i ERRATUM. In Mr Chapman’s paper on “ Foraminifera in the vol. xxiii. p. 395, description of Plate III., Figs. transposed. riving Condition,” 1 and 2 have been i 0 PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. ^Ob4t PROCEEDINGS THE ROYAL SOCIETY EDINBURGH. VOL. XXIV. L r } NOVEMBER 1901 to JULY 1903. EDINBURGH: PRINTED BY NEILL AND COMPANY, LIMITED. MDCCCCIV. CONTENTS. PAGE Election of Office-Bearers, Session 1901-2, 1 Chairman’s Opening Address. By the Hon. Lord M‘Laren, Vice- President, , ........ 2 On an Instrument for Trisecting any Angle. By Jas. N. Miller. Communicated by Professor Chrystal. Issued separately Feb- ruary 12, 1902, ....... 7 Note on the Mathematical Theory of Miller’s Trisector, and its Relation to other Solutions of the Problem of Trisection. By Professor Chrystal. Issued separately February 12, 1902, . 9 On the Relation of Miller’s Trisector to the Quartic Trisectrix, with a Description of a Seven-bar Lima9onograph. By Pro- fessor Chrystal. Issued separately February 17, 1902, . . 17 The Condition of the Iron in the Spleen. By William Brodie Brodie, M.D., F.R.S.E., Muirhead Demonstrator of Physiology in the University of Glasgow. Issued separately February 17, 1902, 21 A Theoretical Representation leading to General Suggestions bear- ing on the Ultimate Constitution of Matter and Ether. By John Fraser, Ordnance Survey. Communicated by Professor Chrystal. Issued separately February 21, 1902, . . 26 On the Existence within the Liver Cells of Channels which can be directly injected from the Blood-Vessels. By E. A. Schafer. Issued separately March 8, 1902, ... 65 Quaternion Binaries : an Extension of Quaternions to give an Eight-element System applicable to Ordinary Space. By Dr W. Peddie. Issued separately March 8, 1902, . . .70 Certain Considerations regarding Algol Variation, with special reference to C.P.D. — 41°#4511. By Alexander W. Roberts, D.Sc., F.R.A.S. (With Two Plates.) Issued separately March 31,1902, . . . . . . .71 Suggested Modifications of the Sign of Equality for use in Chemical Notation. By Hugh Marshall, D.Sc. Issued sep- arately March 21, 1902, . . . . . .85 The Action of Silver Salts on Solutions of Ammonium Persulphate. By Hugh Marshall, D.Sc., and J. K. H. Inglis, M.A., B.Sc. Issued separately March 31, 1902, . . . .88 Magnetic Shielding in Hollow Iron Cylinders. By James Russell. (Abstract.) Issued separately March 31, 1902, . . .94 A New Specifying Method for Stress and Strain in an Elastic Solid. By Lord Kelvin. Issued separately April 1, 1902, . 97 vi Contents. Note on Selected Combinations. By Thomas Muir, LL.D. Issued separately April 1, 1902, ...... A Continuant Resolvable into Rational Factors. By Thomas Muir, LL.D. Issued separately April 1, 1902, .... The Plague Research Laboratory of the Government of India, Parel, Bombay. By W. B. Bannerman, M.D., B.Sc., Major, Indian Medical Service, Superintendent of the Laboratory. Communicated by Professor C. Hunter Stewart. Issued separately May 1, 1902, . . . Observations on the Amount of Dissolved Oxygen in Water re- quired by young Salmonidse. By D. Noel Paton, M.D., F.R.C.P.E. ( From the Laboratory of the Royal College of Physicians of Edinburgh.) Issued separately May 2, 1902, The Theory of Jacobians in the Historical Order of its Develop- ment up to 1841. By Thomas Muir, LL.D. Issued separately June 9, 1902, ....... Functional Inertia, a Property of Protoplasm. By David Fraser Harris, M.D., B.Sc. (Lond.), Lecturer on Physiology and Hist- ology, University of St Andrews. Issued separately June 9, 1902, On the Functional Inertia of Plant Protoplasm. By R. A. Robertson, M.A., B.Sc., Lecturer on Botany, University of St Andrews. Communicated by Dr Fraser Harris. Issued separ- ately June 9, 1902, ...... Molecular Dynamics of a Crystal. By Lord Kelvin. (With Seven Diagrams.) Issued separately June 9, 1902, Does the Spectrum-place of the Sodium Lines vary in different Azimuths'? By the late Professor C. Piazzi Smyth, Astronomer- Royal for Scotland. Communicated by Professor J. G. Mac- Gregor. Issued separately July 18, 1902, The Dissociation of the Compound of Iodine and Thio-urea. By Hugh Marshall, D.Sc. Issued separately July 18, 1902, Some Identities connected with Alternants and with Elliptic Functions. By Professor W. H. Metzler. Issued separately August 14, 1902, ....... The Theory of Orthogonants in the Historical Order of its Develop- ment up to 1832. By Thomas Muir, LL.D. Issued separately August 18, 1902, ....... Experimental Observations on Leucolysis. By Alexander Goodall, M.D., M.R.C.P.E., and Edward Ewart, M.B., Ch.B. Communi- cated by Dr D. Noel Paton. (With a Plate.) Issued separately August 19, 1902, ....... Amorphous Sulphur and its Relation to the Freezing Point of Liquid Sulphur. By Professor Alexander Smith, Ph.D., D.Sc. Issued separately August 19, 1902, .... Application of Miller’s Trisector to the Quinquesection of any Angle. By James N. Miller. Communicated by Dr Knott. Issued separately August 19, 1902, .... Thallic Sulphates and Double Sulphates. By Hugh Marshall, D.Sc. Issued separately August 30, 1902, . . . On Superposed Magnetic Inductions in Iron. By James Russell. {Abstract.) Issued separately October 7, 1902, . PAGE 102 105 113 145 151 196 200 205 225 233 240 244 289 299 302 305 312 Contents. Vll PAGE On the Use of Quaternions in the Theory of Screws. By Dr W. Peddie. Issued separately October 7, 1902, Contributions to Scottish Mineralogy. (Part I.) By J. G. Good- child, of the Geological Survey, F.G.S., F.Z.S. Communicated by F. Grant Ogilvie. Issued separately November 15, 1902, On Causes which determine the Formation of Amorphous Sulphur. By Professor Alexander Smith, B.Sc., Ph.D. Issued separately November 15, 1902, ...... Quaternion Notes. By the late Professor Tait. Communicated by Professor C. G. Knott. (With a Plate.) Issued separately January 17, 1903, ...... The Electric Conductivities and Relative Densities of certain Samples of Sea- Water. By J. J. Manley, Daubeny Curator, Magdalen College, Oxford. Communicated by Sir John Murray, K.C.B. Issued separately January 24, 1903, Freezing-Point Depression in Electrolytic Solutions. By James Walker, F.R.S., and A. J. Robertson, B.Sc., University College, Dundee. Issued separately January 28, 1903, . Note on Pure Periodic Continued Fractions. By Thomas Muir, LL.D. Issued separately February 28, 1903, The Generating Functions of certain Special Determinants. By Thomas Muir, LL.D. Issued separately February 28, 1903, Some Observations on Cyanosis. By G. A. Gibson, M.D., D.Sc., F.R.C.P. Edin., Physician to the Royal Infirmary, Edinburgh. Issued separately April 4, 1903, ..... On the Isoclinal Lines of a Differential Equation of the First Order. By J. H. Maclagan-Wedderburn. Communicated by Professor Chrystal. Issued separately April 4, 1903, On the General Scalar Function of a Vector. By J. H. Maclagan- Wedderburn. Communicated by Dr W. Peddie. Issued separately June 5, 1903, ...... On the Equation of a Pair of Tangents to a Conic. By Professor A. H. Anglin, Queen’s College, Cork. (Abstract.) Issued separately June 5, 1903, ...... Suggestion as to the Cause of the Earth’s Internal Heat. By George Romanes, C.E. With Notes by Professors A. Gray and. C. G. Knott. Communicated by Dr C. G. Knott. (With a Plate.) Issued separately June 5, 1903, .... The Occurrence of the Sperm Whale or Cachalot in the Shetland Seas, with Notes on the Tympano-petrous Bones of Physeter, Kogia, and other Odontoceti. By Sir William Turner, K.C.B., F.R.S. Issued separately June 5, 1903, .... Some Observations on the Young Scales of the Cod, Haddock and Whiting before Shedding. By Mr Alex. Wallace Brown, St Andrews. Communicated by Dr A. T. Masterman. Issued separately August 3, 1903, . . . . On the Series — Issued separately August 3, 1903, . 314 321 342 344 347 363 380 387 393 400 409 413 415 423 437 439 viii Contents. PAGE The Theory of Colour Vision. By Dr W. Peddie. Issued separ- ately August 3, 1903, ...... 448 A Direct Electrical Method of Determining Latent Heat of Evapora- tion at the Boiling Point. By A. Cameron Smith, M.A., B.Sc., Lecturer in Physics, Heriot-Watt College, Edinburgh. Com- municated by Professor F. G. Baily. Issued separately August 8, 1903, ........ 450 The Wild Horse ( Equus prjevalskii, Poliakoff). By J. C. Ewart, M.D., F.B.S., Professor of Natural History, University of Edin- burgh. Issued separately August 8, 1903, . . . 460 Statistical Evidence regarding the Influence of Artificial Propaga- tion upon the Salmon of the American Hi vers. By William Murray. Communicated by Dr D. Noel Paton. Issued separ- ately August 8, 1903, ...... 469 October Salmon in the Sea. By D. Noel Paton, M.D. (With Three Plates.) Issued separately August 28, 1903, . . 486 Note on a Method of Bringing Together the Two Spectra Com- pared in the Ordinary Spectrophotometer. By J. R. Milne, B.Sc. Communicated by Professor J. G. MacGregor. Issued separately August 28, 1903, ..... 496 Note on Resistance Change accompanying Transverse Magnetisa- tion in Nickel Wire. By Professor C. G. Knott, D.Sc., and Mr Peter Ross, M.A. Issued separately August 28, 1903, . 501 On the Abdominal Viscera of Cercocebus fuliginosus and Lagothrix humboldti. By O. Charnock Bradley, M.B. (With Three Plates.) Issued separately October 21, 1903, . . . 505 The Molecular Condition of Iron Demagnetised by Various Methods. By James Russell. Issued separately October 17, 1903, ........ 544 A Special Circulant considered by Catalan. By Thomas Muir, LL.D. Issued separately October 21, 1903, . . . 547 The Theory of Axisymmetric Determinants in the Historical Order of Development up to 1841. By Thomas Muir, LL.D. Issued separately December 28, 1903, .... 555 On the Origin of the Pineal Body as an Amesial Structure, deduced from the Study of its Development in Amphibia. By John Cameron, M.B. (Edin.), M.R.C.S. (Eng.), Carnegie Fellow, Demonstrator of Anatomy, United College, University of St Andrews. Communicated by Dr W. G. Aitchison Robertson. (With a Plate.) Issued separately December 28, 1903, . . 572 Meetings of the Royal Society — Sessions 1901-1903, . . 582 Donations to the Library, . . . . . .612 Obituary Notices, ...... 642 Abstract of Accounts for Session 1902-1903, . . . 650 Index, ........ 661 PROCEEDINGS OF THE ROYAL SOCIETY OE EDINBURGH. vol. xxiv. 1901-2. The 119th Session. GENERAL STATUTORY MEETING. Monday , 28 th October 1901. The following Council were elected : — President. The Right Hon. Lord KELVIN, G.C.V.O., F.R.S. Vice-Presidents. Sir Arthue Mitchell, K. C.B. , LL. D. Sir William Turnee, K. C. B. , F. R. S. Professor Copeland, Astronomer- Royal for Scotland. The Rev. Professor Duns, D.D. Professor James Geikie, LL.D., F.R.S. The Hon. Lord McLaren, LL.D. General Secretary — Professor George Chrystal, LL. D. Secretaries to Ordinary Meetings. Professor Crum Brown, F.R.S. Ramsay H. Traquair, M.D., LL.D., F.R.S. Treasurer.— R. D. Maclagan, Esq., F.F.A. Curator of Library and Museum — Alexander Buchan, Esq., M.A., LL.D., F.R.S. Ordinary Members of Council. The Rev. Professor Flint, D.D. James Burgess, Esq., C.I.E., LL.D. R. M. Ferguson, Esq., Ph.D., LL.D. Robert Iryine, Esq., F.C.S. Professor John G. M‘Kendrick, M.D., LL.D., F.R.S. Professor Schafer, F.R.S. Dr Robert Munro, M.A., LL.D. J. S. Mackay, Esq., LL.D. Sir John Murray, K.C.B., LL.D. R. Traill Omond, Esq. F. Grant Ogilyie, Esq., M.A., B.Sq. Dr Geo. A. Gibson, F.R.C.P.E. PROC. ROY. SOC. EDIN. — VOL. XXIV. 1 2 Proceedings of Royal Society of Edinburgh. [sess. The Hon. Lord M‘LAREN, LL.D., Vice-President, in the Chair. Chairman’s Opening Address. (Read November 4, 1901.) I find that during the past Session forty-one papers have been read. Of these, ten belong to the department of Physics, five to Mathematics, one to Chemistry, two to Geology, seven to Biology, one to Botany, four to Anatomy, three to Physiology, four to Meteorology, four to Astronomy. Since the commencement of the past Session, eighteen Fellows have been added to our numbers. Of these, one is a Professor of Philosophy, two are University Lecturers, and six are Doctors of Medicine. The Professor above mentioned is Sanjiban Ganguli, of the Maharaja’s College, Jaipur. He has written treatises on the Philosophy of Locke and Descartes. In his letter of application for admission to the Society he says, “ May I be excused for the trouble I am going to give you? It has long been my desire to become a member of the Royal Society of Edinburgh. ... To be in touch with the great scholars and learned men of the time is itself a matter of great educational importance.” During the same period sixteen Fellows have been taken from us by death. These include our late Secretary, Professor Tait, in reference to whom the Council adopted the following minute — “ The Council desire to put on record their sense of the great loss which the Society has sustained by the death of Professor Tait. Professor Tait became a Fellow of the Society in 1861, when he came to Edinburgh as Professor of Natural Philosophy. At the beginning of the next Session he was elected a member of Council, and in 1863 one of the Secretaries to the ordinary meetings. In 1879 he was elected General Secretary, in succession to Professor Balfour. “ During his forty years of Fellowship he contributed a very 1901-2.] Chairman's Opening Address. 3 large number of papers — all of them original and interesting — some of them of the very highest importance — with a place assured for ever in the development of science. “His loss will be felt in the Society, not only as a contributor, but perhaps even more as a wise councillor and guide. “ The Council always felt that in his hands the affairs of the Society were safe — nothing would be forgotten — everything that ought to be done would be brought before them at the right time and in the right way. “ This is not the occasion for an analysis of Professor Tait’s work and influence. That will no doubt be given in due time by those specially qualified. What the Council now feel is that a great man has been removed — a man great in intellect and in the power of using it, in clearness of vision and purity of purpose, and there- fore great in his influence, always for good, on his fellowmen. They feel that they and many in the Society, and far beyond it, have lost a strong and true friend.” To this tribute I shall only make one addition, because I under- stand a biographical notice is in preparation, which will be read to the Society at a future meeting. But I may remind the Society that the value of Professor Tait’s scientific work is not to be measured solely by his original papers, most of which were communicated to this Society. He shares with Lord Kelvin the honours of the authorship of the Treatise on Natural Philosophy , which has been recognised throughout the world of science as the standard and classic work of the nineteenth century on the subject, — a work which, if I may say so, combines the most lucid and convincing exposition of principles with masterly analytic investigation of the more difficult parts of the subject. Professor Tait is more widely known to the general reader through his smaller text-books, of which I need only mention those relating to Light and the Properties of Matter. Through these books the results of scientific research have been made known to a large circle of readers who do not make science their vocation, but who desire to know what can be understood without the use of the higher mathematics. I believe that a considerable number of the younger physicists -of our time were aided in their studies by Professor Tait, through 4 Proceedings of Royal Society of Edinburgh. [sess. attending the Natural Philosophy Class, and through correspond- ence in later life. His recommendation was always at the service of the best men for public positions connected with science. I need not say there are many in this Society who have been aided by his instruction and the results of his wide reading in the prosecution of their studies and researches. W. Williams, Principal of the Veterinary College, Leith Walk, was known as an eminent teacher of Veterinary Science to every agriculturist and veterinary practitioner in the country. His two great works — the one on Veterinary Medicine , and the other on Veterinary Surgery — are used as text-hooks, not only in the Veterinary Colleges of Great Britain, hut also in those of America. George Frederick Armstrong, horn at Doncaster in 1842, after being successively Professor of Engineering in JVFGill’s College, Montreal, and the Yorkshire College of Science, Leeds, succeeded the late Professor Jenkin as Eegius Professor of Engineering in our University. He was also appointed Engineering Adviser to the Local Government Board for Scotland. During 1879 he under- took an extensive series of observations and experiments, with a view of determining the diurnal variation in the amount of carbon dioxide in the air, the results of which have been accepted as a standard of reference on the Continent as well as in this country. Sir Thomas Clark, Bart., was head of the widely known publishing house of T. & T. Clark, Edinburgh. He filled with ability the offices of Assistant, Treasurer, and Master of the Edinburgh Merchant Company. In 1885 he was elected Lord Provost of Edinburgh. During his provostship his name was associated with much good civic work. On the occasion of the Queen’s visit to the Edinburgh Exhibition of 1886, Sir Thomas Clark received his baronetcy. Dr Stevenson Macadam was widely known in Edinburgh as a popular and successful lecturer on Chemistry. He taught large- classes at the School of Arts, Surgeons’ Hall, and the Veterinary Colleges. In 1862 he was made President of the Boyal Scottish Society of Arts. He conducted a large consulting business, and was much sought after for expert evidence. A tragic interest attaches to the death of our Fellow Mr Frederick Pullar, who lost his life in one of his beloved lochs,. 1901-2.] Chairmans Opening Address. 5 while attempting to save the life of a young lady. At Airthrey Loch, near the Bridge of Allan, the ice suddenly gave way, and by this mishap three persons were immersed. Mr Pullar having successfully assisted them to land, next went to the rescue of a lady, whom he supported for some time, but his strength failed, and they both sank and were drowned. William Skinner, W.S., performed important work in the educational administration of this city, and held the important office of Town-Clerk from 1875 to 1895. Among the deceased also is Dr Stubbs, Bishop of Oxford, one of our British Honorary Fellows, who is considered to haye been the greatest authority of late years on English Constitutional History. The University of Glasgow having resolved to celebrate the ninth Jubilee of the foundation of the Institution on the 12th of June last, invited the Society to send representative delegates to take part in the proceedings, — or, in the words of the circular, “ Sacra autem paramus saecularia cum haec nostra Academia natalem quadringentesimum celebrare velit. . . . Societatem vestram precamur ut legatos mittat quos gaudiis ceremoniisque nostris inter- futuros mensis Junii die xii laeti laetos excipiamusA To this invitation the Society sent the following congratulatory address, and appointed Dr Robert Munro as their delegate : — Rectori magnifico, Praefecto et Senatui Universitatis Glas- GUENSIS SoCIETAS REGIA EdINBURGENSIS SALUTEM DAT. Maxima cum laetitia, viri illustres, accepimus vestras litteras humanissimas invitantes nostram Societatem mittere legatos ad capessendam partem in celebratione solemnium in memoriam noni Semisaecularis Anni Universitatis vestrae. Occasio sane idonea est ad renovandam memoriam philosophorum qui famam vestrae Universitatis per orbem terrarum divulgaverunt. Communio studiorum inter Universitatem Glasguensem et Societatem Regiam ad benignam confraternitatem inducit, atque eo magis quod multi Professores Glasguenses, nobis electione adjuncti, ditant et ornant Acta Societatis nostrae communicationibus ingeniosis. Legatus noster, qui vicem Societatis implet, ad istam solemni- sationem missus, Robertus Munro, M. A., M.D., est vir antiquitatum 6 Proceedings of Royal Society of Edinburgh. [sess. Scotiae peritissimus, et linguae Scoto-Celticae non ignarus, auctor etiam Operis doctissimi de Domiciliis supra Lacus sitis primitivorum Incolarum Europae, aliorumque operum late cognitorum. Ille vota fausta secum fert. Laetamur valde vestram Universitatem progressu continue auctam esse ope Archiepiscoporum Glasguensium, Begum et Begentium Scotiae, Mariae Scotorum Beginae, et sui filii Jacobi VI. fundatoris alterius ; atque etiam gratulamur vobis de augmentis scientiarum tarn theoreticarum quam meclranicarum apud vestram Universitatem feliciter consummatis, et simul de studiis litterarum humaniorum ibi semper eximie florentibus, Faxit autem Deus Optimus Maximus ut inclyta sedes vestra eruditionis laudes saeculorum decursu acquisitas per omne tempus futurum tueatur. Subscribitur , Kelvin, Presses. P. G. Tait, A Secretis. Die xxiv. Mensis Mail, MG1 MI. On 24th May last the Council beard a statement from Sir John Murray with reference to a Bathymetrical Survey1 of the Fresh- Water Lakes of ' Scotland, to the effect that Mr Laurence Pullar was willing, on certain conditions, to set aside a sum of money to enable the survey to be completed. It had been com- menced by Sir John and Mr Pullar’s son, Mr F. P. Pullar, but which had been interrupted by the unfortunate death of the latter gentleman, who was accidentally drowned. Mr Pullar was prepared to incur the outlay, provided Sir John would himself undertake the superintendence of the survey and the publication of the results. The Council learnt with much satisfaction that arrangements were in contemplation for carry- ing to a successful completion the admirable survey which had .been commenced by Sir John and Mr Pullar’s son, whose death they all deplored. We may now proceed to the ordinary business of the evening, which includes three papers. 1901-2.] On an Instrument for Trisecting any Angle. i On an Instrument for Trisecting any Angle. By Jas. N. Miller. Communicated by Professor Chrystal. (Read November 4, 1901. Issued separately February 12, 1902.) The instrument is delineated in the accompanying drawing. It consists of two thin and flat pieces, A I F D and B H 0, of metal or other material. Those pieces are closely connected together by a cylindrical pin inserted perpendicularly to their flat surfaces through a cylindrical hole at C in each of them, which it fits, and round which they can turn. The piece A I F D has also a small cylindrical hole through it at A. The centres of the holes at A and C, and also the point D, are all in the same straight line A C D. The point B, the centre at C, and the border E 0 of the piece B H 0, are all in the same straight line B C E O. The line AC connecting the centres at A and C is bisected at I. The border I O of the arm I E is straight, and is perpendicular to the line A C. Therefore, A O is equal to C O, and A O C is an isosceles triangle. The point B is at the same distance from the centre at C as that centre is from the centre at A; in other 8 Proceedings of Royal Society of Edinburgh. [sess. words, the straight lines B C and A C are equal to each other. Consequently A C B is also an isosceles triangle. The arm I F may be extended indefinitely beyond F. The piece B H 0 admits also of indefinite extension beyond 0. The drawing exhibits the instrument as in the position in which it trisects the angle B AO, which may he any angle less than 135°, that is, than a right angle and half of a right angle. For, as A B C is an isosceles triangle, so the angles B A C and A B C at its base A B are equal to each other. And, as A C 0 is also an isosceles triangle, so the angles A C 0 and C A 0, at its base A C, are also equal to each other. But the angle A C 0 is exterior to the isosceles triangle A C B, and therefore it, as also the angle C A O, is equal to twice the angle B A C. The angle B A C is therefore equal to a third of the angle B A O. When the instrument is being used to trisect any angle under 135°, as for instance the angle B A O, a sharp cylindrical pin or a needle is inserted through the hole at A into the vertex of the angle. The point B is then moved along a side, as A B, of the angle until the borders E O and I O meet in a point (which may he termed O) in the other side of the angle. The angle is thereby trisected, or may he so, by bisecting the angle C A O. An angle of 135° or larger may he trisected by first dividing it into two or four equal angles. Two or four thirds respectively of one of those divisions of the angle will be equal to a third of the entire angle. If the instrument be made and used with precision, any angle may be trisected with it. The reasons why an angle of 135° or upwards cannot be directly trisected by the instrument are — that, if the angle to be trisected were 135°, then the angles ABC and BAC would each be half of a right angle, and the angle OCA would be a right angle, and consequently the borders I 0 and E 0, inasmuch as they would be parallels, would never meet ; nor would they ever meet if the angle B A O were upwards of 135°. 1901-2.] Prof. Chrystal on Theory of Miller’s Trisector. 9 Note on the Mathematical Theory of Miller’s Trisector, and its Relation to other Solutions of the Problem of Trisection. By Professor Chrystal. (Read November 4, 1901. Issued separately February 12, 1902.) The problem of the trisection of an angle, analytically considered, depends on the solution of a cubic equation, say 4x8 - 3x — cos a = 0, which is in general irreducible, and therefore not soluble by means of quadratic radicals. It follows that the trisection of an angle cannot be effected by means of the ruler and compass alone. This, in fact if not in theory, appears to have been known to the early Greek geometers, and they proposed the use of various curves, the continuous mechanical construction of which must, of course, be postulated, for the solution of the problem. The Quadratrix and the Spiral of Archimedes, both transcendental curves, may be mentioned. It was early discovered that the conic sections could be used for the purpose, and in modern times various solutions have been suggested involving their use.* The earliest example of the use of an algebraic curve of higher degree than the second is the trisection by means of the conchoid of Nicomedes, a circular quartic, represented by the equation (x2 + y2)(x - a)2 = b2x2. This solution is of special interest, because the conchoid admits of a very simple mechanical construction by means of the well-known Trammel of Nicomedes. If, as is commonly believed, this apparatus was invented by Nicomedes himself (< ca . 150 b.c.), this would be the earliest example of the mechanical construction of a curve of higher degree than the second. Among the trisectrix curves of higher degree that have been used by modern mathematicians may be mentioned the Cubic * Those interested in the subject may consult Cantor’s Geschichte der Mathe- matik (Leipzig, 1894) ; Allman’s Greek Geometry (Dublin, 1889) ; Gow’s Short History of Greek Mathematics (Cambridge, 1884) ; Newton’s Arithmetica Uni- versalis, 2nd ed. (London, 1722), Appendix de Equationum Constructione Lineari ; Maclaurin’s Algebra (London, 1756), chap. iii. ; Klein’s Vorlesungen iiber Ausgewahlte Fragen der Element argeomctrie (Leipzig, 1895). 10 Proceedings of Royal Society of Edinburgh. [sess. Trisectrix of Maclaurin { 2x(x2 + y 2) = a(y 2 - 3x2) ; see Basset’s Cubic and Quartic Curves , § 133}, and the Quartic Trisectrix ((x2-\-y2)2 - 3a2(x2 + y2) - 2a3£C = 0), which is a particular case of Pascal’s Limagon. We shall see presently that Mr Miller’s instru- ment is closely related to the last of these curves. The interest excited by the invention of Peaucellier’s inverting linkage in 1864 led to the discovery that the division of any given angle into any given number of parts could be effected by a linkage. Two distinct forms of trisecting compass, one due to Sylvester and the other to Kempe, are described and figured in a little work published by the latter in the Nature Series, under the title How to Draw a Straight Line (Macmillan, 1877). The Sextic Trisectrix. Miller’s Trisector consists of two bars jointed at A, so that OA = AB = 2 a, say. To the middle point C of O A is fixed a perpendicular bar C P. To trisect an angle X O K, O is placed at O and B slid along O X until the intersection of B A P and C P, viz., P, falls on OK; then A 0 B is one-third of K 0 X. To find the locus of P, when B moves along 0 X, we have x = a cos 6 - a tan 2 0 sin 0 = a cos 39 sec 2 0 ; ) y — a sin 0 + a tan 2 0 cos 0 = a sin 30 sec 2 0 . ) 11 1901-2.] Prof. Chrystal on Theory of Miller, s Trisector. We have obviously x2 + y2 = a 2 sec2 2 0 , yjx = tan 30 . Hence, if we have X = J{(x 2 + if - a2) /a2} , Y = y/x, tan 20 = X , tan 30 = Y . Therefore tan 0 = tan (30 - 20) , = (Y - X)/(l + XY) . Hence X = tan 20 = 2 tan 0/(1 - tan 20) , 2 (Y -X)(l +XY) ~~ (1 + XY)2 - (Y - X) ’ The eliminant is therefore X{X2Y2 - X2 - Y2 + 4XY + 1 j = 2 {XY2 - X2Y - X + Y} . This may be written X(X2 - 3)(Y2 - 1) = 2Y(1 - 3X2) ; hence X2(X2 - 3)2(Y2 - l)2 = 4Y2(3X2 - l)2, which gives (x2 + y2 - a2) (x2 + y2- 4 a2) (x2 — y2)2 = ka2x2y2{3(x2 + y2) — 4}2 . . (2). The equation (2) may be considerably simplified. Arranging, in the first place, according to powers of x2 + y2, we have (x2 - y2)2{(x2 + y 2)3 - 9 a2(x2 + y2)2 + 24 cfi(x2 + y2) — 16a6 } = ia2x2y2{2)(x2 + y2)2 - 24a2(^2 + y2) + 16a4} . Hence we get (x2 — y2)2(x2 + y2)3 — 9 a2(x2 + y2)4 + 24 a\x2 + y2)3 - 1 6a6(aj2 + y2)2 == 0 . If we reject the factor (a:2 + y2)2, we get finally for the equation to the locus of P (x2 + y2)(x2 - y2)2 - 9 a2(x2 + y2)2 + 24a4(a;2 + y2) - 16a6 = 0 .... (3). A circular sextic, which we may call Miller'’ s Trisectrix. 12 Proceedings of Royal Society of Edinburgh. [sess. If we turn the axes through 45°, the equation takes the form 4{x2 + y2)x2y2 - 9 a2{x2 + y 2)2 + 24 a\x2 + y2) - 16a6 = 0 . . . (4), which is more convenient for some purposes. Since the curve is symmetrical with respect to the original axes, and also with respect to the new axes, which are the octant lines of the old axes, it is immediately evident that it must consist of eight congruent portions. For the intersection with the cc-axis, taking equation (3), we get, putting a = l, for brevity, x6- 9x4 + 24x2-l6 = 0, that is, (x2- l)(x2- 4)2 = 0. Hence the intersections with the ^c-axis are (-2,0) his, (-1,0), (+1,0), ( + 2,0) is Similarly the intersections with the y-axis are (0,-2) his, (0,-1), (0, + 1), (0, + 2) bis. We may trace one of the eight congruent branches of the sextic by causing 0 to vary from 0 to 7t/4 in the formulae (1). If dashes denote differentiation with respect to 0 , we have x = (2 cos 3 0 sin 20-3 sin 30 cos 20)/ cos2 20 ; y = (2 sin 30 sin 20 + 3 cos 30 cos 20)/ cos2 20 ; y _ 2 sin 30 sin 20 + 3 cos 30 cos 20 # x 2 cos 30 sin 20 - 3 sin 30 cos 20 ’ x - yx If = 3/(2 sin 30 sin 20 + 3 cos 30 cos 20). e X y x' y' y'/x' a? - yx’ly' 0 1 0 0 3 00 1 7T IT 0 2 -6V3 4V3 -2/V3 3 7 r T - oo 00 00 00 -1 3/\/2 1901-2.] Prof. Chrystal on Theory of Miller's Trisector. 13 We thus get the branch AGE, approaching the asymptote C F, whose equation is x + y — 3/^/2 . The same result may he obtained very simply by seeking the intersections of the sextic with the variable circle, x2 + y2 — r2 (5). If we combine this equation with (3), we get (x2-y2)2 = (3r2~4:)2lr2 (6). Since (5) and (6) have only eight finite intersections, we see that four of the intersections of the sextic with the circle must be at infinity. It appears, therefore, that the sextic touches any circle at each of the circular points at infinity. The rest of the finite intersections are given by the equations x2 + y2 = r2 x2 - y2 = ± (3r2 -4 )jr . We have therefore the following parametric representation of points on the sextic trisectrix : x2 = (r - l)(r + 2)2/-2r , y2 = (r + l)(r - 2)2/2r ) xfi==(r+ l)(r - 2)2/2r , y2 = (r- l)(r + 2)2/2r j" ’ Since the second of these is derivable from the first by merely changing the sign of rf we see that the sextic is completely repre- sented by where r may have any real value numerically greater than unity. The formulae for the branch AGE are obviously where r ranges from + 1 to + oo . If we now mirror the branch A G E in the axes of x and y and in the octant lines successively, so as to get the eight congruent branches indicated by the form of the equation, we get the curve drawn in fig. 2, where 0 A = 1, OB = 2, 0 C = 3/^/2, 0 G = 2/^/3. It will be seen that the curve has two pairs of parallel asymp- totes, corresponding to two double points at infinity. There are also eight finite double points : two on each of the 14 Proceedings of Royal Society of Edinburgh. [sess. axes of x and y, and two on each octant line. The coordinates of these points are ( ± 2, 0), (0 ± 2), and ( ± ^2/ JS, ± J2/J3). Since the number of double points is ten, the maximum which a sextic curve can have, it follows that the curve is unicursal. This might have been seen from the original parametric representation of the curve given by the equations (1). In fact, if £ = tan \ 6 , we get a? = (1 - £2)(£4 - 14£2 + V)/(ft + 1)(£4 - 6£2 + !) \ y = 2t(3t2 - 1 )(^2 — 3)j(t2 + 1 )(£4 - 6^2 + 1) j a rational parametric representation of the curve. 15 1901-2.] Prof. Chrystal on Theory of Miller’s Trisector. This last result is merely a particular case of the general theorem that the curve x=f(sin mf, cos m-fi, sin m20, cos m20, ), y = pr(sin miQt cos mf, sin m20, cos m20, ), where / and g are rational functions, and mv m2, integers is a unicursal curve. If n denote the degree of the sextic, d the number of its double points, k the number of its cusps, m the class, t the number of inflexions, r the number of double tangents, we find by means of Pliicker’s equations, since n = 6, d— 10, k = 0, that m = 10, t= 12, r = 24. Of the twelve inflexional tangents only four are real, viz., the four asymptotes, each of which meets the curve in four points at infinity. Since the curve is unicursal, its quadrature can be effected by means of elementary transcendents. Its quadrature depends on the integral I d0j( 4 sin2 20 + 9 cos2 20)/cos2 2 0 , which can be expressed in terms of elliptic transcendents. Trisection of an Angle by means of the Ruler , the Compass , and a Sextic Trisectrix Template. Let a template be constructed, one of whose edges, 0 A (fig. 3), is the a-axis of the sextic trisectrix when represented by the equation (3), and the other the branch A E of the trisectrix. Then we may trisect any given angle UOX (<135°) as follows:— Place the template so that its 0 falls on the 0 of the given angle, and 0 A on one of the arms, say 0 X. Mark the point P where the curved edge of the template meets the other arm O U. With O A as radius and 0 as centre, describe a circle ; and with half 0 P as radius, and the middle point of O P as centre, describe / 1 6 Proceedings of Royal Society of Edinburgh. [sess. another circle. If Q be the intersection of these circles which falls within X 0 U, then X O Q is one-third of X O U. The proof will be obvious from the geometrical property which defines the trisectrix. 1901-2.] Prof. Chrystal on Trisector and Quartic Trisedrix. 17 On the Relation of Miller’s Trisector to the Quartic Trisectrix, with a Description of a Seven-bar Lima- conograph. By Professor Chrystal. (Read November 18, 1901.) If, instead of causing the point B of the trisector to move along an arm of the angle, we cause P, the intersection of the two edges, to move along an arm, and find the locus of the point B, we get another curve associated with the instrument. If 0 X (fig. 4) be the fixed arm, O being the fixed pivot of the trisector, 0 A = A B = a, say, and the angle B P X (4 0) be twice the PROC. ROY. SOC. EDIN. — VOL. XXIV. 2 18 Proceedings of Royal Society of Edinburgh. [sess. angle A O X (2 0), then B is a point on the curve in question. And if we take X 0 Y as axes, we have x = a (cos 2 0 + cos 4 6) = 2 a cos 0 cos 3 0 , | y == a (sin 2# + sin 40) = 2a cos 6 sin 30 j ' ’ from which we get at once the equation (x2 + y2)2 - 3a2(x2 + y2) -2asx = Q .... (10). The curve is therefore a bicircular quartic. This quartic may be readily identified with a well known curve by means of its characteristic property. If we take a point O, such that Ofi = OF = a, join OB, and draw OD parallel to AB, it is seen that 0 A B D is a rhombus. We therefore have OD = DB=a. Hence the locus of D is a circle whose centre is O ; and it follows that the locus of B is a particular variety of Pascal’s Limagon. If we put y> = 2 0, and take O X as prime radius, the polar equa- tion of the curve is r = a (2cos <£ + 1) (11) ; and the Cartesian equation referred to XOY' is (x2 + y2 - 2ax)2 — a2(x2 + y2) . . • . (12). The origin is obviously a real node, whose tangents are y — ± J3x. Prom (10) it is obvious that the lines y±ix = 0 are imaginary cuspidal tangents at the circular points at infinity. The quartic is therefore trinodal, bicuspidal, and also unicursal. Its Pluckerian numbers are n = 4, 8=1, k = 2 ; m = 4, r = 1, i = 2. If we regard the curve as a Cartesian oval, it is easily seen that O is the triple focus ; and that the other three real foci are H(OH = Ja) and O, the latter counting twice. It is thus readily found that we have the property 20B + 4HB = 3 a (13). It will, in fact, be easily verified that 2 J{x2 + y2) + i J{(x - f)2 + ?/2} = 3« . . . (14) rationalises into the equation (12). Taking a hint from Kempe’s trisecting link-work, already re- ferred to, we can construct a seven-bar link- work which will enable us to draw the Quartic Trisectrix, or, indeed, a Limagon of any 1901-2.] Prof. Chrystal on Trisector and Quartic Trisectrix. 19 given eccentricity. In fig. 5, 0 and K are fixed centres. 0 A = LU = LM = 6; OL = AU = AM = £&; AP = c; K N = b‘ AN = OK = 2b. Hence the two contra-parallelograms 0 L M A, OANK, are similar, so that the angles A 0 K and A 0 L are equal. It follows that P traces the Limagon r=2c cosO + b, whose eccentricity is 2c/6. We therefore get a hyperbolic Limagon, the Cardioid, or an elliptic Limagon, according as c> = <\b. The trisectrix is obtained by taking c = b. Other link-motions have been given for tracing the Limagon ; in particular, a five-bar one by Mr Hart ( Proc . Lond. Math. Soc., vol. vi. p. 138); hut they have not the advantage of being so readily adjustable as the above so as to give Limagons of varying eccen- tricity. A working model was shown to the Society. Finally, we may remark that the quartic trisectrix can be generated as an epitrochoid, the radius of the fixed and rolling circles being equal, and the distance of the tracing point from the centre of the rolling circle equal to the diameter. This property gives another simple mechanical process for tracing the curve. 20 Proceedings of Royal Society of Edinburgh. [sess. Use of the Quartic Trisectrix as a Trisecting Template. If we construct a template of the form OXYO (fig. 4), we may trisect any angle U 0 X with it as follows : — Place the template with 0 at the vertex of the angle and O X along one arm, and let its curved edge meet the other arm in B. Describe circles with O and B as centres and 0 F as radius. If A be that intersection of the two circles which falls within U 0 X, then U 0 A is one-third of the angle U O X. This property of the curve is well known. See Basset, Elementary Treatise on Cubic and Quartic Curves (Cambridge, 1901), § 297. From the present point of view the characteristic property of the quartic trisectrix is that it is generated by the intersection of two radii, through the fixed points 0 and O, moving so that twice the angle B O X is equal to thrice the angle BOX. It is obvious that another trisectrix would be obtained by causing B to move so that the angle B 0 X is thrice the angle BOX. The locus of B is then the cubic trisectrix of Maclaurin,* to which we have already referred. * See Maclfturin’s Fluxions (Edinburgh, 1742, vol, i. p. 262). {Issued separately February 17, 1902.) 1901-2.] The Condition of the Iron in the Spleen. 21 The Condition of the Iron in the Spleen. By William Brodie Brodie, M.D., F.B.S.E., Muirhead Demonstrator of Physiology in the University of Glasgow.* (Read January 20, 1902.) Many investigations have been made as to the destruction of red blood corpuscles in the spleen, and there can be little doubt that this occurs. If the hsemoglobin set free is also destroyed in the spleen, iron in some form will be liberated. The presence of iron in the spleen has already been shown by various observers, but the exact nature of the substances with which it is combined is still a matter of doubt. The object of this investigation was, if possible, to trace the stages in the decomposition of haemoglobin as shown by the existence of intermediate compounds, and to ascertain the chemical nature of such compounds. The work already done has been partly histological and partly chemical. The present communication is to be regarded as of a preliminary character, and indicates the lines along which further researches will be carried out. (A.) Histological. This portion of the work was devoted entirely to ascertaining what were the histological elements containing the iron or iron- containing substances other than the hsemoglobin present in the red corpuscles. The method adopted was that of treating sections of material hardened in alcohol with potassium ferrocyanide and hydrochloric acid, and thereafter counterstaining with carmalum. This method leads to the development of Prussian blue, while the cell nuclei are coloured by the counterstain, and the cellular elements so rendered more prominent. Treatment with ammonium sulphide was also adopted, but in this case it was found advisable to avoid the carmalum counterstain, as this stain seems to attack the iron sulphide thus formed and to decolorise it. * This research is being conducted for the British Association for the Advance- ment of Science by a Committee consisting of Professor M ‘Kendrick, Professor Stockman, and Dr Brodie Brodie. 22 Proceedings of Royal Society of Edinburgh. [sess. It is advisable to apply both of these reagents for the detection of iron in the splenic elements ; for although the Prussian blue method yields the more brilliant results, yet there are some structures which give with it a doubtful reaction, and yet, when treated with ammonium sulphide and no counterstain applied, are seen to be distinctly darkened. Up to the present time these observations have been made only on the spleens of sheep and of rabbits, and it may be stated that iron-containing elements have been observed in every case. The amount of iron present and its histological position vary, however, a good deal in different animals. By the position of the iron is meant whether it is intra- or extra-cellular, both of which con- ditions are to be observed, and both usually side by side in the same section. I. Of the extra-cellular forms there are several varieties. (a) Small globules lying free in the spaces of the spleen pulp. They vary in size from very minute points to globules approaching, and in rare cases even exceeding, in size that of the red blood corpuscles. Some of the larger globules seem to be formed by coalescence of the smaller ones, and in all cases the coloration both with potassium ferrocyanide and ammonium sulphide is of great intensity. (b) It is quite common to find in sections of spleen dark brown masses which frequently appear crystalline. These seem to contain iron also, although its presence is not so readily demonstrated as in the previous case, probably owing to the already deep colour of the crystals, which obscures the effect of the ferrocyanide. Occasionally, however, some of these bodies are surrounded by a blue halo ; and in places where a large mass has been shattered by the impact of the knife during cutting, the finer portions of the debris exhibit the dark blue reaction, and around the fine particles there is usually a good deal of diffuse blue staining. (c) This third form is one regarding which I was for some time in doubt as to whether it really contained iron or not. Some of the bodies, however, undoubtedly darkened with ammonium sulphide, while any development of Prussian blue was frequently doubtful. These bodies are of variable shape and size, highly 1901-2.] The Condition of the Iron in the Spleen. 23 refractive, and when unstained, of a greenish-yellow colour. They may often be observed as if they were in process of coales- cence to form larger masses, which, along with the many peculiar forms they assume, give one the impression that they are viscid and semifluid in consistence. They seem to resemble closely the bodies described as stromata exuded from red blood corpuscles by Dr William Hunter.* II. Intra-cellular iron is found in the spleens of both rabbits and sheep. The iron-containing cells of the sheep are much smaller than those of the rabbit. In both cases, however, the iron-containing substance is in the form of more or less discrete granules, and there is also, in some cases, a diffuse staining of the protoplasm. Some cells contain vacuoles, but only in rare instances have I observed blood corpuscles enclosed therein. These are the leading points which I have observed so far as regards the presence of the iron. The presence within cells led me to carry out at the same time investigations into the proteids of the spleen, to ascertain whether iron existed in combination with such proteids. The methods and results were as follows. (B) Chemical. Iron- containing substances in the spleen. Four or five fresh spleens were minced up, pounded with sand and a 5 per cent, solution of chloride of sodium, and the fluid was then squeezed out by pressure in a meat press. The turbid haemoglobin-stained fluid so obtained, when mixed with an equal volume of a saturated solution of ammonium sulphate, gave a copious precipitate, which was filtered off — (Pre- cipitate A). The filtrate was perfectly clear and deeply coloured by haemoglobin — (Filtrate A). The precipitate was redissolved by the addition of water, the resulting solution being very opalescent, hut in thin layers it was sufficiently transparent to admit of any coagulation being readily noticed. By fractional coagulation two substances were separated out, one at about 50° C., the other at from about 60°-70° C. Of * Hunter, Pernicious Anaemia. London, 1901. 24 Proceedings of Royal Society of Edinburgh. [sess. the two heat precipitates, that appearing at 50° C. was the more abundant, and, moreover, the coagulation occurred more sharply. From the solution of the original precipitate (Ppt. A) a white flocculent precipitate was thrown down by the addition of acetic acid. After removal of this precipitate by filtration, the filtrate when heated threw down a coagulum at 50° C. Another portion of the solution of Precipitate A, after being heated to 50°-55° C. and the coagulum filtered off, yielded a precipitate upon the addition of acetic acid. The original precipitate therefore obtained from the splenic extract by half saturation with ammonium sulphate consisted of at least two proteid bodies insoluble in that degree of concentration of the salt, but which were soluble in more dilute solutions. One of these substances coagulates at 50° C., and is not precipitated by acetic acid. The second proteid coagulates at 60°-70° C., and is thrown out of solution by acetic acid. These two proteids probably correspond to those described by Gourlay * as occurring in the spleen, viz., a cell globulin coagulating at 49°-50° C., and a nucleo-proteid coagulating at 57°-60° C. There is a considerable discrepancy between the co- agulation point of Gourlay’s nucleo-proteid and that of my second substance. This possibly may be accounted for by difference in nature and concentration of the saline matters in the solution. Gourlay used a 5 per cent, solution of magnesium sulphate to make his extract, and examined that extract to determine the coagulation points. At the same time I found that this proteid did not coagulate sharply. The fluid began to grow opalescent at 60° C., but distinct flocculi would not separate out, even when kept for a considerable time at 60°-65° C. The formation of flocculi only became apparent after the temperature had been allowed to rise to 70° C. Having thus determined the existence of two proteid bodies in the solution of Precipitate A, these were separated from the main bulk of the fluid by the following method : — The fluid was heated on the water-bath to 50° -55° C., and maintained at that temperature for half an hour. It was then set aside for twenty-four hours, so that the fairly copious coagulum which had formed might settle to the bottom of the vessel. The * Journal of Physiology, Cambridge, 1894, vol. xvi. p. 23. 1901-2.] The Condition of the Iron in the Spleen. 25 fluid was decanted off, and the sediment washed by decantation, collected on a filter, washed first with water, then with alcohol and ether, and finally dried. This substance may be called Precipitate B. The fluid which had been decanted off was filtered, and then coagulated at a temperature of 70°-72° C. The resulting coagulum, when collected and treated just as in the previous case, constituted Precipitate C. The original filtrate from the half-saturation of the splenic extract with ammonium sulphate (Filtrate A) was then saturated by shaking up with crystals of the salt. A copious brown-coloured precipitate resulted, which contained the haemoglobin, the filtrate from this being almost colourless. This precipitate (Precipitate D), after solution in water, was coagulated by heat, and the brown coloured clot was thoroughly extracted by alcohol and ether (2 :1) containing 1 per cent, of oxalic acid. By this extraction the haematin was easily removed. Afterwards the coagulum was collected, washed and dried, and reserved for further examination for the presence of iron, as was also the case with the two substances— Precipitates B and C. It should be stated that in order to avoid any chance of error from the presence of haematin, Precipitates B and C were also extracted with the alcohol-ether-oxalic-acid mixture. These substances — Precipitates B, C, and I) — were suspended in water and then treated (1) with ammonium sulphide, which gave a black colour; (2) with hydrochloric acid and potassium ferrocyanide, the result being the development of Prussian blue ; (3) with hydrochloric acid and ammonium sulpho-cyanide, which gave the usual red reaction. These tests indicate the presence of iron in the proteid bodies obtained from the spleen by the methods already described. Each of the precipitates also gave a small quantity of ash, which on analysis showed the presence of iron. Finally, it may be mentioned that nucleo-proteid matter obtained from the spleen by Halliburton’s sodium-chloride method was also found to contain iron. Thus substances containing iron have been identified histologi- cally, and proteids containing iron have been separated by chemical methods. ( Issued separately February 17, 1902.) 26 Proceedings of Royal Society of Edinburgh. [sess. A Theoretical Representation leading to General Sugges- tions bearing on the Ultimate Constitution of Matter and Ether. By John Fraser, Ordnance Survey. Com- municated by Professor Chrystal. (Read January 6, 1902.) The idea has long been floating around in thoughtful minds that all the different varieties of matter in the world are formed out of one primordial substance or stuff, and that the different qualities they exhibit are caused by a mere difference of number, arrange- ment, and motion of the primordial units forming the atoms of our so-called elementary substances. For instance, the atom of oxygen would contain sixteen times as many of these units as the atom of hydrogen, which would account for the difference of their atomic weights ; and besides, the motions of the units forming each atom would be different for the different atoms, thus accounting for their other different qualities. Nitrogen would contain fourteen times as many of these units, carbon twelve times, and so on according to their atomic weights. It appears to me that the ether of space must be, if such a thing exists, this primordial substance or stuff out of which all bodies are formed. It seems to exist throughout all space. At least, wher- ever a star can be seen, the space between us and that star must be full of this stuff, for it is the vehicle by which the light of that star travels to us; and it seems to fill all ponderable bodies, as water does a sponge, for its vibrations pass through transparent and diathermanous bodies in the form of rays of light and heat, and by the X-rays we know that so-called opaque bodies are penetrated by it. Well, then, what is the constitution of the ethereal units? They must be perfectly elastic, or all motion between their parts would soon come to an end ; and here is the great difficulty — if they are units, they cannot be formed out of anything else ; they must be wholes, and continuous throughout their mass — that is, contain 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 27 no pores within their mass — to account for what passes at present as the cause of elasticity. To give the reader an idea of this great difficulty I will quote from Stallo’s Concepts and Theories of Modern Physics , p. 40, his criticism of the various attempts to get over it. Those of my readers who are conversant with the subject may skip this part ; I quote for the benefit of those who are not. It also lays the subject open in a form which I could not hope to do. “ Chapter IV. — The Proposition that the Elementary Units of Mass are absolutely Hard and Inelastic. “ From the essential disparity of mass and motion and the simplicity of the elementary units of mass it follows that these units are perfectly hard and inelastic. Elasticity involves motion of parts, and cannot therefore be an attribute of truly simple atoms. “The concept ‘elastic atom,’ justly observes Professor Wittwer, ‘ is a contradiction in terms, because elasticity presupposes parts, the distances between which can be increased and diminished.’ “The early founders of the mechanical theory regarded the absolute hardness of the component particles of matter as an essential feature of the original order of nature. ‘ It seems probable to me,’ says Sir Isaac Newton, ‘ that God in the beginning formed matter in solid, massy, hard, impenetrable, movable particles, of such sizes and figures, and with such other properties and in such proportion to space, as most conduced to the end for which He formed them ; and that these primitive particles being solids, are incomparably harder than any porous bodies compounded of them ; even so very hard as never to wear or break in pieces ; no ordinary power being able to divide what God Himself made one in the first creation.’ “Strangely enough, while the requirement by the mechanical theory of the absolute rigidity of the elementary units of mass is no less imperative than that of their absolute simplicity, it meets with an equally signal denial in modern physics ! “The most conspicuous among the hypotheses which have been devised since the general adoption of the modern theories of heat, light, electricity, and magnetism, and the establishment of the 28 Proceedings of Boyal Society of Edinburgh. [sess. doctrine of the conservation of energy, in order to afford consistent ground for the mechanical interpretation of physical phenomena, is that known as the Kinetic theory of gases. “In the light of this theory, a gaseous body is a swarm of in- numerable solid particles incessantly moving about with different velocities in rectilinear paths of all conceivable directions, the velocities and directions being changed by mutual encounters at intervals, which are short in comparison with ordinary standards of duration, but indefinitely long as compared with the duration of the encounters. It is readily seen that these motions would soon come to an end if the particles were wholly inelastic, or imperfectly elastic. For in that case there would be loss of motion at every encounter. The assumed perpetuity of the motion of the particles therefore leads to the necessity of asserting their perfect elasticity. And this necessity results, not merely from the peculiar exigencies of the kinetic theory of gases, hut also from the principle of the conservation of energy in its general application to the ultimate constituents of sensible masses, if these constituents are supposed to he in motion. In the case of the collision of ordinary inelastic or partially elastic bodies, there is a loss of motion which is accounted for by the conversion of the motion thus lost into an agitation of the minute parts composing the colliding bodies. But in atoms or molecules destitute of parts no such conversion is possible, and hence we are constrained to assume that the ultimate molecules of a gaseous body are absolutely elastic. “ The necessity of attributing perfect elasticity to the elementary molecules or atoms, in view of the kinetic theory of gases, has been expressly recognised by all its founders. £ Gases,5 says Kroenig, ‘ consist of atoms which behave like solid, 'perfectly elastic spheres, moving with definite velocities in void space.5 This statement is adopted by Clausius, and emphasised by Maxwell, the first part of whose essay, ‘ Illustration of the Dynamical Theory of Gases,5 is a treatise on the motions and collisions of perfectly elastic spheres. And the highest scientific authorities are equally explicit in declaring that the hypothesis of the atomic or molecular constitution of matter is in conflict with the doctrine of the conservation of energy, unless the atoms or molecules are 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 29 assumed to be perfectly elastic. ‘ We are forbidden,’ says Sir William Thomson, 1 by the modern theory of the conservation of energy to assume inelasticity or anything short of perfect elasticity of the ultimate molecules, whether of ultra-mundane or mundane matter. . . Secchi clearly apprehends the inadmissibility of attributing elasticity to simple elementary atoms. c It is evident,’ he says, ‘that while it is possible to admit its existence in a compound molecule, the same thing cannot be done in the case of elementary atoms. Indeed, elasticity in the received sense presupposes void spaces in the interior of the molecule, whose form is changed by compression so as to return afterward to its original figure. Now, we regard the atoms as impenetrable, and not as groups of solid particles; hence they cannot include void spaces which permit their dilatation and contraction.’ ... The difficulty, then, appears to be inherent and insoluble. There is no method known to physical science which enables it to renounce the assump- tion of the perfect elasticity of the particles whereof ponderable bodies and their hypothetical imponderable envelopes are said to be composed, however clearly this assumption conflicts with one of the essential requirements of the mechanical theory.” [It is easy, however, to see that the assumption of perfect hardness and perfect rigidity in the ultimate particles of the ether may be regarded as identical with perfect elasticity, if that be understood in the sense that what Thomson and Tait* call the ‘ coefficient of restitution ’ is unity for the ether-atoms. To simplify matters, consider two perfectly hard, perfectly rigid spherical atoms of equal mass ; and let their velocities before and after a direct impact be a b and u v respectively. Since no part of the kinetic energy due to their motions of translation can be converted into internal motion, or otherwise lost, we must have u2 + v2 — a2 + b2 (1). Also the third law of motion requires that the sum of their momenta shall be the same before and after impact. Hence we must have u + v = a + b (2). * See Treatise on Natural Philosophy by Thomson and Tait (new ed.), vol. i. part i. §§ 300-306. 30 Proceedings of Royal Society of Edinburgh. The two equations (1) and (2) admit of two, and only two, solutions, viz. — u = a, v — b; and u — b , v = a. The first leads to a dynamic paradox, which it is needless to discuss; the second, viz., that the particles simply exchange velocities, is the well known result of supposing the coefficient of restitution of the particles to he unity. (See Thomson and Tait, l.c.) The dynamics of the more complicated cases that arise when we take the masses of the particles to he unequal and the impact to be indirect is merely a particular case of the usual formulae worked out from Newton’s fundamental law. As our ether particles are assumed to have no internal structure, and to he absolutely undeformable, we are not to expect to find any exact counterpart in ordinary tangible bodies. The nearest approach which Newton found in his experiments on impact was the case of balls of glass, for which he found the coefficient of restitution to be y|-. As a mere matter of logic, we are entitled to attribute to our ethereal particles any properties which do not contradict the laws of motion, and in particular the Law of the Conservation of Energy. This having been seen to, it is merely a matter of phraseology whether we speak of them as perfectly rigid or perfectly elastic. The point is, that their impacts are to be infinitely short in duration, to produce no deformation, to involve no loss of energy.*] The Constitution of the Ether. I now proceed to introduce my theory of the constitution of matter. And first as to the ether : — I regard the ether, as I have already hinted, as the substratum of all matter, and to consist of perfectly hard, globular, smooth, and inconceivably small bodies, of equal mass, and equal in every other respect. It seems to be everywhere present in space and in the pores of other bodies ; and as for its density, instead of being, as is generally conceived, exceedingly tenuous, I, for my part, am of the opinion of Sir J. Herschel, viz., that its density is like that of “an adamantine solid.” In fact, that it is far denser than the densest metal. *The part within brackets has been written by a mathematician, who advised me that it was better put in this way than in the way in which I originally put it. It is, at any rate, far more concise. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 31 As the opinion of a physicist and mathematician like Sir J. Herschel is of far greater weight than that of an obscure individual like myself, I make no apology for quoting at some length his conclusions on this subject, from his lectures on Light, 1873 : — “ As we have attempted to form an estimate of the forces required to account for observed facts on the corpuscular hypothesis, let us now attempt a parallel estimate on the undulatory. And here the way is equally open and obvious. Starting with the observed facts that sound travels in the air at the rate of 1090 feet per second, while light is propagated through the ether 186,000 miles in the same time (that is to say, 901,000 times as fast), we are enabled to say how many fold the elastic force of air, or its resistance to compression, would require to be increased in proportion to the inertia of its molecules , to give rise to an equally rapid transmission of a wave through it. For it results from the theory of sound that in media of different elasticities (so understood), but similarly constituted in other respects, these forces are to each other as the squares of the velocities with which the waves travel. So that the elastic force of the air would require to be increased in the proportion of the square of 901,000 ( i.e ., 811,801 millions) to one to produce an equal velocity. Even this enormous number must be still further increased, since the velocity of sound is augmented by a peculiarity in the constitution of the air which we would hardly be justified in attributing to the luminiferous ether, in virtue of which its elasticity is increased by heat given out in the act of compression, and without which the velocity of sound would be only 916 feet per second, instead of 1090. Thus the number above arrived at has to be further increased in the proportion of the square of 1090 to that of 916, which brings it to 1,148,000,000,000. Let us suppose, now, that an amount of our ethereal medium equal in quantity of matter to that which is contained in a cubic inch of air (which weighs about one-third of a grain) were inclosed in a cube of an inch in the side. The bursting power of air so inclosed we know to be 15 lbs. on each side of the cube. That of the imprisoned ether would be 15 times the above immense number (or upwards of 17 billions) of pounds. Do 32 Proceedings of Royal Society of Edinburgh. [sess. what we will — adopt what hypothesis we please — there is no escape, in dealing with the phenomena of light, from these gigantic numbers, or from the conception of enormous physical force in per- petual exertion at every point , through all the immensity of space .... Every phenomenon of light points strongly to the con- ception of a solid rather than a fluid constitution of the luminif- erous ether, in this sense— that none of its elementary molecules are to be supposed capable of interchanging places , or bodily trans- fer to any measurable distance from their own special and assigned localities in the universe.” This is the famous ‘jelly’ constitu- tion of the ether, which has been much ridiculed, but never yet confuted. The only argument of any weight against it is the difficulty of conceiving how solid, massive bodies like the planets can trace out their orbits, time after time, with unvarying regular- ity through it, without showing the slightest sign of retardation. Well, now, I think if I can rid the theory of this particular stumbling-block — and I think I may promise my readers that I can — it ought to be considered once more as within the bounds of the possible. Futhermore, I may help to greatly strengthen it by pointing out the possible — nay, probable — source of the intense pressure which Herschel found must exist in the medium. When we look abroad on a clear, dark night, we find the sky -studded with sparkling points of light in every direction ; and if we use a telescope, it is found that the higher its powers the more -closely set do these points of light become, until it is difficult to resist the conclusion that the bodies which emit the light are existing in every direction in space, and that the reason the sky does not show up as one square sun, as it were, is that a great deal of the light may be absorbed, or change its character on its practically infinite journey towards us; in short, that the uni- verse of stars is practically infinite. Now, it is practically certain that all those stars are suns ; suns, some of them far larger, and some no doubt smaller, than our own sun, each emitting light and heat, generally, comparable with the sun. Now, what becomes of this light and heat? Is it wasted? No; the principle of the conservation of energy forbids that it should be wasted. What, then, becomes of it? Let us consider. If the ether is formed of discrete particles — and I think it is impossible 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 33 to clearly conceive of it being formed of anything else — then a wave of heat or light (the undulatory theory being conceded) consists of a condensation, or a crowding together, of a more or less number of particles, all at the same distance from the source of the wave, or point of disturbance, followed by a dilatation or spreading out of the particles, — in fact, a spherical shell of con- densed ether, followed by another of dilated ether, spreading out, like the circular waves on a pond surrounding the point disturbed by a stone, in every direction in space. But as they spread out they dwindle in amplitude, and the further they spread the more they dwindle, till at last they can dwindle no further, having reached the point where the shell or condensation would be only one particle thick, or where there would be only one particle left on the front or crest of the wave moving by itself and giving up its motion to its neighbour, and so on continually. At this point the wave would cease to exist as such, but the single-particle motion would go on for ever, or until it was absorbed in doing some work. The spherical shell would be no longer continuous — would break, in fact — but the amount of motion would be the same as at the start.* At the start, the particles partaking in the motion would be all crowded together close to the surface of the luminous body. At the end of the existence of the leave , who shall say how many thousands of billions of miles separate the particles to which the motion has been transmitted? Now, every wave of heat or light in the universe, if not absorbed in doing work, must at last result in this degraded form ; and as the universe of stars is, at least practically, infinite, there must be in space a practically infinite amount of these single-particle motions equal in amount from every direction; motion from one direction being * After the condensation, or shell, was reduced to a single particle in thick- ness, the motion still continuing, as it must, the particles to which the motion has been transmitted can no longer be neighbours, but gradually get separated, till at last they may be any distance apart. So that the motions which generate the pressure in the ether, and in which contiguous particles are engaged, may be generated in opposite quarters of the universe. And these motions can no longer be wave motions, for the latter consist in a simultaneous motion of all the particles, at the same distance from the point of disturbance of the medium, in the same direction ; but the pressure motions being generated at all distances, in opposite directions, act on each particle independently, and thus subject the medium to simultaneous and contrary motions, and which generate the intense pressure. PROC. ROY. SOC. EDIN. — VOL. XXIV. 3 34 Proceedings of Royal Society of Edinburgh. [sess. neutralised by an equal motion from the opposite. Hence the pressure in the medium which Sir J. Herschel found must exist.* Further, to show that such pressure, or something equivalent to it, must exist in space, it has only to be pointed out that if the earth fell from its present distance into the sun, under the action of gravity, the amount of heat developed by the impact would cover the solar emission for about 95 years. The impact of Jupiter, if he fell from his present distance, would cover it for 32,240 years. This enormous amount of heat, or its equivalent in some other form of energy , must reside in the space separating the bodies, or in the bodies themselves. We cannot suppose it to be created out of nothing. It is there, and we cannot get away from it. But we see a practically infinite amount of heat (the radiation of the stars) apparently wasted. What, then, more natural to conceive than that, instead of being wasted, it produces the intense pressure, other- wise not accounted for, mathematically demonstrated by Sir J. Herschel ? Amongst other reasons for Sir J. Herschel’s conception of a solid constitution for the ether, was the necessity of conceiving of some bounding envelope to restrain the pressure; for he says — that under no conception but that of a solid can an elastic and, expansible medium be self-contained. If free to expand in all directions, it would require a bounding envelope of sufficient strength to resist its outward pressure. And to evade this by supposing it infinite in extent, is to solve a difficulty by words without ideas — to take refuge from it in the simple negation of that which constitutes the difficulty.” How, it seems to me that the solid conception does not rid us of this difficulty ; for, granted that the solid constitution does prevent its constituent particles from flying off into space, the mere statement of this fact does not at all explain how it is brought about. But if the theory I am dealing with can give a reasonable and mechanical ex- planation of the fact, it ought to be a great point in its favour. And first as to the ether, — other solid bodies being dealt with in the chapter on the constitution of matter. Although I hold with Sir J. Herschel that the ether is a solid in the sense that he limits * It goes without saying that I conceive this pressure as the source of all the so-called “Forces.” .3901—2.] Mr J. Fraser on Constitution of Matter o.nd Ether. 35 it to, yet it cannot be a solid like the solids of our experience, as the reader will see when I come to deal with the constitution of matter. What, then, is the bounding envelope resisting its outward pressure, spoken of by Herschel? I have pointed out that the probable cause of the pressure was the radiations of a practically infinite number of heated bodies in space — that in every direction in space some of those bodies were situated. Seeing, then, that it is the action of those bodies on the medium which causes pressure, the reaction of the medium on the bodies can only be equal to the action, not greater. In other words, the bodies are capable of resisting the pressure which they themselves produce; and as, ex liypothesi , they exist in every direction in space, there is the bounding envelope.* The Constitution of Matter. I come now to the consideration of the constitution of ordinary matter, that is, matter which we can see, or weigh, or perform other operations upon, — the ether, I need hardly add, eluding all cognisance of this kind. And yet, as I have hinted at the be- * It has been objected to the constitution of the ether herein set forth that it would be nothing more nor less than a gas. This objection I quote from Stallo’s Concepts and Theories of Modern Physics, p. 97 : — “The negative evidence here adduced against the supposition of an atomic or molecular constitution of the light-bearing medium is re-enforced by positive evidence derived from a branch of the atomic theory itself — the modern science of thermo-dynamics. Maxwell has remarked, with obvious truth, that such a medium (whose atoms or molecules are supposed to penetrate the intermolecular spaces of ordinary substances) would be nothing more nor less than a gas, though a gas of great tenuity, and that every so-called vacuum would in fact be full of this rare gas at the observed temperature and at the enormous pressure which the sether, in view of the functions assigned to it by the undulatory theories, must be assumed to exert. Such a gas, therefore, must have a correspondingly enormous specific heat, equal to that of any other gas at the same temperature and pressure, so that the specific heat of every vacuum would be incomparably greater than that of the same space filled with any other known gas.” I submit, it will be found further on that this objection is met by the difference which this theory assigns to the constitution of the ethereal atom from that of the material atom. The ethereal atom is a simple, structureless body, incapable of absorbing or retaining heat, simply passing it on as received from its neighbours on the one side to its neighbours on the other ; but the material atom and molecule are furnished with springs, so to speak, and upon which motion can be impressed so as to be retained by them for some time, until by their vibrations they gradually ■communicate it to the ethereal medium. 36 Proceedings of Royal Society of Edinburgh. [sess. ginning, it is out of the ether I propose to construct all the matter of our experience. Now, in the beginning, to prevent confusion, I propose to call the unit of ether a particle , and the unit of ordinary matter an atom, a molecule being two or more atoms united together, as in the science of chemistry. Well, now, how are we to construct these atoms of chemistry out of the particles of the ether so that they shall exhibit all their chemical and physical peculiarities, such as “attraction of cohesion,” “chemical affinity,” “valency,” etc. ; also, and above all, motion through the ether without resistance and ‘ 1 attraction of gravity ” in pro- portion to mass? If we suppose the ethereal particles to be gathered together into little masses in proportion to the atomic weights, with what qualities are we to endow them so as to enable them to preserve their identity under all conditions ; and also, as I propose to utilise the pressure of the ether for the solution of this enigma, how arrange the particles so that gravity shall be exactly proportional to mass, i.e., so that each particle shall be subjected to exactly the same pressure ? I believe there is one way, and but one way only, by which this may be done, and that is, to endow the constituent ethereal particles of each atom with motion athwart the direction of the pressure in the medium ; in the same way as the planets are endowed with motion athwart the direction of the sun’s so-called attraction, and in such sort that they shall revolve round and round a vacuous space, and so closely set together or otherwise endowed with so much motion as to preclude the possibility of any of the particles of the medium penetrating the vacuum. In fact, a veritable ethereal bubble. Yes ; but a bubble, the skin of which, unlike a soap-bubble, is only one particle thick ; and also unlike a soap-bubble, in that inside there is perfect emptiness — an absolute vacuum ; and also unlike it, in that all the particles forming the film are in rapid motion in great circles round the vacuum. Of course, in a bubble of this kind there would be innumerable collisions between the revolving particles. Indeed, if they happened to be very closely set, a particle might take a very long time in going round the bubble, or for that matter it might never go round it at all ; all that is essential is that the motion of each should be fully trans- ferred to its neighbours, and by them to be retransferred in every 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 37 direction, in great circles, all round the bubble, to preserve it unbroken to all eternity, or so long as the pressure lasted. Now, the first thought to strike one on conceiving an idea of this kind is, how is it possible for motion to be conserved under the above circumstances 1 And to answer this satisfactorily, one must disabuse one’s mind of the idea of ordinary ponderable bodies being engaged in the operation. The bodies engaged are ideal ones — none the less real for all that — ideally smooth, ideally globular, and infinitely hard or rigid, so that motion can be transferred from one to another instantaneously without any loss, which is the same as saying that they are perfectly elastic. Another thing which must not be forgotten is, that they move in pure empty space ; for they only come into contact with the particles of the medium at their outer surfaces, and these are per- fectly smooth, and those contacts are only momentary, for the particles of the medium can never be at rest, but are incessantly bombarding the revolving particles in a direction at right angles to their orbits, just as the planets are acted upon by the sun’s gravity. But an objection will arise — and perhaps this is the best place to deal with it — that, owing to the composition of the velocities of the bombarding and revolving particles, or the particles of the medium and of the bubble, the revolving particles would sustain more impingements on their fronts than on their rears — that is, that the impingements of the particles of the medium on those of the bubble would, instead of being strictly at right angles, be directed more on the front, and thus motion would be lost. This objection, if it could be sustained, would, apparently at least, prove fatal to the theory. But, happily, it cannot be sustained, as I shall show. It is a well known fact to astronomers, that should a planet moving in a circular or other orbit meet with any obstruc- tion to its motion, that the obstruction, instead of slowing down the motion, would ultimately hasten it ; and that after being in perihelion, it would return to that part of its orbit where it was obstructed, to go through the same course again if it were no further obstructed, but if it were, each return to the perihelion after obstruction would be at shorter and shorter intervals, and a less and less distance from the sun, till at last it would graze his 38 Proceedings of Boyal Society of Edinburgh. [sess. surface, and only then would its aphelion distance begin to shorten, when, after a greater or less number of revolutions, grazing his surface at every return, its orbit would become circular once more, just before being absorbed by the sun. In all the foregoing pro- cess the planet, before grazing the sun, would be gaining kinetic energy, but at the expense of potential. In other words, it would be gaining speed, but at the expense of position, for it would be all the time drawing nearer and nearer to the sun ; and although its speed would be increased, yet the sun’s hold on it would be increased in the same degree, until, as already described, it would become ultimately absorbed. Well, now, what would be the difference between our hypothetical planet and a revolving par- ticle ? The main difference would be in one profound particular,, and that is, that in the case of the planet, as the speed increased, the centripetal force increased in the same degree, but in the case of our particle no such increase of centripetal force would take place, for there is no central sun for it ; and supposing our particle to be placed near the centre of the bubble, exposed to the full pressure of the outside ether, it is quite clear that the pressure would be exactly the same in that position as elsewhere. Well, then, in the case of the planet, as before said, as the centrifugal tendency increased, so would the centripetal, so that it could not escape from the sun, but in the case of our particle, no increase of centripetal force would take place, and owing to its increased centrifugal tendency, induced by increase of speed, it would regain its former position, with its former speed, with this difference — that its orbit would thenceforth change from a perfectly circular to a slightly elliptical one, to go through the same course again,, and so on ad infinitum. Having now, as I hope, shown by sound analogy that the orbital motion of these particles might go on for ever, or so long as the pressure existed, we may now inquire into the relative sizes and consistencies of the bubbles, or atoms. It is not for me to say how bubbles of this kind could originate, but we may suppose' that a great number of vortical motions, of various degrees of energy, were impressed on the ether; each vortex then would expand into spherical bubbles of various sizes, and quantities of ethereal particles, according to their various energies and the: 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 39 quantities of ether upon which the motion was impressed. The expansion would continue until all the particles worked their way out to the surface by centrifugal tendency, and afterwards would still continue to expand until this tendency and the pressure of the ether exactly balanced one another.* I said they would work their way out to the surface of the bubble because of the greater centrifugal tendency of the inner particles, owing to their narrower orbits in that position, and which orbits at the surface would, of course, be equalised. In this way bubbles of various sizes and degrees of consistency might be formed. For instance, a bubble — or atom, as I shall call it after this — might consist of 16 or any other number of times the number of units as another, and yet not have a greater surface area than the latter, the constituents of the atom of least mass in proportion to size having a correspondingly greater motion. Their motions must be so rapid as to preclude the possibility of any of the particles of the medium ever penetrating the surfaces of the atoms. To illustrate the possibility of this, suppose a stone to be thrown at a wheel in very rapid rotation, the stone could not penetrate between the spokes if the velocity of the wheel were great enough, for, before the stone could penetrate more than a small distance, some one or other of the spokes would be certain to intercept it. The particles must move rapidly enough to cover the space between them over and over again, in every direction normal to the surface of the atom, and to be, as it were, everywhere at once, in order to prevent penetration. That this motion must be far greater than the motion of the in-pressing particles of the medium may be realised by referring to the speed of a planet round the sun, as compared with the distance it would fall towards him in the * It will, no doubt, be observed that this is a position of unstable equili- brium, which the slightest disturbance would overthrow. How, then, are we to save the situation ? It is saved already. For, as pointed out above, the particles would be more bombarded on their fronts than on their rears, so that although, by falling in towards the centre, they would regain their former speed, their orbits, if ever they were circular, would never be so again, but always elliptical, and no bubble could ever be in so expanded a state as it would be possible for it to be if this excess of bombardment on the fronts of the particles over the rears did not exist. In short, owing to this excess of bombardment on their fronts, no bubble could ever expand to the stage of unstable equilibrium. 40 Proceedings of Royal Society of Edinburgh. [sess. same time under the action of gravity. For instance, the earth’s speed in her orbit is about equal to 18 J miles per second, while the distance she falls towards the sun, under gravity, in the same time, is about the -Jth of an inch. Of course the disproportion would not be nearly so great as this, but something of this order. The centrifugal force increases inversely as the radius, and the radii of these orbits are excessively small, but, as shall presently appear, the force of gravity is a mere differential force, and is not likely in any case to be equal to the total pressure in the medium, whereas the pressure on these particles is equal to the total pressure in the medium for their mass. But this merely by the way, the meaning of which will be more apparent further on. The larger the bubble, or atom, then, in proportion to its weight, the greater the speed of its constituent particles. For instance, two atoms of the same size, but weighing one twice as much as the other; the speed of the particles of the lighter of the two would be twice as great, for they would have twice the area to protect from penetration. Of course, with twice the speed there would be four times the centrifugal tendency, but also there would be four times the centripetal, for, with twice the speed, each particle of the bubble, or atom, would come into contact with twice the number of in-pressing particles of the medium, and their pressure would be continued through twice the distance. Or, the pressure would be twice the quantity through twice the distance, equal to four times; and so on in proportion for other speeds.* Now, it would appear from a superficial view of the foregoing that the pressure on any atom would be proportional to the surface area, and not to the number of particles in it ; but this is an erroneous view of it. The pressure would be at any instant directly proportional to the number of particles in the atom, and which, as they (the particles) are of equal size and mass, is the same as saying that the pressure on any atom would be directly propor- tional to its mass. * Or, to put it in another way, as in twice the distance of a circular orbit there is four times the curvature, twice the pressure for the same time pro- duces twice the velocity = four times the kinetic energy, which would produce four times the curvature in the paths of the particles. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 41 The Cause of Gravity. We may now consider the cause of gravity in bodies. I said at the end of the last paragraph that the pressure on any atom would he directly proportional to its mass, and not to its surface area, as would appear at a first view. I hope that any of my readers will not have the thousandth part of the trouble over the elucidation of this point as I had myself, before the explanation occurred to me. Preliminarily, then, it must be pointed out that each particle of every atom is, instant by instant, taking toll, as it were, of the store of pressure in the medium, and, instant by instant, restoring it again. This modicum of force, or pressure, is used up in curving the paths of the particles in their orbits, and it is restored again by their centrifugal tendencies. At any particular instant there is a certain quantity of force or pressure absent from the medium ; and to show that this quantity will be precisely proportional to the number of particles taking part in the process, I will use a homely illustration. Suppose two runners running round a circle, and one of the two to be running twice as fast as the other. Further, suppose the spectators to be stationed 10 yards apart all round the circle, and as each runner passed to bestow a kick on him, which was immediately returned by the runner, or rather both runner and spectator kicked almost simultaneously, it is perfectly clear that the fastest runner would receive most kicks, but also they would be quicker returned ; and supposing the first kick was not returned by him till he received the second, nor the second until he received the third, and so on, it is clear that he would never be the recipi- ent of but one unpaid kick at any moment. The fastest runner, then, would receive twice as many kicks in the same time as the slow one, but at any instant he would be the recipient of only one unpaid kick, just like the slow one. Or take another case : Suppose two traders to be dealing at the same bank, and one to borrow money at the rate of £100 per week, to be repaid at the end of the week, without interest, when another £100 would be withdrawn; and the other to borrow at the rate of £100 per month, to be repaid at the end of each month, also without interest — a proceeding which no bank out of Utopia 42 Proceedings of Royal Society of Edinburgh . [sess. would permit of, blit the bank I have in my mind’s eye requires only the money advanced to be returned, neither more nor less. Well, then, it is clear that by the transactions of these two men, although one of them was making use of about four times more of the bank’s money than the other, the bank at any particular instant^ owing to these transactions, would be deficient in only <£200, each of the two traders owing it exactly £100. Now for the application of the parable, which the knowing reader already sees. The bank is the store of pressure, or force, in the medium ; the traders are the particles of the atoms ; the quick borrower is the nimble particle, and the slow borrower is the less nimble particle. The quickest moving particles borrow at a quicker rate from the bank of pressure, but also they repay at a proportionally quicker rate. Take, for example, two atoms of the same size but different weights, at any instant the pressure on each would be exactly as their weights ; but during any small portion of time, the pressure on each for that time would be proportional to their areas, providing the time were great enough to permit of the particles to each move once through its own domain or orbit. (Tor each particle would have its own little domain to protect from penetration, moving about in a space proportional to the size of the atom distributed amongst the number of its particles ; for, of course, no particle could complete an orbit right round the atom owing to collisions with its fellows, and would therefore be confined to one particular locality.) Well, then, each particle would be subjected during that time to an amount of pressure equal to that which would be received by a surface equal to the area of its domain. This pressure would not be on the particles all at one time, but would be distributed throughout the space through which they moved, a certain quantity being allotted to each portion of the space equal to that which they themselves would occupy when at rest, or equal to the disc-area of each particle. To sum up, then : Atoms are, some comparatively large, others comparatively small ; some comparatively open-grained, others comparatively close-grained in texture. The particles of the open or spongy kind will have a greater velocity than those of the close or rigid kind, they having, as it were, to perform, each of them, the duty assigned to two or more of those of the rigid kind, or at least to more than one. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 43 The particles of the spongy kind are drawing on the store of pressure at a greater rate than those of the others, but also they are repaying at a proportionately greater rate, so that at any assigned instant they owe no more than the others, or have no more capital withdrawn from the bank than any of the others. In fact, each particle of every mass will, at any particular instant, have precisely the same amount of pressure or force withdrawn from the universal store, and at that precise moment being put to another use, viz., that of changing the direction of motion of the said particle. It ought not now to be very difficult to see what is the cause of gravity or the apparent attraction which one body has for another. Every body in the universe will have, at any instant of time, a quantity of force or pressure withdrawn from the medium, precisely proportional in amount to the number of particles of which its atoms consist. In other words, the quantity of force withdrawn and otherwise utilised will be precisely proportional to the mass. Well, then, from the sun, and from every other body in the universe, there will be a slope or incline of pressure, so to speak, radiating out in every direction through space. In the neighbour- hood of each body the slope will he steepest, and its steepness will be proportional to the mass of the body, and growing less and less steep as the square of the distance from it, or growing steeper and steeper inversely as the square of the distance. It is readily seen that this must be so ; for the body is taking toll of the pressure in the space around it, and as the areas of circles are proportional to the squares of their radii, it is evident that in any plane circular space, say at double the distance from the centre of the body, there will he four times the area from which to withdraw the pressure, and that consequently any particular portion of the space at this- distance will be deficient in only a quarter of the pressure. At three times the distance the space will he deficient in only ^th the pressure, and so on inversely as the square of the distance ; the slope or incline of pressure growing less and less steep as the square of the distance grows. Well, now, suppose two bodies to be placed in the vicinity of one another, under these conditions, what must happen? The one would screen the other from the radiations which presumably 44 Proceedings of Royal Society of Edinburgh. [sess. produce the pressure in the medium ; consequently the space separating them would he deficient in pressure, with the result that, owing to the superiority of pressure in the space outside of them over that between them, they would be pressed together. The one would screen the other, in the same way that a screen placed in front of a fire would prevent the radiations of the fire from pursuing the paths in which they originally set out. In such a case the screen in itself becomes a hot body, and, as such, dis- tributes its radiations impartially in every direction ; so much of them being sent back into the fire again as into the screened space, and quite as much in every other direction as into this space, with the result that the latter is kept cool. Now, then, take such a body as the sun,* and another body such as the earth, and consider the condition of the medium in the vicinity of each. In the vicinity of the sun the decrement of pressure would be very rapid, and in that of the earth not nearly so rapid. It would be as though the earth were placed on the summit of an exceedingly high, steep mountain, and the sun on a gently sloping eminence, and permitted to roll towards one another, the motion of each would be proportional to the steepness of its path. Of course their paths must be supposed to grow steeper and steeper the nearer they approach one another, inversely as the square of their distance apart, and directly as their masses. This, then, answers all the requirements of gravity. Every mass in the universe is making momentary use of a quantity of force or pressure out of the medium surrounding it, and which quantity is propor- tional to each mass ; each mass screens every other from the radi- ations which cause the pressure, with the result that the spaces between them are kept free from these radiations in proportion to the screening masses. Consequently, the pressure in the screened spaces being weaker than that outside them, the bodies tend to approach one another in proportion to the difference of pressure. Now, a few words as to the supposed instantaneous action of gravity. It is objected to all hydro -dynamical theories of gravi- tation that they are “obnoxious to the fatal criticism of Arago ” : — * It must be borne in mind that it is only the radiations causing pressure which would be effective here — not the sun’s waves. These would only be effective after dwindling to their lowest form. