with temperature will be too small when
measured by the heat lost by the radiating body, whereas it will be too large when
measured by the heat gained by an irradiated surface coated with lamp-black or
platinum-black.

Finally we may note that, as both m and n decrease at higher temperatures, the
accuracy of both methods increases as the temperature rises.

Let us now see to what extent the actual experimental work verifies the conclusions
we have just arrived at.

In fig. 10 (Plate 21) the values obtained by the first method are represented by
the results of Dr. J. T. BOTTOMLEY and of SCHLEIERMACHER, whereas the second
method was used by F. PASCHEN and by myself.

To facilitate the comparison of the slopes of the curves, all the results are reduced
to the same arbitrary value at a temperature of 800° C.

We see that Curve No. II., obtained by BOTTOMLEY, lies between the Curves No. I.
and III. obtained respectively by SCHLEIERMACEER and by PASCHEN, or precisely in
what is for theoretical reasons the most probable position of the true curve of
radiation.

I have recently made a series of determinations by aid of the bolometer described
above ; the results within the limits of the probable experimental errors fall on this
same curve.

It seems very probable that Curve II. represents the actual law of thermal
radiation between 400° and 800° C., but even were this not so, we might safely admit
that the true curve must fall somewhere between Curves I. and III., fig. 10. All
expressions therefore giving values outside these limits may be discarded without
further study. The formulae of STEFAN, of DULONG and PETIT, and of ROSETTI, fail
when tested by this criterion. WEBER'S formula, on the other hand, agrees closely
with Curve II. from 400° to 800° C. At all higher temperatures the rate of change
of radiation with temperature calculated by this formula, is certainly too great.

Taking the value of WEBER'S constant at 800° C. as unity and comparing the
calculated results with Curve II., we obtain I'Ol, '96 and '94 for the values of the
constant for 700, 600 and 500 degrees centigrade, showing that the expression
remains nearly correct over this range of temperature.

Table XYI. and fig. 11 give the results of a series of observations on the radiation
from platinum at temperatures ranging from 500° C. to the point of fusion of the
metal.

VOL. cxci. — A. 3 u

514 MR. J. E. PBTAVEL ON THE HEAT DISSIPATED BY

In Table XVII. a few results obtained from the' readings of a thermopile, the
surface of which was coated with lamp black, are recorded. These results, though
much less accurate than those obtained by use of the bolometer, serve as a confirma-
tion of the previous values and show that these values are not affected by the type of
instrument used.

PART III. — ON THE VARIATION OF THE INTRINSIC BRILLIANCY OF PLATINUM

WITH TEMPERATURE.

The flux of light emitted by one square centimetre of platinum at a fixed
temperature has many times been proposed as a standard of light; By VIOLLE*
this temperature is defined as the point of solidification of the metal, whereas
LUMMEB and KuRLBAUMt advocate a lower temperature not far from the melting
point of palladium. It was essentially as a preliminary to the study of these two
standards that the present work was undertaken. The method employed for the
temperature measurements was in every particular the same as used in the first part
of this paper. The experiments were made with wires 1'12 millims. in diameter
enclosed in a gun-metal box 8 X 8'5 X 3 centims., which could be fitted with
diaphragms of any desired shape. The wire under observation was horizontal, the
points of contact of the potential leads being a little more than one centimetre to the
right and left of the edge of the diaphragm, the opening of which was 38 millims. in
length. The walls of the gun-metal casting forming the enclosure were hollow, and
a water circulation maintained the temperature at about 15°C.

As standards of comparison, incandescent lamps were used. These were first left
burning for from fifty to one hundred hours to avoid the initial rapid change in
candle-power and then compared with a number of standard candles ; the usual
corrections being in all cases applied where the candles burnt more or less than the
prescribed forty grains of sperm in ten minutes. The candle-power of these lamps
was finally established from the mean of some hundreds of photometric readings.
While any one of these lamps was in use, the electromotive force at its terminals
was kept constant by the aid of a potentiometer. No photometric reading during
which the electromotive force had varied more than '05 per cent, from its correct
value, was recorded.

Where the lights differ largely in colour, as was the case in most of the observa-
tions recorded in the table, the measurements become influenced mainly by two
physiological phenomena —

(1.) The relative value of two lights of different colour depends on the absolute
illumination of the two surfaces used in comparing them.