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 45 “ If attraction is the result of the impulsion of a fluid, its action, must employ a finite time in traversing the immense spaces which, separate the celestial bodies,” — I quote from Stallo’s Concepts of Modern Physics , and he (Stallo) goes on, — whereas there^is now no longer any reason to doubt that the action of gravity is instan- taneous. If it were otherwise — if gravity, like light or electricity, were propagated with a measurable velocity — there would necessarily be a composition of this velocity with the angular orbital velocities - of the planets, resulting in their acceleration ; the apparent line of attraction would be directed to a point in advance of the real place of the sun, just as the sun’s apparent position is displacedi in the direction of the earth’s orbital motion by the aberration of light.” Similarly, R. A. Proctor, in the Old and New Astronomy , p. 321: — “The direction in which the sun’s light seems to reach, the earth is not the true direction of a straight line joining the centres of the earth and sun, but is inclined to this direction so as to be directed from the earth to a point nearly 9200 miles from the sun’s centre. If gravity, acting outwards from the sun, tra- versed the interplanetary spaces with the same velocity as light, the pull of gravity would not be directed to the sun’s centre (which we have seen (Art. 465) is essential to the fulfilment of Kepler’s second law), but at an angle of about 20'445" to that direction. And precisely as the rays of light reach the earth with this slight slant directed towards her in her advance (as rain falling verti- cally seems to meet with a slight slant, directed towards him, a swiftly advancing traveller), so thq direction of the action of gravity on the earth would be inclined with a slant in the direction of the earth’s motion.” And then he goes on to show that this slight slant pull, if it existed, “ would communicate to the earth an increase of velocity in her orbit amounting to about half a millionth of a foot per second, and which would increase as the square of the time, and be very quickly rendered sensible.” The assumption underlying the above reasoning is that tho centre of the earth’s motion is situated on the line joining her own centre with that of the sun, and that the action of the sun on her is a pulling action. Kow, it is utterly inconceivable how one body could pull another without something of the nature of a cord, passing between the two, and which could be wound up as the 46 Proceedings of Royal Society of Rdinburgh. |_sess. distance decreased, and unwound as it increased. Newton himself disclaimed the idea of attraction as an ultimate physical fact. In his third letter to Bentley he says : — “ It is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter, without mutual contact, as it must do if gravitation, in the sense of Epicurus, be essential and inherent in it. And this is the reason why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else by and through which their action may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting constantly according to • certain laws ; but whether this agent be material or immaterial, I have left to the consideration of my readers.” If the action of the sun on the earth were a pulling or attract- ing action, then, unless it were perfectly instantaneous, the result pointed out by Arago, Proctor, and the rest would certainly follow ; but if gravity be caused by a pressing or pushing force, as in this theory, retardation, at first sight, instead of acceleration, would seem to result. For the radiations to which the pressure in this theory is attributed would, owing to the earth’s velocity in her orbit, meet her partly on her front instead of acting in direction of the sun, or at right angles to her path, and seemingly she ought to be retarded. Of course I mean that portion of the radiation which is effective in producing gravity, or that travelling in the sun’s direction ; this would meet her partly on her front, and apparently retard her ; but besides this, the radiations meeting her directly from the direction in which she was travelling being strengthened by ner motion, added to their own, ought apparently to retard her still more, for the radiations following her up behind being minus her motion, would be so much in defect. Apparently, then, nothing can save her from retardation. But before explaining this, it is first necessary to show how bodies can travel through the ether, • dense as I believe it to be, without loss of motion. I am now about to describe to my readers one of the most 1901-2.] Mr J. Fraser on Constitution of Matter and Ether-. 47 ingenious pieces of natural mechanism, if I may call it so, con- ceivable,— a sort of self-winding clock, which can go for ever, without the slightest loss or gain of motion, — in fact, I believe the only perpetual motion machine in existence. Although it has a sort of artificial flavour, yet every astronomer will recognise at a glance its truth to nature. It is a fact well known to all astrono- mers, that the motion of a planet increases as it approaches the sun. In the language of the text-books, as the distance decreases, the attraction increases inversely as the square of that distance ; and consequently the velocity increases inversely as the square root of the distance. If, in the above sentence, we substitute the word “pressure” for “attraction,” it will be made applicable to the present theory. Well, then, as the pressure on an atom increases owing to its motion through the ether, the velocity of the particles of which it is composed also increases in the ratio of the square root of the pressure, the particles on the front of each atom, or bubble, being depressed by the resistance of the ether to motion. This depression is analogous to the depression of a planet towards the sun by an increase of his so-called attracting power, the result in both the case of the planet and particle being an increase of velocity. Well, then, in the case of the planet, this increase of velocity is sufficient, after perihelion, to carry it back to aphelion against the resistance of the sun’s gravity ; and in the case of the particle, it is sufficient to clear the ether out of its path by in- creased centrifugal tendency, — in both cases the effect being just equal to the cause. In other words, in both cases the increased pressure is producing an increase of velocity which just balances the increased pressure : so that the planet can rise to the height from which it fell, and the atom can continue its motion indefin- itely. To meet possible objections, I should like to point out that the radiations impinging on the front of the atom will be strengthened by its motion, and those impinging on its rear will be corre- spondingly weakened, so that these two circumstances cancel one another, the increase of motion in the particles caused by the strengthening of the radiations being destroyed by the decrease caused by their weakening. So that there only remains to be considered the increase of pressure caused by the inertia of the 48 Proceedings of Royal Society of Edinburgh. [sess. ether, or its resistance to motion ; and this, as we have already seen, is just balanced by the increased centrifugal tendency of the particles consequent on their increased velocity induced by the increased pressure.* We are now in a position to see how a body like the earth, bombarded as she is on every side by an infinity of radiations from an almost infinite universe of centres of energy (suns), can move with a great velocity through, certainly, a sub- stance of some kind (the ether), let it be dense or rare, without the slightest sign of retardation. Granted that she is formed of atoms of the nature above described, no one competent to give an opinion, I think, can deny that she, or any body like her, can go on so moving to all eternity without the slightest loss of motion. As a last word on this point,, let it be remembered that when resistance to motion generates a difference in pressure , the resistance is self -compensatory, and motion is neither destroyed nor generated ; but where a difference in pressure exists , not generated by motion , new motion is generated , and in the direction of least pressure. Before concluding this part of my subject, there is a certain peculiarity arising out of my theory which I should like to point out to my readers. It is, that owing to the radiations producing the pressure not being propagated instantaneously, but with the velocity of light, the centre of the earth’s motion cannot be sit- * I am anxious that this point should be thoroughly grasped ; so, at the risk of some repetition, I will go over it again in other words. Well, then, it seems clear that, granted the radiations producing pressure equal from all directions, if a body moved in any direction it must move against them, and if they have any influence on it they must, apparently, retard it, for the motion of the body added to that of the radiations opposing it strengthens those radiations, and weakens those following the body up behind, so that, apparently, nothing can save it from retardation. But owing to the constitution of my theoretical atoms no retardation, so far as the strengthen- ing of the radiation is concerned, could take place, for the increased pressure on the front of each atom would tend to increase the motion of the particles of which it is composed, and which increase would be neutralised by the decrease of pressure on the rear due to the diminished radiations from this direction, so that, on the whole, the mean motion of the particles of each atom would not be altered. This clears the way so far as the radiations are concerned, 'provided they are equal from every direction ; if they are not, it seems clear that the body must drift in the direction from which the diminished radiation reaches it, and with a velocity proportional to that diminished radiation. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 49 uate on the line joining her own centre with that of the sun, but on the line joining her centre with the position occupied by the sun’s centre about 8J minutes previously ; in fact, with the optical sun’s centre. And the centre of the sun’s motion, or that portion of his motion influenced by the earth, would be situate on the line joining his own centre with the position occupied by that of the earth 8 J minutes before ; that, for the sun, would be the optical earth. The earth would be continually pressed towards the position occupied by the sun when he intercepted the radiations, the lack of which at the moment is producing a difference in pressure. In fact , she never could even begin to revolve round any other but this point .* But this circumstance, so far as I can see, being no mathematician, can make no difference ; for the earth would revolve round the point where, in a manner of speaking, she felt the sun to be ; and the sun, in the same manner of speaking, would revolve round the point where he felt the earth to be. Remembering that no new motion can be generated ex- cept in the direction where there is a deficiency of pressure, it is difficult to see how the non-instantaneity of gravity can produce either acceleration or retardation in the earth’s motion. If any material amount of either acceleration or retardation can be proved to follow from the principles of this theory, I am willing to reconsider it, being persuaded in my own mind that a natural explanation can be found for any discrepancy that may exist ; for the coincidences where a natural explanation was found for every difficulty which occurred up to now, are too numerous to apply to any but a true theory. The Thermo- dynamical and Chemical Aspects of the Theory. It remains now to say a few words on the thermo-dynamical and chemical aspects of the theory • but before doing so, I should like to confess that all the little knowledge of chemistry which I possess has been acquired with a view of trying if its leading principles can be explained by this theory. I have neither the time nor the patience to dip deeply into the subject, knowing, as I * I doubly emphasise this point, for it would be as though the sun existed there and nowhere else. PROC. ROY. SOC. EDIN. — VOL. XXIY. 4 50 Proceedings of Royal Society of Edinburgh. [sess. do, that there are hundreds of competent chemists who, when once the mechanical explanation of, at least, some of the principles of their science is pointed out to them, can almost at once, I have little doubt, follow these principles down into all their minute details, and give a mechanical explanation for each and every one of them. First, then, let me try to point out what I conceive to be the mechanical explanation of the radiation of heat. Let my readers conceive two of our bubble atoms to be brought so close together as to touch. If they were perfectly rigid, being perfect spheres, they could only touch at one point ; and all round that point for a certain distance (depending on the size of the spheres, the distance, of course, being greater for the large spheres and less for the small ones) would be included an area from which all ether would be excluded, because, within that area, the space separating the two spheres would be narrower than the diameter of an ethereal particle; and seeing that these particles are incompressible, they must all have been squeezed out by the act of bringing the spheres together. But we know that our spheres are not perfectly rigid — far from it ; some of them are very open-grained and squeezable ; therefore what must happen when they are brought so close to- gether as to touch 1 What happens to a boy’s “sucker ” when the air is excluded from between it and the flat stone to which he presses it ? Why, it simply sticks to the stone by the pressure of the air above it not being counterbalanced by an equal pressure in the opposite direction, all air being excluded from between it and the stone.* So with our sphere ; the ether being excluded from the area round the point of contact, that point would have momentarily to support the pressure which was formerly supported by all the points within the said area ; but that single point, not being able to support the pressure which was formerly distributed amongst all the points within the area, naturally would get squeezed inwards, the spheres being flattened at their point of contact, thus bringing more and more of their surfaces together, and squeezing, in the process, more and more of the ether out from * The spheres would begin to “ attract ” one another when they were some little distance apart (I use the word “attract” as having a well-established meaning) ; in fact, this process would begin when the film of ether between them became so thin as not to be able to carry the full pressure. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 51 between them ; this process going on till, owing to the flattening, the ethereal particles forming the bubbles would become so crowded together, and their motions consequently so accelerated, that a re- action would take place, and the bubbles would spring apart for a ■certain distance, but would not separate (unless they happened to come together very forcibly, in which case they would, after the above process, fly apart again), but would go through the same process again and again until their motion became exhausted. But how would their motion become exhausted ? Simply because at each spring apart a certain quantity of ether would have to be bodily displaced. The boy’s “sucker ” is held to the flat stone by a pressure of 15 lbs. to the square inch, and this weight would have to be lifted a certain distance in order to separate them, but these atoms are held together by a force of at least 17 billion pounds to the square inch, and no wonder that motion should soon become exhausted in trying to overcome such a force. The above, then, in my view, is the cause of radiation ; at each vibration a quantity of ether has to be lifted, so to speak, bodily lifted, just as the weight of the atmosphere has to be lifted in overcoming the adhesion of the “sucker”; this quantity of ether is crowded to- gether into a smaller space than it would naturally occupy, a part of the condensation dilating into the space left vacant by the approach of the atoms together again, but a great part of it spread- ing out in a spherical condensation or wave all around the point of vibration. Until the above explanation occurred to me it was a constant puzzle how it was that solid bodies, such as the earth and planets, etc., appeared to move freely through the ether without the slightest sign of loss of motion, but that the atoms and mole- cules of which these bodies are made up, which also move through the ether, some of them as projectiles, such as gases, could not so move without loss, appeared to me perfectly incomprehensible. But the reason now appears perfectly plain. Motion of translation through the ether is conserved in the way already set forth, pages 46, 47 above, and motion of vibration is the only sufferer. The gaseous molecules lose nothing by their translatory motion ; only in separating again after each collision do they suffer loss, by having to, as it were, lift the ether bodily out of its place before separat- ing, and so crowding it together in the form of a wave. Let us now consider how atoms generally join in pairs to form 52 Proceedings of Royal Society of Edinburgh. [sess. molecules. Before the union we may conceive them flying about in space in much the same way as afterwards, coming into collision and flying apart, in the majority of cases, with the violence of the rebound ; hut here and there one would catch another up from behind, and the union would be so gentle, and the force binding them together after union would be so great, that they could not separate. In course of time they would all pair. But why pair 'l it will be asked. Why not join in threes and fours, etc. ? For a very good and substantial reason, I reply. When two atoms unite they cling together more or less forcibly or closely, dependent on their sizes and squeezability; the more closely they cling, the more are they flattened together at the point of contact, thus crowding the par- ticles of which they are formed closer together, and so rendering the atoms less squeezable. In this state it is far easier for an unpaired atom to unite with another unpaired one, because the paired ones are made by their union so rigid that it is difficult for another to gain and retain a hold upon them. The area of contact between such would be so comparatively circumscribed, owing to the rigidity of the paired atoms, that on the first opportunity the odd one would unite with another unpaired one, and so on until they were all paired. It is evident that the larger-sized atoms in proportion to mass — the more open-grained, in fact — would cling closer together in forming molecules; their particles having more space between them, could be made to crowd closer together, and so admit of a greater surface-contact. But there would be a limit to this. — See Valency. They might be so large-sized in proportion to atomic weight or mass, and their union would be so close, as to admit of only, owing to their small weight, an almost insensible quantity of vibration. The total heat of a gas is made up of two quantities, viz., the internal vibration of the molecules and their translatory motions within the space occupied by them. The heat of some gases may be almost entirely made up of translatory motion, and comparatively little of it of vibratory. Tyndall found* that the absorption of radiant lieat by the simple gases hydrogen, oxygen, and nitrogen was almost insensible, and also that their radiation was on the same level. How, if we assume that the atoms of these gases are so very large in proportion to * Heat, a Mode, of Motion, p. 346, 6th ed., 1880. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 53 their atomic weights, and their union so close that their molecules are almost unable to vibrate, the motion of each atom being as that of a child in the grip of a giant, we can see the reason for the results of his experiments. Radiant heat , according to this theory, would affect only the vibrations of molecules, and it would not affect their translatory motion at all. Waves of radiant heat synchronising with the vibrations of molecules would be absorbed by them, each wave adding its energy to that of the preceding one ; and evidently, the greater the amplitude of vibration, the greater the absorption.* It may very well be, then, that the small absorptive and radiative powers of the gases hydrogen, oxygen, nitrogen, and dry air is owing to the close union of their atoms preventing vibration. Now, after radiating a quantity of heat away, gases would begin to liquefy. In other words, their molecules would begin to unite. The liquid state being, I take it, where a great number of contiguous molecules would be momentarily united, the least force separating them again, to temporarily unite immediately afterwards with others in their vicinity, and so on; their motion being as yet too violent to admit of permanent union in the solid form. This form is only attained after a further cooling, and it appears to consist in all the molecules being united to others at various points in a sort of honeycomb pattern, and vibrating in and out, concertina-fashion, in all directions. It is not my intention to try to account for the various temperatures at which bodies will change their state from the gas to the liquid or solid ; I shall content myself merely by pointing out, as it appears to me, the general principle governing the change. It appears to me, then, that the amount of grip left by the atoms, after their union to form molecules, will be the chief factor in the problem. It will be evident that the atoms of molecules may be united so closely or loosely, depending on their sizes and squeezability, that various quantities of grip will be left for the attachment of the molecules themselves. The * It is clear that absorption must be brought about in the inverse way to radiation. Radiation means a crowding together of the ethereal particles by the energy resident in the molecule ; absorption is brought about by the energy in the wave ; the wave strikes the atom on one side only, the opposite side being, at least, partly shielded by its partner in the molecule, and in this way motion towards its partner is produced. 54 Proceedings of Poyal Society of Edinburgh. [sess. atoms of small surface and close in the grain will not unite so closely, and thus may leave a greater hold for their molecules to unite. There are some atoms, I take it, so close-grained and small in surface proportional to weight as not even to form molecules before they solidify. For instance, mercury. Mercury is mon-atomic, and the above, I believe, is the reason for this fact. There are some others, though, which, after the union of the atoms, leave so great a hold for the attachment of their molecules, by reason of their great squeezability, that they have not yet been liquefied, so far as I know, much less vaporised. For instance, carbon. It is for mathematicians and physicists to apply this principle to particular cases, for I confess that I myself, being no mathe- matician, am unable to do so. Chemical Affinity. Chemical affinity appears to be caused by the differences in the sizes and closed-grainedness and generally the squeezability of atoms. An atom of one kind may find a better hold on an atom of another kind than on one of its own kind. For instance, take two spheres, a large one and a small one, and press the two together, and if they are at all compressible, the area in contact with the small sphere will be larger than if it were pressed against a sphere of its own size. Again, take two spheres of the same size, but one being more compressible than the other, and press them together. The least compressible one would be in contact with a greater surface area than if it were pressed with the same force against another of its own kind. This, then, appears to me to be the principle governing chemical affinity. Small atoms, finding a better hold on large atoms than on their companions, quit the latter, and attach them- selves as parasites to the former, crowding the large ones out by taking up the available amount of grip, and so causing the large ones to lose their hold on one another ; or, a less squeezable atom finding a better hold on a more squeezable one, quits its companion for the latter. Of course it is only when circumstances are favourable that the change takes place, such as when the elements are in a state of solution, or when in the gaseous state, or in the- nascent state; or sometimes a gas unites with a solid, as when 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 55 oxygen attacks iron ; all depending, I have no doubt, on the strength of the “affinities.” It is evident that bubbles, or atoms, of the kind postulated, will be more or less compressible as they are more or less open-grained ; and which, in their turn, depends on the greater or less speed of the ethereal particles forming the skin of the bubbles. It will be evident, also, that the greater the speed of the particles, the more will the bubbles resist compression, as centrifugal tendency varies as the square of the speed for orbits of the same size. These two statements seem to contradict one another, but really there is no contradiction, for the close-grained atoms resist compressions because they cannot be further compressed, as there must be a limit, owing to being in touch all round. But the open-grained kind can be compressed, if sufficient force be applied, to a very great extent. Of course it will be understood that the compression I am speaking of must be brought about by the close contact of the atoms shielding off portions of one another’s surfaces from pressure, whereby an increase of pressure is applied to other portions of their surfaces, otherwise an increase of pressure applied equally all round would generate a repulsive force equal to the increase in the pressure. (See p. 47.) Well, then, the open-grained kind will be the most compressible, and the close-grained kind the least compressible. This leads me to conduction and valency. The best conductors will, of course, be the most rigid kinds, and these will be found amongst the close-grained. A close-grained atom, if pushed against another one, being more rigid, would transfer the energy of the push more completely to its neighbour than an open-grained one. An open-grained one would retain a large proportion of the energy amongst its own particles by springing in and out on its own centre, and would, by so springing in and out, radiate a goodly proportion away. As an example : if a man wanted to push a heavy weight in front of him by means of a rod, he would use a stiff, rigid rod, and not a thin, flexible cane, because in the former case he would be practically applying all his strength to the moving of the load, while in the latter he would be applying it to the bending of the cane. The more rigid the atom, other things being equal, the better the conductor ; and the more springy, the worse the con- ductor. But a good deal depends on whether the molecules of the bodies 56 Proceedings of Royal Society of Edinburgh. |_SESS- are closely united, or the reverse, — a fact which can he tested by their melting-points, as to whether they may he good or had conductors. — for molecules with a small hold on one another cannot transmit their motion from one to another so well as those which are closely united ; and, moreover, the amplitude of vibration would be greater with the former, thus increasing the radiation. There are two classes of bodies which most strongly differentiate themselves one from the other by being good conductors as a class or bad conductors as a class. These are the metals and the non- metals. The metals are all good conductors, though they differ greatly amongst themselves, while the non-metals are nearly all bad conductors. The metallic atoms, then, must be all close- grained, and this would seem to account for their great specific gravities, while the non-metallic atoms must be all open-grained, which would account for their low specific gravities. Now, it is found that different bodies unite with one another in different proportions, and this is termed “valency.” To account for this in full is most difficult. But I believe that the cause is the different squeezabilities of atoms. For instance, two monads unite with a dyad because the dyad is squeezable to the extent of afford- ing a sufficient hold to two bodies of the monad class. Three monads unite with a triad because it is squeezable to the extent of affording a hold to three bodies of the monad class, and so on. The higher the valency, the greater the squeezability of the nucleus of the resulting molecule. Now, metals are amongst the atoms of highest valency; and this would seem to contradict what I have said about metals as a class being the least squeezable. But we must remember that metals resist compression because they are more or less near the point of absolute unsqeezability, but some of them may be so far from this point as to admit of a considerable amount of compression, more especially when we regard their generally high atomic weights. On the other hand, non-metals, although their absolute squeezabilities are far greater than metals, may practically be far less squeezable, having regard to their generally less atomic weights, owing to the greater velocity of their particles. To illus- trate this point, and give the reader a clear idea of my meaning, let us suppose an atom to be so close-grained that no compression of it was possible. Such an atom would have no valency ! simply because no other could stick to it if that other were as unyielding 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 57 as itself, and at most it could only have a valency of one. Now, let us suppose an increase of speed to be given to the particles ; this would have the effect of opening them out, making the atom less rigid ; it could now he compressed to a certain extent, and we may now regard it as having a valency of one. Imagine the speed to he further increased, this would have the effect of opening out the particles still further, and making it still more compressible, and giving, say, a valency of two. In this way, by increasing the speed we might give it a valency of three, four, five, or six, pro- vided that there were a sufficiency of particles; or, in other words, provided the atomic weight were great enough. But does the reader suppose this process could go on indefinitely — that by continually increasing the speed we could increase the valency? No ; a point would at last be reached, sooner or later, according as the atomic weight were small or great, when a further enlargement would decrease the valency ; simply because the excessive speed of the particles by their centrifugal tendencies would prevent com- pression beyond a certain point by the force available for that purpose. It is not that continued expansion would make it less compressible, it would make it absolutely more compressible ; hut the force required to compress it would be enormously greater than when in the middle stage of the expansive process. Such an atom, when in this last stage of expansion, instead of being sought out for union by an easier compressible one, would seek it out, and attach itself to it, because it would find a better hold on it than on its own kind. The easier compressible kind, or, in other words, those of great valency, do not seek union with those of less valency, but are sought out by the latter, which attach themselves to them like parasites, and so crowd out their legitimate companions. Why, though, it may be asked, does not a dyad atom, say, content itself with the company of only one monad instead of two ? There would surely be a better hold for one than for two? Yes; but it takes the strength of the two monads to divorce a dyad from its legitimate partner, and it takes the strength of three monads to divorce a triad, and so on. In other words, the hold or grip which a monad would take of a dyad would only be a little more than half of that of its own partner, and it would require two monads to attack it to divorce them, and the attack of three monads to divorce a triad molecule, and so on. 58 Proceedings of Royal Society of Edinburgh. [sess. In a compound molecule the component atoms group themselves symmetrically round the nucleus, because, as it seems to me, they find a better hold as far away as possible from where their companions are attached to it, the nucleus being less compressed at these parts. The traps and pitfalls for the unwary in dealing with this subject are almost innumerable. Before recognising the fact that the valency of an atom could be changed, by altering the velocity of its particles, from that of a monad up through all the stages of which it was capable — possibly to a hexad or higher — depending upon its atomic weight, and down again to that of a monad by mere increase of velocity, I was led into a perfect quagmire of con- tradictions; and although I must leave the subject still in a good deal of obscurity, for lack of mathematical and chemical knowledge to disentangle it, yet I hope I may have shed a little light on it, of which people in a better position than I am to avail themselves of it may take advantage, and make it clear as the sun at noon- day. "We may now inquire if the theory can give any account of the fact that when an atom changes its valency it does so by chang- ing two steps at a time. Thus, manganese is a hexad, but it can also act as a tetrad or a dyad, but never as a pentad, triad, or monad. Nitrogen is a pentad, and it can also act as a triad or monad, but never as a dyad, tetrad, or hexad. The first follows the order of the even number, and the last that of the odd number. “There are several of the chemical elements, and these among the most important and widely distributed, whose quantivalence appears to be invariable. This is especially true of hydrogen ; it is likewise true of the alkaline metals, lithium, sodium, potassium, csesium, and rubidium, and it is also true of silver, all elements whose atoms are univalent. It is further true of the trivalent element boron. Again, oxygen is always bivalent, and so are also the metallic radicals of the alkaline earths, calcium, barium, stron- tium, and magnesium ; and so are, moreover, the well known metallic elements, lead, zinc, and cadmium. Lastly, aluminium, titanium, silicon, and carbon are always quadrivalent, although, in the single instance of the molecule, CO, the carbon atom appears to be bivalent.” * * J. P. Cooke, “ The New Chemistry ” pp. 246, 247. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 59' Now, I think the explanation of this variation of quantivalence will be found in the fact that some atoms are more compressible than others. I think it will be found that the atoms of invariable valency are closer-grained, and therefore nearer the limit of squeezability ; while those of variable quantivalence are more open- grained, and therefore more susceptible to compression. A differ- ence in the conditions of temperature or pressure, or both combined, might cause a greater or less number of monads or dyads, etc., to cling to the nucleus of the open-grained kind; while it is quite conceivable that, owing to the rigidity of the close- grained kind, and the comparative slow motion of their particles, a difference in the conditions of combination would have far less effect in causing variable quantivalence. Size of atom might also limit the variability, as it seems evident that only a few large atoms could be in touch with a small one. But why should the variable atoms change their valency two steps at a time ? it will he asked. To answer this, I must again point to the fact that in a compound molecule the component atoms group themselves symmetrically round the nucleus because, as it seems to me, they find a better hold as far away as possible from where their companions are attached to it, simply because the nucleus is less compressed at those points. Well, then, in order to illustrate my conception of this change in valency, suppose the atom which acts as a nucleus to be a perfect sphere, furnished with an equator and poles, and suppose a great circle to pass through each pole, cutting the equator at right angles. Let us now take as an example the atom of manganese mentioned above as our nucleus, and which can change its valency from a dyad to a hexad. First, then, we can have MnF2 ; in which case we will suppose the two fluorine atoms to he attached one to each pole of the manganese nucleus. The molecule in this way will be perfectly symmetrical. The next change will be MnF4, that is, a fluorine atom attached to each pole, and one each attached to opposite sides of the equator where the great circle cuts it. The molecule in this way will be also perfectly symmetrical. We could not possibly make this a triad molecule with a fluorine atom attached to each pole unless we remove the two latter to some other parts of the sphere, and as, I presume, the condition which determined the change was either a decrease of temperature, or increase of pressure, or both, the twa •60 Proceedings of Royal Society of Edinburgh. [sess. polar atoms would cling all the closer, and would refuse to "be removed, therefore any additional one which could find a soft place to cling to would find the softest on some part of the equator, and as on the opposite side of the equator there would be an equally soft place, another would attach itself there. The valency in this way would advance from two to four, that is, two steps at a time. Now for the next two steps, from a tetrad to a hexad — let us suppose the conditions to be favourable for the change, which I again assume to be a decrease in temperature, or increase in pressure, or both, and consider where another fluorine atom could find a place to cling to, so as to be as far away as possible from its companions, and I think it will be found that this place would be where another great circle passing through the poles at right angles to the first would cut the equator ; and as the two points of intersection of the equator by this last circle would be equally favourable for the attachment of an atom, two would consequently attach themselves there, thus increasing the valency by another two steps. In this way we would have four atoms attached to the equator, all at equal distances from one another, and one each attached to the poles, and all separated by equal spaces. Now, I am not chemist enough to know if there are any cases in nature where the valency advances beyond six for artiads, but if there are, and carrying out the above process, I must admit that the next advance would be, not by two steps, but by four. I need not trouble to show how this would be, as readers can easily see for themselves that it must be so, according to the theory. Now the variable jperissads would vary on precisely the same principle as the artiads. Let us take phosphorus trichloride, P.C13, as an ■example ; I assume in this case that the three chlorine atoms would be attached to the phosphorous nucleus at equal distances on the equator, in such a way that the lines connecting the chlorine atoms through the phosphorous nucleus would form an equilateral triangle. It will be seen that the molecule in this way would be perfectly symmetrical. On the formation of P.C15, which I believe takes place under a lower temperature or greater pressure of the chlorine, two more chlorine atoms would become attached to the nucleus, one to each pole ; thus again forming a symmetrical molecule. As in the case of the artiads, if there are any cases in nature where the valency of a perissad increases beyond five by the 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 61 attachment of monads of the same kind to the nucleus, it must be by four steps, in order to preserve the symmetry. We may now, with profit, look a little closer into the reason why atoms of the same hind should group themselves symmetrically round a nucleus. First, then, the nucleus is granular in structure ; and consequently in the vicinity of the points to which the atoms are attached, owing to the vibrations of the said atoms, those portions of the nucleus are alternately in a more or less condensed or rigid condition ; that is to say, when the atom reached the inward limit of its swing towards the heart of the nucleus the portion of the latter to which it was attached would he in its most rigid con- dition, and when it reached the outer limits it would be in its most dilated condition, and the same would he true of the areas of attachment of the atoms to the nucleus. These alternate expan- sions and contractions of the granules would act as a sort of wave motion along the surfaces of the nucleus and atoms, the amplitudes of which would be greatest in the immediate vicinity of the points of attachment, so that if a stray atom sought to attach itself near any of these points, the pressures of these wave motions would cause it to be removed to a point where the said pressures would be equalised all round, and that would he at an equal distance from all the others. It would, in fact, in a manner of speaking, slide downhill to the lowest point, where it would remain. I have now given my notions on this portion of my subject, so far as I have explored it ; I have not, so far as I know, made use of any but well known mechanical principles, so that if my reasoning is anywhere fallacious it can be readily exposed. I have not entered into the subject of the constitution of complex molecules, feeling that I am scarcely competent to do so without further study, and this latter I have little time for. This I must leave, with a great deal more, in the hands of more competent people to deal with.* * If I have made any blunders in my chemistry I hope I may be excused, as I gave fair warning when entering on the subject that I knew very little about it. But to the chemists I should like to say that I have given them an atom, with its great potentialities, and shown them its hooks, clamps, or bonds ; and I do think that it is for them, with the knowledge of the facts at their finger- ends, to clamp the atoms together in the way which best suits the facts ; and sure am I that if they give the subject a little of the thought which I have given, they will soon be able to string all the facts together into one perfect whole. My text-books of chemistry, I find, are somewhat out of date. Since writing the above I have been given to understand that valency does not 62 Proceedings of Royal Society of Edinburgh. [sess. Crystallisation. 1 believe that the principle regulating the formation of crystals will have something to do with the symmetrical arrangement of the components of molecules, especially so with regard to complex molecules. This principle I believe to be the inequalities of 'pressure in opposite directions which must exist within a solid body if the molecules are not symmetrically arranged. For instance, suppose the molecules of compound bodies, on solidifying, froze together anyhow ; it is easy to see that if this happened the spaces separating the molecules, owing to the peculiar shapes of the latter, would not be equal between contiguous molecules, so that the ether within those spaces would, owing to the vibrations existing within every body, be unequally compressed and unequally dilated after each vibration within the spaces. Now, this principle of the in- equality of pressure is, I take it, the very principle which arranges 'x f,C. / the molecules of crystals in symmetrical patterns. For instance, if, when on the point of solidifying, a molecule be not acted on by the ethereal pressure equally in opposite directions , this very in- equality of pressures will turn the molecule about until it is in a position of equilibrium ; that is, until it is in a position where it would be acted on by equal pressures from opposite directions. In this position the molecules would freeze together; and the various shapes of the resulting crystals would be accounted for by the inequalities of size which existed between the constituents of the molecules of the various crystals. For instance, a molecule of ice would have the shape shown in plan in fig. 1 ; this shape de- always vary by two steps at a time. If this be so, there must be some, peculiarity in the circumstances attending the departure from the rule, or the departure, one would think, would have been discovered long ago. Possibly , though, this very departure, when the circumstances attending it are thoroughly inquired into, may serve to confirm my theory. 1901-2.] Mr J. Fraser on Constitution of Matter and Ether. 63 pending on the inequality of size existing between the oxygen nucleus and the hydrogen atoms, and their arrangement. Fig. 2 is a rough attempt to give my idea of the arrangement in plan of the molecules of a piece of ice. This would also give the germ of the ^explanation of the cause of water and some other bodies expanding on solidifying. The molecules simply arranging themselves, by the inequalities of pressure mentioned above, in such a way as to bake up more space than when in the liquid condition. Atomic Volumes and Isomorphism. It is assumed, on apparently good grounds, that the atoms of isomorphous bodies are of the same size. It cannot be disputed, of course, that they occupy the same space in the solid form , but this is not quite the same as saying that they are equal in size. Silver and gold are isomorphous bodies, as their atomic weights and specific gravities prove — the same difference existing between their atomic weights and their specific gravities, so that they must occupy the same space in the solid form. How, I think it would be a very curious fact — for there are great numbers of these bodies — if each set were exactly equal in size with their atomic weights so various, but I think the fact of their occupying the same space in the solid form can be accounted for in another way than by assuming their equality of size. It can be accounted for by assuming that the velocity of the particles of which the 64 Proceedings of Royal Society of Edinburgh. [sess. atoms are composed is the same. For instance, take gold and silver — the velocity of the particles of the gold atom may be the same as that of those of the silver atom, and the gold atom may be larger in size though closer in the grain , which latter quality would admit of only the same speed of particles as that of a smaller atom; and greater size in an atom with only the same speed of particles as a smaller one would permit of greater compression. Thus it may he that the gold atom though of larger size may be more squeezable in exact proportion to its larger size, and thus appear to have the same volume as that of silver. That it is more squeezable appears from its greater valency — that is, if my explanation of valency be accepted. It is this state of various proportions of weight, size, close- grainedness, and velocity of particles which may exist which give my theoretical bubble atoms such a power of diversity in their qualities as to at once constitute them a source of the most mysterious properties, with a resemblance to the ways of nature which to me is irresistibly convincing. I have now gone as far with my theory as I intend at present. The subject is endless; other people with more time at their disposal and with better opportunities may take it up if it so please them. It is now about seven years since I first conceived the idea of the bubble atom ; and since then, during all the time at my disposal, when absent from duty, the subject has seldom been out of my mind. I hope that I may have shed some light on some of the darkest parts of it, but I am conscious that the deficiencies of my education will not admit of my going further with it, at least till those deficiencies are overcome, and of which I see very little hope at present, if ever. I have not touched at all on the subject of electricity, simply because I know very little- on the subject; but sure am I that within the folds of this theory lies hidden the mystery of electricity. {Issued separately February 21, 1902.) 1901-2.] Prof. Schafer on Injection of Liver Cells. 65 On the Existence within the Liver Cells of Channels which can be directly injected from the Blood- Vessels. By E. A. Schafer. (Read February 3, 1902). In a series of papers dating from 1897, resumes of which are given in the Bulletin international de Vacademie des Sciences de Cracovie * Professor Browicz gives the results of observations on the liver cells, both in the normal state and in a pathologically altered condition, and draws from these observations certain important deductions. The most prominent fact which is recorded is the appearance (under normal conditions in the liver of the new- born child, and under pathological conditions in the adult human liver; under normal conditions, and after the intravenous injection of haemoglobin some hours prior to death, in the liver of the dog) of erythrocytes, singly and in groups, and of free haemoglobin and haemoglobin crystals, as well as brown pigment, in the form both of granules and crystalline clumps, within the hepatic cells, both in the nucleus and in the cytoplasm. After affirming the undoubted existence of intracellular biliary passages communicating with intercellular bile ducts, — which intracellular passages he regards, not as accidental and temporary vacuoles, but as preformed channels within the cells, — Professor Browicz states that his observations, especially the presence within the liver cells of * “Intracellular biliary passages in the liver cells,” March 1897 ; “On pathological conditions of the nucleus of the hepatic cell indicating that the nucleus is a secreting organ,” April 1897 ; “On the structure of the liver cell,” May 1897 ; “How and in what form is haemoglobin brought to the liver cells?” June 1897 ; “ On crystallisation phenomena in the liver cells,” April 1898; “On the intravascular cells in the hepatic capillaries,” April 1898 ; “On microscopic appearances in the liver cells after intravenous injection of haemoglobin,” November 1898 ; “ Intussusception of erythrocytes by the liver cell and the appearances of the cells thereby produced,” July 1899 ; “ Channels of nutrition in the liver cell — with a resume of the author’s results since 1897,” Jnly 1899 ; “ Structure of intra-acinous blood-capillaries and their relations to the liver cells,” May 1900. PROC. ROY. SOC. EDIN. — VOL. XXIV. 5 66 Proceedings of Royal Society of Edinburgh. [sess. erythrocytes, demonstrate that the cells must he in communication with the hepatic capillaries. The exact nature of such communica- tion is not determined, but Professor Browicz goes on to show that the connection of the liver cells with the blood capillaries is much more intimate than has been generally supposed ; that the peri- vascular lymphatics which have been described by M‘Gillivray and others as encircling the lobular blood capillaries in all probability do not exist ; and that some of the cells which form the walls of those capillaries resemble the liver cells in containing biliary products in vacuoles, and in minute channels, and display pro- jections which penetrate into the cytoplasm of the adjacent liver cells. The inference which he draws from these observations is thus expressed by him*: — “All these circumstances compel the conclusion that in addition to the intracellular biliary passages, which act as excretory channels, there must also exist special afferent nutritive channels, or canaliculi, in the liver cells. That these cannot be made visible as a system of canaliculi is no draw- back to this conclusion. The intracellular canaliculi must in any case be exceedingly fine, and could only, under the most favour- able circumstances, be here and there visible, considering the very small amount of nutritive and functional material which has at any one time to pass into a liver cell ; the microscopic appearance is in fact only a snap-shot.” I am, I believe, in the position of being able to offer the objec- tive proof of this important deduction. In specimens of injected liver of the rabbit which we possess in the Physiological Laboratory of the University of Edinburgh, one can clearly see the intracellular nutritive canaliculi, the existence of which has been inferred by Professor Browicz (fig. 1). They are everywhere throughout the sections filled with the carmine gelatine injection material with which the vessels have been injected from the portal vein, and here and there is seen what appears to he a direct offset from a blood capillary to the intracellular network. The injection within the cells is not irregularly diffused nor limited to the exterior layers of the cytoplasm, but is confined within sharply defined, somewhat varicose intercommunicating canaliculi, many of which are in the immediate neighbourhood of the nucleus or nuclei, but never, so * Bull . internat. de Vacad. d. Sc. de Cracovie , Juillet 1899, pp. 369, 370. 1901-2.] Prof. Schafer on Injection of Liver Cells. 67 far as I have been able to observe, actually within the nucleus,* and it has the same intensity of colour within the cells as in the blood-vessels. The injection has not passed into the cells after having been extravasated into the intercellular biliary ducts, for the latter contain no trace of injection, and are indeed completely invisible. This is also the case with the perivascular lymphatics of the lobules (if such vessels exist). There is, besides, no diffusion of carmine, and the cell nuclei are wholly unstained. In the substance of the lobules the injection is confined to the Fig. 1. — From liver of rabbit injected from portal vein. The injection material has passed into minute channels within the cells. blood-vessels and to the intracellular channels. It is also seen, but of a fainter colour, i.e., in a diluted condition, in the lymphatics which accompany the branches of the portal and hepatic veins ; and the connective tissue around these vessels, and extending a short distance into the lobules, is stained by carmine. The intracellular channels bear a very close resemblance to those which have been figured by Professor Browicz within a liver cell, from a case of obstruction of the bile ducts ; and although, at first sight, it might seem that these are to be regarded as enlarged intracellular biliary canaliculi, yet, as he has himself suggested, they may at least in part represent the nutritive channels, the existence of which he infers, and into which, in * Professor Browicz is of opinion that the nutritive canals, the existence of which he has, it would seem correctly, assumed, are continued into the interior of the nucleus. 68 Proceedings of Royal Society of Edinburgh. [sess. this case, he supposes that the biliary secretion may have forced its way. This figure of Professor Browicz I here reproduce (fig. 2). I am given to understand that the preparations of injected liver, now for the first time described, have been in use for class purposes in the Physiology Department of the University for a number of years, sections having been regularly cut from them and mounted by members of the class. It was whilst going round the Histology Class this summer that I first noticed the appearances in question. None of my present assistants were able to give me the history of the liver from which the preparations were made, nor had their attention been drawn to the intracellular channels which all the sections show. But I found in the collection of specimens which were left to the University by the late Professor Rutherford several sections, apparently of this same liver (rabbit), and others n c Fig. 2. — Liver cell showing intracellular canaliculi (Browicz).* n, nucleus ; c, canaliculi. purporting to be made from the liver of the cat, some of which are labelled in his handwriting (others in that of some other person), “ injection in hepatic cells ” ; one of these specimens bears the date 1886. I accordingly wrote to Professor Carlier, who was at that time, and for some years subsequently, assistant to Professor Rutherford, for any information he could give me regarding the specimen. In Professor Carlier’s reply he says, “these specimens of liver injected in red were done by Simpson under my direction, and used for class purposes. These canals were first noticed by me and shown to Rutherford, who would not let me publish a note of them.” I imagine this refusal by Professor Rutherford to permit of the publication of Dr Carlier’s observation was due to his having * Taken from Szymonowicz, Lehrl. d. Histologie, 1901. 1901-2.] Prof. Schafer on Injection of Liver Cells. 69 regarded the appearance as purely accidental, as it certainly was inexplicable on the ideas which were then prevalent regarding the structure of the liver and the relation of the liver cells to the blood-vessels. Whatever may have been the reason, it is a matter for regret that so important an observation should have been suffered to lie so long dormant. Por it not only affords an explanation of the presence within the liver cells of the easily squeezable erythrocytes, which Professor Browicz has shown to occur even in the normal liver, and in larger numbers under abnormal conditions, but may also help to account for phenomena in connection with this organ which have hitherto been obscure. It is impossible to overrate the value of a fact which appears to demonstrate the existence within the cells of any organ of vascular canals which are normally too narrow to admit blood- corpuscles, but which are doubtless capable of conveying the fluid portion of the blood directly to the cell protoplasm. I have submitted sections of the liver in question to Professor Browicz, who agrees with me that they unmistakably demonstrate the existence of channels within the cells communicating directly with the blood-vessels. Since the above was written, my attention has been drawn to a paper entitled “ Preliminary note on inter- and intra-cellular passages in the liver of the frog,” by J. W. Praser and E. Hewat Praser ( Journal of Anatomy and Physiology , vol. xxix., 1895, p. 240), in which the authors describe and figure in the livers of frogs injected from the bulbus aortse with Hoyer’s bichromate gelatine, and with Carter’s carmine gelatine, fine passages filled with injection within the protoplasm, and even penetrating to the nucleus of some of the liver cells. The authors conclude that these passages are in direct communication with the blood-vessels, and that their function is nutritive. I have no doubt that, although less completely shown, the canals referred to are of the same nature as those which I have described in this paper. [Issued separately March 8, 1902.) 70 Proceedings of Royal Society of Edinburgh. [sess. Quaternion Binaries: an Extension of Quaternions to give an Eight-element System applicable to Ordinary Space. By Dr W. Peddie. (Read November 18, 1901.) {Abstract.) In this system vectors are regarded as translators only. A special operator, R, transforms them into rotors. A second appli- cation of the operator retransforms rotors into translators. The system is essentially Hamilton’s, with the removal of the restric- tion that vectors shall act as translators in addition, and as rotors in multiplication. The quantities i, j, k being unit rectangular vectors, the fundamental equations may be wiitten R ij = k , V\jk = i , R hi =j ; ^2=J2 = 7,2= _ i . R2=i . In the present paper, the fundamental properties of a quaternion binary B = q + Rr , q and r being quaternions, are investigated, the applications being restricted to the theory of screws, — in particular, of screws upon a cylindroid. The related binary RB, and the corresponding cylindroid, are discussed specially. The relation of the above system to that of Clifford, as developed by Macaulay in his Octonions, is evident from the consideration that Macaulay’s operator fi, while it transforms a rotor into a translator, destroys a translator, so that the transforming action of O is restricted to rotors. In other respects, O and R both act as scalars, the former as an infinitesimal scalar, the latter as a unit scalar. (Issued separately March 8. 1902.) 1901-2.] Dr A. W. Roberts on Algol Variation.. 71 Certain considerations regarding Algol Variation, with special reference to C.P.D. — 410,4511. By Alexander W. Roberts, D.Sc., E.R.A.S. (With Two Plates.) (Read November 4, 1901.) There has recently been discovered at the Royal Observatory. Cape Town, an interesting star of the Algol type of variation, thus making in all ten such stars at present known in the southern sky. The star is C.P.D. - 41° *45 1 1 . R.A. lOh. 16m. 44s. (1875). Dec. - 41° 43'-8. The importance attaching to variable stars of this well-defined type is due in some measure to the fact that we have certain knowledge of the causes which produce their light changes. We are able to go beyond the circumstances of their variation to the conditions, physical or otherwise, which have produced the phenomena observed. With many of the other types of stellar variation, our knowledge ends at our observations. Research, for example, has indicated no satisfactory explanation of long period variation, or of irregular variation ; while of theories to account for new stars there is no end. But the eclipse of one star by another affords a complete and at the same time an exceedingly simple explanation of Algol variation. If a close binary system revolves in an orbit coincident, or nearly so, with the line of sight, it is evident that each component will eclipse or occult the other every revolution. The character, extent, and duration of the eclipse will depend upon the form and position of the orbit in which the stars move, as well as on their relative size and brightness. It is this condition of orbital movement — proximity of the component . stars, and movement 72 Proceedings of Royal Society of Edinburgh. [sess. in the line of sight — which produces variation of the Algol type. That is, every Algol variable is a binary system, the plane of whose orbit is coincident, or practically so, with the plane of sight ; eclipse, either partial or total, taking place when the stars cross the line of sight. It is evident that if both stars are luminous, there will be two eclipses each revolution. We have just said that the character, extent, and duration of either of these two eclipses in any Algol system depend on the elements of the orbit, and on the relative size and brightness of the component stars. If the latter he known, we can determine from simple geometrical principles what the amount of eclipse will be. In practice, however, the problem is the converse of this : — from the form of the light curve, as indicated by observations, we seek to determine the elements and dimensions of the orbit. Beyond the interest which belongs to phenomena the interpre- tation of which we are able to declare, there is this added interest connected with all Algol variables — that, intimately associated with the problem of their variation, are other problems bearing directly on some of the most profoundly interesting matters in astronomical science. Tor example, a full determination of the elements of any Algol binary star yields material for a further determination of the density of the system. It is not necessary to know the actual dimensions of the system ; all we require to know is the relative dimensions in order to arrive at this important result. Then, again, there are certain Algol stars, the components of which revolve round one another in contact. These remarkable systems are at that stage of stellar revolution when bipartition is on the eve of taking place. Only five such systems are at present known; of these, three are in the southern sky. Now, observations of the light changes of these stars, when conducted with the greatest care and precision, and treated as rigorously as one would the most refined measures, will reveal, I think, the figure of equilibrium which two contiguous stars 1931-2.] Dr A. W. Roberts on Algol Variation. 73 will take under the mutual attraction of their component masses. In the case of two southern Algol variables of this class — V. Puppis, Ch. 2852, R.R. Centauri, Ch. 5099, observations made with the new prismatic equatorial, generously given me by Sir John Usher, indicate that the amount of distortion at the junction of the two stars is apparent and measurable; that, indeed, the form of the dual system conforms to that obtained by Prof. Darwin in his classical research on “ Figures of Equilibrium of Rotating Masses of Fluid ” (plate 22, Philosophical Transactions of the Royal Society , vol. 178). These are only two of the many allied lines of research intimately connected with the consideration of Algol variation. Indeed, this whole direction of thought and inquiry brings us to the very threshold of the question of stellar evolution, and to the heart of not a few of the greatest cosmical problems. When the certain future development in spectroscopy brings within the apprehension of this method of research movement in the line of sight of stars as faint as the tenth or twelfth magnitude, and when the bounds of the science of stellar variation will be extended to include changes of only one-tenth of a magnitude in extent, the data at the disposal of astronomers for the determination of conditions under which binary systems are formed will be both extensive in amount and sufficient in character. We may exemplify some of the general conclusions already stated by a more particular consideration of the Algol variable recently discovered at the Royal Observatory, Cape Town. Unfortunately, observations were only begun at Lovedale when the star was well down in the west in the evening, and thus an extended series of observations on any evening was not possible. The most, however, was made of the time available. 74 Proceedings of Royal Society of Edinburgh. The observations secured are as follow : — [SESS. Date. Reduced Date. Mag. Date. Reduced Date. Mag. 1901. h. m. li. in. 1901. h. in. li. m. 1 July 2 7 26 July 1 7 26 io-oo 48 July 16 6 0 July 1 10 0 io-oo 2 3 6 15 1 9 45 10-00 49 6 58 10 58 10-00 3 4 6 0 2 9 30 9-90 50 9 45 13 45 10-00 4 8 12 11 42 9-90 51 17 6 30 2 10 30 io-oo 5 5 8 28 1 15 28 io-oo 52 17 7 30 2 11 30 10.00 6 6 6 10 2 13 10 9-90 53 17 8 15 2 12 15 10-00 7 8 12 15 12 10-45 54 8 48 12 48 10-10 8 8 40 15 40 10-60 55 9 20 13 20 10-20 9 9 5 16 5 10-75 56 9 40 13 40 10-10 10 6 9 30 2 16 30 19-90 57 9 48 13 48 10-05 11 6 9 48 2 16 48 10-95 58 18 6 0 1 13 30 10-00 12 8 8 0 1 18 30 io-oo 59 7 0 14 30 10-10 13 8 30 19 0 10.00 60 7 30 15 0 10-00 14 9 30 20 0 10 00 61 8 30 ! 16 0 10-00 15 9 7 14 1 21 14 9-90 62 19 6 15 2 13 45 10-00 16 8 10 22 10 9-90 63 7 35 15 5 10-35 17 8 57 22 57 io-oo 64 7 45 15 15 10-40 18 10 7 42 1 1 12 io-oo 65 7 55 15 25 10*60 19 9 3 2 33 io-oo 66 8 5 15 35 10-60 20 9 15 2 45 10.00 67 8 20 15 50 10*70 21 11 5 40 1 23 10 10.00 68 8 30 16 0 10*95 22 5 50 23 20 10-00 69 i 8 45 16 15 10-95 23 7 30 2 1 0 io-oo 70 9 0 16 30 10-85 24 8 20 1 50 9-90 71 9 15 16 45 10-80 25 12 5 45 1 2 45 9-90 72 9 30 ' 17 0 10-65 26 7 6 4 6 10-00 73 9 45 17 15 10-68 i 27 8 25 5 25 io-oo 74 20 6 0 1 17 0 10*15 28 8 55 5 55 10-00 75 7 0 18 0 10-15 29 9 25 6 25 10-00 76 8 0 19 0 10-05 : 30 13 5 45 2 2 45 io-oo 77 9 0 20 0 10-10 31 6 0 3 0 io-oo 78 21 6 0 2 17 0 10-60 32 7 15 4 15 10-00 79 6 15 17 15 10-45 33 7 52 4 52 io-oo 80 6 30 17 30 10-30 34 8 55 5 55 ! 10-00 81 6 45 17 45 10-25 35 9 25 6 25 10-00 82 7 0 18 0 10-15 36 10 5 7 5 10-00 83 7 15 18 15 10-15 37 14 6 0 1 6 30 10-00 84 8 30 19 30 10-05 38 6 30 7 0 io-oo 85 22 6 14 1 20 44 10-00 39 7 14 7 44 10-00 86 6 20 20 50 10-00 40 9 6 9 36 io-oo 87 23 8 3 1 2 3 io-oo 41 9 30 10 0 10-10 88 8 15 2 15 10-00 42 15 6 0 2 6 30 io-oo 89 Aug. 14 6 20 2 14 50 1 0 25 43 7 0 7 30 10-00 90 6 35 15 5 10-40 44 7 45 8 15 io-oo 91 6 50 15 20 10-55 45 8 12 8 42 10-05 92 7 5 15 35 10-75 46 8 45 9 15 10-05 93 7 20 15 50 10-85 47 9 45 10 15 io-oo 94 14 7 25 2 15 55 10-75 In the above table, column (2) gives the date when the observa- tions were made in Cape mean time. The fourth column is the 1901-2.] Dr A. W. Roberts on Algol Variation. 75 observed magnitudes. An examination of these indicates that the period of the variable cannot be far from — Id. 2 Oh. 30m. With this period all the observations were reduced to the mean light period of 1901 July 1 and 2. The reduced dates are given in column (3). Arranging the dates in column (3) in order of sequence, we obtain the following representation of observations :■ — Reduced Date. Mag. Reduced Date. Mag. Reduced Date. Mag. 1901. 1). in. 1901. h. in. 1901. h. m. 1 July 1 1 12 10-00 33 July 1 20 50 io-oo 65 July 2 13 48 10-05 2 2 3 10-00 34 21 14 9-90 66 14 50 10-25 3 2 15 10-00 35 22 10 9-90 67 15 5 10-35 4 2 33 10-00 36 22 57 io-oo 68 15 5 10-40 5 2 45 10-00 37 23 10 10-00 69 15 12 10*45 6 2 45 9-90 38 1 23 20 io-oo 70 15 15 10-40 7 4 6 10-00 39 2 1 0 10-00 71 15 20 10-55 8 5 25 10-00 40 1 50 9-90 72 15 25 10-60 9 5 55 10-00 41 2 45 io-oo 73 15 35 10-60 10 6 25 10-00 42 3 0 io-oo 74 15 35 10*75 11 6 30 10-00 43 4 15 io-oo 75 15 40 10-60 12 7 0 10-00 44 4 52 10-00 76 15 50 10-70 13 7 44 10-00 45 5 25 io-oo 77 15 50 10-85 14 9 36 10-00 46 6 25 10-00 78 15 55 10-75 15 9 45 10-00 47 6 30 10-00 79 16 0 10-85 16 10 0 10-10 48 7 5 io-oo 80 16 5 1075 17 10 0 10-00 49 7 26 io-oo 81 16 15 10-95 18 10 58 10-00 50 7 30 io-oo 82 16 30 10-90 19 13 30 10-00 51 8 15 io-oo 83 16 30 10-85 20 13 45 10-00 52 8 42 10-05 84 16 45 10-80 21 14 30 10-10 53 9 15 10-05 85* 16 48 (10-95) 22 15 0 10-00 54 9 30 9*90 86 17 0 10 65 23 15 28 10-00 55 10 15 io-oo 87 17 0 10-60 24 16 0 10-00 56 10 30 10-00 88 17 15 10-45 25 17 0 10-15 57 11 30 io-oo 89* 17 15 (10-68) 26 18 0 10-15 58 11 42 9-90 90 17 30 10-30 27 18 30 10-00 59 12 15 io-oo 91 17 45 10-25 28 19 0 10-05 60 12 48 10-10 92 18 0 10-15 29 19 0 10-00 61 13 10 9-90 93 18 15 10-15 30 20 0 10-00 62 13 20 10-20 94 19 30 10-05 31 20 0 10-10 63 13 40 10-10 32 20 44 10-00 64 2 13 45 io-oo 1 If the foregoing observations be charted down in the form of a light curve, we find that, with one exception, the star remains constant in magnitude for 1 day 16 hours out of the full period of Taken too low down for accurate observations. 76 Proceedings of Royal Society of Edinburgh. [sess. 1 day 20 hours 30 minutes. During the remaining 4 hours 30 minutes the star is variable in magnitude. During 2 hours and 1 5 minutes — half the total time of variation — it is decreasing in magnitude. During the remaining 2 hours and 15 minutes the star is increasing in brightness. There is evidently no stationary period at the minimum phase. Midway in the stationary maximum phase there is an apparent drop of 0T0 magnitude. This is indicated by the two observations — h. m. M. 1901. July 1, 17 0 10*15. 18 0 10*15. Thus we have two minimum phases — one well defined, lasting 4 hours 30 minutes, and the other not so clearly defined. In the first or principal minimum the star diminishes M. 0-85 in brightness ; in the latter or secondary minimum only M. 0-10. The whole course of the light curve is set forth in fig. (1), PL I. The light curve of the star during principal minimum phase is given in fig. (2), PL II. 1901-2.] Dr A. W. Eoberts on Algol Variation. 77 Before dealing with the immediate interpretation of the main features of the light curve of C.P.D. - 41°*4511, it may be of interest to indicate the equations which relate light variation ta orbital movement. And I may be allowed to state these rela- tions in the form I have found most serviceable in my own investigations of Southern Algol variables. Other equations more precise and more elegant will no doubt occur to those interested in such investigations. Still, in practi- cal work, the convenience of any expression is not its least merit. In fig. 3 we have the representation of the real orbit of an Algol binary star. In this typical system let — A. = angular distance of periastron from line of sight. v = true anomaly, POS. 0 = angular distance of star from line of sight, LOS. = (A + v). 1 = inclination of orbit, e = eccentricity of orbit. 8 = projected or apparent distance between centres of component stars. Then (1) S=— ? - — - — \/l - cos 2l cos 20. 1 + e cos v In fig. 4 we have the projection of the system on a plane at right angles to the line of sight. Fig. 4. Let the position of the two discs represent the eclipse of com- ponent S(1) by component S(2). 78 Proceedings of Royal Society of Edinburgh. [sess. Then if 0^ = 3 B0102 = ^>1 boa = 2 OjB = radius of S(1) = i\ 02B = radius of S(2)\=r(2) = total amount of light of S(1) L2 = total amount of light of S(2) it follows that (2) Also of f S2 + r - r cos <&, = — - 32 - f + f C0S $2 = 0 % 2 2r08 L0 = light of system at any time T. M0 = magnitude of star at time T. M = magnitude of star at constant phase, that is, combined mag. of S(1) and S(2) , then in the case where S(1) is eclipsed by S(2) , /(2<^>1 - sin 2<£1) 4- r*(2<£2 - sin 2 2) (3) Also (4) 1^=1 -Li j7rr and in the case where S(2) is eclipsed by S(1) I K _ L | ^(20! - sin 2^) + r\{ 2 cos 2cf> cos 2K L„ = 1 - L,( ~ sm ^ 1 - L„ 2 cf> - sin 2 A M0 = 2-5(10 -Log L0) + M The date when eclipse begins or ends yields yet another relation; for let P = period of variable. Tm = date of minimum phase. T0 = date when eclipse begins or ends. Tr__(±To + Tw)360°. iv p , then (9) 2 r— Jl - cos 2t cos 2K . We may now apply the foregoing formulae to the particular case of the light curve of C.P.D. — 41°*45 1 1, as exhibited in %. (2), PI. II. An examination of the light curve indicates that the star passed its minimum on h. m. July 2, 16 17 (C.M.T.) We also find that the descending and ascending phases are similar, both portions of the light curve being symmetrical on either side of the minimum. 80 Proceedings of Royal Society of Edinburgh. [sess. The orbit of the system is therefore circular. At principal minimum, the star falls from its normal brightness, 10m*05, to 10m,90, that is, it loses 0-543 of its light. At secondary minimum the star falls from 10m,05 to 10mT5, that is, the system loses 0-088 of its light. Assuming, as a first solution, 7*1 = r9 equation (8) becomes Lji /2<£ — sin 2 \ ) = 0-543 \ 7T ) L2i /’2\ ( = 0-088 \ 7T ) + L2 = 1* L1 = 0-860 L2 = 0-140 e£ = 72°-53' 46". From equation (7) cos l = O^M^ 9219 43'. vs 0-995 Since K_ 2h 1 5m x 360° _ 1g0 10, 44h 30m then from equation (9) 2 r= JO-1057 = 0-325 r = 0-163 . The foregoing determination of some of the principal elements of the binary star C.P.D. - 41°*4511 enables us to arrive at a value of the density of the system. In the Astrophysical Journal , vol. x. Ho. 5, I pointed out that a very simple relation existed between the orbital elements of an Algol system and the mean density of the system. 1901-2.] Dr A. W. Boberts on Algol Variation. 81 Thus if p = ratio of diameter of S(1) to diameter of orbit q — ratio of diameter of S(2) to diameter of orbit ml = mass of Sx m2 = mass of S2 Ax = density of S: A2 = density of S2 t = time of revolution in solar days then and if p = q . *0135 _-0135 At + A2 = m1 m1 + m2 m2 m1 + m2 0135 pH2. In the system we are considering, Therefore p = r — 0*163 t = 1’854 days. Aj + A2_ -2 = CP44 that is, the mean density of the system is a little less than half that of the sun. It may be stated that this value is a good deal higher than is found for most of the other close binary systems. In the AstropJiysical Journal , vol. x. Ko. 5, I pointed out that the average density of Southern Algol binary stars is one-eighth that of the sun. A recent investigation, including a larger number of stars, yields a similar value. There is this important reservation, however, that if we consider only these stars that revolve round one another, practically in contact, we will obtain a much smaller mean density than if we take the mean density of all the Algol binaries. The importance of this fact in connection with the evolution of close binary systems will at once be evident. The mean density of binary stars that are in that early stage of their development when fission has taken or is taking place, will be more rare than the mean density of systems that have, through the reflex action of their PROC. ROY. SOC. EDIN. — VOL. XXIV. 6 82 Proceedings of Royal Society of Edinburgh. [sess. tidal forces, gone apart, one or two diameters, in their long spiral journey. We may now collect under one statement what has been ascertained concerning the star C.P.D. - 41°*4511 from an ex- amination of its light curve. (1) The star is a close binary system, revolving in a circular orbit in 44 hours 30 minutes. (2) This orbit is inclined 5° 43' to the plane of sight. (3) The two component stars are apparently equal in size, but not in brightness. (4) If the combined brightness of both stars be considered as equal to unity, then the brightness of one component is 0*860, and of the other 0*140. That is, one star is six times brighter than the other. (5) If the distance between the centres of the two component stars be regarded as unity, then the diameter of either star is 0*325. That is, the stars are separated from one another by a distance equal to twice the diameter of any one of them. (6) The mean density of the system is slightly less than one- half that of the sun. That is, it is two-thirds the density of water. (7) It is impossible to give, and of no value to speculate upon, the actual dimensions and mass of the system. The star lies far beyond the reach of spectroscopic examination, at least with its present limitations. We have no means, also, of ascertaining the distance of the star, but it is confidently hoped that some observatory, possessing the necessary equipment, will undertake the parallax of this and all other similar binary systems. The test of any result, or group of results, determined directly from observations, is the amount of accordance between the theory arrived at and the observations from which it has been deduced. The salient features of the observed light curve of C.P.D. - 41° *45 11 indicate a system such as that set forth. In order to ascertain with what completeness the theory con- forms to all the facts of variation, we compute a theoretical light curve, and compare this theoretical light curve with that actually observed. 1901-2.] Dr A. W. Roberts on Algol Variation. 83 The accordance between them is the measure of the validity and reliability of the conclusions arrived at. Three equations are sufficient for the determination of the theoretical magnitude at any instant : In which, T = interval in minutes to or from the minimum date July 2, 16h. 17m. L = amount of light at time T M = magnitude of star at time T With these equations, theoretical values of the magnitude of the variable for instants when actual observations were made were computed. These computed values are given in the following table, column (5), and the residuals between the computed and observed magnitudes in column (6). The mean observed light curve and the corresponding residuals are given in columns (3) and (4). It will be seen that the theoretical and observed light curves are practically identical. As exhibited in column (7), the divergence between the two curves never exceeds one-hundredth of a magnitude. In fig. (2) the theoretical light curve is shown as a dotted line. [I have not been unmindful in the foregoing investigation of the possibility of different portions of the stellar discs being un- equally bright. I think, however, the assumption that the stars are in that stage of development when a distinct atmosphere has not yet been formed is reasonable. If no light absorbing outer layer exists, then all portions of the stellar discs will be equally luminous.] («) = 3-077 -0-991 cos2(0'135 T)° L= 1 - 0-274(2$ - sin 2) M = 2-5(10 - Log L) + 10m'05 . [Table. 84 Proceedings of Royal Society of Edinburgh. [f Date. Obser. Observed Light Curve. Theoretical Light Curve. Observed Theoreti- cal Curve. Mag. Res. Mag. Res. 1901. h. m. M. M. M. M. M. M. July 2 13 48 10*05 10*05 + 0*00 10-05 + 0*00 + o-oo 14 50 10*25 *26 - -oi •26 - *01 + •oo 15 5 10*35 *37 - *02 •36 - •01 - •01 15 5 10*40 *37 + *03 •36 - *04 - •01 15 12 10*45 *43 + *02 •43 + •02 + •oo 15 15 10-40 *45 - *05 •45 - *05 + •oo 15 20 10*55 *50 + *05 *49 + •06 - •01 15 25 10*60 *54 + *06 , *53 + *07 - *01 15 35 10*60 *64 - *04 *63 - *03 - •01 15 35 10*75 *64 + *11 *63 + •12 - •01 15 40 10*60 *67 - *07 •67 - •07 + •oo 15 50 10-70 *77 - *07 *77 + •07 4. •oo 15 50 10*85 •77 + *08 •77 + •08 4. •oo 15 55 10*75 *81 - *06 *81 - *06 + •oo 16 0 10*85 •84 + -01 •85 + *00 + •01 16 5 10*75 •88 - *13 •88 - •13 + •oo 16 15 10*95 •92 + *03 *91 + •04 - •01 16 30 10*90 *88 + *02 *87 +. •03 - •01 16 30 10*85 •88 - *03 *87 - •02 - *01 16 45 10*80 •77 + 0*03 *76 + 0*04 - •01 16 48 (10-95) •73 (+*022) •72 ( + 0*23) - •01 17 0 10*6 *62 + 0*03 •61 + 0-04 - •oi 17 0 10*05 •62 - *02 *61 - •01 - •01 17 15 10*40 •48 -0*03 •48 0*03 + •oo 17 15 (10*68) •48 ( + 0*20) *48 ( + 0*20) + •oo 17 30 (10*30) •35 -0*05 •35 0*05 + •oo 17 45 10*25 •24 + *01 •24 + *01 + •oo 18 0 10*15 •16 - *01 *15 - •oo - •oi 18 15 10*15 •10 + *05 *09 + •06 - •01 19 30 10*05 10*05 + 0*00 10*05 + 0-00 + 0-00 M. Mean residual (observed curve) =0*040 Mean residual (theoretical curve) = 0*041 The two observations of July 2, 16 h. 48 m., and July 2, 17 h. 15 m., were made when the star was too low down for accu- rate observations to be made. It would be, of course, entirely unnecessary to differentiate the equations which relate light changes to orbital movement, and so form equations of condition with the values given in column (7). The data are too meagre for any such refinement. Indeed, it is evident that the accordance between theory and observation is so complete as to yield not only strong testimony to the certitude, validity, and sufficiency of the conclusions come to, but to render any differential correction unnecessary. (. Issued separately March 31, 1902.) Proc. Roy. Soc. Ed in. Vol. XXIV. Roberts: Variable Stars of the Algol Type— Plate I. Pr rr /r Of hi St A/ *)! s/ h/ I N* 4 V |! Ni *51 // • QO Proc. Roy. Soc. Edin. Vol. XXIV. Roberts: Variable Stars of the Algol Type— Plate II. <0 * 0 ‘ i V Vi T[ :4| Of SC OT s/ s ° 4/ ss OS 'S'/ O'/ sc oe ST or s> o/ S ° // ss OS sh at se or sc or s/ o/ s o p/ ss os ss *V £ 55 O • ^ •* S 3 v> <0 Q fe *o Q sr or s / O/ o &/ SS as SS se or sc oc s/ c, ^ o V.'5 ^ .V ^ C Vi A ^ (i i) U, . SN Qv- t «* v T.K "I ^ <5 <« vjj \j O- $ ao •// S6-°/ ■ . . 1901 — 2.] Dr Marshall on Modifications of Sign of Equality . 85 Suggested Modifications of the Sign of Equality for Use in Chemical Notation. By Hugh Marshall, D.Sc. (Read January 6, 1902.) During recent years the great attention which has been paid to the study of reversible chemical actions has led to a desire for some method of specially indicating in the equations the reversi- bility of these actions. This is generally done by replacing the sign of equality by two parallel and oppositely-directed arrows; the use of this symbol in such cases does not only imply the reversibility of the action, it has the same value as the sign of equality in an ordinary equation. Unfortunately, there are several drawbacks to the use of this symbol : as generally printed, it is rather clumsy in appearance ; the arrows, being widely separated, do not suggest the sign of equality which they replace ; and further, in conjunction with formulse, arrows are frequently employed, especially in organic chemistry, to indicate merely the stages and methods by which a substance can be produced from some other substance as starting-point. In such cases, as no attention is paid to the bye products, there is nothing of the nature of an equation involved. Numerous examples are to be seen in Richter’s Organische Chemie , and amongst them cases in which oppositely- directed arrows, resembling the symbol employed for the equations for reversible actions, are used to indicate that each of the two substances concerned may serve as starting point for the prepara- tion of the other, without involving equality on the two sides of the symbols. Such a double use of a symbol is highly objectionable. These drawbacks might be avoided by employing a modified sign of equality composed of a pair of split or singly barbed arrows, the barbs being only on the outer sides of the symbol. This symbol can be made neat and comparatively small ; it suggests the ordinary sign of equality, and also the idea of direction given by ordinary arrows. It is also possible to modify the symbol in such a way as to indicate clearly the different varieties of chemical 86 Proceedings of Royal Society of Edinburgh. [sess. action which it may be desirable to distinguish, without affecting the general character of the symbol. I would therefore suggest the adoption of some definite scheme similar to that indicated below. (1) The use of ordinary arrows should he restricted to the indication of chemical actions, when it is desired to follow only some of the substances involved in the change, without writing complete equations, in the manner referred to above as being frequently used in organic chemistry. Thus — c6h4< ,ch2 CHo c6h4< •CH, 'CH- >g6h4^. u / CO, C6H4, 'CO >C6H4 {Richter. ) (2) The ordinary sign of equality should be used only in equations which are required merely for purposes of calculation, etc., to represent equalities in quantities of matter (or of matter and energy, as in thermo-chemical equations), irrespective of whether the substances represented on one side of the equation change directly into those on the other side or not. Thus, the equation — KCl + 3H20 = KClOg + 3H2 does not represent a chemical change capable of being directly carried out, but it might be employed for calculations concerning the electrolytic preparation of potassium chlorate. Similarly with tliermo-chemical equations — 2K + 2S + 80 = K2S208 + 454*5 Cal. (3) When it is desired to write a complete chemical equation, not only representing the quantities of matter involved, but also indicating clearly that chemical change can take place directly in accordance with the equation, then the sign of equality should be composed of singly barbed arrows as illustrated below, modifications being introduced for the principal varieties of chemical action which it is desirable to differentiate. (a) For irreversible actions, the sign thus : — KC104==^IvC1 + 202; 4KN03 + 5C == 2K2C03 + 3C02 + 2lSr2. 1901-2.] Dr Marshall on Modifications of Sign of Eguality. 87 (b) For reversible actions such as dissociations, etc., the sign thus : — CaC03^Ca0 + C02; 2H20^2H2 + 02; H2S04 ^ IT + HSO; ^ 2H* + SO/. (c) For reversible actions associated with a definite transition temperature, the sign above which might be noted the appropriate temperature,* thus : — Na2Mg(S04)2,4H20 + 13H20 Na2S04, 10H,O + MgS04, 7H20 ; CuC12,2H20 + 2K2CuC14,2H20 2KCuC18 + Ii,CuCl4,2H20 + 4H20 3KCuC13 + KC1 + 6H20. ( d ) In cases where reversible actions go practically completely in one direction under ordinary conditions, and the reverse action may be rendered negligible, or when there is any doubt as to the reversibility of the action, and it is not desired to make a definite indication either way, the sign === might be used. This would apply to many of the equations representing the actions involved in analytical work, thus : — CuCl2 + H2S — CuS + 2HC1; BaCl2 + K2S04 =^= BaS04 + 2KC1. The symbols printed above f may be considered unduly large for the type employed ; they might easily be made considerably smaller if a still neater form is desired. It is possible that the general use of separate symbols to distinguish between equations representing mere equalities in quantities of matter, and those representing actual chemical changes, might go far to remind students of chemistry that a particular chemical change is not necessarily possible merely because an equation representing it can be produced on paper. * In such cases the barbs on the upper half of the symbol might be re- moved, which would simplify it somewhat without affecting its intelligibility, thus, t From type supplied by Messrs Miller & Richard, letterfounders, Edinburgh. (. Issued separately March 31, 1902.) 88 Proceedings of Royal Society of Edinburgh . [sess. The Action of Silver Salts on Solutions of Ammonium Persulphate. By Hugh Marshall, D.Sc., and J. K. H. Inglis, M.A., B.Sc. (Read February 17, 1902.) In a paper already communicated to the Society by one of us* attention was drawn to the fact that, in presence of a soluble silver salt, ammonium persulphate is decomposed with formation of nitric acid as well as sulphuric acid. The action had at that time been studied only in a roughly quantitative way, and it was therefore thought that a more exact study of the velocity of the reaction, and of the influence exerted by other salts present in solution, might prove of interest, and also throw light on the course which the reaction follows. As stated in the paper already referred to, the reaction appears to be expressible by the equation — (a) 4(NH4)2S208 + 3H20 = 7NH4HS04 + H2S04 + HN03 , but it was found that the rate of change corresponded approxi- mately to that of a monomolecular reaction, indicating that the reaction takes place in several stages, one of which being much slower than the others, determines the velocity. If the above equation is written in the form — (&) NH4' + 4S208" + 3H20 = 10H‘ + 8S04" + N03' , it is evident that the reaction could not be monomolecular. But if the intermediate formation of silver peroxide is assumed (and there are good grounds for the assumption), the reaction is divisible into the two stages represented by the following equations — (c) S208" + 2 Ag' + 2H20 2S04" + 4H* + Ag202 ; (d) 4Ag202 + NH/ + 6H‘=^: 8Ag* + N03' + 5H20 . Neither of these is actually monomolecular. But we may consider the concentrations of the Ag' and the Ag202 as being constant, if the concentration of the total silver is small compared with the * Marshall, Proceedings , vol. xxiii. p. 163. 1901-2.] Action of Silver Salts on Ammonium Persulphate. 89 concentration of persulphate ; for the silver will then be divided in a practically constant ratio between the two phases Ag’ and Ag202. Making this assumption, equation (c) becomes in effect monomolecular, for, if the reaction is irreversible, it is necessary to consider only the members on the left of the equation. The case is quite different with equation ( d ), for the concentrations both of H’ and of NH4* vary, that of the former starting from zero and increasing rapidly, while that of the latter diminishes. Preliminary experiments showed that, as was to be expected, change of temperature has a very marked effect on the rate of change ; at 50° the velocity is about eight times greater than it is at 25°. Also, the rate increases nearly, but not quite, proportion- ally, with increase of concentration of silver. As a result of the preliminary experiments, it was found most convenient to study the reaction in solutions which were 0*5 normal as regards ammonium persulphate and 0’0005 normal as regards silver; the solutions were kept in closed flasks placed in a thermostat at 25°. The choice of salts whose influence on the action could he studied is rather a limited one, halides being, of course, entirely excluded on account of the precipitation of insoluble silver salts. Six solutions were prepared as indicated in the following table, the quantities stated being dissolved, in each case, in one litre of solution. I. J(NH4)2S208 + ^oAgNOs II. J(NH4)2S2Os + -joVoAgNOg + JNaNO, III. J(NH4)2S208 + WV» AgN03 + |(NH4)2S04 IY. !(NH4)2S2Os + T J*o AgN03 + piI2XOg V. i(NH4)2S208 + W^AgY 03 + |HN 03 YI. J(NH4)2S208 + ^AgN0g + iH2S04 The course of the reaction was followed by removing, at intervals of twenty-four hours, 5 c.c. of each solution and titrating with jST /5 alkali, using methyl orange as indicator. In presence of silver the indicator is rapidly bleached by the residual persulphate, but the difficulty was overcome by the addition of a drop of sodium chloride solution, which precipitated the silver. In the case of solutions Y. and YI. the initial acid concentration was determined 90 Proceedings of Boyal Society of Edinburgh. [sess. in a similar manner and taken into account in determining the rate of increase. The figures obtained are given in the following table ; in the case of solutions Y. and VI., A indicates the initial figure. Acidity , in c.c. of Y/5 alkali required to neutralise 5 c.c. of the solution. Time in days. Sol. I. Sol. II. Sol. III. Sol. IY. Sol. Y. Sol. YI. 0 A A 1 2*63 2-09 2-21 2-20 A + 2-74 A + 2-66 2 4-86 3*98 4*01 4-11 4*90 4*79 3 6-61 5*58 5-68 5-77 671 6- 77 4 8-24 6-89 7*19 7-17 8-38 8-30 5 9*54 8-18 8-40 8-42 9-63 9*37 6 10*56 9-25 9-29 9-46 10-60 10-34 7 11-43 10-14 10-22 10-40 11-37 11-29 8 12-11 10-94 11-10 11-21 12-05 12-03 9 12-65 11-53 11-67 11-80 12-53 12*53 10 13-17 12-10 12-23 12-33 12-90 12-94 00 15*34 15-34 15-32 15-31 14-56 14-82 These figures show that the addition of a neutral salt simply retards the action somewhat, hut the end point is the same as with persulphate alone. In solutions III. and IY. the hTH4' concentra- tion is doubled, or nearly so, and yet the velocity is not increased ; hence equation b cannot determine the reaction-velocity. This must therefore he determined by equation a, that is, by the action of Ag* on S208", as already deduced on other grounds. The retardation caused by the addition of the various salts would thus be explained, for such an addition would tend to diminish the concentration of the persulphate ions. The effect of acids, as shown in Y. and YI., is specially note- worthy. At first it would appear that the action is accelerated ; but the end point is not the same as in the other case, indicating the simultaneous occurrence of some other action which results in the production of less acid. The equation (e) 2 Ag202 + 4H‘ = 4Ag* + 02 + 2H20 would represent such an action, and it is evident that in this case an increase of acid concentration would tend to accelerate the change. 1901—2.] Action of Silver Salts on Ammonium Persulphate. 91 Kegarding the action as determined by the equation (c) S208" + 2Ag’ + 2H20 =5*: 2 SO/ + 4H’ + Ag202 the expression for the reaction velocity is - 1 where C denotes the concentration of persulphate ions, the con- centration of the silver ions being taken as constant. As the solutions to begin with were only semi-normal as regards ammonium persulphate, the concentration of the persulphate ions may be assumed, with a fair degree of accuracy, to be that of the unde- composed persulphate. Integration of the above equation gives or, changing to common logarithms (which gives a new constant K), L C0 Tr tlog'°crR; where C is the initial persulphate concentration, and Gt is the con- centration after a time t. If x is the increase of acid concentration in the time t, and a is the increase in infinite time, then C0_ a Ct a - x ' Hence ilog— a =K. t *a-x Values of K for the different solutions. Time in Days. Sol. I. Sol. II. Sol. III. Sol. IV. Sol. V. Sol. VI. 1 •082 •064 •068 •067 •090 *086 2 •083 •065 •066 •068 •089 •071 (?) 3 *081 •065 •067 •068 •098 *088 4 •084 •065 •069 •069 •093 •088 5 •085 •066 •069 •069 •C94 •087 6 *084 •067 •068 •070 •094 •087 7 *085 •067 •068 •071 •094 •089 8 •085 •068 •070 •071 *095 •091 9 •084 •067 •069 •071 •095 •090 10 •085 •068 •070 •071 •094 •090 92 Proceedings of Royal Society of Edinburgh. [sess. At first sight these results appear very satisfactory, the nearly constant values obtained for any single solution indicating that the reaction was really monomolecular, while the different values for different solutions showed the effect of the various substances added. When the results are examined critically, however, especi- ally those from the two acid solutions Y. and VI., difficulties are encountered. In the two cases mentioned the end points are different from the remaining ones and from one another. The discrepancy is greater with nitric acid than with sulphuric acid, as would be expected from the greater ionisation of the former. Though the discrepancies are not very great, they indicate a con- siderable amount of action other than that resulting in the formation of nitric acid. Examination of the two equations (b) NH4* + 4S208" + 3H20 = 10H' + 8S04" + N08' (/) 4S20s" + 4H20= 8H’ + 8S04" + 202 shows that the second action produces only one-fifth less acid than the first. Assuming that in solution I. the ultimate acidity (15 '34) was wholly produced by the first action, the acidity which would have been produced by the second action alone would have been 3 ‘07 less than this. In solution Y. therefore it may he assumed that one-fourth of the total persulphate had decomposed without producing nitric acid; in the case of the sulphuric acid solution VI., the proportion would he about one-sixth of the total. The higher rate of increase in the acidity in solutions Y. and YI. would therefore seem to be due, not to an acceleration of the action producing silver peroxide, but to the superposition of another action (apparently negligible in the other cases) similar to that which takes place in solutions of persulphate free from silver. It might have been expected that the acid gradually produced in, say, solution I. would result in bringing about the second of the above actions ; but, if the effect in that case had been appreciable, it is to be presumed that in solutions II. , III. and IV., to which neutral salts had been added (and in which, therefore, the concentration of the hydrogen ions would be diminished), the effect would have been less marked, with the result that the final acidities in these cases would have been higher ; but this is not the case. Even in solution I. the concentration of hydrogen 1901-2.] Action of Silver Salts on Ammonium Persulphate. 