* ' Comptes Rendus,' 1884, p. 1032, vol. 98.
t ' Elektrotechnische Zeitschrift,' 1894, p. 475.

A PLATINUM SURFACE AT HIGH TEMPERATURES. 515

(2.) LEPINAY and NICATI* have shown that the intensity of a light is not the
same when measured by the distinctness with which the details of an
object can be seen as when measured by the apparent brightness of a
surface. The form of photometer used is sufficient indication that it is
the latter method which has been adopted for the experiments recorded
here.

In 1879 ViOLLEt published a research on much the same subject as is here dealt
with. His results and mine though in substantial agreement as to the intrinsic
brilliancy of platinum at its melting point, differ considerably at lower temperatures.
In the paper alluded to no information is given as to the methods or instruments
used, it is therefore impossible to suggest to what the divergence may be due. The
author sums up his results by giving for the relation between the temperature, t, and
the light emitted, I, the formula

Log I = 8"244929 + 0'011475£ - 0'00000297t2.

This expression by a modification of the constants could doubtless be made to agree
with the present results ; but the formula given below, owing to its simplicity, will
be found of more practical value.

If t denote the temperature in degrees centigrade, and b the intrinsic brilliancy of
the surface in candle-power per square centimetre, the relation may be expressed as

(t — 400) = 889-6 V£

The temperature calculated by this formula ^ will be found in column 3 of Table XVIII.
The crosses on fig. 13 represent the intrinsic brilliancy as calculated. The circles
show the values obtained experimentally. § Where the formula is used over so wide
a range of temperature the agreement for the lower values is but approximate ; but
this expression will be found to give very correct values of b when used for a limited
interval above and below any temperature for which the constant has been
determined.

Finally, before bringing this paper to a close, I desire to express my deep indebted-

* ' Annales de Chimie et de Physique,' .1883, vol. 30, p. 145.

t ' Comptes Rendus,' vol. 88, 1879, p. 171.

J When the platinum is observed in a dark room, the first grey light becomes visible at a tempera-
ture of 417° C. It is as the limiting temperature, so far as visible radiation is concerned, that 400° C.
is used in this formula.

§ I have recently made a number of observations on the intrinsic brilliancy of the crater of the
electric arc. As these results refer to a study at present far from complete, it will be sufficient to state
here that the intrinsic brilliancy is about 11,000 candle-power per square centimetre. Inserting this
value for b in the formula given above, we obtain 3830 as the temperature of the crater. This fact may
be taken as a confirmation of the formula for temperatures above the melting point of platinum,
though I do not think that it would be wise to attach much importance to a coincidence of this kind.

3 U 2

516

MR. J. E. PETAVEL ON THE HEAT DISSIPATED BY

ness to Professors DEWAR and FLEMING, and to Dr. A. SCOTT, who throughout this
work have most kindly helped me with their advice on the more difficult theoretical
points, and with their long experience of all the details of experimental work.

NOTE ADDED HTH SEPTEMBER, 1898.

It has been suggested that the standard resistance referred to on p. 502, owing to
the large currents which were used, might in reality rise to a temperature much in
excess of that indicated by the thermometer, and thus by the consequent change in
resistance materially affect the accuracy of the observations. The following experi-
ment was carried out to ascertain to what extent this actually took place.

A platinum resistance was constructed similar to the standard manganin resistance
referred to above. This new resistance was calibrated as a platinum thermometer.
It was then placed in the same oil bath as used for the manganin standard. An
electric current of increasing intensity was passed through it, and after each succes-
sive increase of current the temperature of the oil was read on an ordinary mercury
thermometer, and the temperature of the platinum strip was calculated from^ its
resistance. The stirrer was, of course, kept in motion during the entire experiment.

The results, reduced so as to apply to the manganin standard, are given in the
following table : —

Current through the standard.

amperes.
18-6
39-6
57-0
74-3
102-0

Probable rise of the manganin strip

above the temperature indicated

by the mercury thermometer.

0-3
1-4
3-1
4-9
9-3

This standard resistance was rarely used for currents above 50 amperes, and never
for currents above 70 amperes. For larger currents it was replaced by a '001 ohm
resistance. However, taking an extreme case, and supposing 74 '3 amperes to be
passing (the temperature coefficient being '0003 per cent.), it is clear that the rise of
temperature would not involve an error of more than '0015 per cent.

A PLATINUM SURFACE AT HIGH TEMPERATURES.

517

TABLE I. — On the Volume Specific Resistance of Platinum and Palladium at

their Melting Points.

Platinum.

Palladium.

Number of
radiator.

Volume
specific
resistance.

Number of
radiator.

Volume
specific
resistance.

C.G.S. units.

C.G.S. units.

1

62,720
C.3,140
61,760
62,370
62,830

Mean volume specific
resistance of plati-
num at its melting-
point = 62,600
C.G.S. units.

Rn

53,960
55,400

Mean volume specific
resistance of palla-
dium at its melt-
ing-point = 54,700
C.G.S. units.

TABLE II. — Emissivity in Dry Air at Three Atmospheres Pressure (See fig. 2.)

Temperature. Emissivity.

1

Temperature.

Emissivity.

°C.

C.G.S. units.

°C.

C.G.S. units.

50
322
800

•0008234
•001337
•002082

24th February, 1898.
Temperature of the
enclosure 9° C.

765
996
1130

•002006
•002545
•002758

26th February, 1898.
Temperature of the
enclosure 10* C.

1087

•002658

1370

•003520

518

MR. J. E. PETAVEL ON THE HEAT DISSIPATED BY

• . '

..

aT'l
xn

CO CO CO CO -* Ol

P O rH rH rH' OJ
O P Q O O O
p p p p p p

f»

hrH

'fl

-

• "'!

•* 00
O -^P CO OJ I> rH
OJ CO 1O CO rH CO

§O t-H ^M rH cVj
O O O O O

o

erf

1 03 m

rH OS OJ t^. rH I>-
OO lO CO !>• OS O

rH

- S a

a
5

rH CO ^^ ^) P- £j

<D

t>^ OJ

f J-s 1

OJ OO O
O CO 00 CO rH OS

1 O f> '•"

*? i 1 1

II 2 g

|

ff .S £

T3

"— I CO rH OO OO
rH rH OJ

* *" 1 1

03
03

£
O

, 2

1!

S.

. CO rH •* OS CO OS
O O5 I>- t- CO O rH
0 i— 1 1/J l> O OJ

rH rH

^ « 1

^T "a- s*

Si

<B

03

1 •

f| -

_. ip Oi r>. 00 1O

o. OJ do do i?- >b <M

o Ci CO CO rH 00 ^

rH 1/7. D~ CO O

o o

.** CH Q .

1- II

P -*S

^ '

; aj II

of

l^ CO CO CO CD C"
rH OS CO 00 t>. Tfl
<M CO OJ CD O -^1
<— ' O rH rH OJ OJ
p p O O O O

Sj

Q

.£

02 CD

11

03 *

./ 25 ** °* os oj 01
i 2J o CD oo oo -3

S OJ Tf OJ CO 0 ^

i 3 S § § § g

21

a

» 1

a S

03

'a

a ° > •
ll'll

4 1 8 S 8 S 13

> ^^ OJ 00 "*}H ^Jl OS

.S -S

1 1

H f § -f

-™-1 r— „

iC 0
Trt I> Q< «C

w

d
i— i

r— 1

1
^ CD

i i rH IS
J; rH OJ O T? t^ ^Jl

a. y^ rH os co T-H f»
§co os cb o -^i 6b
rH OJ OJ OJ

23 r-T ° "«

^^ N— J r^ m

- . r « s

|ob fl S

d -isf I §> -

W

fO

&

llifll
1 §«§ 11

prl O ^""^ 03 QJ

EM

—

1 | 8 S3 g g 2

0 t> OJ OO Tfl rfl OS
P i-H CO 10 ti O5

^, .£ ^.. ^ .£ ^
g" J f £ g 'S g

1 1 1 S N 1 1

OO
Oi
00

r— '

W

11 § « „ S co

gl! y* "7* 00 CO r- 1 tC
a CD OS CD O •* do
•-1 OJ OJ OJ

1 1 s i : 1 1 1

03 5 03 S '^ «*H

-3 „ § I S ° °

.a

1

Sc . bb
^ CD -*s rt

"8 "43 P P i9

IP1!

4 I i i s 2 |

> o , i £S ^J ^ ^5

0 "3 S* 3
| 5 1 OJ 1
g ° p, OJ g"

* ~ s 5 9 ii *

d

1*11

Ew CM JO IQ
23 s* i— i •<* t» jo
<-P rH OS CO rH I>

a to os cb o -* do

-I OJ OJ OJ

&. -f" M ll -»

1 -i? II ll ll

A PLATIKUM SURFACE AT HIGH TEMPKBATUBES.