93 ions would probably not be very considerable until the greater portion of the persulphate had been decomposed, owing to the considerable quantity of neutral salt present ; so that the former equation ( b ) might perhaps be replaced by the following one — (V) NH4* + 4S208 + 3H20 = 2H' + 8HSO/ + NOf While the results of the experiments are in some respects in- complete and inconclusive, there seems little doubt that the speed of the reaction is determined by the formation of silver peroxide, the oxidation of the ammonium ions taking place much more rapidly. It has been assumed that silver peroxide is actually formed in the above decompositions, but another, and, on the whole, prefer- able hypothesis is possible. Silver peroxide has never been prepared quite pure, and its exact nature is somewhat doubtful. When prepared from silver nitrate it cannot be obtained free from nitrate ; it dissolves in nitric acid, and can be reprecipitated from the solution. It is therefore permissible to assume that silver peroxide is really an exceedingly feebly basic oxide, whose salts are hydrolysed with great ease. If we make the assumption, which involves the possible existence of perargentic ions (‘ diargention ’) in minute quantity, the interaction of persulphates and silver salts may be represented as follows — (c) 2Ag+S208"-== 2Ag" + 2S04". In absence of oxidisable substances hydrolysis takes place, with separation of perargentic hydroxide, oxide, or basic salt. In presence of ammonium ions, however, the formation of nitric acid takes place, with £ reduction 5 of the perargentic ions as follows — (d') 8Ag” + KH/ + 3H20 8Ag* + 10ET + N08\ {Issued separately March 31, 1902.) 94 Proceedings of Boyal Society of Edinburgh. [sess. Magnetic Shielding in Hollow Iron Cylinders. By James Russell. (Read February 3, 1902.) {Abstract.) This investigation deals with the shielding which exists within hollow iron cylinders when placed in a uniform transverse magnetic field. This transverse field divided by the internal magnetic field is defined as the shielding ratio. For the thin iron cylinders ex- perimented with, it appears from various mathematical contributions to the subject that the shielding ratio minus unity may for present purposes be taken to be proportional to certain geometrical data and to the permeability (the permeability being large). By means of a rotating inductor within the shield, connected with a ballistic galvanometer, experimental determinations of the shielding ratio are made under various conditions of magnetisation ; and an endeavour is also made to show how far and under what conditions these results approximate to theoretical formulae which assume the permeability to be uniform all round the shield, and the absence of retentivity and coercive force in the iron. Two iron shields were experimented with, the hysteretic constants {rj) being •0015 and *0028 respectively. The following results have been arrived at, viz. : — I. When no other magnetising force is acting upon the iron than that due to the transverse field increased by increments from zero, the shielding ratio minus unity is proportional to what may be called the ratio permeability (B/H), and not to the differential permeability (dB/dR). If, however, the transverse field is de- creased by steps from a maximum, the theoretical conditions are not fulfilled, and the shielding ratio increases very rapidly and becomes infinite, but does not change sign if account be taken of the negative residual field within the shield when at each step the transverse field is withdrawn. II. When a circular magnetising force is acting upon the iron 1901-2.] Mr James Bussell on Magnetic Shielding. 95 in addition to that due to the transverse field, the order and manner in which the one field is superposed upon the other affects the shielding ratio to an enormous extent. The conclusions arrived at are not in harmony with the inductions drawn from the investi- gations of Stefan and Du Bois, which “lend support to the assumption made throughout that shielding against moderate dis- turbing fields depends within due limits upon the constant permea- bility for small increments or decrements of force superimposed upon any condition of magnetisation.” (See Electrician , “ Magnetic Shielding,” Du Bois, vol. 40, p. 654, 1898.) It is found that when upon a pre-existing induction due to the transverse field, increments of the circular field are superposed (T C conditions), the shielding ratio minus unity is proportional to the differential permeability as impressed upon the iron by the circular field. The shielding ratios plotted as ordinates against the corre- sponding values of the circular field as abscissae rapidly increase, attain a maximum, and then decrease, finally approximating to a minimum asymptotic value. If, however, upon a pre-existing in- duction, due to the circular field increasing from zero, the transverse field is superposed at each increment (C T conditions), the shielding ratio curves take much lower values. The superposition of repeated reversals of the transverse field (C T T conditions) still further lowers the shielding ratio, and in this case there appears to be no initial rise as the curve leaves its origin in the vertical axis. On the other hand, the superposition of repeated reversals of the circular field (T C C conditions) increases the shielding ratio so long as the values of (dB/dU.) are high. When the values of dB/dK are a maximum, the shielding ratio under the T C C con- ditions are six to seven times greater than under the C T T con- ditions. The curves obtained under all the conditions of super- position of fields appear to approximate to the same minimum value as the circular magnetising force is further and further increased. A distinction is drawn between the permeability of the iron to the circular field and the permeability of the iron to the trans- verse field. Thus, while under the T C conditions the shielding ratio minus unity is proportional to the permeability ( dB/dR ) of the iron to the circular force, the same does not hold good under, 96 Proceedings of Royal Society of Edinburgh. [sess. say, the C T T conditions. The effect of repeated reversals of the transverse field is shown to increase the permeability of the iron to the force which remains constant, viz., the circular field, and, in virtue of the phenomena associated with hysteresis, to lower the permeability of the iron to the transverse field, and consequently also to lower the shielding ratio. Further, when, upon a pre-existiug induction due to the trans- verse field, the superposed circular field carries the iron round a complete cycle, the shielding ratio curves resemble the theoretical curves obtained from the corresponding values of dR/dK impressed upon the iron by the circular field. III. The effects of a longitudinal magnetising force acting upon the iron in addition to that due to the transverse field have not yet been fully investigated. Preliminary experiments show that in this case also the order and manner of field superposition largely determine the shielding ratios. Also that when, upon a pre- existing induction due to the transverse field, increments of the longitudinal field are superposed, the differential permeability appears to be the determining factor. {Issued separately March 31, 1902.) 1901—2.] Lord Kelvin on Stress and Strain. 97 A New Specifying Method for Stress and Strain in an Elastic Solid. By Lord Kelvin. (Read January 20, 1902.) The method for specifying stress and strain hitherto followed by all writers on elasticity has the great disadvantage that it essen- tially requires the strain to be infinitely small. As a notational method it has the inconvenience that the specifying elements are of two essentially different kinds (in the notation of Thomson and Tait e, /, g, simple elongations ; a, b, c , shearings). Both these faults are avoided if we take the six lengths of the six edges of a tetrahedron of the solid, or, what amounts to the same, though less simple, the three pairs of face-diagonals of a hexahedron,'" as the specifying elements. This I have thought of for the last thirty years, but not till a few weeks ago have I seen how to make it conveniently practicable, especially for application to the generalised dynamics of a crystal. § 1. We shall suppose the solid to be a homogeneous crystal of any possible character. Cut from it a tetrahedron ABCD of any shape and orientation. Let the three non-intersecting pairs (AB, CD), (BC, AD), (CA, BD) of its six edges be denoted by (%, Sp-), (32. 3gO, (Sr, 30 . . . (1). This notation gives (p, p')> fo 2')» (r, r') (2) for the six edges of a tetrahedron, similar to ABCD, formed by taking for its corners (a, /3, y, S) the centres of gravity f of the four * This name, signifying a figure bounded by three pairs of parallel planes, is admitted in crystallography ; but the longer and less expressive ‘ £ parallele- piped” is too frequently used instead of it by mathematical writers and teachers. A hexahedron, with its angles acute and obtuse, is what is commonly called, both in pure mathematics and crystallography, a rhombohedron. A right-angled hexahedron is a brick, for which no Greek or other learned name is hitherto to the front in usage. A rectangular equilateral hexahedron is a cube. t For brevity I shall henceforth call the centre of gravity of a triangle, or of a tetrahedron, simply its centre. PROC. ROY. SOC. EDIN. — VOL. XXIV. 7 98 Proceedings of Royal Society of Edinburgh. [sess. triangular faces BCD, CDA, DAB, ABC respectively, so that we have p — a/3, q = f3y, r = ya, p = yS, f = aS, r' = f3§. Consider now, in advance, the amounts of work done by the six pairs of balancing forces constituting the six stress-components described in § 2, when the strain-components vary ; for example, the balancing pulls P, parallel to AB, when a/3 increases from p to p + dp, all the other five lengths q, r, p, q, r remaining constant. For the reckoning of work we may suppose the opposite forces, P, to he applied at a and (3, instead of being equably distributed over the faces ADC, BDC. Hence the work which they do is P dp ; and other five pairs of balancing pulls, Q, R, P', Q', R', do no work. § 2. Parallel to the edge AB apply to the faces ADC, BDC equal and opposite pulls, P, equally distributed over them. These two balancing pulls we shall call a stress or a stress-component. Similarly, parallel to each of the five other edges apply balancing pulls on the pair of faces cutting it. Thus we have in all six stress- components parallel to the six edges of the tetrahedron, denoted as follows : — (P,F) (Q,Q’) (B, B’) .... (3); and we suppose that these forces, applied as they are to the four faces of the solid, are balanced in virtue of the mutual forces between its particles, when its edges are of the lengths specified as in (1). Let _p0, pf , g0, gf , r0, rf be the values of the specifying elements when no forces are applied to the faces. Thus the differences from these values, of the six lengths shown in formula (2), represent the strain of the substance when under the stress represented by (3). § 3. Let w be the work done when pulls upon the faces, each commencing at zero, are gradually increased to the values shown in (3). In the course of this process we have dw = Ydp + Ydp + Qdq + (fdq + Rcfo* 4- R'dr' . . (4). Hence if we suppose w expressed as a function of p, p , q, q', r, r , we have dw -p. die , dw _ dw dw „ dw T>, ^=p> #'=p> dj=q’ s?=Q’ d/=R • (4)- This completes the foundation of the molar dynamics of an 1901-2.] Lord Kelvin on Stress and Strain. 99 elastic solid of the most general possible kind according to Green’s theory, expressed in terms of the new mode of specifying stresses and strains, without restriction to infinitely small strains. § 4. To understand thoroughly the state of strain specified by (1) or (2), let the tetrahedron of reference, A0B0C0D0, for the condition of zero strain and stress, be equilateral (that is to say, according to the notation of § 2 (1) let J of each edge = Po ~ % ~ ro ~Po = $o = ro)m AqBoCoDo inscribe a spherical sur- face touching each of the six edges. Its centre must be at K0, the centre of the tetrahedron ; and the points of contact must be the middle points of the edges. Alter the solid by homogeneous strain * to the condition (p, q , r, p , q , r) in which A0B0C0D0 becomes ABCD. The inscribed spherical surface becomes an ellipsoid having its centre at K, the centre of ABCD, and touch- ing its six edges at their middle points.! This ellipsoid shows fully and clearly the state of strain specified by p , q, r, p ', q', r'. It is what is called the “strain ellipsoid.” J § 5. Two ways of finding the ellipsoid touching the six edges of a tetrahedron are obvious. (1) Through AB and CD draw planes respectively parallel to CD and AB; and deal similarly with the two other pairs of non-intersecting edges. The three pairs of parallel planes thus found constitute a hexahedron which contains the required ellipsoid touching the six faces at their centres; or (2) draw AK, BK, CK, DK, and produce to equal distances KA', KB', KC', KD' beyond K. We thus find four points, A', B', C', D', which, with A, B, C, D, are the eight corners of the hexahedron which we found by construction (1). A circumscribed hexahedron being thus given, the principal axes of the ellipsoid, and their orientation, are found by the solution of a cubic equation. § 6. Another way of finding the strain- ellipsoid, which is in some respects simpler, and which has the advantage that in its construction it does not take us outside the boundary of our * Thomson and Tait’s Natural Philosophy , § 155 ; Elements , § 136. t Thus we have an interesting theorem in the geometry of the tetra- hedron If an ellipsoid touching the edges of a tetrahedron has its centre at the centre of the tetrahedron, the points of contact are at the middles of the edges. X Thomson and Tait’s Natural Philosophy , § 160; Elements , § 141. 100 Proceedings of Royal Society of Edinburgh. [sess. fundamental tetrahedron, is as follows: — In the equilateral tetra- hedron A0B0C0D0 describe, from its centre K0, a spherical surface touching any three of its faces. It touches these faces at their centres ; and it also touches the fourth face, and at its centre. Hence, if we solve the determinate, one-solutional, problem to draw an ellipsoid touching at their centres any three of the four faces of any tetrahedron ABCD, and having its centre at K, this ellipsoid touches at its centre the fourth face of the tetrahedron ; and it is the strain ellipsoid for the homogeneous strain by which an equilateral tetrahedron of solid is altered to the figure ABCD. § 7. To bring our new method of specifying strain and stress into relation with the ordinary method for infinitesimal strains and the corresponding stresses : — Let X denote the length of each edge of the equilateral tetrahedron of reference, A0B0C0D0; and let h be the edge of the cube of which A0, B0, C0, D0 are four corners (this cube being the hexahedron found by applying either of the constructions of § 5 to the tetrahedron A0B0C0D0). The twelve face-diagonals of this cube are each equal to X, and there- fore X = h J 2. Let now the cube be infinitesimally strained so that its edges become 7i(l + e), 7i(l + /), h( 1 + g) ; and so that the angles in its three pairs of faces are altered from right angles to acute and obtuse angles differing respectively by a , b, c from right angles. This is the strain (e, /, g, a , b, c) in the notation of Thomson and Tait referred to in the introductory paragraph above. By the infinitesimal geometry of the affair, we easily find the correspond- ing alterations of the face diagonals, which according to our present notation are (p - 1)A, {p - 1)A, (q - 1)A, etc., and thus we have as follows: — V -1=K/+!7 + “) P - ^=W+9-a) q-1 =^g+e + b)\ g'-l=i (g+e-l) W r - 1 *= -|( e + /+ c) r -l=±(e+f-c) , for the relation between, the two specifications of any infinitesimal strain. Adding these, and denoting e +/ + g by s, we find p+p +q + q' + r-\-r' - Q = 2 s .... (6). 1901-2.] Lord Kelvin on Stress and Strain . 101 And solving for a, b, c, e , /, g, in terms of p, q, r, p', q\ r, we have a=p-p; b = q~q'; c = r-r' : e = s-p-p+ 2 ; /= s-g-^' + 2;^ = s-r-/ + 2:J ' § 8. The work required to produce an infinitesimal strain e, /, g, a, b} c, in a homogeneous solid of cubic crystalline sym- metry is expressed by the following formula : — 2 w = H(e2 +/2 + g 2) 4- ’2$$(fg + ge + ef) 4- n{a? + b 2 + c2) (8). This may be conveniently modified by putting * = J(H + 2»); ^ = • . • . (9), where & denotes the bulk modulus and nv n the two" rigidity- moduluses. With this notation (8) becomes 2w = k(e +f + gf + § n^_(f - gf + {g- ef + (e -ff] + n(a 2 + b2 4- c2) . . . (1 0). The rigidity relative to shearings parallel to the pairs of planes of the cube, or, which is the same thing, changes of the angles of the corners of the square faces from right angles to acute or obtuse angles, is nY The rigidity relative to changes of the angles be- tween the diagonals of the faces from right angles to acute or obtuse angles is n. The compressibility modulus is k. Using now (7) in (10) we have 2 w = ks 2 + ^nx[{q + q - r - r'f -1- (r + r -p -pf -\-(p+p -q- qff] + n[{p-p')‘2 + {q-qf + {r-r'f) (11). (. Issued separately April 1, 1902.) 102 Proceedings of Boyal Society of Edinburgh. Note on Selected Combinations. By Thomas Muir, LL.D. (Read January 6, 1902.) (1) Having formed from n things all possible sets of r , we may subject each of the Cn>r sets to the test of fulfilling one or more conditions, and so obtain a reduced number possessing a special characterisation. The present note deals with a few instances of this in which the reduced number is of the same algebraical form as the full number,* that is to say, is a combinatorial Ca>& . (2) First let us take the condition that the set must not contain two things which in the original row of n things occupied con- secutive places, and let us denote by Cn>r the reduced number due to the imposition of this condition. Denoting the things by «i, a3> • • • j CLn we see that for r = 2 all the possible combinations are got by taking (1) aj along with a3 , a4 , ab , ... , an (2) a2 a4, a5, . . . , an (3) as a5, . . . , an and that therefore CWj2 = {n — 2) + (n — 3) 4- ... +1, ’ =i(w-l)(*-2), = CU,2. The general theorem which we propose to establish, viz. C n,r r+1, r is thus seen to be true for the case where r = 2. But if it be true for any particular value of r , say r = s , it can be shown to be * An analogous peculiarity appears in the well-known theorem that the number of combinations of n things taken r at a time when repetitions are allowed is the same as the number of combinations of n-r+ 1 things with repetitions debarred, i.e ., is equal to Cn_r+i,r» 1901-2.] Dr T. Muir: Note on Selected Combinations. 103 true for the next case, vis., where r = s + 1. For we shall get the combinations desired by taking a1 along with s things taken from aB , a4 , ... , an a% . . . ... ... ... $4 , • • • J &n the total number of them thus being Cn— 2, s "1" ^n— 3, s which by hypothesis is equal to ^n—s—l, s d" C'n— s— 2, s and consequently equal to c„ a.,* u — (s+l)+l, s+1 as was to be shown. (3) As a second instance let us take the condition that each set of r things is to be obtainable by deleting from the n things a set of n - r of them so chosen that they form J(w - r) pairs of con- secutive things. The full list of pairs of consecutive things is («i » ®2) » ( a2 » az) > (a3 > K As each of these n— 1 pairs has an element in common with the pair which follows, we cannot in selecting \{n - r) pairs for deletion include among them two consecutive pairs. It follows therefore that if the desired number of sets of r things be denoted by K'w, we have ~^-n,r Cn_ 1 1 1 (n—r) , Cn— 1— i(n— r)+l, i(n—r)t "^y § 2 , , \(n—r) > (n+r),r* (4) The proposition of § 2 may also be written P — C KJn+r— 1 , r ) and we thus see that any known identity connecting several C’s has corresponding to it an identity connecting as many C'’s. For example, corresponding to the identity Cn>r — CK t n-r Cn,r ~ C‘2n— 3r+l , w— 2r+l ; we have 104 Proceedings of Royal Society of Edinburgh. [SE£ and corresponding to the identity Qn,r = ^n— 1 , r d* r_i we have ^n,r ~ ^n—l , r d" h?i_2 r—1 • Similarly, since the proposition of § 3 may be written ^n,r ~~ r r we have the identities ~^n,r = l^-J(?i+3}-) , (n—r, j ■K-w,r Kji-2, r d* i, j»_i • (5) In the proposition of § 2 the condition imposed was such that the selection of any particular element to be one of a set meant the rejection of the element immediately following it in the original row. A manifest extension of this condition at once suggests itself, viz., that the selection of any particular element shall exclude the k elements immediately following. The more general proposition thus obtained — and obtained in exactly the same way and almost as easily — is If Cnkr denote the number of combinations of n elements taken r at a time subject to the condition that no element is to be taken along with any one of the k elements immediately folloiving it in the original roiv, then ^n,r = ^n-k(r+ 1) , r • Derived from this in the same way as the proposition of § 3 was derived from that of § 2 we have the second general proposition — If K kr denote the number of different sets of r things obtainable from a row of n things by leaving out of n - r of them which form (n - r)/k k -ads of consecutive things , then Ki = C|(n+&Mr- Of course each of these may be utilised as the less general pro- positions were dealt with in § 4. ( Issued separately April 1, 1902.) 1901-2.] A Continuant Resolvable into Rational Factors. 105 A Continuant Resolvable into Rational Factors. By Thomas Muir, LL.D. (Read January 6, 1902.) (1) In a paper in the Proceedings of the London Mathematical Society , vol. xxxiii. p. 229, Professor Elliott has occasion to consider the equation x p - 1 X p- 1 - P X which, it is stated, presents difficulties when a direct algebraical solution is attempted, but is seen indirectly to be (a:2+ l2) {x2 + 32) (x2 +p2) = 0 , x(x2 + 22) (x2 + 42) {x2 + p2) = 0 , according as p is odd or even. As the determinant involved is one of a large family possessing the property of resolvability into quadratic factors, I have thought that it would be interesting to give a direct mode of effecting the resolution of it, and thus indicate how a generalisation may be arrived at. (2) Confining ourselves, merely for the purpose of convenience in writing, to the case of the 8th order, viz. x 7 - 1 x' '6 - 2 x 5 - 3 x 4 -4 x - 5 2 . x 1 7 x .106 Proceedings of Royal Society of Edinburgh. [: let us first perform the operations* colj - col3 + col5 - col7 , col2 - col4 + col6 - col8 , col3 - col5 + col7 , col4 - col6 + col8 , col5 - col7 , colg — colg ,, with the result X 7 -7 X 6 — X - 7 X 5 7 - X -7 X 4 X 7 - X -7 X 3 -7 X 7 - X -7 X 2 - X - 7 X 7 - X -7 X 7 - X -7 X 7 - X - 7 1 x • Next let us increase the 8th row by the 6th, the 7th by the 5th, and so on, and we obtain x 7 | - 7 x 6 . . . . x 5 . . . [ - 1 x 4 . . I . . , -2 x 3 . . .-3x2. - 4 x 1 -5 .;x I * These are equivalent to multiplication row-wiso by unity in the form 1901-2.] A Continuant Resolvable into Rational Factors. 107 which, clearly, is equal to x 5 -1 x 4 .-2x3 . - 3 x 2 - 4 x 1 . -5 x . But as the second factor here is exactly of the same form as the determinant with which we commenced, we are thus led at once to the final result x 7 — 7 x X 7 1 X 5 X 3 1 I X 1 -7 X 1-5 X -3 X 1 -1 X I (x2 + 72) • (x2 + 52) • (x2 + 32) • (x2 + l2). In the case of an odd-ordered determinant the penultimate result is x 6 | x 4 x 2 — 6 x 1 — 4 x - 1 x 1 .-2 x which finally gives ( x 2 + 62) ( x 2 + 42) ( x 2 + 22) x . (3) Now with this process as a guide, let us deal with the much more general determinant x bx “ft " X ^2 -ft x h - /?5 x bx “ft * h ~ ft x h - /?2 x b7 — ft X Multiplying row-wise and then column-wise as before we obtain a result which is seen to he resolvable into a continuant of the 2nd order and a continuant of the 6th, provided we can satisfy the equations 108 Proceedings of Royal Society of Edinburgh. [sess. Si~ Ss~ ^6 = ^ J 02 At b§-\- by = 0 j S3 — So ~ ^4 8" ^6 ~ ^ j 04 “ A — ^3 8" ^5 = ^ > /?5 - - 62 + &4 = 0 , 06 “ + ^3 = o ; that is to say, provided we have ^6 = Si S3 » 06 ~ ^1 ^3 > 04 = ^1 “ ^5 » b2 = Si~Si* ~co to II CH 1 O-Jj • Making these substitutions in the original determinant we thus know that it resolves into x b1 -Si x and Si x Si Si &3 — ?q x -ft A A &5“A - S3 X The conditions for the similar resolvability of the latter factor are of course Si ~ So ~ S3 ~ Si > ^1 — ^5 = ^3 ~ ^7 5 Si~ Sz~ S3 ~ S3 5 ^1 ” ^3 = ^3 " ^5 • These conditions, however, simply mean that the series b1, b3, b5, b7 and the series Si , S31 Sv Si are ea°h t° be i*1 equidifferent progression ; so that if the two common differences of the progressions be - d and 3 respectively, the result which we have reached takes the form x -s+ 33 x b I -S X ’ b x -d 33 x b — d -0 + 28 * 23 - 2c? it* -0 + 3 & - 2c? x -3d 3 £ b-3d x b - d -0 + 3 x x b - 2c? -0 + 23 aj -0 a* I a; b - 3d I -0 + 33 x I, 1901-2.] A Continuant Resolvable into Rational Factors. 109 where it will be seen (1) that the elements of the main diagonal of the continuant are each equal to x ; (2) that the elements in the odd-numbered places of the upper minor diagonal and those injthe even-numbered places of the lower minor diagonal are b, l-d, b-2d, b-3d, ... -d, -2d, -3d, ... respectively ; (3) that, when counting in the opposite direction, the elements in the odd-numbered places of the lower minor diagonal and those in the even-numbered places of the upper minor diagonal are 0, 0-8, 0-28, 0- 38, . . . -3, -23, -33, ... with their signs changed ; and (4) that the factors are (x2 + b0) (x2 + b^~d- 0^8) (x2 + b^2d- 0^28) (4) What the corresponding theorem is in the case where the continuant is of odd order will readily be seen from the example x 3b - (3 x c-v2d -b x 2b — (3 — d x c + d -2b x b — [3 -2d x c -3b x = {x2 + 3 b(0 + c + 2d)} . {x2 + 2 b(0 + c + d)}-{x2 + b{0 + c)} • x . (5) When in §3 0 = b — half the order-number of the continuant,, and 3 = d = 1 , we obtain a continuant having factors resembling those of Professor Elliott’s continuant, viz. x p - 1 X p-l - 1 X p-l - 2 x p - 2 -p + 2 x 2 -p + 1 x -P + = (x*+l2)(x2 + 2 2)...(x2+p2), 1 X 1 p X 110 Proceedings of Royal Society of Edinburgh. [sess. where each integer except the first in the upper minor diagonal appears twice. Another having a like interest is x p i % P~i - 1 X p - 1 - 1 J X p - 1 J -p+ 1 x 1 -p + J a? -p which is the case of § 4 where b = d= 1 , /? = c = J . (6) The theorem which is the basis of the two identities of § 3, 4 is the following : — If in the continuant of the nth order x \ ~ Pn-l X b2 - Pn-2 X = x(x2 + l2) (x2 + 22) . . . (x2 +p2), -@2 X bn-l -Si X the difference betioeen the element following x in any odd row and the element preceding x be constant , equal to hx say ; and the corre- sponding difference in the case of any even row be also constant , equal to /5: say ; then x2+¥i is a factor of the continuant , the co-factor being the similar con- tinuant of the (n — 2)ih order whose minor diagonals are got from those of the original by striking out bj , b2 from the one and Si 3 S2 from lhe other. (7) If the last row of the continuant he an exception in this matter of a constant difference, the continuant instead of resolving into two becomes transformed into another continuant. Thus, for the case of the 6th order, wdiere in the given con- 1901-2.] A Continuant Resolvable into Rational Factors. Ill tinuant the last horizontal difference is not /3 but J3 — bQ we have* x b1 b2~ ft X b2 b3 - b± x b3 b4-/3 x b4 h~bi x \ b6~ft x x h b2~P x h x \ \-ft x b6 b5-b1 x h-p h x The two general theorems, one of which is illustrated by this example, are : — 1. If in a continuant of the (2n)^ order the elements of the main diagonal be all alike , those of the upper minor diagonal be hi } b2 , b3 , ... , b2n_i and those of the lower minor diagonal be so related to the latter that the horizontal differences are tq and /31 alternately save the last which is not /31 but /31 — b2n , then the upper minor diagonal may be changed into h3 , h4 , h5 , , h2n_! , b2n , tq without altering the value of the continuant. 2. If in a continuant of the (2n + 1)*71 order the elements of the main diagonal be all alike , those of the upper minor diagonal be bj ? b2 , b3 , • . . , b2n and those of the lower minor diagonal be so related to the latter that the horizontal differences are tq and /31 alternately save the last which is not \ but tq — b2n+1 , then the upper minor diagonal may be changed into bg , b4 , bg , ... , b2n , b2n4_j , /?4 , without altering the value of the continuant. When in the one case b2n = 0 , and in the other b2n+1 = 0 , the theorems degenerate into that of the preceding paragraph. ^ (8) Another point of interest attaches to this transformation, however ; because it may happen that while the original continuant does not satisfy the conditions of § 3 requisite for resolvability into quadratic factors, the derived continuant may. * It should he noted that in the derived continuant the vertical differences are £ - b2 , , /3 , bx , j8. 112 Proceedings of Royal Society of Edinburgh. [sess. In illustration of this we may take a continuant which has points of resemblance with Professor Elliott’s, viz. x ip - 1 x ip - 1 - 2 x ip -2 - 2p + 3 x 2p + 3 -2p + 2 x 2p + 2 -2^5+1 in which the horizontal differences are all 4 p except in the last row where the difference is 4 p - (2p + 1) , and for which therefore we may substitute x ip - 2 - 1 x ip -?> - 2 x ip - 4 - 2p + 3 aj 2p + 1 -2p> + 2 x ip -2p + l x But this, if the last column be divided by 2 and the last row multiplied by 2, is a continuant with the constant horizontal difference ip - 2. It therefore follows that x2 + (ip - 2)2 is a factor of it ; and as the co-factor is a continuant of the same form as the original, the process may be continued with the final result {x2 + (ip - 2)2}{x2-\- (ip - 6)2} . . . {x2 + 62}{x2 + 22} . The analogous result for the case of an odd-ordered con- tinuant is x ip + 2 — 1 X ip + 1 - 2 x ip - 2 p + 2 x 2p + 2 - 2p + 1 x 2p + 3 -2 \p x — {x2 + (ip)2} [x2 + (ip - i)2} .... {x2 + i2}x. {Issued separately April 1, 1902.) 1901-2.] Plague Research Laboratory of Government of India. 113 The Plague Research Laboratory of the Government of India, Parel, Bombay. By W. B. Bannerman, M.D., B.Sc., Major, Indian Medical Service, Superintendent of the Laboratory. Communicated by Prof. C. Hunter Stewart. (Read February 17, 1902.) Before beginning the account of the present state of the Plague Research Laboratory, it may be interesting to glance rapidly at the history of Mr W. M. Haffkine, C.I.E., its originator and present Director- in-Chief. After several years of work in the Pasteur Institute, this scientist succeeded in elaborating a vaccine which effectually protected the lower animals from attacks of cholera artificially induced in them, and he was naturally anxious to try its efficacy in human beings. Mr Haffkine had originally intended to visit Saigon for this purpose, but ascertaining from Lord Dufferin, who was then British Ambas- sador in Paris, that he 'would have a better field for his researches in India, he, through his influence, obtained the necessary intro- ductions, and began work in Calcutta in 1893. It is therefore to this eminent statesman that India owes the presence of the man who has saved thousands of her children from cholera and plague. The results obtained by the inoculations against cholera were so encouraging that the tea-planters in Assam invited Mr Haffkine to visit the tea-gardens to operate on their coolies, and Government were induced to help financially. Three years after his arrival in India, plague broke out, and the Government of India sent Mr Haffkine to Bombay with instructions to investigate the cause of the outbreak and to devise if possible some method of dealing with this new and terrible disease, similar to that which had so impressed them in the case of cholera. He reached Bombay on the 7th of October 1896, about a fortnight after plague had been officially declared to exist there, and next day began work in a room in the Petit Laboratory of Grant Medical College. His laboratory consisted of one room and a corridor, and his staff of one native clerk and three peons or messengers. It. PROC. ROY. SOC. EDIN. — VOL. XXIV. 8 114 Proceedings of Royal Society of Edinburgh. [sess. was here that Mr Haffkine made the discovery of the stalactite growth assumed by the plague bacillus when grown in nutrient broth, which will ever be connected with his name as a reliable and easy means of diagnosing the organism. In December 1896 Mr Haffkine was successful in protecting rabbits against an inoculation of virulent plague microbes, by treating them previously with a subcutaneous injection of a culture in broth of these organisms sterilised by heat. The rabbits treated in this way became immune to plague. On the 10th of January 1897 Mr Haffkine caused himself to be inoculated with 10 c.c. of a similar preparation, thus proving in his own person the harmlessness of the fluid. Shortly after this, various medical men and prominent citizens of Bombay were publicly inoculated, to encourage others to submit to operation ; but it was not till the publication of the result of inoculating half the prisoners in the House of Correction at Byculla, in Bombay, that the measure became popular. Increased popularity involved more wholesale manufac- ture of the prophylactic, and this necessitated removal to ‘ The Cliff’ bungalow, Malabar Hill, which was placed at Mr Haffkine’s disposal by the municipality of Bombay. The staff had now been increased to two military assistant surgeons, two clerks, three peons and four hamals or house-cleaners. The Cliff was occupied from April to November 1897, when a second move had to be made to another bungalow in Nepean Sea Koad. At this time H.H. Sir Sultan Shah, Aga Khan, K.C.I.E., the head of the Khoja Mussul- man community, who had been early convinced of the efficacy of inoculation, fitted up at his own expense Khushru Lodge, one of his bungalows in Mazagon, for Mr Haffkine’s use. The staff under Mr Haffkine was again increased, and now consisted of one com- missioned officer of the Indian Medical Service, four medical men of those sent out by the Secretary of State for India for work there, four local medical men, three clerks and six servants. The laboratory remained there for more than a year, but in March 1899 the demands for vaccine from all parts of the world became so pressing that more accommodation was an urgent necessity. This was found in the Old Government House at Parel (fig. 1), and it is the laboratory as it has been developed there that I propose to describe to you to-night. The Old Government House is a huge rambling building, standing 1901-2.] Plague Research Laboratory of Government of India. 115 in extensive grounds in the suburb of Parel, which has passed through many changes of fortune. It was originally a Roman Catholic seminary in the time of the Portuguese, but was forfeited to Government on account of suspected intrigues between its possessors and the independent rajahs in the Western Ghats. About the year 1765 it became the official residence of the Governors of Bombay, and continued to fulfil this function till 1883, when it was finally abandoned after the death of the Governor’s wife from cholera thought to have been contracted there. It stood vacant for some fifteen years, till the advent of plague suggested its use as a convenient hospital, and within its walls hundreds of plague patients were treated in 1897-98. In July 1899 the move from Mazagon was begun, and by the 10th of August everything was in readiness for its formal opening as a laboratory by His Excellency Lord Sandhurst, Governor of Bombay, which took place in the presence of a large concourse of the leading European and Native inhabitants of Bombay. The staff at present numbers fifty-three, consisting of the following, viz. : — Dir ector-in- Chief (Mr W. M. Haffkine, C.I.E.), who is also scientific adviser to the Government of India. Superintendent , a commissioned medical officer of the Indian Medical Service, responsible for the general conduct of the laboratory. Three European Medical Men , two being commissioned officers of the I.M.S., and one engaged on account of special training in laboratory work, both chemical and bacteriological. Three Native Medical Men , twelve clerks, seven decanters, one engineer, and twenty-five laboratory attendants. During the busy months of the year this staff is altogether employed in the manufacture and despatch of the plague vaccine, hut when the demand declines other subjects of investigation are taken up, such as snake-bite, malarial and relapsing fevers, Malta fever, an undetermined fever occurring in the famine relief camps of Guzerat, the manufacture and curative action of serums for plague and snake-bite, and vaccines for typhoid and cholera. The manufacture of the plague prophylactic, which at present forms the main work of this large establishment, has gradually 116 Proceedings of Royal Society of Edinburgh. [sess. reached a stage of comparative perfection, so that we are now certain of sending out from the laboratory vaccine which is pure- and sterile, and capable, when used in the doses prescribed, of pro- ducing a uniform protective effect in those on whom it is used. Unfortunately this new laboratory was not in existence when the Indian Plague Commission visited India, and in consequence there will be found in their report, recently published, various criticisms of the processes then employed which are now no longer applicable,, as I hope to show presently. Fig. 1. — Old Government House, Pare! A description of the preparation of the prophylactic will serve to illustrate the size of the laboratory and the difficulties that had to be overcome by Mr Haffkine and his staff in suiting themselves to the conditions of Oriental life, and the religious and other pre- judices of the natives. The plague prophylactic is essentially a culture in broth of the plague bacillus, thus differing entirely from the curative serums which are produced by the action of the bacillus on a living animal. The preparation of the culture medium or bouillon is the first 1901-2.] Plague Research Laboratory of Government of India. 117 process, and as this differs from the methods emp’oyed in European laboratories, it is necessary to describe it somewhat in detail. The necessity for this departure from established custom arose from a consideration of the religious prejudices of the natives of India, •certain articles such as beef and pork being abhorrent to the majority of them. How the peptone of commerce is derived from •one or other of these substances, and it could not in conse- quence be used to enrich the bouillon in the ordinary way. The Government of India therefore deputed the late Colonel Warden, Fig. 2. — Media-preparation Room. I.M.S., who was then the Chemical Examiner in Calcutta, to try to discover a method of making peptone from goats’ flesh, and the process now to be described, though not fundamentally novel, is the result of his labours, modified as experience dictated. The operations connected with this process are carried on in a spacious stoned-paved room on the ground floor, fitted up with eight large autoclaves (fig. 2). Lean goats’ flesh finely minced is the basis of the bouillon. The meat, placed in glass jars, has 80 c.c. of hydro- •chloric acid of the B.P. standard added to it for each kilo, and the jars are then immersed up to the neck in water at 70° C. 118 Proceedings of Royal Society of Edinburgh. [sess. (158° F.), kept warm in a jacketed tub. The jars are left there for a week, by which time the meat has become dissolved so that the contents look like brown porridge. The albumen has become converted into acid albumen with a little albumose and peptone. The fluid is now transferred to large glazed earthenware jars, and water is added to make the bulk equal to three litres of fluid for each kilo of meat originally present. The large jars are now placed in the autoclaves, and kept there for three hours under a steam pressure of three atmospheres, or 44 T lbs. to the square inch, which gives a temperature of about 143° C. (290° F.). The fluid after this treatment is dark brown and somewhat viscid, containing quantities of undigested fibrous tissue, etc. It is undoubtedly rich in albumoses and peptones, but the amount of each has not yet been determined. This fluid is filtered and made up to the original bulk of three litres per kilo. Bone charcoal is now added in the proportion of 150 grammes per litre, and the whole well stirred for a quarter of an hour. This improves the colour of the bouillon and also its cultural properties. The charcoal is then roughly strained out by means of a cloth, and the fluid neutralised by the addition of caustic soda. It is also diluted by the addition of from two to two and a half volumes of water for each volume of peptone solution. In this dilution the amount of hydrochloric acid used, when neutralised as above, gives a bouillon containing 0*5 per cent, of chloride of sodium or common salt. If the bouillon is required for ordinary laboratory purposes, each volume of the peptone solution is diluted with from three to five volumes of water, and in this case more salt has to be added to get the 05 per cent, required for the nutriment of bacteria. This mixing and neutralising is done in a wooden tub holding about 60 litres, for it is quite a common thing for us to produce 50 litres or more of bouillon daily. The neutralising is in the first place purposely carried too far, for during the heating process the reaction goes back, and a perfectly neutral fluid is necessary for the proper growth of the bacillus pestis. The fluid is then heated for an hour under a pressure of 15 lbs. to the square inch (121° C_ or 250° F.) to throw down the phosphates, etc., refiltered and distri- buted in quantites of 2 litres into 4-litre flasks. These flasks are specially made for us in Paris, of pale green glass, flat-bottomed 1901-2.] Plague Research Laboratory of Government of India. 119 and oblately spheroidal in shape. Two litres of fluid in these flasks form a layer of barely two inches thick, thus affording a very large surface for aeration of the growing bacilli. This medium has been found excellent for the growth of all sorts of bacteria, and has proved much cheaper than the ordinary method of making meat extract, and then adding commercial peptone, for not only is the expense of buying peptone avoided — and we used to use as much as twenty-five shillings’ worth in a day — but from one kilo of meat we make as much as 9 or 10 litres of bouillon, as against 2 by the old method. This medium is known in the laboratory as ‘ Warden’s bouillon,’ but the process of gradual digestion in the water bath and treating with charcoal are modifications devised by Dr F. Maitland Gibson, one of the permanent staff. A similar bouillon was also made by the latter official having for its basis wheat flour, from which the starch has been removed by washing, but it is not at present widely used. It was made to meet the objections of some vegetarian sections of the native community, and though not now much in demand, has served the purpose of reconciling some of the leading native practitioners to the use of the prophylactic. Into each flask thus made ready for final sterilising are dropped four drops of cocoanut oil, whicli produces an abundance of droplets on the surface, without forming a continuous layer of oil on the bouillon. The necks of the flasks are plugged with sterilised cotton wool in the ordinary way, and the whole batch of fifty or sixty sent out to the corrugated iron shed containing the big steriliser (fig. 3). This steriliser is merely a jacketed clothes disinfector made strong enough to stand a pressure of 15 lbs. to the square inch, and with a trolly fitted with shelves on which sixty flasks can be safely placed. The flasks are kept under this pressure for one and a half hours, one hour having in practice been found insufficient to sterilise their contents when working on this wholesale scale. The flasks containing the sterilised bouillon with the layer of oil on the surface are now taken to the incubation room to be sown with plague bacilli. It is therefore necessary to explain where the germs come from and how a pure culture of them is produced. It is unfortunately easy at any time of the year to find cases of 120 Proceedings of Royal Society of Edinburgh. [sess. virulent plague in Bombay or its vicinity, and there has not, there- fore, arisen any necessity to have recourse to cultivation in animals to obtain plague microbes of a suitable potency. A little of the juice of a plague bubo is asperated out by means of a sterilised glass tube drawn out to capillary size at one end, and known as a Pasteur’s pipette. When this juice is withdrawn with suitable precautions, it is found to yield in almost every case a pure culture of plague bacilli when sown on agar jelly. If the resulting culture appears to be pure, then portions of it are removed Fig. 3. — The Steam Steriliser. to small trial flasks of bouillon prepared exactly as the large ones are. These are placed in the dark and kept at a suitable tempera- ture, and in forty-eight hours thereafter, if the growth sown in them has been pure, an abundant crop of stalactites makes its appearance in the hitherto perfectly clear bouillon, forming, when viewed by transmitted light, one of the most curious and beautiful sights a bacteriologist could wish to see. The stalactites have been likened to silk threads hanging down into the bouillon from the surface. Having proved the absolute purity of the growth in this way, the operator may proceed with confidence to transfer the contents of 1901-2.] Plague Research Laboratory of Government of India. 121 these trial flasks to the £ ballons Pasteur ’ which are used for sowing the large fermentation flasks. This operation is performed in what was originally the chancel of the Portuguese church, and the flasks when sown are at once removed to the body of the church, which, for the past hundred years, has served as the Governor’s banqueting-hall. This splendid hall, measuring some eighty feet in length, is now furnished with six rows of stout teak- wood tables running lengthways down the room. Each row of tables serves to accommodate three rows of Fig. 4. — The Incubation Room. the large 4-litre flasks. The view which I now show you was taken a few days before the visit of H.E. Lord Curzon, the Viceroy of India, on the 8th of November 1899. At this time there were in the room 1238 flasks, containing 2058 litres of bouillon sown with plague bacilli, the weight of the fluid amounting to over two tons. If we suppose that all these flasks ultimately reached the decanting room, their contents would furnish 411,600 adult doses of vaccine. The newly-sown flasks remain in this room for six weeks at a temperature of 26’6° C. (80° F.). For a few months in the cold weather the hall has to be heated by gas stoves, but as a rule this is not necessary. During this six-weeks period the flasks are 122 Proceedings of Royal Society of Edinburgh. [sess. watched carefully, and any showing signs of contamination are at once removed and destroyed. Every second day the contents of each flask are shaken to break up the stalactites and prevent the formation of growths in masses, which might afterwards block the needle of the injection syringe. Once a week every flask and table is wiped down with corrosive sublimate solution, and the floor scrubbed with the same antiseptic, as a precaution against the countless moulds which flourish in the steamy Bombay climate. At the end of six weeks the growth of stalactites- has completely ceased, and though bacilli remain alive in the flasks for months sometimes, yet no perceptible thickening of the fluid takes place after the lapse of this period. The flasks containing the most turbid fluid are chosen, therefore,, and sent into an adjoining room, where they are tested for purity. The number of flasks thus sent out daily varies from twenty to- thirty, according as the demands for material or other circum- stances necessitate. The testing is done by withdrawing a drop of fluid from each flask, with the usual precautions, and smearing this on the surface of a couple of dry agar tubes. The requisite dryness of the agar jelly is got by keeping the tubes in a closed jar containing quicklime for a couple of days or so, and the agar is ready for use when it has shrunk away a little from the glass at the bottom of the tube. If the agar is thus dried, the fluid con- taining the bacilli is at once absorbed by it, and two days after, a characteristic growth of plague microbes is found covering the entire surface. Four days after these tubes are sown they are examined by the testing officer, along with the Superintendent, and the result noted in the register kept for the purpose. In case either of the examiners should have any doubt as to the appearance of any of the tubes, a portion of the contents is at once placed in a small test flask and the formation of stalactites watched for. Only when an unanimous verdict is given by both examiners is a flask passed as fit for use. The flasks, immediately after testing, are passed on into the sterilising room, where they are taken in hand by one of the native medical men. They are then firmly fixed on heavily-weighted retort stands, and immersed in water kept at 60° C. (140° F.) in a jacketed tub. The flasks are arranged in a ring round the circumference of 1901-2.] Plague Research Laboratory of Government of India. 123 the tub, and in the centre is placed a similar flask containing as much cold water as the others do prophylactic. By means of a hole in the lid of the tub a thermometer is lowered into this centrally-placed water flask, and it is assumed that by the time its temperature has reached the desired height, the temperature of the vaccine flasks will be similar. Fifteen minutes later the flasks are removed and at once plunged into a trough containing cold water circulating through it, which rapidly cools them down to the atmospheric temperature, prolonged heating being one of the dangers to he avoided. As soon as the contents of the flasks have been cooled down, carbolic acid is (fig. 6) added in the proportion of 2^0' the hulk of the fluid, or in other words, 0-5 per cent. The vaccine is then ready to he decanted into bottles for distribution. This is done in a long room contrived out of the east verandah of the banqueting room, by converting the arched openings along the side into large windows (fig. 5). This department of the work is presided over by another native medical man, who supervises the seven decanters and is responsible for all that is done in the room. The flasks to be decanted are kept in a sloping position by means of rings attached to large heavily- weighted retort stands, and are well elevated above the tables to admit of the adaptation of a glass syphon. This syphon has a portion of rubber tubing inserted in its length for the application of a clip to control the flow, and as a means of starting the syphon-action. The syphon apparatus is sterilised in a large autoclave under three atmospheres of pressure, the open ends of the tubes having been closed with cotton plugs in the usual way. The only part, therefore, that has to be sterilised by the decanter before he introduces the end into the flask is the outside of the tube, and this he does in a bunsen burner immediately before use. The syphon is started by succes- sively pinching the rubber tubing with a sort of milking action, one finger and thumb grasping it alternately below the other. In this way suction by the mouth is avoided, and the lower end of the tube is in no danger of contamination. The syphon action having been started, the lower end of the tube is sterilised carefully in a bunsen flame, and the fluid filled into the bottles held below. These bottles have been specially made for the purpose, and have the neck flush with one corner, so that no difficulty may arise in ex- 124 Proceedings of Royal Society of Edinburgh. [sess. tracting the last drops of the contents when the syringe is filled from it. They each hold from 25 to 30 c.c., i.e., five or six doses, and are closed with rubber stoppers. The bottles before being filled are plugged with cotton wool and sterilised by dry heat at a temperature of 150° C. (300° F.) kept up for two hours. This is done in an oven holding 1400 bottles at one time, which number usually suffices for a day’s work, though in times of pressure the stove has to be filled twice a day. The rubber stoppers are sterilised by soaking in a 5 per cent, solution of Fig. 5. — The Decanting Room. formaline for three days, though by experiment it was found that three hours was sufficient for this purpose. Each decanter has supplied to him a tray full of bottles and a dish of stoppers immersed in formaline solution, and he manipu- lates these latter by means of spring forceps. The wool plug is withdrawn, the neck of the bottle sterilised in the flame of the bunsen burner, the end of the syphon inserted into its mouth, the clip loosened, the bottle filled, and the stopper inserted in a wonderfully short time, and without the slightest danger of con- tamination from the operator’s fingers. Before the filling of 1901-2.] Plague Research Laboratory of Government of India. 125 each bottle the flask from which it is to be filled is shaken round so as to distribute the bacterial sediment equally through the fluid, and thus ensure an equal distribution of this valuable material among all the bottles of the batch. The decanters are trained at first with water, then with flasks of sterile broth, and not till they are able to decant and bottle aseptically a flask of such highly putrescible material, are they allowed to touch the carbolised Fig. 6. — Assistant Surgeons, carbolising, testing, and decanting the Prophylactic. vaccine, which, of course, is much less liable to contamination than the uncarbolised bouillon. To test the purity of the vaccine at this stage, specimens from the first and the last few drops taken from the flasks are dropped into test tubes containing 10 c.c. of sterile broth. Four days after, these tubes are ex- amined to see whether any appearance of growth is visible or not, thus serving at once as a test of the sterility of the vaccine and as a check on the skill of the decanter. In this department — 126 Proceedings of Royal Society of Edinburgh. [sess. as indeed in all the others — a careful record is kept by which the identity of the operator is known and any shortcoming on his part brought home to him. After decanting, the empty flasks and syphons are sent to the cleaning department, while the batch of bottles is labelled with date and number of flask from which they are taken. These bottles are kept separate for ten days, and then from each batch two samples are taken at random for examination. These are opened and a drop from each is placed (fig. 6) in a couple of test tubes containing 10 c.c. of sterile bouillon. If after four days of incubation no sign of growth is apparent, the batch to which it refers is passed into the packing department. Two standard bottles from each brew are sent up to the Superintendent to have the dose fixed. This is at present done by estimating the opacity of a layer of constant thickness, and the results are checked occasionally by actual experiment on human beings. This may not appear a scientific procedure, but it has proved in practice surprisingly accurate, and no other workable method has as yet been devised, though many have been tried. Inoculation of animals has hitherto proved unreliable for this purpose, and even if effective would be unmanageable on the scale on which we work. For instance, it is quite usual for the laboratory to turn out twenty-five different brews in a day for weeks on end, each of which would, if tested on animals, require the sacrifice of three lives, or a total of seventy-five animals daily. This would necessitate the employment of two or perhaps three skilled bac- teriologists devoting their whole time to this and nothing else, to say nothing of the number of assistants and animal keepers, and the multitude of animals that would be required. The dose having been fixed, the labels can now be filled up for each bottle. Another long label with directions for opening and handling is pasted across the stopper to preserve the contents from the investigations of the curious. The bottles are then packed in pairs in wrappings of corrugated paper and stowed in boxes holding 1000 doses, for transmission to any part of the world. To give an idea of the care with which these various processes are carried out, I would refer to the number of tests the material is subjected to at every stage of its manufacture. The elaborate precautions taken to ensure the purity of the microbial growth to begin with, the daily supervision of the fermenting flasks, the trial for purity before the sterilisation 1901-2.] Plague Research Laboratory of Government of India. 1 27 •of the vaccine, the double test in the decanting room, and the final trial which takes place ten days after bottling. Each department keeps an accurate record of all operations performed, with the result of the tests, and the whole is copied into a large register forming a history of each brew from the actual making of the broth to the despatch of the vaccine to the operator. By referring to this register we have been able frequently to convince an inoculator that the appearance of abscesses in his patients has been due to his own shortcoming and not to ours. That the processes are suitable to the conditions of the country and climate is manifest from the results obtained, as the following extract from the half-yearly report to Government for the first part of 1900 will testify. The report is written by Mr Hafikine, as Director-in-Chief, and is the last that I have access to. Referring to the question of the sterility of the prophylactic, he says: “Of the 2139 brews manufactured during the first half of the year, 152 were rejected upon the results of the examination made before they had reached the bottling room, and one was rejected subsequently on account of its weakness. The arrangement followed in the laboratory was that the material, bottled and closed with india-rubber stoppers, and made ready for despatch, was kept in the laboratory at least a fortnight after its preparation ; and ten days after the bottling, samples of each brew were re-tested for sterility. “In this way it was detected that out of 1987 brews passed into the bottling room as sterile, 4, or about 1 in 500, got contaminated subsequently, and were rejected; the 1983 others had remained sterile, and were declared fit for distribution to operators. “These details warrant the presumption that no contaminated bottles are issued from the laboratory. “ The above results were obtained by an effective supervision over each part of the preparation, and by the duties being dis- tributed amongst an adequate staff of officers.” The above remarks refer only to the operations as carried on in the new laboratory, which was not in existence when the Indian Plague Commission visited Bombay. This fact should he kept in mind by anyone studying the elaborate report of that Commission. Having described the process of manufacture of the plague vaccine, I should like to lay before this Society a short account of what can be accomplished by its means. 128 Proceedings of Royal Society of Edinburgh. [sess. The vaccine was first used, as we have seen, in January 1897,. and from that time up to date close on 2,500,000 doses have been issued from the laboratory. It would, of course, be impossible,, within reasonable compass, to give even a resume of the results achieved in the numerous places from which reports have been received, but those wishing to study the matter for themselves will find full information in a pamphlet published by the Government of Bombay, and obtainable from any of the agents for Indian Govern- ment publications in London.* In all cases without exception there has resulted a striking reduction in the case-incidence, and a very remarkable lessening of the case-mortality ; in other words, fewer cases of plague occur among the inoculated, and those persons who are attacked in spite of inoculation suffer much less and recover in much greater proportion than the uninoculated. From the mass of reports received, I have selected two to lay before you to-night, because they are peculiar in being, I suppose, the most complete and thorough of any that have ever been compiled from among human communities, relative to the action of an experimental remedy. The conditions existing in these two places enabled the operators to reach that degree of accuracy which is usually attained in laboratory experiments only. The first experiment was undertaken at the request of the authorities of the Baroda State, and with their assistance. It happened that Mr Haffkine and I visited the town of Baroda to carry on inoculations there during a severe epidemic of plague, in February 1898, and while there were told by one of the native officials of a small village six miles off where plague was raging, and where it would be possible to have the people under close ob- servation. Mr Haffkine at once agreed to go down there, for he saw the value a crucial test would have at a time when people were en- quiring what remedies were available for use in epidemics of plague. The village of Undhera, six miles from Baroda city, inhabited by agriculturists, was attacked by plague in the end of December 1897. According to a census taken by the Baroda authorities, on the 5th of January 1898, there were 1031 souls in the village. * Statistics of Inoculations with Haffkine’s Anti-plague Yaccine, 1897-1900. By W. B. Bannerman, M.D., B.Sc., Major, I.M.S., Superintendent, Plague Research Laboratory, Bombay. Printed at the Government Central Press, Bombay, 1900. 1901-2.] Plague Research Laboratory of Government of India. 129 Between that date and 12th February, when the inoculations were performed, seventy-six persons died of plague, ten left the village, and five were born, thus leaving 950 persons alive on the latter date. This mortality is equivalent to a death-rate of 766 per mille per annum , and will serve to show how virulent the epidemic was. On the 12th of February Mr Haffkine and I visited the village, accompanied by the chief plague officials of Baroda State and some half-dozen local medical men who had just been instructed in the art of inoculation. The witnesses of the experiment were therefore fairly numerous and representative. The village was divided into two halves, Mr Haffkine operating in one section and I myself in the other. Our mode of operation was as follows : — We went from door to door, calling out the people by name by reference to the census paper. As each household was collected in the street, half of the members were inoculated, and half left untreated to serve as controls. An endeavour was made to divide equally the men, women and children in each household, and to have an equal number of sickly in the two groups. That the division effected was a fair one will be manifest from the following table, which shows the sex and age distribution of the inhabitants of the twenty-eight houses in which plague cases occurred after the inocu- lation. A subsequent examination also showed that there were, at the time of operation, more sickly individuals among those inocu- lated than in the uninoculated group, so that the balance was in favour of the latter in this respect. Ages. Inoculated. Not Inoculated. Five years and under, | Between 6 and 59 f years inclusive. \ Sixty years and over, | Males, . . 4 \ q Females, . 9 J Males, . - 34 ) _4 Females, . 20 \ Males, . . 3 \ . Females, . 1 J Total, . 71 Males, . . 5 ) 1n Females, . 5 i Males, . . 18 ) Females, . 33 \ 0 Males, . . 1 j q Females, . 2 / Total, . 64 No difficulty was experienced in inducing the people to be inoculated ; in fact, one had to refuse to go beyond the half in each family in spite of the protests of its members. When we summed up at the end of the day, it was found that we had inoculated 513 persons, or 76 more than the half. The village was then left in PROC. ROY. SOC. EDIN. — YOL. XXIY. 9 130 Proceedings of Royal Society of Edinburgh. [sess. charge of a native medical man who had a small hospital within a hundred yards of the place. Six weeks after (4th April 1898), an investigation was made by the late Surgeon-General R. Harvey, M.D., C.B., Director-General of the Indian Medical Service, Mr Haffkine, Captain Dyson, I.M.S., and myself, assisted by the local authorities of Baroda. Each house in which a plague case had occurred since the 12th of February was visited, and the occur- rences among the members ascertained by personal enquiry from the survivors, by reference to the hospital register, and from the census papers, in which had been entered the doses of prophylactic administered. I myself carried the census lists, and from it called out the inhabitants of the plague-infected houses by name, and I shall never forget the experiences of that day. One incident I particularly remember — the finding in two huts in succession that all the uninoculated had died and that only the inoculated members of the households came out to answer to their names. The attitude of the people was also very striking, for when asked about the results of the experiment, they said that “about fifty uninoculated had been attacked and they are all dead, while only a few inoculated have been ill, and they have all recovered.” This turned out to be an example of oriental imagination, as the following will prove, hut it serves to show the impression the operations had left on the minds of the villagers. That the neighbours were also favourably impressed was shown by the arrival of a deputation, requesting us to come next day to inoculate them, which we accordingly did, operating on 700 in three hours. The following table is the result of this investigation. Plague continued in the village for forty-two days after the inoculations were performed, and affected twenty-eight families. Among the Inoculated. (a) There were no deaths from causes other than plague. ( b ) There were no deaths during the first three days after inoculation, the first fatality being recorded on the 21st, eight clear days after. (c) From the 15th February till the end of the epidemic there were eight attacks of plague, of which three had a fatal termination. Among the Uninoculated. (a) A child aged one year died of bronchitis on 21st February 1898. (b) Three died of plague during the first three days following the inoculation, and are therefore omitted from the calculations as having been attacked before the inoculations were carried out. (c) From the 15th of February till the end of the epidemic, twenty-seven more attacks of plague occurred, of which twenty-six died. 1901-2.] Plague Research Laboratory of Government of India. 131 One of the three fatal cases among the inoculated had no interval between the inoculation fever and the manifestation of plague; the two others had a-pyretic intervals of six and eight days respectively. The events in each of the twenty-eight families affected were noted on ‘ investigation sheets/ one for each household, in which the names of every member present at the time of inoculation were entered as either ‘inoculated ’ or ‘ not inoculated.’ The occurrences in each household being entered on each sheet below the names, it was then easy to compile the following table : — Ward No. House No. Inoculated. Not Inoculated. Number Inoculated in the Family. Number of Attacks. Number of Deaths. Number not Inocu- lated in the Family. Number of Attacks. Number of Deaths. 1 8 4 1 1 1 1 63 3 2 1 1 1 67 3 2 1 1 2 24 1 1 1 1 3 1 2 2 1 1 3 15 2 1 3 3 20 3 1 4 1 1 3 29 3 2 1 1 3 39 4 i 1 3 1 1 3 42 1 2 1 3 48 5 3 1 1 3 49 1 5 1 1 4 7 1 i 1 1 1 4 8 2 l 1 4 10 4 l 1 4 12 2 2 1 1 4 1 13 2 1 1 1 4 18 3 2 2 2 4 26 1 2 2 2 4 30 1 3 1 1 4 31 4 l 1 2 4 34 2 3 1 "i 4 35 1 l 1 1 1 l 4 53 2 2 1 i 4 80 4 4 1 i 4 84 5 5 2 2 4 89 2 1 1 1 4 90 3 3 1 1 Total, I 71 8 3 64 27 26 If one contrasts the fourth and fifth columns, which show the numbers of attacks and deaths among the inoculated, with the 132 Proceedings of Royal Society of Edinburgh. [sess. seventh and eighth, which set forth the same for the uninoculated, one gets a good idea of what would happen in a plague-stricken place if all the inhabitants were to he inoculated. For if the uninoculated half of the villagers had also been operated on — and there would not have been the slightest difficulty in doing this — the mortality among them would have been reduced 89*6 per cent, from what it actually was, or, in other words, they would have had ten times fewer deaths. For if the inoculated had suffered to the same extent as their uninoculated relatives, they should have had 29 deaths from plague, instead of 3 only. The proportional number — 2 9 ‘—is reduced by 26, which is equal to a diminution of 89-6 per cent, of mortality attributable to inoculation. As throwing light on the time necessary for the prophylactic to act, the subjoined figures, showing the number of days which elapsed between the date of inoculation and the occurrence of a death from plague in those families is instructive. The small figures show the number of deaths on any one day. In the inoculated, deaths took place — ------- - 91- -121-141- - - - - ___ and in the uninoculated, deaths took place — _ _ 32 41 53 _ 72 - 103 113 121 - - 151 161 - - 191 201 211 - - 241 ------- 321 - _ - - - - - - - 421 days after date of inoculation. Eight days, therefore, elapsed after operation before any deaths occurred among the inoculated, while during this time 11 deaths from plague were registered among the unprotected. It would seem, then, that the inoculation acted at once, and, as will appear later, this is borne out by many other observations. The following table shows the occurrences in the twenty-eight affected households, arranged according to sexes : — Inoculated. Not Inoculated. Numbers. Oases. Deaths. Numbers. Cases. Deaths. Males, 41' 4 3 24 7 7 Females, 30 4 40 20 19 Total, . n 8 3 64 27 26 1901-2.] Plague Research Laboratory of Government of India. 133 If, now, we take the population of these same twenty-eight houses, and arrange them in three groups according to age, we find that the following was the incidence of plague in each : — Ages. Population. Cases. Deaths. Five years and under i inoculated, . ] not inoculated, 13 10 1 3 1 3 Between 6 and 59) f inoculated, . 54 5 2 years inclusive | [ not inoculated, 51 22 21 Sixty years and over -| f inoculated, . [ not inoculated, 4 3 2 2 2 The report of this investigation, submitted to the Government of India by Surgeon-General Harvey, naturally produced a very great impression on the authorities, and induced them to encourage the introduction of this measure into all plague-infected places, and no one has been able to throw any sort of doubt on the accuracy of the figures. The Indian Plague Commission remark * that they have scrutinised the investigation sheets, and find that the “ pro- portions of the various ages and sexes were evenly distributed among the two classes. The correctness of Mr Haffkine’s figures is further confirmed, after making allowance for the deaths excluded by him from his calculations, by figures separately given to us by Major Bannerman and Mr H. D. Mehta. The experi- ment at Undhera appears to us to he the most important experiment regarding the effect of anti-plague inoculations that has as yet been carried out. It derives its importance firstly from the fact that the inoculated and uninoculated persons under observation were fairly numerous, and that their numbers were determined with a considerable degree of accuracy ; secondly, because the com- munities of the inoculated and the uninoculated were comparable, * Report of the Indian Plague Commission, vol. v. p. 214. Note. — This quotation was taken from the “proof copy” of chapter iv. sent out to India, and published there by the Government for general in- formation. Since this paper was sent to press it has been compared with the complete report of the Indian Plague Commission recently published, and it is found that though the meaning remains the same, yet the actual wording has been modified to give expression to Dr Ruffer’s doubts regarding the accuracy of the statistics aroused by the high death-rate among the uninoculated section of the Undhera community. The other members of the Commission agree in considering the statistics substantially accurate. 134 Proceedings of Royal Society of Edinburgh. [sess. and lived under similar conditions; thirdly, because the com- munities were equally exposed for a period of six weeks to an out- break of plague which carried off 6*2 per cent, of the uninoculated and, lastly, because the results appear to have been accurately verified. From this experiment we learn two lessons. Firstly, we learn that the number of plague attacks among the uninoculated is much larger than that among the inoculated ; in this particular case, as the table given above will show, plague attacks were proportionately four times more numerous among the uninoculated than among the inoculated. The second lesson we learn is that the percentage of deaths among the attacked is very much higher among the uninoculated than among the inoculated ; in this particular case, as the table again shows, the percentage of deaths among the uninoculated was proportionately ten times as great as among the inoculated.” The only sentence in the above remarks which seems to indicate that the members of the Commission were not quite satisfied is that in which they say that the numbers in the two groups of * inoculated ’ and ‘ uninoculated ’ were deter- mined with “ a considerable degree of accuracy.” This remark detracts considerably from the value of the approval thus bestowed,, and it will be well to examine whether the errors are likely to be serious. The two objections put forward are — (a) that the census was- made a month before the inoculations were performed, and ( b ) that the number of uninoculated “ appears to have been ascertained not by a direct enumeration hut by computation.” Now with regard to the first objection, it is certain that no one would come to the village while plague was raging in it, therefore the population would not be increased in the interval. We know that ten persons left the village during this time, and it is unlikely that more did so, for we used the census papers at the time of inoculation, as a means of getting at all the people, and if any had been absent it would have become apparent to us as we proceeded from house to house. Now beyond these ten who had left and a few men absent for the day at market, none remained on the list un- accounted for. I consider, therefore, that the census was accurate in a high degree and not merely in a considerable degree. But even supposing a large number of men had departed from the 1901-2.] Plague Research Laboratory of Government of India. 135 village during the month previous to inoculation, this would not in any way vitiate the results, for, as already described, it was the population actually present on the 12th of February that was divided into two sections, equal in number as far as possible. As regards the second objection, that the number of uninoculated was got by subtracting the number inoculated from the figure repre- senting the total population, I cannot, at this distance of time, give a decided opinion, but I am certain of this, that after the in- oculations were done, we checked the numbers inoculated and uninoculated, by counting those marked as operated on, in the census lists. Now, as I have already shown, these lists had been used to get at all the people, and no mistakes had been detected in them as we went round, which would inevitably have been the case had such existed. As the people were correct according to the lists — and granting our arithmetic also correct — it would seem an accurate enough mode of arriving at the number of the uninoculated to subtract the total of the inoculated from the ascertained population, even had we omitted to count the actual names, which I do not admit. In conclusion, it may be pointed out that these criticisms cannot apply to the population of the twenty-eight plague-infected houses, for the inhabitants of these were all seen by us both on the days of inoculation and of investi- gation. I consider, therefore, that the figures are accurate to a high degree, and deductions founded on them worthy of serious thought. The second example which I propose to give you is of more recent date, the outbreak having taken place in February 1900, in a ‘chawl’ (or barrack-like house) in the Colaba Ward of Bombay city. The chawl was a single-storey building containing ten rooms, only eight of which concern us here, arranged in two rows back-to- back, separated from one another by 7-foot partitions, but having the roof space common to all. All these rooms communicated with one another by doors, with the exception of two at one end, which were self-contained, but did not, however, escape infection on that account. Three rooms did not furnish any cases, though some uninoculated persons were living in them. The inhabitants all belonged to a low caste of Hindus called Mangs, and would therefore mix freely with one another. The chawl was situated 136 Proceedings of Royal Society of Edinburgh. [sess. in a square close to the sea, and near it was another building inhabited by a set of Gaolis or milkmen, and a cowshed. When plague broke out among the Mangs, the Gaolis noticed that rats were dying in their quarters, and therefore wisely removed to the cowshed close by. They had no cases among them, though, with the exception of one family, they were not inoculated. Thus the isolated Mang community, numbering sixty-one persons, afforded a good opportunity for an exhaustive enquiry ; and this was accord- ingly carried out by Dr E. L. Hunt, one of the medical men sent out to Bombay by the Secretary of State for India for temporary plague duty, and acting at the time as Inspector of Inoculation, assisted by Dr Arjani, Section Medical Officer of A. Ward, acting under the direction of Mr Haffkine himself. Of the sixty-one inhabitants, fifty-seven were either seen personally by one or other of the above-named investigators, or, if they had died meanwhile, the papers relating to them were secured from the adjacent plague hospital where they had been treated. The remaining four were not seen personally, as they had left the locality, three having gone to Poona, and one, a professional beggar, having dis- appeared in the city of Bombay. Only one of these four had been inoculated, and none of them were attacked. With the exception of these four, about whom the statements of neighbours had to be accepted, it was possible to furnish documentary evidence for all, showing when and by whom the inoculations had been performed, the date, duration and termination of the illness, with all the hospital notes and charts, and the condition of the survivors in July 1900, five months after the epidemic ceased. All this evidence was submitted to Government and published by them in full. The account nowr given is a condensation of this report. The first case of plague developed symptoms on the 1st of February 1900, on the 2nd of the same month seven were attacked, on the 3rd, six, and then one case occurred on each of the following dates, viz., 4th, 5th, 7th, 9th, 10th, and 12th February. Altogether twenty attacks took place, with twelve deaths. The inoculations had been performed on various dates from 11th December 1899 to 29th January 1900. The single case among the inoculated was in the person of a female, inoculated on 18th December, and attacked on 9th February : she recovered. 1901-2.] Plague Research Laboratory of Government of India. 137 The following table shows the distribution of the inoculated and uninoculated, room by room, and the occurrences in each group : — N umber of Room Number of Inmates and whether they No. were Inoculated. Attacks. Deaths. 1 ! 1 1 not inoculated, ..... 2 inoculated, ...... o j 1 8 not inoculated, ..... 1 "3 2 i 3 inoculated, ...... q J r 6 not inoculated, 4 "k 3 1 i 4 inoculated, ...... 1 A 1 r 1 not inoculated, ..... 1 1 4 1 L 2 inoculated, ...... r 1 4 not inoculated, . . . . "4 "4 0 1 inoculated, ...... fi J f If not inoculated, ..... 3 1 6 1 l 5 inoculated, ...... 7 J F 3 not inoculated, ..... i 7 1 L 3 inoculated, . . o J f 3 not inoculated , ..... 8 1 9 ] L | r 2 inoculated, ...... 3 not inoculated, ..... 1 inoculated, ...... 4 not inoculated, ..... 1 inoculated, ...... 2 1 1 10 j The inhabitants of the rooms numbered 1 and 10, which adjoined one another at one end, but did not communicate directly with any infected room, may be excluded from the above, as no cases of disease occurred among them. No cases occurred in room 8 either, but as it communicated with room number 3, in which five cases occurred, its occupants are included in the follow- ing calculations. In the eight rooms taken into account, there were living fifty-three persons, of whom thirty-two had not been inocu- lated, leaving twenty-one who had been thus operated on. Each room contained a mixed population of inoculated and uninoculated persons, and we find that — The 32 not inoculated had. 19 cases, with 12 deaths. The 21 inoculated had 1 case, with no deaths. If the inoculated had suffered to the same extent as those not inoculated, they should have had twelve cases with eight deaths, but they only had one case, which recovered. This is equal 138 Proceedings of Royal Society of Edinburgh. [sess. to a reduction of 91*7 per cent, in the attacks in favour of the inoculated. The fact that twenty-one of the inhabitants were away at work all day, and only returned to the chawl to sleep at night, makes no difference between the two groups, for on enquiry it was found that no cases of plague had occurred in the various places where these people worked. The infection was undoubtedly localised in the chawl, and no circumstance existed that in any way favoured the one group over the other. The following statement brings this out clearly : — Of those going out to work during the day, 16 not inoculated had 10 cases. Of those staying at home all day, 16 not inoculated had 9 cases. Of those going out to work during the day, 5 inoculated had no cases. Of those staying at home all day, 16 inoculated had 1 case. The fact that the uninoculated persons were equally divided between those who stayed at home and those who worked outside, and that the two groups returned almost the same percentage of cases, indicates that the source of infection was common to both, and must therefore have been in the dwelling-house itself. The age-distribution of the 53 inhabitants of the eight infected rooms was as follows : — TT , A e ( 1 1 not inoculated had 7 cases with 2 deaths. UptolOyearsofage | 4inocukted „ 0 „ 0 „ From 1 1 to 50 years j 20 not inoculated had 12 cases with 10 deaths, of age (17 inoculated 1 „ 0 ,, AinTrn f (1 not inoculated had 0 case with 0 death.. Above 50 years of age j Qinoculated _ Q ^ 0 >f From the above it will be seen that the fatalities occurred mainly among those between the ages of eleven and fifty years, and that,, though the children show a number of attacks, yet they had a smaller death-rate. The sex-distribution given below shows that the men were more susceptible than the women, but had greater power of recovery. All the women who were attacked died, except the one who had been inoculated. 1901-2.] Plague Research Laboratory of Government of India. 130 Males Females 21 not inoculated had 14 cases with 7 deaths. 6 inoculated ,, 0 ,, 0 ,, 1 1 not inoculated had 5 cases with 5 deaths. 15 inoculated ,, 1 „ 0 ,, The above particulars, carefully got together by painstaking enquiry, prove (a) That the infection was localised in the chawl. ( b ) That all the inhabitants were living under precisely the same conditions, with the single exception that some were inocu- lated and others not. (c) That the lessened incidence of plague in the inoculated was due solely to the effect of the material injected. These two demonstrations, which are accurate to a high degree,, prove unquestionably the two propositions formulated above (p. 128), and which may be again restated with the added weight of the words of the Indian Plague Commission.* “ 1. Inoculation sensibly diminishes the incidence of plague attacks on the inoculated population, but the protection which is afforded against attacks is not absolute.” “ 2. Inoculation diminishes the death-rate among the inoculated population. This is due not only to the fact that the rate of attack is diminished, but also to the fact that the fatality of attacks is diminished.” Two further questions now arise, viz. : — (a) Is the plague vaccine harmful when applied to those in the incubation stage of the disease ? and (b) When does protection begin to be effective, and how long: does it last ? As regards the first of these queries, we find Calmette asserting, before the International Congress of Hygiene and Demography at Paris, in September 1900, that “a person in the period of in- cubation for a slight attack of plague would find the disease considerably aggravated if he submitted during this period to a preventive inoculation of Haffkine’s vaccine. The case would almost certainly end fatally.” f Some months later the same- * Report of the Indian Plague Commission, vol. v. p. 262. t Abstract in Brit. Med. Journal of 27th October 1900. 140 Proceedings of Royal Society of Edinburgh. [sess. scientist emphasises this expression of opinion in the Harben lecture * delivered by him in London, and condemns the practice of inoculating with Haffkine’s fluid those who have been in contact with a plague case. These opinions he founds on laboratory ex- periments only, never having had an opportunity to use the plague vaccine during an epidemic among human beings. It is well known, however, that the immediate effect produced by the action of a microbial virus varies with each species of animal operated on, and that quite as various degrees of immunity are produced in them by this means, so that it is impossible without trial to pre- dict what the exact action on any fresh species may be. We must take experiments on animals as an indication merely of what may be expected if the same procedure be applied to man, and it is quite legitimate, therefore, to set aside this dictum of Calmette’s if we find from the examination of a sufficient mass of evidence derived from human beings that there is no harm apparent to those inoculated during the incubation stage of plague. Inoculations have been carried out on a large scale in various parts of India, hundreds of thousands of persons (over 200,000 in Bombay city alone) having been operated on during the last four years. From the reports sent in from certain prisons and small villages, where accurate statistics have been kept, it has been found possible to compile the following table showing the case- mortality in persons inoculated during the incubation period of plague. These figures include all instances known to us in India where statistics have been kept with sufficient accuracy to admit of the extraction of this information.! * Brit. Med. Journal , 24th November 1900. t For full table, showing separately for each place the results obtained, vide Brit. Med. Journal of 14th September 1901. [Table. 1901-2.] Plague Research Laboratory of Government of India. 141 Attacks. Deaths. Case- Mortality per cent. Cases which had plague actually evident at time of inoculation, or which de- veloped it the same day, 43 21 48-8 Cases which developed plague on 1st day after inoculation, ..... 40 23 57-5 „ „ 2nd ,, 40 22 55-0 y> > ? ? ; 3rd , , 38 21 55*3 ,, „ ,, 4th ,, 27 10 37-0 ,, „ „ 5th ,, 37 18 48-6 „ „ „ 6th ,, 26 10 38-5 ,, ,, ,, 7th ,, 29 14 48-3 „ „ „ 8th ,, 24 9 37 5 ,, ,, ,, 9th ,, 24 15 62-5 ,, ,, ,, 10th ,, 30 9 30-0 Total within the first ten days after inocu- lation, 358 172 48-04 Cases which developed plague subsequently, 566 230 40-6 Total plague cases among the inoculated, . 924 402 43-5 Total plague cases among the uninoculated portion of the population during the same epidemics, ..... 5079 3726 73-3 We have here records of over 6000 attacks of plague, with a case-mortality in the inoculated of 43*5 per cent., and in the uninoculated of 73 -3 per cent. If Calmette’s contention were correct, we should see a case-mortality of over 73-3 per cent, among those who were inoculated either with the plague symptoms already manifest, or who developed plague within the next few days. But what, on the contrary, do we find ? According to the table, we see that on no occasion does the case-mortality even approach to such a figure ; even those inoculated with the disease already manifest have a case-mortality of 48 ’8 per cent, only, instead of 73 3 per cent. It appears that in these statistics we have a sufficiently large body of trustworthy evidence to enable us to set aside Calmette’s warnings — founded as they are on laboratory experiments on animals only — and to encourage us to inoculate all persons during a plague epidemic, whether they have been exposed to infection or not. One is further encouraged in this by the opinion of almost all 142 Proceedings of Royal Society of Edinburgh. [sess. those who have had practical experience of the measure — and in this number I include myself ; — for instance, Dr Alice M. Corthorn, M.B., London, who with her own hand performed some 32,000 inoculations in the towns of Dharwar and Gadag, says : * “I think that these and the Dharwar figures prove that, so far from its being inadvisable to at once inoculate the contacts of a case of plague lest they be incubating the disease, it is desirable at once to inocu- late all who have been exposed to infection ” Again, in a report on plague at Sydney, Australia, Dr Ashburton Thompson, the Chief Medical Officer of the Government, and Presi- dent of the Board of Health, reports : f “ Among the inoculated public, thirteen were attacked, .... all these patients not merely recovered, but had conspicuously light attacks. The cases occurred almost entirely among the earlier 200, while the virulence of the infection was at its highest” (page 12). “It will be noticed that attacks which occurred at, or before the lapse of about ten days from, inoculation were not aggravated by it” (page 13). It may he stated that of the thirteen cases noted by Dr Thompson, eleven occurred within periods varying from the actual day of inoculation to the seventh day after the operation. Although I have not been able to find that the Indian Plague Commission considered this actual point, yet they incidentally remark, | when considering how soon protection is acquired, that “the case- mortality of the first three days compares favourably with that of uninoculated patients,” and in another place recommend “the encouragement of inoculation among persons left in the houses with the sick,”§ thereby certainly implying that there could be no risk in such a procedure. We may then confidently say that the plague vaccine is harmless to those incubating the disease. With regard to the first portion of the second of the queries noted above, an answer is at once furnished by the table. For it is evident * Report on Anti-plague Inoculation Work in the Dharwar District, by Alice M. Corthorn, M.B., B.Sc., Medical Officer on Plague Duty, Bombay. Government Central Press, 1899. t Report on an outbreak of Plague at Sydney, 1900, by the Chief Medical Officer of the Government and President of the Board of Health. Sydney, William Applegate Gullick, Government Printer, 1900. X Report of Indian Hague Commission, vol. v. p. 258. § Ibid., p. 338. 1901-2.] Plague Research Laboratory of Government of India. 143 that a measure which reduces the mortality in a plague-stricken community from an average of 73*3 per cent, to 57*3 in those develop- ing plague the very day after inoculation, is one which acts with astonishing rapidity. On this point Haffkine, in his evidence before the Indian Plague Commission, says : * “ The time necessary for the plague prophylactic to produce a useful effect is shorter than in any preventive treatment known, this period being in the anti-cholera inoculation four days, in vaccination against small-pox seven days, in the inoculation against anthrax twelve days, in the inoculation against rabies fifteen days, and in the present treatment apparently less than twenty -four hours.” In the case of Undhera village, already given, we have seen f that the number of attacks among the inoculated at once becomes less than among those not so operated on. As the villagers were divided into two groups as evenly as possible, there is no reason to suppose that there could have been fewer persons in the incubation stage of plague among those inoculated than among their uninoculated relatives. Why, then, did they have but two cases in the first week instead of eleven, which is the number they ought to have had if they had suffered in the same proportion as the uninoculated ? The answer plainly is, that the inoculation had aborted the disease in these cases. Further, if we study the figures in the last column of the table, we see that there is really very little difference from day to day in the effect produced during the first ten days after inoculation, although there is a steady increase of effect observable as time goes on. Up to the ten-day limit the case-mortality is 48*04 per cent., while after this period it is 40*6 per cent. These daily variations, and the comparatively small difference in the rates between the beginning and the end of the ten-day period, seem do point to a sudden very considerable measure of protection being secured after the lapse of twenty-four hours only. The reply to the first part of the query is therefore : — Protection begins to be effective after the lapse of twenty-four hours, and goes on slowly and steadily increasing for some considerable number of days thereafter. It is proper, however, to point out that the Indian Plague Commission, from a consideration of the case-mortality * Report of Indian Plague Commission , vol. i. p. 6, para. 47. t Supra, p. 132. 144 Proceedings of Royal Society of Edinburgh. [sess. figures at their disposal — which were, however, some three times les& than the 924 cases which I have been able to collect — arrived at the conclusion that “ Inoculation does not appear to confer any great degree of protection within the first few days after the inoculation has been performed.” * Nevertheless, we have seen that in the case- of Undhera this degree of protection was sufficient to at once differ- entiate the group of inoculated from their unprotected relatives. The last point to be decided is — How long does the pro- tection last ? To this, time alone can furnish a complete answer. If, five years after Jenner made his immortal discovery, he had been asked — How long does the protection conferred by vaccination last? he would probably have given it as his opinion that it was lifelong. Time has proved it otherwise, how- ever, and one would be rash to predict what it may evolve in the case of plague inoculation. The Indian Plague Commission are of opinion that “Inoculation confers a protection which certainly lasts for some considerable number of weeks. It is possible that the protection lasts for a number of months. The maximum duration of protection can only be determined by further obser- vation.” f Mr Haffkine, in his “Discourse on Preventive Inocula- tion,” delivered at the Royal Society of London on 8th June 1899, J says: “As to the duration of the effect of the plague inoculation, the statement which can be made for the present is that it lasts at least for the length of one epidemic, which, on the average, extends from over four to six months of the year.” Lately, statistics have been received in the Plague Research Laboratory from the Collector of Dharwar, which seem to show that the effect has not worn off after eighteen months or two years ; but the matter is one which cannot be definitely settled for the present. The final conclusion then appears to be, that in this discovery of Haffkine’s we have within our reach a means of controlling an epidemic of plague, and converting it into a manifestation of sporadic cases only, easily controlled and exciting little alarm. * Report of the Indian Plague Commission, vol. v. p. 262. f Ibid. X Lancet, 24th June 1899. (. Issued separately May 1, 1902.) 1901-2.] Dr Noel Patou on Dissolved Oxygen in Water. 