519

TABLE Ills. — Emissivity in Dry Air at Atmospheric Pressure. (See fig. 2.)

Temperature.

Emissivity.

3rd February, 1898.
Temperature of the
enclosure 20° C.

Temperature.

Emissivity.

3rd February, 1898.
Temperatnre of the
enclosure 20° C.

°C.

1281
1172

1044

877
700

C.G.S. units.
•002717
•002378
•002031
•001760
•001380

°C.
528
387
293

C.G.S. units.
•001141
•000992
•000878

Itadiator. °C.
B3 1779

•004960

TABLE IV. — Emissivity in Air at Atmospheric Pressure satm'ated with moisture.

(See fig. 2.)

Tempera* ure.

Emissivity.

Temperature.

Emissivity.

°C.

C.G.S. units.

°C

C.G.S. units.

1331

1196
1036

•002977
•002550
•0021 15

3rd March, 1898.
Temperature of the
enclosure 11° C.

142

236
377

•000833
•000966
•001106

28th February, 1898.
Temperature of tin-
enclosure 10° C.

888
751

•001800
•001553

534

•001290

533

•001257

Radiator. °C.

203
74

•000845
•000666

B10 1500
BU 1500

•003820
•003945

B, 1779

•005228

TABLE V. — Emissivity in Dry Air at 1 centim. Pressure. (See fig. 2.)

Temperature.

Emissivity.

Temperature.

Emissivity.

eo.

C.G.S. units.

°C.

C.G.S. units.

922
1002
1137

•001137
•001365
•001590

1 1th February, 1898.
Temperature of the
enclosure 13° C.

1291
715
268

•001145
•000870
•000490

22nd February, 1898.
Temperature of the
enclosure 9° C.

1244

•001995

TABLE VI. — Emissivity in Dry Hydrogen at Three Atmospheres. (See fig. 3.)

Temperature.

Emissivity.

3rd March, 1898.
Temperature of the
enclosure, 10° C.

Temperature.

Emissivity.

2nd March, 1898.
Temperature of the
enclosure, 10° C.

°C.

119
183
271

408
770

C.G.S. units.
•003336
•003086
•004029
•004535
•005916

°0.
971

188

C.G.S. units.
•006720
•003679

141
680
1281

•003499
•005721
•008270

8th March, 1898.
Temperature of the
enclosure, 9° C.

520

MR. J. E. PETAVEL ON THE HEAT DISSIPATED BY

TABLE VII.— Emissivity in Dry Hydrogen at Atmospheric Pressure. (See fig. 3.)

Temperature.

Emissivity.

7th March, 1898.
Temperature of the
enclosure, 9° C.

Temperature.

Emissivity.

7th March, 1898.
Temperature of the
enclosure, 9° C.

°C.

123
222
291
384
526
692
690
886

C.G.S. units.
•002577
•002970
•003209
•003509
•003955
•004385
•004373
•004962

•c.

1020
1324
1305

C.G.S. units.
•005386
•006459
•006435

Radiator. ° C.
R8 1779
R- 1779

•00942
•00940

TABLE VITT. — Emissivity in Dry Hydrogen at 6 centims. Pressure. (See fig. 3.)

Temperature.

Emissivity.

8th March, 1898.
Temperature of the
enclosure, 9° C.

Temperature.

Emissivity.

8th March, 1898.
Temperature of the
enclosure, 9* C.

°C.

195

487
725 '
741
954

C.G.S. units.
•002213
•002853
•003413
•003350
•003954

°C.
1120

427

C.G.S. units
•004520
•002668

1265

288

•004988
•002345

7th March, 1898.
Temperature of the
enclosure, 9° C.

TABLE IX. — Emissivity in Dry Carbon Dioxide at Three Atmospheres Pressure.

(See fig. 4.)

Temperature.

Emissivity.

Temperature.

Emissivity.

°C.

G.G.S. units.

°c.

C.G.S. units.

371

•001428

28th February, 1898.

821

•002322

1st March, 1898.

277
157

•001271
•001031

Temperature of the
enclosure, 9° C.

1020
1160 .

•002808
•003230

Temperature of the
enclosure, 9" C.

61

•000783

j

1269

l^Ql

•003594

./VVinQC

554

•001776

1st March, 1898.

685

•002028

Temperature of the

enclosure, 9° C.

A PLATINUM SURFACE AT HIGH TEMPERATURES.

521

TABLE X. — Emissivity in Dry Carbon Dioxide at Atmospheric Pressure. (See fig. 4.)

Temperature.

Emissivity.

Temperature.

Emissivity.

°C.

1278
1267
1167
1026

C.G.S. units.
•002892
•002816
•002541

•002157

17th February, 1898.
Temperature of tlio
enclosure, 11° C.

°c.
1249
1361
1419

C.G.S. units.
•002809
•003197
•003407

18th February, 1898.
Temperature of the
enclosure, 11° C.

762

•001582

90
241
483

•000655
•000888
•001155

28th February, 1898.
Temperature of the
enclosure, 9n C.

TABLE XL — Emissivity in Dry Carbon Dioxide at 6 centims. Pressure. (See fig. 4.)

Temperature.

Emissivity.

21st February, 1898.
Temperature of the
enclosure, 10" C.

Temperature.

1

Emissivity.

21st February, 189S.
Tompeiature of the
enclosure, 10° C.

°C.
188
312

988
708

C.G.S. unit-.

•000422
•000530
•001447
•000975

°C.
1097
1200
1374

C.G.S. units.
•001712
•001971
•C02540

1334
521

•002385

•000715

18th February, 1898.
Temperature of the
enclosure, 10° C.

TABLE XII. — Emissivity in Oxygen at Three Atmospheres. (See fig. 4.)

Temperature.

Emissivity.

Temperature.

Emissivity.

"C.

815
413
723

C.G.S. units.
•002092
•001429
•001914

28th February, 1898.
Temperature of the
enclosure, 9° 0.

"C.

1135
1291
61

C.G.S. units.
•002882
•003463
•000848

28th February, 1898.
Temperature of the
enclosure, 9° C.

TABLE XIII. — Emissivity in Steam at Atmospheric Pressure. (See fig. 5.)

Temperature.

Emissivity.

3rd March, 1898.
Temperature of the
enclosure, 107" C.

Temperature.

Emissivity.

3rd March, 1898.
Temperature of the
enclosure, 107° C.

°C.
962
1293
764
1122

C.G.S. unils.
•002392
•003489
•001742
•002790

°C.
925
754
534

C.G.S. units.
•002174
•001714
•001268

VOL. CXCI. — A,

522

MR. J. E. PBTAVEL ON THE HEAT DISSIPATED BY

TABLE XIV. — Emissivity in Air at Atmospheric Pressure. Wire horizontal in
cylindrical iron enclosure. (See fig. 7, Curves II. and III.)

Temperature.

Emissivity.

29th January, 1898.
Temperature of the
enclosure, 455" C.

Temperature.

Emissivity.

29th January, 1898.
Temperature of the
enclosure, 22° C.

°C.

858
919
1007
1142

C.G.S. units.
•00216
•00234
•00260
•00309

°C.
61
106
188
381

C.G.S. units.
•000676
•000848
•000991
•001225

629

•00155

778

•00189

912

•00239

TABLE XV. — Emissivity in Air at Atmospheric Pressure. Gun-metal Enclosure.

(See fig. 7, Curve I.)

Radiating Wire Horizontal. Diameter of Wire '112 centim.

Temperature.

Emissivity.

, Temperature.

Emissivity.

°C.

C.G.S. units.

°C.

C.U.S. units.

64'
89

•000814
•000899

Temperature of the

enclosure 17" C.

771

785

•001952
•001972

Temperature of the
enclosure IT C.

118

•000962

804

•002034

133

•000976

861

•002147

219

•001133

942

•002312

276

•001159

945

•002344

351

•001272

1016

•002532

385

•001309

1115

•002832

420

•001393

1150

•002915

481

•001477

1171

•003018

510

•001540

1274

•003387

615

•001693

137 L

•003754

TABLE XVI.— Emissivity in Air at Atmospheric Pressure. Gun-metal Enclosure.
Radiating Wire Horizontal. Diameter of Wire '0600 centim. Kadiator No.
(See fig. 7, Curve V.)

Temperature.

Emissivity.

°C.

C.G.S. units.

1012
890

•002782
•00254(5

Temperati
enclosui

775

•002379

664

•002203

564

•002056

Temperature.

Emissivity.

Temperature of the
enclosure 16" C.

°C.