145 Observations on the Amount of Dissolved Oxygen in Water required by young Salmonidse. By D. Noel Paton, M.D., F.R.C.P.E. ( From the Laboratory of the Royal College of Physicians of Edinburgh .) (Finally adjusted for publication March 31, 1902.) That the absence of dissolved oxygen from water is fatal to fish is well shown by an observation of Wilh. Thorner ( Forschungs - berichte iiber Lebensmittel , 1897, 4, 172), on the condition of the water of the Hase at Osnabriick, where a great destruction of fish had occurred. Besides a complete absence of free oxygen in the water, nothing hurtful to fish could be detected. So far, no very satisfactory experiments have been recorded on the extent to which the dissolved oxygen in water must be diminished in order that it may act prejudicially on such active fish as the migratory salmonidse. Konig (Die Verunreiniguvg dev Gewasser, Bd. ii. p. 37, 1879) records some experiments, conducted by himself, in which fish — the kind of fish is not specified — were kept in water with 2*95 and with 1*38 ccm. of oxygen per litre without being harmed. F. Hoppe-Seyler and Duncan (Ztscli.f. phys. cbem., Bd. 17, p. 165, 1893) showed that trout kept for from 1 \ to 2J hours in water with from *98 to 1*710 cc. of oxygen per litre all showed marked symptoms of dyspnoea. Unfortunately these experiments were not continued for a longer period, and the observations on the influence of water containing greater and lesser quantities of oxygen were made upon tench, and the results cannot be applied to salmonoids. The present observations were made in connection with an investigation of the influence of estuarial pollution — and more especially the pollution of the Tyne Estuary — on salmon. Apparatus. — A Wolff’s bottle with three necks, holding 6000 ccm., was fitted with three rubber stoppers. Through one a long thick tube projected downwards to near the bottom of the jar, and PROC. ROY. SOC. EDIN. — VOL. XXIV. 10 146 Proceedings of Royal Society of Edinburgh. [sess. was closed above with a piece of rubber tubing firmly clamped. The opposite stopper was bored for a small glass tube passing to within an inch of the bottom, and closed above by a piece of rubber tubing securely clamped, and served for the removal of samples of the water. This stopper also carried a thermometer. The central stopper was fitted with a short piece of tubing connected with a cylinder of nitrogen. Procedure. — The bottle was completely filled with nearly boiling water from the tap, the stoppers were securely put in, and the water was driven out of the first opening by a stream of nitrogen through the central tube till about 2500 cc. were left. This was allowed to stand all night, and next morning the oxygen generally amounted to 3 cc. per litre. In this, rainbow trout of about nine months old, weighing from 1'5 to 5 grms., were introduced. For these I am indebted to Mr Armistead, of the Solway Fishery Company. Water for analyses was taken by connecting a 50 cc. pipette with the small rubber tube in the small glass tube, and forcing up the water by allowing nitrogen to pass from the cylinder into the bottle. A control observation was generally made by keeping fish in a flask of tap water For the determination of the dissolved oxygen Professor Ramsay’s method was used ( Journal of the Society of Chemical Industry , 30th Nov. 1901). The method is a rough one, but it was found to be sufficiently accurate for the present inquiry in the hands of one accustomed to use it. The standards had been checked at University College, London, just before this series of observations were made. Experiments. Experiment I. — A mixture of a number of samples of water sent from the Tyne Estuary was made, and divided into two equal parts. One was left standing for two days without agitation, and the other was frequently shaken with air. On the second day the former contained no oxygen detectable by Ramsay’s method ; the latter, immediately after agitation, 4 cc. per litre (on standing for a few hours this fell markedly). Tap water was boiled vigorously, and as it cooled, siphoned 1901-2.] Dr Noel Paton on Dissolved Oxygen in Water . 147 into the large Wolff’s bottle filled with nitrogen, which contained only small traces of oxygen, and kept for two days, when the water contained 2 cc. oxygen per litre. On the second day a young rainbow trout was placed in each of these vessels. A. placed in Tyne water (400 cc.) oxygenated at 4.56 p.m. B. ,, ,, (550 cc.) not ,, at 4.59 ,, C. ,, water (2000 cc.) under nitrogen at 5.3 ,, The fish in B. died at 8.45 (in 4 hours), that in C. died during the night, while that in A. was alive and well next morning. The water then contained 2 cc. oxygen per litre. This observation seems to show that there was not any chemi- cal substance prejudicial to fish life in the water of the Tyne Estuary at a time when many salmon were being found dead in it. Experiment II. — 1200 cc. water were put in the Wolff’s bottle as already described. At 10 a.m. next day the water contained 3 cc. oxygen per litre. At 10.30 a trout of 4 ‘5 grm. was in- troduced. It died at 6.30 p.m. — after eight hours. At 10 a.m. next day the oxygen was 0 cc. per litre. At 10.40 another fish of 4 ‘5 grm. was put into the bottle, and at 1.25 it was lying on its side gasping. It was taken out and placed in tap water, where it recovered. Experiment III. — A. 2000 cc. of water were placed in the Wolff’s bottle, and B. 1600 cc. in a large flask. At 10.30 a.m. next day the water in — A. contained 3 cc. oxygen per litre. T. 14° C. B. ,, 6 cc. ,, ,, ,, Three small trout, weighing 2 grm., 3*7 grm., and 1*5 grm., were placed in the bottle, and three, weighing 3 grm., P5 grm., and about 3*5 grm., were put in the flask, at 10.30 a.m. At 2.45 the small trout in A. was dead. (4 hours.) At 5.10 both remaining fish were sluggish. Oxygen = 2 cc. T. 14° C. At 6.10 the second small fish died. (7 hours.) 148 Proceedings of Royal Society of Edinburgh. [sess. The remaining fish was living next day. (24 hours.) Oxygen =1 cc. per litre. T. 13° C. After 48 hours it was still alive. Oxygen = 1 cc. per litre. T. 11° C. The fish in the flask seemed perfectly well after 48 hours. Oxygen = 4 cc. per litre. T. 13° C. At 10.20 on the third day two small fish from the flask were placed in A. One died at 10.30, and at 10.45 the last fish from the flask was placed in the bottle. It was lying on its side gasping at 11.25, and though placed in oxygenated water at 12 it died. (75 minutes.) The other was removed. The water in the bottle was shaken up with air at 12.30, when there was 4 cc. oxygen per litre. At 1 p.m. two fish of 3 grm. and 1*5 grm. were put in the water. At 11.45 p.m. both were living, but at 10 a.m., after 22 hours, they were both dead, and the water contained 0 cc. oxygen per litie. Experiment IV. — The Wolff’s bottle A. was filled to 2400 cc. and a flask B. was filled with 2800 cc. water. The water from the bottle used in the last experiment was filtered, and after the amount of oxygen had been found to be 2 cc. per litre it was divided into two portions of 700 cc., of which C1 was left un- oxygenated, and C2 was shaken with air, and a fish placed in each at 12.40. At 10.30 next morning, T. 13° C. — A. contained 3 cc. oxygen per litre. T> £ -1-** 55 U ,, ,, ,, C 2 vy* 15 5 5 5 5 5 5 At 12.40— Three fish were placed in A. Two fish ,, ,, B. A. On the morning of next day the two smaller fish were dead. Oxygen = 1 cc. per. litre. T. 14° C. The remaining fish was quite well on the following day at 12.55, when the water contained 1 cc. oxygen per litre. After 72 hours it was put in oxygenated water and was very lively. 1901-2.] Dr Noel Pat-on on Dissolved Oxygen in Water. 149 On the same day at 11 a.m. a fish from the flask B. was put in the bottle and died in 40 minutes. At 11.45 three small fish were put in the bottle, and at 12.30 two were dead and the last died at 12.55. (Oxygen =1 cc. per litre.) B. The smaller fish died next morning. Oxygen = 3 cc. C1. The fish lived in this w^ater till 3.20 p.m. on the second day (55 hours). Oxygen per litre — First day 1 p.m. 2 cc. Second day 10 a.m. 3 cc. Third day 10 ,, 1 to 2 cc. ,,' 12 noon 0 cc. At 11.45 a small fish was introduced and it died at 2. C2. The fish lived till 6 p.m. on the next day (30 hours). Oxygen per litre — First day 1 p.m. 5 cc. Second day 10 a.m. 0 cc. At 3.20 on the second day the water from these two beakers was combined — oxygen = 2 cc. per litre — and a fish from the tank put in. At 4 o’clock it was almost dead. Experiment V. — The Wolff’s bottle was put up with 2600 cc. water and the flask B. was charged with water from the last experiment to the same amount. Next day at 10 a.m., T. 13° C. — A. Oxygen = 3 cc. per litre. P- 55 ^ 5 5 5 5 5 5 At 11.30 three trout were put in each. A. At 3 p.m. one fish was dead. 3.30 p.m., oxygen = 2 cc. At 8-30 p.m. the other two fish were dead. B. The fish in the flask were well and lively at 4 p.m. on the third day. Oxygen = 5 cc. 150 Proceedings of Royal Society of Edinburgh. [sess. Experiment VI — 2000 cc. of the water from the bottle were partly oxygenated and returned to the bottle under nitrogen at 10-50 and three fish introduced. One died during the night, one about 11.30 on the next day, and the last fish, which seemed quite well, was taken out at 12.30. Oxygen in ccs. per litre — At 4 p.m. on first day =4 cc. At 10.30 a.m. „ second day =1 cc. Conclusions. , These experiments along with those of Hoppe-Seyler and Duncan seem to show that a fall in the amount of dissolved oxygen in water to below one-third of the normal amount, 2 ccm. per litre, is prejudicial and generally fatal to young salmonoids. When the surface of the deoxygenated water is exposed to an atmosphere containing oxygen as in Experiment IV., the fish frequently seem able to live by constantly coming to the thin layer of more oxygenated water at the surface. Some individuals, e.g., the largest fish in Experiment III., are able to sustain life for very prolonged periods in water containing only minimal traces of dissolved oxygen. Such fish are seen to lie very quietly at the bottom of the bottle. (. Issued separately May 2, 1902.) 1901-2.] Dr Muir on the Theory of Jacobians. 151 The Theory of Jacobians in the Historical Order of its Development up to 1841. By Thomas Muir, LL.D. (Read January 20, 1902.) It is not improbable that determinants in which the number of a row is distinguished by differentiation with respect to a definite variable, and in which the number of a column is distinguished by a particular function set for differentiation, may have appeared long before the time of Cauchy and Jacobi, the likelihood probably being the greater the fewer the number of functions and variables involved. There can be little doubt, for example, that expressions like du dv du dv dx dy dy dx may be found repeatedly in the writings of mathematicians belong- ing to the eighteenth century. It would appear, however, that the first who got beyond the second order, and clearly associated the expressions with determinants, was Cauchy. Cauchy (1815). [Theorie de la propagation des ondes a la surface d’un fluide pesant d’une profondeur indefinie. Mem. presentes par divers savants a V Acad. roy. des Sci. de I’ Inst, de France . . . . I. (1827) : or (Euvres V ser. I. pp. 5-318.] Cauchy was a competitor for the prize for mathematical analysis in the ‘concours’ of 1815, and gained the prize. His work, however, like others belonging to that interesting political period, was not printed until long afterwards. In the form which it takes in the collected works the essay proper extends to only 108 pages, the remaining 210 being occupied with notes : this was probably due to the circumstances under which the paper was first written. In the same way is explained the writer’s action in referring in it to himself by name, the object being to preserve his anonymity. 152 Proceedings of Royal Society of Edinburgh. [sess. There is only one passage in it which directly concerns the student of determinants, but it is interesting from more than one point of view. The exact wording of the passage is as follows : — “ Cela pose, concevons que le sommet de la molecule m, auquel appartenaient, dans le premier instant, les trois coordonnees a , b , c, se trouve, au bout de temps t , transport^ en un point dont les coordonnees soient x , y , z. Les trois aretes de la molecule qui aboutissaient au sommet dont il s’agit, et qui, dans l’origine, se trouvaient paralleles aux trois axes des coordonnees, auront alors cesse de l’etre, et les projections de ces memes aretes sur les axes dont il s’agit, projections qui dans Torigine etaient respectivement egales, pour la premiere arete a da , 0 , 0 , pour la seconde, a 0 , db , 0 , pour la troisieme, a ... . 0 , 0 , dc , seront alors devenues pour la premiere arete ~da , da , — da , da da da pour la seconde , ~db , ~ db , ob db db pour la troisieme — dc , — dc , ~dc . F be dc dc Il est aise d’en conclure (voir la Note I.) que le volume de la molecule, qui etait primitivement egal a da db dc , sera devenu, au bout du temps t , dx dy dz dx dy dz dx dy dz .da db dc da dc db 06 da dc dx dy dz dx dy dz + dc db da dc da db dx dy dz db dc da da db dc ; et, comme ces deux volumes doivent etre equivalents, on aura, par suite, dx dy dz dx dy dz dx. dy dz dx dy dz da db dc da dc db 06 da 0c db dc da dx dy dz dx dydz_ ^ dc db da dc da db 1901-2.] Dr Muir on the Theory of Jacobians. 153 Si, pour plus de simplicity on fait usage de la notation adoptee par M. Cauchy dans son Memoir e sur les f auctions symetriques ,* 1’equation prendra la forme suivante : le signe S etant relatif a la permutation des trois lettres Here we have clearly the Jacobian of x, y , z with respect to «, b, c : and we have it expressed also in the determinant notation then in use. The second point of interest is centred in the note to which the formal statement of a theorem, and extends to only ten lines, is as follows : — “ Si Ton rapporte la position des sommets d’un parallelepi- pede a trois plans rectangulaires des x, y, et z ; que l’on designe par A, B, C, les longueurs des trois aretes de ce parallelepipede qui aboutissent a un meme sommet, et par les projections respectives des memes aretes sur les axes des x, y, et z>, le volume du parallelepipede aura pour mesure Here we have one of those so-called “ applications of deter- * There is a curious oversight here. In a footnote, Cauchy says “ Le Memoire dont il estici question a ete imprime en partie dans le xviie. Cahier du Journal de Vltcole Poly technique." Now, as a matter of fact, there is no memoir bearing this title. The well-known memoirs contained in Cahier xvii. are headed “ Memoire sur le n ombre des valeurs . . . .” and “Memoire sur les fonctions qui . . . .” The second part of the latter, it is true, bears the approximate designation, “ Des fonctions symetriques alternees . . . but the notation in question occurs in both parts. It is also not clear what was intended by the words ‘ imprime en partie ’ in Cauchy’s footnote. author directs his reader. This note, which consists merely of the 154 Proceedings of Royal Society of Edinburgh. [sess. minants to geometry ” which are often supposed to belong to a much later date. Jacobi (1829). [Exercitatio algebraica circa discerptionem singularem fractionum, quae plures variabiles involvunt. Crellds Journ.,, v. pp. 344-364 : also abstract in Nouv. Annates de Math., iv. pp. 533-535.] To the great mathematician whose name was ultimately associ- ated with determinants of this special form, they first appeared in a totally different connection. He was considering a problem of the partition of a fraction with composite denominator into others whose denominators are factors of the original, and the paper to which we have come concerns given fractions of the form {ax + by - ty1 ( b'y + ax - t'y1, {ax -\-by-\-cz — ty1 (b'y + c'z + ax - t'y1 (cz + ax + b"y - fy1 , The expansions of these clearly contain a variety of terms, the reciprocal of each linear expression contributing negative powers of its first term and positive powers of the others; and the ‘ discerptio singularis 5 consists in obtaining fractions which produce, each of them, the aggregate of the terms of a particular type found in the expansion. Thus, to take the simplest example, viz., (ax + by)~l (b'y -{-ax)-1 , it is seen that the expansion of ( ax + by)~ 1 will contain one term with negative power of x. and others with a negative power of x and a positive power of y, that the reverse will be the case in the expansion of (b'y + ax)-1, and that the product of the two expan- sions will therefore contain a term in xr1 y~x, a series of terms with negative powers of x and positive powers of y, and a series of terms with positive powers of x and negative powers of y. How Jacobi establishes the identity (ax + by) 1 (b'y 4- ax) 1 b ax -f by 1 a! \ y b'y + ax) y where on the right there are three parts ; and as the first is a term 1901-2.] Dr Muir on the Theory of Jacobians. 155 in x~l y~x, the second equivalent to a series of terms consisting of negative powers of x and positive powers of y, and the third equivalent to a series of terms consisting of negative powers of y and positive powers of x , it is ciear that the three portions of the expansion of (ax + by)~l(b'y + axf1 have been isolated and summed. Now it will be noticed that a common factor of the three parts is the determinant | ah' j , or as Jacobi, following Cauchy, writes it (ah'). The corresponding factor in the next case, where there are three linear expressions and three variables, is found to be ( a b'c") ; and Jacobi then makes a generalisation regarding the first of the partial fractions in each case, viz., to the effect that the coefficient of /y» l/y» I vi 1 'yt 1 tAJ * * • V — 1 in the expansion of u~xufuf .... u~\ i.e. of (ax + bx1 + cx2 + . . . )_1 (b'xl + cx2 + . . . . )-1 (c",x2 + .... )_1 ... . is (a b'c". —the result being, so to speak, the discovery of the generating function of the reciprocal of a determinant. Shortly after this follows the passage which is interesting in the history of Jacobians. It stands as follows “At theorematis, de quibus in hac commentatione agimus et quorum modo mentionem injecimus, latissimam conciliare licet extensionem. Ponamus enim, u - t , u1 — t' , . . . iam series esse quaslibet, sive finitas sive infinitas, ad dignitates integras positivas elementorum x , xY , ... procedentes, quarum serierum t, t\ . . . sint termini constantes. Sint porro in seriebus illis u, ult u2, ... termini, qui primas ipsorum x, xl9 x2, dignitates continent, respective ax , b'x1 , c"x2 , . . . , ac pon- amus, uti in casu lineari, fractiones (u — t)~l , (u1 - £')-1 , (u2 — t")~l , . . . evolvi respective ad dignitates descendentes terminorum ax, b'xY, c"x2, .... Yocemus porro A deter- minantem differentialium partialium sequentium : Proceedings of Royal Society of Edinburgh. 156 du du du du dx ’ dxf dx2 ’ ’ ' ’ dx ’ v^n- 1 dux dux dux dux dx ’ dxf dxf ’ ' ’ a^_i’ du2 du2 du2 du2 dx ’ dx1 ’ dxf ’ dxn_x ’ dUn- 1 dun_ i dun_ i d“„-i dx 5 dxx ’ dx2 ’ dxn_x' Erit e.g. pro tribus functionibus u , u1 , u2 , tribusque variabil- ibus x , y , z : du dux du2 du dux du2 dux . 0W2 du dx dig dz dx dz dy dy dx dz du2 du dux du + — ■ dux du2 du i - dux du2 dz ~ dy dx dy dz dx dx ' 32/ quarn patet expressionem casu, quo u , , u2 , sunt expres- siones lineares, in expressionem ipsius A supra exhibitam redire. Quibus positis dico, siquidem a? = £> , x2 = p2, ...., Wn-i — Pn-i satisfaciant aequationibus u = t, u1 = t', u2 = t" , , un_x — t(n~x) , producti A^ (u - t) {ux - t') (u2 - t") . . . (un_x - ’ dictum in modum evoluti, partem earn, quae omnium simul elementorum x , xx , ... dignitates negativas neque ullius positivas continet, ut supra in casu multo simpliciore, fieri (x -p) (xi-p1) (x2 - p2) (xn_x Pn—i) ' It will be observed that Jacobi looks temporarily upon the ordinary determinant {aVc . . .) as the particular case of the Jacobian in which the involved functions are linear in all the Variables concerned. Jacobi (1830). i [De resolutione aequationum per series infinitas. Crelle’s Journ ., vi. pp. 257-286.] 1901-2.] Dr Muir on the Theory of Jacobictns. 157 Although the general subject here is new, there is a certain link of connection with the preceding paper, in that one of the results of that paper is employed, and also that Jacobi is using onco more the method of ‘generating functions.’ Passing over the first two cases, let us note how he proceeds with a set of three equations and three variables. As a preliminary he introduces after the manner of the preceding cases the determinant of the partial differential coefficients, the sentence in regard to it being [page 263] — “ Ut similia eruamus de tribus functionibus, tres variabiles x , y , z involventibus f(x, y, z) , y, z) , if/(x, y, z) adnotetur aequatio identica : z d-Q'W'fo) - dx dij + ztyQWty) - = 0 dz ’ quam differentiationibus exactis facile probas. E qua, positu brevitatis causa v = / (*)[>'($ /)■/''( z )-4> (? +f{z)W(x)^'{y) - r (y)'p' {x)\ , flnit sequens : d/fo'O/V'OO - ■ftW'O/)] , z/WWW - dx dy , z „ dz V' Here the concluding identity has to be noted. He then establishes certain results concerning the coefficient of x~Y y~Y z _1 in the ex- pansion of y , or, as he writes this coefficient, [^] x-'y-'z-1 • Thus prepared he attacks the given equations (p. 284) — c r = ax +by +cz + dx2 + exy + . . . , t = cl x + b y -f- c z + d x 2 -t- e xy -1- . . . , v — ax + b"y + cz + d”x2 + e"xy + . . . ; obtains first the derived set 158 Proceedings of Royal Society of Edinburgh . [sess. s = A x + ax2 + j3xy + yy2 + . . . t = A y + ax2 + p’xy + yy2 + . . . u = A z + ax2 + fi'xy -f y'y2 + . . . where the values of s , A , a , . . . , t , A , a, ... u, A , a", . . . are sufficiently suggested* by giving one of them, viz., A =(a Vc) • and then seeks to find any function of the roots, F(^, y , z) say, in the form of a series proceeding according to powers of the constants s , t , u , the result being that the coefficient of sptqur in the said series is shown to he in general 'Fix, y, z) V XpY qZr x 1 y 1z 1 where X , Y , Z are the variable members of the derived set of equations, and y is the determinant of their partial differential co- efficients with respect to x, y, z. Xo other case is dealt with, but the paper closes with the sentence — uQuae autem hactenus de duabus, tribus aequationihus inter duas, tres variabiles propositis protulimus, eadem facilitate ad numerum quemlibet aequationum et variabilium extenduntur.” Jacobi (1832, 1833). [De transformatione et determinatione integralium duplicium commentatio tertia. Crelle’s Journ., x. pp. 101-128.] [De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant : una cum variis theorematis de transformatione et determinatione integralium multi plicium. Crelle’s Journ., xii. pp. 1-69.] The latter of these memoirs is by far the more important ; in fact, * In later notation the derived equations would of course be written — (4,4, 4) = 0, erit d$l 8^ 8^_2; = + ^ . 0&A 9^2 • • • 9^n-2 0F 0$ 0F 0<3> 0vx dv2 dvn) 0F 0$ 0F 0$ . bL-Mn - 9£»- 1 . 3v»-l0Vn ~ His third theorem is a special case of his first, and may therefore be passed over. Then we have Theorema 4. Supponamus , £2 , . . . , £«-i datas esse sub forma fradionum V + d£»-i - 'V 2% 9^2 ^ ~ 0vx 0v2 dvn-i un ^ “ U 9^1 3v2 S^n-l &„_i ubi in altera summa inter indices permutandos etiam referri debet index 0 seu index deficiens. From this is deduced Theorema 5. Si loco functionum u , u i ’ u9 ponitur - , — , ^ , • . . , ^-1 , designante t aliam functionem t t t t quamlibet, expressio dux du2 dur U 0Vj 0V2 dvY obit in -2 tn*-* 0%T 0%2 duri dv1 dv2 dv„ sive in differentiations bus instituendis denominatorem com- mumen t ut constantem considerare licet. The last of the series is Theorema 6. Sint u , ux , u2 , . . . , un_L expressions linear es aliarum functionum w , w2 , w2 , ... , wn-1 , datae per aequationes huius modi 1901-2.] Dr Muir on the Theory of Jacobians. 161 vk = akW + akw1 + akw2 + . . . + aj? l)lOn_x ft du-, duc 2±«-x ™2 dv2 du^ dv„ Z± a af) . . . cLn-i^ dW»-l to which is added the remark that if there were one additional independent variable v we should similarly have du dux ^ ~dv 0iq dun_x dvffx dw dw1 ~ dv dvx Sv’n-l) SvnJ Jacobi (1841). [De determinantibus functionalibus. Grelle’s Journ ., xxii. pp. 319-359.] Up to this point, as will have been evident, the special deter- minants which we are considering have turned up merely inciden- tally in the coarse of other work. Now, however, we come upon a separate and direct investigation of their properties, the memoir under consideration being the second of the three portions into which Jacobi divided his formal exposition of the theory of determinants. From the mere fact that separate treatment is be- stowed by him on only one other special form, it is clear that the subject of the memoir had come to be considered of particular importance. The same is rendered still more strikingly apparent when it is recalled that of the 87 pages occupied by the whole exposition, as many as 41 are devoted to this second portion con- cerning a subordinate form, while only 34 are assigned to what we are bound to consider the main portion, viz., that dealing with determinants in general. At the outset the preceding memoir 1 De formatione et pro- prietatibus determinantium ’ is referred to, and intimation made that there is now about to be considered the special case where the elements are partial differential-quotients of a set of n functions each of the same n independent variables, and that in this case the special name functional determinants may with convenience be used. Jacobi takes pains, however, to explain that this relation of general to particular may appropriately be taken in reverse order, going, in fact, so far as to say that from the properties of functional determinants the properties of what he calls algebraic PROC. ROY. SOC. EDIK — VOL. XXIV. 11 162 Proceedings of Royal Society of Edinburgh. [sess. determinants may be deduced. He is careful to note also another relationship of the same kind, his statement being that in various questions relating to a system of functions the functional deter- minant is the analogue of the single differential-quotient in the case of a function of one variable. The subject of the notation of partial differential-quotients is then entered on at some length (pp. 320-323), and the decision made to use 0 in the manner which soon afterwards came to be familiar. The insufficiency of this notation is not forgotten, how- ever, although its advantages over the different devices of Euler and Lagrange are recognised, his illustrative example being the dz case of — where z is a function of x and u, and u is a function of dx x and y. He puts the whole matter in a nutshell when he says that it is not enough to specify the function to be operated on and the particular independent variable with respect to which the differentiation is to be performed, but that it is equally necessary to indicate the involved quantities which are to be viewed as con- stants during the operation.* * I may state in passing that in 1869 when lecturing on the subject I found it very useful to write 4>x, y, z , Ts, t, u, v , .... in place of m y, *) , /(s, t,u,v), . . . . and then indicate the number of times the function had to be differentiated with respect to any one of the variables by writing that number on the opposite side of the vinculum from the said variable ; thus 1 2 3 4>x,y,z meant the result of differentiating once with respect to x, thrice with respect to y , and twice with respect to z. Using this notation to illustrate Jacobi’s example, we see that if it were given that Z — 4>X , u we should have dz_ _ i dx ~ x > u ’ but that if it were given that z=4>x>u then we should not be certain as to and u = if/x,y dz the meaning of ^ , as it would stand for dx i l l l x,u or x>u + x,u * 'I'XiV according as u or y was to be considered constant. 1901-2.] Dr Muir on the Theory of Jacobians. 163 The dependence or independence of equations is the next pre- liminary subject (pp. 323-325), the starting-point being the definition of an identical equation as one in which every term is destructive of another, and from which, therefore, it is impossible to express one of the involved quantities in terms of the rest. On this the definition of mutually independent equations is made to hang, such equations being defined as those of which no one at the outset is an identical equation nor can be transformed into an identical equation by aid of the others. Then taking m+ 1 equations, u = 0 , uY = 0 , . . . . , um = 0 involving n + 1 quantities x, x1} . . . , xn he contemplates the possi- bility of solving u = 0 for x in terms of xlt x2i ... , xn and the sub- stitution of the expression in place of x in the remaining equations. The latter equations as altered he supposes to be dealt with in the same way, and the process continued until k + 1 quantities have been eliminated and m-h equations left involving xk+l , xk+2 , . . . , xn. Reasoning from this, he concludes that a number of given equations are mutually independent or not according as by their help the same number of involved quantities can or can not be ■expressed in terms of the remaining quantities. In this connec- tion he does not omit to draw attention to the existence of ex- ceptional cases, such as that in which two of the quantities, xh, xk , say, occur indeed in all the equations, but always in the form xh + xk\ and this leads him, for the sake of greater definiteness, to introduce the qualifying phrase £ with respect to certain quantities 7 in using the expression ‘ mutually independent.7 His words are — “Aequationes u = 0, u1 = 0, . . ., um — 0 quibus totidem quantitates x, xl3. . . , xm quas involvunt, determinantur, harum quantitatum respectu dico a se independentes.77 From the independence of equations he naturally passes (§ 4) to the independence of functions , with the remark that exactly similar propositions are found to hold in regard to the latter, — a state- ment which it is not hard to believe when we recall that any function, x2 + y2 - 4 xy say, may be denoted by a functional symbol, / say, the equation f =x2 + y2- 4 xy thus resulting ; and that any non-identical equation connecting two or more quantities 164 Proceedings of Royal Society of Edinburgh. [sess. implies that any one of the latter is a function of the others. Functions of several variables are said to be mutually independent when no one of them is constant or can he expressed in terms of the rest. This is extended and made more definite by saying that if functions of x , x1 , . . . xn involve also the quantities a, ax , a2 , . . . the functions are said to he mutually independent with respect to the quantities x , xx , . . . , xn if no equation subsists between the functions and the quantities a , ax , «2, . . . these definitions will suffice to indicate the analogy above referred to, and the deduced propositions (pp. 325-327) need not he entered on. All this introductory matter having been disposed of, Jacobi proceeds (§ 5, p. 327) to deal with the subject proper, his starting- point being the fact that if there he n + 1 functions /, f\ , f , • • . , fn of the same number of variables x, xlt . . . , xn there arise in connection with these the \n-\- 1)2 quantities V m dxk The determinant formed therefrom, viz., y +v Wi . . . . ¥. ^ dx dxx ' ’ dxn he calls the ‘ determinant pertaining to the functions /, fx , ... , fn of the variables xfxlt...txnJ’oT the £ determinant of the functions f, fu ... 5 fn with respect to the variables x, x1 , ... , xj The case where n = 0 is then referred to in a line, after which cases are taken up where it is the functions that are specialised. The first of these is that in which fm+l = *^m+l j fm+ 2 = *^m+ 2 5 • • * ) f n ~ ■> it being pointed out of course that the order of the determinant is then lowered, being equal to xry V * V\ Vm ~cx dxx * dxm Another is that in which the functions fm+1 , /m+2 , ... § fn do not involve the variables x , xli ... , xm, the peculiarity then being that the determinant breaks up into two factors similar to itself, being equal to 1901-2.] Dr Muir on the Theory of Jacobians. 165 'Sp _J_ Of bfm ^ + bfm+ 1 ^ 5/m+ 2 bfn ^ ~dx 0% 0arm "_0^m+1 0^m+2 ' ' ‘ ' dxn' The important proposition regarding a vanishing functional determinant is then dealt with (§ 6), viz., the proposition “ func- tionum a se non independentium evanescere Determinans, function es quarum Determinans evanescat non esse a se independentes.” The proof of the first part of it opens with the assertion that since the functions are not mutually independent, there must exist an equation n(/,/i = 0 such that on substituting for /, /l5 . . . , fn their expressions in terms of x , xx , . . . , xn we shall obtain an identity. From this by differentiating separately with respect to x, x± , . . . , xn there is obtained the set of equations dfdu dfdn m dfnf n dx 9/ dx 0/j 3^i '0/M ’ ^ an an .... n 3/ 9^1 3/x * ’ 0«1‘ 0/n ’ A 0/ 0n 0/, 3n 0/n 0n ~”0^n 0/ + bxn' 0/l + + dxT‘dfn ' Then it is recalled that in a set of linear equations + a12^2 + • • • + alnxn = 0 anlXY + + * * • + amxn—Q the determinant of the coefficients must vanish unless all the unknowns vanish. And as the vanishing of an 3n an 9/ ’ 9/r " ' ’ ’ bf n would imply that the expression n(/, fv... , fn) was free of /, /i > • • • , fn , the conclusion is reached that the determinant of the coefficients of these differential-quotients must vanish, i.e ., that the functional determinant +Qf_ t ^ ~ dx a^ bfn dxn = 0. 166 Proceedings of Royal Society of Edinburgh. [sess. The proof of the converse proposition, Jacobi owns, is ‘paullo prolixior.’ It is of the kind improperly known as £ inductive,’ and the first part (§ 7) of it goes to show that if the proposition holds in the case of ^ ^ ~ it will also hold in the case ' dx1 ox2 dxn o tZ±l-°£ . dA dx dx1 the lemma that- As a preliminary, there is established If it fi , f 2 , . . . , fn are mutually independent, then V + A n i have of course df A dx A + d£p + ■ 0/ A - + - .Ml. dx 1 . . dA ' dxn A A dx + + ■ + Mia = dxnAn 0 >• dx A + I;a. + - + = dxn 0 But since fY , f2 , ... , /„ are mutually independent it is possible by solution to obtain x1 , x2, ... , xn each in terms of the remaining variable x and f1 , f2 , . . . , /„ ; and as a consequence it is possible by substituting for x1 , x2 , . . . , xn to obtain / in terms also of x, f , f2 , . . . , fn . Differentiating / with respect to x , x1 , . . . , xn , and using brackets to indicate that the / within them is to be viewed as a function not of x, x19 , but of »,/n • • • ,/», we have 1901—2.] Dr Muir on the Theory of Jacobians. 167 ¥_ dx ¥1 dx Vn dx (bf\ •lr + (bf_\ ,A + * ■ , • J- (¥\ A Wi> W2y ' dx • -q- wJ " dx (bf ^ ,¥ . (df_\ t . 4- (¥\ Mr Wi ) }dxx \^/2y '3a, T • T WJ 1 dx1 M . /df ^ 4- . . . . 4- (¥) ,dA. hxn W2y ’dxn i . -f- WJ dxn From these, on multiplying both sides of the first by A, both sides of the second by A1? and so on, and then adding in columns, there is obtained, with the help of the n + 1 equations immediately preceding, Y+dl.dA ^ dx dx1 dxn \dx ) which is what was to be proved.* A + 0 + 0 + • • • + 0 ,. Now suppose that in any way it has been established that if ’ ' * * =6, the functions involved are not mutually independent, and that the next higher case is to be investigated, viz.y where "V ±~ • • • • • ^ = 0 . Of the n + 1 functions in- ^ dx dx1 oxn volved in the latter determinant the last n of them must either be independent or not. If they are not independent, there is nothing more to be proved. And if they be mutually independent, then the lemma gives !)-»• * Using this theorem upon itself we have §£. . . . . bfn _ ^ dx dxx dxn \dx J\dx1) ^ dx2 dfn dxn provided that on the right / is expressed as a function of x , , /2 , . . and /i as a function of x, ,/2 , ... , fn ; and ultimately . jh .... ^ dx dxj dxr, (A m\.. . . (A\ \dx) wj \dXn) provided that in every instance on the right /& is expressed as a function of x , x1}x2, ... t xk , fk+1 , ... ,fn. A theorem like this ultimate case Jacobi enunciates and proves quite independently at the end of his memoir (v. § 18). The one, however, is seen to include the other if we note the simple fact that Y , 3/;3/l dfn _ y ■ dfndfn-x ... . df dxdx-L dxn ^ dxn dxn-i dx 168 Proceedings of Royal Society of Edinburgh. [sess. and therefore (SH 0 for it is impossible that the functional determinant A could he 0 , as the functions involved in it are by hypothesis mutually indepen- dent. The vanishing of however implies that x is not in- volved in /: and this, if we bear in mind the meaning of the brackets, further implies that / is a function of only fx , /2 , . . . ,/„ — that is to say, that f,fl,f2i . . . , fn are not mutually indepen- dent. There now only remains to establish the proposition for the case where the determinant is of the second order, — that is to say, where Vffi dx dx. m = o ^ dxl dx Here the function f1 must involve both variables, or only one of them, x say, or be a constant. If it be not a constant, xx is expressible in terms of fx or at most in terms of fx and a?, and thus, by substitution, / is expressible in terms of the same. In this way we have 0/-l fbf\ + Wjdx a* ' \dx) r dxx 1 + • • • + dfi ~ rn dxn dx + dx1 1 + • • • + dfn Tn dXn in which the determinant of the coefficients of the unknowns is the functional determinant ^ 4. " ~ dx dxl dxn Of course a condition of solution is that this determinant does not vanish, and therefore that the functions/, j\ , . . . , fn are mutually independent. But this being the case it follows that x, . . . , xn are expressible in terms of f,f19 ... , fn , and, as a consequence of substitution, that any other function of x, xY , . . . , xn is ex- pressible in terms of the same, thus giving us dcj> dcf> dx d(f> dx-^ d dFjdf, dxk df'dxk dffdxje + dfn'dxk 5 which may be viewed as giving an expression for - — • Using dxk this expression (n + l)2 times we obtain for ^ dx dx^ dxn an equivalent determinant each of whose elements is the sum of n+ 1 products, and which from the multiplication-theorem we know to be equal to 174 Proceedings of Royal Society of Edinburgh. y +dl .Ai . . A . y+¥ .A . ... . 8Z». “ 0/ ' dfY dfn dx ' dx1 dxn. It thus follows that the result sought is y 0F 0FX 0FW y +WV 1 A = (_ 2 \^+ 1^™^ 0fl! 0*U^ 03?},, dxn ~ y~ ; + 0F aFj 0F„ ’ ^J±3/ s/i 3/» a theorem which Jacobi again takes pains to have noted as the analogue of the theorem which holds when F (/, ic) = 0, viz., df _ _3F_^0F dx 0/ * dx By way of corollary it is remarked that as the equations F = 0 , F1 = 0 , , F„ = 0 cannot be more appropriately viewed as giving the f’s in terms of the a?’s than as giving the a;’s in terms of the f s, we therefore have the twin result v +dl . • ••• • dA V + Z Z] . . ■ 'CX" = I _ 1 V«+l . ~ W 3/l , 1, . . . , <]>p of the x’s. Here, of course, we have dfi 0/i dcf> dfi 01 df i dcf>p dxk ~ 0c p * dxk + d^dxk + * * * + dp * dxk and therefore by (n + l)2 substitutions there is obtained for ^ +df_ df, dU 2j ~dx dx, dxn an equivalent determinant, each of whose elements is the sum of p + \ products. This latter determinant, however, we know from 1901-2.] Dr Muir ou tlie Theory of Jacobians. 175 Binet’s multiplication-theorem is equal to 0 whenp di 00n dx dxj dxn when p = n; and, when p>n , is equal to a sum of such products, viz., 3/l . . 0/n Y +00 90i fyn 1 0<£ i dcfrn ^ “ dx dx1 dxn ) where the different terms included under the S are got by taking all the different sets of n + 1 ’s from the p + 1 available. This tripartite result Jacobi carefully enunciates at length in the form of three propositions. He notes, too, that the first is practically a result already obtained, because the functions in that case are not independent ; that the second has for its analogue or ultimate case the theorem df _ df dy . dx dy dx ’ and similarly that the third is the extension of a result actually used in the proof, viz., df _ df dc/> df d(f>i df dcf>p dx 0 dx S (/q dx ^ dp dx He even enunciates formally a variant of the second proposition, calling the variant “ Proposition iv,” viz., If f , fx ,...., fn be functions of y , yx , . . • , yn , and it be possible to express both the f }s and the j’s in terms of n + 1 other quantities x, xl5 , , . , xn} then V + M . ... Y + — ^ ^ ~ ^Xl ^Xn ^ ~dV ’ tyi _ y < _ byf ^ ~ dx " dxl dxn The analogue also is again referred to in the form df df _ dx dy ~ dy 7 and the special case, already twice obtained, where f=x,f1 = xl; 176 Proceedings of Royal Society of Edinburgh. [sess. Still further importance is given to the second proposition by assigning the next section (§ 12, pp. 341-343) to the consideration of certain deductions therefrom. First there is taken the special case where 4> ~ **' J 4*1 = **1 > * * * 5 4>)n — % m 5 and where therefore 34>l _ 'ST' + d4>m+ 2 “ dx dx± dxn ^ ~ dxm+l dxm+2 dxn * the result clearly being that If f, f15 . . . , fn be functions of x , Xj , ... , xm, 4>m+1 n <£m+1 , . . . , 4>a be functions of x , Xj , . . . , xn , then the functional determinant of f , ft , ... , fn with respect to x , xx , ... , xn V + ¥ t ¥l ^ ~dx dxY tym ' 4fm+l m+l d4>n' dxm+l Here the first factor would reduce to 0^n‘ Y + ^ . ¥1 ¥3 J ~ 0£ 0aq 0a?m if it were possible to put 4>m+ 1 fm+l ? 4*m+2 ~Jm+2 1 ’ ‘ > 4*n ~fn } hence there follows the further proposition, which Jacobi speaks of as “ prae ceteris memorabilis,” that the functional determinant of f , f 1 , ... , fn with respect to x , xx , ... , xn is equal to m. , (Vi) . . . ( of m\ V S/m+1 ^/m+2 W \dx J ^m+2 if the brackets in the first factor be taken to mean that the functions therein occurring , viz. , f , fT , . . . , f m are considered to be expressed in terms of x1, x1, ... , xm , fm_i , fm+2 , . . . s fn . An extreme case of this, viz., where the first determinant factor is of the order 1, has already been given. In order that we may be able to substitute 1 2± 0r vu'm+ 1 iwi 0Xm+ 9 ^4>m+2 Y + ^4*171+2 7)r U sLmi ~ 7)r ‘ 0r (^n {Jd"m+2 $4>n Hn fan’ in the former of these two propositions it is necessary that from the equations which give m+1 , <£m+2, . . . , 4>n in terms of x, 1901—2.] Dr Muir on the Theory of Jacobians. 177 x1 , ... , we obtain xm+1 , xm_2 , ... , xn in terms of the other sds and , <£m+2 , . . . , n. Consequently, we have the proposition, If f , fx , ... , fn xm+1 , xm+2 , . . . , xn be ex- pressed in terms of x, x19 ... , xm , <£m+1 , <£m+2 n the functional determinant of f , fx , ... , fn with respect to x , xx , ... , xn is equal to Y + dl.dA ^ dx dx} 3/m ^ ¥m+l dxm 8c pm+l bxm+-y bxm+2 " “ 3^m+l 3 fm 5 1 J •^m+2 > • • • j «^n be expressed in terms of x , x^ , . . . , , fm+^ , J m+2 > • * • 5 /n > the functional determinant off,f , . . . , fn with respect x, is equal to +1 • ffm dxm 2 dx^ 3/ m+i 0JC, 3/w m+2 dxn Wn Leaving this, Jacobi harks back (§ 13) to an earlier proposition with a view to a generalisation now possible, viz., the proposition where the functions are given implicitly in terms of the independent variables by means of n + 1 equations F = 0 , F1 = 0 , . . . , Fn = 0 . the extension arises from the number of equations now given being n + m, viz., Y = 01 , Fx = 0 , . . . , Fn+m = 0 , PROC. ROY. SOC. EDIN. — YOL. XXIV. 12 178 Proceedings of Royal Society of Edinburgh. [sess. and each of the F’s being a function not only opM , x1 , ... , xn , f, fi, ... , fn hut also of fn+ 1 i f n+2 J * ■ • 5 fn+m • If the last m of the equations were solved for the last m of the f s, and these f s thus eliminated from the other + 1 equations, the functional determinant desired would, by the proposition sought to be generalised, be equal to where the brackets are used to indicate that the enclosed F’s are in the] altered forms resulting from the substitution referred to. Multiplying enumerator and denominator by Z+ w+l 11+2 .... dl n+m ~ ¥n+ 1 ’ ¥n+ 2 ”” ¥n+m we obtain a new enumerator which by a later proposition is equal to y + aF 8Ft dFn 8F„+1 S 0Fw+2 . . _ 8F n+m dx ' dxx dxn dfn+1 ’ m+2 3 fn+m ’ and a new denominator which for a similar reason is equal to V+— —i dFn+m 2j-df' by the use of the equations f\ — al 5 fl = a2 5 • • ■ fn — an » we see that in addition to x , aq , ... , xn may involve aT, '2 > * * • 5 J and that therefore we have V deft rf dcf> . 3A I dcf> , % dx • • + 0C f> bfn dx dx da1 dx T da 2 dan * 0a; 5 df def> + d . dA I def) % + . dx1 . . _L_ def> a/» dx1 dxl da dx1 i • 0a2 dan ‘ 0aq ’ df_ def) 1 0c p ,dA 0c f) dA • • + def) . dA dxn dXn T 0a1 dx„ + da2 dxn dan ' dxn ' By substituting in the functional determinant V + jf . A .... A ~ dx dx1 dxn ’ 180 Proceedings of Royal Society of Edinburgh. [sess. the equivalents here given for df df_ df | , dx ’ dxf ' ’ ‘ ’ dxn ’ there is obtained a determinant which is expressible as the sum of n+ 1 determinants, all of which vanish except the first. We thus arrive at # ; = a > fz = a2 > fd = a3 » • • • > fn = an the function f is changed into \, then exactly as before it can be shown that y +d-i . dA . . . . A = y +dl . dh . Ms A ^ ~ dx dxx dxn ^ ~ dx dxx dx2 dxn ' More questionable is the logic of his second step, which is to the effect that from this and the previous result it follows that y + A §/» — y + A df dfn ^ ~ dx dxx dxn 1 — dx dxx dx2 dxn His third step is simply the assertion that by proceeding in this way we may prove generally that if by use of the equations f=a, /i = ai5 • • • • = + a fi becomes changed into then ■y _j_ df df cf n __ ^ ^ dcj> dcf>i dn . ^ “ dx dxx dxn J “ dx dx1 dxn ’ and the matter is concluded with the further assertion that if in the elements of the second determinant there be substituted for a , cq , . . . , an the functions which they represent, that determin- ant will be identically equal to the other. Considerable space is next given (§ 15) to the discussion of the case where the number of variables, x , xx , ... , xn+m which the 1901-2.] Dr Muir on the Theory of Jacobians. 181 functions involve is m greater than the number of functions. First it is noted that if the functions be not mutually independent, they are not independent with respect to any n + 1 of the variables, and therefore each functional determinant formed with respect to n + 1 of the n + m + 1 variables must vanish. Then the converse proposition is taken up, viz., that if all these determinants vanish, the functions are not independent. The method of proof is that known as mathematical induction, that is to say, the assumption being made that the proposition holds for n functions f, flt ... fn_x it is shown to hold for n-+ 1. Clearly we may start by viewing f,flt ... , fn-i as being independent, for if they be not, there is nothing to prove; and this being the case, the various determinants of these functions with respect to n of the variables cannot vanish. Denoting the first of the said determinants, viz., 'V + xL . A J — dx dx1 Wn- 1 dxn_l by B, and choosing from the given vanishing determinants of the (n + l)th order those having n of their variables the same as those of B, viz., the m + 1 determinants, 'V 4- ^ .dA . . ¥n 2 — dx dx± dxn_ i * 0 V yJ+%- .A . . dx1 bjn- 1 dxn__1 . dxn+ 1 yj±if ■ dx . A . . dx1 'b.tn— l dxn_ i . 8A ^*®n+n i we see that from a previous proposition these are respectively equal to where the operation indicated within brackets is meant to be performed on fn as expressed in terms of />/ i , fn- \ j •X'n ) 'Ayn+ 1 5 • ) •/Ln+rn • As B does not vanish, it follows from this that = 0, ¥n.\_ ox. = 0, • • 182 Proceedings of Royal Society of Edinburgh. [sess. and consequently that/,, involves only/, j\ , ... , fn_x ; that is to say, that /, f , ... , fn are not independent. This result being obtained, it only needs to be noted that the proposition being manifestly true in the case where the number of functions is one , must be true generally. As an addendum, it is noted that since the vanishing of the m + 1 determinants 'V + ^ .... ^n~i d/w ^ ~ dx dxx dxn_1 dxn ’ 'V + *!L . .... d/tt-i s/n ^ “ dx dx1 0cc„_1 0£n+1 ’ y + ^ .... e/»-i . ^ ~dx ' 0a5x 0Bn_1 * 0awm when the determinant, B, and the n 2 elements common to all these does not vanish, implies that/,, is a function of /, f\ , . . . , /w_x : and since this mutual dependence of /, /x , ... fn_l , /„ implies the vanishing of all the functional determinants formed with respect to any n + 1 of the n + m + 1 independent variables, we are led to the conclusion that, provided B does not vanish, the vanishing of all these functional determinants of which the number is (?z + m+ l)(n + m) • • • (m+\) _ (n + m + \)(n + m) • • • (n+1) 1-2-3 (» + l) 01 1-2-3 • • • (m+1) is a consequence of the vanishing of a certain m + 1 of them. In order that the connection between the members of this group of functional determinants formed from the differential- quotients of /, f\ , . . . , fn with respect to any n + 1 of the variables x , xx , ... , xn+m may he better looked into, several identities regarding square arrays of functional determinants are next given (§ 16). Taking in addition to /, f , ... , fn the m arbitrary functions fn+l 5 / n+2 j • • • J f n+m of the same n + m+1 variables, and denoting the determinant y\±dl.dA ^ dx dxx bfn-l_ bxn_i 0/„ dXy by bk 1931-2.] Dr Muir on the Theory of Jacobians. 183 where i, k may each have the values 0, 1, 2, . . . , ra, we see that from a previous result we have B* Afn- if fn+i within the brackets involves /, f\ , ... , fn_x in place of x , x1} ... , xn_v From this it immediately follows that jJ±bb[ .... C = »Y + 1 .... ^«~1 . ” J—-L a)/V» fW S')* ‘ - v/i+1 2 From equating cofactors of tyn m bfn+j ...... dfn+m dx 0aq dxm ’ Jacobi proceeds to equate cofactors of ¥n+ 1 ty n+m dx in the same fundamental identity, the resulting theorem now being 2 ± »aa ■■■■ /ca = ( - • lr and then he adds, 3/n dxn+m ’ u Eodem modo obtinetur generaliter [m- 1) = ± ^ ± 0a*., ^4 186 Proceedings of Royal Society of Edinburgh. [sess. qua in formula signo ± substituendum est aut ( - l)n(mrfl) aut ( - l)m(n+1) prout i par aut impar est.” * The second derived identity, — that is to say, the case of the final general identity where i = 1, — Jacobi proceeds to utilise for the purpose of proving his proposition regarding the effect of the vanishing of certain m+1 functional determinants. The path which he follows to reach his result is not a little surprising. Instead of saying that the vanishing of b, blt b2, ... , bm, — for these are the m + 1 determinants in question,— entails the vanishing of the left-hand side of the identity, viz., •• • p (m-i) rii ? * The fact that these identities can be derived in the way here indicated from another which the preceding footnote has shown to be true, not merely of functional determinants but of determinants in general, is convincing proof that they also ( i.e.} the derived identities) are not restricted to any special form of determinant. Using the fundamental identity as enunciated in the footnote, and taking the special case of it where n~ 4 and m = 2 , and where therefore the given determinant may be written | axb2c3d4e5 f3g7 ( , we have | ax b2 c3 d4 e5 | | ax b2 c3 d4e6 \ | ax b2 c3 d4 e7 \ I ai h c3 d4f5 | | axb2 c3 d4f6 | \ axb2 c3 d4f7 | ! ai c3 d4 g$ | I a4 b2 c3 d4 g6 | | ax b.2 c3 d4 g7 | \ a4b2c3d4\ . | ax b2 c3 d4 e5f3 g7 | , How in each of the determinants forming the first row on the left here, e occurs as an element, in the second row f2 similarly occurs, and in the third row g3 , while on the right these only occur in | a^b^d^^^ | . Consequently, equating cofactors of e4f2 g3 we have | a2 b‘3 C4 ^5 l I a2 b$ C4 d6 | - I ax b3 c4 d5 I - ( a4 b3 c4 d6 [ [ ai b2 c4 d5 | 1 ct4 b2 c4 d3 | which when put in the form I a5 b2 C3 d4 I I a6 b2 c3 d4 | | cl4 b5 c3 d4 I | &6 c3 d4 | | a4 b2 c5 d4 | | ax b2 c6 d7 \ ) b3 c4 d7 | I ai b-3 c4 d7 | I ttx b2 c4 d7 | | a7 b2 c3 d4 1 | ax b2 c3 d4 1 | ax b2 c7 d4 | 2 | axb2c3d4 | . 1 a4b5c3d7 | | axb2c3d4 | . | a4b5c6d7 | is a case of the first derived theorem. The original theorem, it should be noted, is true for all values of n and m ; the derived holds only when m he 6, b i> 6„in 2±W r ... C1’ by*.X points out that as none of the (T s involves f n the same is true of the A’s. On the other hand, the b' s do involve fn , but only one of them, viz., bk , involves the differential-quotients of fn with respect to xn+7c , this differential-quotient being in fact the last element of all and having B for its cofact n\ In this way it appears that on the left-hand side of the identity the cofactor of — dx is Ak-B . If in the same manner we take ^ + n+k 0/ dx„, dx, Vi denote * the cofactor of dx*. m+1 dfn_ dxn. and in it by pk , it must follow that on but while cx occurs in each element of the first row on the left, and d2 similarly in the second row, e3 does not so occur in the third, and consequently the cofactor of cxd2e3 on the left takes a different form from that given by Jacobi. The first derived theorem in its general form may be enunciated as follows : — If there be two determinants D and A of the nth order such that the last n - m columns of D are the same as the first n - m columns of A, and if there be formed a square array of new determinants by supplanting each of the first m columns of D by each of the last m columns of A, the determinant of this square array of the n\th order is equal to (_l)m(w+l) Dm— iA . To illustrate the second derived theorem we may equate cofactors of fxg2 where we formerly equated cofactors of e-J^g^ the result clearly being | ax b2 c3 d4 e5 | | a2 b3 c4 d5 | 1 ax b3 c4 d5 | | ax b.2c3 dA e6 ( I a2 b3 c4 de | I h c4 d6 1 | ax b2 c3 cl4 e7 | | a2 b3 c4 ^7 I | ax b3 c4 d7 | - I (hb2c3d4 |2 . | a3b4cyd3e7 1„ The next of the series would be got by equating cofactors of gx. * This is not the same as putting, with Jacobi, V_L§^_ 3/l dU _ §/» dfy dfn dxm ■ dxm+x ■ ‘ ' dxn+m ^dxn + ^1dxn+1+ * * ' + dxn+m for the determinant on the left being of the (%+l)th order there should be n + 1 terms on the right instead of m + 1. 188 Proceedings of Royal Society of Edinburgh. [sess. the right-hand side the cofactor of — — is ( - l),K'S)B/a.. . bxn+k The connection between the A’s and the fs is thus k = so that A. b + + . + hmb,m = ( — ])?».(»+i)Bm 1(/a& + /*1&1+ . . . + f^mbm). The left-hand member here, however, being equal to the left-hand member of the identity with which we started, it follows that the two right-hand members must also be equal, and therefore that fib + + . . . + pmbm = B v ±df ^ ill dA dxm bfn dx„ Of course this shows, exactly as the original identity did, that if b = bl= . . . . =bn = 0 and B ^ 0 then 'V + ^ ?fi .... _ q ^ ~ cxm dxm+1 bxm+n — that is to say, the functional determinant of /, f1 , ... , fn with respect to another set of n+ 1 variables vanishes also. Jacobi, however, does not at once say this, but drawing his reader’s attention to the fact that the new set of variables contains n-m taken from x , aq , ... , xn_x and m + 1 others, viz., xn , xn+1 , , xn+m ) he affirms that the identity reached shows how the functional determinant of /, f , . . . , fn with respect to any set of n + 1 variables is expressible in terms of the m + 1 functional deter- minants whose variables are **T ? • . . . , xn_ -1 : i '^n 5 3 , xl , . . . . , xn_ -1 ’ X , £^1 , . . . . , xn. -1 » 'A • His words are — “Unde formula docet quomodo e functionum /, f , . . . , fn Determinantibus bk per idoneos factores multiplicatis et additis proveniat earundem functionum Determinans quarum- cunque variabilium respectu formatum atque per ipsum B multi plicatum. Hinc bene patet, quod § pr. demonstravi, 1901-2. j Dr Muir on the Theory of Jacobians. 189 quomodo omnibus bk evanescentibus neque ipso B evanescente, simul cuncta ilia Determinantia evanescant.” * Continuing the work of deduction, Jacobi lastly equates the 'df cofactors of — in the two members of the identity OX yb + y1b1+ • • • +fimbr B-2±ai +dJL . dA dfn dxm dx m+ 1 02L noting that this differential quotient does not occur at all on the right-hand side, nor in the /x’s on the left-hand side, but in bk occurs with the cofactor df df df2 0/w_1 -2 dxn+k dxx dx. dx, n- 1 The result is the proposition! — “Sit jxk functionum /, f , . Determinante df df dxm. dxm , fn_ l Determinans quod in V ± f— ¥n dx „ per ¥n dx„ multiplicatur, ubi m < n , erit + /h 2j ± + /V_2, ± The case where m — n. is specially noted ¥ dfdf .. Ai dxn dx1 dx2 df df df 2 . . A=1 dxn+1 dxY dx 2 ¥ df\ df 2 dxn+m dx1 dx2 " a~=° -i * Of course this theorem also is not limited to determinants having differential-quotients for their elements. The general enunciation may be put as follows : — If m determinants of the nth order all have the same n - 1 columns in common , and vanish independently, then every determinant of the nth order whose n columns are chosen from the m + n - 1 different columns must vanish likewise. (Vide Proc. Roy. Soc. Edin. , xviii. pp. 73-82.) f This proposition, and that from which it is derived, are again propositions which hold regarding determinants in general, the class to which they belong being that which concerns aggregates of products of pairs of determinants, — a class, the first instances of which occur in Bezout (1779). In connection with Jacobi’s remark regarding the case where m — n, it is worth while to note Sylvester’s enunciation in Philos. Magazine (1839), xvi. p. 42. 190 Proceedings of Royal Society of Edinburgh. [sess. Leaving now these general theorems which involve two suffixes m and n , and which concern groups of functional determinants, Jacobi returns (§ 17) to the consideration of the properties of a single functional determinant, the specialisation being made, not by giving a particular value to m the second suffix introduced, but by leaving it unrestricted, and putting the original n — 1. In the theorem a) 2±M. B then stands for ±¥ dx dx1 djf dxn ra+l b{m = B" m Y. and if for "V +— . . Making in dx k ^ ~ dx dxk+1 this the further specialisation, /= 0, so that x has to be considered as a function oi x1, x2, ... , Xm_x we have + *+i df dx dx dx; &+i and consequently i.e. ^n), and the difference between any one of the old forms and the corresponding new by F with the appropriate suffix. He thus has n + 1 equations 0 = F = f - (•£ 3 A J % . . . , aj„) , 0 = = A - a (A ^i, ^2, . • • j xn) 3 0 = f2 = /2 - A (A A, aJ2» • . . , ®n) , 0 = Fn = fn ~ /„(/,/!, • ' ‘ J A-i > A, connecting 2n + 2 variables X j X j . . . » xn , j , y ] ? • • • , A now viewed as independent. By a previous theorem there is thus obtained y +dJ .dh ... dl* y dA ( pn+i dx ' * * ' dxn Z-i — ' 3F 9Fj 3Fj * ±Tf ' W ^ Vn PROC. ROY. SOC. EDIN. — VOL. XXIV. 13 194 Proceedings of Royal Society of Edinburgh. [sess. Now the numerator here reduces to one term, viz., 0F df\ djf dx * dx1 dxn 5 and the denominator in similar fashion to 0F 0Fj dff ¥ ’ ¥ l * ’ ’ ’ ¥n ’ as Jacobi might briefly have justified by a reference to Prop. Ill of § 5 of his “ De formatione et proprietatibus Determinantium ”• — this proposition being that which concerns a determinant whose elements on one side of the ‘diagonal,’ as it afterwards came to be called, all vanish. Further, the factors of the reduced numerator are equal to and those of the reduced denominator to 1,1,1,-. Our final result thus is y + ¥ m ¥i J—J ~dx dx1 where the brackets on the right are meant as a reminder that fi is there expressed in terms of/,/x, ... ,/*_!, xir xi+1, . . . , xn . The last section (§ 19) is occupied with a theorem of the Integral Calculus, viz., + dl.sA dx dx1 • dx • dx1 • • - • dxn t it being noted that the cases where the number of variables to be changed are 2 and 3 had been already dealt with by Euler and 1901-2.] Dr Muir on the Theory of Jacobians. 195 Lagrange.* Then come the final words, “Et haec formula egregie analogiam differ entialis et Determinantis functionalis declarat,” — a not inappropriate ending in view of the author’s attitude through- out the memoir. * The memoirs referred to seem to be — Euler, L. — De formulis integralibus duplicatis. . . . Nov. Comm. Acad.. Petrop. (1769), xiv. i. pp. 72-103. Lagrange, J. L. — Sur l’attraction des spheroides elliptiques. Nouv. Mem. Acad. . . . Berlin (1773), pp. 121-148 ; or CEuvres, iii. pp. 619-658. Equally worthy of note is one of date December 1839, but not published till 1840, viz. — Catalan, E. — Sur la transformation des variables dans les integrales multiples. Mem. couronnes par V Acad, de Bruxelles , xiv. (2) pp.. 1-47. ( Issued separately June 9, 1902.) 196 Proceedings of Royal Society of Edinburgh. [sess. Functional Inertia, a Property of Protoplasm. By David Fraser Harris, M.D., B.Sc. (Lond.), Lecturer on Physi- ology and Histology, University of St Andrews., (Read February 3, 1902.) [Abstract.) 1 Dead ’ matter has two forms of inertia — that of rest (mass), and that of motion (momentum) — and it seems to me that living matter possesses a property, not hitherto recognised as such, which might he called functional inertia, or metabolic inertia of proto- plasm whether animal or vegetable. It is owing to the inertia of matter at rest that the heavy gate, swung even on almost friction- less hinges, cannot he instantaneously set in motion ; and when it has been set swinging, it is owing to its inertia of motion (momentum) that it continues to swing for some time after we have ceased to push it. By functional inertia of protoplasm I mean the power or property of protoplasm to remain in the meta- bolic status quo ante ; if resting — anabolising — to remain doing so, even after the reception of a stimulus tending to bring about the opposite metabolic state, this might be called anabolic inertia ; if active — katabolising — to remain so even after the reception of a stimulus tending to induce anabolism, or after the cessation of the stimulus that elicited the activity — katabolic inertia or functional momentum. Functional inertia is the power or property which bioplasm has of maintaining its functional status quo for a longer or shorter time according to the function considered ; it is that power of continuing to exhibit the particular phenomena it has been exhibiting even after the death of the organism of which it is a constituent. The inertia of livingness expresses itself under several more or less distinct modes or categories — viz., ‘latent period ’ if we study time-relation to stimulus, the bio-chemical, or pre-eminently the metabolic, refractory period (physiological insusceptibility) if we study affectability (‘irritability’), rhythm if we study alternation of metabolic phase, and finally the psychical if we study conscious correlates. As examples of katabolic inertia 1901-2.] Functional Inertia, a Property of Protoplasm. 197 under category of latent period, we have the latent period of stimulation of the cardiac vagus, the relatively long time (18 seconds) before stimulation of splanchnic nerve is followed by inhibi- tion of intestinal peristalsis, the time during which the frog-heart will beat at accelerated rate (30 seconds) after the withdrawal of accelerating stimulus. Thus, as exhibited under the time category, functional inertia is that property of protoplasm in virtue of which it does not respond to a stimulus — the duration of the ‘ latency ’ or non-response being longer or shorter according to the function and kind of protoplasm considered, being in some cases of vegetable protoplasm an hour or more. It is thus the opposite property to ‘irritability ’ or affectability. In fact, functional inertia as non- response is the property underlying physiological insusceptibility it is thus the functional counterpart of affectability. It is by reason of the property of irritability that living matter responds to stimuli, but it is in virtue of its other property of functional inertia that it does not do so instantaneously. Just as non-living matter cannot be instantaneously caused to change its state, neither can bioplasm ; by the inertia of its mass or its motion, c inert ’ matter tends to remain in the status quo ante ; by the inertia of its livingness — in relative rest or in relative activity — bioplasm tends to remain in its metabolic status quo ante . As general examples of katabolic inertia, there is that large class embracing all cases of local life of organs, tissues or cells after somatic death, the post-mortem expression of the functional inertia of katabolism, e.g., the muscle and nerve of the familiar nerve-muscle preparation which still ‘ act ’ though removed from their nutrient lymph, the excised (isolated) bloodless frog-heart beating on a glass plate for hours after the death of the animal, the excised medullated nerves still giving evidence of conductivity for many hours after isolation, the vivisected non-nucleated portion of Lacry maria olor and other Protista,* the ciliated epithelium from frog living for days as an isolated patch, and the cilium detached from the cell exhibiting movements ‘till it perishes.’ In this class I do not include the cases of organs surviving by reason of perfused defibrinated blood, for here their metabolism is being constantly supported by nourishment applied under artificial conditions, and * Verworn, General Physiology, p. 570. London, 1899. 198 Proceedings of Royal Society of Edinburgh. [sess. as long as it is applied, so long will the organ live ; indefinitely, in fact, if the proper conditions he observed for a sufficiently long time. But as examples of katabolic inertia I would include all cases of organs isolated from all nervous and vascular connec- tions, surviving by reason of the perfusion of salt solution. Here no ‘food’ is introduced, the HaCl is powerless to prolong life indefinitely — no longer than the time when the katabolic inertia of the protoplasm shall have been spent. Of course organs and tissues still in situ in the dead body can exhibit their katabolic inertia ; examples are numerous — e.g ., the respiratory centres emitting impulses even after the animal’s evisceration (marmot, Marckwald), the post-mortem continuation of intestinal peristalsis, post-mortem parturition (uterine katabolic inertia), the nerves remaining excitable three to four days after severance from the central nervous system, the growth of hairs, and all cases of local cell-life after death — spermatozoa in the vesiculse seminales, cilia in trachea, etc., and amoeboid leucocytes generally. All cases of post-mortem bio-chemical change are illustrations of katabolic inertia, e.g., the excised liver continuing to convert glycogen to dextrose, the isolated and bloodless muscle expiring C02 even in N or H, the tissues continuing to produce heat, to reduce deoxidisable material brought into contact with them, and the post-mortem formation of enzymes generally, the salivary glands secreting on nerve-stimulation in decapitated animals. By its katabolic inertia an organism ‘ lives ’ until it dies when deprived of oxygen, water or food. Examples from Pathology can be given : the ‘ mo ibid ’ tendencies to deposit fat or to produce sugar (Diabetes) in spite of all drugs, etc., are due to protoplasmic inertia ; and Professor Adami finds the principle of this inertia under- lying the perverted ‘habit of growth’ in cancer cells.* Examples of Anabolic Inertia are as follows : — The latent period of contraction of muscle ; post-stimulant latent period after splanch- nic inhibition of intestine, and after cardio-inhibition ; latent period of secreto-motor effects, of photo-mechanical effects on cilia. Under physiological insusceptibility we have such cases as the need for a series of repeated stimuli before certain reflexes are produced, the lengthening of the reaction-time (‘ personal equation ’) in persons of * Brit. Med. Jour., 16th March 1901, pp. 624, 626. 1901-2.] Functional Inertia, a Property of Protoplasm. 199 sluggish temperament, the insusceptibility that exists through a whole series of increments of stimulus well known in the Weber- Fechner law — all these are examples of anabolic inertia. The recurrence of the positive and negative after-images is an example of retino-cerebral functional inertia; the positive (katabolic) and the negative (anabolic) with their respective conscious correlates; the analogy being the oscillations of the jelly. Rhythm of discharge elicited by a single stimulus is due to functional inertia of proto- plasm, e.g ., in Wundt’s tetanus, Ritter’s tetanus, strychnine tetanus ; also rythmic discharges from spinal cord cells when the rate of stimulation is higher than the rate of discharge. Anabolic inertia is the property underlying insusceptibility to drugs. In the more particularly psychic sphere, the same double-phased inertias are seen. Katabolic psychic inertia or mental momentum is seen in the hypnotic state, where the suggestion is rigidly followed out, and in the ‘dogged’ temperament; whereas there is very little of it in the child-mind or in a volatile nature. Anabolic psychic inertia is the principle underlying bigotry, dislike to change, obstinacy of the uncultured and ‘ fixed delusion ’ of the lunatic ; and in the race, the lingering of superstitions, and the power of inheritance (persistence of mental type). ( Issued separately June 9. 1902.) 200 Proceedings of Royal Society of Edinburgh. [sess. On the Functional Inertia of Plant Protoplasm. By R. A. Robertson, M.A., B.Sc., Lecturer on Botany, University of St Andrews. Communicated by Dr Fraser Harris. (Read February 3, 1902.) {Abstract.) The response by a plant protoplast to an inducing or to an inhibitory stimulus is preceded by a period of non-responsiveness, and the withdrawal of the stimulus is succeeded by a period of continued response or inhibition as the case may be. During these intervals, the familiar latent period and period of after- effect, or temps de memoir of Massart, the protoplast manifests the property of functional inertia (Harris, Brit. Med. Assoc., 1900), and in the case of the inducing stimulus the two phases follow in the order of (1) the Anabolic and (2) the Katabolic, while in inhibition the order is reversed. This property is ex- hibited by the protoplasm of growing and of adult organs, as well as by that of isolated and excised organs ; further, it varies in amount and appears under different aspects. Taking stimuli and irritabilities in the most general sense (Pfeffer, Physiology , p. 11), we have to regard the ordinary growth of a plant as a continuous manifestation of irritability in response to a particular combination of external stimuli, the so-called tonic conditions. These external influences are effective only when they operate within definite limits of intensity — the maxima and minima. The living molecules have their own rate of vibration, and no amount of pushing, i.e., stimulation, will induce them to swing faster or farther : they thus exhibit a refractory period, and this is one aspect of their functional inertia. A change in one or more of the external conditions induces a growth- variation, but not all at once ; there is a period of non-responsiveness, of accommodation, or of anabolic inertia, and the duration of this period varies with the stimulus, e.g., the time-value of the ana- bolic inertia is less for a temperature change than for one of the oxygen pressure or food supply (Pfeffer, Physiology , p. 512). 1901-2.] The Functional Inertia of Plant Protoplasm. 201 The post-stimulant continuance of periodicity of growth or of movement — the opening and closing of flowers, movements of leaves — whether diurnal or seasonal, is to be credited to this property of functional inertia. Thus the periodicity of growth induced by the alternation of day and night is retained for a time in continuous darkness, and seasonal periodicity is exhibited by deciduous trees when removed to countries where the vegetation is evergreen, while in cases of experimentally induced periodicity (Darwin and Pertz, Annals of Botany , vol. vi. p. 245), the periodicity continues for a time after the stimuli are withdrawn. All these are examples of the katabolic phase of the inertia. For particular cases the time-value of the inertia varies : thus the diurnal periodicity is lost after two days by some plants, while it is retained for as much as two weeks by others. The latter have relatively more inertia than the former. Seasonal periodicity exhibits similar variations, and in the practical horticultural operations of f forcing ’ and ‘ cold storing,’ we see the property under other aspects. In heterauxesis, induced by unilateral stimu- lation, by gravity, light or heat, there is a latent period before the geotropic, heliotropic or thermotropic curvatures begin to be mani- fested. This latent period — that of anabolic inertia — may be of any length from a few minutes for geotropism to a few hours for thermotropism. In Photomechanical induction (Wortmann, Bot. Zeit ., 1883-1885 ; also Lewes, Annals of Botany , vol. xii. p. 420), functional inertia takes the form of a physiological insus- ceptibility, in that a definite time-exposure to light is necessary to produce the maximum effect, and no additional exposure will cause the molecules to swing sooner, farther or faster. To wound-stimuli, growing as well as adult organs exhibit non- responsiveness in varying amount. Thus traumatropic curvature in growing roots is only manifested after a latent period — of ana- bolic inertia — of an hour or so, and this period may be artificially lengthened to as much as eight days (Spalding, Annals of Botany , vol. xiii. p. 423). Here, as in other cases, where a sense organ has been demonstrated (Czapek’s, Darwin’s, and other experi- ments), transmission-time has to be deducted from the latent period to get the duration of the anabolic phase. Similar evidence is adducible from the researches of Townsend 202 Proceedings of Royal Society of Edinburgh. [sess. (Annals of Botany , vol. xi. p. 515) on the correlation of growing organs where the latent period is twenty-four hours ; for cases of wound fever in adult organs (Richards, Annals of Botany, x. 534, and xi. 29), for healing reactions in adult leaves (Blackman and Matthaei, Annals of Botany, vol. xv. p. 533), where the latent period is very prolonged ; and if fertilisation be regarded as of the nature of a traumatic as well as a chemico-vital stimulus, by those of Farmer and Williams. The latter show that a membrane forms around the egg ten minutes — phase of anabolic inertia — after entrance of the spermatozoid (Phil. Trans. Roy. Soc., 1898, p. 625). In other plants the anabolic phase is more prolonged. In contact-stimulation either of growing organs, where the re- sponse is a growth reaction, as in tendril climbers, or in adult organs, where it is a movement of variation depending to a certain extent on turgor changes, as in leaves of sensitive plants and of insectivorous plants, functional inertia finds expression in the phenomena of latent periods and of physiological insusceptibilities. Here the latent period — period of anabolic inertia— may vary from a few seconds, as in leaves of Dionsea, Mimosa and tendrils of Passiflora, to a few hours, as in the tendrils of Ampelidese. In the same genus, and to the same stimulus, the anabolic phase varies : thus in different species of Drosera, according to Darwin (Insectivorous Plants), it varies from ten seconds to twenty minutes. Again, physiological insusceptibility is seen in relation to the character of the stimulus. For tendrils, according to Pfeffer, the stimulus must be contact with a rough surface, i.e a series of simultaneous impacts at discrete points of the sensitive organ ; for Dionaea, two successive delicate impacts ; for Drosera, a prolonged series of successive impacts amounting to a pressure. Matters are complicated in these cases by the existence of sense organs, as in Mimosa, e.g., where, according to Ewart (Annals of Botany, vol. xi. p. 448), the leaf has special sense organs for light intensity and a different set for light direction. The list of examples can be indefinitely extended in the case of mature organs. Thus the researches of Ewart (Jour. Linn. Soc., vol. xxxi. p. 364) on assimilatory inhibition furnish a mine of illustrations of inertia in the two phases — (1) Katabolic and (2) Anabolic ; those of Arber on the assimilation of Halophytes (Annals of Botany, 1901-2.] The Functional Inertia of Plant Protoplasm. 203 xv. pp. 39, 669) supply examples of physiological insusceptibility. On swarming and other phenomena of individual cells, the experiments of Darwin, Detmer, Ewart, Farmer, Pfeffer, Mottier, and Vines, and many others may be referred to. Interesting cases of physiological insusceptibility are seen in nectaries (Pfeffer, Physiology , p. 286) in Pfeffer’s Chemotactic application of the Weber-Fechner law, and in a special set of phenomena in the life of parasitic fungi illustrated in the works of Eriksson, Marshall Ward and others — all tending to prove the universality of this property in plants. Isolated Organs. As the wheel, in virtue of its inertia of motion, continues to rotate for a time after the driving gear is slipped, so isolated organs or organoids may for a time manifest functional activity. This is an expression of their katabolic inertia. Isolated chloro- plasts continue to assimilate for five hours (Ewart, Jour. Linn. Soc., vol. xxxi. p. 424), isolated scutellar epithelium secretes enzymes, corrodes and dissolves starch (Brown and Morris, Jour. Chem. Soc. Trans., vol. lvii. p. 494), isolated endosperm of Bicinus grows for six weeks and carries on metabolic changes (Van Tieghem, Ann. d. Sc. Nat., 1876, p. 83), the nuclei of the staminal hairs of Tradescantia carry on Karyokinetic division after the cell protoplasm has been killed (Demoor in Pfeffer, Phys., p. 52), non-nucleated fragments of Zoo-spores swarm and non- nucleated cytoplasm streams (Pfeffer, l.c., p. 51), translocation goes on in the ears of cereals (Pfeffer, l.c., p. 585), fruits ripen and oak galls continue their metabolism (MacDougal, Phys., p. 64), after being removed from the plant. In the acquirement of new characters by living matter, it is suggested that functional inertia is a factor of importance. Some such fundamental property appears necessary to secure the sum- mation of effects and the more or less indelible stamping of these on the protoplasm. In this connection evidence is supplied with reference to the acquirement of polarity by Detmer ( Pract . Phys., p. 507), and periodicity by Darwin and Pertz ( Annals of Botany, l.c.). The following also bear on this point : Dubourg’s experi- ments on yeast ( Compt . Rend., vol. cxxviii., 1899, p. 440), the 204 Proceedings of Royal Society of Edinburgh. [sess. acclimatization of plants, artificial production of new varieties of bacteria by cultures and variation of external conditions, Stahl’s experiments ( Bot . Zeit ., 1884) on plasmodia, and De Tries’ researches on biastrepsis of Dipsacus ( Annals of Botany , voi. xiii. p. 395). In all these is noticeable the element of time — accom- modation to the new conditions — preceding the ultimate response ; this represents the time-value of the inertia in respect of the particular stimulus. There still remain some interesting cases where the inertia amounts to a physiological insusceptibility, and where two or more stimuli are required to elicit a manifestation of irritability. In this category come the cases of latency described by Pierce {Bot. Central ., Bd. 89, 1902, p. 36) and by Terras {Trans. Bot. Soc. Edin ., xxi., 1900, p. 318). These lead up to the case of the resting seed, which may be regarded as the most extreme in this connection. Here at least three stimuli all of a tonic character are required to induce a manifestation of irritability. The functional inertia of the seed may be said to be infinite in- respect of any one of these alone, but relatively small when they all act together. Hot only must these stimuli be applied simul- taneously, but they must act within definite limits of intensity. Any one alone is insufficient to elicit a response, whether applied within the limits or far in excess of them. On this point the experiments of Brown and Escombe {Roy. Soc. Proc., vol. lxxii. p. 161), and of Thiselton-Dyer {Annals of Botany , vol. xiii. p. 599), have a distinct bearing. Conclusions. Plant protoplasm, like animal protoplasm, possesses the property of functional inertia. This property finds expression in the phenomena underlying the latent period, and temps de memoir in the existence of stimulatory limits, in the acquirement, maintenance and post-stimulant continuance of periodicity, whether diurnal, seasonal or other ; in the phenomena of polarity ; in the possession of a limited power of independent activity exhibited by isolated organs, protoplasm, and cell organoids. On it as one factor depend the possibility of educating protoplasm and the acquirement of new characters as seen in nature, and also the results of experiment. [Issued separately June 9, 1902.) 1901-2.] Lord Kelvin on Molecular Dynamics of a Crystal. 205 Molecular Dynamics of a Crystal. By Lord Kelvin. (With Seven Diagrams.) § 1. The object of this communication is to partially realise the hope expressed at the end of my paper of July 1 and July 15, 1889, on the “Molecular Constitution of Matter”:* — “The mathe- matical investigation must be deferred for a future communication, when I hope to give it with some further * developments. ” The italics are of present date. Following the ideas and principles suggested in §§ 14-20 of that paper (referred to henceforth for brevity as M. C. M.), let us first find the work required to separate all the atoms of a homogeneous assemblage of a great number n of molecules to infinite distances from one another. Each molecule may he a single atom, or it may he a group of i atoms (similar to one another or dissimilar, as the case may be) which makes the whole assemblage a group of i assemblages, each of n single atoms. § 2. Remove now one molecule from its place in the assemblage to an infinite distance, keeping unchanged the configuration of its constituent atoms, and keeping unmoved every atom remaining in the assemblage. Let W be the work required to do so. This is the same for all the molecules within the assemblage, except the negligible number of those (§ 30 below) which are within influential distance of the surface. Hence -JwW is the total work required to separate all the n molecules of the assemblage to infinite distances from one another. Add to this n times the work required to separate the i atoms of one of the molecules to infinite distances from one another, and we have the whole work required to separate all the in atoms of the given assemblage. Another procedure, sometimes more convenient, is as follows : — Remove any one atom from the assemblage, keeping all the others unmoved. Let w he the work required to do so, and let 2? o denote the sum of the amounts of work required to do this for * Proc. Roy. Soc. Edin ., and vol. iii. of Mathematical and Physical Papers , art. xcvii. 206 Proceedings of Royal Society of Edinburgh. [sess. every atom separately of the whole assemblage. The total amount of work required to separate all the atoms to infinite distances from one another is This (not subject to any limitation such as that stated for the former procedure) is rigorously true for any assemblage whatever of any number of atoms, small or large. It is, in fact, the well-known theorem of potential energy in the dynamics of a system of mutually attracting or repelling particles ; and from it we easily demonstrate the item ±nW in the former procedure. § 3. In the present communication we shall consider only atoms of identical quality, and only two kinds of assemblage. I. A homogeneous assemblage of N single atoms, in which the twelve nearest neighbours of each atom are equidistant from it. This, for brevity, I call an equilateral assemblage. It is fully described in M. C. M., §§ 46, 50 . . . 57. II. Two simple homogeneous assemblages of -|-N single atoms, placed together so that one atom of each assemblage is at the centre of a quartet of nearest neighbours of the others. Tor assemblage II., as well as for assemblage I., w is the same for all the atoms, except the negligible number of those within influential distance of the boundary. Neglecting these, we there- fore have = N?tf, and therefore the whole work required to separate all the atoms to infinite distances is — JN w (1). § 4. Let <£(D) he the work required to increase the distance between two atoms from D to oo ; and let /(D) he the attraction between them at distance D. We have /(D)=-A^(D) (2). For either assemblage I. or assemblage II. we have w = (D) + (D') + <£(D") + etc (3); where D, D', D", etc., denote the distances from any one atom of all neighbours, including the farthest in the assemblage, which exercise any force upon it. § 5. To find as many as we desire of these distances for assemblage I. look at figs. 1 and 2. Fig. 1 shows an atom A, and neighbours in one plane in circles of nearest, next-nearest, next- 1901-2.] Lord Kelvin on Molecular Dynamics of a Crystal. 207 next-nearest, etc. Fig. 2 shows an equilateral triangle of three nearest neighbours, and concentric circles of neighbours in the same plane round it. The circles corresponding to r4 and r8 of § 7 below, are not drawn in fig. 2. In all that follows the side of each of the equilateral triangles is denoted by A. § 6. All the neighbours in assemblage I. are found by aid of the Fig. 1. (a) The atoms of the net shown in fig 1. The plane of this- net we shall call our “middle plane.” Let lines be drawn per- pendicular to it through the atom A, and the points marked.^, c, to guide the placing of nets of atoms in parallel planes on its two sides. (b) Two nets of atoms at equal distances A^/-! on ^he ^wo sides of the “ middle plane.” These nets are so placed that an atom of one of them, say the near one as we look at the diagram, is in the guide line b ; and an atom of the far one is in the guide line c. 208 Proceedings of Royal Society of Edinburgh. [sess. (c) Two parallel nets of atoms at equal distances, on the two sides of the “middle plane,” so placed that an atom of the -near one is in the guide line c, and an atom of the far one is in the guide line b. (d) A third pair of parallel planes at equal distances, SX^/f, from the “middle plane,” and each of them having an atom in guide line A. (e) Successive triplets of parallel nets with their atoms cyclically arranged Abe Abe . . at greater and greater distances from A on the near side of the paper, and Acb Acb . . at greater and greater distances on the far side. § 7. Let qv q2 , q% . • • be the radii of the circles shown in fig. 1, and rv r2, r3 . . . be the radii of the circles shown in fig. 2 ; and for brevity denote X^/f by k. The distances from A of all the neighbours around it are : — In our “ middle plane ” : 6 each equal to q1 ; 6, q2 ; 6, q3 ; 12, 24; 6, ?6; ... . 1901-2.] Lord Kelvin on Molecular Dynamics of a Crystal. 209 In the two parallel nets at distances k from middle : 6 each equal to x/(k2 + ?’12); 6, Ji^ + rf); 12, J(«2 + rf); 1 2, + rf) ; 6, J^ + rf); 12, J(# + r*); 6, J^ + rf). In the two parallel nets at distances 2k from middle : the same as (B) altered by taking 2/c everywhere in place of k . In the two parallel nets at distances 3k from centre : the same as (A) altered by taking *J(9k2 + q2\ sJ(9K2 + qf), etc., in place of qv q2, etc. In nets at distances on each side greater than 3k : distances of atoms from A, found as above, according to the cycle of atomic configuration described in (e) of § 6. § 8. By geometry we find — A ; #2— V3* = 1‘732a ; q3=2\; q±= n/7a = 2,646a ; q5 = 3a : I = ViA= '577a ; r2 = 2\AA— 1 '154a ; rs=V|A = 1*527a; r4= VV-A = 2’082a ; H4). = 4ViA = 2*308A; r6= VV9A = 2'517a ; r7 = 5V|A = 2’887A. J § 9. Denoting now, for assemblage I., distances from atom A of its nearest neighbours, its next-nearests, its next-next-nearests, etc., by Dj, D2, D3, etc., and their numbers by jv j2 , jz, etc., we find by §§ 7, 8 for distances up to 2A, for use in § 12 below, Dj = A, D2 = 1-414 A, D3= 1*732 A, D4 = 2A, ii = 12>i2=6; is=18; J* = Q- § 10. Look back now to § 5, and proceed similarly in respect to assemblage II., to find distances from any atom A to a limited number of its neighbours. Consider first only the neighbours forming with A a single equilateral assemblage : we have the same set of distances' as we had in § 9. Consider next the neighbours which belong to the other equilateral assemblage. Of these, the four nearest (being the corners of a tetrahedron having A at its centre) are each at distance f-^/f-A, and these are A’s nearest neighbours of all the double assemblage II. Three of these four are situated in a net whose plane is at the distance x\/§A on one side of our “ middle plane ” through A, and having one of its atoms on either of the guide lines b or c. The distances from A of all the atoms in this net are, according to fig. 2, + V), n/(tV2 + ^2). etc (5). The remaining one of the four nearests is on a net at distance l^f- A from our “middle plane,” having one of its atoms om the PROC. ROY. SOC. EDIN. — VOL. A XIV. 14 210 Proceedings of Royal Society of Edinburgh. [sess. guide line through A. The distances from A of all the atoms in this net are, according to fig. 1, + + . . . (6). All the other atoms of the equilateral assemblage to which A does not belong lie in nets at successive distances k, 2k, 3k, etc., beyond the two nets we have already considered on the two sides of our “ middle plane ” ; the atoms of each net placed of course according to the cyclical law described in (e) of § 6. §11. Working out for the double assemblage II. for A’s nearest neighbours according to § 10, we find four nearest neighbours at equal distances = *61 3A ; twelve next-nearests at equal distances A ; and twelve next-next-nearests at equal distances x/3g1A = 1T73 A. These suffice for § 12 below. It is easy and tedious, and not at present useful, to work out for D4, D5, D6, etc. § 12. Using now §§ 9, 11 in (3) of § 4 we find, — for assemblage I., ic = 120(A) + 60(1 ’414A) + 180(1‘732A) + 60(2A) + . .! for assemblage II., f ' io = 40(*613A) + 120(A) + 120(1 -173A) + These formulas prepare us for working out in detail the practical dynamics of each assemblage, guided by the following statements taken from §§ 18, 16 of M. C. M. § 13. Every infinite homogeneous assemblage of Boscovich atoms is in equilibrium. So, therefore, is every finite homogeneous assemblage, provided that extraneous forces be applied to all within influential distance of the frontier, equal to the forces which a homogeneous continuation of the assemblage through influential distance beyond the frontier would exert on them. The investigation of these extraneous forces for any given homo- geneous assemblage of single atoms — or groups of atoms as ex- plained above (§ 1) — constitutes the Boscovich equilibrium- theory of elastic solids. It is wonderful how much towards explaining the crystallo- graphy and elasticity of solids, and the thermo-elastic properties of solids, liquids, and gases, we find; without assuming, in the Boscovichian law of force, more than one traijsition from 1901-2.] Lord Kelvin on Molecular Dynamics of a Crystal. 211 attraction to repulsion. Suppose, for instance, that the mutual force between two atoms is zero for all distances exceeding a certain distance I, which we shall call the diameter of the sphere of influence; is repulsive when the distance between them is <£; zero when the distance is = £ ; and attractive when the distance is >£ and < I. § 14. Two different examples are represented on the two curves of fig. 3, drawn arbitrarily to obtain markedly diverse conditions of equilibrium for the monatomic equilateral assemblage (I.), and also for the diatomic assemblage (II.). The abscissa ( x ) of each Fig. 3. diagram, reckoned from a zero outside the diagram on the left, represents the distance between centres of two atoms; the or- dinates ( y ) represent the work required to separate them from this distance to oo . Hence — — represents the mutual attraction at ax distance x. This we see by each curve is - go (infinite repulsion) at distance LO, which means that the atom is an ideal hard ball of diameter LO. For distances increasing from LO the force is repulsive as far as L61 in curve 1, and 1'55 in curve 2. At these distances the mutual force is zero ; and at greater distances up to L8 in curve 1, and L9 in curve 2, the force is attractive. The force is zero for all greater distances than the last mentioned in the two examples respectively. Thus, according to my old 212 Proceedings of Royal Society of Edinburgh. [sess. notation, we have £=1*61, I = T8 in curve 1 ; and £=1-55, 1 = 1*9 in curve 2. The distances for maximum attractive force (as shown by the points of inflection of the two curves) are 1’68 for curve 1, and T76 for curve 2. According to our notation of § 4 we have y = <£(D), if x = D in each curve. § 15. The two formulas (7), § 12, are represented in fig. 4 for curve 1, and in fig. 5 for curve 2; with x = \ for Ass. I., and iC='613A. for Ass. II. In each diagram the abscissa, x, is distance between nearest atoms of the assemblage. The heavy portions of the curves represent the values of w calculated from (7). The light portions of the curves, and their continuations in heavy curves, represent 4 and 12 g>(x) respectively in each diagram. The point where the light curve passes into the heavy curve in 1901-2.] Lord Kelvin on Molecular Dynamics of a Crystal. 213 each case corresponds to the least distance between neighbours at which next-nearests are beyond range of mutual force. All the diagrams here reproduced were drawn first on a large scale on squared paper for use in the calculations from (7) ; which included accurate determinations of the maximum and minimum values of w and the corresponding distances between nearest neighbours in each assemblage. The corresponding densities, given in the last column of the following table of results, are calculated by the formula J 2/A.3 for assemblage I., and 2J2JX8 for assemblage II. ; “ density ” being in each case number of atoms per cube of the unit of abscissas of the diagram. This unit is (§ 14) equal to the diameter of the atom. For simplicity we assume the atom to be an infinitely hard ball exerting (§ 13) on neighbouring atoms, not 214 Proceedings of Poyal Society of Edinburgh. [sess. in contact with it, repulsion at distance between centres less than £ and attraction at any distance between £ and I. Assemblage I. Assemblage II. Distances be- tween centres of nearest atoms for maximum and minimum values of w. Maximum and minimum values of w. Densities. Distances be- tween centres of nearest atoms for maximum and minimum values of w. Maximum and minimum values of w. | Densities. Law of Force according to Curve 1. 1*16 8 ’28 (max.) ’904 1-00 11-52 (max.) •652 1-23 1 5 ’22 (min.) •759 1-10 *76 (min.) •490 1-61 14 ’76 (max.) •338 1-61 4*92 (max.) •158 Law of Force according to Curve 2. 1-00 11’58 (max.) 1-414 1-00 12 ‘36 (max.) •652 1-07 3 ’78 (min.) 1-146 1-15 0*16 (min.) •433 1*22 10 ’44 (max.) •774 1-53 5-20 (max.) •184 1"28 9 ’36 (min.) •671 1-53 15*60 (max.) •393 § 16. To interpret these results, suppose all the atoms of the assemblage to be subjected to guidance constraining them either to the equilateral homogeneousness of assemblage I., or to the diatomic homogeneousness of assemblage II., with each atom of one constituent assemblage at the centre of an equilateral quartet of the other constituent assemblage. It is easy to construct ideally mechanism by which this may be done ; and we need not occupy our minds with it at present. It is enough to know that it can be done. If the system, subject to the prescribed constrain- ing guidance, be left to itself at any given density, the condition for equilibrium without extraneous force is that w is either a maximum or a minimum ; the equilibrium is stable when w is a maximum, unstable when a minimum. It is interesting to see the two stable equilibriums of assemblage I. according to law of force 1, and the three according to law of force 2; and the two stable equilibriums for assemblage II. with each of these laws of force. § 17. But we must not forget that it is only with the specified 1901-2.] Lord Kelvin on Molecular Dynamics of a Crystal. 215 constraining guidance (§ 16) that we are sure of these equili- briums being stable. It is quite certain, however, that without guidance the monatomic assemblage would be stable for the small density corresponding to the point m of each of the diagrams, because for infinitesimal deviations each atom experiences forces only from its twelve nearest neighbours, and these forces are each of them zero for equilibrium. It may conceivably be that each of the maximums of iv, whether for the monatomic or the diatomic assemblage, is stable without guidance. But it seems more probable that, for assemblage I. and law of force 2, the intermedi- ate maximum m ' (close to a minimum) is unstable. If it is so, the assemblage left to itself in this configuration would fall away, and would (in virtue of energy lost by waves through ether, that is to say, radiation of heat) settle in stable equilibrium corresponding to the maximum m (single assemblage), or either of the maximums m" (single assemblage), or m" (double assemblage). It is also possible that for law of force 1 the maximum m for the single assemblage is unstable. If so, the system left to itself in this configuration would fall away and settle in either of the configurations m (single assemblage) or m" (double assemblage)* Or it is possible that with either of our arbitrarily assumed laws of force there may be stable configurations of equilibrium with the atoms in simple cubic order (§21 below) : and in double cubic order ; that is to say, with each atom in the centre of a cube of which the eight corners are its nearest neighbours. § 18. It is important to remark further, that certainly a law of force fulfilling the conditions of § 13 may be found, according to which even the simple cubic order is a stable configuration ; though perhaps not the only stable configuration. The double cubic order, which has hitherto not got as much consideration as it deserves in the molecular theory of crystals, is certainly stable for some laws of force which would render the simple cubic order unstable. Meantime it is exceedingly probable that there are in nature crystals of elementary substances, such as metals, or frozen oxygen or nitrogen or argon, of the simple cubic, and double cubic, and simple equilateral, and double equilateral, classes. It is also probable that the crystalline molecules in crystals of compound chemical substance are in many cases simply the 216 Proceedings of Royal Society of Edinburgh. [sess. chemical molecules, and in many cases are composed of groups of the chemical molecules. The crystalline molecules, however constituted, are, in crystals of the cubic class, probably arranged either in simple cubic, or double cubic, or in simple equilateral, or double equilateral, order. § 19. It will be an interesting further development of the molecular theory to find some illustrative cases of chemical compound molecules (that is to say, groups of atoms presenting different laws of force, whether between two atoms of the same kind or between atoms of different kinds), which are, and others which are not, in stable equilibrium at some density or densities of equilateral assemblage. In this last class of cases the molecules make up crystals not of the cubic class. This certainly can be arranged for by compound molecules with law of force between any two atoms fulfilling the condition of § 13; and it can be done even for a monatomic homogeneous assemblage very easily, if we leave the simplicity of § 13 in our assumption as to law of force. § 20. The mathematical theory wants development in respect to the conditions for stability. If, with the constraining guidance of § 16, w is either a maximum or a minimum, there is equilibrium with or without the guidance. For iv a maximum the equilibrium is stable with the guidance ; but may be stable or unstable without the guidance. A criterion of stability which will answer this last question is much wanted ; and it seems to me that though the number of atoms is quasi infinite the wanted criterion may be finite in every case in which the number of atoms exerting force on any one atom is finite. To find it generally for the equilibrium of any homogeneous assemblage of homogeneous groups, each of a finite number of atoms, is a worthy object for mathematical con- sideration. Its difficulty and complexity is illustrated in §§ 21, 22 for the particularly simple case of similar atoms arranged in simple cubic order; and in §§ 23-29 for a still simpler case. § 21. Consider a group of eight particles at the eight corners of a cube (edge X) mutually acting on one another with forces all varying according to the same law of distance. Let the magni- tudes of the forces be such that there is equilibrium ; and in the first place let the law of variation of the forces be such that the 1901-2.] Lord Kelvin on Molecular Dynamics of a Crystal. 