522

489

C.G.S. units.
•001990
•001938

Temperature of the
enclosure 16° C.

383

•001779

194

•001462

i

A PLATINUM SURFACE AT HIGH TEMPERATURES.

523

TABLE XVII.— On the Law of Thermal Radiation.

Heat received by the Bolometer ; Sensitive Surface coated with Platinum Black ;
Temperature of the Bolometer, 17° C. (See figs. 10 and 11.)

4th May, 1898.

9th May, 1898.

Temperature.

Mean corrected
readings.

Temperature.

Mean corrected
readings.

c.

°C.

511

4-9

1197

115-2

665

12-9

1121

911

716

166

1014

62-4

768

22-6

P39

46-8

830

29-8

765

22-3

899

40-6

657

11-9

968

53-4

1050

74-3

1117

96-4

1224

144-0

. ,. .. , . ii T • •• ii- • L f Radiator No. R.,. . . . 4C

Radiation received from palladium at its melting point . . < ^ j^-' .^j-

„ n. .. • T f ii' i j.- f Radiator No. R.in

Radiation received irom melting platinum < x. „-•>

I ii 1>o- "vt

407
396

. . 830
804

Heat received l>y o Thermopile, the Sensitive Surface of which was coated with

Lamp Black.

Corrected

Temperature.

galvanometer

deflection.

1166

97-8

1059

72-7

900

41-8

635

11-2

3X2

524 HEAT DISSIPATED BY A PLATINUM SURFACE AT HIGH TEMPERATURES.

TABLE XVIII. (See figs. 12 and 13.)

•

Temperature.

Intrinsic brilliancy
of surface in candle-
power per sq. centim.
of projected area.

Calculated
temperature
(£-400) = 889-6 v'k

•

November, 1897. Enclosure
at about 15° C, Projected
area of radiator '4256 sq.
centim.

"c.
1036
1076
1140
1189

•OG77
•131
•250
•437

1002
1063
1128
1189

1132

•231

1119

1240

•735

1251

1286

1-14

1307

1360

1-96

1381

Temperature.

Intrinsic brilliancy
of surface in candle-
power per sq. centim.
of projected area.

Calculated
temperature
(f-400) = 889-6'v/&:

Radiator No. R>,

1500

4'15

i

„ No Rn

1500

4,-qt;

1509

,, No. R21
No. R23

1779
1779

20-53

18-00

J
1766

Pctavel.

Phil. Trans., A, vol. 191, 1898, Plate 18.

Fij*. 2.

006

0J L^05 ' ZOO 300 400~^~5Od^~6dd^JOd^~600'' 900 " 1,000^1,100 tfOO 1300 1,400 /.iOO

Temperature in degrees Centigrade

i,acxr

01

•009
•008

Fie. 3.

I* J^

eM

S

I

005

VI

re:

if

ooe

001

0 TOO £00 MO 400 iOO 600 700 000 300 I.OOO I,IOO I.ZOO

Temperature in degrees Centigrade.

700

PetavcL

Phil. Trans., A, vol. 191, 1898, Plate 19.

Fig. 4.

/

/

/

/

5

i

s

*

/

/

/

/ '•_/'

,'

1

I. \

'

j

r A /

,,' i '

/

I

>

I

>

,f«/

/

L^

v^

J!

'&.

/

f

<y«

fifc^

^

^7

\

%

S

i

*/
g

3CIJ

/

/

A

j!

-?

]

.jft17

f/

-

/

Pl-

/

*<

/

t

?

A

j's

""(f*

/

:z

5*

k

I

I

•OOI

o

i

•f

&/

/

2

— -j-

•f/

'/

/

1

,

^/

x'

"n „ v

, ^6 /

i

y/

/

/

x

yf

X

o-'

" ^6'

X

&A

X

,''

Ifl-'^

X

/,

''

^l

X

?

&

7

X

5

<*>

£

* /

•£.-

f

x

Cf

^K

,

0*

X

'

•*

/

lOCJ-^

^J

0

C'

2

X

^

X

a

^

IOO 2OO JOO •WO SOO 60O TOO 800 9OO

Temperature in degrees

Centigrade.

^

/oo

so

<U ^J

I- a

503 <«W

Temperature of platinum wire.

Petavel.

Phil. Trans., A, vol. 191, 1898, Plate 20.

Fig. 7

•005

•OO4

Q

1

-003

.5
^

-002

•OOI

500 1,000 1,600

Temperature in degrees Centigrade

2DOO

Petavel.

Phil. Trans., A, vol. 191, 1898, Plate 21.

Fig. 5.

1

Em

AS/1

iCy

in

SU

am

£nc/t

aura

At

K>7°1

ant.

Pro.

taur

76 t

ms.

/

/

1

Emiss/riey in COS unite.

i \

/

/

/

/

/

/

y

/

rt

/

*

$

W-

//

I

y

?

y

A

^/

/"

/

*

—

tOO MO 4OO XO 600 70O BOO SCO IOOO I.IOO

Temperature in degrees Centigrade.

IjOO

Fii?. 10.

5

Hfeti

od

T

ftl"

Che

'leaf,
ctlel
T&

y,v,

m/!

•^/

^

p

^/d^

wp-

Ji//?l

i^?l?

^

Cif

rye .

o

X

J

?rm
ttn.

iche
n/e^

f.

>x

^

—

3

£

i

Man

hod

Bv ft

/rtni

the

mo

'•ire,

/ /jy

d r

C/-/7

O^)//<

o/-

<i5

•jlom

»/f»r

,y

^^ -•

—

SL=-

Vt\

rve.
, A

r . o
7" A

by,

fZ7

Ffi

eta.

ISC/

'el,
en

^

^

•x'"

x-

r,-

^

2g

^t-

Ca

t*5'

[:-•

^-<

k*«"

*&

r

£ar\

23

r=^

r-*

* —

nrj^

^=-

oo

MO

600 TOO tH*J

Temperature in degrees Centigrade.

Petavel.

Phil. Trans., A, vol. 191, 1898, Plate 22.

Fig. 11.

730

iOO 6OO TOO iOO 3OO I.OOO l/OO 1200 WOO I,*OO I,i00 IJbOO 1,700 I.6OO

Temperature in degrees CenCigra.de.

Fig. 12.

-\600

4000

l/OO " IfOO <30O

Temperature in degrees Centigrade.

Petavel.

Phil. Trans., A, vol. 191, 1898, Plate 23.

Fig. 13.

4/00

Temperature in degrees Centigrade.

[ 525 ]

INDEX

TO THE

PHILOSOPHICAL TRANSACTIONS,

SERIES A,

FOR THE YEAR 1898 (VOL. 191).

A.

Absorption, Change of, produced by Fluorescence (BuBKE), 87.

Aneroid Barometers, Experiments on. — Elastic After-effect; Secular Change; Influence of Temperature (CiiliEE), 441.

B.

Bolometer, Surface, Construction of (PBTAVBL), 501.

Brilliancy, Intrinsic, Law of Variation of, with Temperature (PETAVEL), 501.

BUBKE (JOHN). On the Change of Absorption produced by Fluorescence, 87.

C.

CHKBK (C.). Experiments on Aneroid Barometers at Kew Observatory, and their Discussion, 441.
Correlation and Variation, Influence of Random Selection on (PiiABSON and FILON), 229.
Crystals, Thermal Expansion Coefficients, by an Interference Method (Tui'TON), 313.

D.

Differential Equations of the Second Order, &c., Memoir on the Integration of; Characteristic Invariant of (FoKSVTH), 1.

52G INDEX.

K.

Electric Filters, Testing Efficiency of; Dielectrifying Power of (KELVIN, MACLEAN, and GALT), 187.

Electricity, Diffusion of, from Carbonic Acid Gas to f.it; Communication of, from Electrified Steam to Air (KELVIN,

MACLEAN, and GALT), 187.
Electrification of Air by Water Jet, Electrified Needle Points, Electrified Flame, &o., at Different Air-pressures; at

Different Electrifying Potentials ; Loss of Electrification (KELVIN, MACLEAN, and GALT), 187.
Electrolytic Cells, Construction and Calibration of (VELEY and MANLEY), 3G5.
Emissivity of Platinum in Air and other Gases (PETAVEL), 501.

Equations, LAPLACE'S and other, Some New Solutions of, in Mathematical Physics (FOBSYTH), 1.
Evolution, Mathematical Contributions to Theory of ; Influence of Random Selection on the Differentiation of Local Kaces

(PEARSON and FILON), 229.

F.

FILON (L. N. G.) and PEARSON (KARL). Mathematical Contributions to the Theory of Evolution.— IV. On the Probable
Errors of Frequency Constants and on the Influence of Random Selection on Variation and Correlation, 229.