217 equilibrium is stable. Build up now a quasi infinite number of such cubes with coincident corners to form one large cube or a crystal of any other shape. Join ideally, to make one atom, each set of eight particles in contact which we find in this structure. The ■whole system is in stable equilibrium. The four forces in each set of four coincident edges of the primitive cubes become one force equal to the force between atom and atom at distance A. The two forces in either diagonal of the coincident square faces of two cubes in contact make one force equal to the force between atoms at distance A J2. The single force in each body-diagonal of any one of the cubes is the force between atom and atom at ■distance A^/3. The three moduluses of elasticity (compressibility- modulus, modulus with reference to change of angles of the square faces, and modulus with reference to change of angles between their diagonals) are all easily found by consideration of the dynamics of a single primitive cube, or they may be found by the general method given in “On the Elasticity of a Crystal according to Boscovich.” * (In passing, remark that neither in this nor in other cases is it to be assumed without proof that stability is ensured by positive values of the elasticity moduluses.) § 22. Now while it is obvious that our cubic system is in stable equilibrium if the eight particles constituting a detached primitive oube are in stable equilibrium, it is not obvious without proof that this condition, though sufficient, is necessary for the stability of the combined assemblage. It might be that though each primitive cube by itself is unstable, the combined assemblage is stable in virtue of mutual support given by the joinings of eight particles into one at the corners of the cubes which we have put together. § 23. The simplest possible illustration of the stability question of § 20 is presented by the exceedingly interesting problem of the equilibrium of an infinite row of similar particles, free to move only in a straight line. The consideration of this linear problem we shall find also useful (§§ 28, 29 below) for investigation of the disturbance from homogeneousness in the neighbourhood of the bounding surface, experienced by a three-dimensional homogeneous assemblage in equilibrium. First let us find a, the distance, or one of the distances, from atom to atom at which the atoms must * Proc. R.S.L. , vol. 54, June 8, 1893. 218 Proceedings of Royal Society of Edinburgh. [sess.. be placed for equilibrium ; and after that try to find whether the equilibrium is stable or unstable. § 24. Calling /(D) (as in § 4) the attraction between atom and atom at distance D, we have for the sum, P, of attractions between all the atoms on one side of any point in their line, and all the- atoms on the other side, the following finite expression having essentially a finite number of terms, greater the smaller is a : /(a) + 2/(2a) + 3/(3a)+ . . . . =P . . . (8). Hence a, for equilibrium with no extraneous force, is given by the functional equation /(a) + 2/(2o) + 3/(3a)+ .... =0 . . . (9) which, according to the law of force, may give one or two or any number of values for a : or may even give no value (all roots imaginary) if the force at greatest distance for which there is force at all, is repulsive. The solution or all the solutions of this equation are readily found by calculating from the Boscovich curve representative of /(D) a table of values of P, and plotting them on a curve, by formula (8), for values of a from a = I (the limit above which the force is zero for all distances) downwards to the value which makes P = — go , or to zero if there is no infinite repulsion. The accompanying diagram, fig. 6, copied from fig. 1 of Boscovich’s great book,* with slight modifications (including positive instead of negative ordinates to indicate attraction) to suit our present purpose, shows for this particular curve three of the solutions of equation (8). (There are obviously several other solutions.) In two of the solutions, respectively, A0, A', and A0, A", are consecutive atoms at distances at which the force between them is zero. These are configurations of equi- librium, because A0B, the extreme distance at which there is mutual action, is less than twice A0A', and less than twice A0A". In the other of the solutions shown, A0, Ap A2, A3, A4, A5, A6 are seven equidistant consecutive atoms of an infinite row in * Theoria Philosophise Naturalis redacta ad unicam legem virium in natura existentium, auctore P. Rogerio Josepho Boscovich, Societatis Jesu,. nunc ab ipso perpolita, et aucta, ac a plurimis prsecedentium editionum mendis expurgata. Editio Yeneta prima ipso auctore prsesente, et corrigente. Yenetiis, MDCCLXIII. Ex Typographia Remondiniana superiorum per- missu, ac privilegio. 1901—2.] Lord Kelvin on Molecular Dynamics of a Crystal. 219 Fig. 220 Proceedings of Royal Society of Edinburgh. [sess. ■equilibrium in which A5 is within range of the force of A0, and A6 is beyond it. The algebraic sum of the ordinates with their proper multipliers is zero, and so the diagram represents a solution ■of equation (9). § 25. In the general linear problem to find whether the equilibrium is stable or not for equal consecutive distances, a, let (as in § 4) be the work required to increase the distance between two atoms from D to oo . Suppose now the atoms to be displaced from equal distances, a, to consecutive unequal distances — .... a + ut_ 2 , a + Uj_x , a + uit a + ui+1 , a + ui+2 , • (10). The equilibrium will be stable or unstable according as the work required to produce this displacement is, or is not, positive for all infinitely small values of . . . . , ui , ui+1 , . . . . Its amount is W0 - W ; where W denotes the total amount of work required to separate all the atoms from the configuration (10) to infinite mutual distances. According to § 2 above W is given by • • . + + u\ + wi+x + ....). . (11); where wi =cfi(a + ui) + (2a + ui_1 + ui) + 4>(3a + ut_2 + ^_i + u^) + . .. + (a +wi+1) + cf>(2a + ui+1 + ui+2) + c£(3a + ui+1 + ui+2 + ui+3 ) + . . . + (12). Expanding each term by Taylor’s theorem as far as terms of the second order, and remarking that the sum of terms of the first order is zero for equilibrium * at equal distances, a, and putting "(D) = -/'(D), we find W0- W‘=\%{f\a){u? + u\+l) +f('2a)[(ui_i + Uiy + (ui+1 + ui+2)2] +./,(3a)[(wi_2 + + u{)2 + (ui+1 + ui+ 2 + wi+3)2] + etc. etc. etc. etc. } (13); * It is interesting and instructive to verify this analytically by selecting all the terms in H/> which contain uh and thus finding This equated to zero, for zero values of . . . m-\ ,% , Ui+ 1, . . . gives equation (9) of the text. 1901 — 2.] Lord Kelvin on Molecular Dynamics of a Crystal. 221 where 2 denotes summation for all values of i, except those corre- sponding to the small numbers of atoms (§§ 28, 29 below) within influential distances of the two ends of the row. § 26. Hence the equilibrium is stable if f'(a), f'(2a), f'(3a), etc., are all positive ; but it can be stable with some of them negative. Thus, according to the Boscovich diagram, a condition ensuring stability is that the position of each atom be on an up-slope of the curve showing attractions at increasing distances. We see that each of the atoms in each of our three equilibriums for fig. 6 fulfils this condition. § 27. Fig. 7 shows a simple Boscovich curve drawn arbitrarily to fulfil the condition of § 1 3 above, and with the further simplification for our present purpose, of limiting the sphere of influence so as not to extend beyond the next-nearest neighbours in a row of equidistant particles in equilibrium, with repulsions between nearests and attractions between next-nearests. The distance, a, between nearests is determined by f(a) +:2/(2a) = 0 (14), being what (9) of § 24 becomes when there is no mutual force except between nearests and next-nearests. There is obviously one stable solution of this equation in which one atom is at the zero of the scale of abscissas (not shown in the diagram) and its nearest neighbour on the right is at A, the point of zero force with attraction for greater distances and repulsion for less distances. The only other configuration of stable equilibrium is found by solution of (14) ac- cording to the plan described in § 24, which gives a= '680. It is. shown on fig. 7 by A*, Am, as consecutive atoms in the row. § 28. Consider now the equilibrium in the neighbourhood of either end of a rectilinear row of a very large number of atoms which, beyond influential distance from either end, are at equal consecutive distances a satisfying § 27 (14). We shall take for simplicity the case of equilibrium in which there is no extraneous force applied to any of the atoms, and no mutual force between any two atoms except the positive or negative attraction ^(D). But suppose first that ties or struts are placed between consecutive atoms near each end of the row so as to keep all their consecutive distances exactl}T equal to a. For brevity we shall call them ties, 222 Proceedings of Royal Society of Edinburgh. [sess. Fig. 7. 1901—2.] Lord Kelvin on Molecular Dynamics of a Crystal. 223 though in ordinary language any one of them would be called a strut if its force is push instead of pull on the atoms to which it is applied. Calling A1} A2, A3 . . . the atoms at one end of the row, suppose the tie between Ax and A2 to be removed, and A1 allowed to take its position of equilibrium. A single equation gives the altered distance A1A2, which we shall denote by + Let an altered tie be placed between A4 and A2 to keep them at this altered distance during the operations which follow. Next remove the tie between A2 and A3, and find by a single equation the altered distance a + j[X0. After that remove the tie between As and A4 and find, still by a single equation, the altered distance a + Yx 3, and so on till we find 1^7 or 1a;8 or Yxh small enough to be negligible. Thus found, xxv jj&g, jajg, .... jXt give a first approximation to the devi- ations from equality of distance for complete equilibrium. Eepeat the process of removing the ties in order and replacing each one by the altered length as in the first set of approximations, and we find a second set 2xv 2x2, 2x3 ... . Go on similarly to a third, fourth, fifth, sixth .... approximation till we find no change by a repetition of the process. Thus, by a process essentially con- vergent if the equilibrium with which we started is stable, we find the deviations from equality of consecutive distances required for equilibrium when the system is left free in the neighbourhood of each end, and all through the row (except always the constraint to remain in a straight line). By this proceeding applied to the curve of fig. 7 and the case of equilibrium a =*680, the following suc- cessive approximations were found : — Xi ^3 a4 a?6 x7 1st Approximation . + •018 - *009 + •004 - -002 + •001 rH O O 1 ■ooo 2nd + •026 - *014 + •007 - -003 + •002 3rd + •031 - -018 + *009 - -005 + •003 4th ,, + •034 - -020 + -on - -006 5th + "036 - -022 + •012 - *007 6th + "03/’ - *023 + •013 7th ,, + •038 - -024 8th + -039 Thus our final solution, with a = ‘680, is x± = +-039, x2 = --024, a3= +-013, x4= -‘007, ||= + -003, xQ= - -001, x7= -000. 224 Proceedings of the Royal Society of Edinburgh. [sess. § 29. It is exceedingly interesting to remark that the deviations, of the successive distances from a are alternately positive and negative, and that they only become less than one-seventh per cent, of a for the distance between A* and Ag. Thus, if we agree to neglect anything less than one-seventh per cent, in the distance between atom and atom, the influential distance from either end is 7 a, although the mutual force between atom and atom is null at all distances exceeding 2*2 a. § 30. If, instead of /(D) denoting the force between two atoms in a rectilinear row, it denotes the mutual force between two parallel plane nets in a Bravais homogeneous assemblage of single atoms, the work of §§ 27, 28 remains valid; and thus we arrive at the very important and interesting conclusion that when there is replusion between nearest nets, attraction between next-nearests, and no force between next-next-nearests or any farther, the disturbance from homogeneousness in the neighbourhood of the bounding plane consists in alternate diminutions and augmenta- tions of density becoming less and less as we travel inwards, but remaining sensible at distances from the boundary amount- ing to several times the distance from net to net. (. Issued separately June 9, 1902.) 1901-2.] Professor C. Piazzi Smyth on Sodium lines . 225 Does the Spectrum-place of the Sodium lines vary in different Azimuths? By the late Professor C. Piazzi Smyth, Astronomer-Royal for Scotland. Communicated by Professor J. G. MacGregor, D.Sc., F.R.S. (Read May 5, 1902.) The above question having been set before me by my friend Prof. P. G. Tait, and in such a guise that its practical solution, if amounting to anything sensible, might have astronomical applica- tions, I set myself to examine it with the highest dispersion power in my possession, viz., a fine Rutherfurd Diffraction grating, of 17,296 lines to the inch, ruled over a surface P6 inch square; a telescope 52 inches long, with magnifying powers from 20 to 50 ; and a collimator 32 inches long, armed with a very substantial slit-apparatus by Mr Adam Hilger, and some other fittings. These were all laid out in horizontal plane on the levelled top of a table, which revolved on three wheels below, in a circle divided to every ten degrees of astronomically determined azimuth, on the floor of an upper chamber. Some preliminary trials were made with sodium light, de- rived from salt burning in a Bunsen-burner gas flame; first, by placing burners on either side of the slit, and sending their lights into that by metal reflectors placed oppositely to each other and at 45° each to the axial direction. This plan, therefore, gave two images of the salt-lines (say D1 and D2), one above the other in the field of view. The second plan consisted in sending the light of a single burner direct into the slit, and noting by micrometer the absolute spectrum place of one of its D lines, while the table was turned to successive steps of azimuth all round the circle. No change of spectrum place, as depending on azimuth, could be established by either of these methods ; but the images of the lines were so barbarously coarse and hazy, that it was hazardous to attempt to say from them within what fraction of the distance from D1 to D2 the negative could be considered absolutely proved. I therefore arranged a variety of the apparatus for trying the PROC. ROY. SOC. EDIK — VOL. XXIV. 15 226 Proceedings of Royal Society of Edinburgh. [sess. experiment with sodium lines rendered luminous in an end-on gas-vacuum tube by electric induction sparks. The lines so pro- duced were not quite so bright as I could have wished, chiefly owing to the small size of my galvanic battery of induction coil, whose sparks were only one inch long ; but the definition of the lines was as perfect as could be desired, and fully worthy of the high fame of the Rutherfurd grating. The distance D1 to D2 measured, in the second order of spectrum of that grating, 266 units of the micrometer, with an average error on each occasion of not more than two of those divisions ; and the final conclusion derived from three different methods of trying the experiment at eight points equally distributed round the azimuthal circle, was, — that there is no change of spectrum-place in a sodium line, depend- ing on the spectroscope looking in any one azimuth rather than another, to the amount of g-Jo- of the distance between D1 and D2. Wherefore, if theory continues to assert, on its own very secure grounds, that there is some effect of that kind, it must be so small that the next search for it must be conducted with a very much more powerful apparatus than anything I possess. That is, if I made no mistakes in trying the case ; and in order that that may be fully judged of by others, I append herewith the original obser- vations, together with the following few words on the principles followed in summing them up, viz. : — As all the electrically illumined series herein alluded to were made by the method of absolute place on one spectral image alone, it became necessary to guard against any other source of disturb- ance to absolute place than the problematical one inquired about, while these observations were going on. Except when some positive change was made by hand, and in that case duly noted, to some part of the apparatus capable of altering the micrometer zero, there was nothing that occurred to the visible spectrum-place of any spectral line, except to exceedingly small amounts, and in a manner that allowed the assumption, and consequent correction in a second column, that it varied|as the time. The duration, there- fore, of any whole series of measures (as starting from any given azimuthal point, and returning to it again after passing through 360°) was made as short as possible, >iz., five to seven minutes; while continued observations, in one and the same azimuth, were 1901-2.] Professor C. Piazzi Smyth on Sodium lines. 227 subsequently kept up through several hours, as a test that no larger and cumulative time-change existed. Most of the observations were made with the grating in the angular position most nearly normal to the collimator, when the distance from I)1 to D2 marks 266 of the micrometer on the second order of spectrum. But the negative results thence derived have been tested and confirmed by other observations made after turning the grating through its position of simple reflection, to the second order of spectrum on the opposite side thereof, where the distance of the lines is necessarily less wide, and, in fact, measured only 243 of the micrometer. Again the poles of the induction coil have been changed, so that the direction of the illuminating current through the capillary has been made to pass straight to, or straight from, the slit, besides the light having been sent to the slit sometimes direct in the axis of the collimator and sometimes at right angles, by means of a totally- reflecting prism and an altered place of standing for the tube- holder ; but none of these things have had any certainly visible effect on the absolute spectrum-place of any line in the field of view. See the following observations on May 18 and May 20, as extracted from my observing ledger. [Tables. ft May 18> 1882.^— 3 p.m.— Tait Experiments by C.P.S.— As to effect of change of Azimuth on Spectrum-place of Sodium lines, viewed by electric illumination of 1 inch induction sparks, in an end-on Gas- vacuum tube. 2nd 228 Proceedings of Royal Society of „ Edinburgh , bo ^3 o CM o b£ "-+0 C3 o a o (D Oh o [sESS. 1 ' . ofl CD O O -+J &Q ;© VT1 W PQ CO vo CO CO CM CM Micrometer Reading. CO CO ^ CM CO (M CO CM VO VO CM CM i 00 OO 00 00 00 OO UO OO OO OO 00 CXD 1 — 1 rtico COCOCOCOCO COCOCOCOCO CO'cfl Object Observed. QQ CfiflQfl flfl Azimuthal Direction Observed in, or Light South of slit. 5 5 5 5 5 5 Light South of slit. East North West South ,, S. West „ „ S. East ,, ,, N.East ,, ,, N. West „ „ S. West „ „ S. West „ ,-Q o o £ o o O COCO-rH-rHCN^COCOCO: OOQOOOOOOOOOOOOOOO to to (O I® to o 50 CO 50 ii ii ii ii ii ii ii i: II 3 w P* s- ^ o. ^ • L° . O o o ® --2 bcxia _ S3 CD -**l ^ ce nd , a + o O |-1 ^ S 2^ 5 o 2 S3 ^3 -5 0 a. in 1 J C£ CD pu ® O 1 02 JS -2 w P ® 2 «c S £ £ ^ c 3 a 1 c6 <*_ d c_j tc y 7 C 3 I 'is - — i o >-r ■ ® *o ■c> 17— < .2 *- cc >7 _ 'y c6 - £ b— 1 O O Combination of both sets in abnormal position of grating. Azirn. to South — 618 for place of D2. „ S.W. - 617 ,, West = 619 „ N.W. = 621 ,, North = 618 ,, ,, „ N.E. 4= 620 „ East = 616 ,, S.E. = 617 ,, South = 618 When dist. from D2 to D1 = 243 of the same units ; but error of obs. large. Corrected for Time Change. GO Oi 30 SO 00 !>• r— l O *0 1 — l rl H rH 1— • H (N (N H rl so so so so so so so so so so Temp. >p cm 0 00 CO CO Difference or Dist. D1 to D2. 10 1-1 v** CM CM Micrometer Reading. UO O CO OO 00 N O CO ® O N H ffiO H CO HHHHO HhNhN fh CO COX CO CD CO CO CO CO CO -O CO CO cox Object Observed. QQ QCGPP GOQflfl GO Azimuth Observed in, or -4-5 4-j rS rS « - f "ao ~ ^ ^ m fee g tc - - - - % % j 2 z " z be - 5 hG Time of Obs. h. m. 2 17 p.m. 2 20 ,, '2 32 „ 2 40 ,, After above set changed back to Normal position of Grating; also changed direct mode of sending light into slit, for a side method, by a prism (totally reflecting) of hard crown glass. 1901-2.] Professor C. Piazzi Smyth on Sodium lines. 231 co Id £C o<^ Q2 & ~Oi o a Ph o o +3 O 05 a) efi m X. -M &0 '&0 P g B ^2 S.-s !*== o O o o o (M CM t/c •g GO £ Qfl flCflClfl QQflflQ QQQQQ Qfi CO r CO* ^ 5zi M OQ GO* ^ &3 0Q co ^ W cd CO gr. - W : Hcq^^H W p o 2J8 .So H s . ^ g o 2 CO 232 Proceedings of Royal Society of Edinburgh. r L SESS. For effect of changing Poles of Induction Coil. Time. Az. of Light. Obj. Obs. Pole. Microm. Reading. h. m. 5 0 p.m. S. ; D2 Pos. 916 6 1 „ ? 5 D2 Neg. 914 Therefore no sensible effect. 5 2 „ D2 Pos. 914 For effect of long continued interval of Time. Time. Az. of Light. Obj. Obs. Microm. Reading. 1 Temp. h. m. 4 40 p.m. 5 55 ,, 7 0 ,, 8 23 ,, 10 13 ,, 12 45 ,, S. ’ ? 5 ? D2 D2 D2 D2 D2 D2 915 914 916 914 915 913 0 64*7 64-4 64-0 63*5 62-8 61*9 Therefore no effect worth noticing. 15 Royal Terrace, Edinburgh, May 25, 1882. ( Issued separately July 18, 1902.) 1901-2.] Dissociation of Compound of Iodine and Thio-urea. 233 The Dissociation of the Compound of Iodine and Thio-urea. By Hugh Marshall, D.Sc. (Read June 2, 1902.) When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2 2CSX2H4 + 2HX + 0 === (CSN2H4)2X2 + H20 . Of these salts the di-nitrate is very sparingly soluble, and is pre- cipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts. The di-chloride and di-bromide were prepared by Claus * by the direct action of the corresponding halogen on an alcoholic solution of thio-urea ; he found, however, that under similar conditions the di-iodide was not produced. McGowan f investigated a number of the salts and prepared them in various ways. Among other experiments, he tried the action of the chlorides of iodine on alcoholic solution of thio-urea ; this he found gave rise to the di-chloride of the base and free iodine ; he states that “ in aqueous solution the iodine separates more slowly.” By shaking together the di-chloride and potassium iodide with alcohol, there were formed potassium chloride, thio-urea, and free iodine; here again “in aqueous solution the separation of iodine proceeds more slowly.” He succeeded in preparing the di-iodide, however, by rubbing together solid thio-urea and iodine with a small quantity of alcohol, washing away excess of free iodine by means of benzen, and drying on filter paper. In * Ann. Chem. Pharm., clxxix. 139 (1875). t Journ. fur prakt. Chem., xxxiii. 188 (1886). 234 Proceedings of Royal Society of Edinburgh . [sess. this way long colourless crystals were obtained, the properties of which are described as follows : — “ Once prepared, it is somewhat stable. When heated it fuses with separation of iodine. Water, alcohol, and ether dissolve the di-iodide, again with separation of iodine ; hence the explanation of its non-formation in solution. On warming the aqueous solu- tion, or adding an alkali, sulphur is liberated “ Dilute hydrochloric acid does not decompose the di-iodide in the cold, as the latter substance is nearly insoluble in it ; but, on warming, iodine separates in considerable quantity, without libera- tion of sulphur. This shows that the di-iodide decomposes more easily into iodine and thio-urea than into hydriodic acid, sulphur, thio-urea, and cyanamide. “ If a drop of alcohol and, immediately, some nitric acid are added to a portion of di-iodide, there instantly results a precipitate of the di-nitrate.” Recently I had occasion to examine a product, obtained from thio-urea, which appeared to be one of the above series of salts. The results obtained on adding potassium iodide to its aqueous solution seemed anomalous, as the amount of iodine liberated by a given quantity of substance varied greatly, and in an erratic manner, in different experiments. It was also found that in cases where the amount of iodine liberated was comparatively small, slight warming of the solution deepened the colour quite distinctly, while subsequent cooling changed it back again. This behaviour indicated that the action was a reversible one, and that what McGowan described as slower decomposition in aqueous solution might really be less complete decomposition. In that case the forma- tion of the di-iodide from thio-urea and iodine should be effected more easily in the presence of water than in presence of alcohol. Experiment showed that this was the case, and the preparation is exceedingly simple. An examination of the properties of the di- iodide obtained in this experiment had most interesting results, as its behaviour in solutions of various kinds is very peculiar. So far, only qualitative experiments have been performed, but it has been considered advisable to publish a description of them in a prelim- inary communication, as it is probable that a full investigation of the points involved will take some time. 1901-2.] Dissociation of Compound of Iodine and Thio-urea. 235 The di-iodide was prepared by gradually adding 5 parts of iodine crystals to 3 parts of powdered thio-urea (1 mol. I2 to 2 mols. CSNoH4) mixed with about 25 parts of water. The iodine rapidly dissolves in the solution of thio-urea, forming at first a colourless solution, and the liquid becomes warm ; as more iodine is added, the whole of the thio-urea goes into solution, and prismatic crystals of the di-iodide soon separate, especially if the liquid is cooled — a precaution which is advisable in view of the slight stability of all these salts. After all the iodine has been added and the liquid thoroughly cooled, the mother liquor possesses only a pale brown colour, unless excess of iodine has been employed. It is now only necessary to filter, draining thoroughly, and dry the crystals between filter paper ; they are practically colourless. On standing for some time the mother liquor deposits sulphur, along with clear yellow crystals of some compound which has not yet been analysed. When water is added to the crystals of di-iodide, they dissolve, forming a yellowish solution, which is distinctly acid to litmus. Whether this acidity is the result of hydrolysis, or is due to a slight amount of decomposition of a more profound character, it is impossible to say at present. The yellow-brown coloration, due to the liberation of iodine, is not very strongly marked, and becomes less intense on dilution of the solution with more water. This is not merely an apparent diminution, due to the increased quantity of solution ; for, if a concentrated solution is divided into two equal portions in similar tubes, and one of them is diluted with water, the colour of the latter appears much feebler than the other when viewed along the axis of the tube. If the solution is gently warmed, the colour is intensified and changes back on cooling ; but this experiment must be carried out cautiously and quickly, otherwise decomposition takes place. The addition of thio-urea to the solution diminishes or destroys the colour. The decrease in the quantity of free iodine, which takes place on dilution, points to the occurrence of some action other than mere dissociation into iodine and thio-urea, when the di-iodide is dissolved in water ; for, if we represent the latter action by the equation (CSN2H4)2I2 ^ 2CSK,H4 + 12 236 Proceedings of Royal Society of Edinburgh. [sess. the quantity of free iodine ought to increase, corresponding to' increase in the degree of dissociation, with diminution of the con- centration. As the balance shifts in the opposite direction, we must assume that the undissociated di-iodide is being used up by some other reaction, which does not result in the liberation of iodine, so that the dissociated products will reunite to restore the equilibrium. The nature of such a second reaction is suggested by the saline character of the compound, and the phenomena are explicable on the assumption that in aqueous solution the di-iodide undergoes ionisation, and that the increased ionisation produced by dilution is the cause of the diminished dissociation. In addition to the first balanced action we may therefore assume the second one, represented by the equation (CSN2hM2=^ (CSN2H4y • + 21' . This assumption that the di-iodide and its analogue are true salts is justified by the readiness with which they undergo double decomposition with other salts, as exemplified by the precipitation of the sparingly soluble di-nitrate. Further, the aqueous solution of the di-iodide dissolves free iodine, like solutions of metallic iodides ; it precipitates lead iodide and silver iodide from solutions of lead and silver salts ; with mercuric chloride it gives a pre- cipitate of mercuric iodide, soluble in excess of the di-iodide solution. If the above explanation of the diminished dissociation is a correct one, it is evident that any diminution in the amount of ionisation will result in an increased liberation of iodine. A number of experiments bearing on this point were tried in order to test the theory, and in all cases the results were of a confirmatory nature. In the first place, the experience of previous investigators with organic solvents, already referred to, was satisfactory in that respect, for alcohol and ether are deficient in ionising power. Other organic solvents which dissolve the di-iodide give dark solutions ; those tried included methyl alcohol, acetone, and glacial acetic acid. It has been shown by Carrara * that, as a rule, methyl alcohol * Journ. CJiem. Soc. (Abs.), 72, ii. "471 (1897). 1901-2.] Dissociation of Compound of Iodine and Thio-urea. 237 is intermediate between water and ethyl alcohol as regards ionising power, whilst acetone is generally inferior to the latter in this respect. Experiments with the di-iodide gave results entirely in agreement with Carrara’s experience. Equal parts of an aqueous •solution of the di-iodide were diluted to the same volume with water, methyl alcohol, ethyl alcohol, and acetone, respectively. The solution which was diluted with water became nearly colour- less ; that with methyl alcohol assumed a deeper tint than formerly, that with ethyl alcohol still deeper, and that with acetone was darker than any of these. When ether was added to an aqueous solution in quantity in- sufficient to form a separate layer, a deepening of the colour was observable; when the quantity was sufficient to form a separate layer, the latter was dark brown. On removing the layer and adding more ether, another dark layer was formed ; this could be repeated several times without a very marked diminution in the depth of colour of the ethereal layer, in accordance with the assumption of a moderate dissociation of the compound in the aqueous solution. According to the ionisation hypothesis, the addition, to a solution of a salt, of any other salt having an ion in common with the first one results in diminishing the extent to which that salt is ionised. Such was found to be the case with the di-iodide ; the addition of potassium or sodium iodide, solid . or dissolved, to its solution resulted at once in the production of a dark brown colour. The corresponding chlorides and bromides had no appre- ciable effect ; but, on the other hand, the di-bromide corresponding to the di-iodide produced a distinct deepening of the colour, though the effect was very much less than that caused by a metallic iodide. Dilute hydrochloric acid has no appreciable effect on a solution of the di-iodide, but concentrated acid produces a dark coloration. This may result from various causes, one of which probably is the union of the hydrochloric acid with the thio-urea in solution to form a definite compound, the equilibrium consequently being affected. The phenomena here described are, so far as I can discover, quite novel, and are of particular interest from their bearing on the 238 Proceedings of Royal Society of Edinburgh. [sess. ionisation theory of salt solutions. They are easily explicable on the basis of that theory, but not very easily otherwise. They bring into striking prominence the contrast between ionisation and ordi- nary dissociation, and a case such as this illustrates the advantage of using the former term in place of the expression ‘ electrolytic dissociation,’ which is still frequently employed. It will be observed that thio-urea is to a certain extent analogous to a metal ; a molecule of it corresponds to a half atom of a dyad metal. Like a metal it unites directly with the halogens to form salts. It can c reduce ’ metallic salts from a higher to a lower stage; thus Rathke * found that it acts upon cupric sulphate to form cuprous sulphate — momentarily — and the sulphate correspond- ing to the di-iodide 2CuS04 + 2CSX2H4 Cu2S04 + (CSX2H4)2S04 . The cuprous sulphate unites with more thio-urea to form a complex compound. Other cupric salts behave in a similar manner. When thio-urea is added to a concentrated solution of ammonium persulphate there is a considerable evolution of heat, and, on cool- ing, the above sulphate separates out, while ammonium sulphate remains in solution : — 2CSX2H4 + (NH4)2S208 =£= (CSX2H4)2S04 + (NH4)2S04 , corresponding to such an action as Zn + (XH4)2S208 == Zn S04 + (XH4)2S04 . The exact constitution of the class of salts here dealt with does not appear to have been very fully investigated, and in what pre- cedes I have simply adopted the formulae generally employed. Adopting the imide formula for thio-urea, the most plausible assumption is that the salts may be represented by the graphic formula — XILX XELX i i H-N =0-S-S-C= N-H . The difficulty of satisfactorily investigating this and other points Berichte, xvii. i. 297 (1884). 1901-2.] Dissociation of Compound of Iodine and Thio-urea. 239 connected with the salts is considerable, owing to the instability of the hydroxide or of the corresponding amine base. In discussing the properties of the di-iodide in solution, I have referred only to the double balancing action 2C8N2H4 + 12 ^ (CSN2H4)2I2 === (CS N2H4)2‘ ‘ + 21' , but this will not represent the complete condition of equilibrium ; some of the liberated iodine will form periodide, and, from indica- tions I have observed, I suspect that the di-iodide can unite with thio-urea to form a definite compound. (. Issued separately July 18, 1902.) 240 Proceedings of Royal Society of Edinburgh . [sess. Some Identities connected with Alternants and with Elliptic Functions. By Professor W. H. Metzler. (Read June 2, 1902.) As is well known, the usual form of the Addition-Theorem for Elliptic Functions of several arguments expresses these functions as the quotient of two determinants. When two or more arguments become equal, both numerator and denominator of this quotient vanish, and in seeking to remove the common vanishing factor, Cayley, in his paper * “ ISTote sur l’addition des fonctions ellip- tiques,” in connection with the cases of three and four arguments, brought to light some identities connecting certain alternants. Cayley gave these identities without proof, saying, “ Je n’ai pas encore trouve la loi generale de ces equations.” Dr Thomas Muir, in his very interesting paper f of the same title as this one, gives several demonstrations of these identities, and ends by saying, “ The problem of finding for determinants of a higher order than the fourth, identities similar to Cayley’s, I do not at present enter upon ; like Cayley, ‘ Je n’ai pas encore trouve la loi generale de ces equations.’ I content myself with stating the problem in as simple a form as possible.” The problem for deter- minants of order five, as stated definitely by Muir, is to determine a, ft, y, 8, e, p, r] , 0 , SO that (A) | a0 b1 c2 D eE | . |A°B2C4D6ES| = | a° 61 c2 D2 eE2 | • | A0 B“ C8 Dy E5 | ± | i12| 01359 | + | a0 b1 c2 D4 eE4 | {vx | 01256 | +v2 | 01346 | +v3 j 02345 | + v4 | 01247 | +v5-| 01238 | } | a0&4c2D6eE6 | • | 01234 | , PROC. ROY. SOC. EDIN. — VOL. XXIV. 16 242 Proceedings of Royal Society of Edinburgh. [sess. where | 0 a /3 y 8 | denotes the determinant | A° B“ O'5 DT Es | . The problem now is to determine X, /x, .... to satisfy the re- lation. To show that any one of the products cannot exist, it will be sufficient to show that it contains a term which is contained in no other product. The term B5 C2 D3 E12 d2 cb is contained in the product | a?Wc2 D4eE4 | . | 01238 | , and the only other product in which it might seem to appear is | a°51c2D2eE2 | • | 012510 j, but a litfle investigation will show that it does not. It follows, therefore, that vb = 0 . Similarly, on account of the terms A4 B C6 D4 E7 e2 d a , B2 C7 D8 E5 e2db, BC7 D8 E6 e2dh, A4 B C6 D5 E 6 e2dc, which the products | a0 bl c2 D4 eE4 | • | 01247 | a9 c2 D4 eE4 | • | 02345 | , | b1 c2 D4eE4 | • | 01346 | , | a« b1 c2 D4 eE4 | • | 01256 |, respectively contain, and which are contained in no other products, we have v4 = v3 = v2 = v1 = 0 . Since v4 = vb = 0, it follows that-/x9 = /x]0 = /xn = gl2 = /x13 = 0 , for these terms are the only ones which contain eleventh and twelfth powers of the large letters. Again, on account of the term^B C6 D6 E10 d2 c b, which is found in | a0 b1 c2 1)2 eE2 | • | 01368 | , and in no other product, /x2 = 0. The term B C8 D3 E10 e d2 b is contained in the product | a0 b1 c2 B6 eE6 | • | 01234 | , and since /x2 and v2, the coefficients of the only other products which could contain this term, vanish, it follows that p = 0 . The term B2 C6 D5 E9 d2 c b is contained in 1901—2.] Professor W. H. Metzler on Alternants. 243 the product | a 0 b 1 c2 D2 eE2 | - | 02457 | , and in no other product, therefore /x7 = 0 . The term B2 C4 D7 E9 e c2 b is contained in the product | a0 b1 c2 D eE | • | 02468 | , and since p , /x7 , and v3 , the coeffi- cients of the only other products which could contain this term, vanish, it follows that X — 0 . Thus it is seen that the relation (C) does not exist, and it also appears why these identities cease with determinants of order five. It may be added here that the object for which Cayley desired these identities can be obtained in other ways.* * Of. Clebsch, Geometrie, i. ; Funfte Abtheilung, vii. Laeroix, Calcul diff. et int., 6th ed., Paris, 1862, p. 68. Bertrand, Calcul int ., pp. 578- 583. Koenigsberger, Elliptische Functionen, ii. pp.1-17. Story, American Jour. Math., vol. vii. No. 4. Abel, “ Recherches sur les fonctions elliptiques,” Journal fur Mathematik, Bd. ii. Kronecker, Sitzungsberichte Der Akademie, Berlin, 1883, S. 717-729 ; 1883, S. 949-956 ; 1886, S. 701-780. {Issued separately August 14, 1902.) 244 Proceedings of Royal Society of Edinburgh. [sess. The Theory of Ortkogonants in the Historical Order of its development up to 1832. By Thomas Muir, LL.D. (Read May 19, 1902.) The special form of determinant to which we have now come is connected with a problem in coordinate geometry — the problem of transformation from one set of rectangular axes to another set having the same origin. The actual appearance of determinants in any of the attempts to solve the geometrical problem did not take place until comparatively late in its history, probably because the connection between the two subjects was less patent than in other cases, the problem when transformed into algebraical language being not a mere matter of elimination of unknowns from a set of linear equations. The earlier portion of the history of orthogonal substitution, although of considerable interest, is thus not sufficiently germane to our subject to warrant detailed treatment of it. For those interested in this earlier portion it will suffice to give the following chronologically arranged list of papers from 1770 to 1840:— 1748. Euler. Introductio in Analysin Infinitorum. Tomi duo. Lausannae et Genevae (v. ii. Appendix de Super- ficiebus*). 1770. Euler. Problema algebraicum ob affectiones prorsus singulares memorabile. Novi Commentarii Acad . Petrop., xv. p. 75; or, Commentationes Aritli. Collectae , i. p. 427. 1772. Laplace. Eecherches sur le calcul integral et sur le systeme du monde. Hist, de Vacad. roy. des sciences (Paris), 2e partie, pp. 267-376. 1773. Lagrange. Nouvelle solution du probleme du mouvement de rotation d’un corps de figure quelconque qui n’esfc anime par aucune force acceldratrice. Nouv. mem. de Vacad. roy. . . . (Berlin.), pp. 85-120. * Or in Labey’s French Translation, ii. pp. 370-378. 1901-2.] Dr Muir on the Theory of Orthogonants. 245 1775. Euler. Eoimulae generales pro translatione quacunque corporum rigidorum. Novi Comm entarii Acad. Petrop., xx. p. 189. 1776. Euler. Nova metliodus motum corporum rigidorum determinandi. Novi Commentarii Acad. Petrop., xx. p. 208. 1776. Lexell. Tlieoremata nonnulla generalia de translatione corporum rigidorum. Novi Commentarii Acad. Petrop. y xx. p. 239. 1784. Monge. Sur l’expression analytique de la generation des surfaces courbes. Mem. de Vacad. roy. des sciences (Paris) [pp. 85-117], p. 114. 1802. Hachette et Poisson. Addition au memoire precedent, Journ. de Vec. polyt., Cahier xi. pp. 170-172. 1806. Carnot, L. N. M. Sur la relation qui existe entre les distances respectives de cinq points quelconques pris dans l’espace. 1810. Lacroix, S. E. Traite du calcu'l differ entiel et du calcul integral. 2e edition, i. p. 533 .... 1811. Lagrange. Mecanique analytique. 2e edit., i. p. 267. 1818. Gauss. Determinatio attractionis .... Commentarii Soc. Gott., iv. ; or, WerJce, iii. pp. 331-355. 1827. Jacobi. Euleri formulae de transformatione coordinatarum. Crellds Journ. , ii. pp. 188-189. 1827. Jacobi. Ueber die Hauptaxen der Elachen der zweiten Ordnung. Crellds Journ. , ii. pp. 227-233. 1827. Jacobi. De singulari quadam duplicis integralis trans- formatione. Crellds Journ. , ii. pp. 234-242. 1828. Cauchy. Sur les centres, les plans principaux et les axes principaux des surfaces du second degre. Exercices de Math., iii. ; or, CEuvres (2), viii. pp. 8-35. 1828. Cauchy. Discussion des lignes et des surfaces du second degre. Exercices de Math ., iii. ; or, CEuvres (2), viii. pp. 83-149. 1829. Chasles. Sur les proprietes des diametres conjugues des hyperboloides. Corresp. math, et phys., v. pp. [137-157] 139-141. 246 Proceedings of Boy al Society of Edinburgh. [sess. 1829. Clausen. Ueber die Bestimmung der Lage des Haupt- Umdrehnngs-Axen eines Korpers. Crelle’s Journ ., v. pp. 383-385 ; or, Nouv. Annates de Math., v. pp. 81-83. 1829. Cauchy. Sur l’equation a l’aide de laquelle on determine les inegalites seculaires des mouvements des planetes. Exercices de Math., iv. ;' or, CEuvres (2), ix. pp. 172-195. 1831. Jacobi. De transformatione integralis duplicis indefiniti .... in formam simpliciorem Crelle’s Journ., viii. pp. 253-279, 321-357. 1832. Grunert. Ueber die Yerwandlung der Coordinaten im Raume. Crelle’s Journ., viii. pp. 153-159; or, Nouv. Annates de Math., v. pp. 414-419. 1832. Encke. Ableitung der Formeln von Monge flir die Trans- formation der Coordinaten in Raume. Berliner Astron. Jahrbuch (1832), pp. 305-310; or, Corresp. math, et phys., vii. pp. 273-277. 1832. Jacobi. De transformatione et determinatione integraliinn duplicium commentatio tertia. Crelle’s Journ., x. pp. 101-128. 1833. Jacobi. De finis quibuslibet functionibus homogeneis secnndi ordinis .... Crelle’s Journ., xii. pp. 1-69. 1833. Grunert. Supplemente zu Kltigers Worterbuch: Art. “ Coordinaten.” 1835. Jacobi. Observationes geometricae. Crelle’s Journ., xv. pp. 309-312. 1839. Catalan. Sur la transformation des variables dans les integrales multiples. Mem. couronHes par V Acad, de Bruxelles, xiv. ii. pp. 1-47. 1839. Reiss. Sur les neuf angles que forment reciproquement deux systemes d’axes rectangulaires. Corresp. math, et phys., xi. pp. 119-173. 1840. Rodrigues. Des lois geometriques qui regissent les deplace- ments d’un systeme solide dans l’espace, . . . Liou- ville’s Journ., v. pp. [380-440] 404-405. Of these, only five need be taken account of, because of their 1901-2.] Dr Muir on the Theory of Orthoyonants. 247 connection with determinants, viz., two by Jacobi in 1827, one by Cauchy in 1829, three by Jacobi in 1831-3, and one by Catalan in 1839. Jacobi (1827). [Ueber die Hauptaxen der Flachen der zweiten Ordnung. Crelle’s Journ ., ii. pp. 227-233.] Without unnecessary preliminaries Jacobi enunciates the problem which he wishes to solve, viz., the transformation of an expression of the form Ax 2 + By2 + Cz2 + 2 ayz + 2 bzx + 2 cxy , where a?, y , z are the coordinates of a point referred to an oblique coordinate-system, into an expression of the form L£2 + M^2 + ]ST£2 , where £, y, £ are the coordinates of the same point referred to a rectangular system having the same origin. This implies that the things directly sought are the nine coefficients which give each of the original coordinates in terms of the new. Jacobi, however, prefers to begin with a related set of unknowns, taking the equations which give the new coordinates in terms of the old. These being assumed to be £ — ax + Py +yz, 7] = a'x + py + yZ , £ = a "x + p'y + y"z, the equivalent set giving the old in terms of the new is of course c 1 (py'-p'y)i + (P'y-Py")y + (Py'-:fi'y)£) A-y = (ya - y"a)£ + (y"a - ya)y + (ya - ya)£ > A‘Z ■■= (cl ft" - CL "p)£ + (a"/3 - a p')rj + (a p - a P)£ j where A = OLpy" + (3y a" + ya p' - a p'y - p/'a - ya "p. Denoting the known angles between the original axes by A, y, y, there is obtained at once the set of six equations 248 Proceedings of Royal Society of Edinburgh. [sess. a + ,2 a + „2 a = 1, ^ + + (3"2 = 1, 0 y + ,2 y + y = 1, f3y + Pi + P'y = cos X , ya + y’a + rr rr y a = COS fJL , a (3 + a/3' + a "/3" — COS V j and, since the expression L(aX + f3y + yz)2 + M (ax + (3'y + y'z)2 + N(a"x + f3"y + y"z)2 has to be identical with Ax2 + By2 + Cz2 + 2 ayz + 2 bzx + 2 cxy , we have thus by implication another set of six equations, viz. : La2 + Mo.'2 + Na"2 = A, u2 + M/3'2 + N/3''2 = B, Ly2 + My'2 + Ny"2 = c, L/?y + M/3'y' + N/8"y" = a , Lya + My’ a’ + ±3y"a" = b, La f3 + Ma 73' + Na "f3" = c. What, therefore, remains to be done is the solution of these twelve equations in the twelve unknowns a , P , y : a, /3', y : a", /3", y : L , M , N . Jacobi’s mode of accomplishing this is very interesting. He notes first that A may be looked upon as known, by reason of the fact that it is expressible in terms of X , y, , v , the connection in modern notation being A2 a + a'" + a a/3 + a J3' + a" (3" ay + ay + a"y" a/3 + a/3' + a" (3" f32 + f3'2 + /3"2 y/3 + y'/3' + y"(3" . / > , n n rt . n' ’ . n" " 2 \ /2 • "2 ay+ay+ay py+ py + p y y +y +y ,, 1 COS V COS [X cos v 1 cos X cos y, cos X 1 In the next place he draws attention to the resemblance between the two sets of six equations, and points out that as a consequence any equation legitimately obtainable from the second set is matched by an equation which might in like manner be obtained from the first set, but which is much more readily got by using the substitution 249 : ( f 1901-2.] Dr Muir on the Theory of Orthogonants, l=m = n=a=b = c = i , a ,b , c = cos X , cos /x , cos v He then from the second set of six equations forms three groups La • a + Ma • a' + lSTa • a = A La -p + Ma'./T + Na"-/5" = C La • y + Ma' • y + Na" • y L/5 • a + M/5' • a' + N/5" • a" == c , L p-p + M/5' • /5' 4- N/5" • /5" = B , L/5 • y + M/5' • y + N/5" • y" = a , Ly • a 4- My' • a' 4- Ny" • a" = 5 , j Ly./5 + My' • (3' + Ny"-f3" = a, V Ly • y + My' • y + Ny" • y" = C , ) and solves the first group for La, Ma', Na"; the second for L/5 y M/5', N/5"; and the third for Ly , My', Ny"; the results being La = (/5'y" - /5"y')A + (y'a " — y"a')c 4- (a'/5" — a"/5 ')& Ma' = (/5"y - /5y" )A 4- (y"a — ya" )c + (a"/5 - a/5" )6 Na" = (/5y' — /5'y )A 4- (ya' - y'a )c 4- (a/5' — a'/5 )& L/5 = (/5'y" — /5"y')c + (y'a" — y"a')B 4- (a'/5" - a (3')a ^ M/5' = (/5"y -/3y")c 4- (y"a - ya" )B + (a"/5 - a/3" )a > N/5" = (/5y' - /5'y )c + (ya' -y'a )B 4- (a/5' -a'/3)a) Ly = (/5'y" — /5"y')& 4* (y'a" — y"d)a 4- (a'/3" — a"/5')C j My' = (/5"y-/5y")6 4- (y"a -ya")a 4- (a"/5 - a/5" )C V Ny" = (/5y -py)b 4- (ya' - y'a )a 4- (a/5' - a'/5 )C ) Making the substitution above referred to he derives the corre- sponding results which are obtainable from the first set of six, viz. : A • a = (/5'y" - /5"y') + (y'a" - y"a') cos v 4- (a'/5" - a"/5') cos /x | A • a' = (/5"y - f3y" ) 4- (y"a - ya" ) cos y 4- (a"/5 - a/5" ) cos /x > A • a" = (/5y' - /5'y ) 4- (ya' - y'a ) COS v 4- (a/5' -a /5 ) COS /X ) A • (3 = (/5'y" - /5"y') cos v 4- (y'a" - y"a') 4- (a'/5" - a"/5') cos X ^ A • /5' = (/5"y - /5y" ) cos v 4- (y"a - ya" ) 4- (a"/5 - a/5" ) cos X > A • /5" = (/5y' - /5'y ) cos v 4- (ya' - y'a ) 4- (a/5' - a'/5 ) cos X ) A • y = (/5'y" - /5"y') cos /x 4- (y'a" - y"a') cos X 4- (a'/5" - a"/5') A • y = (/5"y - /5y" ) cos /x -I- (y"a — ya" ) COS X 4- (a" /5 — a/5" ) A • y" = (/5y' - /5'y ) COS /x 4- (ya' - y'a ) COS X 4- (a/3' - a'/5 ) 250 Proceedings of Royal Society of Edinburgh. [sess. He then takes each of these nine equations along with the one from which it was derived, and by subtraction obtains nine new equations, which he groups as follows : — 0 = (L — A) * {fi'y" — fi"y') + (L cos v — c )’{y,a — y"a) + (Lcos fi — b)‘ {a' fi" — a'fi') 0; = (L COS V - C ) * {fi'y" - fi"y') + - y'a) + (L COS A - Cb) ’(a'/3" - a'fi') 0 = (L cos /a — b ) * {fi'y" — fi'y') + (L cos A — a) * (y'a" — y "a) + (L — C)'(a'fi" — a"fi') 0 = (M - A)’ {fi"y — fiy") + (M cos v - c ) '(y'a - y a") + (L cos jx - b ) •(a,/8 - afi") 0 = (M cos v - c ) • (£"7 - $y") + (M-B)*(7"a - ya" ) + (M cos A - a)’ (a" 13 - a/3") 0 = (M cos /a - b ) * (£"7 - £7") + (M cos A - a) * (y'a - 7a") + M - C) * (a"j8 - aj8") 0 = (N — A) ' (fiy' — fi'y) + (N COS v — C ) ’ {ya! — y'a) + (N COS jx — b)' ( afi’ — a fi) 0 = (N cos v - C ) '{fiy - fi'y) + (N-B)'(7a' - y'a) + (N cos A - a)’ {afi' - a' fi) 0 = (IT COS fx — b ) ‘ {fiy' — fi'y) + (N COS A — CL )’{ya — y'a) + (IT — C )' {afi! — afi), How from the first of these groups of three it is possible to eliminate /3'y" - fd"y , y'a" - y'a , a! ft" - a" ft' ; from the second, ft"y — ft'y" , y 'a — ya" , a" ft -aft"; and from the third, fty — ft'y , ya - y'a , aft' - aft ; and this being done there is obtained the set of three equations 0 = (L - A)(L - B)(L - C) + 2(L cos A - a)( L cos g - b)( L cos v - c) - (L - A)(L cos A - a)2 - (L - B)(L cos g-b)2 - (L- C)(L cos v - c)2 , 0 = (M - A)(M - B)(M - C) + 2(M cos A - a)( M cos g - b)( M cos v- c) - (M - A)(McosA - a)2 - (M - B)(McoSya - b)2 - (M - C)(M cosv - c)\ 0 = (H - A)(H - B)(H - C) + 2(H cos A - a)(H cos g - 6)(N cos v - c) - (H - A)(H cos A - a)2 - (N - B)(Hcos/x - b)2 - (N - C)(Ncos v-c)2; from which it is clear that the unknowns L , M , H are the three roots of the equation in x , 0 = (x - A)(x - B)(x - C) + 2(x cos A - a)(x cos g - 6)(x cos v-c) - (x - A)(x cos A - a)2 - (x - B)(x cos g - b)2 - (x - C)(x cos v - c)2, and therefore may be considered as expressible in terms of the nine knowns, A,B,C,a,5,c,A,/x,v. To obtain the remaining unknowns — which, be it noted, are not a 1 ft > 7 ' O' ’ a , p , y a-", F, y" P'y" - fi'y > P"y - Py" . Py - P’y . t rr rr / ya -ya, y'a — ya , ya - ya , aft" — a" ft' , a" ft — aft" aft' — aft , but 1901-2.] Dr Muir on the Theory of Orthogonants. 251 — recourse is had to the two original sets of six equations. In the first equation of each set a 2 occurs, in the second /32, and in the sixth a/3. Eliminating these in succession we have (L - M)a'2 + (L - N)a"2 = L - A , (L - M)/3'2 + (L-N)/T2 = L-B, (L — M)a(3' + (L — N)a"/3" = L cos v — c ; and thence {L-M)(L-N)(a,y8,,-a/'/5')2 = (L - A)(L - B) - (Lcosv-c)2; so that one of the nine unknowns A)(L - B) - (L cos v - (L - M)(L - N) ^)2 the others being like it, and indeed derivable from it, although Jacobi does not say so, by cyclical permutation of triads of letters. The solution thus reached we may formulate as follows : — The Cartesian equation Aic2 + By2 + Cz2+ 2ayz + 2bzx + 2cxy = 0, where the axes are inclined to one another at angles X, /x, v, may be transformed into + + - 0, where the axes are rectangular , by means of the substitution (L - B)(L - C) - (L cos a. - a)2 a2(L - M)(L - N) h / (M - B)(M - C) - (M cos l A2(M - N)(M - L) f v f (N - B)(N - C) - (X cos a. - a)2 ’I i r a2(n-l)(« -m) . j c f (L - C)(L - A) - (L cos ju - bf ~\ \ \ a2(L-.M)(L-N) f (L - A)(L - B) - (L cos v - c)2 ^4 l a-(L-M)(L-N) £ + ivhere L, M, hi are the roots of the equation x - A X COS V - c x cos fx - b X COS V — c X -B x cos X — a x cos ix -b x cos X — a x~C 1 COS V COS fX A2 t= COS V 1 cos X COS V cos X 1 252 Proceedings of Royal Society of Edinburgh. [sess. The paper closes with a reference to the case where cos X — cos //. = cos v — 0, and to the case where a=b—c— 0 ; the equation for the determination of L, M, N being in the former case x3 - ( A + B + C)x2 + (AB + BC 4- CA - a2 - b2 - c2)x — ABC + A a2 + B62 + Cc2 — 2 abc = 0 , and in the latter case A 2x3 - (A sin2A + B sin2//, + C sin2v)a:2 + (AB + BC +■ CA)aj - ABC = 0, “ welche beide Gleichungen schon sonst gegeben sind.” Jacobi (1827). [De singulari quadam duplicis integralis transformatione. Crellds Journ ., ii. pp. 234-242.] Although the title of this paper is quite unlike that of the pre- ceding, it will be seen that the two are in essence most closely related. The double integral referred to is where // sin if/-dif/-dcf> p = a + a cos2t f/ + a" sin2i f/ cos 2 + 2 b' cos \ ft + 2 b" sin if/ cos <£ + 2b'" sin if/ sin + 2c sin2 if/ cos sin cf> + 2c,/cos^sin^sin + 2c" cos if/ sin if/ cos cf> , — that is to say, where p is a quadratic function of cos if/, sin if/ cos , sin if/ sin ; and the purpose of the paper is to show that the integral can be transformed into ll sin P-0P-00 G + G' cos2P 4- G" sin2P cos2# + G ' sin2P sin2# ’ where the denominator is a quadratic function of cos P , sin P cos # , sin P sin # , but contains only the squares of these quantities. The transformation is avowedly suggested by Gauss’ solution of a simpler problem of the same kind, viz., the transformation of f 0E J VL(A - cos E)2 + (B - b sin E)2 + C2] into the form 1901-2.] Dr Muir on the Theory of Orthogonants. 253 / 0P V(G + G'cos2P + G"sin2P) * As in the preceding paper, Jacobi does not begin with the substitution which is really sought, but with the reverse sub- stitution,— that is to say, the substitution necessary for the trans- formation of // sin P-0P-8# G + G' cos2P + G" sin2P cos2# + G'" sin2P sin2<9 into // sin if/‘dif/-dcf> P ’ — knowing that from the latter substitution, when found, the former will be obtainable. This substitution he takes in the form cos P sin P cos 0 a + a cos if/ + a sin ifr cos + a" sin if/ sin cf> 8+8' cos if/ + 8" sin ifr cos + 8"' sin if/ sin ’ P + p' cos if/ + /3" sin i/r cos <£ + /3"' sin if/ sin 0 8 + 8' cos if/ 4- 8" sin if/ cos + 8"' sin i//- sin <£ ’ sin P sin 0 = Z±X G0S^ + y" sin if/ cos + y"' sin if/ sin 8 + 8' cos i ft + 8" sin if/ cos + 8"' sin if/ sin ’ the three new facients, cos P , sin P cos 6 , sin P sin 6 being ex- pressible as fractions whose numerators and common denominator are linear functions of the original facients. It rests with him therefore to prove that the sixteen quantities a, a , a , a p, p, p, r r rr /// 7> 7* 7 » 7 8, 8', 8", 8'" und the four G, G', G", G'" are so determinable that the performance of the substitution may bring back the original integral. By reason of the fact that cos2P + sin2P cos20 + sin2P sin2# = 1 for all values of P and 0 , it follows that the expression (a + a cos if/ + a" sin ifr cos + a” sin ifr sin <£)2 + ({$ + P' cos if/ + p" sin if/ cos cf> + p'" sin if/ sin <£)2 + (y + y cos if/ + y" sin if/ cos cf> + y" sin if/ sin <£)2 - (8 + 8' cos if/ + 8" sin if/ cos + 8"' suiif/sm)2 254 Proceedings of Royal Society of Edinburgh. [sess. must vanish for all values of if/ and cf> , and that therefore a number of relations must exist between products of pairs of the coefficients. These relations Jacobi might have obtained by giving special values to if/ and : for example, by putting if/ = 0 and if/ = 7r he might have obtained (a +y + y- i) + 2(aa' + /3/3' + yy - SS') + (a + ^ + / - f) = 0 and (a +/32 + y- S2) - 2(aa' + $8' + yy' - SS') + (a'2 + ^ + / - S'2) = 0 and thence act! + /3f3' + yy — 88' = 0 i 2 n2 2 ,.2 . ,2 2 ,2 ~,2v and a + /3 + y — 8 = — (a + /3 +y — 8 ) . As a matter of fact, however, taking a hint from Gauss, he con- cludes that since cos2i fr + sin2^ cos 2cf> + sm2i[/ sin2<£ = 1 , the expression must he of the form k(cos2if/ + sin2i f/ cos 2cf> + sin2i jr sin 2 - 1) and that therefore by equalisation of coefficients a2 + + f - S2 = -ky 2 CL + p2 + ,2 r - S'2 = k, a + / 3"2 + „% y - 8"2 = k, 2 a + r2 + »//2 7 - S"| = k , aa + pp + ti - 88' = 0, rr aa + PF + rr 77 - 88" = 0, nr aa + ppr + rrr 77 - 88'" = 0, rr nr a a + p'p" + rr r»r 7 7 - 8"8'" = 0, a!" a! + + y"Y - 8'"8' = 0, r rr a a + PP' + / // 77 - 8'8" : 0, where k is arbitrary.* Again, since by making the substitution in the denominator G + G' cos2P + G" sin2P cos20 + G'" sin2P sin20 The fact that these equations imply | afty'd"' | '= ±&2 is not alluded to. 1901-2.] Dr Muir on the Theory of Orthogonants. 255 a multiple of the original denominator p must be obtained, it follows that the expression G' (a + a cos if/ + a" sin if/ cos + a" sin if/ sin <£)2 + G" (P + P' cos i J/ + P" sin if/ cos + P'" sin if/ sin <£)2 + G'" (y + y cos if/ + y" sin i ft cos cj> + y" sin if/ sin <£)2 + G (8 + 8' cos if/ + 8" sin if/ cos + 8"' sin if/ sin cf> )2 must also he a multiple of p. Putting it equal to lcp , and equalising the coefficients, we obtain another set of ten equations Gra2 + G"p 2 + n'" 2 G Y + GS2 = ah , 2 Ga + G"P'2 + G"V2 + G8'2 = ah , rv "2 G a + G"p"2 + r\ /// "2 G y + G8"2 = ah , rv ///2 G a + G "P"'2 + r\rn rn 2 G y + G8'"2 = a"'h , G^a^ + G "pp' + G""yy + G88' = b'h, Gr'aa" + G"pp" + G-'yy" + G88" = b"h , G'aa" + G "pp"' + G'W" + G88'" = b'"h, G'a"a"" + G "p"p"' + G"yy" + G8"8'" = e'h, G'aa' + G"P'"P' + r'yftr fff t G y y + G8'"8' = c"h, G'aa + G"P'P" + G"’y'y + G8'8" = e'h . We have thus a score of equations from which to determine the score of unknowns, a , p , y , 8 , a , . . . , G , G' , G" , G"'. Prom this point onward the procedure closely follows that of the preceding paper. Noting that the specialising substitution -G = G' = G" = G'" = 1,\ - a = a = a" = a" = 1,1 V = b" = V" = 0,j c = c = c"' I 0,) changes the second set of ten equations into the first, he confines himself at the outset to the second set. From this four sets of equations are selected, e.g. , the set a.’G a + P-G"p + y-G y + m = ah , a *G a + P\G"P + y'-G"'y + S'-GS - b'h, cL'G’a + P" - ( a " + x)(a" + x)b'u - (a" + x)(a + x)b"2 - (a + x)(a" + x)b'"* ) + 2 cc'c"(a - x) + 2 cb"b'”(a + x) + 2 cb'"b\a" + x) + 2 c"b'b"{a!" + x) V"2 - 2 b'V'cc" - 2 b"b"'c"c'"\ - 2 b'"b'c'"c , 17 , ,2 /2 ,„2 f/2 7 n,2 +b c +b c +b PROC. ROY. SOC. EDIN. — YOL. XXIY. 258 Proceedings of Royal Society of Edinburgh. [sess.. just as if he had expanded the determinant according to products of the elements of the principal diagonal. Interrupting the process of solution for a moment Jacobi draws attention to the fact that elegant relations between the sixteen quantities a , a , a" , a" , . . . and the sixteen A , A' , A", A"' , . . . have been handed down by Laplace, Vandermonde, Gauss, and Binet, — an interesting remark as showing what writings on deter- minants were then known to him. Upon the subject of these relations, however, he does not enter, contenting himself with giving two sets of equations derivable from them with the help of the sixteen results a — - JcA , /5 = -&B, .... The first set resembles the half-score of equations obtained near the outset, being 2 2 ft 2 , ,2 — a +a/ + a + a ' =/r, — a (3 + a/5' + a"/3" 4- a" /5'" = 0 , — y8 + y'8' 4- y"8” + y'"8'" — 0 . The other set consists of sixteen of the type a/5' - a'/5 = - (y"8"' - y"'8")e , where e = ± 1 , and is in effect a prolix way of stating the fact, nowadays familiar, that any two-line minor of | a/3'y"8'" \ differs from its complementary minor only in sign, if it differ at all. Further he inserts at this stage the reverse substitution of that with which he started, viz., - 8' + a cos P + /5' sin P cos 0 + y sin P sin 0 COS if/ ~ g _ a cog P _ gin p C0S Q _ y gin p gin 0 * . f f - 8" + a" cos P + /5" sin P cos 6 + y" sin P sin 0 smi/rcos - g _ a cosP - p sin P cos ^ - y sin P sin - 8"' + a"'cos P + /5"'sin P cos 0 + y'"sin P sin 0 smi/fSin — g _ a cos p _ p sin p cos 0 - y sin P sin 0 ’ to which is added the fact that the common denominator here is the quotient of k by the common denominator in the original substitution. These results, he states, are easily proved, — doubtless 1901--.] Dr Muir on the Theory of Ortliogonants. 259 by solving the three equations of the original substitution for cos if/ , sin if/ cos cf» , sin if/ sin , or by taking the results as already found, and verifying them by substituting the values of cosP, sin P cos 6 , sin P sin 0 . On returning to the main line of investigation, viz., the solution of the set of twenty equations, Jacobi unfortunately does not proceed with the same fulness of explanation as before the interruption. In fact, the values of the remaining sixteen unknowns are merely put on record without any indication of the mode in which they have been obtained, “ brevitati ut con- sulate,” the first four of the sixteen being a (ft — G )(ft — G )ft — Gr ) — C (ft — G ) — c' (ft” — G ) — C (ft — G ) + 2 C C C ' h (G + G)(G — G^fG^ — Gr ) a (ft” — G')(ft' — G )(ft + G f — c (ft + G ) — b"' (ft” — G') — b /L(ft” — GA) + 2 b"b" e _ — (G' + G)(G' — G”)(G’ — G'”) a"- (d" - G’)(a + G')(a' - G') - c"~(a + G') - b'\d" - G') - b"'\d - G') + 2 V'b'c Jc (Or + G)(G' - G")(G' - G'") a'"’ _ (a + G')(d - G')(a" - G') - c"'\a + G') - h"'\d - G') - b%T - G') + Clh'b"c"' h (G + G)(G — G )(G — G ) and the others obtainable therefrom by the change of 2 ,2 ,,2 ,,,2 a ) aj a) a t G, G , G , G , into P*. f, G , G", G' , G'", y » yf , y'" > y"'1 , G , G'", G", G' , — 8 5 — b , — S” , — S ' , — G, — G, G, G”. The difficulty of the double sign which appears in every case is got over by merely fixing at will the sign of a , /3 , y , S , — the reason being that there are rational expressions for aft , aa” , aa'” , /3j3' , . . . , yy , . . . , SS' , . . . . and indeed also for a a' , aa' , a!’ a" , similar to those just given for a2, . . . For example, *a = b'(a" - G ')(a'" - Gr') - c"b"\a - G') - c"b"(a" - G') - 5V2 + b"c'c" + b'"c'c" . h (G' + G)(G' — G")(G — G”') 260 Proceedings of Royal Society of Edinburgh. [sess* There is nothing to suggest that the numerators of all these expressions are determinants, and still less that in the case of a aa P T* aa ~F’ aa PT a a a a X* PT a a a PT h the numerators are* the ten principal minors of * For the modern reader the following substitute for the missing demonstra- tion will suffice : — If the cofactors of the elements in the four-line determinant above given]be denoted by [11], [12], . . . , then from the equations — (a + Gr)a + b'a + b" a' + V"a!n = 0 — bf a + (a! — Gr)af + d" a" + c" a!" = 0 — b" a + c'V + (a"-G')a" + d «" = 0 + c'V + c' a" + (a'"-G')«"' = G- we have a _ a _ a" _ a" m ~ m ~ [is] ” iu]» [21] [22] ’ • • * ’ a [31] ’ [41] Multiplying in these lines by «, a, a", a" respectively we see that 2 2 2 2 a _ a _ a" _ a!" [TT] ” [22] “ [33] “ [iT] and therefore that each of them is equal to a — a r/2 #//2 a -a and thus equal to [11] -[22] -[33] -[44] - 1c [11] - [22] - [33] - [44] But by the rule for differentiating a determinant the denominator here is the differential -quotient of the determinant with respect to G' ; and this because of the theorem clx {{x-rx)(x- r2)(x-r3) ... } (?\ - r2)(r1 - r3) . . . is equal to - (G' - G")(G' - G'")(G' + G) : consequently k _ a2 (G'-G")(G'-G'")(G' + G) “ [11] _ 1901-2.] Dr Muir on the Theory of Orthogonants. 261 a + G' V b" V" V a - G' c" c b" c ' ci' — G c bm c" c a" - G' . The next and concluding paragraph of the paper is of course occupied in showing that by making the substitution whose coefficients have just been obtained, the given integral can he transformed as desired. It is worth noting here that although this paper and the previous one are contiguous in the original volume of publication, and the problem solved in the second is in essence quite similar to that solved in the first, there is not a word to indicate that the author viewed them in this common light. Cauchy (1829.). [Sur l’equation a l’aide de laquelle on determine les inegalites seculaires desmouvements des planetes. — Exercices de Math., iv. ; or, CEuvres , 2e ser. ix. pp. 172-195.] The equation as it arises with Cauchy would be more fitly described as the equation whose roots are the maxima and minima of a homogeneous function of the second degree with real co- efficients, and with variables subject to the condition that the sum of the squares equals unity. Denoting the function by + AyiJy2 + AZ!z2 + • • • + 2Axyxy + 2Axzxz + • • • • , or for shortness’ sake by s , he of course begins with the known equations for determining the extreme values in question, viz., the •equations 0s ds ds dx _ dy dz x ~ y ~ 7 ~ An elementary algebraical theorem gives each of these ratios 0S 'dx 0S 0S + y^~ + + • • Jdy dz x1 + y2 +z2 + 262 Proceedings of Royal Society of Edinburgh. [sess. and therefore by the fundamental theorem regarding the differentiation of homogeneous functions and by the above- mentioned condition \= 2s . He thus obtains the set of equations or ^ = sy’ te = S2’ !0Z (Rxx - S)X + RxyV + RxzZ + • * • = 0\ A y& + (A yy-s)y + Ayzz + ••• = ()! Azxx + Azyy + (Azz -s)z + • • • = 0 [ and therefore concludes that, on eliminating x , y , z , . . . from the set, the resulting equations in s , S = 0 say, has for its roots the maxima and minima values of s. The third chapter of the Cours dJ Analyse is then referred to and taken as warrant that “S sera une fonction alternee des quantites comprises dans le Tableau Axx s Axy Axz .... A A q A ■^xy -“2/2/ ° .... Axz Ayz Asz-s .... and the developments of the function are given for the cases n = 2 , n = 3 , n = 4 exactly in the form adopted by Jacobi. The question of the particular values of the variables x , y , z , . .. * which correspond and give rise to each of the n extreme values of s is next taken up, the equations for the determination of them being clearly the set from which the equation S = 0 was obtained (a set, be it remarked, which of itself can only give the ratio of any two) and the additional equation x2 + y2 + z2 + .... = 1 . A series of identities connecting these n2 values is however first obtained. Denoting by xr , yr , zr , . . . the values of x , y , z , . . 1901- .] Dr Muir on the Theory of Orthogonants. 263 which corresponds to the extreme value sr of s, he has, by a double use of each equation of the set, the n pairs of equations {-h~xx ~ ®i)^i d" -h-xyl/ 1 d" A xzZx + ... — 0 ) (Aaa — *.2)X2 d- A xyy2 + A xzz2 + ... = 0 j A Xyxi d- (Ayy ~ sfyx +. A y^l d- ... = 0 A xyX2 + (A yy ~ S g)?/ % + A + ... = 0 I A3afc1 + Ayzy1 + (Azz - s1)^1 + ... = 0 \ Axzx2 + A yzy2 + (Azz — s2)z2 + ... = 0 i From the first pair Axx can be eliminated, from the second pair A yy , and so on. Consequently there is in this way obtained the n equations (s2 - + Axy(x2y1 - xxy2) + Axz(x2zx - xxz2) + • • • = 0 AXy(y2x1-yix2) + ('q2-si)yit/2 + AyZ(y2z1-y1z2) + ■•• = 0 AUz2x1-ZiX2) + Azy(z2y1-z1y2) + (s2-s1)z1z2 + ••• = 0 and from these by addition (x1x2 + y1y2 + z1z2+ ... )(s2-s1) = 0, the conclusion being “ Done, toutes les fois que les racines sx , s2 seront inegales “entre elles, on aura xxx2 + yxy2 + zxz2 + ... = 0 ; “ et, si Inequation S = 0 n’offre pas de racines egales, les “ valeurs de a?, y , zt . . . correspondantes a ces racines “verifieront toutes les formules comprises dans le Tableau “ suivant : r\ + V \ + ••• = 1 ,x1x^ + y1y2+ ... = 0, , X]Xn + Vlyn + • = 0 Vi + yyj j + = 0 , x2 + y.y1 + ••• = 1, , x2xn + yyUa + - - • = 0 + VnVi + ■■■ = 0,xnx2+yny2+ ... = 0, , x* + y* + ... - 0”. This interlude over, the fundamental set of equations is returned to, and, the first of them being deleted, there is got from the remainder 264 Proceedings of Royal Society of Edinburgh. [sEsa. x y z P = ~p~ = ~ p~ = ■*- XX -L xy -1- xz where the denominators are seen to be what we now call certain ‘ principal minors ’ of S ; or, as Cauchy says, where Vuv is “ce que devient S, lorsqu’on supprime dans le Tableau “ les termes qui appartiennent a la meme colonne horizon- “tale que le binome A uu-s, avec ceux qui appartiennent “ a la meme colonne verticale que Avv - s , ou bien encore “les termes compris dans la meme colonne verticale que “ Auu — s , et ceux qui sont renfermes dans la meme colonne u horizontal que Avv - s .” The ratios x : y : z : . . . . having thus been got, there only remains, for the determination of x , y , z , ... , to use the equation «2 + ?/2 + z2 + ... - 1 . But before doing so it is temporarily convenient to introduce an alternative notation, viz., denoting the signed minors P - P - P xx > yy ) L zz » • • • • by X , Y , Z , . so that the values of these corresponding to xr , yr , zr , .... , and therefore to sr , may be denoted by Xr , Yr , Zr , . . . We thus have from the additional equation x y z \ X = \ = Z = ' ' ' ‘ = * fX- + Y2 + Z2 + . . . and therefore ^ BB?/i --1 = ..... = + 1 Xj Yj z, + y22+ z22 + • • • *2 _ y%_ _ % _ _ + l x2 y2 z2 ~ VX22+Y22+Z22+ • • • _ .'/« . . _ _ + f x. y» z» ' _ \/x„2 + y„2z„2 + • • • • Of course this supposes that the special values of Xx , Y1 , Zx , . . . occurring in the denominators do not vanish ; and Cauchy’s con- clusion therefore is 1901-2.] Dr Muir on the Theory of Orthogonants. 265 “les expressions *^i » Vi ’ % 5 • • • • x2 ’ V2 ’ % ’ • ‘ ' ' “seront, aux signes pres, completement determinees . . . . , “ a moins que des racines de l’equation S = 0 ne verifient en “ meme temps la formule P« = 0 The next step is to prove that the roots of the equation S = 0 are all real so long as the coefficients of the quadratic s are real. If ihe contrary he supposed, viz., that one of the roots sp is of the form \ + fi J - 1 , this will of course entail the existence of another sq of the form A. - /x J — 1 . Also, Xp being the same function of sp , that Xs is of sq , it will follow that Xp and Xq will be of the form M + NV^I , M-N^l and therefore that X,X2 = M? + jSt* This means that XpXq will be positive or zero, and similar reason- ing would prove the same regarding YpYq , ZpZq , . . . None of them, however, can be positive ; for since % + VpVq +•••* = 0 , it follows from the values obtained for xp , xq , . . . , that XpXg + YpYq + • • • = 0 . And since they are all zero, and each the sum of two squares, we are forced to the conclusion that = Xg = 0, Y„ I Yg = 0, Zp = Zq = 0 , which is the same as to say that the roots xpi xp satisfy the equations 0 266 Proceedings of Royal Society of Edinburgh. [sess. The supposition therefore that the equation of the nth degree S = 0 can have a pair of imaginary roots leads us to assert that a perfectly similar equation, Pa.x = 0, of the (n - l)th degree, will have the same pair of roots. It is thus seen that the supposition and reasoning, if persevered in, will ultimately land us in an absurdity, when we reach, as we are bound to do, one of the equations of the first degree Rxx ^ = 0 5 Ryy ® = 0 , ’ “ Done l’equation S = 0 n’a pas de racines imaginaires.” The next object being to fix the limits between which the roots of the equation S = 0 are comprised, a theorem necessary for the accomplishment of this is first attended to Formally enunciated in modern phraseology it is : — S being any axisymmetric determinant , E the determinant got by deleting the first row and first column of S, Y the determinant got by deleting the first row and second column of S, and Q the deter- minant got from E as E from S, then , if E = 0, SQ = -Y2. As the mode of proof employed by Cauchy applies equally well when S is not axisymmetric, let us take | a1b2c2d4: \ for the given determinant, and write the proof as it would nowadays be given. To begin with, if A1 , A2 , ... be the complementary minors of the elements nq , a2 , ... in | afi.f %d | we have al A1 - a2 A2 + as A3 - a4A4 = \afiffif, j &iA4 — b2A2 + &3A3 — 64A4 = 0, ^iAj c2R2 + C3A3 c4A4 0 , d^ A4 — d2 A2 + 6?3A3 — d^ A4 = 0 . J Flitting A4 = 0 , and leaving out one of the last three equations, we- obtain — a2 A 2 + oqAg — a4 A4 = | af)2czd± | , 'j “ c2A.2 "H C3A3 c4A4 0 , - d2A2 + dB A3 - d± A4 = 0 j from which by solving for A2 there results A _ ~ 1 uf)2cfi^ | • [ cfi4 2~ I I 1901- .] Dr Muir on the Theory of Orthogonants. 267 that is, I a2C2,^4: I ’ I ^16*3^4 I = “ I a1^2C3^4 I ' i C3^4 I and this, when the original determinant is axisymmetric, becomes I ^lC3^4 I 2 “ “| af-2C^i i * I C3^4 ! ’ or, as Cauchy writes it, -Y2 = SQ. The first four cases of S = 0 are then considered, viz., the series- of equations S^O, S2 = 0, S3 = 0, S4 = 0, or, as] at a later date they would have been written, Auu - i >“ = o, Azz - S A zu = °, Ku A; ^ ~ S Ayy ~ 8 A^ Ayu A yZ Kz- s Azu = 0, Ayu Azu Auu - s A»-s ^xy Kz Ku A Xy Ayy - S Ayz Ayu = 0, A „ K Kz- s Azu A xu A yu Ku Amm — s where each determinant is the complementary minor of the element in the place (1, 1) of the next determinant. The root in the first case is evidently Auu . In the second case the solution is s = i{ + Auu + V ( Azz - A uu)2 + (2AZU)2 f, where the reality of the roots , s2 is manifest ; and as their sum is Azz + Auu , it follows that s2- Auu may he substituted for A zz.-s1 in with the result that we have (Amm - s2)(Amm = — Azf and are able to conclude that the roots s1 , s2 of the equation S2 = 0 lie on opposite sides of the root Auu of the equation Sx = 0 . * We know from a later theorem (Jacobi, 1833) that when Ax is~not 0 the identity is AiB2 | = | I • I c3d4 | . 268 Proceedings of Royal Society of Edinburgh. [sess. Coming now to the case of S3 = 0 we proceed differently, ths three roots being localised by observing the changes of sign in S3 as we pass from one value of the variable s to another. Four values of s which suffice for the purpose are - oo , s4 , s2 , + oo . No reasoning is necessary to show that, when s is = — oo , S3 is positive, and when s = + co , S3 is negative. When s = s4 we have 52 = 0 , and therefore know from our auxiliary theorem that S4 and 53 must have different signs, — a fact from which we deduce that S3 is then negative. Similarly, when s = s2, it is seen that S3 is positive. We thus have the set of values s = - oo } s1 , s2 } + oo , and S3 = + j — , + , + , which shows that one value of s which makes S3 = 0 lies between - oo and s1 , another between Sj and s2 , and the third between s2 and + oo . In other words, the roots s', s", s'" of S3 = 0 are such that between each consecutive two of them there lies a root of S2 = 0. The case of S4 = 0 is treated similarly, the five values given to s in S4 being / rr rn -CO , S , S , S , +00. As before, there is no difficulty about the first and last of these, the value of S4 being seen to be positive for both. When s is put =s' we know that S3 vanishes, and that therefore S2 and S4 must have different signs. The sign of S2 is settled from recalling that s' lies between — co and s4 , and that for these values of the variable S2 is equal + oo and 0 respectively : consequently the putting of s = s' makes S4 negative. Similar reasoning enables us to complete the set S = -co , s' , s" , s'" , + CO 1 S4= + , +i from which we learn that one value of s which makes S4 = 0 lies between - co and s', a second between s' and s", a third between s" and s'", and the fourth between s'" and — co . Having reached this point Cauchy adds — (t Les memes raisonnements, successivement etendus au cas ou la fonction s renfermerait cinq, six, . . . , variables, fourniront evidemment la proposition suivante : 1901- .] Dr Muir on the Theory of Orthogonants. 26$ Theoreme I. — Quel que soit le nombre n de variables x, y,w z , . . . V equation S = 0 et les equations de meme forme R = 0, Q = 0, .... auront toutes leurs racines reelles. De 'plus, si I’on nomme s', s", s'", . . . , s(n-1} les racines de V equation R = 0 rangees par ordre de grandeur , les racines reelles de V equation S = 0 seront respectivement comprises entre les limites oo , s , s , s 00 . Considerable space (pp. 188-192) is next given to extending this theorem to the case where several values of s satisfy at the same time two consecutive equations of the series S = 0 , R = 0 , P = 0, .... Then follows a series of noteworthy deductions, which bring us round to the solution of a general problem of a quite different, character, viz., the problem of transformation which we have seen Jacobi attacking in detail. Denoting, as before, the extreme values, all different, of the quadratic function A^2 + A yiJy2 + • • • + 2Axyxy + • • by s± , s2 , ... , sn , and by xr, yr, zr, ... the values of the independent variables which give rise to sr , we know that we have (A» -*iK + A „/j1 + + • • = 0 + {^-yy ~si)Vi + Ay^Z 1 + • . = 0 + Kvi + (A« - siK + • • k o xf + Vi + ^2 + • = i (A« - S2)x2 + A xyy2 + AX7z2 + • • = 0 AxyX2 + (K ~ hYdz + Ayzz2 + • • = 0 Axzx2 + A yzy2 + {Kz - hK + • • = 0 ry » 2 + v-2 + z22 + • . = 1 ) V 270 Proceedings of Royal Society of Edinburgh. [sess. (Rxx - sn)xn + h.xyyn + A xzZn + * ' * =0 A xyXn (A yy Cll)yn 4" A yiZn ■+* ' =0 AxzXn + KyzVn + (A zz-Sn)zn + • • ■ =0 xf + y2 + z2 + • • • = 1 •and that, further, when r and s are unequal xrxs + yrys + zrzs + • • = 0 . Recalling this, Cauchy says that if a new set of n variables be taken £ 5 V » £ 5 • • • • related to the old by the equations X = xf "4* X^X] -J- ~h ' ' ' y = y i£ + vw + y£ + • • • z M zf + z2r] + z3£ + • • • it is at once verifiable that x2 + y2 + z2+ • • • = £2 + ^ + i» + . . . In the second place, if we take any one, say the first of the set of •equations connecting , xx , y1 , zlt ... , the corresponding equation of the set connecting s2 , x2 , y2 , z2 , ... , and so forth, writing them in the form A^^q + A xyy^ + AX7z^ + • • • = s^x^ , RxxX2 + A ^2 + A xzZ2 + • • • = S2£2 , multiplication by £ , g , £ , ... respectively, followed by addition, gives Axxx + A xyy + Axyz + • • • = sxx^ + s9x2tj + s3x3£ + • • • \ A xyx + A yyy + A yzz + • • • = slVf + s2y2rj + s3y3C + f Axzx + A,;Jzy + Azzz + • • • - sfat + s^2rj + s3z3£ + • f In the third place if the equations giving x , y , z , ... in terms -of £ , y , t , ... be taken, multiplication by irr , , . . . respec- tively, followed by addition, gives 1901—2.] Dr Muir on the Theory of Orthogonanls. 271 £ — + V\V + z\z + rj = x2x + y2y + z2z + l = + yzy + zzz + In the fourth place, if we take the second of these derived sets of ■equations, multiplication by x , y , z , . . . respectively, followed by addition, gives Axxx2 + A wy2 + • • ■ + 2 A XIJxy + • • • • = s,£2 + s2f + ss£2 + ... With these results before him Cauchy is led to formulate the following proposition previously given “dans le dernier volume -des Memoires de V Academie des Sciences ” : — “Theoiteme II. Etant donnee une fonction homogene et du second degre de plusieurs variables x , y , z , ... , on peut toujours leur substituer d’autres variables £ , rj , £ , ... liees a x, y , z , .... par des equations lineaires tellement clioisies que la somme des carres de x , y , z , ... soit equivalente a la somme des carres de £, rj , l, ... , et que la fonction donnee de x , y , z , ... se transforme en une fonction £ , rj , £ , . : . . homogene et du second degre, mais qui renferme seulement les carres de £ , 77 , £ , ... ” The validity of this rests on the supposition that the equation R, = 0 has all its roots unequal ; but Cauchy is careful to point out that even if this were not the case, the requisite inequality ■could be brought about by giving an infinitely small increment e to one of the coefficients Axx , Axy , ; and as e could be made to approach indefinitely near to zero without the theorem ceasing to be valid, the validity would remain even at the limit. After a reference to the special case of three variables, the paper closes with the announcement that Sturm had arrived independently at the theorems marked I. and II., and had offered his paper on the subject to the Academy on the same day as Cauchy’s.* * A short account of Cauchy’s memoir is given in the Bulletin des Sciences ■Math., xii. (1829), pp. 301-303, by C. S(turm), who says, “M. Cauchy a bien voulu observer, en terminant son article, que j’etais parvenu, de mon cote, a 272 Proceedings of Royal Society of Edinburgh. [sess. Jacobi (Deer. 1831). [De transformations integrals duplicis indefiniti ’ d

aid' + PP' + yY = o, 2 a + d + / 1 a' a + P'P + YY = o, "2 . a + d + y"- = aa + pp + YY = 0; that from these and the given substitution we obtain the reverse substitution s H ax + fty + yz , j s = a'x + ft'y + yz, s' = d'x + /3"y + y'z ; J and that this latter substitution when taken along with the original condition gives the second set of six relations a + a" + a"" = 1 , fty + ft'y' 4- ft"y" = 0 , ft + ft' + ft" — 1 , ya + yd + y"a" = 0 , y + y + y"~ ==• 1 , aft + a! ft' + aft" = 0 . Further, it is pointed out that if we put € for a(ft'y" - ft"y) + ft(y d' ~ y"a!) + yW ft" ~ a"ft') the ordinary solution of the given substitution results in es §= x(ft'y" - ft"y) + y( ya - y"d) + z( aft" - a" ft') \ eS = x(ft"y - fty" ) + y{y" a - ya" ) + z( a" ft - aft" ) l €S" = X(fty - fty ) + y(ya - y'a ) + z( aft' - aft' ) j , and that a comparison of this with the reverse substitution as already obtained produces ea = fty" -Fy, ea | ft"y - fty", ea" - fty' -Py eft = ya" - y'a-', *P = y"a - ya-", eft" = ya' -y'a "s II W - a." ft', £ 7 = a" ft - aft", ey" — aft' - aft In the next place it is noted that with the help of these the left side of the identity (y"a - yd')(aft' - a ft) - (ya - y'a) (a" ft - aft") = ae 1901-2.] Dr Muir on the Theory of Orthogonants. 275 becomes first and then and that consequently •W-jSV) e2 • ea \ e2 = 1 . Lastly, attention is very pointedly drawn to the fact that if the nine quantities a, a , a, ft , ft', ft", y, y, y" be such as the foregoing results imply, and any three quantities X, Y, Z be connected with other three P, Q, R by the equations X = aP + a'Q + a"R ] Y = ftF + ft'Q + ft"R> Z = yP + y Q + y"R j then it follows that P = aX + ftY + yZ ] Q = a'X + ft'Y + y'Z [ R = a"X + ft'’ Y + y 'Z j and X2 + Y2 + Z2 = P2 + Q2 + R2 .* (0) The next preliminary step is to formulate the equations which result from the identity of (Ax + By + C z)to + • • • • with G st + G's'^ + G"s 'v . These are f A :?= Gatt + G'ab +G "a'c B = G/3a +G ' g!b +G"j3 ”c C = G yCL +G 'y'b +G "y"c A' = GaClt + G' o!b' -\-G''a"c' B? — G^a' -r G7/3 V + G" fi"c! C' = Gya' 4- G'y'b' -\-G"y"c' A' — Gaa" + G'ab" + (Y a'c" Bw = Gj8a" + G f&'b" G” f& "c" G" — Gycc' + G'y'b" + G"y"c' . Along with the twelve relations previously obtained, they give in all twenty-one equations for the determination of the three G’s, and the eighteen coefficients of the substitutions. The actual process of solution consists in a long series of deductions from the last-obtained set of nine equations, the repeated use of the twelve other equations being disguised by * In leaving these preliminary deductions, it may be worth remarking that the like results which flow from the second given substitution and its associated condition are not taken entirely for granted by Jacobi, but are given with equal fulness, the two series indeed appearing in parallel columns. t v. next page. 276 Proceedings of Royal Society of Edinburgh. [sess. employing the theorem above called (0). Thus from the first column of equations this theorem gives Ga = a A + a A' + a"A!‘ G a = bA + b A + b A G a = cA + c A + c A ; j the second column gives similar expressions for G f3 , G'/T , G "13" ; and the third column for Gy , G'y , G"y". The whole set is in later notation Ga G'a G"a GjS G'/T G ,fp Gy G^yr Gr,yr/ where 9, h, h a a’ a" b b' b" , „ c c c ) aTa/_a" A A' A" A A' A" a a a" b V b" c c c B JB' B' B B' B" B B' B" a a a" b V b" c c c" C C' C" C C' C" C C' C" > denote gp + h cr + Jct . Similarly by tak- ing the same set of nine equations in wws there is obtained ( Ga G b G ' c Ga G'b' G V Ga G'b" G 'c" a p y a P y a' P' y A B c A B c A B c a P y a P y a' P' n y A' B' G' A' B' C' a: B' C' a P y a P y a' P' y" A" B" G" A" B/# C" A" B" G" From these two sets of equations it is clear how the coefficients of one of the substitutions may be obtained when the G’s and the coefficients of the other substitution have become known. Separating the latter of these new sets of nine in a similar fashion into column-sets of three, but solving this time in the ordinary way, Jacobi obtains a further set, which, if only to save space, we may write in the form t Jacobi writes the nine equations in one column : they are better arranged in three, however. Cayley at a later date would have preferred to write more luminously ABC ) ( (G , G' , G "\a , a' , a\a , b , c ) ,t ,e) . . ■ ) b b — (G , G? , Gf,^a , a' , a"§af , V , c') (G , G' , G"^ , /S' , /3 "\ar ,b' , c') ... 1901-2.] Dr Muir on the Theory of Orthoyonants. 277 A a A a A a" ( | aB’C" | | &B'C" | . ) j cB'C" j TT G' ' Al G M G A/T G" = | aC'A" | | hC'A" | | cC'A" | Ay G Ay G A/ G" | «A'B" | | bAB" | | cAB" | where A = | AB'C" | , or, as Jacobi of course writes it, A = A (B'C" - B"C') + B(C'A" - C"A') + C(A'B" - A"B') . From a set giving the Italic coefficients in terms of the Greek coefficients we have thus got a reverse set. The other reverse set obtainable in the same way need not be given; but it is easily seen that the two have the same practical value as the two from which they are derived. To make another advance, either of our latest sets of nine is taken and separated into row-sets of three, and theorem (0) applied. The result which Jacobi gives in nine separate equations of the type B'C" - B"C' _ a a a b aTc_ A G G' G" may be written more compactly and more instructively in the form | B'C" | | C'A" | | A'B" | 1 ( ^ aCL a'b a"c + G" Ba £'& . y3 "c ya ,y'b ,y"G G G' ^G" A A A G +G' G + Gr + Qir | B"C | | C"A | 1 A"B I acd a'b' G G' a"c' _i &a' j8 'b' /3 V yd' y'b' y V A A A + G" gT + G, G„ G G' ^ G" |BC' 1 1 CA' | | A"B' | acd' a'b " [ G G' a"a" Ba" ’ B'b" B"d' ya" y'b" y'c" A A A + G" Gr G' + G" G G' G" Any one of the nine here, however, may be matched by one deduced directly from the set of nine which we obtained at the very outset. Thus* * Nowadays we should rather put | B'C" | — I G/3cd + G'/3'b' + G"j8"c' Gfial' + G A'b" + G" f$”c" \ I G yo! + G'y'b' + G"y"c' GyCt" + G'y'b" + G"y"ti" , I G/3 G'/3' G"J3" I i a! b' c' I I Gy Gy G"y" I I a" b" c" \ , — GG' ■ | fiy' \ ’ | a'b" | + GG" * | I3y" | * | a' d' | + G'G" * | Ay" \ * [ b'd’ | . 278 Proceedings of Royal Society of Edinburgh. [sess, B'C" - B"C' = (G /3a + G'p'b' + G "p"c)(Gya" + G'yb" + G "y"c") - (G ^a" + G'p'b" + G"P"c")(Gya + G'y'b’ + G"y"c) , =-■ G'G'X/3'y" - p"y')(b'c" - b"cr) + G"G (p"y - py" )(c 'a - c"a) + GG' (Py -P'y )(ab"-a"b') — G G att 4- G^Ga b + GG'a'c . With this we have to compare A A , A „ aa + o + ^j,a c , the result being that we obtain GGG" = A , and thus reach the first resting-stage on our journey. At the outset of the next stage it is found desirable, for brevity’s sake, to introduce six additional letters to denote certain functions of the known quantities A , A', A", .... viz. p for a2 + B2 + c2 , q for A' A" + B'B" + C'C", p for A'^ + B/2 + C'2, q or A"A + B"B + C"C , p" for A"2 + B"2 + C"2, q" for A A' + BB' + C C\ These are said to entail the six identities p'p" - q = (B'C" - B"0')2 + (C'A" - C"A')2 + (A'B" - A"B')2, VV - S'2 = (B"C -BC")2 + (C"A -CA")2 + (A"B -AB")2, VP - ?"2 = (BC' - B'C)2 + (CA' - C'A)2 + (AB' - A'B f, qq"~pq = (B"C - BC" )(BC' - B'C ) + (C"A - CA" )(CA’ -C'A ) + (A"B - AB" )(AB' -A'B ), q"q -p'q' = (BC' -B'C )(B'C" - B'C") + (CA' - C'A )(C'A" - C"A') + (AB' -A'B )(A'B" - A"B'), qq' -p"q" = (B'C" - B"C')(B"C - BC" ) + (C'A" - C"A')(C"A - CA" ) + (A'B" - A"B')(A"B - AB"), and A2 = 19V V -M2 ~PP2 - P V2 + ^qqq • The original set of nine equations, giving A, A', A", ... in terms of the three G’s and the coefficients of the substitutions, is then returned to, and the following equations derived, — 1901-2.] Dr Muir on the Theory of Orthogonants. 279 2 2 2 2 2 2 p = G a 4- G7 b 4 G" c , ‘2 fi ,^fiy2 y/fi 'l p = G a 4* G b + G c , 9 2 9 ,9 9 9 p" = G a" + G r + G c , 2 2 2 q = G a a" 4- G' bb" + G" c'c", = G d'a 4* G b b + G c c, 2 , fir tfi t q G ad + G bb + G cc ; the first three being got by use of the second part of theorem (0), but all of them readily verifiable by merely substituting the said values of A, A', A", ... In exactly the same way from another set of nine equations, viz., those beginning ^ = (B'C"-B"C> + (B"C - BC")a' + (BC'-B'C)a", there is obtained pp" - q2 a 2 4- fP + c2 9 > A2 G G'2 G" pp- q2 fi 7 fi 2 a 4- b + C 9 T 9 i A G Gr G" pp - q r/2 b" „2i a + C 2 ~ 2 .9 + 9 ? A G G G" qq -pq da" b'b" cc" o A G 4 ¥2 4* qq - pq a"a 4- b"b n C C A2 ) + (C'A"-C"A '),j + (A'B"-A"B» + [(BTC - BC" )x + (C"A -CA ")y + (A"B - AB" )?J\io + [(BC' - B'C )x + (CA' - C'A )y + (AB' - A'B )z]w" — G G st + G "Ga'u + GGV'y . This means, in modern phraseology and nomenclature, that the linear orthogonal substitutions lohich cha7ige X y z A B C w A' B' a w A'' B" C" w (jst + G ' su + G'tf'v will at the same time change x y z | B'C" | | C'A" | | A'B" | | B"C | j C"A | | A"B | | BC' | | CA' | | AB' | w w into G'G "st + G"Gs u + GG ^ v . In parallel columns with these results regarding the p’s and g’s Jacobi places a series of others perfectly similar to them, the twin series originating in the fact that in squaring | ABC" | , as we should nowadays put it, the multiplication may be performed either row-wise or column-wise. The chief points in the second series we may state rapidly in modern compact form as follows. By way of defining the new letters introduced we start with l n m n m V ml'n . ) A2+ A'% A"2 AB + A'B' + A"B" AC + A'C' + A'C" BA + B'A' + B"A" B2+ B'2+ B'2 BC + B'C' + B"C' CA + C'A' + C"A" CB + C'B' + C"B" C2+ C'2+ C"2 282 Proceedings of Poycd Society of Edinburgh. [sess. whence it follows that the determinant of either matrix is equal to | AB'C" | 2 , and the secondary minors equal to ( I B'C" | B'C"| | A'B"| + ... | C'A" 1 1 A'B" | + . . . i a'b" i2+ ... ( , ) ( l n' m' n' m V m! l' n i2 + 1 B"C I2 + 1 BC'|2 | B'C" 1 1 C'A" I + 1 B"C 1 1 C"A \+\ BC' 1 1 C A' | | C'A" I2 + | C"A |2 + ICA'I2 Then from the original set of nine equations we have G a + G' a'~-fG" a" G a/8 + G' a'jS' + G" a' 3" G ay + G ay + G"a'y"' G~0~ + G' 3' +G 3"^ G ^y + G' 3'y* ^G"2 3" y" •2 2 2 2 ,2 2 G y + G y + G" y and from this, in passing, by the addition of diagonal elements, 2 2 2 l 4- m + n = G + Gr + Cl ' . Next, as the matrix on the right ( . ) ( there follows G a Gc'^a' G” a a P y Gr2ft G,2d' G"2/T 2 - .9 ..9 .. a p y Gy Gy Gy a" /?" y" < , - . ' , ) ( ) ( n ) G a G a G a l n m a a a G2£ G’2P G'2/3" ' = n m l' ft ft' ft" G27 G'V G'V m l n y y' y" ^ la + nft + m'y la + nft' + m'y la" + n ft" + my" na+mft + l'y na! +mft' + l'y na" + mft" + ly" m a + lift + ny m a + V ft' + ny m a" + V ft" + ny" whence, by summing the squares of the elements of each row separately, we have /N„ 2 ,2 ,-,,4 ,-/2 72 ,2 ,2 \ G a + G a + G a = / + n -J- Wi , j r^n o2 '4 nr2 a /;4 n"2 2 . 7'2 , '2 L up + 6 p + G p = m + / + , j u y + G y + G y — n + m + Z Among the results obtained up to this point, there are sufficient to determine the twenty-one unknowns, and to this Jacobi now definitely devotes a section (§ 14). First the G’s are dealt with. There having been obtained 1901-2.] Dr Muir on the Theory of Orthogonants. 283 9 9 9 G + G' + G" _ ,9 ..9 o o 2 2 G G + G 2G2 + 6 6 = l + m + n = p + p +p" , 2 , fr 2 = ( mn - V ) + ■ • • • = {pp -q) GG'G = A + • • * 2 2 2 it is perceived at once that G] G' , G"“ are the roots of the equation 39 2 2 ,2 # - af (Z 4- m + rc) + x{mn + nl + Im - V - m ' - n ) , , ,2 ,2 ,2 - (/77m + Tim n -ll - - nn ) = 0 , or 3 2 t n tun 2 ,2 |,2 x - x (p+p +p') + x(p'p" +p"p + pp -q — q — q ) n f ft 2 ,2 9 ,/2 - (j)p p" + 2qq q" - pq -pq -p q = 0; which respectively are the same as 2 2 2 (a? — — m)(a? — n) — l' (x — l) — m {x — m) — n (x-n) - Tl mn — 0 , 2 /2 /r2 tf f n (x -p)(x —p)(x - p") —q{x -p) - q (x - p) - q (x -p") - 2 qqq" = 0 ; and either of which is ^3 _ a;2(A2 + B2 + Q2 + A'2 + B'2 + Q’2 + A"2 + B"2 + Q-2) ( ( B'C" - B"C')2 + (O' A" - C"A')2 + ( A'B" - A"B')2 ] + xl +(B"C -BC")2 + (C"A -CA")2 + (A"B -AB")2 V ( + (BC' -B'C )2 + (CA; - C'A )2 + (AB' -A'B )2 j - { A(B'C" - B"C') + B(C'A" - C"A') + C(A'B" - A"B')}2 = 0 . As an alternative to this, however, it is pointed out that we might, by putting the equations G2a = la + n /3 + m'y G 2/3 = n'a + m/3 + l'y G2y = m2a + Z'/3 +ny f 0 = (l — G2)a H- n’p+ m'y in the form J 0 = n'a + (m - G2)/3 + Ty [0 = m a + l f$ + {n — G2)y f eliminate a, /3 , y and obtain a cubic in G2; then by similar action ,2 ,,2 obtain the same cubic in G and the same cubic in G . In this 22 way the left-hand side of the equation, whose roots are G , G'", o G"J, would naturally recall determinants, although Jacobi does not say so; and after Cayley (1841) it might have been written t x n m p - x q" q n m — x l' or q" p — x q m V n — x q q p" - x 284 Proceedings of Royal Society of Edinburgh. [sess. In the next place, four equations having been found in a, a > 2 a" . viz., mn - V A2 a + a + a = n2 2 tjr a + .2 .2 G a + p/,2 //2 lx a — 1 2 1 ,2 1 „2 ~~2a G + g/2 a a fj ay f P 7 /3'2 Pi 72 /r2 p'Y 7 5 7 3 y" and the corresponding denominators, (G2-G'2)(G2-G"2), (G'2-G"2)(G/2-G2), (G"2-G2)(G"2-G'2) 1901— .2] Dr Muir on the Theory of Orthogonants. 285 2 ,2 As an alternative to this process for finding a , a , . . . there is given another, which in some respects is the more interesting of the two. Beginning with a different set of equations, viz., the set (l - G2)a + rifi + m'y = 0 j n'a + (m - G2)/3 + Ty = 0 m'a + T/3 + (n - G2)y = 0 ) Jacobi drops out the first and finds a : /3 : y , drops out the second1 and finds /3 : y : a , drops out the third and finds y : a : /3. Then since these three sets of ratios are the same as the three sets a2 : a fi : ay , jS2 : /3y : /3a , y2 : ya : y/5 ; and as the expressions found proportional to aft , ay in the first set are respectively equal to the expressions found proportional to the same unknowns in the other sets ; it follows that a2 , a/5, ay P, Pi O T are proportional to (m - G ){n - G ) - 1 , l m - n (n - G ) , riV - m'(m - G2) , (n - G2)(Z - G2) - m 2 , mV - /'(/ - G2 ) ,. (I - G2)(m - G2)'- «2; and therefore that a2 aB — 9 o 5 or — 9- (rn — G 2)(n — G") - V I'm - n'(n - G ) a2 + ft2 + y2 or (m — G"j)(?2> — G“) + (w — G )(Z — G ) + (Z — G )(m — G ) — l' — m' — tz' Here, however, the numerator is equal to 1 : and the denominator, being obtainable by differentiating 9 9 2 - l)(x — m){x - ri) - V (x - 1) - - m) - n'~(x - ??,) - 2 Vmri with respect to x, and substituting G2 for x in the result, must be what is obtainable in the same way from ( x — G")(.q? — G' )(x — G ) and therefore must be equal to (G2-G,2)(G"-G//2) There thus result the same values for a2 , a/3 , . . . as before. '286 Proceedings of Boy al Society of Edinburgh. [sess. The values oi a2 , aa , ... are throughout given side by side with those for a2 , a/3 , .... ; thus — 2 1 (G~ - m)(G~ - n) - r _ (Q2 - j/)(G' -p") - ^ 9900) 9 9 9 9 * (G - G' )(G - G" ) (G - G' )(G - G" ) At this point “ Problema I.” stands fully solved : one or two interesting addenda, however, are given in a concluding section {§ 15). From the equations Ga = Aa + B/3 + Cy , s = ax + py + yz , G 'b = Aa + B/3' + Cy , and s' = ax + /3'y + y'z , G"c = Aa" + B/3" + Cy" , s' = a'x + (3"y + y'z , Gc& — by multiplication and addition* there are obtained Ax + By + C z — Gas + G'&s' + G "as" , A'x + B y + C'z = Ga's + G'b's + G "c's" , - A"x + B "y + C "z = Ga"s + G'b"s + G "c's" ; •and then from these by the second part of theorem (0) (Ax + By + Gz)2 + (A'x + B'y +■ G'z)2 + (Ai'x + B"y + G"z)2 9 9 9 9 9 9 = Gs + G' s' + G" s" , which may also be written in the form zj a a z z, 2 2 z, ’/ lx + my + nz + 2T yz + 2 m'zx + 2 rixy = G s + G' s' + G" s" . To this ot course may be appended the derivative from it by the substitution of — — ^ , .... for A , .... viz., {(B'C"-B"C> + (C'A" - C"A')y + (AB'-A'B»! + ((B"C -BC")* + (C"A-CA ")y + (A"B -AB")z}2 + {(BC' - B'C )x + (OA' -C'A )y + (AB' -A'B)z}2 2 ,2 2 222 999 = G G" s + G" G s' -f GG' s" . * We may formulate for use here the following theorem in modern dress : — If | a/3'y" | he an orthogonant , then 1901-2.] Dr Muir on the Theory of Orthogonants. 287 Further, it will be observed that only one substitution is here involved, and that consequently in connection with the other substitution there must be analogous results, beginning with pio +p'w -\-p"w" + 2 qw'io" + 2<2 'w"w + 2 q'ww' -■ 9 9, 9 9 2 2 GY + G' u + G" v . All of them, manifestly, may be described as transformations simultaneous with the main transformation, and, like one which appeared earlier in the paper, may be usefully enunciated in modern form as follows : — The linear orthogonal substitutions which change ■r !]_ A B A' B A" B C C’ C" into Oast 4- G’s'u + G "s"v will at the same time change x y l n m n m V m l' n y A A A A tA into G s + G' s' + G' s" » mn — l' I'm — nn nl' — mm I'm — nn' nl — m mn - ll' nl' — mm mn - ll' Im - n'“ into G' G" s + G"2G s + G G' s"~ > w p Q (2 q" p' q <1 2 P into G *t* G ' u + G" v > The second result, however, is seen to follow from the first, and a fourth from the third by the previously enunciated theorem of this kind. A,B,C a,/3,7 A , B , C . a , 0 , y ^ A ,B,C < d' , 0' , y" a , j8 , 7 x ,y , z a, 1 8,7' x , y , z a" , 0' , 7" x , y , z _ a , b . a ~ X ,y , z • 288 Proceedings of Royal Society of Edinburgh. [sess. Jacobi (1832). [De transformatione et determinatione integralium duplicium commentatio tertia. — Credo's Journ ., x. pp. 101-128.] This memoir, although classed by its author with the two others of which we have given an account, is of much less interest on the purely algebraical side. In fact it consists almost entirely of the transformation of integrals like VR sin <£ d cty , [[ da cos,?=^/fr sin y cos 0 = n sin cos if/ Vr sin 7] sin 0 = p sin § sin if/ Jr where R = m2 cos 2cf> + n2 sin 2 cos 2if/ + p2 sin 2cf> sin2if/ . When, however, an advance is made from R to U, i.e. to a 2 cos2<£ + b2 sin 2 cos2i {/ + c 2 sin 2cf> sin 2if/ + 2d sin 2 cosi f/ sin if/ + 2e cos sin c£ sin ^ 4- 2 f cos sin cos i f/ , the underlying algebraical problem becomes of more importance ; for example, such a problem (p. 122) as the finding of the co- efficients of the substitution u = gx + hy + iz A v = g'x + h'y + tz , t io = g'x + h"y + x'z , J which transforms ax2 + by2 + cz2 + 2 dyz + 2 ezx + 2 fxy , ax2 + b'y2 + cz2 + 2 d'ryz + 2 ezx + 2 fxy, into u2 + v2 + w2 , u2 v 2 io2 m\ n2 pl ’ respectively. Still there is nothing calling for more than this passing mention. (. Issued separately A ugust 18, 1902.) 1901-2]. Experimental Observations on Leucolysis. 289 Experimental Observations on Leucolysis. By Alex- ander Goodall, M.D., M.R.C.P.E., and Edward Ewart, M.B., Ch.B. Communicated by Dr D. Noel Paton. (With a Plate.) (Read July 7, 1902.) In examining stained "blood films we have frequently observed necrobiotic changes in the white cells. These changes have been referred to by different authors. Cabot (1) writes, “frequently in leukaemia and occasionally in other conditions one sees leucocytes apparently moribund. That they are not always artifacts is shown by the fact that in normal blood they do not appear when treated by the same technique, that reveals them in the blood and hardened clot of leukaemic cases, as well as by the fact that Botkin and others have produced similar appearances by keeping the leucocytes a few days in an aseptic state.” Ehrlich and Lazarus (2) describe fraying and budding of the protoplasmic border in large lymphocytes. Gulland (3), in a recent paper, has described degenerative changes in the cells from the blood in pleural effusions. Ewing (4) describes degenerative changes in the leucocytes, and gives references to the more important literature on the subject. A paper by Leishman (5) on phagocytosis suggested to us a method by which these changes might be investigated. We found that all degrees of necrobiotic changes could be produced by the action of organisms ; but, after familiarising ourselves with their appearance, we noticed that an occasional cell in our “ controls ” showed similar changes, and on further investigation we have been able to demonstrate them even in healthy blood. Our procedure was as follows: We measured with a Thoma- Zeiss leucocytometer (usually to the mark *5) a quantity of sterile normal salt solution warmed to 98‘4° F., and placed this on a slide previously warmed and labelled. A similar quantity of the blood to be examined was added, and a loopful of organisms was gently stirred into the mixture. A cover-glass was then applied, PRO. ROY. SOC. EDIN. — YOL. XXIV. 19 290 Proceedings of Royal Society of Edinburgh. [sess. and the slide was placed in an incubator at blood beat, supported on two matches over a piece of damp blotting-paper, and left, generally, for half an hour. For making several experiments we mixed a larger quantity of blood and saline in a warmed glass cell, then measured a quantity (up to the mark 1 on the leucocytometer) on to each slide to be inoculated. After a slide bad been incubated for the required time the edges of the cover-glass were moistened with a needle dipped in normal saline solution. The cover-glass was then slipped off, and from it and the slide several films were made. After trying many methods we found Gulland’s (6) formalin-alcoliol method of fixing, and eosin and methylene-blue staining, to give us by far the best definition of both cells and organisms. A necessary preliminary observation was to note the effect on the leucocytes of incubating blood along with an equal quantity of normal saline solution alone. On carefully comparing films made from the incubated mixture with ordinary films of the same blood, we found that the white cells remained practically unchanged for two hours. After that period the experiments became unsatisfactory, owing to drying of the blood. To exclude errors of technique, in every case we prepared ordinary blood films, and also films made from the blood incubated with saline solution only with those made from inoculated blood. We made experi- ments with a variety of organisms, and found that, while as regards chemiotaxis and phagocytosis the results were very different, leucolysis was always caused, though the extent of the changes varied considerably. Normal blood incubated with Staphylococcus pyogenes aureus at 98 '4 F. for thirty minutes. Polymorphonuclear leucocytes. — -The great majority of these contained organisms. The numbers in each cell varied from one to forty or fifty. In parts of the film the only organisms to be seen were within the white cells, evidence apparently of chemiotaxis. Whether containing organisms or not, nearly all the polymor- phonuclear cells showed marked changes. The first stage which seems to occur is vacuolation of the protoplasm and faint staining of the granules. A second stage is then reached, in which the 19j1— 2.] Experimental Observations on Leucolysis. 291 protoplasm swells, loses its outline, shows more marked vacuolation, *md instead of granulation a diffuse pale pink colour (with eosin and methylene-blue). Further appearances vary according to the number of organisms taken up by the cells, Many are so engorged that they show only an indistinct nucleus and a faint •cell border, and some are represented only by fragments of a nucleus surrounded by a hunch of cocci. Other cells containing only a few organisms or none at all show all gradations, from wacuolation, raggedness, and swelling of the protoplasm with wacuolation and indistinctness of the nucleus, to disintegration into a granular debris. Lymphocytes. — -The large lymphocytes also exhibited phago- cytosis, but there was no definite evidence of cliemiotaxis. The number of organisms ingested was much fewer than that taken up by the polymorphonuclear cells, the average being four or five, and the maximum never having been noticed to exceed twenty. It was quite exceptional to find a large lymphocyte which did not show degenerative change. Fraying or budding of the protoplasmic border is usually the first change to occur. Along with this, err in some cases preceding it, there is vacuolation of the protoplasm. The cell then swells up, nucleus and protoplasm appear to fuse, vacuolation becomes extreme, and total disintegration follows. The small Lymphocytes. — Yery rarely a small lymphocyte which had ingested one or two organisms was seen, but this was quite •exceptional. The small lymphocytes showed the least degree of necrobiotic change ; and in some films where every other white cell had been disintegrated, normal lymphocytes could be found. The necrobiotic changes they show are fraying and budding of their border, and occasionally vacuolation of their protoplasm. The nucleus is very resistant, but eventually becomes vacuolated and loses its shape before disintegrating. It is interesting to note that Gulland (3) finds in pleural effusions that “the lymphocytes do not appear to degenerate wffth anything like the same frequency as the polymorphonuclear cells.’’ The Eosinophiles. — The normal scarcity of these cells, the readi- ness with which they are broken up, and their resemblance to the neutrophiles in the later stages of disintegration make the study of the eosinophiles particularly difficult. The examination of sputum 292 Proceedings of Royal Society of Edinburgh. [sess.. containing a large number of these cells from cases of asthma gave us some help in recognising the changes which they undergo. As- regards their behaviour to the organisms, some of the cells showed phagocytosis and took up from one to ten staphylococci, but many did not. Many show vacuolation of their protoplasm and then break down, and the last stage at which they can be differentiated is shown by an indefinite nucleus, surrounded by a few scattered granules. It now occurred to us that the presence of organisms in the cells might be accounted for by invasion on the part of the organisms, rather than by phagocytosis. In order to clear up this point we repeated the experiment with dead staphylococci, and were able to conclude that the attack was on the part of the leucocytes. We took several scrapings from a twenty-four-hour-old culture of staphylococci and mixed them with normal saline solu- tion in a test-tube, and boiled the emulsion for ten minutes. We then centrifuged the emulsion and added some of the deposit of dead organisms to a mixture of normal saline solution and blood, and incubated as before. We found that the polymorphonuclear cells had engulfed numerous dead organisms ; the large lymphocytes had also taken them up, but again in smaller numbers than the polymorphs. None were noticed in the small lymphocytes. Many of the eosinophiles contained nine or ten. Leucolysis was again found and the films were very interesting, as the great majority of cells showed only the earlier changes. Vacuolation, and oc- casionally some of the further changes, were found in polymorpho- nuclear cells, large lymphocytes, and eosinophiles ; the small lymphocytes seemed unaffected. In the case of other organisms, for purposes of comparison, we shall only refer to incubations made for half an hour and with cultures twenty-four hours old. Streptococci. — The changes found were identical with those shown in the case of staphylococci. Pneumococci. — Briefly, these excite less phagocytosis and cause slightly more leucolysis than staphylococci. Polymorphs, large lymphocytes, and eosinophiles take up from one to five cocci, and occasionally a small lymphocyte contains one or two. After being engulfed, the capsule of the organism seems to swell, so that a very clear picture is obtained in stained films. Many of the organisms 1901-2.] Experimental Observations on Leucolysis. 293 are swollen and disintegrated. We made a series of observations on the blood of pneumonic patients incubated with pneumococci. The first group examined consisted of non-fatal cases, with leuco- ■cytosis of from 12,000 to 30,000, (a) before, and ( b ) after the ■crisis. (a) Before the crisis we found rather less phagocytosis than normal blood showed, and a greater amount of leucolysis. (b) After the crisis the leucocytes behaved very much as those of normal blood. Broken down cells were more numerous. A second group of fatal cases with leucocytosis showed the same changes as the non-fatal before the crisis, while one fatal case with only 6000 white cells per c.mm. showed very little phagocytosis and very marked leucolysis as compared with normal blood. Bacillus anthracis. — Marked chemiotaxis and phagocytosis were exhibited. Many of the long filaments were found coiled in a series of zigzags into a circle inside a phagocyte or all that remained ■of one. Again, a row of disintegrated polymorphonuclear cells might be found arranged along a filament. The large lymphocytes -again showed less phagocytosis than the polymorphonuclear cells. Practically, every white cell, with the exception of a few of the small lymphocytes, showed marked necrobiotic change. Bacillus mycoides. — The changes only differed slightly in degree from those found in the case of B. anthracis , and were rather less marked. Most of the cells showed necrobiotic changes — more than would have been expected from the action of a non-pathogenic organism. Bacillus diphtlierice. — The polymorphonuclear cells and large lymphocytes were all utterly disintegrated, and were only repre- sented by debris in the centre of little clumps of bacilli. Some of ;the small lymphocytes contained bacilli, and only a few were iound unaffected. Spirillum cholerce. — The white cells reacted exactly as they did to B. diphtlierice. Bacilhis typhosus. — Evidence of chemiotaxis was not noticed. The polymorphonuclear cells and large lymphocytes showed only slight phagocytosis, and they exhibited only a moderate degree of necrobiotic change. The small lymphocytes showed very little -change. 294 Proceedings of Boyal Society of Edinburgh. [sess. Bacillus coli communis. — The leucocytes seemed indifferent to- the presence of these organisms. There was no sign of chemio- taxis, hut phagocytosis seemed to occur just as the bacilli were met with, and no cell was seen containing more than two or three. Beyond an occasional vacuole, no necrobiotic changes occurred. As regards their leucolytic power, we would place the organisms examined in the following order : B. diplitherice and Spirillum cholerce ; Bacillus anthracis and Bacillus mycoides ; pneumococcus , staphylococcus , and streptococcus ; Bacillus typhosus \; dead staphylo- cocci ; and lastly, Bacillus coli communis. We now turned our attention to ordinary blood films in health and in a variety of diseased conditions. We found that although they may he scanty, leucocytes showing necrobiotic changes may always he found in the blood. It is difficult to estimate their numbers even approximately, hut we may say that at least one moribund cell may generally be found in every two or three films of normal blood, while in certain conditions the whole process of leucolysis may he traced in a single film. To avoid fallacy in estimating the amount of leucolysis in a given specimen, it is necessary to note the degree of necrobiotic change as well as the number of cells affected. Unless this is noted, the picture of a large number of cells showing early stages is apt to give the impression of great leucolysis, while a slide showing only a few necrobiotic cells might indicate that leucolysis was slight, whereas- the true inference would be just the reverse. Where the changes are advanced, we may assume that a number of leucocytes have- altogether disappeared. We have examined the blood in a great variety of conditions, and find degenerated leucocytes constantly present. General conditions of Disease. — Cells showing necrobiotic changes are generally rather more numerous than in health, and seem to vary in proportion to the general nutrition of the patient, though only within very narrow limits. Blood Diseases. — A considerable number of white cells showing necrobiotic change can always be found. The proportion usually varies with the gravity of the anaemia. In pernicious anaemia the changes are marked. Toxcemic conditions. — In these conditions wre found not only the 1901-2.] Experimental Observations on Leucolysis. 295 most marked necrobiotic changes, but also the greatest variation in their amount. In cases showing leucocytosis the young forms suffer most. We find leucolysis much more marked in cases of malignant cachexia without increase in the number of white cells than in cases showing leucocytosis. Again, in severe pneumonia cases the amount of necrobiotic change was always greater when the total number of leucocytes was comparatively low. In scarlatina with marked leucocytosis the changes are but slight. The explanation seems to be that a high birth-rate keeps the leucocyte population fairly healthy, in spite of infant mortality and adverse circumstances. There are several directions in which our observations might be extended, but meanwhile we would state the following conclusions : 1. Necrobiotic changes occur in the , circulating leucocytes in health. 2. These changes are much more evident in conditions of impaired nutrition and toxsemia, notably in cancerous cachexia. 3. In toxic conditions usually associated with leucocytosis the extent of the necrobiotic changes in the white cells varies in inverse ratio to the number of leucocytes in the circulating blood. 4. These necrobiotic changes can be rapidly induced “ in vitro ” by the action of certain organisms or their products. The rapidity and extent of the changes depend on (a) The kind of organism. (b) The virulence of the culture. (c) The number of organisms employed. We wish to express our indebtedness to Dr James for kindly placing his cases at our disposal, and to the Superintendent of the Royal College of Physicians’ Laboratory for affording us facilities in connection with the bacteriological part of the work. [References. 296 Proceedings of Royal Society of Edinburgh. [sess. REFERENCES. (1) Cabot, Clinical Examination of the Blood. (2) Ehrlich and Lazarus, Histology of the Blood , Myer’s translation, figure 1. (3) Gulland, Scottish Medical and Surgical Journal , June 1902. (4) Ewing, Clinical Pathology of the Blood , 1901, p. 112. (5) Leishman, British Medical Journal, Jan. 11th, 1902. (6) Gulland, Scottish Medical and Surgical Journal , April 1899. APPENDIX. We have tabulated a variety of conditions with simply a note of the changes seen in one film, and venture to think that some idea of the frequency of necrobiotic cells may be indicated in that way. For convenience of reference we may recapitulate the appearances at different stages of the necrobiotic change, but it must be under- stood that these changes are not necessarily synchronous as regards the different kinds of cells. Polymorphonuclear Cells. Lymphocytes. Eosinophiles. . Stage 1. Yacuolation of pro- toplasm and faint staining of gran- ules. Vacuolation of pro- toplasm and fray- ing or budding of its border. Vacuolation of pro- toplasm. Stage 2. Swelling, marked vacuolation and diffuse pale stain- ing of protoplasm. Swelling of the whole cell, nuclear outline lost. Scattering of gran- ules. Stage 3. Vacuolation of nu- cleus ; disintegra- tion. Great vacuolation. Disintegration of protoplasm, then of nucleus. Breaking up of nu- cleus, disappear- ance of granules. 1901-?.] Experimental Observations on Leucolysis. 297 ^ 1 w i ! © Condition. i- c Ph d o g o aj o g 'S 03 Ph a/ O «> >» » o © O &G . ^ o =2 o a ® S p J,® -+J £ o O 82 m £ >> I. General Conditions of Disease. 1. Asthma, . 4,200,000 8,250 80 3 1 2 2. Diarrhcea, 3,282,000 5,400 38 1 3 1 3. Tubercular glands, . 4,550,000 8,750 60 4. Scleroderma, . 4,928,000 6,250 80 1 5. Mitral disease, 2,800,000 4,840 53 1 1 6. Aortic disease, 2,500,000 6,250 26 3 II. Blood Diseases. 1. Chlorosis, 3,583,400 3,125 30 2 3 1 2. . 3,000,000 4,062 15 l”' 1 3. . 1.874.000 2.730.000 5,902 19 1 1 4. • 5,312 20 1 2 3 1 3 5- • • 4,400,000 6,250 60 3 1 3 6. . 4,270,000 4,685 45 7. Pernicious anaemia, . 580,000 5,312 17 12 3 13 8. 990,000 6,500 22 Q 1 9. 728,000 8,487 16 2 3 1 2 io. 1,472,000 2,500 22 1 1 11. Purpura, 2,800,000 14,000 20 12. ,, 875,000 14 12 3 1*2 3 13. Haemorrhage, . 3,300,000 5, 'ooo 45 1 2 3 1 2 3 „ . . 2,850,000 3,000,000 4,375 20 1 15. Chronic malaria, *3 1 16. Metrorrhagia, . 17. Acute lymphaemia, . 2,800,000 2,870,000 7,000 18 1 2 12 3 159,062 42 1 3 1 18. 19. Myelocytliaemia, 3 1 2,940,000 116*562 55 12 3 12 3 1*2 3 20. 338,000 1 1 3 III. Toxemic Conditions. 1. Gastric cancer, 2,800,000 ~ 7,812 15 1 2 3 1 2 3 1 2. 2,000,000 5,937 22 | 1 2 3 1 2 3. 11,000 3 1 1 4. Intestinal cancer, 3,490,000 4,062 40 1 1 5. Uterine cancer, 3,000,000 14,000 40 1 1 6. Scarlatina, 26,000 7. . 13,000 1*” 8.,, • . 30,000 3 1 9. . 12,000 1 2 3 1 io. „ . . 32,000 1 1 11. Acute rheumatism, . 2,760,000 17,500 52 l" 3 1 12. Pneumonia, 4,200,000 30,000 79 1 3 1 13. „ . . 5,000,000 22,000 85 1 14. „ • • 5,000,000 13,000 77 1 2 3 123 15. „ . . 16. Empyema, 3,800,000 6,000 70 12 3 1 2 3 4,600,000 26,000 80 1 2 3 1 2 3 12 Eosinophiles — Stages. 298 Proceedings of Royal Society of Edinburgh. [sess.. REFERENCES TO THE PLATE. [Nos. 1 to 12 are all drawn from a case of gastric cancer with cachexia.] 1. Normal polymorphonuclear leucocyte. 2 to 5. Polymorphonuclear leucocytes, showing necrobiotic changes. 6. A collection of blood plates. 7 to 10. Large lymphocytes, showing necrobiotic changes. 11 and 12. Small lymphocytes, showing necrobiotic changes. [Nos. 13 and 14 are eosinophiles from asthmatic sputum.] 13. Early necrobiotic change. 14. Late necrobiotic change. [Nos. 15 to 19 are drawn from films made from healthy blood incubated with Staphylococcus pyogenes aureus .] 15 to 18. Polymorphonuclear leucocytes, showing phagocytosis, and leucolysis. 19. A large lymphocyte, showing the same. [Nos. 20 to 23 are drawn from films of healthy blood incubated with pneumococci.] 20 and 21. Polymorphonuclear cells. 22. A small lymphocyte. 23. An eosinophile. 24. Shows a group of Bacilli diphtherias ingested by a phagocyte which they have destroyed. 25. Is drawn from a film of healthy blood incubated with B . anthracis , showing phagocytosis and leucolysis. (All from films fixed with formalin-alcohol, stained with eosin and methylene-blue. x 1000.) (. Issued separately August 19, 1902.) Proc. Roj/.Soc. Edin. Vol. XXIV. EXPERIMENTAL OBSERVATIONS ON LEUCOLYSIS JK .'A TAf 6 L'.th-Eairi1' 1901-:.] Prof. Alexander Smith on Amorphous Sulphur. 299 Amorphous Sulphur and its Relation to the Freezing Point of Liquid Sulphur. By Professor Alexander Smith, Ph.D., D.Sc. (Read June 2, 1902.) Rhombic and monoclinic sulphur are two physical states of the element with a transition point at 96°. The relation of amorphous sulphur to liquid sulphur is entirely different. The former is produced by heating liquid sulphur, and is found in increasing proportion as the temperature is raised. At 448° it reaches about forty per cent. As the temperature falls, the proportion of amorphous sulphur recedes, although sudden cooling arrests the regression, and furnishes solid specimens containing the larger proportions proper to higher temperatures. The case seems to be one of chemical equilibrium, S (liquid) S (amorphous), in which the action, as written, is endothermal. If this interpretation is correct, the variability of the freezing temperature of sulphur, observed by Brodie, Gernez, Schaum, and others, may be due simply to depression of the freezing point by varying proportions of amorphous sulphur, the latter acting as a foreign dissolved body. Two series of experiments, in which the freezing points of specimens of previously heated liquid sulphur were accurately determined by means of a Beckmann thermometer and suitable apparatus, confirmed this view. As soon as the temperature had been read, the mass was poured into a cold dish, and on the following day was pulverised and extracted with carbon disulphide. A correction was made for the solubility of the amorphous sulphur after separate investigation of this point. The following table gives one series, the quantities of amorphous sulphur in the second column being those associated with 100 gr. of soluble sulphur : — 300 Proceedings of Royal Society of Edinburgh. t A A at 118*25 0*764 1*00 41*9 117*10 1*546 2*16 44*7 116*20 2*251 305 43*4 115*24 3*100 4*01 41*4 114*43 3*716 4*82 41-5 Mean, 42*6 When the temperatures and quantities of amorphous sulphur are plotted, the points are found to lie upon a straight line, which intersects the temperature axis at 1 19*25°. The depressions of the freezing points below this point (the freezing point of pure soluble sulphur) are therefore proportional to the amounts of the dissolved body. Raoulfs law thus expresses the relation, and the freezing point seems to be determined solely by the quantity of the dissolved amorphous sulphur. In the third column are given the depressions below 119*25°, and in the fourth the depressions which would be produced by 32 gr., one atomic weight, of amorphous sulphur dissolved in 100 gr. of soluble sulphur. The latter should be constant. The mean atomic depression is 42*6. Another series of four observations gave exactly the same mean result. Now, the molecular depression can be calculated by means of Van Jt Hoffs formula, . 0*0198 T2 Am = , where T is the freezing point of the solvent in the absolute scale (119*25° + 273°), and q is the heat of fusion of the solvent (9*368). From these data AM = 325°. Dividing this by the atomic depres- sion, the quotient 7*6 represents the number of atoms in the molecule. Inasmuch as the quantities of amorphous sulphur are probably somewhat underestimated, and the atomic depression is therefore too large, we may take it that the molecule contains eight atoms. The equation for the change is thus S# (liquid) S8 (amorphous). 1901-2.] Prof. Alexander Smith on Amorphous Sulphur. 301 The value of x is unknown. Soluble sulphur has been shown, to have the formula S8 in solution, so that the change seems to be in the main an intramolecular rearrangement. Other phenomena connected with the relation of amorphous sulphur to liquid sulphur, e.g., the speed of the transformation, and the proportions existing at various temperatures when equi-. librium has been reached, are being investigated. ( Issued separately August 19, 1902.) .302 Proceedings of Royal Society of Edinburgh. [sess. Application of Miller’s Trisector to the Quinquesection of any Angle. By James N. Miller. Communicated by Dr Knott. (Read July 7, 1902.) The diagram outlines the Trisector as in the position in which the angle B AL, which may be any angle of less than 225°, is by its aid divided into five equal angles. That instrument was described to this Society in a communica- tion submitted to them on the 4th of November last. It consists of two pieces, AIFD and B H G. Those pieces are conjoined, as fhe blades of a pair of scissors are, by a small cylindrical pin inserted in a small cylindrical hole at C in each of them, which it fits, and round which they may turn. As a step towards this division of the angle, a perpendicular, K Q, to its side, A L, is drawn from the point K in it, which is at the same distance from its vertex as the point I is from the 1901-2.] Application of Miller s Trisedor. 303 •centre of a small cylindrical liole made at A through the piece AIFD. The method of using the Trisector so as to divide the angle, as proposed, into five equal angles, is to insert through that hole at A a sharp-pointed cylindrical pin, fitting it, but not too tightly, into the vertex of the angle, and then to move the point B of the piece B H G along the other side, B A, of the angle until the edges or borders I R and E G, which are straight, meet in a point which may be termed 0 in the perpendicular K Q. The straight line A 0 is then drawn from the vertex A, and may be produced to N ; and the angles C A 0 and 0 A L are bisected by the lines A M and AP. The angle BAL is, by this operation, divided into five equal angles, namely, BAG, CAM, MAO, 0 A P, and PAL. For As the sides A C and B C of the triangle BAG are, by the •construction of the Trisector, equal, so that triangle is isosceles, and its angles B A C and ABC are equal. But as the point C is, by construction, in the straight line B C E G, so the angle A C 0 is exterior to the isosceles triangle B A C, and is therefore equal to twice that angle BA C. Again, as by construction the sides C I and A I of the right- angled triangles C 0 I and A 0 I are equal, and as they have a common side I O, so their corresponding angles I C 0 or AGO and I A O or C A O are also equal. But the angle A C O is equal to twice the angle BAG. Therefore the angle C A O is also equal to twice that angle. Further, as the right-angled triangles 0 1 A and OKA have equal sides, A I and A K, and the same hypotenuse A 0, so their corresponding angles I A 0 or C A 0, and OAK or 0 A L, are equal to each other. The angle 0 A L is therefore equal to twice the angle B A C. Moreover, as the angles CAM and MAO are halves of the angle C A 0, and the angles 0 A P and PAL are halves of the angle O A L, so each of them is exactly equal to the angle BAG. The angle B A L is therefore divided into five equal angles — namely, B A C, CAM, MAO, 0 A P, and PAL. 304 Proceedings of Royal Society of Edinburgh. [sess. An angle of 225° or upwards cannot, however, be thus divided into five equal angles. But if half of it be so divided, then the entire angle will be equal to five angles, each of which is two-fifths of that half of it. An angle of not much less than 225° may be most conveniently divided by this latter process into five equal angles. ( Issued separately August 19, 1902.) 1901-2.] Dr Hugh Marshall on Thallic Sulphates. 305 Thallic Sulphates and Double Sulphates. By Hugh Marshall, D.Sc. (Read Jane 16, 1902.) In a note on “ the hydrolysis of thallic sulphate ” which I com- municated to the Society some time ago,* I commented on the conflicting nature of the statements, made by various investigators, as to the exact composition and nature of thallic sulphate. As the subject appeared of some interest, I subsequently commenced an investigation of the salt and of the double salts derived from it. The results so far obtained are in some respects rather striking, but the investigation is not yet completed, and my reason for now publishing a general statement of these results is, that apparently others besides myself are working in the same field. A paper has just been published by James Locke, f in which the author describes a caesium thallic sulphate which he obtained while endeavouring to prepare caesium thallic alum. No alum could he obtained, the most hydrated salt corresponding to the formula CsT1(S04)2 , 3H20. This question of the formation of thallic alums was one to which I turned my attention early in the investigation. I employed more particularly ammonium and rubidium sulphates, and was quite unsuccessful, being unable to obtain even mixed alums in which thallium (TT”) had partly replaced other triad metals.. Mixed solutions prepared from ammonium chrome alum, ammonium sulphate, thallic sulphate, and dilute sulphuric acid were allowed to crystallise. Crystals of chrome alum and of a hydrated ammon- ium thallic sulphate formed separately ; the latter were colourless and of prismatic habit, and much less hydrated than an alum ; the chrome alum crystals were free from thallium, except a trace of thallous sulphate replacing ammonium sulphate. Similar results * Proc. R.S.E., xxii. p. 596 (1899). f Abstract in Chem. Central- Blatt, 1902, i. p. ] 266, from Amer. Chem ► Journ ., xxvii. p. 280. PROC. ROY. SOC. EDIN. — VOL. XXIV. 20 306 Proceedings of Royal Society of Edinburgh. [sess. were obtained when common ammonium alum was employed in place of the chrome alum. These results seem to indicate conclusively that thallic sulphate does not form alums.* Thallic Sulphates. My experience in trying to prepare thallic sulphate from thallic hydroxide and sulphuric acid was very similar to that of Willm. f Well-formed crystals of a basic salt were easily obtained from the solution, but in none of the experiments was a normal salt obtained, no matter what acid concentration was employed. Willm gives the formula of the most hydrated salt obtained by him as T120(S04)2,5H20 . Several analyses made by me gave somewhat higher percentages of water than corresponds with this formula, and seemed rather to indicate a hexahydrate. As one molecule of water is retained above 100°, the salt is probably a hydroxy- sulphate, and, adopting Willm’s proportion of water, the formula might then be written T10HS04,2H20 . As stated by Willm, the addition of concentrated sulphuric acid to a solution of this salt in dilute sulphuric acid produces a granular precipitate, still consisting of basic salt, but generally less hydrated than the above. Only on one occasion did I succeed in obtaining a different sulphate. This separated from a small quantity of solution in large clear crystals, very soft and entirely different from any others. The substance, after being powdered and placed in a stoppered tube, gradually deliquesced to an oily liquid. From determinations of T1 and S04 the composition of the salt corresponds to the formula HT1(S04)2,4H20 or 5H20. The water was not determined directly. This acid salt may be looked upon as the acid from which are derived a series of well-defined double salts. An attempt was made to prepare thallic sulphate by electrolysis of an acid solution of thallous sulphate (kept saturated with the latter salt) in the manner employed by me for the preparation of oobaltic sulphate. £ A sparingly soluble thallous-thallic sulphate was obtained, which will be described later. * Statements to the contrary are occasionally met with in text-books. t Ann. Chim. Phys. [4], v. p. 28 (1865). X Chem. Soc. Trans., lix. p. 760 (1891). 1901-2.] Dr Hugh Marshall on Thallic Sulphates. 307 Ammonium Thallic Sulphates. As the experiments with thallic sulphate alone were so far not very satisfactory it was decided to examine the double salts, in the hope that they would give information which might help to eluci- date the mystery concerning the formation of the normal salt. A number of double salts have been described by previous investiga- tors, but as the ammonium derivatives had apparently escaped attention a beginning was made with these. Solid ammonium sulphate was added to the mother liquors from a preparation of basic thallic sulphate, and the liquid was warmed. As the crystals dissolved,,., a firie granular precipitate began to form and increased to a considerable quantity, remaining even when the liquid was heated to boiling. The substance, drained and dried between filter paper, gave on analysis results showing it to be an anhydrous double sulphate, corresponding to the formula NH4T1(S04)2. Though sparingly soluble in the acid liquid containing a large excess of ammonium sulphate, this double salt is easily soluble in dilute sulphuric acid. (Like all the thallic salts it is hydrolysed by water.) When this solution is allowed to evaporate at the ordinary temperature it deposits fairly large colourless crystals of prismatic habit, apparently monoclinic ; their composition is represented by the formula NH4T1(S04)2,4H20 . The quantity of water found on analysis is rather less than this, but, as the crystals effloresce at the ordinary temperature, this is to be expected. When an acid solution of thallic sulphate, or of the above double sulphate, is saturated with ammonium sulphate at the ordinary tem- perature, crystals of a totally different appearance are gradually formed. They are translucent, soft, and rather indefinite as regards crystalline character, and resemble fine crystal aggregates rather than distinct crystals. Their composition corresponds to the formula (NH4)3T1(S04J3 . When treated with dilute sulphuric acid they yield a solution which, on spontaneous evaporation at the ordinary temperature, yields the above-mentioned salt NH4T1(S04)2,41I20. 308 Proceedings of Royal Society of Edinburgh. [; Potassium Thallic Sulphates. Strecker* (who is one of those who succeeded in obtaining normal thallic sulphate) states that, on addition of potassium hydrogen sulphate to thallic sulphate solution, he obtained a white granular precipitate of a basic salt corresponding to the formula 2K2S04, T120(S04)2 ; this salt was very sparingly soluble in dilute sulphuric acid. I had no difficulty in getting what appeared to be the same substance, but its composition is not exactly that given by Strecker. The substance examined by me proved to be a hydroxy-basic salt, the appropriate formula being K.,T10H(S04)2 . It loses water only at a high temperature, and as the amount is less than 2 per cent, this had been overlooked by Strecker. When heated, the salt becomes very dark and then white again ; in the latter state it still contains thallic sulphate. As mentioned by Strecker, the salt is very sparingly soluble in dilute sulphuric acid; I found, however, that it dissolved easily in dilute nitric acid. The nitric acid solution, when allowed to evaporate at the ordinary temperature, deposits fairly large trans- parent crystals, exactly similar in appearance to those of the hydrated ammonium salt. They proved to be the corresponding hydrated potassium normal salt, KT1(S04)2 , 4H20 . The action of the nitric acid seems to be expressible by the equation — K2T10H(S04)2 + HN03 KT1(S04)2 + KN03 + H20 . As in the case of the ammonium salt, the crystals effloresce on exposure to the air, so that at a higher temperature the nitric acid solution would probably deposit the anhydrous salt KT1(S04)2 r corresponding to that obtained from the warm solution of the- ammonium salt. If the crystals of the hydrated potassium normal salt are gently warmed with dilute sulphuric acid, they give a granular precipitate of the original basic salt. As the ratio K : T1 is in the former salt 1 : 1 and in the latter 2 : 1, it follows that there must, under these conditions, be an accumulation of excess of thallic sulphate in the solution. Doubtless in consequence of this excess of thallic * Annalen, cxxxv. p. 207 (1865). 1901-2.] Dr Hugh Marshall on Thallic Sulphates. 309 sulphate, the mother liquor from the granular precipitate, when allowed to stand at the ordinary temperature, deposits colourless prismatic crystals of the hydrated normal salt ; it is therefore probable that the normal salt could be obtained directly from a thallic sulphate solution by using potassium sulphate in small quan- tity only. A simple method of preparing the basic salt consists in heating together thallous sulphate, potassium persulphate, and potassium carbonate, with a sufficient quantity of water ; evaporating, and then digesting the residue with dilute sulphuric acid. After re- peatedly exhausting with dilute sulphuric acid to remove any excess of potassium sulphate, the granular precipitate is drained and dried. Rubidium Thallic Sulphates. By mixing rubidium sulphate and thallic sulphate solutions and allowing the mixture to crystallise, either a granular anhydrous double sulphate RbTl(S04)2, or the hydrated salt RbTl(S04)2, 4H20 may be obtained, depending on the temperature and the acid concentration employed. The hydrated salt crystallises in clear colourless prisms resembling those of the corresponding ammonium and potassium salts. Thallous Thallic Sulphates. When attempting to prepare thallic sulphate by electrolysis (y. supra) a large quantity of a light-yellow finely crystalline powder was obtained ; analysis showed this to be the anhydrous double sulphate Tl5Tl"'(S04)4 . (The corresponding iodide T13I4 or T15T1Is is known.) The same double salt was also obtained as a product of decomposition from thallic sulphate residues which had stood for a considerable time exposed to the air. The dry salt underwent a distinct change of colour when kept in a closed tube, becoming much paler ; this is possibly due to its undergoing a transition into a mixture of other salts. A quantity of this double salt was dissolved in warm dilute nitric acid and allowed to crystallise ; the first crops of crystals, in fern- like aggregates, were found to be pure thallous nitrate. Thereafter, 310 Proceedings of Royal Society of Edinburgh. [sess. finely crystalline deposits of double sulphates separated. One of these on analysis gave for the ratio Tl* : Tl"* the value 2, corre- sponding to the formula T14T12(S04)5 . As succeeding crops gave a varying ratio, it is probable that these substances were really mix* tures of two double salts, T13T1(S04)3 and T1T1(S04)2, corresponding to the anhydrous ammonium double sulphates. By mixing thallous sulphate and thallic sulphate solutions in molecular proportions, and allowing to crystallise, small anhydrous- crystals of the salt T1T1(S04)2 were obtained. This salt was pre- pared some time ago by Lepsius, who describes the crystals as exhibiting faces of the cube and the octahedron, and calls them an anhydrous alum.* Those obtained by me showed strong double refraction when examined in polarised light, and appeared to be rhombic. From the results outlined above, especially in connection with the potassium compounds, it is evident that these thallic salts exhibit several interesting peculiarities. Apparently a £ thallic sulphate ’ solution contains a large proportion of basic salt, even when the acid concentration is very great, and it would seem to be largely a matter of solubility whether normal or basic salts are obtained from it. Possibly the difficulty of obtaining the normal salt by itself is due to the fact that it unites with sulphuric acid to form the complex acid HT1(S04)2: , so that in the solution thero exists the balancing system — 2T10HS0, + 2H2S04 ^ HT1.2(S04)3 + 2H20 + H2S04 ^ 2HT1(S04)2 + 2H20. Under some conditions this may practically become— T10HS04 + H2S04 === HT1(S04)2 + H20 . In a similar way the decomposition of the normal potassium double sulphate when treated with dilute sulphuric acid may most simply be regarded as that represented by the equation — 2KT1(S04)2 + II20 ^ K2T10H(S04)o + HT1(S04)2 . * Central -Blatt, 1891, i. 694. This is probably the source of the statement that thallic sulphate forms alums. 1901-2.] Dr Hugh Marshall on Thallic Sulphates. 311 Looked at from this point of view it appears not unlikely that at some suitable temperature normal thallic sulphate might be most easily obtained from solutions containing a minimum of free sul- phuric acid, and experiments in this direction are being carried on. In the present paper I have omitted analytical results, etc., reserving details for the full report which I hope to communicate on completion of the investigation. {Issued separately August 30, 1902.) t. 312 Proceedings of Royal Society of Edinburgh. [sess. On Superposed Magnetic Inductions in Iron. By James Russell. (Read July 7 and 21, 1902.) [Abstract.) A communication was made to this Society on February 3, 1902, on “Magnetic Shielding in Hollow Iron Cylinders.” At an early stage of this investigation, and when dealing with more magnetis- ing forces than that due to the transverse field acting upon the, shield, the subject of cross magnetisation necessarily came to the front. This paper deals, First , with the superposition of two magnetising forces at right angles to each other, and the co- ordination of the two components of the resultant magnetic induction, under the various conditions of field superposition ; and Second , with the magnetic ceolotropy of demagnetised iron. The same hollow iron cylinders were used as in the shielding experiments. I. Of the two magnetising forces at right angles to each other, let Hj be the force first acting, H2 the force superposed. Each force acting alone produces the normal B-H induction curve. Let Bj and B2 he the two components of the resultant induction in the directions of Hx and II9 respectively. When H2 is superposed upon a pre-existing induction due to Hj, the Bx component o the resultant induction always lies above the B2 component. Repeated reversals of H2 accentuate this result; Bx is further increased ; and concurrently with this, B2 is further lowered. For low fields, the Bj component is considerably above the normal induction curve, but as the fields are increased a point is reached where the curves cross, the Bx component then falling below the normal curve. Also the superposition of H2 lowers the B2 component below the normal induction curve, ivith this excep- tion, that at low values of Hx the superposition of the second force H2 increases the B2 component above the normal induction curve. This is, however, a relatively small effect. 1901-2.] Mr James Bussell on Magnetic Inductions. 313 When the superposed force H2 carries the B2 component round a complete magnetic cycle, the component due to Hj kept at a constant value responds and likewise passes through a complete magnetic cycle or series of cycles. A connection appears to exist between the change which takes place in the Bx component and the permeability impressed upon the iron by the magnetising force which is superposed. This is especially well marked when Hj is constant and the superposed force cyclic, the maximum and minimum values of Bx corresponding with the maximum and minimum values of dB2/dH2 respectively. The whole phenomena, however, are exceedingly complicated — permeability, retentivity, coercive force, and vibration effects all contributing to the final result under the various conditions of field superpositions. II. During the early stages of induction, the experiments de- scribed show that iron is more permeable to a reapplication of a magnetising force in the same direction (positive or negative) as that used in the immediately preceding process of demagnetising by decreasing reversals, than it is to a force (positive or negative) at right angles to that used in the immediately preceding de- magnetising process. The difference for the two qualities of iron used was found to be of the order of 30 per cent., but it vanishes as the magnetising force is increased. An explanation of this magnetic aeolotropy of demagnetised iron based upon the molecular theory of induction is given. [Issued separately October 7, 1902.) 314 Proceedings of Royal Society of Edinburgh. [sess. On the Use of Quaternions in the Theory of Screws, By Dr W. Peddle. (§§ 1-2 communicated on February 6, 1902 ; §§ 3-7 on July 21, 1902. Other illustrative sections communicated on the earlier date, and since found to have been given in substance by Joly, are omitted.) 1 . A quaternion r = Sr + Y r denotes the sum of a scalar and a vector, the former being an essentially undirected quantity. In many cases, however, and specially in the theory of screws, we have to deal with two co-directed quantities. In the usual nota- tion, the components of a translation, A, which are parallel to,, and perpendicular to, a rotation, /x, are represented respectively by the first and second terms in the identity A = (SAyx_1 + YA/x_1)/x.. The axis of the corresponding screw of pitch SA /x_1 has the direc- tion of /x, and YA/x_1 is the perpendicular upon it from the origin, A certain advantage in point of unity would arise from taking Sr and TYr to represent respectively the magnitudes of the transla- tion and the rotation in a screw whose axis passes through the origin, and has the direction of Y r. When such a use is made of a quaternion, it is necessary to attach a special symbol. Thus, Mr may be taken to denote the motor S?**UYr + TYr-UYr whose axis- passes through the origin and, whose pitch is Sr /TYr. 2. To determine the form of the general expression for a motor,, not passing through the origin, we may consider three motors pass- ing through the origin Mr, = Sr,.UYr, + TWyUYr, , M>2 = Sr2-UYr2 + TV?yU Vr2 , Mr 8 = Sr3-TJYr3 + TYr3-UYr3 ; and we may assume these to be rectangular. The sum of these motors is Mr, + Mr2 + Mr3 = (Sr,.UYr, + Sr2U Vr2 + S»y UVr3) + (T YrrUYr, + T Vr2-UYr2 + TVr3.UVr3> = (Srj-UYr, + Sr2-UYr2 + S%ITYra) + p 1901-2.] Dr Peddie on Quaternions in Theory of Screws. 315 where p is the resultant rotation. The first three terms may be written as S?’1*UY/,1 + (1 + m)Sr2-UVr2 + (1 + w)Sr3-TJYr3 - raSr2-UVr2 - wSr3*TJYr3 in which we may choose .. Sr, TYr2 S?^ TV?'3 S n> LVrj Sr3 1 V r^ If we put Srl=plTWr1 , S?2 = Vr2 , Sr3=^3TVr3 these give 7\ -Po Pi ~ V% m = , n = — — — : P2 Pz and the vector S^U-Y^ + (1 + m)S?yUYr2 + (l + w)Sr3.UVr3 = /n>,Q., - pt) UVrJ 1 + T2p ' 2?xT"V rA +p.2 T3Yr2 +^3T°Y?3 'Fp ' "" The last term represents the component of translation in the direc- tion of p. Hence the pitch, tt, of the resultant motor is 7 r = p1 COS2^ + £>2COS22 + 2^COS203 where the cosines are the direction cosines of the axis with refer- ence to the axes of the three component rectangular screws. This- is a well-known relation. Since the product of the second term within the bracket into p- represents the component of translation perpendicular to p, that term must itself represent the vector perpendicular upon the re- sultant axis, 3 say. Thus we have /x = Mrx + Mr2 + Mrs = (1 + 3 + tt)(V rx + Y r2 + Vr3) . 316 Proceedings of Royal Society of Edinburgh. [sess. We easily obtain the following expressions, P = 2-Vr,- ^■S(rVYr-r) 7r~ ' 2 ' 2-T2Vr " ’ * '2fp-p)YrYr 6- 2-T2Vr The symbol M is obviously not distributive. 3. We may proceed to resolve a motor in any direction by the usual method of resolution. Thus we may decompose it into three rectangular components having the same pitch. Any one of these may be regarded as the component of the given motor, and will correspond to the actual motion when the other components are regarded as being balanced by constraints. The motor ml = Sr'.uw + TV/-UY/ , = (1+7T)W, = - (1 + 7r)S V/TJ Vr-U V r - (1 + 7r)YYr' U V/’-U VY , represents in this way a motor - (1 + ir)SVr'UVr-TJYr, in the direc- tion of Vr, and whose vector displacement is -7rYYr'UYr, pro- vided that the rotation YYr'UYr-UYr is prevented by constraints. Thus the quaternion r is the symbol of a complex of motors, each of which is determinate when the direction of its axis is given. The complex is formed by the generators of a hyperboloid, whose axis is Yr, when the angle between Y r and Y r is fixed. The screws corresponding to the one set of generators have the opposite pitch to that of the other set. Each quaternion thus represents pairs of reciprocal screws passing through each point of space. And a set of three non-coplanar quaternions furnish a set of six reference screws through each point of space. By proper choice of the origin that set might be made the canonical co-reciprocals for any one point in space. By taking the three rectangular coordinate screws with both positive and negative pitches we get a set of six canonical co-reciprocals, to which any screw may be referred. 4. In the case of a rigid body moving with two degrees of free- dom, the motion may be represented in terms of two rectangular coordinate screws. We may write these in the form 1901-2.] Dr Peddie on Quaternions in Theory of Screws. 317 M?q = - (1 + p^VrfJVr-VVr - ( 1 + pJVYrfJVr- UVr , M>2 = - (1 + _p2) S Vr2 UYr •UYr - (1 + j>2)VVrsUVr. UVr . When the condition VV^UVr-f- VVr2UVr=0 holds, the motion is free. When the condition does not hold naturally, we may suppose that it holds in consequence of constraint which prevents rotation except around an axis parallel to Y r. Let 0 be the angle between Y r and Yrlt and put TY?q = ^cos^, TVr2 = ?7sin0. The condition for complete freedom gives £=77. When constraint has to be applied to make V?’ the resultant axis, the coordinate motors are M'^ = Mrx + VV^UVr-UVr and MV2 = Mr2 + VVr2UVr-UVr „ and the pitch of the resultant is „_&icos2fl + Win20 ** $cos20 + rjsm20 If we write y = xtsm0 , and 2 = TS, where ^ ( p ^ -p2y)cos6smd fcos 20 + yjsin 20 we find that the resultant screws for different values of 0 are the generators of the ruled surface + yy2) = {jpf -p2y)xy . When ^-y, this becomes the well-known equation of the cylindroid. To find the nature of the more general surface, consider an ellipse whose semi-axes are 1/^/1 and l/Jy . Take the former of these as the a>axis, and take the origin at one extremity of the diameter. Intersect the elliptic cylinder whose normal section is the above ellipse by a plane which passes through the se-axis and makes an angle with the plane of the ellipse. Take as the 2-axis the generator of the cylinder which passes through the origin, and draw perpendiculars to it through the points of intersection of the cylinder by the inclined plane. Consider further a circular cylin- der of radius 1 /J£ surrounding the elliptic cylinder. We have 2 = yt&ncf) = \/ — y' tariff) 1 318 Proceedings of Ptoyal Society of Edinburgh. [sess. where x, y\ are the coordinates in the circular cylinder correspond- ing to x, y, in the elliptic cylinder. If y' = xt&nO\ we get k = - tan<£sin2 0 Jv = 2 ^tan<£^ = 2 Jtan sh r'2 Ji V2 -where r "2 = x2 + y2 = x2 + y2g/^ • Hence 2 z(fx2 + rjy2) = -- tan cf>xy . v t Thus the ruled surface is the elliptic cylindroid , for which 2tan<£ = J€(pf~2hv)- A very simple model of the elliptic cylindroid may he made by means of two elliptic rings, hinged together at one extremity of their major axes. Points at the same distance from the hinge on each are joined by a pair of elastic cords crossing each other. As the rings are opened out, the cords lie on different elliptic cylin- droids, one for each angle between the rings. 5. If TV/ be the magnitude of the free screw, while TVristhat •of the constrained screw, we have as the components of these TVr'cosi// , TVr'sin ' and tcosxf/ , rjsimf/ respectively, while TV?* = icos2if/ + r]sm2i[/. If we write x = pcosif/, y = psim]/ we see that TVr-p2— 1, provided that £x2 + yy2 = l . Thus the magnitude of the rotation Vr is the reciprocal of the square of the parallel radius vector in the ellipse £x2 + gif — 1 regarded as having its x and y axes respectively parallel to Vr, and Vr2. But this ellipse is the ellipse, when referred to its centre, which is the transverse section of the elliptic cylinder above considered by the plane z = 0. It may be termed the amplitude conic. The pitch is given by pfx2 + pf) =pfx2 +p2yy2 . 6. If we eliminate 0 between the equations p{£c,os20 + ys\xi26) ^pfco^O -\- p2g^m26 2(£cos20 + ^sin20) = (pf - p.2g)sin0cos0 wve get (p - 2h)(p - P2)(p f - Ihvf + Pi = 0 1901-2.] Dr Peddie on Quaternions in Theory of Screws. 319 where (pf -p2y) = k(p1 - p2) . Taking p and zjk as current coordinates, we get a plane repre- sentation of the elliptic cylindroid by a circle. If we draw a line, parallel to the z-axis, in that plane at a distance (Pi+p^/Z from the centre, the distance of a point in the circle from that line is the pitch of the corresponding screw ; and k times the distance of the point from the ^-axis gives the value of z. Otherwise, we might take p and z as current coordinates and get the plane representation by an ellipse. The representation of other quantities, such as the angle between two screws, is not so simple as in Ball’s plane representation of the •ordinary cylindroid. The whole class of elliptic cylindroids, for which, with different values of £ and y, the condition pl£-p2y=Pi ~P% holds, have the same representative circle as the ordinary cylindroid has, so far as p and z are concerned. From the relations x' = £cos0, f = ysm6, where x, y are the coordinates of the extremity of TVr', we see that the locus of the extremity of TV/ is the ellipse This ellipse is therefore the amplitude conic for TV/, i.e. for the axes of the unconstrained screws, the name being more suitable in the present case since the radius is the rotation vector. The locus of the extremity of TVr is the sextic A simple geometrical construction for both loci is got by drawing through a fixed point a line of constant length which carries at its end a circle of constant size whose centre is in the prolonga- tion of the line. Through the junction a line, whose direction is fixed, is drawn, and the whole system revolves round the fixed point. The intersection of the circle by the line whose direction is fixed traces the free locus; and the foot of the perpendicular, 320 Proceedings of Royal Society of Edinburgh. [sess. drawn from that intersection, upon the revolving line traces the constrained locus. 7. The extension to similarly constrained screws in a rigid system having otherwise three degrees of freedom is easy. The pitch is given by p(&2 + w + &) = ipp2 + wi y2 + &f| or jpSpp = Spi/^p where p = xi + yj + zk , 4>p = - + k£Sk)p , XP= - (ipftt +jp2Sj + kp3Sk)p , and = X^> * The cone Sp^ - 7r<£)p = 0 gives the directions of screws of given pitch. To determine the locus of screws of constant pitch, let a be a unit vector along a generating line of the ruled surface, and let cr he a vector to a point upon the line. We have crSpp = - Pi£V iaSia - ppf^j aS/a — p3£ V koSka + Vo. where v is a variable scalar. So o"Sp<£p — Vif/aa + Va , = V (if/ — p)aa + Va , a — “ (^-p)_1Vo-aSppl = - -p)a-ypS(r(iJ/ —p)cr = = (Pl£ -P)(P2V -P)(P£ ~P) • When = 0 this becomes S'2ppSo-i/ro- =PiP-iPffC , the surface upon which lie the rotors of the system. If we choose Sp^p = it becomes Scrif/a- —PiPiPz • ( Issued separately October 7, 1902.) 1901-2.] J. G. Goodchilcl on Scottish Mineralogy. 321 Contributions to Scottish Mineralogy. (Part I.) By J. G. Goodchild, of the Geological Survey, F.G.S., F.Z.S. Communicated by F. Grant Ogilvie, M.A., B.Sc., F.R.S.E. (Read July 7, 1902.) Introduction. — The following observations are based chiefly upon the results of an exhaustive survey of the specimens in the Scottish Mineral Collection in the Edinburgh Museum of Science and Art, which were made while I was studying the material for completing The Mineralogy of Scotland. The survey has been continued since the publication of that work, in connection with the revision of the arrangement of the specimens in question. A large number of facts of interest have come to light in the course of this work. Further- more, the forms of several hundreds of crystals in the Collection have now been determined, and freehand drawings of most of these crystals have been made, and placed alongside of the speci- mens to which they refer. It is proposed from time to time to lay some of the more interesting of the results arising from this work before this Society, especially as little or none of them have hitherto been published, and as the crystals, which present considerable interest, do not appear to have been previously figured. It will be assumed here that the principal object in studying minerals (apart from the work necessary for museum purposes) is to obtain as much information as possible relating to their genetic history, and also to contribute to our knowledge of chemical — or perhaps it would be better to say, molecular — physics. The fact that a certain mineral occurs in this or the other parish, or that its crystals show such and such a combination of forms- — facts not without interest in themselves — are of far less importance than the larger questions referred to, and are of value chiefly in proportion to which they help us to know more about the genetic or develop- mental history of the native compound under consideration. Albite. — Amongst the crystals just referred to, those of the PROC. ROY. SOC. EDIN. — VOL XXIV. 21 322 Proceedings of Royal Society of Edinburgh. [sess. Anorthic Felspars form a very prominent feature ; and of these felspars the one which presents the greatest number of interesting features is the Aluminium-Sodium Polysilicate, Albite. As regards the mode of occurrence of this felspar the fact may be stated that it is known to have been formed under at least five different sets of conditions. The first of these, and probably the most common, is as an original constituent of certain eruptive rocks of deep-seated origin. In this connection it is found as one of the constituents of certain granites, notably, in Scotland, in that of Peterhead, where it is a common associate of a potash felspar, of Quartz (usually the smoky variety), and of a variety of Biotite containing rather a higher percentage of iron than is usual in that species. In the case of such plutonic rocks as these, it would probably be safe to say that Albite never shows all of its ordinary natural crystalline boundaries, because it has grown up from the magma concurrently with the enlargement of its associated minerals, through which cause its natural tendency to take a definite shape has been hindered more or less. In other words, in these plutonic rocks Albite is allotriomorphic, and has been obliged, in growing, to adapt its boundaries to those of its neighbouring crystals which were doing the same. Albite occurs in granites also under somewhat different circumstances, and is of much commoner occurrence thus. Its history in these cases is not difficult to make out. It is clear that during the later phases of consolidation the granite masses gave off considerable quantities of vapour, in much the same way as lavas do. From this cause many cavities, analogous to the vapour cavities of lava, were formed — usually on the outer parts of the granite, where it may be presumed that the pressure was liable to be occasionally lessened. A druse formed under these conditions presented a very suitable nidus for the development of crystals. Accordingly, we find in them some of the most interesting minerals that occur in native connection with the granites. Amongst these may be mentioned Topaz, Beryl, Cairngorm, Pock Crystal, and other gem stones. The commoner rock constituents of the granite itself also appear in these cavities. A definite order of succession prevails amongst these. As a rule, the potash felspar crystallises first, then the quartz crystallises upon that, and later than both 1901-2.] J. G. Goodchild on Scottish Mineralogy. 323 commonly occurs the mineral under notice. One is almost justified in regarding these occupants of the druses in granites as represent- ing such of the substances which existed in the solutions from which the granite originated, which were not, so to speak, needed for the building up of the granite itself. If eruptive rocks are really formed (as I think they may have been) through the slow solution of various pre-existing rock materials, sediments amongst others, by the action of thermal waters holding alkalies in solution, the formation of these druse minerals can be easily enough accounted for. They could, in that case, simply be regarded as the latest manifestations of the same hydro-thermal causes as those to which the formation of the parent eruptive rock was due. It is noteworthy that in all the cases in which the druse is simply lined with the crystals, they have developed fully, and have retained their own crystal boundaries. But wherever the crystals have grown so as to touch those adjacent, allotriomorphism has commenced, and the crystal boundaries of both minerals in contact disappear. In a few cases it can be shown that a certain amount of absorption of the substance of the crystal affected has taken place ; and it is noteworthy that nearly all such absorption takes place along the crystallographic axis which has previously coincided with the direction of greatest elongation during growth. This appears to be the case with all the minerals which tend to assume a more or less elongated form. Allotriomorphism affects elongated crystals of Beryl, for example, almost wholly at their ends, and hardly at all round the prism zone, the faces of which usually remain idiomorphic. In the case of the granite druses which contain Albite, it is not at all uncommon to find that the associated potash felspars yield on analysis a variable, though usually small, percentage of soda. This soda may be regarded as occurring in the condition of Albite, intergrown with the potash felspar in some way which has not as yet been accurately determined. As already mentioned, the commencement of the growth of the Albite crystals does not appear to have been begun until after the formation of the Quartz and the Orthoclase crystals had been completed. This is shown by the fact that the Albite may occur upon the crystalline faces of either of those of its associates. In 324 Proceedings of Royal Society of Edinburgh. [sess. the cases where the Albite has grown upon the potash felspar, it not uncommonly selects some definite form, such as the unit prism {110}, and grows exclusively upon that. There would thus appear to he some greater surface tension between certain faces of potash felspar and the solutions tending to deposit Albite than there is in the case of other faces. Gravitation has nothing to do with this selective action, because the faces in question may lie in any direction with respect to the horizon. The feature in question is well shown upon specimen 316/33 of the Scottish Mineral Collec- tion,* which is from Sterling Hill, Aberdeenshire. I shall endeavour to give fuller details further on in support of the view here advanced, that variations in the magnitude of the surface tension between the solutions and certain faces upon which deposition is in progress, are responsible for the remarkable selective action which has evidently regulated the growth of many crystals. The next point for consideration in connection with the Albite crystals relates to their growth. It is probable that few, or no, Albites are entirely free from traces of Calcium ; though, in general, the proportion is below two per cent. The usual theory regarding this calcium is that it exists in combination within the Albite as units (or, as some would regard it, as molecules) of the Aluminium- Calcium Polysilicate, Anorthite. It is usually considered that there is a progressive change in composition from Albite through Oligoclase, Labradorite, etc., to the pure lime felspar Anorthite ; the minerals just mentioned being only two c species ’ arbitrarily chosen out of an almost perfect series of intermediate gradations between the two extremes. It will be convenient to assume, for the object at present in view, that the Scottish Albites under notice are of this compound character, and are not absolutely pure. We may here recapitulate the points under consideration : — The stages so far noticed in the developmental history of Albite are represented by (1) the chemical atoms composing each of the Units of Substance ; (2) the separate units of Soda Felspar and Lime Felspar ; (3) the substance formed by the admixture of these two. Beyond this point many who have speculated upon the subject * The registration marks, or other means of identifying the specimens referred to, will be given in every case cited in these papers, so that the statements may be independently verified when anyone wishes to do so. Mineralogists should agree to give similar references in all cases. 1901-2.] J. G. Goodchild on Scottish Mineralogy. 325 would be inclined to place the Units of Structure, by whose arrangement the sub-individuals which now come to be considered have been built. Be this the correct view or not is immaterial. But (4), the stage of the sub- individuals, is evident enough. The whole of the plagioclase felspars, from Albite to Anorthite, have long been known to be made up of laminae of varying thickness, but which are usually very thin, and rarely exceed a millimetre in thickness. Each of these laminae generally shows the commonest unit forms of the complete crystal, i.e. {010, 110, llO, 001, 101 and T01 }, but rarely other forms than these. Each of these sub- individuals has grown next to another sub-individual, which has been, as it were, rotated, through 180° around an axis coincident with {010}. (5) These Polysynthetic Twins are firmly united into what one may regard as a crystal unit, or, as it will be called here, Individual, which, in addition to the crystal forms just mentioned, may show others which are never developed upon its components. (6) The developmental history does not stop at this. In many cases one individual is again united with another, either individual being invariably rotated, in relation to the other, through 180°, around one or other of several different axes, thereby giving rise to Twins, which are usually doublets, but may be triplets or higher combinations than those. It is remarkable that the plane with reference to which the two individuals become symmetrical after the revolution referred to in no case coincides in direction with a plane of symmetry for the individuals themselves. This, of course, is not manifest in the Anorthic System, though it is clearly enough seen in the others. (7) In some few cases Albite twins may be twinned yet again, whence arise Compound Twins. Lastly, (8) Albite twins, simple or compound, may group together into either Parallel Growths, in which some face may be developed either so as to be coplanar throughout the entire group, or else into more or less Irregular Aggregates, in which any such coplanar growth of a particular face does not take place. One remarkable feature connected with these aggregates is the tendency manifested by many of them to arrange themselves in such a manner that further enlargement may, so to speak, facilitate the growth of the whole colony into one large crystal. This curious fact, which may almost be regarded as foreshadowing future events 326 Proceedings of Royal Society of Edinburgh. [sess. in the developmental history of the crystals, has not received the amount of attention it seems to deserve. I am not aware, at present, that it has been noted at all. Yet it is quite a common feature of crystal aggregates of many different species. Perhaps it has been passed over owing to a certain reluctance on the part of both mineralogists and biologists to admit the existence of too close a parallel between organic processes and the forces which regulate the growth of a crystal. It may be remarked now that there is not a single feature to which attention has been called in the case of Albite that does not find its parallel in the case of many other minerals. Polysynthetic twinning, for example, is by no means confined to Anorthic minerals. Much of what passes for simple parallel growth in some Orthorhombic species may really be of this nature ; and it may be found in crystals belonging to other systems as well. In Albite the striations, which are the outward manifestations of polysynthetic twinning, are due to alternate salient and retiring angles formed by the anorthic angles of the adjacent sub-individuals. These, in the aggregate, often give rise to definite angles which quite commonly agree with the true crystallographic angle proper to two adjacent faces. It is a curious fact to be noted in this connection, that many species of minerals show these aggregate forms. In Gypsum, for example, the unit prism {110} is fre- quently represented by an aggregate formed by the edges of several other forms in the prism zone, amongst which, occasionally, the unit prism itself may be conspicuous by its absence, and yet the angles -agree with those of the form which is simulated. Many other instances of the same kind could be cited. In all of these, as in the case of the coplanar faces of the aggregates noted above, there seems to have been what might be regarded as a kind of agreement amongst the components of the aggregate to grow out together to one'definite plane, and there to stop. It may be remarked here that the different grades in the developmental history of such a crystal as Albite have not yet all received suitable designations. It may be safely remarked that such terms will be needed as soon as mineralogists come to recognise the important bearing these facts must have upon many problems which await solution relating to crystal growth. 1901-2.] J. G. Goodchild on Scottish Mineralogy. 327 Reverting to the modes of occurrence of Albite, we find that this mineral occurs, sometimes in large masses, as one of the constituents of the pegmatites which form part of the Archaean gneisses of Scotland, especially in the Hebrides and in Shetland. These gneisses appear to have originally been various eruptive rocks of deep-seated origin, such as granites, diorites, gabbros, and others. They have all undergone more or less deformation, as a result of differential movements throughout the mass, caused by crust-creep, which is probably one of the manifestations of the same terrestrial forces to which the elevation of continental areas and the depression of ocean basins have nearly always been due. The rock materials have been crushed and drawn out into a laminar arrangement by the earlier movements, which took place under great superincumbent pressure, and in the presence of water. When this movement has been repeated (if the rocks have not been annealed by exposure to a high temperature in the mean- time) the constituents have become granulitic, just as metals would under like circumstances. At a later period reconstruction has taken place, in the presence of concentrated sea water, I think, and the constituents, perhaps as a consequence of a temporary and local relief of part of the pressure, have recrystallised. The process takes place through the formation of ‘ augen,’ in which each ‘ eye ’ consists of a separate mineral, which has enlarged apparently by the migration in the direction of each ■* eye 5 of constituents derived from the rock adjacent. If, as seems probable, the process took place under conditions which permitted of the introduction of additional alkalies derived from the sea water carried in by osmosis, the percentage of alkali silicates, Albite amongst others, may well have been increased. But these are, at present, little more than hypotheses, which, though they appear to harmonise with the facts, may not in all cases be the true explana- tion. Be that as it may, the largest masses of Albite occurring in Scotland are found under the conditions above mentioned. They are never idiomorphic, and there is, therefore, no means of deter- mining what their developmental history as crystalline masses has been. They all, however, show traces of polysynthetic twinning. Under subsequent strain arising from movements after the final stage of consolidation, a further change has affected the Albite 328 Proceedings of Royal Society of Edinburgh. [sess. augen (as well as the minerals associated with them). This appears to have resulted in a slight separation of the polysynthetically- twinned laminae, whereby a peculiar chatoyance has been developed. Excellent examples of this structure may be seen in the Scottish Mineral Collection in the Edinburgh Museum of Science and Art. The subject is referred to here as one illustration that may be chosen out of many in which a characteristic lustre has been developed by the formation of structures allied to the twinning referred to. As examples, reference may be made to Stilbite, Heulandite, Cerussite, and many others. In all of these closely- adpressed laminae occur as original features. In other cases a slight separation along the cleavage planes has arisen through the opera- tion of changes subsequent to the completion of the crystal growth, as in the case of some Pyroxenes. It may be as well to remark that polysynthetic twinning in Albite cannot well have arisen through causes of a mechanical nature operating since the crystal was completed. Albite has lately been shown by Mr Clough of the Geological Survey * to occur as a constituent of certain schistose rocks in Cowal. There is clear proof that the rocks in question were originally sediments of the normal kind, and consequently there has been much discussion as to how the Albite got there. I should like to offer the suggestion that the Albite may, possibly, have originated through the transference, by osmosis, of sodium, from sea water, while these rocks were undergoing the dynamic changes which have led to their present schistose character. Geologists are apt to overlook the fact that there is a continual transfer of the alkaline carbonates from the land to the sea, and that the proportion of these returned to the land in the form of wind-borne sea-spray cannot bear more than a small proportion to that of the quantity travelling in the opposite direction. If part of the alkali is returned to the sediments beneath the ocean floor by osmosis or other causes, one can readily understand how the balance is maintained, and also how the cycle of change from eruptive rocks to sediments, and from sediments back again, by the restora- tion of the alkalies divorced from their parent rocks by sub-aerial * Geological Survey Memoirs in The Geology of Cowal, by C. T. Clough (and others). 1901-2.] J. G. Goodchild on Scottish Mineralogy. 329 waste, may be continually in process of completion. Whether this is so or not, the possibility of such a process being always in action deserves to be borne in mind, and especially when we are dealing with the genesis of the distinguishing minerals of the eruptive rocks — the felspars. In some few cases Albite, or a substance agreeing with it in composition, has arisen through changes which have operated upon some zeolitic minerals. A specimen of the so-called Erythrite, pseudomorphous after Prehnite, (316/44), is exhibited in the Scottish Mineral Collection, from Bowling, Dumbartonshire. Possibly other instances may be brought to light when the subject is further looked into. The Albite crystals in the Collection just mentioned, of which Albite. Fig. 1. — Orthographic projection on Z. Poly synthetically twinned sub-individuals, again twinned in accordance with the Albite Law. Forms: — Right positive hemiprism and its left analogue { 110 } ; together with the right and left liemi-brachyprisms { 130 } ; the basal pinacoid { 001 } ; the brachypinacoid { 010 } ; the right and left negative hemipyramids (ill, Il2, 113} ; and the negative macrodome there are many fine examples, do not appear to have been figured.* I therefore think it well to give the following examples here. Their registration numbers, by which the originals in the Museum may be readily identified, are fig. 1, 316/22; fig. 2, 316/7a; and * After a careful and prolonged search through the Scottish Mineral Collection, I have only been able to find a very small percentage of any of the crystals which correspond to the figures in The Mineralogy of Scotland. fig. 3, 316/7. 330 Proceedings of Royal Society of Edinburgh. [sess. Has Fig. 2. — Orthographic projection on b {010. } A simple crystal drawn so as to show the general habit of each individual of the Albite twins. The forms shown are the brachypinacoid { 010 } ; the unit prisms { 110 } ; and the brachyprisms { 130, 150 } ; the basal pinacoid { 001 } ; the negative macrodomes (101, 20l and 405), the latter being unusual ; and the right lower hemipyramid (111). Fig. 3. — Clinographic projection of a complex group of five individuals. Each is a polysynthetically-twinned aggregate of many sub-individuals, and the individual crystals are again combined in accordance with the usual Albite Law. The forms present are : — The brachypinacoid, b { 010 } ; the basal pinacoid, c { 001 } ; the right .unit hemiprism, m (110) ; and its left analogue, M (110) ; with the left brachvprism, 2 (130) ; the negative macrodomes, x (101), y (201), and E (203) ; together with a negative hemipyramid of the unit series, which appears to be A {443}. Forsterite. — When Mr Clough of the Geological Survey was examining the metamorphic marbles of the Glenelg district he dis- covered several minerals of considerable interest. To these was afterwards added the mineral named above, which was identified by Messrs Teall and Pollard, and was described by them in the Quarterly Journal of the Geological Society of London, vol. Iv. p. 372. The marble had been considerably altered by both dynamic causes and by hydrothermal action ; so that the impurities originally present in it, together, possibly, with other constituents carried into the rock by the heated waters, eventually combined 1901-2.] J. G. Goodchild on Scottish Mineralogy. 331 and formed the assemblage of minerals which usually occurs in calcareous rocks under these circumstances. Speaking generally, these, in Scotland, usually include Quartz, Tremolite, Diopside, Phlogopite, Andesine, Anorthite, Spinel, Wollastonite, Graphite, Sphene, Pyrrhotine, the variety of Garnet known as Cinnamon stone, together with Tdocrase, and often with Zoisite, Spodumene, and others. Latterly Brucite has been detected in one of the Scottish marbles. Many of these minerals, and especially the six mentioned first, are closely related to the original constituents of eruptive rocks. Hence it may be concluded that these latter rock- forming minerals themselves may really he, like those developed by metamorphic action within the marble, also of hydrothermal origin, as so many observers have from long since maintained. Olivine has not yet been observed to have been formed in this way. But the mineral under notice, Forsterite, which is an Orthorhombic Magnesium Orthosilicate, is very closely allied in both composition and crystalline habit to Olivine, and has indeed often been termed White Olivine. Since Mr Teall’s paper appeared I have made a careful examination of other Scottish marbles whose history had been much the same as that of Glenelg. One set of specimens of metamorphosed Durness (Cambrian) dolomitic limestone, from Ledbeg, in Sutherlandshire, showed on a weathered surface a large number of crystals of this magnesium silicate. It had been placed with Diopside, and appears to have been figured as that species. Other examples of the same mineral in the Collection had been regarded as Chondrodite, Totaigite, etc. Some crystals were forthwith detached from the matrix, mounted on small corks as usual, and then examined with the goniometer. The faces were too dull to give more than rough approximation, but these agreed with Forsterite. As this mineral has rarely been figured, I give two characteristic examples here (figs. 4 and 5). The crystals are all attached by the c {001} axis; and they are all flattened parallel to b {010}. Their general habit is rhomboidal. The forms shown are the following : — The Brachypinacoid, b {010} ; the Unit Prism, m {110} ; the Unit Pyramid, e {111} ; the Brachydome, k {021}; and the Brachypyramid, Z {131}. The registration numbers of the specimens figured are 375/7 and 375/8. It may he observed 332 Proceedings of Royal Society of Edinburgh. [sess. here that the first number in the registration mark is that adopted for the species by Dana, in the Sixth Edition of his System of Mineralogy. It would be extremely convenient if, now that the numbers have been so used, mineralogists in general, including the author of the above named work — the most generally used of all — were to agree to continue the practice. It often saves much time to denote a mineral by its number. It may be well here to recapitulate the nature of the parallel between the minerals thus developed by metamorphic- action with- in marbles and those which constitute the commoner minerals of the eruptive rocks. Quartz is common to both ; Tremolite is a mere variety of Amphibole ; Diopside is a variety of Augite ; Andesine runs parallel, as it were, to Labradorite, the commonest felspar of the more basic eruptive rocks; Phlogopite is a near ally of Biotite ; Spinel may he regarded as representing Magnetite ; and, lastly, Forsterite is regarded as one of the Olivine Group. These facts need to be steadily borne in mind in all considerations respecting the origin of the eruptive rocks ; especially if we regard all these latter as simply having crystallised from aqueous solutions which were at a high temperature. It is, however, an interesting fact in this connection that some closely-allied minerals have evidently been formed under quite different conditions. For example, Pectolite, which in origin is almost to be regarded as a Zeolite, has certainly been formed from the decomposition of parts of an eruptive rock of basic composi- 1901-2.] J. G. Goodchild on Scottish Mineralogy. 333 tion, by the action of non-thermal waters, percolating downwards from the surface. Its near ally Wollastonite, however, has not yet been proved to occur under any conditions but those connected with the metamorphism of lime-bearing rocks by the action of heated waters. Epidote and Zoisite occur in like manner. Epidote in a large number of cases is certainly due to the thermo-meta- morphism of one of the Green Earths, which, primarily, was formed by exactly the same process as the Zeolites, and is confined to rocks of eruptive origin. Zoisite has in many cases, and probably in all, been formed within calcarebus rocks, by exactly the same processes as those which give rise to Wollas- tonite. Pyrrhotine and Pyrites, again, form a similar parallel. All (or nearly all) Pyrrhotine occurs in direct association with calcareous rocks which have been affected by hydrothermal action. Many other cases of the same kind could be cited. Malformation , or irregular development, of crystals. — In the case of the great majority of crystals, whether formed by natural agency or by artificial means, there is manifested a tendency towards the irregular or unequal development of the size of the faces of one or more of the forms present. Mineralogists commonly refer to this as ‘distortion,’ a term which must strike many persons as singularly inappropriate, seeing that the crystal never was of ideal habit, and that it has not been forced out of its original shape. ‘ Malformed ’ will probably be thought by many to be a better term. The malformation itself is clearly due to some special cause in operation during the growth of the crystal, and the malformation in question is therefore a record of part of the history of the conditions under which the growth was carried out. That being the case, it would seem desirable to preserve the record, in case the feature in question should some day prove of value in throwing light upon its own history. It seems to be the usual practice, however, in drawing crystals, to depict them in ideal symmetry. Nearly all the figures reproduced in Heddle’s Mineralogy of Scotland are thus represented. The objections to the practice are, that in restoring ideal symmetry, an important piece of evidence relating to the history of the crystal has been wiped out; and, what is of still more importance, the '334 Proceedings of Royal Society of Edinburgh. [sess. identification of the specimen from which the figure is reputed to have been drawn is rendered, in many cases, quite impossible. It would be much better, in all cases, if each crystal were drawn as nearly as possible as it appears ; and that the whereabouts of the particular crystal so drawn should be indicated in such a manner that the evidence for the forms shown could be easily examined by anyone who wished to do so. This would also act as a check in future upon anyone who should trace a modified copy of a foreign crystal in order to have it re-engraved as one of British origin. As examples of actual crystals, malformed by unequal growth in different directions, the following figures from crystals of Analcime in the Scottish Mineral Collection are given here. Their registration marks are given next the figures, as in the other cases. Figs. 8 and 9 show forms which are rare in Analcime. Fig. 7. 1901-2.] J. G. Goodchild on Scottish Mineralogy. 335 Cerussite. — An exceptionally fine series of crystals of Scottish Cerussites, nearly all from the Leadhills district, is exhibited in the Scottish Mineral Collection. Few, if any, of these appear to have been figured, In going through the Collection, chiefly with the object of determining and recording the various combinations represented, I made freehand drawings of a large number of the crystals. The individual crystals so drawn are clearly indicated by means of pointers attached to the specimens in which they occur, or else are mounted with seccotine on small corks, which, in turn, are fastened to the card on which the sketch has been made. Registration marks readily enable one to refer to, or to find, the specimens so figured. This plan has been followed throughout the whole of the Collection. Cerussite, the Orthorhombic Lead Carbonate, is always a mineral of secondary origin, and has generally arisen from the decomposition of other ores of lead through the action of waters percolating from the surface. Galena is the^usual parent of Cerussite. The active solvent agent appears to have usually been either car- bonic acid or else one or other of the humus acids. It is worthy of remark, therefore, that Cerussite arises only under humid climatal conditions. In those regions where there are deserts, or where, in general, arid conditions obtain, it is probable that the carbonates are not so commonly formed as decomposition products of metal- liferous deposits as are the chlorides and bromides. A review of the regions of the world where the chlorides and bromides of the metals are of most common occurrence will at once show that these regions are geographically regions of inland drainage, where the rate of evaporation either balances, or is sometimes in excess of, the precipitation. Under these conditions the chlorides which are carried inland from the sea, first in the form of sea-spray, and then of salt dust, eventually form the nuclei of raindrops, of flakes of snow, or of dew and hoarfrost, and are left upon the land as evaporation carries off their solvent. The surface soil thus becomes heavily impregnated with these desiccation products, which, under normal geographical conditions, would have been returned to their native home, the sea, but which, under arid climatal conditions, slowly percolate into the crust of the earth, and set up chemical changes which may eventually become of considerable importance. 336 Proceedings of Royal Society of Edinburgh. [sess. On theoretical grounds it has been stated that one might expect to find that the conversion of the Lead Sulphide, Galena, into the Carbonate, would have left some traces of the sulphate, Anglesite, seeing that the process involves the oxidation of the sulphur present in the original ore of lead. In actual experience this does not appear to be in all cases supported by the facts, as Cerussite may occur quite commonly without the slightest trace of the sulphate. What really happens is, that the solutions arising from the decom- position of the Galena pass almost directly into the carbonate, if any carbonate of lime happens to be present. Hence Anglesite is rarely, or perhaps never, found in any of the lead veins traversing the Car- boniferous Limestone areas of the North of England. It is only in the veins traversing non-calcareous rocks that one may look for Anglesite with any hope of success. In the Leadhills district, where the rocks are non-calcareous, both Cerussite and Anglesite are found in close contiguity. There is a slight difference in their mode of occurrence. Both are confined to metalliferous veins ; but the Anglesite usually occurs actually grown upon the Galena by whose decomposition it has arisen ; while the crystals of Cerussite are more commonly found growing upon other constituents of the mineral veins, although in no case at any great distance from the parent mineral. It rarely occurs quite alone, but more usually in close association with other products of decomposition. In the Leadhills mines its common associates are Pyromorphite, Vana- dinite, Chrysocolla, Malachite, and Limonite. Some of the rarer minerals associated with it are, Leadhillite, Linarite, Caledonite, and others. Chalcopyrites, as well as Blende, and one of its decomposition products, Hemimorphite, the Orthorhombic Hydrous Zinc Subsilicate, are also common in the same veins. The history of the Leadhills metalliferous veins appears to be somewhat more complex than that of the great majority of mineral veins. They occur, as usual, in fault fissures; which appear to have acted as channels of communication between the inner portions of the Earth’s crust and its surface, at many different geological periods. Hence the veins in question have probably been filled by the uprise of thermal waters, and subsequently depleted to some extent, by the action of surface waters, on several different occasions. The varied and complex nature of the 1901-2.] J. G. Goodchild on Scottish Mineralogy. 337 contents of these veins at the present day may therefore be safely regarded as the net outcome of all these ebbings and flowings in the past. The mode of occurrence of much of the Chalcopyrites, Galena, and Blende, appears to suggest that, whatever may be the age of some of the others, these minerals may have been intro- duced in their present state in comparatively-recent geological times. Perhaps they are the result of that phase of volcanic activity which formed the closing episode of the outbreaks to which the vast pile of eruptive rocks of the Hebrides and Arran, etc. are due. There is good reason for believing that most of the present contents of the North of England lead veins date from this same period, and that they are all of later date than the last great movements of upheaval of the land.* There may well have been remnants of older deposits in the Leadhills veins, formed at any one of the periods of volcanic history prior to this, and later than the formation of the fissures in which the veins occur. The associations of the Cerussite at Leadhills are of interest, and throw some light upon the history of the mineral. Cerussite is, of course, of later origin than the Galena. It is later, also, than the associated Chrysocolla, because, wherever the two occur together, the Cerussite occurs invariably upon this mineral. It is later than part of the Pyromorphite, for the same reason. But the growth of the associated Pyromorphite must have gone on concurrently with that of the Cerussite, because the two minerals are occasionally intergrown, and Pyromorphite, in the form of tufts of minute crystals, is sometimes sprinkled over the finished Cerussite. Exactly the same relationship as that between Cerussite and Pyromorphite can be made out between the former and the Monosymmetric Hydrous Copper Carbonate, Malachite. Its relation with the rarer minerals, Leadhillite, Caledonite, Linarite, and Lanarkite, are of the same nature. But the Cerussite is older than much of the Calcite, and also of the Limonite. All this points to Cerussite as being one of the latest-formed minerals in the veins. Perhaps that fact may serve to explain why it is that no clear evidence seems yet to have been obtained of the * Trans. Cumb. and Westd. Association , part vii. p. 108 (1883) ; and Proc. Geol. Assn., vol. xi. No. 2, p. 57. PROC. ROY. SOC. EDIN. — YOL. XXIY. 22 338 Proceedings of Royal Society of Edinburgh. [sess. dissolution of any part of the Cerussite. Certain of the contours of the crystals seem at first sight to be rounded ; but this feature is, on closer examination, seen to be due to the co-existence of a number of adjacent faces, usually brachy domes, which form but small angles with each other. It is possible, of course, that these may represent later crystalline growths upon a surface rendered convex by the action of solvents. Scottish Cerussites are usually colourless, or, at the most, of a pale cream-colour. A few are tinted green, but whether the colouring matter in this case is due to admixture with compounds of Vanadium, or of copper carbonate, has not yet been determined. Some Cerussites are lead grey, ranging to sooty brown. The colouring matter in this case may prove to be lead dioxide. (A compound or mixture largely consisting of lead dioxide has been stated to occur at Leadhills. Some mineralogists have regarded this as a definite compound, and have elevated it to the rank of a separate mineral species, under the name of Plattnerite. It seems to me to be a decomposition product, of indefinite composition, which may have arisen through the action of surface waters upon what has been called plumbo-calcite.) It is remarkable that so little of the Cerussite shows any trace of coloration by the Vanadium compounds, although these have imparted brilliant yellow, green, and other tints to the associated Hemimorphite and Pyromorphite. The Vanadium in the Leadhills mines may possibly have originated through the decomposition of the ferro- magnesian silicates present in the Arenig lavas there, as small traces of Vanadium can usually be detected in the fresh minerals in question. Possibly the phosphoric acid, to which the Pyro- morphite is partly due, may have arisen from the decomposition of the Apatite crystals in the same rocks. The point • of attachment of the Scottish Cerussites appears to be usually c {001}. The direction of greatest elongation varies in different crystals, and does not appear to follow any definite law. But elongations parallel to a {100} and to c {001} are commoner than others. The subjoined figures (10 to 17) will illustrate the diversity of habit referred to. Traces of a kind of polysynthetic growth in directions parallel to b {010} are so very general that they may be regarded as of 1901-2.] J. G. Goodchild on Scottish Mineralogy. 