Fluorescence, Change of Absorption produced by (Bur.KE), 87.

FOBSYTH (A. K.). Memoir on the Integration of Partial Differential Equations of the Second Order in Three Independent
Variables when an intermediary Integral does not exist in general, 1.

G.

GALT (ALEXANDER), KELVIN (Lord), and MACLEAN (MAGNUS). Electrification of Air, of Vapour of Water, and of
other Gases, 187.

H.

Harmonic Analysis, applied to Tidal Theory (HouGH), ISO.

Helium Stars, Condensation of, in Galactic Zone— possible presence of Oxygen in Spectra of (JIcCLEAN), 127.
HOUGH (S. S.). On the Application of Harmonic Analysis to the Dynamical Theory of the Tides.— Part II. On the
General Integration of LAPLACE'S Dynamical Equations, 13^.

K.

KELVIN (Lord), MACLEAN (MA«sus), and GALT (ALEX.). Electrification of Air, of Vapour of Water, and of other Gases,

187.

LEES (CHABLES H.). On the Theimal Conductivities of Single and Mixed Solids and Liquids and their Variation with
Temperature, 399.

M.

MCCLEAN (FRANK). Comparative Photographic SpecUa of Stars to the** Magnitude 127

MACLEAN (MAGNUS), KELVIN (Lord), and GALT (ALEX.). Electrification of Air, of Vapour of Water, and of other Gases,

Io7t

MANLEY (J. J.) and VELEY (V. H.). The Electric Conductivity of Nitric Acid, 365.

> BAMAY (WILLIAM)' and SHIELDS <JOHN>. 0" the Occlusion of Hydrogen and Ozygen by Palladium,

INDEX. 527

N.

Nitric Acid— Preparation and Purification, Properties of Anhydrous Acid, Hydrates of, Specific Resistances, Conductivities
and Temperature Coefficients (VELBY and MANLEY), 3C5.

O.

Occlusion of Hydrogen and Oxygen by Palladium — Nature of, Heat of, Effects of Pressure (MoND, RAMSAY, and SHIELDS),

105.
Ocean, Free Oscillations of the (Houori), 139.

P.

Palladium, Occlusion of Hydrogen and Oxygen by; Density of (MOND, RAMSAY, and SHIELDS), 105.

PEARSON (KARL) and FILON (L. N. G.). Mathematical Contributions to the Theory of Evolution. — IV. On the Probable

Errors of Frequency Constants and on the Influence of Random Selection on Variation and Correlation, 229.
PETAVEL (J. E.). On the Heat Dissipated by a Platinum Surface at Hi<*h Temperatures, 501.
Photometry, Photographic Method in (BuBKE), 87.

Platinum, Emissivity of; Specific Resistance of, at Melting- point (PETAVEL), 501.
Probability— Theory of Errors in Frequency (PEARSON and FILON), 229.

R.

%

Races, Local, Influence of Random Selection on Differentiation of (PEARSON and FILON), 221.

RAMSAY (WILLIAM), MOND (Luowio), and SHIELDS (JOHN). On the Occlusion of Hydrogen and Oxygen by Palladium,
105.

S.

SHIELDS '(JOHN), MOND (LuDwm), and RAMSAY (WILLIAM). On the Occlusion of Hydrogen and Oxygen ny Palladium

105.
Stars, Photographic Spectra of— Classification of— Distribution (McCLEAN), 127.

T.

Thermal Conductivities of Solids, Liquids, and Mixtures. Variation with Temperature (LEES), 399.
Thermal Expansion Coefficients for Platinum, Iridium, Aluminium, and Small Crystals (TuTTON), 313.
Tides, Dynamical Theory; LAPLACE'S Theory (HOUGH), 139.
TUTTON (A. E.). A Compensated Interference Dilatomcter, 313.

V.
VELEY (V. IT.) and MANLEY (J. J.). The Electric Conductivity of Nitric Acid, 365.

LONDON :
HARRISON AND SONS, PRINTERS IN ORDINARY TO HER MAJESTY,

ST. MARTIN'S LANE, w.c.

BINDING LOT •*!

Q Royal Society of London
4L Philosophical transaction)

L82 Series At Mathematical and

v.191 physical sciences.

Phyiical &
Applied Sci.
Serialt

PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET

UNIVERSITY OF TORONTO LIBRARY