339 universal occurrence. They remind one of the polysynthetic twinning seen on Albite, and, as a matter of fact, there is nothing to show that they are not really twin laminae. This remark applies to the so-called parallel growths in orthorhombic crystals in general. In addition to this feature of somewhat doubtful nature there is present in nearly all Cerussites a more or less evident tendency to form stellate aggregates by the twinning of three (? of six) individuals on m {110} as a twinning plane. The angles between any two adjacent brachydiagonal axes does not 340 Proceedings of Royal Society of Edinburgh. [sess. exactly form a divisor of 360°. Hence the compound crystal built up by growth between adjacent individuals always shows re-entrant angles. I have never seen a crystal of Cerussite that did not show some trace of the twinning referred to. See figs. 10-12, 14-16. One remarkable feature connected with these compound Fig. 15. Cerussites is that the c face {001} is (or appears to be when studied under a high-power lens) coplanar to all the individuals, no matter what amount of irregularity of growth may characterise the other parts of the crystal. This feature will recall what was stated above in connection with Alhite. If these cases stood alone one might pass them by as accidental. But the same features are seen in the case of many other minerals, and they must be due to some cause which has not as yet been satisfactorily explained. One is almost tempted to regard it as a kind of 1901-2.] J. G. Goodchild on Scottish Mineralogy . 341 foreshadowing of coming events, and as if the crystal aggregate were being laid down upon predetermined lines with the ultimate intention of building one large crystal upon the foundation con- stituted by an aggregate of many smaller ones. Be that as it may, the fact is worthy of attentive consideration, which may eventually lead to a fuller knowledge than we possess at present of the laws regulating growth in the inorganic world. Coplanar faces are shown upon figs. 11, 12, 14, 15. An inspection of the accompanying figures of Cerussite (which are all taken from registered specimens in the Scottish Mineral Collection) may serve to make these points clearer. I will only add, in concluding this paper, that the symbols employed are those used by Dana. These are excellent in their way, but if we were to adopt a uniform set of symbols for each of the more commonly occurring forms it would be a great improvement upon the system (or want of system) at present in use. I would suggest that the Unit Octahedron {111} in all minerals should be lettered o ; that the Unit Macrodome (or its equivalent form) {101} should be denoted by d; and the Unit Brachydome (or its equivalent form) {011} should always have the symbol e. {100, 010, 001} and {110} are already provided for by a nearly universal employment for them, respectively, of the symbols a, b , c, m. We might go further in the same direction with advantage. ( Issued separately November 15, 1902 ) 342 Proceedings of Royal Society of Edinburgh. [sess. On Causes which determine the Formation of Amor- phous Sulphur. By Professor Alexander Smith, B.Sc., Ph.D. (Read July 21, 1902.) While the investigation of the freezing-point of sulphur, de- scribed in these Proceedings (vol. xxiv. p. 299), was in progress, some experiments on the maximum quantities of amorphous sulphur, formed at various temperatures from 130° to 448°, were being made. The results showed remarkable irregularities. It was also found that the addition of foreign bodies, which are • for the most part without chemical action on sulphur, greatly magnified these dis- turbances, and led to the discovery of their source. The following is a condensed statement of the principal facts. When sulphur, purified by recrystallisation from carbon disul- phide, was heated for one and a half to two hours at 448° in an open tube, varying proportions of amorphous sulphur seemed to be formed. The tube was transferred rapidly to, and broken into a vessel of ice, so that no opportunity might he given for reversion to the soluble form. The variations, which ranged from 32 per cent, down to 10 per cent, of amorphous sulphur, appeared to depend in part upon the length of time intervening between the purification of a large lot and the use of a particular portion of it. Portions taken early gave, as a rule, less of the amorphous body. Heating for a longer time than two hours seemed also to reduce the proportion. Part of a specimen gave, after two hours’ heating, 31*8 per cent., while another part of the same specimen, after eight hours’, gave only 2 '6 per cent. It seemed evident that sulphuric acid or sulphur dioxide, pro- duced by the action of the air in variable hut minute amount, was responsible for the differences. The more prolonged heating eliminated most of the sulphur dioxide. Passing certain gases through the sulphur during the heating increased the yield of amorphous sulphur. Thus, dry air gave 34*5 per cent., dry sulphur dioxide 36-5 per cent., dt-y hydrogen chloride 1901-2.] Causes of Formation of Amorphous Sulphur. 343 38*5 per cent. Under these conditions, long continuance of the treatment did not cause any reduction in the yield. Addition of powdered glacial phosphoric acid had the same effect as these gases (37 ’3 per cent.). On the other hand, certain other gases, when used from the beginning of the heating, seemed to prevent the formation of the amorphous sulphur. Nitrogen gave 4 ’7 per cent., carbon dioxide 4’5 per cent., hydrogen sulphide 0'8 per cent., ammonia 0 per cent. These gases also reduced the quantity to the same minima when, by use of air or sulphur dioxide, the larger amounts had already been formed. This remarkable fact was confirmed by repeated trial. In the case of phosphoric acid, however, treatment with carbon dioxide did not reduce the proportion, presumably because this acid is non-volatile. Conversely, air and sulphur dioxide restored the amorphous sulphur, which had already been destroyed by treatment with carbon dioxide. A little powdered potassium hydroxide had the same effect as the second class of gases. From these observations it may he inferred that absolutely pure sulphur would altogether lack the power to form amorphous sulphur. This work is being continued. The obviously cognate question of the supposed rule that sulphur precipitated from acid solutions is amorphous, and from neutral solutions is crystalline, will be examined. It is not advisable to offer any theory until further work has been done. ( Issued separately November 15, 1902.) 344 Proceedings of Royal Society of Edinburgh. [sess. Quaternion Notes. By the late Professor Tait. Communicated by Professor C. G. Knott. (With a Plate.) (Read June 16, 1902.) The following quaternion notes are interesting as the last piece of mathematical work written down by Professor Tait. During his last severe illness he referred several times to the importance of working out still further the theory of the linear vector function ; but he had not been able to put pen to paper for months previous to July 2, 1901, just two days before his death. On that date he wrote down the notes now printed, probably at intervals through- out the day. They would serve, in all likelihood, chiefly as nuclei round which his thoughts would circle ; and they will now serve to indicate in some measure the directions in which his thoughts were moving. The foolscap sheet on which the notes were written was handed to his eldest son, with the request that it should be preserved, as he believed it contained the germ of an important advance. A facsimile of this sheet, on a reduced scale, is repro- duced in the accompanying Plate. With the exception of one line, the notes have to do with the linear vector function. This one line p= +?M_iVaUp.T^ represents motion in which the acceleration consists of a resistance- retardation directly as the square of the speed, and a component perpendicular to the plane containing the radius vector and a given constant vector, with a value proportional to the speed and to the sine of the angle between the radius vector and the constant vector. Had the second term on the right contained Up instead of Up, we should have had the case of a rotating projectile moving through air, gravity being neglected. This possibly was the intention. I think it may be assumed that Professor Tait was for the time engrossed with a fresh consideration of his well known golf-ball problem. 1901—2.] Late Professor Tait on Quaternion Notes. 345 The remaining notes should be read in connection with former brief abstracts read before the Society (May 18 and June 1, 1896 ; March 1, 1897 ; May 1, 1899). In the first line D is a contraction for Sa fiy, so that in the expanded form we have <£pSa/?y = ^aS/fyp + g.2fiSyap + gsySa/3cf> where a/3y are the directions of the real roots g1g2g 3. This may be written in the form D.p = (AaSa. + B/3S/3. + CySy.)(A' V fiySfiy. 4* . . . )p = D(AA'aS /3yp+ . . . ) = u)Top where (op = AaSap + (3 p + . . . and TSp = A'Vfiy&fiyp + BWyaSyap + . . . both being two self-conjugate functions or pure strains. Expanding and comparing term by term, we find D.AA' = p1} D.BB' = 02, D.CC' = p3. When ABC are given, A'B'C' follow. Hence there is an infinite number of ways in which (with real roots) may be decomposed into two pure strains. The next section begins with cf)p = oi& p (two pures) ~9P then so that the roots are being sought. Put 6T^p = (T to£7“cr = gZ5~}i(r Z!3^u)ZlJ\ a = gcr Hence G7*oou7- has the same root values as wS7, but different directions. The linear vector function is now expressed in the form p = C7p -f Vep where 67 is the pure non-rotational part of (f> . PROC. ROY. SOC. EDIN. — YOL. XXIV. 23 346 Proceedings of Royal Society of Edinburgh. [sess. Take any two vectors i, j, and consider the expressions S .ufyj = SJGX; + S.iej S .ji = S.jtti + S .jei S i&j is the same as S j&i because T3 is self conjugate. But S iej and Sjei are of different signs necessarily. Hence if S .ij and S .jcf>i are of the same sign, neither S .iej nor S.jei can he greater than Si&j. The remaining expressions treat of bi-vectors, and their inten- tion is to extend the investigation to the case where has only one real root. The root-vectors and the corresponding root values are a , p. + ic, p - 1€ aI1(i 7-7 7 -7 g , /i + ik, h - ik i being now the symbol for J — 1 and ghk scalars. The expression for D.<£p becomes pSae/3 = ga$e/3p + (he + k/3)B/3ap + (h/3 - ke)Saep . This form has already been partly discussed in the paper of June 7, 1897, and lines of development are indicated in the earlier paper of March 1, 1897, with which indeed these notes seem to have a very close connection. ( Issued separately January 17, 1903.) FACSIMILE REPRODUCTION OF PROFESSOR TAIT’S NOTES. Proc. Roy. Socy. of Edin. ] [Vol. XXIV. V <*' 1 v 2 V, V * . ♦ * "A ^1. i* Al ^ s.0: ^ * ’ ? ^ v O X, <3 ■V* > o <0 Si. k <5. v> & ?/ r k & * 'H IN i) $ # T ^ t i J? • <*L t ^ * Co * ^ ^ rT 11 « i." *1 u 'A 1 > 5" 'fa l S ~k* a k , 3 \ 5 * 5 a ■t -\A s. d" ^ § 3 u " A * •3 t .iff ft. U v. .T? I?1 4U*<* t • °!cii 1 '•4. *Cr + * 'M ^ £ 3 I i»h •h JP a UJ t <31 11 S*. u» C/> ^ '•A T» 1902-3.] Mr Manley on Electrical Conductivity of Sea- Water. 347 The Electric Conductivities and Relative Densities of certain Samples of Sea-Water. By J. J. Manley, Daubeny Curator, Magdalen College, Oxford. Communicated by Sir John Murray, K.C.B. (Read November 3, 1902.) Introductory. Several years ago, Dr Yeley and myself were engaged in making a prolonged series of measurements of the electric con- ductivities of nitric acid solutions of various strengths. During the course of the work we were repeatedly impressed by the fact that a very slight variation in the strength of a solution produced a very marked change in the value of the electric resistance, as measured in a somewhat high resistance cell. Some little time after the work upon nitric acid had been completed, the refractive indices of some samples of sea-water were determined ; and it was shewn in a paper * communicated to the Society that the optical method for distinguishing between the various samples was quite as delicate and as reliable as the relative density method. By the experience gained during the work upon the nitric acid solutions, the present author was led to conclude that the electric conductivities of different samples of sea-water would, in all probability, vary much more than either the corresponding refractive indices or relative densities. Accordingly, it was determined to make and compare a series of measurements both of the electric conductivities and relative densities of a few samples of sea-water, and to establish if possible some simple relationship between the two. With this object in view, a bottle of water was collected from off the head of the pier at Sandown, I.W., at 8.30 p.m. on Sept. 8th, 1900. The tide at the time was an inflowing one, and at almost half the full height. Before collecting the sample, the bottle was thoroughly washed out several times with * Proc. Roy. Soc. Edin., Jan. 1900. 348 Proceedings of Royal Society of Edinburgh. [sess- the sea-water, then filled and hermetically sealed ; it was then- packed and kept in complete darkness until its contents were- examined in the laboratory. It should here he stated that when this research was commenced I was quite unaware that Professor Knudsen had published in the Berichte der Kommission zur ivissenschaftlichen Untersuchung der ddnischen Fahrwasser (Copenhagen, 1900) the results of some very interesting experiments conducted by him, partly upon the open sea and partly in the laboratory. An abstract of his paper by Schott appeared in the Annalen der Hydrographie for March 1901, and to this Mr H. N. Dickson very kindly drew my attention. Knud sen’s experiments, like my own, had for their object the determination of the relationship between the total salinity and the electric conductivity of different samples of sea- water. The apparatus and methods adopted by Knudsen differ, however, in some essentials] from those chosen by myself; our conclusions also are somewhat at variance. Consequently, in view of the importance of the subject, it is hoped that a short account of my methods and results may not prove uninteresting. The work has been done in the Daubeny Laboratory of Magdalen College, Oxford, at intervals during the years 1901-2. I desire to express my indebtedness to Mr R. T. Glinther and Mr H. Hilton, Fellows of Magdalen College, for the very valuable assistance which they have given me during the course of this research • and also to the Government Grant Committee of the Royal Society, for the continued loan of certain pieces of apparatus. Preparation of the Samples of Water for Examination. In addition to the natural sea-water, four others were prepared from portions of it ; two had a smaller and two a greater total salinity than that of the natural sea-water. The two weaker samples were respectively prepared by adding 10 c.c. of distilled water to 190 c.c. of the sea-water, and 5 c.c. of distilled water to 195 c.c. of the sea-water. The two samples having a greater total salinity were obtained by concentrating separate quantities in a porcelain dish over a water bath. In all, five samples of water 1902-3.] Mr Manley on Electrical Conductivity of Sea- Water. 349 were available for the determination of electric conductivities and relative densities. The Wheatstone’s Bridge. For the determination of the electric conductivities of the five samples of sea-water, use was made of a 2-metre bridge having a calibrated platinum-iridium wire of 1*5 mm. diameter. As the structure of the bridge has been already described in a paper communicated to the Royal Society,* it need only be stated here that the bridge wire is very highly insulated, that special arrangements are made for avoiding thermo-electric currents at the juncture of the slider with the bridge wire, and that the index line is so placed that the introduction of errors due to parallax in reading the position of the slider is practically impossible. By calibrating the instrument it was found that the joint resistance of the two ends of the bridge was equal to that of 10 mm. of the wire ; hence the practical length of the wire is 2010 mm. The Resistance Coils. The standard resistance coils were made by Messrs Elliot Bros., and their values ranged from 0T to 500 true ohms. Mr Griffiths •of Cambridge very kindly standardised them before they were used in a former investigation,! and a comparison of the coils with each other before the work herein described was commenced, shewed that the relative values of the individual coils had not undergone any sensible change. The temperature of the coils was .kept practically constant during any given sets of measurements by placing the box containing them within another and consider- ably larger one, the space between the two being packed with ■cotton-wool. The Leads and Switchboard. The resistance coils were joined with the Wheatstone’s Bridge by a special type of switchboard, in which the eight mercury cups were highly insulated by mounting them upon pillars of ebonite. The connecting leads were made from No. 18 copper wire ; each * Phil. Trans., vol. 191, p. 376. t Phil. Trans., vol. 191, p. 374. 350 Proceedings of Royal Society of Edinburgh. [sess. consisted of seven strands. The four leads used for joining the coils- with the bridge were placed in the positions which they actually occupied during the conductivity measurements, and their resist- ances were determined both individually and collectively by a fall of potential method. The several results were very concordant, and gave a mean value of *0296 true ohm at the ordinary temperature of the laboratory. This result was checked by calculating the value of the resistance from the known dimensions of the leads and the generally accepted value for the specific resistance of copper in the form of wire ; the number deduced differed only very slightly from that obtained experimentally. The temperature of the laboratory varied but little during the time required for the whole of the conductivity measurements hence the change in the resistance of the leads due to the variations in the temperature of the room would be insignificant and inappreciable. The Electric Current and Telephone . The necessary alternating current was obtained by using a type of small induction coil enclosed in a sound proof box, and made by Fritz Kohler of Leipzig, and a sodium bichromate battery consisting of three 1 -quart cells arranged in series, and each having,, when newly made up, an E.M.F. of about 2-3 volts. For detecting the balancing point upon the bridge wire, a 2-pole telephone of French make was found to be exceedingly sensitive. The Electric Conductivity Cell. The cells which are so largely used for the determination of relative and molecular conductivities are ill adapted for accurate measurements. One of the chief difficulties with which one has to contend when using them, arises from the evaporation of a portion of the solvent into the relatively large air-space immediately above ; not infrequently a condensation of the vapour follows, whereby drops are formed upon the upper portions of the cell from time to time these will trickle down into the main body of the liquid, and suddenly disturb a balance that may have been, obtained ; in consequence very uncertain data are procured for the calculation of the desired conductivity. The difficulty may be 1902-3.] Mr Manley on Electrical Conductivity of Sea- W ater. 351 obviated by almost filling the cell with the solution ; but by adopting such a course we should do away with one of the chief merits claimed for the method, namely, the shortness of time required by the contents of the cell for assuming the temperature of the surrounding hath. The form of the cell employed in this investigation is indicated in fig. 1. The cell is constructed from a length of almost uniform glass tubing. To the lower end is fused a narrow tube carrying a tap, bent upwards and ending in a capillary ; the presence of this tube makes it possible to thoroughly rinse out the apparatus with portions of the sea- water before charging for the conductivity measurement. The lower electrode e is fixed, but the upper one e is movable, being fused to one end of a glass tube t, which just slides smoothly within a second tube T rising from the upper end of a ground and well-fitting glass stopper S. At m a fine index line is etched upon the outer tube ; two similar lines, at a distance of 10-24 cms. from each other, are also etched upon the inner and sliding tube which carries the movable electrode e. This electrode can be held at any desired height by the clamp c which is shown upon an enlarged scale (fig- 1)- The cell was calibrated with mercury according to the method devised by Bunsen ; it proved to have a remarkably uniform cross- sectional area, the mean value being 4*965 sq. cms. The platinum electrodes were platinised before the cell was used, in the manner recommended by Ostwald. Fig. 1. The Thermometers. The thermometers which were used during this investigation were graduated to Ty of a degree centigrade. The errors of the 352 Proceedings of Royal Society of Edinburgh. [sess. several instruments were suitably determined with the aid of a standard instrument from Kew, and the proper corrections applied. Tank for the Conductivity Cell. In order to maintain the temperature of the cell and its contents constant for any required period, they were surrounded by a large zinc tank having a capacity of many gallons, filled with water, and furnished with a large ring stirrer. Placed centrally within this was a small and cylindrical water-tight chamber, supported upon a tripod. The mouth of the chamber was covered with a tightly Fig. 2. fitting cork bung, which had been heated for some time in molten paraffin wax, so as to render its insulating properties as, effective as possible ; suitable holes had previously been made in the bung for the reception of the conductivity cell, the wire leading to the, lower and fixed electrode and a thermometer. Pig. 2 shews the cell charged and placed in position, ready for a conductivity determination. It was found that the contents of the cell usually attained the temperature of the tank water in about an hour, but in no case were any final measurements of an electric conductivity made until after three, and in some cases four, hours had expired. 1902-3.] Mr Manley on Electrical Conductivity of Sea- Water. 353 Preliminary Treatment of Waters. On a former occasion when the relative densities of certain samples of sea-water were being determined at the temperature of 24° C., considerable trouble was experienced owing to the for- mation of minute gas-bubbles upon the interior of the walls of the pyknometer ; their complete removal was absolutely necessary, but this, as a rule, could only be accomplished after much loss of time. Other observers have doubtless met with the same difficulty ; but I am not aware of any account of the following simple device by which it may be entirely avoided. A stoppered 500 c.c. flask is thoroughly washed out with portions of the water to be examined and 200 c.c. of the water poured in • a thermometer is introduced, and the flask is held for short and successive intervals at some little distance above a small Bunsen flame, the contents being continuously shaken. As soon as the temperature of the water has been raised to 26° C. the thermometer is withdrawn and the stopper inserted ; the water is then vigorously shaken for a time, allowed to stand until free from air-bubbles, and is finally poured quietly into some clean and dry vessel, from which it can be conveniently drawn direct into the pyknometer. By this preliminary treatment we not only ensure with complete certainty the entire absence of the vexatious air-bubbles from the walls of the pyknometer, but also secure a practically uniform and very convenient standard aeration for all samples of water. It is almost unnecessary to point out that the presence of air-bubbles within the conductivity cell would be equally objectionable, and ■consequently this process has been employed in all the determina- tions with great success. Method of determining the Conductivities. The condirctivities of the five samples of sea-water were cal- culated from the data obtained by the following method. The cell was first washed out some three or four times with portions of the water to be examined ; the tap was then closed and the cell filled up to a certain line near the top. Next, the movable electrode, together with the glass tube supporting it, was thoroughly rinsed with other portions of the water and immediately placed in its 354 Proceedings of Poyal Society of Edinburgh. [sess. proper position within the cell, the supporting glass stopper being screwed in gently but securely. The cell was then introduced into the inner chamber of the tank as shown in fig. 2, and the cork bung which carried it pressed firmly home. A thermometer was passed through a hole in the cork cover, and adjusted so that its bulb was as central as circumstances would permit. The cell leads were then connected through the switchboard with the right-hand gap of the bridge, the standard resistance coils being joined in a similar manner with the other gap. From time to time the tank water, which usually had a temperature of about 10° C. at the beginning of a series of resistance measurements, was thoroughly stirred ; and likewise the sea-water in the cell, by gently raising and lowering the upper electrode. After the expiration of the three or four hours already mentioned, the sliding electrode was moved up or down until the upper of the two lines etched upon the glass tube coincided with the index line m (see fig. 1) ; it was then clamped (Position A). The slider was then placed in contact with the central point of the bridge wire and the induction coil brought into action ; then the standard resistance having been adjusted until the sound in the telephone had been reduced to a minimum, the slider was moved to and fro until it was thought that a true balance had been obtained. The position of the slider was now read, frac- tions of a millimetre being estimated in tenths; after the first reading had been recorded the slider was displaced, sometimes to the right, sometimes to the left, and a second and perfectly inde- pendent determination of the balance point effected. In an exactly similar manner two other determinations were made, and the mean of the four, which generally differed but little among themselves, accepted as the true value for the bridge reading. The ther- mometers placed in the chamber, tank water and resistance box were now read ; and immediately after, the positions occupied by the cell and resistance coils were interchanged by means of the switchboard; a second set of four independent balancings was then obtained, the mean value being, as before, accepted as the true one. Finally, the total value of the standard resistances was noted. After the completion of the observations just described, the sliding electrode was at once unclamped and moved upwards until the second and lower fixed line etched upon its stem coincided 1902-3.] Mr Manley on Electrical Conductivity of Sea- Water. 355 with the index line m ; it was then secured in this position with the clamp (Position B). Data for the new resistance were then collected by proceeding in the manner given above. From the data obtained with the movable electrode in the Positions A and B we can calculate (1) the resistance B of the cell leads plus that of a short column of sea-water of unknown though constant magnitude ; and (2) the total resistance B', due to the two factors enumerated under (1), together with that of a column of sea-water of known length and cross-sectional area. Hence the resistance X2 of the added column of sea- water will be equal to the difference between B and B', or Xx = B' - B. The resistance B was calculated from the formula 2010 -{r-r} B 2010 + {r-/} “ S’ in which 2010 is the practical length of the bridge wire in milli- metres ; r and r the two mean readings obtained, the former with the standard coils S in the left-hand gap of the bridge, and the latter after reversing the positions of S and the conductivity cell. With the aid of the proper temperature coefficient (vide infra)r X1 is readily reduced for the standard temperature of 24° C. Xow since the dimensions of the added column of the sea- water are known ( vide supra), the specific resistance p24 of the water at the standard temperature may be calculated ; the reciprocal of 1 this, — , will be the specific conductivity sought. P24 Two other sets of observations made at higher temperatures were similarly treated and reduced, and the means of the three independent results taken as the true values for p24 and — . P24 All the samples of water were examined in precisely the same way. Determination of Temperature Coefficients. The temperature coefficient for each of the five samples of water was deduced from the three resistances Xx, X2, and X3, which the added column of sea-water was found to possess at the respective temperatures of tv t2, and t3 ; the values selected for those tempera- tures being approximately 10° for tv 15° for t2, and 20° for t.d. 356 Proceedings of Royal Society of Edinburgh. [sess. The method adopted for the determination of the resistance X1? at the temperature /15 has already been described in the preceding section ; the determination of the two other resistances was pro- ceeded with immediately after the completion of the first, the same method and precautions being observed throughout. The temperature of the tank water was in each case suitably increased by the introduction of steam from a boiler. The constants a and in the formula X, = X0{ 1 + at + were then calculated in the usual manner. Determination of the Relative Densities. These were determined in the manner already described in a former communication,* but with the following additional pre- cautions : — 1. After the pyknometer had been cleaned with hot and fuming nitric acid, it was repeatedly washed out with newly redistilled water only ; because, according to my own experience, it is far •easier to completely remove water vapour than either alcohol or ether vapour. 2. The air used for sweeping out the water vapour from the heated pyknometer was well dried and filtered by passing it through (1) a large tube filled with calcium chloride, and (2) a tube plugged with the finest cotton-wool, through which a strong current of air had previously been forced in order to remove any loose filaments, which might otherwise easily find their way into the pyknometer and so cause 'trouble and inaccuracy in weighing. 3. In calculating the relative densities given under d 24 in Tables II. and III., the usual corrections for reducing the brass weights to vacuo were applied, due allowance being made at the same time for the weight of the volume (49 c.c.) of dry air which filled the generally so-called c empty ’ pyknometer when the instrument was first weighed. The values enumerated under B in Table I. were obtained by reducing the ordinary weighings in the usual manner. 4. To eliminate the corrections due for the buoyancy of the air * Proc. Hoy. Soc . EMn. , 1900, p. 36. 1902— 3. J Mr Manley on Electrical Conductivity of Sea- Water. 357 Regnault’s method was followed, viz., that of suspending from a hook attached to the pan in which the weights are placed a glass counterpoise, having as nearly as possible the same weight, area and volume as the pyknometer. It was prepared from a suitable test-tube, the open end of which was drawn out in the form of a capillary tube, and bent hookwise. As the counterpoise when first prepared was found to be considerably lighter than the- pyknometer, water was introduced into the former until its weight was almost but not quite equal to that of the latter ; the capillary was then sealed before the blow-pipe. 5. All the weighings of the pyknometer and contents were performed according to the method suggested by Gauss; that is, the object was weighed first in one pan and then in the other. Since any two individual weighings gave but slightly different values, the mean was in every case accepted as the true one. 6. The volume of the pyknometer which was used is approxi- mately 49 c.c. As the weight of this volume of distilled water does not greatly differ from that of the same volume of sea-water, it is wholly unnecessary to apply a correction for the air displaced by the brass weights. Such a refinement would only alter the values obtained for the relative densities of the samples of water in the sixth decimal place. Experiments ivith the Counterpoise. With the object of testing the theoretical advantage gained by the use of a counterpoise, the pyknometer was charged and then carefully wiped with a fine white linen cloth; the counterpoise was also wiped at the same time ; they were then placed in clean glass beakers within the balance-case. After the expiration of five minutes the weight of the pyknometer and its contents wa& determined by the reversal method of Gauss, ( a ) with, and ( b ) without, the use of the counterpoise. In suspending the pykno- meter or counterpoise from the arms of the balance, use was made of a small ivory lifter, in order to avoid any direct handling and consequent errors. Experiments showed that it was immaterial whether the weighings without the counterpoise were made first or last ; hence it was concluded that a normal amount of moisture condenses upon the pyknometer during the five minutes allowed 358 Proceedings of Royal Society of Edinburgh. [sess. for rest after the instrument has been wiped. The weighings were repeated at different times during each of the days March 19th to 25th inclusive, the pyknometer and counterpoise having been previously wiped and allowed to stand for five minutes, as already described; the barometer was read on each occasion. The proper corrections due for the buoyancy of the air were then applied to those weighings, which were performed without the aid of a counterpoise. The results obtained are given in Table I. Table I. Date. Time. Barometer. Reduced Wei A. With Counter- poise. ght of Water. B. Without Counterpoise. 1902. , 10 a.m. 29 ‘77 inches 50-0569 grms. 50-0561 grms. \ 11 a.m. Falling 71 „ 66 Mar. 19th < 1 p.m. slowly 71 „ 69 3 3 ( 7 p.m. 29 '59 inches 71 ,, 71 3? r 10 a.m. 29-20 „ 69 ,, 75 Mar. 20th j 1 p.m. 29-18 ,, 69 „ 75 3 3 / 10 a.m. 29-10 ,, 68 ,, 71 Mar. 21st 1 3 p.m. 29-08 ,, 68 „ 73 3 3 [ 10 a.m. 29-10 ,, 65 „ 67 Mar. 22ncl 4 Noon 29-10 ,, 65 „ 68 3 3 l 8 p.m. 29-18 „ 65 ,, 69 3 3 ( 10 a.m. 29-38 ,, 64 „ 55 3 3 Mar. 24th \ 1 p.m. 29-30 ,, 62 „ 61 3 3 \ 8 p.m. 28-90 ,, 62 „ 66 3 3 Mar. 25th 10 a.m. 29-40 ,, 63 „ 53 3 3 On glancing down the column given under A, we notice (1) that, with the single exception of the observation made at 10 a.m. on March 24th, identical results are obtained at any time during a given day, even with a changing barometer ; and (2) that there is a very gradual and almost uniform falling-off in the weight of the pyknometer and its contents ; the loss, which is probably due to a continuous evaporation of water, is so slight that it only becomes a measurable quantity after the lapse of about twelve hours. On comparing the results given in column B, we note that there is 1902-3.] Mr Manley on Electrical Conductivity of Sea- Water. 359 on the whole an absence of that uniformity which obtains in column A, and this even after the application of the troublesome corrections due for the changes both in the temperature and pressure of the air. The irregularity is probably caused by variations in the amount of moisture condensed upon the exterior surface of the pyknometer. When a proper counterpoise is used, such variations are of no consequence, for both the pyknometer and counterpoise are equally affected. Attention may also be drawn to the fact that, whilst the extreme difference in the values given under A amounts to *0009 grm., that for those under B is equal to *0022 grm. In the former we have a definite suggestion of loss through evaporation, but in the latter such probable loss is not even hinted at, except in the most uncertain manner. Judging then from these results, it may, I think, be safely asserted that when it is desired to obtain a value for the relative density of a water which shall be as nearly as possible correct to the fifth decimal place, the use of a counterpoise for automatically eliminating certain incidental errors is absolutely essential. It is important to note that the smaller the differences in the ‘total salinities’ of any waters under examination, the more necessary does it become to introduce the use of a suitable counterpoise. Instead of a specially constructed counterpoise, use may be made of a second and exactly similar pyknometer, which for convenience should be from 0T to 0*2 grm. lighter than the pyknometer proper. After the weight has been suitably adjusted, the two open ends should of course be sealed up before the blow- pipe flame. Such a counterpoise possesses the further advantage of having in all probability been made from the same kind of glass as that which constitutes the pyknometer itself ; hence the conditions for the condensation of moisture upon both will be equally favourable. Several such counterpoises were actually tried, and the results obtained with them fully justified this con- clusion. Experimental Results. The results obtained, according to the methods which have been described at some length, are here grouped together in Table II. Under p2 4 and — are given the respective values in true ohms r24 360 Proceedings of Royal Society of Edinburgh. [sess. and mhos for the specific resistances and conductivities of the waters at the temperature of 24° C. These were calculated with the aid of the constants a and /3 from the resistances found for the introduced column of the several samples of water at the lower and experimental temperatures. The relative densities, d2i, were determined at the temperature of 24° C. Table II. Water. cl2i. I P 24* 1 P-24 a x 105. 13 x 107. No. 1 1 -02528 20-8125 ohms *048048 mho -2814 + 4450 No. 2 598 19-9563 ,, •050109 ,, -2842 + 4342 Natural 659 19-1148 ,, •052315 ,, -2706 + 3619 No. 3 707 187744 „ •053264 „ -2686 + 3477 No. 4 937 17-2480 ,, •057978 ,, - 2695 + 3415 Discussion. A close examination of the results given in Table II. failed to reveal the existence of any of those simple relationships between the resistances and relative densities which I had hoped to find, and such as Knudsen observed for the samples of water investi- gated by him ; and all attempts to found a general equation based upon the data contained in the above-named table, whereby one might be enabled to calculate either the correct density or resist- ance of a similar water, proved utterly fruitless ; even the best formulae obtainable gave but poor approximations to the actual values as determined experimentally. In the matter of the uniformity of the temperature coefficients, it would appear that Knudsen was more fortunate than myself ; and that notwithstanding the fact that the variations introduced for the total salinities were very great in comparison with those dealt with here. It is, however, somewhat surprising that so large a divergence should exist between the maximum value found for the temperature coefficient by him, and the mean of those given under a in Table II., the former being only -0113, whilst the latter is *0275. Attention may also be drawn to the fact that the Danish observer assumes a straight line equation for the temperature 1902-3.] Mr Manley on Electrical Conductivity of Sea- Water. 361 coefficient of a water, and accordingly obtains his data from the resistance measurements made at two temperatures only. Now, the physicist learns from his daily experience in such matters that straight line equations are not in the habit of appearing upon the scene of his labours with that unfailing regularity for which he could sometimes wish ; and even on the rare occasions when such an one does present itself, it is generally viewed with much suspicion, and is subjected to such rigorous treatment by the mathematician that it is speedily robbed of its more primitive form. But from this digression we hasten to a further brief consideration of the facts contained in Table II. On reviewing the data contained in the columns a and /3, one is quickly convinced that the assumption made by Knudsen would be wholly inadmissible here, for in every case it is obvious that the equation obtained is that for some distinctly curved line. Hence it follows that variations in the temperature and specific conductivity of a water are not always directly pro- portional to each other. Some further Experiments and Results. The conclusions drawn from the data obtained for the water collected at Sandown were confirmed by the results obtained from an examination of samples of water from two other localities. For these I am indebted to the kindness of the Director of the Marine Biological Laboratory at Plymouth and to Mr H. E. W. Phillips. The sample procured by the former was collected midway between Stoke Point and Bolt Tail, and the second sample came from about one mile off the shore of Cardigan Bay. The Plymouth water was first examined in its natural state, and afterwards five samples of very slightly different strengths were prepared from it by adding successive portions of 1 gram of re-distilled water to 1000 grams of the sea- water. Two additional and considerably weaker samples were also prepared by mixing 50 and 100 grams of distilled water respectively with two 3 000-gram samples of sea- water. In Table III. the prepared waters are numbered from 1 to 7 in the order of decreasing total salinity. The symbols d24, p24, etc., which appear at the heads of the columns, have the several meanings ascribed to them on page 360. PROC. ROY. SOC. EDIN.— VOL. XXIV. 24 362 Proceedings of Royal Society of Edinburgh. [sess. It will be observed that the temperature coefficients of the Ply- mouth waters distinctly differ from those of the waters fromSandown. Determinations of the amount of chlorine per litre of the water were made for certain of the above-named samples; these were effected (1) in the usual manner with standard silver nitrate solution and potassium chromate as indicator, and (2) according to the method devised by Gay-Lussac. The individual results Table III. Plymouth Water. C?24* P24- 1 P'24 ax 105. 0 x 107. Chlorine per litre. Natural No. 1 „ 2 „ 3 „ 4 „ 5 ,, 6 „ 7 1-02676 71 67 64 61 57 1-02543 1-02424 18*4461 ohms •4645 „ *4979 „ •5052 „ •5155 ,, •5523 „ 19*2608 ,, 20‘0775 „ *054212 mho •054158 „ •054060 „ •054039 ,, •054009 ,, •053902 ,, •051919 „ •049807 „ -2470 - 2518 -2551 -2428 + 2810 + 2970 + 3082 + 2694 20*4297 grms. 20-3714 „ 19-4465 ,, 18-5162 „ Cardigan ) Bay V Water. ) 1-02491 19-6037 „ 0-51011 „ / The specific \ resistance of this water was I 4 measured at r 24° C. a and /3 1 were not deter- 1 ' mined. 18-9336 „ were almost identical, and their mean values are given in the last column of the immediately preceding table. An attempt to discover some definite and simple relationship between the electric conductivity of a water and the amount of chlorine contained in it met with no greater success than that which attended the effort to correlate the conductivities and relative densities. It is a matter for regret that there should be no simple pro- portionality between the electric conductivities on the one hand, and the relative densities or total amount of chlorine present on the other; and the more so on account of the fact that the electrical method has proved to be so much more sensitive than any of those other methods which are at present almost exclusively used in sea-water determinations. (. Issued separately January 24, 1903.) 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 363 Freezing-Point Depression in Electrolytic Solutions. By James Walker, F.R.S., and A. J. Robertson, B.Sc., University College, Dundee. . (Rsad December 15, 1902.) The following research was undertaken for the purpose of de- termining the freezing points of solutions under conditions which would involve a different systematic error from that encountered when the ordinary method of procedure is followed ; and, secondly, for the purpose of obtaining ionisation values for electrolytic solu- tions by the cryoscopic method which should be made by com- pensation as far as possible independent of any systematic error in determining the freezing points. The freezing point of a solution is the temperature at which the solution is in equilibrium with ice, and can only he strictly ascertained when the system of ice and solution is entirely isolated, so that the heat is neither gained nor lost by communication with the exterior. Such a condition is impossible to realise in practice, but the effects of heat transference may be so minimised as not to exceed the experimental errors due to other causes. The first theoretical discussion of the subject was given in a paper by Uernst and Abegg ( Zeitschrift f. pliysik. Chem ., xv. 681), and further contributions have since appeared in the following papers in the same journal : — Abegg, xx. 207 ; Raoult, xxvii. 617 ; Wildermann, xix. 63; xxx. 577; Loomis, xxxii. 578. Briefly, the account of the matter given by Uernst and Abegg is as follows. If the system is not isolated from the exterior, and if only liquid is present, the thermometer will settle at a temperature determined by the environment, which is called the convergence temperature. Ice being now added, the temperature would, if there were no heat communication with the exterior, reach the equilibrium tempera- ture of the ice and solution — that is, the thermometer would register the true freezing point. But the actual temperature registered is neither of these temperatures, hut a compromise be- tween them. On the one hand there is heat exchange with the 364 Proceedings of Royal Society of Edinburgh. [sess. exterior tending to the attainment of the convergence temperature, and on the other hand a compensation for this by the formation or fusion of ice, heat being thereby produced or absorbed, and the system tending in consequence to its true freezing point. The thermometer after a time becomes stationary at an apparent freezing point, which lies between the true freezing point and the convergence temperature, and may differ from the former by a very considerable amount. The divergence between the true and apparent freezing points depends on two factors. First, it depends on the difference between the true freezing point and the con- vergence temperature, and, second, on the rate at which ice is formed and dissolved. If the convergence and true equilibrium temperatures are identical, the thermometer will evidently give the true freezing point. The true freezing point will also be registered if the rate of formation or fusion of ice is infinitely great, for then the compensation for heat exchange will be instantaneous and complete. In most of the accurate work hitherto done, attention has been paid to the first mode of securing the exact freezing point — that is, by suitably tempering the bath in which the apparatus is immersed, the convergence temperature has been nearly adjusted to the freez- ing point which it is desired to determine. The external bath is in such a case at a temperature a little below the true freezing point, so that the heat gained from the room through the thermometer, stirrer, etc., is lost to the external bath, and so the influence of the exterior is on the whole reduced to zero. It must be remembered, however, that this mode of external compensation is only theoreti- cally perfect if the ivhole environment is at the same temperature, i.e ., the freezing point. In actual practice, the external bath is below the convergence temperature, whilst the upper parts of the thermometer, along which conduction takes place, are above that temperature. When the freezing point experiment takes place, therefore, although ice on the whole may neither be formed nor dissolved, local formation of ice near the external bath, and fusion of ice near the thermometer, stirring apparatus, etc., must occur, so that constant stirring must be resorted to in order to eliminate to some extent this error. The stirring in itself, of course, generates heat, and has an appreciable effect on the apparent freezing point, 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 365 but this is allowed for in fixing the temperature of the external bath, which is adapted to a certain uniform rate of stirring. If we consider the other factor which determines the difference between the true and apparent freezing points — namely, the rate of formation or fusion of ice — it is evident that this rate depends,, first, on the quantity or proportion of ice present ; second, on the fineness of its state of division— together, on the surface of contact between the ice and solution. For a given quantity of solution,, the more ice we take and the more finely we divide it, the more rapidly will the equilibrium temperature be restored after any disturbance, and the more closely will the apparent and true freezing points coincide. In the present research it has been specially our effort to reduce the error arising from the finite time required for this restoration to a minimum by employing an amount of ice as great as circumstances would permit. In practically all previous researches ice has been introduced into a solution of known concentration by slightly overcooling the solution, and then inducing crystallisation by inoculation with a minute fragment of ice on the end of a wire or capillary, A certain quantity of ice then separates in correspondence with the degree of overcooling, but this quantity is of necessity small if the equilibrium concentration of the solution is to be accurately known, for as ice separates from the solution the concentration increases, and it is the equilibrium concen- tration alone with which we are concerned. In Raoult’s ex- periments, the ice which separated was only about one-half per cent, of the weight of the solution. In many other cases the amount of ice did not exceed 02 per cent. In our own experiments the minimum proportion of ice was 12 per cent., the average being about 15 per cent. Of course, when such large proportions of ice are present, it is impossible to ascertain the equilibrium concentration of the solution from the initial concentration. We therefore adopted the expedient previously employed by Roloff (Zeitsclirift f. physikal. Chem., xviii. 572) for concentrated solutions, and filtered off the solution from the ice after equilibrium had been attained, the concentration being then determined by analysis. The ice which we used was prepared by freezing distilled water. 366 Proceedings of Royal Society of Edinburgh. and thereafter reduced to a fine state of division by planing with an ordinary hand plane, which was, of course, kept scrupulously clean. As the setting of the plane was never altered, and the amount of dry ice in the wet mass obtained was found to he subject to little variation under our experimental conditions, we were confident that the surface of contact between the ice and the solution varied little in the various experiments. This point is of some importance, as it is well known that the ice which separates from many solutions on overcooling is by no means in the same state of division as the ice which separates from pure water under the same conditions, so that the error due to the lag in attainment of equilibrium may vary considerably owing to this circumstance. The external bath by which our apparatus was surrounded was invariably an ice bath. It was impossible, therefore, for ice to form within our apparatus, whether it contained ice or solution. By this means a possible source of error was avoided, the formation, namely, of a skin of ice round the bulb of the thermometer, which, according to the experiments of Wildermann, greatly affects the apparent temperature of equilibrium (ibid., xv. 358). Apparatus. The general arrangement of our apparatus is shown on p. 367. The vessel used to contain the experimental liquid was a Dewar vacuum tube, which was closed by a rubber stopper perforated by three holes. Into these holes were fitted three tubes, which served as collars. The central tube C was of copper and was wide enough to admit the thermometer T. The other two tubes were of glass, and through them passed the filter F and stirrer S respectively. The stirrer was made of light glass tubing bent horizontally to a circular form at its lower extremity. On the inner surface of the circular portion was a number of small holes, through which, by means of a hand blower attached to the upper portion of the stem, fine jets of air could be blown, thereby supplementing the mixing brought about by the vertical motion of the stirrer as a whole. The filter tube was made of narrow glass tubing widened to a conical shape at its lower end. The filter, which consisted of a small disc of filter paper supported between two discs of muslin, 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 367 was tied by means of cotton thread over the widened end of the filter tube. The thermometer was attached above C to the stage of a micro- scope used for accurately reading the position of the mercury on the scale. The microscope with the thermometer was movable vertically on a stand, fine adjustments for reading being made with the screws of the microscope stage. The hammer of an electric bell, covered with chamois leather, was fitted for tapping the thermometer (compare Barnes, Trans. Roy. Soc., Canada, vol. vi. p. 37). In order to obtain ionisation values for a strong electrolyte, our intention was to make on the same day two experiments, one with a non-electrolyte, the other with a strong electro- lyte, the freezing-point depression being made as nearly as possible the same in the two cases. From the concentration of the two solutions it would then be possible to calculate the ionisa- tion value of the strong electrolyte, due allow- ance being made for the small difference in the freezing-point depression according to Blag- den’s law. Since, however, the accurate analyt- ical estimation of a suitable non-electrolyte wTas found to be a matter of considerable difficulty, we chose instead acetic acid, which is very feebly ionised, and whose small ionisation values are accurately known. Our day’s work finally re- solved itself into determining the freezing-point depression and equilibrium concentration of an approximately decinormal solution of acetic acid, and immediately thereafter the same magnitudes for a solution of a good electrolyte of approximately the same freezing point. We first charged the Dewar vessel with 80 c.c. of distilled water cooled to 0°, and thereafter added 45 grams of freshly-planed ice made from distilled water. This ice was, of course, a wet mass, but the quantity of dry ice it contained could be estimated from the final concentration of the solution in the manner to be after- wards described. The stopper, with stirrer and filter attached, 368 Proceedings of Royal Society of Edinburgh. [sess. was then placed in position, and the vessel immersed in crushed ice contained in a large glass jar protected by thick felt. The ice was heaped up above the level of the stopper so as to surround the tubes which acted as collars. The thermometer was now removed from the ice and water in which the bulb was constantly kept when it was not in use, and introduced through the tube C, being at the same time attached to the stage of the observing microscope. A thick felt cover was now placed on the top of the ice, and the stirrer was moved up and down a few times for the purpose of thoroughly mixing the ice and water, the process being assisted by the hand- blower, which also served to saturate the water with air. The position of the thermometer relatively to the vessel was always the same as measured on a millimeter scale. The reading on the thermometer was generally constant as soon as the stirring was over. Observations were made from time to time during at least fifteen minutes with intermittent stirring, if the reading changed appreciably the thermometer was so far raised out of the solution as to alter slightly the position of the mercury on the scale, and then replaced in the liquid. In every case, with one exception, the reading was then constant for at least fifteen minutes. The rubber tube of the band-blower was now removed from the top of the stirrer, down which was delivered, by means of a gradu- ated pipette, a sufficient quantity of cooled normal acetic acid solution to give approximately the required freezing-point depres- sion. The stirring tube was then washed down with a little cooled distilled water. The blower was now reattached and the solution thoroughly mixed by means of a rapid stream of air bubbles and the vertical motion of the stirrer. The thermometer at once adjusted itself to a new position of equilibrium. Stirring and reading were then continued intermittently as before, and after the lapse of at least half an hour the liquid was filtered off. During the stirring the filter was kept at the top of the liquid, but before filtration it was lowered so as to be on a level with the bulb of the thermometer. The upper end of the filtering tube was connected to a tube passing through the stopper of a filter flask, which was in its turn connected with a 100 c.c. pipette filled with water and used as an aspirator. Two separate portions, of about 30 c.c. each, were then filtered off and analysed separately. 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 369 As a check on the readings, a thermometric reading was made after the filtration had been completed. Notwithstanding the changed conditions owing to the filtration, this reading did not differ from the equilibrium reading when the vessel was fully charged by 0‘0005°. The apparatus was now removed from the ice bath, thoroughly cleaned, and used for a second precisely similar experiment, hydrochloric acid or some similar substance being used instead of acetic acid. The following numbers will serve as an example of our experi- mental data: — I. — Barometer 762*7 mm. Boom temperature 15 *7°. 80 c.c. water and 45 grams wet ice. Time, h. m. Thermometer. Micrometer. 11.0 5-06° 8*0 11.15 9-0 11.35 8*9 Added 11 ’5 c.c. normal acetic acid. 11.45 5-24° 25-0 11.55 22*0 12.0 22-0 12.5 22-0 12.19 22*0 Filtered off two portions. II.— ■Barometer 763'0 mm. Room temperature 15 '3°. 80 c.c. water and 45 grams wet ice. Time. Thermometer. Micrometer. 12.58 5-06° 8-0 1.2 8-7 1.47 8-9 Added 57 c.c. normal hydro- chloric acid. 1.55 5*25° 2*0 1.59 2-0 2.31 2*0 Filtered off two portions. The readings on the thermometer aie the divisions on its scale to which the zero of the micrometer scale was adjusted, so as to measure back to the meniscus. Since the two scales read in opposite directions, the freezing-point depression may be obtained by adding each thermometric reading to its appropriate micro- meter reading, and then subtracting the one temperature from the other. Thus in the first example, the lowering of the freezing point is (5‘24° - S’OG0) + (22*0 — 8’9) divisions. From the cali- bration table it is then found that this corresponds to a depression in terms of the thermometer scale of 0T844°. Each division on 370 Proceedings of Royal Society of Edinburgh. [sess. the scale of the micrometer in the eyepiece of the microscope corresponds on the average to 0 '000384°. Sources of Error. To estimate the degree of ionisation with the accuracy we desired, it was necessary for us, with depressions of about O' 2°, to secure an error not exceeding 0*0005°. With regard to the accuracy attain- able in cryoscopic measurements, we cannot do better than quote the words of the originator of the method, as given in his last paper on the subject (Raoult, loc. cit.). “The final result of this investiga- tion is, that with the apparatus above described, and with due observance of all precautions, it is fairly easy to determine the freez- ing-point depressions of dilute solutions with an accuracy of O'OOl °. “ It is a matter of great difficulty, however, to carry accuracy any further, chiefly on account of chance variations in the zero point ; but if one is so fortunate as to possess a thermometer in which these variations are small and regular, one may indulge the hope of reaching an accuracy of 0'0002°. In my experiments on the molecular depression of dissolved oxygen I actually attained this degree of accuracy.” In the following paragraphs we discuss the various sources of error to which our experiments were liable, and the degree of uncertainty to be expected in each case. The Thermometer. — We experienced some trouble in obtain- ing a thermometer suited to our experiments. The Beckmann thermometers belonging to the Chemistry Department of this College were found to be unreliable when readings were pushed beyond the thousandth part of a degree, owing to the tendency of the mercury to stick at certain places of the scale, no amount of tapping or other mechanical agitation of the thermometer being sufficient to ensure consistent readings. A thermometer, however, belonging to the Physiological Department proved on investigation to be entirely satisfactory, and we have here to express our thanks to Professor Way mouth Reid for kindly placing this instrument at our disposal during the experiments. The thermometer, made by Goetze of Leipzig, was a fixed zero instrument, but as there was a small bulb at the top of the stem, it was possible with care 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 371 to detach a small amount of mercury, so that the zero point was only a few tenths of a degree from the bottom of the scale, which was just visible above the copper tube C. Very little of the mercury column was thus exposed to the room temperature. The scale divisions of the thermometer corresponded to 0'01°, and each of them was divided by the micrometer scale of the microscope into 26 parts on the average. Since it was easily possible to estimate the quarter of a division by the eye, the merely optical accuracy of the thermometric reading corresponded to O’OOOl3. In the interval of 0*3° in which all our observations lay, we carefully determined the value of each thermometer division in terms of micrometer divisions. The values varied from 25 to 27 micrometer divisions for each scale division. The difference between two observations of the value of the same thermometer division did not in any case exceed 0*3 micrometer divisions. The relation of the above part of the tube to the whole scale was then determined by the ordinary calibration with mercury threads. The tube was found to be of remarkably uniform bore, with the exception of the extreme upper part, which we never used in our experiments. Finally, the value of a degree of the thermometer was determined for us by Professor Kuenen and Mr Robson by comparison with a platinum resistance thermometer. The value of the degree of our thermometer was found to be identical with that of the air thermometer within the error of experiment, which was estimated at 0*4 per cent. As has already been mentioned, very little of the stem of the thermometer which contained mercury was exposed to the room temperature. Special experiments showed that at 0°, i.e ., at the point where the greatest column of mercury was exposed, a variation of 2° in the room temperature was barely perceptible on the freezing points as read off on the thermometer, i.e., probably did not amount to 0,0002°. The temperature of the basement room in which we worked varied very slowly, never more than 1° during a complete depression experiment, so that the effect of variation of room temperature on the exposed stem may be neglected. Since at the freezing point of water, O’ 2° more of stem was exposed than at the freezing point of the solutions we mostly 372 Proceedings of Royal Society of Edinburgh. [sess. examined, the apparent lowering of the freezing point will he too great by 0*0003° if the room temperature is taken at 15°. This value was obtained from direct experiment with an exposed stem of 4°. The above error is only of importance however where absolute values of the freezing-point depressions are desired. Tor relative values even on different days it is only the variation of this error caused by change of the room temperature that concerns us. Since the extreme variation of the room temperature during the whole course of our experiments was only 3°, this source of error falls out of account. The effect of pressure on the reading of our thermometer was unusually small, being considerably less than 0*0001° for 1 mm. of mercury. Since in each case the thermometer was in the same position with regard to the vessel, the pressure under which the freezing point of the solution was measured was somewhat higher than the pressure under which the freezing point of water was observed, the vessel in the former case containing more liquid. The maximum error from this source, however, was only 0*000 15°, by which the depression would be too small. On only one day was the change in* the atmospheric pressure sufficiently great to affect the freezing- point depression perceptibly. The zero of our thermometer when in use varied very regularly. In no case did the zero change by as much as 0*001° in twenty-four hours. The maximum difference observed between the zero taken in the morning experiment and that taken in the afternoon was 0*0003°, except on one occasion, when the thermometer accidentally received a sharp blow on the bulb, when the zero was found to have suddenly changed by 0*002°. Convergence temperature. — The convergence temperature in our apparatus was found to be about +0*04°, the temperature of the external bath being 0°. The maximum freezing-point depression which we observed was 0*19°, so that in the most unfavourable case the equilibrium temperature was 0*23° below the convergence temperature. By direct experiment we found that at — 0*25° the liquid in the Dewar vessel gained 0*16 gram-calories per minute, i.e ., enough heat to melt 2 milligrams of ice per minute or 0*12 grams per hour. Since the quantity of solution employed was 120 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 373 grams, the ice melted in one hour would in the most unfavourable case only reduce the concentration by 0*1 per cent, and the freezing-point depression by a like amount. Most of the heat was apparently conducted by the thermometer, etc., the Dewar vessel forming a very efficient protection. Dissolved air. — Kaoult showed that water saturated with air had a freezing point 0*002° lower than water which was air-free. It is therefore of some importance to have the water and solution always saturated with air before the freezing point is observed. Owing to the method of air-stirring which we employed, the liquids whose freezing point we observed were always fully charged with air. Analysis. — The solutions investigated were either acids or chlorides. The acids were estimated by means of N/20 baryta solution with phenol phthaleine as indicator, and the chlorides by Volhard’s method with N/50 solutions of silver nitrate and ammonium thiocyanate. A decinormal solution of hydrochloric acid was taken as ultimate standard both for the baryta and for the silver nitrate. The baryta solution, however, was also checked with pure succinic acid, and the silver nitrate solution with pure potassium chloride. As an example of the constancy of composition of the different portions filtered off from the experimental solutions, and of the magnitude of the analytical error, we here give the analysis of the solutions for which the thermometric readings were detailed on p. 369. The concentrations of the solutions are given in terms of normal solutions. Acetic acid. Hydrochloric acid. First filtrate . . 0*09697 0*05225 Second filtrate . . 0*09709 0*05213 Mean, 0*09703 Mean, 0*05219 Here the divergence of any result from the mean does not greatly exceed 0T per cent. At the outside we may take 0*2 per cent, as the limit of error from sampling and analysis. Quantity of ice. — From the results of analysis the quantity of ice in contact with the solution at equilibrium could be roughly calculated in the following manner. In the above experiment with acetic acid, 11*5 c.c. of normal acetic acid solution were 374 Proceedings of Royal Society of Edinburgh. [sess. added to the original water (p. 369). But the concentration of the solution at equilibrium was 0*097 normal. There was there- fore present at equilibrium 1T5/ 0*097 = 119 c.c. of acetic acid solu- tion. Now the total weight in the vessel was made up as follows : — Water . . • . 80 g. Wet ice . 45 Acetic acid . 11*5 Wash water . 1 Total, . 137*5 g. If from this total we subtract 119 g. as the weight of the final solution, we obtain 18*5 g. as the quantity of ice present at equi- librium. Experiments with M atonic Acid. In order to test the validity of the method for determining ionisation values, we made several experiments with malonic acid, which obeys Ostwald’s dilution law, that is, gives an ionisation constant when the values are determined from the electrical conductivity. The following numbers were obtained in determining this constant at 0° : — Malonic acid, C2H4(COOH)2, at 0°. Poo =228. V n m k 10 25*5 0*112 0*00141 20 36*0 0*158 0*00148 33*3 43*0 0*189 0*00147 50 52*8 0*232 0*00140 Mean, 0*00144 In the above table, v represents the dilution, /x the molecular conductivity in reciprocal Siemens units, m the degree of ionisa- tion, and k the ionisation constant calculated by Ostwald’s formula o 7c— m (1 - m)v Freezing-point experiments with approximately decinormal solu- tions resulted as follows : — I. Depression. Concentration. Acetic acid .... 0*1775 0*09337 Malonic acid . . . 0*1779 0*08445 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 375 II. Depression. Concentration. Acetic acid . 0*1853 0 0974 Malonic acid 0*1825 0*08665 The first experiment may be used to exemplify the mode of calculating the ionisation value from the cryoscopic data. An acetic acid solution of the concentration 0 '09 33 7 normal gives a depression of 0*1775°. Now we know from conductivity measure- ments that such a solution is ionised to the extent of 0'012. A solution of a non-ionised substance which would give the same depression would therefore have the concentration 0-09337 x T012 = 0*0945 normal. But the depression given by the malonic acid solution is 0'1779°. According to Blagden’s law, that the de- pression is proportional to the concentration, a solution of a non- ionised substance which would give this depression would have the concentration 0*0947 normal. We may therefore say that a 0*08445 normal solution of malonic acid has the same freezing- point depression as a 0*0947 normal solution of a non-electrolyte, i.e ., that these solutions contain the same number of molecules. The degree to which the malonic acid solution is ionised is there- fore 0 ----- = 0*121. In this calculation the change of 0*08445 & the degree of ionisation of the acetic acid with concentration has been neglected, since it is so small that the final result is not thereby appreciably affected. A change on the concentration of the acetic acid solutions which is less than 20 per cent, will have no noticeable effect on the calculation. The second experiment with malonic acid yields the ionisation value 0*120. Calculating now the ionisation value m from Ostwald’s dilution formula, we obtain the numbers 0*121 and 0*120 respectively for the two experiments. Here the concordance is absolute, but this absolute agreement is accidental, as these ionisation values are affected with an uncertainty of approximately 5 units in the third decimal place. In any event the accordance of the numbers gives ample evidence of the validity of our method. Further to test the matter, we made experiments with approxi- mately 0*05 normal solutions of the same substances. Here the relative error of our method is greater owing to the smaller depressions observed. 376 Proceedings of Boy al Society of Edinburgh. [sess. I. Acetic acid . Malonic acid Depression. Concentration. 0*0950 0-04965 0-0977 0-04483 Acetic acid . Malonic acid 0-0986 0-0517 0-0935 0-0431 Calculated from, these numbers we obtain the ionisation values 0'158 and 0*156 respectively. From the conductivities we obtain the values 0-162 and 0’164. Although these numbers are some- what more divergent than in the previous instance, still they do not lie beyond the estimated error of the experiments. Experiments with Strong Electrolytes. Having thus shown that in the case of malonic acid our method gave results sufficiently in accordance with the electrical method, we now proceed to give the freezing-point experiments for strong electrolytes for which previous observers had obtained results which were not accordant with the ionisation values obtained from the conductivities. Hydrochloric Acid, HC1. Concentration. Depression. Cone, of Acetic Acid. Depression. Ionisation. 0-05141 0-1847 0-0970 0-1848 0-909 0-05219 0*1870 0-0970 0-1844 0-908 Potassium Chloride, KC1. 0-0527 0*1841 0-09757 0-1856 0-858 0*05275 0-1839 o-iooo 0-1907 0-850 Sodium Chloride, HaCl. 0-0519 0-1829 0-09668 0-1840 0-874 If now we compare these ionisations with the values deduced from the conductivity method, we obtain the following table for 0'05 normal solutions : — - HC1. KC1. NaCl. From conductivity at 18° * 0-937 0-895 0-874 ,, ,, o°t From freezing point 0-956 0-892 0-877 0-908 0*854 0-874 * Kohlrausch und Holborn, Leitvermogen der Electrolyte, pp. 159, 160, 200. t J. G. MacGregor, Trans. Roy. Soc. Canada , 1900-1, pp. 6, 7. 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 377 Except in the case of sodium chloride it will be seen that here there are great divergences between the ionisation values calculated from cryoscopic and electrical observations. The error of the freezing points can scarcely amount to a unit in the second decimal. The error in the conductivities is much smaller than this, so that we are obliged to conclude that the two methods lead to essentially different results in the case of hydrochloric acid and potassium chloride. It is true that, owing to the uncertainty in the maximum molecular conductivity of all acids, the electrical ionisation value for hydrochloric acid is affected by a greater error than the corresponding numbers for the neutral chlorides, hut the divergence from the cryoscopic value is much too great to be accounted for in this way. It might be urged that, as we have taken acetic acid as our standard of reference, the differences might arise from some peculiarity in the behaviour of this substance. When we consider, however, that we obtain correct values for the ionisation of malonic acid by the use of the same method, it must be conceded that such an objection loses much of its force. Again, we find, in accordance with other observers, that by the cryoscopic method sodium chloride appears to be less highly ionised than potassium chloride, whereas by the conductivity method the reverse is apparently the case. Here, if the standard substance is merely uniform in its behaviour, whatever be its peculiarities in other respects, we have a well-marked divergence between the cryoscopic and electrical methods. Equivalent Depression for Acetic Acid. The following table shows the depressions actually observed for approximately decinormal solutions of acetic acid under the conditions prevailing on different days, and for the purpose of comparison a column has been added which gives the values for accurately decinormal solutions calculated by means of Blagden’s law from the two columns immediately preceding it. This table, like all others in this paper, gives the depressions as they were obtained directly from the thermometer, no corrections of any kind, except calibration corrections, having been applied. PROC. ROY. SOC. EDIN. — YOL. XXIV. 25 378 Proceedings of Royal Society of Edinburgh. [sess. Acetic Acid. Barometer. Temp, of Room. Ice present. Concentra- tion. Depression. Depression for decinormal solution. 763 mm. 13° 18*5 g. 0*0970 0*1848 0*1905 758 15 18 0*09668 0*1840 0*1903 763 15 18 0*09757 0*1856 0*1902 772 15 20 0*1000 0*1907 0*1907 771 15 15 0*09337 0*1775 0*1903 769 15*5 17*5 0*0974 0*1853 0*1903 763 15*5 18 0*0970 0*1844 0*1901 Mean 0*1903 The last column shows the extremely uniform character of the results obtained under a very considerable variety of conditions. The extreme difference of the depressions is only 0‘0006°, and the greatest divergence from the mean is only 0*0004°, the divergence from the mean in six observations out of the seven being 0*0002° or less. If we take the ionisation of decinormal acetic acid to he 0*012, we obtain the number 0*188 as the depression which would he shown by acetic acid if it were undissociated in decinormal solution. The equivalent depression for acetic acid, i.e ., the value of the depression divided by the concentration, would thus for the undisso- ciated compound he 1 *88. The mean value found by Loomis for nine non-electrolytes of small molecular weight is 1 *85, the greatest value being 1*87 and the smallest 1*81. Comparison with other Observers. When we compare our depressions for strong electrolytes with those of other observers, we find that there is on the whole an excellent agreement. The comparison is best made by ascertaining the equivalent depressions for the various solutions, and then by interpolation reducing these values to the equivalent depression for exactly 0*05 normal solution. Depression. 0*1847 0*1870 Hydrochloric Acid. Concentration. 0*0514 0*0522 Equiv. Depression. 3*593 3*583 Equiv. Depr. for N/20. 3*594 3*585 Mean, 3*590 1902-3.] Freezing-Point Depression in Electrolytic Solutions. 379 Potassium Chloride. Depression. 0*1841 0*1839 Concentration. 0*0527 0*05275 Equiv. Depression. 3*494 3-486 Equiv. Depr. for N/20. 3-497 3*489 0*1829 Sodium Chloride. 0-0519 3-524 Mean, 3*493 3*526 Taking the results of Loomis and Barnes from the tables com- piled by MacGregor ( Trans . Nova Scotia Institute of Science , 1899-1900, pp. 218-220), we obtain the following comparison for equivalent depressions in 0*05 normal solution : — Loomis Barnes Walker and Robertson . Hydrochloric Potassium Sodium Acid. Chloride. Chloride. 3-59 3-50 3-531 3*597 3-504 3-536 3-590 3-493 3*526 The divergence of our results from the mean of those of Loomis and Barnes is for hydrochloric acid 0T per cent., for potassium chloride 0*25 per cent., and for sodium chloride 0*2 per cent. In each case our result is the smaller. This may be due to a differ- ence in the degree value of the thermometers employed by the different observers — our own is uncertain to 0*4 per cent. — or it might be due to the difference of the systematic error of the methods employed by Barnes and Loomis on the one hand and ourselves on the other. Whatever be the cause of the slight difference, the agreement as it stands is of a very satisfactory nature, and tends to increase confidence in the results of exact cryoscopy. {Issued separately January 28, 1903.) •380 Proceedings of Royal Society of Edinburgh. [skss. Not© on Pure Periodic Continued Fractions. By Thomas Muir, LL.D. (MS. received October 27, 1902. Read November 17, 1902.) (1) There is a short paper* in a recent volume of the Comptes Rendus of the French Academy of Sciences which deserves notice if only in order that the attention of the author and others may he drawn to previous work on the same subject and to more effective methods of treatment. (2) It is well known that by the solution of a quadratic equation we can show that the periodic continued fraction , 1 1 1 CL-, + — — * Ct% + Cfc3 + • • + On + • • • * \/ { (al> • • • ,an) ~ (®2 > • l) j2 * + 4( > a±) = etc., on the right are con- 1 1 -1 «8 1 -1 a4 $2 , . • • ) an in the reverse order we have 1 1_ 1_ + ••••+ a2 + + • • • * V{(a,l,-M%|^Can-i,...,a2)}2 + 4(aM,...,a2)(aw_i5...,ai)+{(an,...,a1)-(a„_i,...,a2)} • 2(as»-i,...la1) the numerator here being the same as before because of the fact * Ceelier, Sur le developpement de certaines irrationelles en fraction con- tinue. Comptes Rendus . . (Paris), cxxviii. pp. 229-231 (year 1899). 1902—3.] Dr Muir on Pure Periodic Continued Fractions. 381 that a continuant is unaltered by reversing the order of its diagonal elements. We are thus led at once to the results 1 a, + . * + an + * 1 an + , * an- 1 ' + cq + * , . . . , qw_i) , (H) (a 2 j • • • i ) (al , . . • , an- 1) • ( a2 » • • • ’ an-i) {an a>2 ^ ~ 1 ) 1 _1_ 4- « . . ! Cl. + *2 Cl,, + 1 1 (HD an_i + • • • + ci, 2 (3) From (I) it is easy to formulate the conditions which must hold in order that a given irrational expression ( J H + R)/D may be transformable into a pure periodic continued fraction: it is more convenient, however, to put this quasi-converse proposition in the following form : — The irrational expression ( N/A2 + 4a/3 + A)/2a is transformable into a pure periodic continued fraction , if positive integers a^ , are determinable so that . , a, and A _ (aT , . . . , a n) — (a2 , . . . , a>n-i) a (a2 , . . • j a n) P = (a1 , . . • , aw_i) a (a2 , . . • , a n ) (IV) (4) Tf these conditions he satisfied we obtain by division A (oq , . . . , ~ > • • • > ^-i) j8 (ci]_ , ■ . . , and therefore *v/A2 + 4ctj8 + A 2j8 / 7 K 1 ^ | (a2» ...) ) j (®i ? vl 2(a1} J (ai> 2(a1,...,an-i) {VKv ■ , «w) - (a25 • • ■ > + 4(al ’ • ' : l)(d2» •••> *) + {(alJ — (a2’ ■ 2(Ol, . . . , CCn—l) l 1 :nn+- * 1 + • • • +(*!+•• 1 382 Proceedings of Royal Society of Edinburgh. [sehs. We thus have the theorem : — If the irrational expression (V/A2 + 4a/3 + A)/2a be representable by a pure periodic continued fraction , so also is (JA2 + 4a/3 + A)2//3i and the cycle of partial denominators for the one is the reverse of that for the other. (V) With this may be compared the theorem which it is the object of M. Crelier’s paper to establish. (5) Irrational expressions of M. Crelier’s form, ( J A2 + a/3 + A) -r a , are not all representable as pure periodic continued fractions. In fact, if we have an expression of this kind which is so representable, say JA2 + af3 + A = a l_ JL_ _1_ a f a9 + as + • • • + an + • • • * then, t being any integer less than A/a, it follows that («,-*) + ! L L J^+^ + A-at * * * d* an + «i + • • • = a * J( A - at)2 + a{fi + 2tA - at2) + (A - at) a where the period of the continued fraction now begins at a different point, hut the equivalent irrational expression is still of the form referred to. An example of this is — N/52 + 3 x 18 + 5 111111 3 “ +l + l + l + 2 + 3 + 5 + --- * * Even here, however, the twin expression ( J52 + 3 x 18 + 5) -f 18 is representable as a periodic continued fraction whose period is the reverse of that just written, viz., we have n/52 + 3 x 18 + 5 1111111 18 ” I + 3 + 2 + 1 + 1 + 1 + 5 + .-. * * (6) The one thing necessary to have recalled in connection 1902-3.] Dr Muir on Pure Periodic Continued Fractions. 383 with the whole matter is the following theorem,* viz. : If a quadratic equation have real irrational roots , and these he trans- formed into continued fractions of the type a, + — — a2 + a3 + * • * > or its reciprocal , where ax , a2 , • • • are positive integers , then both continued fractions are periodic , and f the one period is the reverse of the other. Applying this to the quadratic equation ax 2 - 2 Ax - /3 = 0 , we first note that its roots are sj A2 + a/3 + A ~ sj A2 + af3 + A — a ’ a ’ and that the second of these is transformable into - P jA2 + af3 + A ’ and we thus find that we need not go further. (7) There is an important matter connected with the funda- mental result of § 2, which, apparently, has not hitherto been noticed. This result, if we write pn for (cq , . . . , an) , qn for (a2 , . . . , an) , etc., is 1 ^{'Pn Qn— l) fPn—lQn "h Qn—l) jj. 1 + cjg + • • • + an + • • • 2 qn * Now the number under the root-sign being non-quadrate, its square root must be expressible as a recurring continued fraction, and the problem is thus suggested of finding this fraction. By a * Unfortunately omitted from most modern text-books. See Serret’s Cours d'algebre suptrieure, 4e ed. (1877), i. p. 49. t It would be better to insert here the words “ if the periods be different ’’ ; for they may be really identical, the mode of writing in the case of certain periods being deceptive on this point. Thus, when A = 7 , a = 2 , 0 = 15, we , v/72 + 2‘15 + 7 * , 1 , 1 1 1,aTO 2 ,7+l + lS+i+-- y7*+2-15 + 7_1+J_ 1 l+_ 15 16 1 7 where the periods, although seemingly different, are not really so. A still better instance is got from A — 7 , a -= 1 , 0 = 30 . 384 Proceedings of Royal Society of Edinburgh. [sess* * well-known property of continuants (v. Math. Gazette , ii. pp. 58-59) (Pn ~ R-l)2 + fyn-lQn = (Pn + + gw-i)2 + 4 {Pu + q«>- 1) + 1 i(Po> + q co-i) * l + 2(^>w + _ i) + • • • * whence, by addition of y>w - - 1 to both sides, 2 I flq + — , , ~ , \ = 2 pi* + 1 ( * &2 + aw + • * ’ ) Up + 1 p a2 » • • . , am , x) A — / (a2 J • • . , am , X ) x(ax , a2 , . • • , am) + (« i , a2 , , am_x) x(a2 , . . ■ j am) + (a2 i \ 5 a3 3 • • • j am- 1/ _ X7Tm + 7Tm_i , say. %Pm + Pm— 1 If now we insert the value of x given in § 2, writing it for the moment in the form (N/H + R) + D, we obtain * Serret’s Cours d'algebre suptrieure , i. (1877), p. 46, or Chrystal’s Algebra , ii. (1889), p. 430. 386 Proceedings of Royal Society of Edinburgh. [sess. x = 7rmN/H + (7T.tMR + ^t?h— i.P) . PmsI H + (pmR + Pm-lD) and this after rationalisation of the denominator will be found to be simplifiable by the removal of the common factor D from numerator and denominator, \/H being thus left in the numerator with the coefficient 1. This simplification is the main matter deserving of note, but the full result is — If the irrational expression which is the equivalent of 1 a2 + • • • + an + • • • be denoted by ( JH + R)tD, then the like expression for I _1_ J_ 1 1 a2 + • ■ * 4- am + aj + a2 + • • • + an + • • • * * is (VH + R')-D', where R (in— 1) Pn + In-1, In $ 27Tmpm ) '^‘m—lPm "h ^"mPm— 1 > and V' = ( $ 2 pm, 2pmpm_i, 2 p^_i), 7rm/pm and pn/qn being the m<;i and nth conver gents to ax + — — a2 + • • • + am and an+i — respectively . (VIII) 1 a2 + --- + an D', it will be observed, is got from R' by writing pm , pm-1 for _ _ * 7r m , ^m— 1* * It is extremely probable that among the papers of the late Mr C. E. Bickmore, Fellow of New College, Oxford, who was an earnest and capable student of this branch of algebra, there are to be found quite a number of results as yet unpublished and worthy of being made known. In a letter dated 29th April 1895, in which some noteworthy theorems are given from a paper prepared for the Oxford Mathematical Society, he says: “Our Society is too poor to print anything.” It would appear that French mathematicians are not similarly hampered (v. Lond. Math. Soc. Proc., xxxiv. p. 129). ( Issued separately February 28, 1903.) 1902-3.] Dr Muir on Generating Functions of Determinants. 387 The Generating Functions of Certain Special Determinants. By Thomas Muir, LL.D. (MS. received September 8, 1902. Read November 3, 1902.) (1) From a general theorem, known since 1855,* and perhaps earlier, regarding the reciprocal of the series 1 + axx + a2x2 + asxs 4- it follows that 1-fe + OCT2 = Po + ft* + ft*2 + ft*3 + • • • (A) where /30 = 1 and Pn = b 1 = b a ac b 1 c b a, . ac b 1 . c b a . ac b . . c b .... This at once leads to bx - acx 2 „ ^ ^ „ l-bx + ae* = ft* + £2* + ft* + > (b2 - ac)x 2 - abcx 3 1 -bx + a^~ = ^ + ft*3 + ' and generally, by subtracting 1 + p1x-\-p2x2 + . . . + to - ac/?r_13r+1 _ „ o 1 — bx + (zcic2 •r+Y *r+ 1 + or - acBr_,x 1 - bx + acx 2 = + &+135 + &+2^2 + • • * Fatjre, .... Theoreme sur la somme des puissances semblables des racines. Nouv. Annales de Math., xiv. pp. 94-97. 388 Proceedings of Royal Society of Edinburgh. [sess. From the original and all these derived results we thus obtain by addition (l+/31g + fe«2+ )-aca?(\+plx + B#?+ . . .) llfo-+ aez2 “ Po + zPix + + and therefore 1 acx2 (1 -Aa + OCB2)2 = + 2 A* + + W3 + whence with the help of the original identity we have 2 t)x (1 - bx + acz2)2 = + + + + Further, from (I) and (II) there results ( 1 -bx + «CT2)2 = A + + • • • (I) (II) (HI) (2) Again, it is clear that a c b - + , + a - a2x + asx 2 - a4£3 + . a 1 + ax 1 + cx 1 -bx + acx 2 + c - c2»; + c3£2 - c4sc3 + . + (b - 2aca?) (po + p-jX + P2x2 + . . . ') = (a + c + fy30) + ( - a2 - c2 + bpi - 2 acpo)x + ( as + cs+bp2-2acpi)x2 + where the coefficient of xr is ( - )r~1(ar + cr) + bp^ - '2acpr_2 , or, since Pr = bpr_ x - acpr_2 , ( - )r~1(ar + cr) + pr - acpr_2 . Now this is exactly equal to the determinant which is con- structed by taking pr and putting c in the place (1, r) and a in the place (r, 1), viz. : b a c c b a *. c b a c b b a c b (IV) 1902-3.] Dr Muir on Generating Functions of Determinants. 389 This determinant, however, clearly contains the factor a + 6 + c, or a say being 1 a • . . . ... c 1 b a 1 c b a . . 1 c b . . . 1 , b a 1 . c b r 2c3d4 | — 1 1 . 1 1 a , 1 \ \ h h4 1 c1 c2 c3 c4 1 d1 d2 d3 d4 (5) Returning again to the result (VIII), and multiplying both sides by dx , we obtain with the help of (A) § 1 1 dx{\ — acx2) 1 - bx + acx 2 + (1 + ax)( 1 + cx)( 1 - bx + acx2)2 = 1 + PXX + P2X2 + /?3«3 + . . . . - d H-J IT l . 1 1 x2 - d .111 1 b | 1 b a 1 b a . 1 c b 1 c b a 1 . c b | x3 - . and, therefore, by theorem (e) of § 4 = 1 + (b + d)x + b + d a+d x2 + b + d a + d d c + d b + d c + d b + d a + d d c + d b + d x6 + (XV) From the same two sources we similarly have / x{l - acx1) 1 -bx + acx 2 (1 + ax)( 1 + cx)( 1 -bx + acx2)2 x 2 + I / 1 e + / 1 1 1 si 1 b a 1 c b (XVI) This is due to Dr F. S. Macaulay, v. Math. Gazette , ii. p. 61. ( Issued separately February 28, 1903.) 1902-3.] Dr G. A. Gibson on Some Observations on Cyanosis. 393 Some Observations on Cyanosis. By G. A. Gibson, M.D., D.Sc., F.It.C.P.Edin., Physician to the Royal Infirmary, Edinburgh. (Read December 15, 1902.) The one constant factor in the production of cyanosis is diminu- tion of oxygenation. This may be the result of respiratory affections which hinder the access of air or diminish the area of the aerating surface, or it may, on the other hand, be due to circulatory disorders lessening the amount of the blood flowing to the lungs. These latter affections furnish the most conspicuous examples of cyanosis, and in them the symptom is entirely pro- duced by hindrance to the blood flow. Such is in reality the original conception advanced by Morgagni* and supported by numerous observers. The other view, apparently suggested by Corvisart, f that the condition is due to admixture of arterial and venous blood, is absolutely untenable, as in the vast majority of cases there is no possibility of such intermingling. In general terms the blood in cyanosis may be said to be of high specific gravity — from 1070 to 1080. The amount of haemoglobin rises considerably — often reaching above 100 per cent. The erythrocytes increase in number so as frequently to exceed 7,500,000 per cubic millimetre, while the leucocytes, although, not so commonly altered in number, often reach such a figure as 12,000 per cubic millimetre. The increase of these constituents of the blood was first observed by Toeniessen. I Malassez § showed that there are more red corpuscles in the blood of the superficial than of the deeper parts of the body. Toeniessen and Penzoldt || suggested that this might he due to loss of fluid from the surface, while the interior receives a con- stant supply of fluid from the alimentary tract. Eight years ago, * De Sedibus et Causis Morborum , tome 1, p. 154, 1761. f Maladies du Coeur , p. 281, 1806. X Ueber Blutkoryerchenzahlung bei Gesunden und Kranken Menschen, S. 29, 1881. § Archives de Physiologic, ii. Serie, tome 1, p. 49, 1874. || Berliner Klinische Wochensclirift, S. 45, 1881. PROC. ROY. SOC. EDIK — VOL. XXIV. 26 394 Proceedings of Royal Society of Edinburgh, in describing some observations on the Cyanosis of Congenital Heart Disease, in The Lancet ,* an explanation of the manner in which cyanosis is produced was advanced by me. From that paper it seems advisable to quote the hypothesis to which my observations led me : — “ Starting with the conception that cyanosis is produced by obstruction to the circulation and venous stasis, the question now arises why this condition should be associated with an increase in the number of the blood-corpuscles. It is not only in the cyanosis of congenital lesions that the increase is found, but in all cases where cyanosis is really present on account of failure of the cir- culation. To this point Toeniessen and Schneider have particularly called attention, and of the accuracy of the observation anyone can convince himself by investigation of the blood. The work of Malassez seems to show that the blood in the superficial parts of the body contains a larger number of red corpuscles than that from the deeper layers, and Penzoldt and Toeniessen believe that this increase is caused by the loss of fluid from the surface, while the blood of the interior is constantly receiving fluid from the alimentary canal. Even if this be correct, it cannot be accepted as an explana- tion of the great increase in the number of the corpuscles found in cyanosis, as it would involve the postulate that in some cases where the number of corpuscles is nearly doubled the quantity of the fluids of the blood must be reduced nearly to one-half. Cohnheim’s celebrated experiment of tying the crural vein of the frog, which is followed by a considerable increase of the corpuscles in the vessels, with the transudation of serum into the surrounding tissues, may be regarded as an explanation of the moderate increase in the cases accompanied by anasarca, but it has no special bearing upon cases of congenital cyanosis in which there is no drain of fluid into the tissues. It may possibly be held that in such cases the lymph vessels are unusually active and that the fluid constituents of the blood are as rapidly absorbed as they transude. Such an opinion can scarcely be seriously entertained. The backward pressure on the venous system which causes the transudation must tell on both terminations of the absorbents. It may be admitted freely that the increased pressure on the peripheral veins may tend * The Lancet, vol. i., 1895, p. 24. 1902-3.] Dr G. A. Gibson on Some Observations on Cyanosis. 395 to raise that impelling the fluids into the commencement of the lymphatics, but it must not be forgotten that an elevation of pres- sure in the great veins will hinder the return of the lymph by pressing upon the openings of the lymph vessels into the veins. It is probable that the increase in the red corpuscles may be to some extent compensatory in cases of cyanosis. To say this, how- ever, is not enough ; it affords no rational explanation of the process by which the increase is brought about. Nature does not work by such direct methods as would require to be invoked if the increase of the corpuscles were regarded as a simple compensatory change, balancing the diminished power of oxygenation. Compensation in valvular lesions, for example, is produced by the definite structural changes constituting hypertrophy, caused by increase of work, and compensations in cyanosis must have some reasonable explanation also. It seems to me that such an explanation may be found in a consideration of the functions of the red corpuscles under changed conditions. In venous stasis the corpuscles are insufficiently oxygenated, they cannot perform such an active part as oxygen carriers, and they cannot yield so much oxygen to the tissues. It must further be remembered that in cyanosis there is less metabolism in the tissues, and therefore less waste produced. In a word, the functions of the corpuscles being lessened, the tear and wear which they undergo is reduced, and the duration of their individual existence increased. The number of the corpuscles must in this way be proportionately augmented, and this must lead to the numerical increase, as well as to the high percentage of haemoglobin, until a balance is struck between the production and the destruction of the blood- corpuscles.” In some subsequent remarks upon this subject * it was frankly stated that this explanation could only be seriously entertained if the increase in the blood elements was found to exist throughout the different divisions of the circulatory mechanism — arteries, capillaries, and veins — but that the observations of many workers on the state of the blood in high altitudes were strongly in support of it. The object of the present communication is two-fold : in the first place, to show what relationship exists between the conditions * Diseases of the Heart and Aorta , p. 211, 1898. 396 Proceedings of Royal Society of Edinburgh. [sess. of the blood in the different divisions of the circulatory apparatus, and, in the second place, to indicate the results which have been obtained by the use of oxygen in cases of cyanosis. The two following cases are of interest from the first point of view. Case 1. — A tradesman, aged 48 years, was under treatment in my ward for arterio-sclerosis with interstitial myocarditis. He was very cyanotic. The heart was considerably enlarged and there was escape at both mitral and tricuspid orifices. The radial arteries, like those throughout the rest of the body, were rigid, but the pressure was not high. There were oedema of the lungs and albuminuria, with great anasarca. The patient became maniacal, and then passed into a comatose condition from which he was rescued by venesection. The opportunity was taken of obtaining two or three drops of arterial blood, and the condition of matters was as follows. Artery : haemoglobin, 125 per cent ; erythrocytes, 5,130,000; and leucocytes, 5980 per cubic millimetre. Vein: haemoglobin, 135 percent.; erythrocytes, 5,300,000; and leucocytes, 6250 per cubic millimetre. On the following day another drop of arterial blood was obtained and compared with that from the subcutaneous capillaries. The results were the following. Artery : haemoglobin, 120 per cent.; erythrocytes, 5,268,000; and leucocytes, 6100 per cubic millimetre. Capillary: haemoglobin, 130 per cent.; erythrocytes, 6,830,000; and leucocytes, 11,620 per cubic millimetre. After this blood-letting the patient, by means of various cardiac remedies, rapidly recovered and went home in a few weeks with no symptoms of circulatory disorder and with blood practically normal. Case 2. — A pithead boy, aged 16 years, is at present under treatment in my ward for a group of symptoms which seem only explicable by the diagnosis of mediastinal pericarditis. On admission, in September 1902, the boy was deeply cyanosed. His heart was but slightly enlarged and had no abnormal sounds. The pulse was of low pressure, and manifested in the most beautiful manner the features of the pulsus paradoxus. The patient had enormous enlargement of the liver and considerable increase in the size of the spleen, along with hydrothorax, ascibes, albuminuria, and anasarca. The subcutaneous blood was frequently and regularly examined, and showed an amount of haemoglobin varying 1902-3.] Dr G. A. Gibson on Some Observations on Cyanosis. 397 between 80 and 100 per cent., while the number of red corpuscles fluctuated between 8,000,000 and 8,800,000 per cubic millimetre, and the number of white corpuscles oscillated between 5000 and 8500 per cubic millimetre. On 20th October a drop of blood was obtained from a small twig of the temporal artery and another from a small tributary of the temporal vein in order to compare with the blood in the capillaries, and the following results were obtained. Artery : haemoglobin, 93 per cent. ; erythrocytes, 7,000,000 ; and leucocytes, 6875 per cubic millimetre. Capillary : haemoglobin, 93 per cent. ; erythrocytes, 8,500,000 ; and leucocytes, 8500 per cubic millimetre. Yein : haemoglobin, 115 per cent.; erythrocytes, 10,000,000; and leucocytes, 7400 per cubic milli- metre. Under treament by means of hydriodic acid, cardiac tonics, and diuretics internally, along with inunction of diluted red iodide of mercury ointment, the patient has improved very greatly, and although the liver and spleen are still enlarged the other symptoms have almost entirely disappeared. Along with the disappearance of the symptoms the condition of the blood has come back practically to normal. Uow we learn from these observations that in cyanosis there is a general increase throughout the whole circulation in the amount of haemoglobin and in the number of the erythrocytes, while the state of matters as regards the leucocytes is less constant. It seems to me that this universal, if not uniform, increase goes a long way in support of the explanation which I ventured to advance in the paper to which reference has already been made. The whole sub- ject, however, requires experimental investigation, and it was my intention to have entered upon a series of experiments— an intention which has been frustrated by lack of time. One of my friends has undertaken a research upon the subject, and it is to be hoped that in the course of a few months definite information will be available. It is probable that the universally recognised increase in the number of the hsemocytes in those who live in high altitudes may be found to furnish an available opportunity of working out the problem. Another point arises from the consideration of these observations. It has been said that the increase in the haemoglobin and erythro- cytes is universal but not uniform, and in this interesting fact lies the exposure of a possible source of fallacy in attempting to estimate 398 Proceedings of Royal Society of Edinburgh. [sess. blood conditions. No method based upon the assumption that the condition of the blood obtained from one part of the body can be regarded as an expression of its exact conditions throughout the whole system can, for the future, be regarded as of scientific value. The final part of this preliminary inquiry has been devoted to an attempt to ascertain whether the inhalation of oxygen has any beneficial effect upon cyanotic conditions. Up to the present time it must be confessed that no case which has been under my care has shown any effect upon the blood from the use of this method. Of the cases which have been so treated the following two may be taken as types of all the others. Case 3. — A miner, aged 39 years, has recently been under my care on account of chronic bronchitis, emphysema, and pulmonary collapse, with great displacement of the heart to the right side. He was deeply cyanosed and had clubbing of the fingers and arching of the nails. The heart was slightly enlarged, and two- thirds of its dulness lay to the right of the mid-sternal line. A tricuspid systolic murmur was present. The pulse was frequent, varying between 80 and 100, and was of low pressure. There were anasarca with great enlargement of the liver and considerable enlargement of the spleen, as well as albuminuria. On 20th October 1902, the haemoglobin was 110 per cent., the red corpuscles numbered 6,400,000, and the white 12,500 per cubic millimetre. On the 26th, the haemoglobin was 112 per cent., the red corpuscles numbered 7,500,000, and the white 14,000 per cubic millimetre. On this date the inhalation of oxygen was begun, and on the 31st, five days later, the haemoglobin was 100 per cent., while the red and white corpuscles numbered respectively, 8,000,000 and 9960 per cubic millimetre. Case 4. — A miner, aged 33 years, came under my care on account of mitral obstruction and incompetence. He suffered much from palpitation. His heart was slightly enlarged. There were pre- systolic and systolic mitral murmurs, and the pulse was of very low pressure with considerable irregularity. There were no changes in the respiratory system. The abdominal organs were unchanged and there was no albuminuria. The condition of the blood may be stated in tabular form as follows : — 1902-3.] Dr G. A. Gibson on Some Observations on Cyanosis. 399 Haemoglobin. Erythrocytes. Leucocytes. Per cent. Per cm. Per cm. Oct. 20th 131 .... 7,500,000 .... 11,400 Oct. 31st 120 .... 7,000,000 .... 9,680 Nov. 17th 110 .... 7,250,000 .... 10,000 Dec. 3rd 110 .... 6,500,000 .... 11,000 Dec. 3rd oxygen inhalation was begun. Dec. 5th 115 .... 7,200,000 .... 9,800 Dec. 8th 110 .... 6,000,000 . . 10,000 The results of these two observations seem to indicate that the inhalation of oxygen is barren of result. It is generally understood that the ordinary air of the atmosphere contains considerably more oxygen that can be absorbed in any circumstances, and the results of our observations, now fairly numerous, are strongly in favour of this conception. It is a pleasure to express my warm thanks to my house physicians, Dr L. C. Peel Ritchie, Dr H. H. Bullmore, and Dr A. F. R. Conder, for their unwearied devotion in conducting the examinations of the blood of many patients in pursuance of these observations, and to acknowledge the assistance which we have received in our investigations from Dr A. Goodall and Dr A. Dingwall Fordyce. (Issued separately April 4, 1903. ) 400 Proceedings of Royal Society of Edinburgh. [sess. On the Isoclinal Lines of a Differential Equation of the First Order. By J. H. Maclagan-Wedderburn. Communicated by Professor Chrystal. (Read January 5, 1903.) A differential equation may be regarded from two points of view, one purely analytical, the other geometrical. From the analytical point of view, a differential equation of the first order is merely a functional relation between x , y , and p (where p^dy/dx ), and the problem of solving the equation is to find a function of x, say f(x), such that if f(x) and df(x)/dx are substituted for y and p in the equation, the result is an identity in x. In the geometrical method, on the other hand, x and y are treated as the co-ordinates of a point in a plane and p as a direction. The differential equation then attaches to every point in the plane a certain direction, which may be conveniently represented by an arrow drawn through the point. The problem of integration then resolves itself into finding a family of curves, such that, at every point ( x', y'\ the direction of the curve at that point is the direction obtained by substituting x and y' in the differential equation and solving for p. These curves are called the integral curves of the equation. This method owes its development chiefly to Lie. An instructive example of a differential equation from this point of view is furnished by a well-known experiment in magnetism. A magnet exerts on another magnet, placed in its neighbourhood, a force whose direction and magnitude depend, in a given medium, solely on the strength of the two magnets and on their relative position, and, if one of the magnets is very small, the force on it due to the other is merely directive. We have here, then, a physical representation of a differential equation. If now we cover a magnet with a sheet of paper and sprinkle iron filings on it, each filing becomes a magnet by induction, and therefore sets itself longitudinally in the direction of the force at the point where it falls, and, if the paper is gently tapped, the filings arrange them- 1902-3.] Isoclinal Lines of a Differential Equation. 401 selves in curves, namely the lines of force. These lines of force are the integral curves of the differential equation. A differential equation of the first order = 0, and these curves have the property that all the integral curves, that intersect any particular curve of the second family, have the same direction ( i.e ., the same p) at the points of intersection. The latter family has been called the “Loci of Contacts of Parallel Tangents,” by Hill (Proc. Lond. Math. Soc., x ix., 1888, p. 561), but, at the suggestion of Professor Chrystal, I propose to use in this paper the more convenient term “Isoclinal Family.” This family gives a method of describing any integral curve ; for if, beginning at any arbitrarily chosen point, we draw an infinitesimal line in the direction specified by one of the isoclinal curves passing through that point, we in general come to another isoclinal giving a new direction differing infinitesimally from the original direction, and so on. Of course, as the starting-point is arbitrary, the integral curve must be developed on both sides of it ; also it must be noticed that the direction specified by the isoclinal is not in general the direction of the isoclinal itself. How it is evident that there is in general one and only one isoclinal that passes through the point (x + dx , y+pdx) and is also contiguous to the isoclinal p through (x , y) and similarly for the point (x - dx , y - pdx) ; also in general the isoclinal (p + dp) lies wholly on one side of the isoclinal p in the neighbourhood of (x , y), and at such a point the y of the integral curve and its first differential co-efficients are in general synectic functions of x. This is, however, not in general the case in the neighbourhood of the envelope. For, in general, one of a family of curves does not cross the envelope of the family in the neighbourhood of the point of contact. Thus the three contiguous curves p - dp ,p and p + dp all lie on the same side of the envelope, and all touch it. Moreover the direction p will not in general be that of the envelope, but will cross it. But as there is no contiguous isoclinal across the envelope to indicate a new direction, the integral curve cannot cross the envelope, and must therefore have either a cusp or a stop point. If the envelope is the curve indicated in figure 1 by E and the isoclinal p touches 402 Proceedings of Boy al Society of Edinburgh. [sess. it in P, and if PQ be an infinitesimal element of the integral curve at P, we can in genexal find two and only two isoclinals contiguous to p which pass through Q, and they in general touch the envelope on opposite sides of P. Therefore if we start from P to draw the integral curve, we have a choice of two directions at Q, each differing infinitesimally from p ; and these in general give two distinct branches of the integral curve as the sign of the variation of p is different in the two. The envelope is therefore a locus of cusps for the integral curves. Now Cayley has shown that if (xyp) = 0 is an integral algebraic function of p, the isoclinal family has in general an envelope which is given by the ^-discriminant of . The p-discriminant is therefore in general a locus of cusps on the integral family. Conversely, if E is a locus of cusps for the integral family, it is part at least of the envelope of the isoclinal family ; for two contiguous isoclinals intersect at Q. If, however, the direction p is also the direction of the isoclinal at P, and therefore of the envelope, the direction p does not cross the envelope and there is no discontinuity in the variation of p for the integral curve. The direction of the isoclinal family is given by therefore the condition for the p-discriminant being an envelope locus for the integral family is x+Py = ° .... (2) and this condition is in general both necessary and sufficient. Several other cases now present themselves. The ^-discriminant 1902-3.] Isoclinal Lines of a Differential Equation. 403 may be a locus of double points on the isoclinal family — in general a node locus. In this case the isoclinals, and therefore the integral curves, in general cross the ^-discriminant. For, if P and Q have the same meaning as formerly, we find the same phenomena appearing on the integral curve as in the case of an envelope, with this difference, that we also have a point Q' on the other side of the ^-discriminant from Q at which also two integral curves or two branches of the same integral curve diverge. As the latter is a higher order of singularity it is less general. (It must be noticed that the two isoclinals contiguous to p which pass through Q' are not in general the same as the two which pass through Q.) The conditions for a node locus on the isoclinal family, and therefore for a tac-locus on the integral family, are , = 0 .... (3) in addition <£ = 0 and <£p = 0, and these are in general sufficient. Now, in general, if an isoclinal pass through a point P, there is one and only one curve contiguous to it which passes through a given point which is contiguous to P, but if P is a tac-point of the integral family, and Q is a point contiguous to P on both the integral curves that touch at P, since the rate of variation of p is in general different for the two integral curves, two contiguous isoclinals must pass through Q ; and similarly for Q' on the other side of P from Q, and therefore two isoclinals must pass through P ; and as the p of both integral curves is the same at P, these two isoclinals must be branches of the same curve, therefore the conditions (3) are also in general necessary. The branches of the isoclinal curve at a node divide the plane / 404 Proceedings of Royal Society of Edinburgh. (i) into two regions, one of which contains the ^-discriminant and one which does not. There are three species of tac-loci according as the direction of the integral curve lies in the former or the latter region or along the isoclinal : (i) if it lies in the region not containing the ^-discriminant the two integral curves have opposite curvature ; (ii) if it lies in the region containing the ^-discriminant the curvature is the same for both in direction hut not in general in magnitude ; (iii) if the direction of the integral curve lie along the isoclinal, there is, as will be shown later, an inflexion on one of the branches of the integral curve. Figure 3 gives a geometrical representation of the three cases. If the double point is a point of the first order, the directions of the tangents to the isoclinal are given by the quadratic Fig. 3. ^2^+2'^ + ^ = 0 (4) therefore in any particular case it is easy to decide to which species the tac-locus belongs. (See examples (1), (2), (3) and (7).) If, however, the roots of (4) are equal, i.e., if (5) the p-discriminant is a cusp locus for the isoclinal family. Similar reasoning to the above shows that it is also a cusp locus for the integral family ; and in every case, except when the direction of both families is the same, the curves contiguous to y passing through Q both lie on the same side of P, and therefore the curvature of both branches is in the same direction, i.e., the cusp is a ramphoid cusp. (See fig. 4.) If, however, at any point the direction of the integral curve is the same as that of the cusp locus, there is in general a tac-point on the integral family, the contact being of higher order than the first. (See examples (4) and (5).) If the roots of (4) are imaginary, the ^discriminant is a locus of conjugate points for both families. If at any point P, which is not on the envelope locus of the 1902-3.] Isoclinal Lines of a Differential Liquation. 405 isoclinal family, the direction of the integral curve is the same as that of the isoclinal through P, the contiguous points Q and Q' are in general both on the same side of the isoclinal, but on opposite sides of the point P; therefore the sign of the variation of p changes on passing through P, i.e ., there is in general an inflexion on the integral curve. The condition for this is x +Py = 0 • This is equivalent to the usual condition for an inflexion ; for 4>xdx + 4>ydy + cfipdp — 0 but along an integral curve dy =pdx, hence djP_= _(k±liy==o dx cf>p if p 4= 0, which, along with 4= 0 is the usual condition for an inflexion. Even if p = 0 the geometrical reasoning shows there is still an inflexion, unless the ^-discriminant is an envelope locus for the isoclinal family. (See example (3).) This may obviously be generalised as follows : if the direction of the integral curve lie along the isoclinal at any point, and if the tangent at that point meet the isoclinal in n contiguous points, then it will also meet the integral curve in (n + 1) points. (See example (6).) The following are a few examples in illustration of the above. Example (1) : p2 -l- (3x + 2 y)p - \ x 2 + 3 xy + y2 = 0 . 406 Proceedings of Royal Society of Edinburgh. [sess. The p-discriminant is easily found to be x2 = 0 and the directions of the isoclinal curve at the origin are given by y 2 + 3 xy - \x2 = 0 , hence y = - ix or \x . The direction of the integral curve at the same point is along the x-axis, i.e., in the region not containing the ^-discriminant, hence the integral curves should have opposite curvatures. A first approximation at the origin gives p2 + 3px - \ x 2 = 0 , hence 2/ = i( - 3 ± i)x2 . The two "branches have opposite curvatures as predicted. Example (2) : p2 + 2(x + y)p + \x2 + 2 xy + y2 = 0 . The direction of the integral curves at the origin is along the x-axis. The jp-discriminant is x2 = 0, and the directions of the isoclinals are given by y— - f-z or -\x . Hence the integral curves have a tac-point of the second species. A first approximation gives p2 + 2px + \x2 , hence Example (3) : p2 4- 2{x + y)p + 2 xy + y2 + x3 = 0 . The /9-discriminant is x2(x - 1) = 0 where x2 corresponds to the tac-locus. The directions of the isoclinals are given at the origin by y{y + %x) = 0, hence, as the direction of the integral curve is along the z-axis, there is an inflexion on one branch. 1902-3.] Isoclinal Lines of a Differential Equation. To find the integral curve we have 407 p = -(x + y) ± Jx\x - 1) = -x-y±(x- \x 2) to a first approximation. Hence to same order of approximation y= -i«3, y= - x2 . Example (4) : (y-p) 2 = «3- The ^-discriminant is a locus of cusps for the isoclinal family, and is therefore a locus of ramphoid cusps for the integral family. To find the integral family we have p-y= ± x* hence y — aex± exj e~xx% which to a first approximation is y — a + ax + \x2 ± f x% , a ramphoid cusp, except when a = 0, i.e ., when the direction of the integral curve is the same as that of the isoclinal. Example (5) : p2 — 2 xp + x2 — y3 = 0 . The ^-discriminant y — 0 is a locus of cusps for the isoclinal family and at the origin the direction of the integral curve lies along it, hence the origin is a tac-point. A first approximation gives putting p = x y = ^x2 p = x + m y = \x2 + v ; and neglecting terms not required for the second approximation we get 67 = + J-2" hence 408 Proceedings of Royal Society of Edinburgh. [sess. Example (6) : y~p = xn . The a?-axis has contact of ( n - l)th order, hence, as the direction of the integral curve at the origin is also along the ic-axis, the tangent there has contact of the nth order. Integrating we get y = - ex I xne~xdx to a first approximation. Example (7) : The />-disciiminant is y2(y2 - -^f) = 0. The directions of the integral curves at the origin are p = 0 and '£>— - 1 . The direction of the isoclinal curve corresponding to^> = 0 is V= ±x; the corresponding part of the ^-discriminant is therefore y2 = 0, and the integral family has a tac-point of the second species. The integral curves at the origin are y — - x + \x2 + J#3 , y = \x2 ± \xz . It must be noticed that, although the direction of the integral curve lies along one branch of the , isoclinal family, it is not the branch to which that direction belongs, and so there is no inflexion. ( Issued separately April 4, 1903.) ] 902-3.] J. H. Maclagan-Wedclerburn on Vector Functions. 409 On the General Scalar Function of a Vector. By J. H. Maclagan-Wedderburn. Communicated by Dr W. Peddie. (MS. received 6th February 1903. Read March 2, 1903.) The general scalar function of a vector p may evidently be written in the form 2 (^Sc^pSc^p . . . Sa np) (1) n=l where ax . . . an are constant vectors, each term being homogeneous in p. If the factors of each term of any one of the brackets be permuted in every possible way, i.e., in j ways (where r is the degree of the set of terms chosen), it can be written in the form sP^Pr-1 (2) where p’'-1 is a vector function of p of the (r— l)th degree, the index (r — 1) serving merely to indicate the degree of the function. This notation will be found to lead to no confusion, and to have great advantages. Since the variable p occurs in along with each vector a, and also these vectors have been arranged in every possible way, it follows that, if in each term of we replace s of the p’s by any other vector pn~ao* = that So-^p11 = Sp<£pn-V (4) Joly, who has investigated linear and vector functions of three variables, has called such a function completely self-conjugate (see Appendix to Joly’s edition of Hamilton’s Quaternions , page 467). The following useful properties follow immediately — dp” = 1i{n + \)i$$pndpjdx = -(n+l)pn . . (7) SpVSp<£pn= — (n + l)3pp” (8) PROC. ROY. SOC. EDIK — VOL. XXIV. 27 410 Proceedings of Royal Society of Edinburgh. [sess. It is evident also that (p + pn -f npn~lcr + 3 , ^cru . . (9) a particular case of which is cf>(xp)n = Xncf>pn (10) Prom (7) it follows that Str V S pp" — - (n + 1 )S (rcfipn = — (n + l)Sp<£p™_1o- . By repeated application of the oper ator - So- V we get ( - So-^/1)rSp14>Pin = (?z + l)^(w - 1 ) .... (n — r + 2)Sp<£p,l_V’ , (11) where the. suffix indicates that Vj operates on pl only. If we replace cr by p the expression becomes ( - Sp V = (n +^)n • • • • (n ~ r + 2)Sp<£pn . . (12) which is Euler’s theorem for a homogeneous function of the (n + l)th degree. From (9) and (11) it follows that S(p + o-)cf>(p + pn_2(r2 . . . = (1 - So-V1 + 2 ! 1 .... )S Pl(f>p1 = e-ScviS p^pf (13) From this it follows that if F(p), a scalar function of p, can be expanded in a convergent series of the following form, n—n F(p) = d + Syp + 2 S pcj>pn , n= 1 then (see Tait’s Quaternions , page 399), F(p + cr) = e-s1p + y) + d = 0 • (15) where the t£’s are completely self-conjugate vector functions. It 1902-3.] J. H. Maclagan-Wedderburn on Vector Functions. 411 will be shown later that this admits of being expressed in a more ■concise form. Putting U p = p in (15) we have Tn+1pSp n + T”pSp 4*n-iP 71-1 + • . • • + d — 0 , which shows that any vector in general meets the surface in (n + 1) points. Hence also the cone of asymptotic directions is S p4>pn = 0 . Differentiating (15) and putting dp = T3 — p, the equation of the tangent plane at any point p is S(ET- p)((n + l)i>npn + n^n-iPn~1 . . ■ • ) = 0- 'This can also be written S(C7 - p) V Fp = 0 . In this connection it is useful to note that since by (7) VF(p) = — %{n + l)npn it is easy to pass in any particular case from Car- tesians to quaternions. As was indicated above, the general equation of the (w+l)th degree admits of being put in a more useful form than (15). If, •on the analogy of (2), we form the general homogeneous quaternion scalar expression of the nth degree, 2Hw+1Sgr , where q is a constant and r a variable quaternion, it can obviously in the same way as before be put in the form (17) where rn = - {n + l)V<£r* + (n + 1)S rn = — (n + 1)S crcf^r11 . 412 Proceedings of Royal Society of Edinburgh. [sess. It will now be shown that if Sr is constant, say equal to unity, and Vr = p , then (17) is the general scalar expression of the- (n-r l)th degree in p. For, since p can be expressed in terms of any three non-coplanar vectors, it may be easily shown that the general expression of the (n + 1 )th degree involves (n + 2)(w + S)(n + 4) “ 31 independent constants, but this is also the number of independent constants involved in (17), for it requires in general multiples of four independent quaternions to express any other quaternion.. (See Salmon, Geometry of Three Dimensions , page 233.) If the restriction that Sr = 1 be removed, then (17) represents what Hamilton has termed a full surface. The following are a few illustrations of the use of this form of the general equation to a surface. To find the tangent plane to Sr<£rn = 0 at the point Vr, we have S drrn = 0 S(p-r)<£rw = 0 hence Sp<£r1? = 0 where Sp = Sr = l, and Yp is the vector of any point in the- tangent plane. Differentiating a second time, we get %Srn-1rn = 0 . If rn = 0 for any value of r, the point is a conical point,, hence S dr$rn~xdr = 0 , therefore the equation to the tangent cone at the conical point is Sp<£rM-1p = 0 . Similarly for an s-ple point we have Sp<£r,l~s+1ps_1 = 0 . Since the above was in print, I have found that equation (11) has been given by Tait (3rd ed., p. 420, Ex. 34) for the case r = n+ 1.. See also Kimura, Annals of Mathematics , Yol X., 1896, p. 127. (. Issued separately June 5, 1903.) 1902-3.] Equation of a Pair of Tangents to a Conic. 413 On the Equation of a Pair of Tangents to a Conic. By Professor A. H. Anglin, Queen’s College, Cork. (Read February 16, 1902.) {Abstract.) The equation to the pair of tangents from the point ( x , y) to the conic <£(x, y) = 0 is usually obtained in the form (x’ ?/') = T2 where T = 0 is the equation to the chord of contact. This equation admits of reduction ; and we propose to obtain the reduced form independently, and to supply its geometrical interpretation. Taking the case where the equation to the conic is ax 2 + 2 hxy + by 2 = 1, the line lx 4- my + n — 0 will be a tangent if am 2 - 2 him + bl2 = {ab - h2)n2. If the line pass through (x, y) we have V(y - v) = ~x) = nl(xy' - x’y)- Hence, for any point on either tangent from (x\ y) to the conic, substituting for Z, m, n in the above relation we get a{x - x)2 + 2 h(x - x')(y - y) + b(y - y)2 = (ab - h2)(xy — xy)2, the required equation to the pair of tangents. If Q be the point (x, y'), P any point ( x , y) on either tangent, and C the centre of the conic, this equation is the result of equating two expressions which are equal to the square of 2 A CPQ. If the case of the most general form of equation ax2 + 2 hxy + by2 + 2 gx + 2 fy + c = 0 be deduced from the preceding, we shall get a(x - x)2 + 2 h(x - x')(y - y') + b(y - if)2 + (ab - h2)2 x y 1 x y 1 x y" 1 0, where (x , y") is the centre, and A the discriminant of the conic ; and from which the geometrical interpretation is obvious. 414 Proceedings of Royal Society of Edinburgh. [sess-. If we investigate the general case independently, we shall get as the condition that the line lx + my -j- n = 0 may be a tangent A l2 Bm2 + Cn2 + 2F mn + 2G nl + 2fIZra = 0 with a known notation ; and replacing Z, m, n by the above ratios- we have the required equation to the pair of tangents in the most reduced form. We may observe that on substituting the values of x", y" in the first form of equation, the quantity ab - h2 will be a factor throughout, which on being removed gives the more reduced form last obtained. Since the above results hold alike for oblique and rectangular axes, we may deduce corresponding equations in the areal and trilinear system by changing to oblique Cartesians with two sides of the triangle of reference as axes. In the case of the general equation ux2 + vy2 + ivz2 + 2 uyz + 2 vzx + 2 id xy = 0 in areals, we shall find the result corresponding to the first form of equation to be K2 + u — v - w )(x - x)2 4- — • (xy z")2 = 0 with a known notation; while if it be treated independently of Cartesians, we shall get as the condition that the line Ix + my + nz = 0 may touch the conic UZ2 + Y m2 + W n2 + 2U W + 2Y nl + 2 W'zl L 0 where U, Y, W, etc., have their usual meanings ; and replacing Z, m, n by yz - yz, zx — zx, xy ' — xy respectively, we have the- required equation to the pair of tangents in the most reduced form.. A like observation to that made above also applies to these two- forms of equation. ( Issued separately June 5, 1903.) 1902-3.] Mr G. Romanes on Cause of Earth's Internal Heat. 415 Suggestion as to the Cause of the Barth’s Internal Heat. By George Romanes, C.E. Communicated by Dr C. G. Knott. (With a Plate.) (Read January 5, 1903.) During the discussions that have arisen as to the internal heat of the earth, the writer has never seen any reason given for supposing that there was a time when the earth was a highly heated fluid mass, and he believes that view to have originated by analogy from the case of the sun ; and no other cause of the heat seems to be generally assumed than the collisions of the parts that came together to form the earth’s mass. He has expected to find some one maintaining that gradual gravitational compression of the mass was the main source of the earth’s internal heat, but till recently he has never tried to find out if it could possibly be a sufficient cause. It has always seemed to him that the formation of the earth’s mass must have been accomplished under circum- stances so different from the case of the sun that an analogy could scarcely be drawn between the two cases ; indeed, it is obvious that the amount of heat produced by the formation of planets from nebulae will depend principally on their masses, and will be in a higher ratio than that of the masses. The earth must have drawn its substance, with extreme slowness, from a wide ring of nebulous matter circulating round the sun, of which its orbit would be a nearly central line ; and if we may suppose all, or nearly all, of this matter circulating in the same direction, there would be few sufficiently violent collisions to liquefy the matter, until the masses, into which this ring of matter would tend to collect, were so large as to be able to cause very high velocities of approach by reason of their gravitating influence ; or when small masses did happen to collide with a great velocity, there would be little or no chance that they would remain together after collision, and the heat evolved would soon be dissipated by radiation. Now supposing the whole mass forming the earth and moon 416 Proceedings of Royal Society of Edinburgh. [sess. came together as loose nebulous matter before any considerable condensation took place — which seems probable — this matter must have had a motion of rotation about its centre of inertia, the various parts circulating in every conceivable direction, hut the balance of the whole in the plane and direction in which the earth and moon are now rotating. Collisions of small masses would, in this case, be comparatively frequent, and would tend to bring the colliding parts more and more nearly towards the centre of inertia, but the aggregation of the mass would be so slow that the heat caused by these collisions would be readily dissipated by radiation, and not until the central mass began to approximate the size of the moon would it have power to produce such velocities of impact as would liquefy the body striking and the part struck, and such heating would always be superficial, would come to a maximum at once, and would soon be lost by radiation on that account. On the above views as to the formation of the earth there is no need to suppose that the central parts were originally different, either in kind or density, from the other parts. Now, as the average density of the earth is known to be 5*527 times the density of water, and as the density of the rocks on the surface does not average more than half of this, it is obvious that much energy must have been spent in compressing, and therefore in heating, the mass. Assuming, then, that the matter of which the earth’s substance is composed had originally an average density of 2*76,* let us consider what it is likely to be at the different depths now. If the radius be divided into ten equal parts, and the average densities of the corresponding spherical shells be taken successively at 3*3, 4*4, 5*5, 6*6, 7*7, 8*8, 9*9, 11, 12*1, and lastly 13*2 for the central sphere, we shall find that this gives the average density of the whole mass correctly, and we may consider the amount of energy required to compress a cubic foot of matter of density 2*76 to density 13*2 under a final force equal to the pressure to be found at the centre from the foregoing data. The writer has attempted this by means of a graphic process, and has deduced from the above data a curve showing the intensity of gravity at all points of the earth’s radius, and also a curve whose ordinates are proportional to the pressure at each point * This is about the density of granite. 1902-3.] Mr G. Eomanes on Cause of Earth's Internal Heat. 417 resulting from the density and force of gravity jointly at all points above it. An inspection of the curve showing the pressures resulting from this assumed regular increase of density shows that these increase in a faster ratio than the depth for about three- quarters of the distance to the centre, after which the increase goes on more and more slowly to the centre. If the increase in density were adjusted to be more nearly in proportion to the pressure, it would have to commence more slowly than it has been here supposed to do, and this would entail a higher density at the •centre, because the average density must come out 5 '5 27 in any case ; hence the arrangement chosen probably avoids exaggeration of the density and pressure at the centre. The following is the calculation of pressure at the centre : — 20,900,000x 345 292 2 X 153 6880 x 106 lbs. per square foot. The first factor is half the radius of the earth in feet multiplied into the weight at the surface of a cubic foot of density 5* *527, and represents what the pressure in pounds per square foot would be if the density were uniform throughout, and the second factor is the ratio, found by graphic process, between what the assumed regular increase of density would cause the pressure to be and what uniform density would cause it to be. If, now, we assume that each cubic foot has been subjected to half * this final pressure as an average in compressing it from density 2*76 to density 13*2, we shall probably again avoid exaggeration. This gives 6880 x 10° 10*44 13*2 = 2720 x 10° foot-pounds, as the energy ex- pended in compressing and heating each cubic foot (i.e., each mass of 172*5 lbs.)f near the centre. The energy required to raise one cubic foot of water through lbs. Joule’s Equivalent. 1° Tahr. is 62*5 x 772 = 48,250 foot-pounds, which is only about X6, Ijoo1* the en^rgy which, we have found, may have been * If we were dealing with a fluid this would be too high, but with rock the case is different, for the compression will lag very greatly behind the corre- sponding pressure, and the amount of lag will always be decreasing. t This is equivalent to 15,768,000 foot-pounds of energy expended on each pound of matter, which, again, is equivalent to a fall to the surface from over three times the earth’s radius above it. 418 Proceedings of Royal Society of Edinburgh . [sess. expended on every mass of 172J lbs. near the centre of the earth. If the capacity of rock for heat be taken at *21 that of water, then ‘21 x 9 *76 = 96>6QQ° Fahr. 1S ^ie r^se temperature corresponding. with the energy expended in compressing the rock. It is not meant to be implied that the temperature at any given time can be deduced from the above figures, for considering the extreme slowness with which rock yields to pressure — as shown by the slowness of all earth movements — the action must have continued for an indefinite time, and it would tend to cause a continual flow of heat from the centre outwards, which being of necessity excessively slow, a very high temperature may be safely inferred as obtaining from this cause at the centre of the earth. A similar calculation as to the average pressure in the earth 20,900,000 x 345 92 ^ „ gives — 0 x 253=: 2161 x 10° lbs. per square foot, and the energy expended in doubling the density of each cubic 2161 xlO6 1 foot comes out ~ x ^ = 540‘25 x 10b foot-pounds, or about one-fifth as much as may have been expended on an equal mass at the centre. As to the effect of compression in heating, there can be no doubt ; but such a mode of heating could scarcely cause liquefaction — after all cavities were filled up — for it consists of packing the molecules more closely together, which in the case of rock is the reverse of what is required to enable it to liquefy. However, owing to the flow of heat from the centre, the parts nearer the surface ought to be hotter than would be inferred from considering them in the same way as has been done above for the central parts. As bearing on the above views, the following may be considered. The contortions and heating of the metamorphic rocks are probably both simultaneous results of a gradual yielding to pressure under circumstances which prevented the heat getting away except very slowly. Although the heating of these rocks has been very great, they do not appear to have been liquefied, possibly from not having been able to escape from under the pressure when at their hottest. 1902-3.] Mr G. Romanes on Cause of Earth's Internal Heat. 419' The outflow of molten lava from a volcano is not a proof that the rock from which the lava was formed had as high a tempera- ture as the lava, because in escaping from under a vertical pressure of some miles of rock much energy must have been converted into the form of heat in squeezing the more or less plastic rock into the fissure which formed the vent of the volcano. As yet we have not considered the effect of an extensive atmosphere in accelerating the final aggregation of the earth’s mass, and consequently in raising the temperature of the surface- portions. As the small masses of matter, which we have supposed to have originally constituted the nebulous ring from which the earth was formed, could not have held any gas by gravitation, the atmosphere, and probably the ocean also, must have been formed from matter either chemically combined in or occluded in the pores of these nebulous masses, and the heat of collisions was probably the cause of the liberation of the gases from which the atmosphere and probably the ocean also was formed. An atmosphere so formed round the growing earth indicates intense heat to cause its formation, but probably not intense heat of the whole at the same time, as it would be developed very gradually. It would retard and soon bring to the surface all matter that grazed it, and1 so cause a more rapid development of heat and retain it longer than would be the case with a small body like the moon, which appears not to be able to retain any gas on its surface by reason of its comparatively feeble gravitating influence. To trace all the effects of an atmosphere in this way is quite beyond the writer’s power, but he thinks there would be a time when the atmosphere contained an immense amount of water vapour, in consequence of the surface heat, and in proportion as this was the case storms and geological action of all kinds would be correspondingly energetic. He thinks also that although volcanic * action must have had great influence on the gradient of temperature near the surface in many places, yet gradual gravitational contraction of the mass has been the chief cause in determining this gradient, because the heat so developed was greatest nearest the centre, and consequently must * Perhaps volcanic action should he considered as a manifestation of gravitational contraction. 420 Proceedings of Royal Society of Edinburgh. [sess. take an incalculably long time to escape by conduction and radiation. This contraction must have been going on during the whole period of the earth’s existence, and it is impossible to say when the heating effect ceased to balance the loss of heat by conduction and radiation ; or, in other words, when the earth ceased to grow hotter because it was contracting, and began -contracting chiefly because it was growing colder. NOTES ON MR ROMANES’ PAPER BY PROFESSORS A. GRAY AND C. G. KNOTT. In communicating the paper Professor Knott pointed out that Mr Romanes’ problem was a particular case of the general theory first enunciated by Helmholtz that gravitational contraction is necessarily accompanied by evolution of heat sufficient to account for the high temperatures of cosmic bodies. Mr Romanes worked by a process of averages; but with the same assumed law of density the problem could be worked out rigorously. The most -direct way of doing this was to find an expression for the potential of the whole mass of the earth in the given state. For this pur- pose it is not necessary to estimate the pressures at the different -depths ; but when this is done the results are of the same order of -quantity as those obtained by Mr Romanes by his tentative .process. The problem attacked by Mr Romanes suggested another problem which could be easily solved along the same lines — namely, the rate of gravitational contraction necessary in the present state of the earth to produce a given amount of heat. The earth is known to be losing heat annually at a definite rate. What rate of contraction is required to generate an amount of heat equal to the amount that is being lost by radiation 1 It was found that a very slight contraction indeed was sufficient to restrain the cooling ■of the earth due to loss of heat, a contraction which was the fraction of a foot in a thousand years. Subsequently in reporting on the paper Professor Gray carried -out independently calculations which led to results similar to 1902-3.] Notes on Mr G. Romanes Paper. 421 those just indicated. He also found that when Laplace’s^ formula * for the variation of density with depth is used, the results differ but little from those obtained when the linear law of change adopted by Mr Romanes is made the basis of the- calculation. The results of the mathematical discussion are as follows^ With the assumption of the linear law of density, namely— Px = P + c(a-x); where p is the value at the surface, a the earth’s radius, x the- distance of the point considered from the centre, and c a con- stant, which can be determined in terms of the mean and surface- densities, the expression for the exhaustion of potential energy in building up a sphere of radius, a, by matter brought from infinity is — where h is the gravitation constant ; and the rate of change of this quantity per unit decrease of radius, the mass and p being constant, is — With a = 6*367 x 108 cms, p = 2'76, ac= 11*04, *=6*66 x ICR8 in C.G.S. units, the values of E and - dE/da are — = 2*498 x 1039 in ergs, - dE/da = 5*117 x 1030 in ergs per cm. Now the earth loses about 70 calories per sq. cm. in one year, that is 3*57 x 1020 from the whole surface, which is equivalent to- 1*5 x 1028 ergs. Thus a contraction of 1 centimetre would supply the heat radiated in 340 years ; or a contraction of 1 foot would supply the heat radiated in 10,000 years. Professor Gray concludes with these sentences : “ That a contraction of the order of magnitude of this small amount does take place is, in my opinion, not impossible. A great part of the material of the earth is no doubt in a plastic state, but a good deal * Not much importance can be given to Laplace’s law above any other. Wiechert supposes that the earth consists of an iron central portion of nearly uniform density and a radius of about four-fifths of the earth’s radius. See G. Darwin’s paper on the “Theory of the Figure of the Earth” ( Monthly Notices , R.A.S., vol. lx., 189S). dE _ 2t r27r da 315 ' ^(252oV + 130a5cp + 13 a6c2). 0 1 0 422 Proceedings of Royal Society of Edinburgh. [sess. is no doubt also truly solid, and the amount of this is being added to in consequence of the loss of heat from the surface. It seems probable also that any melted rock or other liquid matter which exists contracts on solidifying. If the earth were solid or liquid throughout the contraction due to the lowering of temperature which would be produced by the amount of heat emitted, accord- ing to any possible estimate of the specific heat and coefficient of expansion of the material, would be only a small fraction of that calculated above for a year. But if there is a slow solidification of liquid matter still going on with the much greater contraction of the material which generally accompanies change of state, a con- traction of the requisite amount might very easily be produced. It is to be noticed that such gradual solidification and consequent contraction supposes cooling to take place on the whole ; but the investigation shows that a very slight contraction would be effective in restraining very greatly the fall of temperature in the interior ■of the earth.” {Issued separately June 5, 1903. ) Vol. XXIV. :rnal heat. fc£ g/3. -.aJL S} -Zs-eS7^^ kPreszttr^ Oft, &Gc~■/ 96.309 ■*-s3 */ 366-3 208- S f6,423 s- £3 '? 20ress~r^ cZ. ^ ZS cx^. iSzWJ&„^ Ic'etmZo oZfeg /f£.eZC (PreTeTt-cT&s , . CrT-aovzZy CZtstre^ 3?*^ST3iy y CU/ S/Zy 3-3 J--6Z /9 S*f6 /$■ S-4-6 9-17 S-7'9 Xi *74 2S-2Z6 43-Soz. 3/ -2 S-747 jrl S-67 3//9J- 74-9*7 S9-6 S--6/o ^r £ f-4s 3S-9YO 770-9/7 9Z-o S-3oy /••r S--/o 39 270 7S0-ZZ/ 730-6 A- 939 f’S 4- ft 37- 773 /f 0-003 770-/ 6-207 99 3-d/ 37 7"? 227-721. Zod-d: 3-4/0 f/ 2 93 32-23 O 23?- 9 32 243-d Z-44-Y '*' /■ S? 32- ?£? Z32-S2/ Z7/-4 7-320 /3-Z ' o-6 6 ’ ?• 7/Z ZT7-1 SfaZ£f>^ar~. - Z?/-S23 j ^2£2) O'OOO S-sz7 x a?T z-7C3?= /sz-76 (^//— ) ColZcssZtz^ppzs of ^eszst&f/ &si*Z fb^^&jficssc^re. 077, yyfo2e,JfcazSf of Jfizrifcs 2)msity ?/““* 5%^? m JfcoomsZeyff -& JftxxSS &£as ^Jpresscore, tfTlssZf jbrof? orZcono0 -Zrfcxss y-2SosJz'-T& 3-3 277 $9/7-3 9-27 $,290 4-4 1/7 9S4-V J7 2 29,799 S-S •7-7’ 7759 9*9-0 S9-S SS272 6- 6 77-6’ 727 93S-Z 93-0 779/2. 7-7 ‘ff 700- y 730-6 97,0-77 t-* 67 S3 6 ■ S’ 770/ 97,309 9-9 a~7 366-3 20 8- 3 f6,6S3 7/ 79 209-0 263-2 SO, 9/4 72/ '? Z4--7 277-S 22,937 73-2 / 73 -2 ZSfZ 3.797 SS27S SVS, Z7S .. ■ e?/.&S /ze/0Z') C&GU. ' /ooo rffAts J-fZy-S' f/^Tf £~' &&Y S~ Jfrl' -&£z t&ts**-**^ yLt~4£& yy ^ ~Ci&> £Z%?~ ^ «2

of his kind offer, but owing to the size of the animal, the weight of the head, the difficulty of taking a steamer into the Yoe where the whale was lying, and the storms of the winter, it was not possible to secure more than the lower jaw, the teeth, and the tympano-petrous bones. In flensing the carcase the point of a massive explosive harpoon was found imbedded in the head of the whale, and had probably been the cause of death. The harpoon fiad penetrated the great chamber for the lodgment of the spermaceti, which, in consequence, had to a large extent drained away, so that the captors obtained only a small quantity of this valuable fat. The blubber yielded about 450 gallons of oil. 424 Proceedings of Royal Society of Edinburgh . [sess. When the stomach was opened, numerous beaks of cuttlefish were found in it. But in addition a quantity of fish-hooks of various sizes were observed, ranging from those used for catching haddocks to the largest size made for use in Shetland and the Faroe- Islands for the capture of halibut. It would seem, therefore, as if the animal had also lived on fish, and had cleared the fishermen’s lines, and swallowed the fish and the hooks to which they were attached. Mr Anderson writes me that a careful search in the intestine of the Shetland specimen for ambergris was made, but without success. He states that the fishermen told him that in flensing the whale they found hair in places rooted in the skin ; but as no portion of the skin was preserved, neither he nor I was able to put this statement to the test of observation. The lower jaw enables one to form a good idea of the magnitude of the animal, and in the following table I have placed side by side measurements of the Shetland specimen, one caught some years ago in Loch Scavaig, Isle of Skye, now in the Anatomical Museum of the University, and one in the Museum of Science and Art, Edinburgh. 1 Shetland. Skye. Museum of Science and Art. Length of mandible, Ft. In. In. 16 3 (195) In. 1904 In. 196 Length of symphysis, 10 3 (123) 116 120 Width between inner border of condyls, 5 i (604) Width between outer border of condyls, .... 6 7 (79) Greatest girth of ramus, . 4 9 (57) 56 54 The length of the mandible in Physeter macroc&phalus was taken in the way adopted by Sir William H. Flower. 2 A lino 1 See my paper in Proc. Boy. Soc. Eclin ., vol. vii. p. 635. 2 “Osteology of the Cachalot or Sperm Whale,” Trans. Zool. Soc. Lond.y vol. vi. part vi. p. 320, 1868. 1902-3.] Sir William Turner on the Sperm Whale. 425 was drawn between the backs of the two condyls, and from its mid-point a straight line was drawn to the tip of the jaw. It should be stated that the jaw had probably been two inches longer than the measurement obtained, as the most anterior tooth socket on each side had been broken across. When these measurements are compared with those given by Sir William Flower in his account of the mandibles of sperm whales in the Museum of the Royal College of Surgeons of England, it will be seen that the Edinburgh specimens compare favourably with the largest mandible from Tasmania in that Museum, and show that they were from animals which had reached adult life and were of great magnitude. Sixty-four teeth were sent to me. Of these, forty- two were undoubtedly mandibular teeth, from their conical shape, the size and depth of the pulp cavity, and the polished, partially worn and somewhat flattened surface of the summit of the crown. The longest of these teeth was 193 mm. (7 '6 in.), and the one with the greatest circumference was 200 mm. (almost 8 in. in girth). The shortest tooth which showed evidence of being worn was 109 mm. (4’3 in.) in length and only 80 mm. (3*3 in.) in circumference. Of the remaining twenty-two teeth it was difficult to say definitely whether seven were or were not mandibular, though probably some of them were; they varied from 104 mm. to 85 mm., and the greatest circumference was 77 mm. Five had shallow pulp cavities, but in the two others the cavity was obliterated by dense tooth tissue, perhaps crusta petrosa. No specimen had the summit of the crown polished or worn, but the tip was roughened and somewhat jagged. Obviously they had never cut the gum or been subjected to friction ; if mandibular, they had been at the hinder end of the dentary arcade, where the dental groove in the jaw is shallow and the sockets for the teeth are comparatively faintly marked. The remaining fifteen teeth had in part, if not altogether, belonged to the upper jaw. They varied considerably in size and form ; eight of the smallest teeth were curiously bent, and these, Mr Anderson states in one of his letters, were found in the upper jaw. Three were attenuated in form, pointed at opposite ends, and PROC. ROY. SOC. EDIN. — YOL. XXI Y. 28 426 Proceedings of Royal Society of Edinburgh. [sess. Rudimentary Teeth of Sperm Whale — (natural size). [For these and the other Drawings I am indebted to Miss Anna Dowden.] 1902-3.] Sir William Turner on the Sperm Whale. 427 varied in length from 80 to 69 mm. The remaining four were pointed at the crown, but were broadened and flattened at the fang ; the longest of these teeth was 88 mm., the shortest was 64 mm., and the broadest part of the fang ranged from 22 to 32 mm. in different teeth. In five of these teeth a shallow pulp cavity was present, but in the others it was obliterated by hard dense tooth substances, wdiich was frequently irregularly tuberculated. The crowns in four teeth were pointed at the tip, but in the others the tip was roughened and somewhat jagged. In no speci- men was the crowm polished or worn, and the presumption is that they had not cut the gum. The bent teeth were most interesting in their form. The largest specimens were curved to about \ of a circle, the two smaller to about ^ a circle, but owing to a twist in the tooth the tips of the crown and fang were not in the same plane. Several had odontomatous excrescences on the concave aspect of the tooth, resembling the maxillary tooth figured by Sir W. H. Flower. These teeth were obviously rudimentary and functionless. The sockets of the teeth were elongated antero-posteriorly. Those which had the largest teeth were 8 inches in length and about 3 \ inches in greatest breadth. In each alveolus the hinder end was the deepest part, from which it gradually shallowed forward. The fang of the tooth was lodged in the deep posterior end. The teeth in the two halves of the mandible behind the 8th pair were not set directly opposite to each other, and for some distance backwards a tooth on one side was opposite the interval between two teeth on the other side, so that the teeth were not symmetrically arranged ; the dental formula, judging from the alveoli, was 23 in the right and 22 in the left half of the mandible. In two previous communications which I have made to this Society on the occurrence of the sperm whale in the Scottish seas, I referred to all the cases of which I had been able to find a reference.1 As the specimen now described gives an additional example, I append the following table, so as to make the record complete. 1 Proc. Roy. Soc. Edin . , Feb. 6, 1871, and Jan. 29, 1872. vol. vii. 428 Proceedings of Royal Society of Edinburgh. Locality. Date. Authority. Hoxay, Orkney. Limekilns. . . . Cramond. . . . Monifieth. Ross-shire. . Cramond. . . . Hoy Sound, Orkney. Oban Thurso Loch Scavaig, Skye. Roeness Yoe, Shetland 9th or 10th cent. ? George Petrie. February, 1689. . Sir R. Sibbald. 1701. . . James Paterson. February, 1703. . Sir R. Sibbald. 1756. . . Sir W. Jardine. 1769. . . James Robertson. About 1800. . . George Low. May 1829. . . Sir William Turner. July 1863. . . J. E. Gray and Sir W. H. Flower. July 1871. . . Sir William Turner. August 1901. . Sir William Turner. It is well known that the sperm whale in its customary habitat moves about as a rule in herds or ‘ schools.5 The specimens captured on the coasts of Scotland have, on the other hand, been solitary animals. The majority of these were males and of great size, a fact which supports the statement made by Mr Thomas Beale that the full grown males go singly in search of food. In connection with this record of the occurrence of the sperm whale in the Scottish seas, it will be of interest to note the cases in which specimens have been observed to the north of Scotland, or on the opposite coast of Scandinavia. Professor Gustav Guldberg of Christiania, in a recent publication,1 has collected evidence of the capture of several examples of this whale in the North Sea. Thus, in 1770, one fifty-two feet long was stranded on the small island of Hjarno in Horsens Fjord ; bones of another specimen were found at an uncertain date on the island Lesso, in the Cattegat. Prof. Collett states that in 1780 one was obtained on the west coast of Norway, in Sond Fjord, and another in 1849 near the island of Smoelen, off Christiansund. Professor Sars mentions that in the summer of 1865 one was got as far north as the Lofoden Islands, within the Arctic Circle. The Bergen Museum obtained in 1888 the tooth of a sperm whale found in the sand on the coast of Jaderens, in the south-west of Norway. Guldberg also records two specimens seen in 1895 ; one, an old solitary male, 19 metres long, was caught in the neighbourhood of the North-West Coast Islands, and its skeleton is now in the 1 Nyt Magazin f. Naturvidenskab , B. 39, H. 4. I am indebted to Prof. Guldberg for a copy of this memoir. 1902-3.] Sir William Turner on the Sperm Whale. 429 Tonsberg Museum. The other, one of a herd of four sperm whales, was captured off the Faroe Islands. It was a male, twenty metres long, obviously full grown, and its skeleton is in the Museum at Copenhagen. In the early summer of 1896 a herd of seven was seen near the coast of East Finmark, and of these two were captured; one was a young male 12*8 metres long, and its skeleton is in the Natural History Museum at Berlin ; the other, a female, was 10 metres long. A third specimen was taken in the same summer in Baadsfjord ; it was a male, 15 metres long, and the skeleton is in the Bergen Museum. In 1899 a herd of sperm whales was seen in the neighbourhood of the Faroes, but they escaped capture. In the summer of 1901 a sperm whale between 60 and 70 feet long was seen north-east of the Faroe Islands, and after an exciting chase was captured. The sex is not stated, but from its length it was probably a male. In August 1901 Captain Albert Gron observed a herd of about ten sperm whales in the neighbourhood of the Faroes, one of which was harpooned, but escaped ; it is not unlikely that the Shetland specimen, which had been struck by a harpoon, and was found floating dead during the same month, was this animal. From these examples it is evident that the seas to the north and east of Shetland have of late years been frequented by the sperm whale in considerable numbers. The presence of quantities of beaks of cuttlefish in the stomach of the specimen from Shetland corroborates the observations made of late years that sperm whales live largely on Cephalopoda. But the hooks found in considerable numbers in its stomach point also to a fish diet. Mr Frank T. Bullen, in the Cruise of the Cachalot , states that during the cutting up of a sperm whale, in addition to dismembered squid of large size, a number of fish, such as rock cod, barracouta, sehnapper, and the like, were found in its stomach. Professor Guldberg relates that in the stomach of the old male captured in 1895, in addition to the remains of cuttlefish, a portion of the spine of a large fish, a fish-hook and some stones, there was about a square foot of the skin of a seal, with its hairs and four of its claws. In the stomach of the young male caught in 1896 off East Finmark a gelatinous and cartilaginous mass several feet long was found. It consisted of the half digested portions of a 430 Proceedings of Royal Society of Edinburgh. [sess. cartilaginous spinal column, which from its magnitude had evidently been a part of a large cartilaginous fish, possibly, accord- ing to Professor Collett, Selache maxima. These observations extend very materially our knowledge of the range of the food of the sperm whale, as, in addition to cephalopoda and smaller species of fish, the animal apparently at times attacks and devours large cartilaginous fish and seals. It is difficult to ex- plain the purpose which was served by the stones found in the stomach of one of the above specimens, hut I should state that some species of seals are in the habit of swallowing stones, and afterwards of ejecting them, a habit which has led the seal hunters to speak; of the stomach in which stones are found as the ballast bag. As the stones were present in the stomach of the same sperm whale in which the remains of the seal were found, it is possible that they had formed part of the contents of the stomach of the seal which the whale had eaten. It is interesting to note that Professor Benham, in his account 1 of the stomach of a full grown specimen of the small species of cachalot Cogia (. Kogia ) breviceps , stranded near Dunedin in August 1900, found a great quantity of cuttle beaks, lenses of eyes, the remains of the pens of some Loligo-like species, and some partially digested red membranes with horny, conical teeth-like structures growing from thick white patches, recalling gizzard teeth of Aplysia. Tymjpano-'petrous Bones of Physeter, Kogia , and other Odontoceti. As the tympano-petrous bones had been sent to me in good order, and as, through the courtesy of the Superintendent of the University Museum of Zoology, Cambridge, S. F. Harmer, Esq., I had the opportunity of examining the corresponding bones in Kogia , the small sperm whale of the southern hemisphere,2 it may he useful to give a description of these bones in Physeter and Kogia , and contrast them with those of other Odontoceti, In Physeter the tympanic bulla was 62 mm. long and 48 mm. in greatest breadth, though, if the ridge on the outer surface be 1 Proc. Zool . Socr Lond., May 21, 1901, vol. ii. 2 Mr Harmer tells me that these bones of Kogia are from the New Zealand specimen, the capture of which is recorded by Professor Benham in Proc. Zool. Soc. Lond., May 21, 1901. 431 1902-3.] Sir William Turner on the Sperm Whale. included, the breadth would be increased to 52 mm. From its posterior end an irregular, strongly denticulated piece of bone, which was closely fused by a broad base with the bulla, projected backwards as a pair of fork-like processes; between the two forks, and in relation to their deep aspect, was a furrowed surface, convex from before backwards, and concave from side to side, with which the posterior end of the petrous-temporal articulated. In the bulla itself the inferior surface was bilobed at and near the posterior end, and the outer lobe was larger and projected farther back than the inner; a distinct but not a deep groove, which extended obliquely across a part of this surface, marked the separa- tion between two lobes ; this surface was smooth in the greater part of its extent. The outer surface was also smooth, and ended in a sharp border which bounded the mouth of the tympanic bulla. A strong ridge sprang from this surface close to its posterior end, 432 Proceedings of Royal Society of Edinburgh. [suss. and formed a well marked convex projection in the transverse plane of the bulla, which was continued by one end into the inferior surface of the outer lobe of the bulla, and by the other it reached its tympanic mouth • a deep fissure separated it from the outer limb of the fork of the denticulated process. The inner surface of the bulla was continued into the thick rounded border of the tympanic mouth, and was divided by a deep depression into an anterior and a posterior lobe. The anterior end of the bulla opened freely into the cavity, and was bounded by an arched border, the piers of which were thickened. The petrous bone was 65 mm. long and 39 mm. in greatest breadth. It articulated between the limbs of the fork by a pos- terior process, having a concavo-convex curve. Its tympanic wall had two foramina, one much larger than the other, but none of the tympanic ossicles had been preserved. The internal meatus was large and its walls were cribriform for the transmission of the branches of the auditory nerve. In Kogia an irregularly shaped mass of bone formed a well marked projection, marked with a reticulated arrangement of 1902-3.] Sir William Turner on the Sperm Whale. 433 delicate furrows, situated behind and somewhat overhanging the tympanic bulla. It was so light in weight that it obviously was composed of very cancellous tissue. From its position and size it was possibly a mastoid, but the absence of the skull prevented me from precisely localising it. The long diameter of the bulla was 30 mm. and its greatest breadth was 18 mm. It was united at its posterior end to the possible mastoid element by a very Kight and left Tympanic Bullse and Petrous Bone — Kogia — (natural size). constricted neck. The posterior end consisted of two lobes sepa- rated by a shallow groove ; the outer was the longer, and projected towards but did not reach a depression in the possible mastoid. The inferior surface was smooth, and a faint furrow, continuous with the groove between the two lobes, passed obliquely across it on to the outer surface of the bulla. Kogia did-not have a definite ridge on the outer surface similar to Physeter and most other Odontocetes, though a scarcely perceptible elevation on one side 434 Proceedings of Royal Society of Edinburgh. [sess.. of the faint furrow apparently represented the ridge : this surface ended in a sharp border which bounded the mouth of the bulla. The inner surface curved to the opposite side of the mouth, formed a rounded obtuse border, which was divided by a shallow de- pression into an anterior and a posterior lobe. The anterior end of the tympanic opened into the cavity of the bulla and was bounded by a strongly curved smooth border. The petrous bone articulated with the mastoid-like element close to the constricted neck by a smooth plate of bone. The petrosal was 28 mm. long and 19 mm. in greatest breadth. Two distinct foramina were in the wall which bounded internally the cavity of the tympanum, but the stapes was not attached to the boundary of the foramen ovale. The internal auditory meatus was distinct, and its wall was cribriform for the transmission of the branches of the auditory nerve. For a number of years I have collected, as opportunities offered, the tympanic and petrous bones of the Cetacea, and I have presented the specimens to the Anatomical Museum of the University. In supplement to the description of their characters in the Physeterinae, I may record the measurements and state some of the most salient points connected with these bones in the Odontoceti now (with the exception of Kogia) in the Collection. Tympanic. Petrosal. Length. Breadth. Length. Breadth. Physeter macrocephalus, . mm. 62 mm. 48 mm. 65 mm. 39 Kogia breviceps, 30 18 28 19 Hyperoodon rostratus, . 57 43 64 44 Ziphius cavirostris, Mesoplodon bidens (Sowerby), 59 38 58 36 45 31 46 25 ,, layardi, Platanista. gangetica, 45 32 48 34 54 26 Monodon monoceros, 51 27 55 28 Phocoena communis, 29 18 30 14 Globicephalus melas, 48 33 45 30 ,, macrorhynchus, 45 22 40 29 Grampus griseus, . Lagenorhynchus albirostris, . 43 24 39 26 34 21 27 21 clanculus, 28 17 31 22 Delphinus delpbis, . 31 18 29 19 ,, tursio, . 39 25 42 24 1902-3.] Sir William Turner on the Sperm Whale. 435 It will be observed that it is the rule for the tympanic bulla to be both longer and broader than the petrosal bone, but in Hyperoodon the contrary is the case in a marked degree. The bones differ in size in the several species clanculus, Phocoena , D. delphis , and Kogia ; the smallest species have the smallest bones, and they attain their greatest magnitude in Physeter, Ziphius , and Hyperoodon. But notwithstanding the huge bulk of the sperm whale, its tympano-petrous bones are only slightly larger than those of the much smaller Hyperoodon , and are much less than those of the whalebone whales ; even Balcenoptera rostrata , the smallest of our northern species of Mystacoceti, though scarcely half the bulk of the sperm whale, exceeds it in the size of its tympanic bullse. In the Ziphioid genera Hyperoodon and Mesoplodon , which are the nearest allies of the Physeterinse, the tympanic bulla is bilobed posteriorly, the outer lobe is larger than the inner, and the inter- mediate groove is relatively shallow, as in Kogia and Physeter. In Ziphius , on the other hand, which gives its name to the group, the posterior end of the bulla is not bilobed, but an imperfect groove is present at the posterior part of the inferior surface of the bulla ; and although it marks off a definite lobe on the outer side, the bone immediately internal to the groove is not prolonged into a lobe, and the bulla differs therefore materially in its character from that of Mesoplodon ,l In Platanista the bulla is bilobed, and as usual the outer lobe is larger than the inner. A broad groove separates the lobes and extends to near the anterior end of the inferior surface, which is attenuated to a fine point. About the middle of this groove a roughened ridge, 18 mm. long, divides it into two parts. In the Delphinidse the bilobed character of the bulla is strongly marked in Monodon , Phocoena , Globicephalus melas and macrorhynchusy Grampus griseus , Lagenorliynchus albirostris and clanculus , Delphinus delphis and tursio : and in these specimens the groove between the lobes, more especially in Monodon , is much deeper 1 I have given figures of the tympano-petrous bones of Ziphius and Mesoplodon in my Memoir on the Cetacea of the Challenger Expedition, Reports, Zoology, part iv. , 1880 ; and in the same Memoir I have figured a tympanic bulla dredged from a depth of 2335 fathoms, which was probably that of Kogia. 436 Proceedings of Royal Society of Edinburgh. [sess. than in the Ziphioid genera, in the Physeterinoe and Platanista, a character which differentiates the Dolphins from the other groups. The position and appearance of the ridge on the outer surface of the bulla are also of importance. In Physeter it is noteworthy from its size, projection, situation near the posterior end of the bulla, and extension on to the inferior surface, where it is continued into the outer lobe. In Kogia , on the other hand, it is absent. In Hyperoodon and Mesoplodon it is not so prominent as in Pliyseter , and does not quite reach the inferior surface, and is not so near the posterior end of the bulla. In Zipliius , again, it projects slightly, and is limited to that part of the outer surface which is in proximity to the sharp outer border of the mouth of the tympanic bulla. In Plalanista also it has only a feeble projection, is near the mouth of the bulla, and about midway between the anterior and posterior ends of the surface. In the Delphinidse specified in the preceding list it is also placed approximately near the middle of the outer surface, is not very projecting, and is limited to the outer surface, where it lies close to the sharp outer border of the mouth of the bulla. The petrous bone in Kogia, Hyperoodon, Mesoplodon, and Zipliius has a smooth plate-like surface of articulation, and is adapted to the tympanic by the apposition of two plane surfaces ; but in Physeter and the Dolphins as is well seen in Globicephalus, the articular surfaces are ridged and furrowed, and are adapted to each other somewhat like the denticulated margins of the bones in the vault of the cranium. ( Issued separately June 5, 1903.) 1902-3.] Mr A. W. Brown on Young Scales of the Cod. 437 Some Observations on the Young Scales of the Cod, Haddock and Whiting before Shedding. By Mr Alex. Wallace Brown, St Andrews. Communicated by Dr A. T. Masterman. (Read May 18, 1903.) During the winter of 1902-3, I conducted observations upon the scales and their condition, in several of the gadoid fishes. Investigation was commenced in October 1902; but it was not until the beginning of March 1903 that the first appearance of the young scale took place. In stained specimens, it can be recognised as a deeply staining “ nucleus,” lying beneath the old scale, just under its centre. Such an appearance was found in cod, haddock and whiting of all ages from one to three or four years ; and, in all, the young scale is clearly recognisable, underlying the old. As soon as these fishes have spawned, they appear to shed their scales, the epidermis first peeling off. An examination of a few large haddocks, 8 lbs. weight and over twenty-seven inches in length, showed that in January the ovary was black, shrunken, and not in spawning condition. I am inclined to think that these fish are past the age for spawning. I examined very carefully this class of haddock right on till April. In every case I found that the scales showed evidences of hard wear, and in some cases were frayed. In these fishes no trace of the replacing scales were found ; and the probable conclusion is that no further shedding of the scales takes place after the close of the reproductive period. It has been suggested that the annual rings of growth may be traced upon the gadoid scales ; but I find that upon the cod, haddock, whiting, green cod and pollack, of one to three years in age, scales may be obtained from different parts of the body showing 90, 60 or 30 rings, according to the part selected. 438 Proceedings of Royal Society of Edinburgh. [sess. I have been enabled to trace back the first appearance of the new scale to the month of February, when it may be recognised as a dark tip growing upon a small papilla. By the middle of April, the epidermis on the head commences to peel off, and, probably somewhat later, over the body. The details of this process will have to be followed in sections ; but sufficient evidence is to hand to make it probable (1) that gadoid fishes shed their scales immediately after spawning ; (2) that after the age limit of spawning is reached no further shedding of scales takes place ; (3) that the concentric rings of the scales do not represent annual increments, but must have other causes. ( Issued separately August 3, 1903.) 1902—3.] On the Series y = 1 + F([a] [/3] [y]) . p-j + , etc. 439 On the Series y = 1 + F([a][/i] [y]) • ppr, + F([a] [£] [y]) • xm F([a][/^][y+ l])r9jj + • • • * and its Differential Equa- tion. By the Rev. F. H. Jackson. (Read June 1, 1903). (i.) F([a] [/?] [y ]) denotes the convergent infinite product T (py~a - 1) (py -a+1 ■a- P + 1 1) (pt-a+K-1 _ 1) . (py-P- 1) (py-fi+1- 1) . . - 1) . . (py-*- P+«~ 1 - 1) . (py - 1) ( py+ 1 - 1) . . . . . . (py~P+K~1 - 1) . . . (^+*-1-1) * 1 which may be expanded in an infinite series p > 1 1 + py-*-P pa - 1 ’ pP—1 p -1 - py- l + p2(y-a-/3) p°- - 1 • jpct+1 — 1 ‘ 790 — 1 * P&+1 - 1 p - 1 • p2 - 1 ■ py - l ■ py+1 - 1 p > 1 When p = 1 the series reduces to the particular hypergeometric series F(a/3yl), and the infinite product reduces to r(y-q — ff)r(y) _ T(y-a)T(y- {3) The object of this paper is to obtain the differential equation which has for a solution the series 2/ = l + F([a][/3][y])-g + F(H M W) • F(W M [r + 1]) • jg + . . (1) X < 1 P > 1 orifp=l y-a-/3>0 Proc. Lond . Math. Soc., vol. xxviii. p. 475. 440 Proceedings of Royal Society of Edinburgh. the general term being [sess. F(fa][/J]M) • F(HM[y + 1]) .... F([«][ffl[y + r]) • (2.) The following notation will be employed : w denotes Pn~ 1 V -1 [*]• p - \ ' p1 - 1 ’ p3 - 1 . . . . p1 — 1 5) ( p-lf In 1 Pn — 1 ■ p"-1 —1 ■ pn-2 - 1 . . . pn-r+l- 1 . l r J 5? P ~ 1 ‘ P1 — 1 ’ P3 — 1 ... pr - 1 . { a l „ L K = CO pa-n+l _ ^ • pa- n-\- 2 _ ^ pa — n-\-K ^ ’ pn~ FI --- ^ l » J pa-\- 1 — ^ • pa+2 _ ^ Ps r— 1 1 + e • 1 rv\R'~TK. 1 P 1 .... p 1 vn(a-n) ■ p* - i p* - 1 1 It is assumed that px when x is not integral means its absolute- value. The four infinite products in is finite single valued,, etc., unless a is a negative integer. In the exceptional cases ( a l is infinite or indeterminate. When n is a positive integer reduces to pa ][ ' pa ~ 1 1 • pa~^ 1 pa — n-\- 1 ^ p — 1 ‘p 2 - 1 p3 - 1 .... pn - 1 / a ( \ n f DW denotes the operator d d(xrr~1) I d(xPr-*) ' ' d f mm I d{x&) I d(xP) (dxl f / ■ (3> fY> tf] 1902-3.] On the Series y— 1 + F([a] [/3] [7]) . p-p + , G^G- 441 The following theorems will be required in subsequent work : j ct -j- h | 1 n j = 0 -+r)i « | n- r}{ b {* r j 00 = 2^ 0 pr(a-n+r) | a (. n-r ) PIP -1]P“2]. W! . . . \b-r + 1] J . . (4) P> 1 f a ) _ / a \ . pn - l PH -1 - 1 . . . . jprc-r+1 _ J } n-r j 1 n ) p&-n-\- 1 _ l * Pa -»+ 2 - 1 . . . pd-n+r _ ^ This can be proved by substituting for j ^ | and j n ^ r j the infinite products which they represent. (3.) d If we operate on x with the operator we obtain ~ !] d d{xP) [m] [m - d-+i] 0 which may be written [m] ^pr(b-n+r) j ^ | | m“ * | which is /jjjp[i»-i] [w] . | ^ | If y = A^M + A2«[7W2l + A3«[m3l + . . . . we shall have ™[r] b I ^[r]n 2^-B+r) {n-r)wm)-y- ±p-^ \n : r } 0 0 - Ajfmj] | !l + nh~1 J.a;P[mi-i]-A2[m2]| b + vh~1 J. j-Mwa-U (5) It is not possible to choose values of m19 m2 ... . •A-l ? -^-2 • • • • to make the right side of equation (5) identically zero ; but if we choose m1 = m2 — 1 m2 = m3 - 1 mr = mr+ 1 — 1 Ar+iK+i] j b + mr^ " 1 } = A, { “ + "* } ... .(6) the series on the right become A ia + mA a/mj + Ai|“ + ffli+1| «^+« + . . . . 1 { w j 2 ( n I — A2[m1 + 1] [ | — Ag[/7'1 + 2 ^ "'l ^ | xA“.+i] - . . d1] 1902-3.] On the Series y = l+F([a][/3][y]) • pj, + , etc. 443 In these series p[m1 — 1] is the lowest exponent and occurs only in a single term ; then, since is arbitrary, denote it by A (not zero) and choose ml so that the coefficient of viz. r i f b -+■ TYi-t — 1 { p. W{ n f=0 which is pmi - 1 y pb+ml-n_\ m p — 1 X 1 , pb+m i — 1 . K= CO therefore . pb+m i - n+K - 1 _ 1 . pn+1 _ J -tb+m.+K-l 1 • pl - 1 . . . . pn+K - 1 . . . pK - 1 pn(a -n) — 0 mY— 0 m1 = n — b mx — n — b - 1 m1—n — b - k - 1 make zero factors in the numerator, and are possible values of m1 , when n is not a positive integer. If, however, n be integral, K] b + m1 - 1 ) n | reduces to pm1 _ 1 pb+ml - 1 _ | . pb+m1 - 2 _ ] p —Ip — 1 * p1 — 1 . . and the possible values of m1 are 0 1 -b •2 - b . pb+ml-n _ J . . . pn - 1 n - b If we use the value m1 — 0 , the relation (6) shows that the series V is y ■ • (O and the differential equation of which this is a solution is 444 Proceedings of Royal Society of Edinburgh. [sess. { „ ! 4 FTi DW • 9 - Zpr(b-n+r) { n-r\ B {■' l i n ) 1 + j a + 1 1 n b t [T]T n) a + l ( I n ( ] i a + 2 ) 1 n j n i f b +■ 1 ( 1 m 1 x[2] + . - ia I n 1 + i"‘[ 1 a -r 1 ) n m :a:3} 14 < 1 6+1 1 1 « ) ‘xPW in w W :Pl 2] ol! + • • ] (8) When p = 1 the right side of the equation identically vanishes, being of the form f{x) -/( x ?) . (4.) i « i I n j The function yjT\ Can *’C transf°rme(l into F([«] [/?] [y]) by 1 n \ any of the four following substitutions : (1) y - a - (3 - 1 7-0-1 -0 (V 7-1 y-p-i a (3) y-a-fi-l 7 - a — 1 — a (4) 7-1 y — a — 1 / 3 and since a l J « i n-r ) j n I pn - 1 • p71-1 - 1 • pn~2 - 1 ... . pn~r+ 1 - 1 ( b I ( b I J,a~n + l- 1 . pa- »■+* — 1 pCt - n+r _ | i n J \n] Substitution (1) transforms this expression into 1902-3.] On the Series y — 1 + F((a] [5] [y]) . ^ + , etc. 445 . m/i+lW + 2\ r/8 + r-ll [y - a] [y — a + 1 ] [y - a, + 2] . . . . [y — a + r — 1] and the differential equation takes the form F(w m m) { 2( - [|Sfe .. [/3 + r- 1] a?[r] . . . [y - a + r- 1] [r]\ ,.,[7+r-l] [r]! [y-a]ty-a+ 1J . . D« -y\ - V ( - 1 '•jpKr-ffi- 9 — + !]• [y] [7 + 1] • • 3™ A \ 1 + F([a — 1] [/3] [7]) * [jjj F([- - 1] [jSf] M) - F([- - 13 D3] [y + xm - A -J 1 + F([a- 1] [/?] [y]) • + The second substitution gives (9) -a)+ r.r-l["a] \a ~ LJL r-r - 1 [a] [a - 1] .... . [a — r + 1] [y - a] [y - a + 1 ] . . . [y - a + 7 1] . . . [a - r + 1] X^r ' py [y — a — /3] . .. [y-a-/? + r-l] 1] ' [r] D(-+iV xW ) FI"' } = A 'j 1 + F([tt] [/? + 1] [y+ 1]) • j-jj, + F([«]DS + l][y+ 1] • F([«] D8+ 1] [y + 2] • g + . . . } — A | 1 + F([a] [/3 + 1] [y + 1]) | (10) Substitutions (3) and (4) simply interchange a and ft in the two equations given above. These differential equations have a solution ™[1] r[2] ?/=l + F([a][fl[y]) . [-IJ, + F([a]M[y])F([a]M[y+l].p]! + If we make any of the following substitutions, 446 Proceedings of Royal Society of Edinburgh. a b n (1) T-/9-1 y - a- P — \ ~P (2) y-/S-l 7-1 a (3) y - a — 1 1 P 1 1 — a (+) y — a — 1 7-1 P *h“ and the equation takes the forms F(W[)8][y] I 2( >' 2 [y] [y + l][y + 2] . . Ly + r - 1] [»■]! D< ’y J „ nr-i[S][S+l][/3 + 2]... f/J + r — 1] xrt'i , „ -2(-1)>^ — ®- 2 [y - a] [y - a +1 ] . . . [y - a + r - 1 ] • W V -M 1 + 1 I A 1 + F([a+l][/5][7+l]) [1]! ^ f 1 XPl i] r T F([a+l]M[y+l]) [1]! + (12) F(HMH ( r_r- 1 fal la - 11 [a - 21 fa-r+11 „ ) | ZP'iy + 2 [y-a-/3][y-a-/3+. 1] . . [y- a - ft + r - 1] ‘ [r]!0^ } 2>ry+ ’ 2 = A < 1 + 1[aira-l][a-2] . . . a-r+1] _ &*rl b-flb-0 + ih .[y-P+r- 1] [r]l 1 Df+B F([a]D»-l][y] P]i + . . . 1 + 1 F([a][/?-l][y]) [1] xpW ) n- «131 and two others obtained by interchanging a and (3 in the above. 447 T[l] 1902-3.] 0^ the Series y = 1 + F([a] [/3] [y]) . |-yj + , etc. (5.) If p = 1, equations (9) and (10) reduce to , p.fi+ip+2 . . . fi + r-l F(a^7)-2(-1' a. y — a + 1 . . . y — a + r — 1 r dr+12/ - V / _ IV /?♦/?+ 1-^ + 2 . . . /3 + r- 1 af ^ y. y+l.y + 2 . . ,y + r— 1 r! flter+1 r! = 0... (14) PY ,-^a. a— l.a-2 ... a — r + 1 xr dr y \afiy)2-iy — a, y — -- 456 Proceedings of Royal Society of Edinburgh. [skss. proximate weight having been already placed in the pan, the agate is raised and the calorimeter swings free of the mercury cups. The weight is then noted to the nearest ^ centigm. The agate is then lowered, a 50-gm. weight removed from the pan, and the 1 » i right-hand hook sinks into the rest and the electrodes into the mercury cups. The current has been meanwhile flowing in an alternative circuit of equal resistance, indicated in the diagram as connected to the two-way switch. 1902-3.] Determining Latent Heat of Evaporation. 457 Evaporation. — At an observed time the current is switched into the calorimeter. Observations of the current are now taken every minute. This is best done by keeping the galvanometer key always closed (if only a small current is flowing through the galvanometer). The deflection on the galvanometer scale corre- sponding to one division on the potentiometer dial is found once for all, and the actual deflection at the minute used as a correction on the dial reading. With a current varying uniformly in one direction, it is not difficult to anticipate the correct dial position, so that the correction may be often zero. No attempt is made to keep the current constant by altering the series resistances during ■an experiment. Simultaneously with these readings it is necessary to check the balance for the Clark from time to time, say every five minutes. When the point of balance for the Clark is not that originally arranged (that corresponding to the temperature), no attempt is made to restore the original balance, but the point of balance merely recorded and allowed for in calculating the current. The variation in the reading is never greater than that corresponding to ’0003 volt. The time of switching off is noted. Second Weighing. — The probable weight has been approximately arranged, and, (the agate being raised), as soon as evaporation has entirely ceased (the balance become steady), the exact weight is noted as before. Advantages of the Method. 1. In nearly all methods the total heat is measured and the latent heat deduced from an assumption of the mean specific heat. In this method the latent heat is determined directly. The same applies to the method of Dieterici ( Weid . Ann., 37 (1889), pp. 494-508), which, however, is only applicable to the temperature 0°, and to the method of Griffiths {Phil. Trans., 186 (1895), A., pp. 261-341), which is only possible for temperatures not very far from those of the room. 2. The latent heat is obtained by an electrical measurement, which may be made more accurately than is possible when the results depend on the reading of an ordinary mercury thermometer. PROC. ROY. SOC. EDIN. — YOL. XXIY. 30 458 Proceedings of Royal Society of Edinburgh. [sess. As a result of several very trustworthy determinations within the last decade, the value of J in electrical units for various calories is known to at least 1 in 2000. Other electrical methods are that of Griffiths already referred to, and that of Ramsay and Marshall {Phil. Mag., [5] 41 (1896), pp. 38-52). The method of the latter experimenters is a comparative one, benzene being taken as the standard substance. Its latent heat was determined by Griffiths- and Marshall for this purpose by experiments between 20° and 50% extrapolated to the boiling point 80° *2. 3. No corrections for heat losses are required. Preliminary Tests. Leakage and Electrolysis. — The copper leads carrying the mercury cups pass through wood covered with a layer of water, and so are very imperfectly insulated. The method of testing this was to raise the electrodes out of the mercury cups, so breaking this path. The current which then passed could not he detected by the apparatus used. This was with the full battery e.m.f. of over 4 volts, while the P.D. between these leads during an experiment is little over 1 volt. That electrolysis does not occur was proved in the first experi- ments by severing one of the wires in the heating coil and watch- ing for the formation of a bubble or the blackening of the ends as the voltage was raised. No appearance of electrolysis was found under about 1*7 volts. The above experiment on the leakage current applies equally to the inside of the glass vessel, since in the early forms of apparatus the copper leads stood in a consider- able depth of water, about J inch or so. The Temperature of the Boiling Liquid. — This was compared with the temperature of the vapour by means of a thermometer placed with its bulb first in the liquid and then j ust over the sur- face. A difference of -05° was noted. Conclusion. The experiments made so far have had the end in view of testing the reliability of the method, and for this purpose water has always been used, in order to enable a comparison to be made 1902-3.] Determining Latent Heat of Evaporation. 459 with Regnault’s result. Taking this as 536 ’3 (calories at 15°) and Joule’s equivalent J = 4*187 (calorie at 15°, Griffiths, Thermal Measurement , p. 108), this gives the latent heat in joules per gm. = 536*3 x4*187 = 2245*5. Compared with this, the numbers 2249, 2264, 2266 have been obtained in the order given. These numbers seem to indicate a value somewhat higher than Regnault’s, but they were obtained with forms of apparatus which subsequent experience showed to be unsatisfactory in several respects. It is hoped that with the present form more concordant results may be obtained. Experiments for this purpose are being continued. The apparatus has been made and used in the workshops and laboratory of the electrical department of the Heriot-Watt College, and I gladly avail myself of this opportunity of expressing to Professor Baily my thanks for the facilities he has afforded for the work during the spare time of the last two summer vacations. ( Issued separately August 8, 1903.) 460 Proceedings of Royal Society of Edinburgh. [sess. The Wild Horse (Equus jprjevalskii , Poliakoff). By J. C. Ewart, M.D., F.R.S., Professor of Natural History, University of Edinburgh. (Read June 15, 1903.) In the time of Pallas and Pennant, as in the days of Oppian and Pliny, it was commonly believed true wild horses were to be met with not only in Central Asia, but also in Europe and Africa. But ere the middle of the nineteenth century was reached, naturalists were beginning to question the existence of genuine wild horses; and somewhat later, the conclusion was arrived at that the horse had long “ceased to exist in a state of nature.”1 This view had barely been accepted by zoologists, when it was announced from St Petersburg that a true wild horse had at last been discovered in Central Asia by the celebrated Russian traveller, Prjevalsky. An account of this horse was communicated by Poliakoff, in 1881, to the Imperial Russian Geographical Society.2 The material at Poliakoff’s disposal being limited, zoologists were not at once disposed to admit that Prjevalsky ’s horse, as it came to be called, deserved to rank as a distinct species. Some believed the new horse had no more claim for a place amongst wild forms than the mustangs of the Western prairies or the brumbies of the Australian bush ; while others asserted it was merely a hybrid between the Kiang ( Equus hemionus) and a Mongolian or other Eastern pony. Even after the brothers Grijimailo in 1890 3 added somewhat to Poliakoff’s original description from material (four skins and a skeleton) brought from the Dzungaria desert, naturalists were still sceptical. The greatest English authority on the structure and classification of the Equidse during the latter part of the nineteenth century was the late Sir William Flower. Writing in 1 Bell’s British Quadrupeds. 2 A translation of Poliakoff’s paper will be found in the Annals and Magazine of Natural History , 1881. See also Tegetmeier and Sutherland’s Horses, Asses, and Zebras. 3 See Proceedings of the Roy. Geog. Soc., April 1891. 1902-3.] The Wild Horse (Equus prjevalskii). 461 1891, Flower says: “Much interest, not yet thoroughly satisfied, has been excited among zoologists” by Poliakoff’s announcement; but he added : “ Until more specimens are obtained, it is difficult to form a definite opinion as to the validity of the species, or to resist the suspicion that it may not be an accidental hybrid between the Kiang and the horse.” 1 Since Flower expressed this opinion, quite a number of specimens illustrating the form and structure of Prjevalsky’s horse at various ages have been added to the St Petersburg Zoological Museum; and in 1902, Mr Hagenbeck of Hamburg (commissioned by his Grace the Duke of Bedford) imported from Mongolia between twenty and thirty living Prjevalsky ‘ colts.’ Though about half of these colts found their way to England, and though Dr W. Salensky, Director of the Zoological Museum of St Peters- burg, published last year an elaborate monograph2 on Prjevalsky’s horse, English zoologists are not yet satisfied that we have in this member of the horse family a true and valid species. As far as I can gather, it is generally believed in England that Prjevalsky’s horse is a hybrid — a cross between a pony and a Kiang. Beddard, however, admits it may be a distinct type. He says : “ This animal has been believed to be a mule between the wild ass and a feral horse ; but if a distinct form — and probability seems to urge that view — it is interesting as breaking down the distinctions between horses and asses.” 3 It must be admitted that in its mane and tail Prjevalsky’s horse is strongly suggestive of a hybrid, but in the short mane and mule- like tail we may very well have a persistence of ancestral characters — in the wild asses and zebras the mane is always short, and they never have long persistent hairs at the proximal end of the tail. Though a superficial examination may lead one to think with Flower that Prjevalsky’s horse is an accidental hybrid, a careful study of the soft parts and skeleton inevitably leads to quite a different conclusion. Though failing to understand why so many zoologists persisted 1 Flower, The Horse, pp. 78, 79. 2 Wissenschaftliche Resultate der von N. M. Przewalski nach Central Asien. Zool. Theil : Band i., Mammalia ; Abtli. 2, Ungulata. St Petersburg, 1902. 3 Beddard, Mammalia, p. 240. Macmillan, 1902. 462 Proceedings of Royal Society of Edinburgh. [sess. in considering the horse of the Great Gobi Desert to be a mule, I decided to breed a number of Kiang-horse hybrids.1 With the help of Lord Arthur Cecil, I succeeded early in 1902 in securing a male wild Asiatic ass and a couple of Mongolian pony mares — one a yellow-dun, the other a chestnut. “Jacob,” the wild ass, was mated with the dun Mongol mare, with a brownish- yellow Exmoor pony, and with a hay Shetland-Welsh pony. The chestnut Mongol pony was put to a light grey Connemara stallion. Of the four mares referred to, three have already (June) foaled, viz., the Exmoor and the two Mongolian ponies. The Exmoor having foaled first, her hybrid may be first considered. It may he mentioned that the Exmoor pony had in 1900 and again in 1901 a zebra hybrid, the sire being the Burchell zebra 1 Sir William Flower, the late President of the London Zoological Society, having more than hinted in 1891 that Prjevalsky’s horse was a mule, one would have thought an effort would have been made forthwith to test this view in the Society’s Garden. 1902-3.] The Wild Horse (Equus prjevalskii). 463 Matopo,” used in my telegony experiments. In the case of her Kiang hybrid the period of gestation was 335 days (one day short of what is regarded as the normal time), but she carried her 1900 .zebra hybrid 357 days, three weeks beyond the normal time. The Exmoor-zebra hybrids are as nearly as possible intermediate between a zebra and a pony ; the Kiang hybrid, on the other hand, might almost pass for a pure-bred wild ass.1 In zebra hybrids the ground colour has invariably been darker than in the zebra parent ; but the Kiang hybrid is decidedly lighter in colour than her wild sire, while in make she strongly suggests an Onager — the wild ass so often associated with the Kunn of Cutch. Alike in make and colour the Kiang hybrid differs from a youug Prjevalsky foal.2 I have never seen a new-born wild horse; but if one may judge from the conformation of the hocks, from the coarse legs, big joints, and large head of the yearlings — from their close resemblance to dwarf cart-horse foals, — it may be assumed they are neither characterised by unusual agility nor fleetness. The Kiang hybrid, •on the other hand, looks as if built for speed, and almost from the moment of its birth it has by its energy and vivacity been a source •of considerable anxiety to its by no means placid Exmoor dam. When four days old it walked over twenty miles ; on the fifth day, instead of resting, it was unusually active, as if anxious to make up for the forced idleness of the previous evening. In the hybrid the joints are small, and the legs are long and slender and covered with short close-lying hair. In the wild horse the joints are large and the ‘ bone ’ is round as in heavy horses. As to its colour, it may be especially mentioned that the hybrid has more white around the eyes than the wild horse, but is of a darker tint along the back and sides and over the hind quarters. Too much importance, however, should not be attached to differences in colour ; for though the two hybrid foals which have already arrived closely agree in their coloration, subsequent foals may differ considerably, and it is well known that young wild horses irom the western portions of the Great Altai Mountains differ in tint from those found further east. 1 In Mendelian terms, the Exmoor pony proved recessive , the wild ass dominant. ‘2 For a skin of a very young Prjevalsky foal I am indebted to Mr Carl Hagenbeck of Hamburg. 464 Proceedings of Royal Society of Edinburgh. Of more importance than the coat-colour is the nature of the hair. A Prjevalsky foal has a woolly coat not unlike that of an Iceland foal. In the hybrid, the hair is short and fine and only slightly wavy over the hind quarters. It thus differs but little from a thoroughbred or Arab foal. The mane and tail of the hybrid are exactly what one would Kiang-Pony Hybrid, art. two days. L expect in a mule; the dorsal band, 75 mm. wide over the croup in the sire, has in the hybrid a nearly uniform width of 1*2 mm. from its origin at the withers until it loses itself half-way down the tail. The tail, which differs but little from that of a pony foal, is of a lighter brown colour than the short upright mane, while the dorsal band is of a reddish-brown hue. In the wild horse the dorsal band is sometimes very narrow (under 5 mm.) and indistinct. In the Kiang sire there are pale but quite distinct stripes 1902-3.] The Wild Horse (Equus prjevalskii). 465 above and below the hocks, and small faint spots over the hind quarters — vestiges apparently of ancestral markings ; but in the hybrid there are neither indications of stripes across the hocks or withers, nor spots on the quarters.1 In having no indications of bars on the legs or faint stripes, across the shoulders, the hybrid differs from Prjevalsky colts ; it also differs in having a longer flank feather, and in the facial whorl being well below the level of the eyes. As in the Kiang and some of the wild horses, the under surface of the body and the inner aspect of the limbs are nearly white. In the hybrid the front chestnuts (wrist callosities) are smooth and just above the level of the skin ; but instead of being roughly pear-shaped as in the Kiang, they are somewhat shield-shaped as in the Onager. In the wild horse the front chestnuts are elongated. In the Exmoor dam the hind chestnuts (hock callosities) aro 27 mm. in length and 10 mm. wide. In the sire there is a minute callosity inside the right hock. In the hybrid the hind chestnuts are completely absent. In the absence of hock callosities the hybrid differs from the wild horse, in which they are relatively longer than in Clydesdales, Shires, and other heavy breeds of horses. In the hybrid, as in the sire and dam, there are smooth,, rounded fetlock callosities (ergots) on both fore and hind limbs. In the wild horse the hoof is highly specialised, the ‘ heels ’ being bent inwards (contracted) to take a vice-like grip of the frog. In the hybrid the hoof closely resembles that of the pony dam ; it is shorter than in the Kiang, and less contracted at the £ heels r than in the wild horse. The Kiang hybrid further differs from a young wild horse in the lips and muzzle, the nostrils and ears, and in the form of the head and hind quarters. The wild horse has a coarse, heavy head, with the lower lip (as is often the case in large-headed horses and in Arabs with large hock callosities) projecting beyond the upper. The nostrils in their outline resemble those of the domestic horse, while the long pointed ears generally project obliquely outwards, 1 The complete absence of stripes in the Kiang hybrid is all the more interesting, seeing that the dam’s previous foals were zebra hybrids. Evidently the Kiang hybrid lends no support to the telegony doctrine. 466 Proceedings of Royal Society of Edinburgh. [sess. as in many heavy horses and in the Melbourne strain of thorough- breds. Further, in the wild horse the forehead is convex from above downwards as well as from side to side, — hence Prjevalsky’s horse is sometimes said to be ram-headed. In the hybrid the muzzle is fine as in Arabs, the lower lip is decidedly shorter than the prominent upper lip, the nostrils are narrow as in the Kiang ; and even at birth the forehead was less rounded [Darurin- Wilmot. Exmoor Pony and lier Hybrid Foal, cel. nine days. than is commonly the case in ordinary foals. The ears of the hybrid, though relatively shorter and narrower than in the Kiang, have, as in the Kiang, incurved dark-tinted tips, and they are usually carried erect or slightly inclined towards the middle line. In the wild horse the croup is nearly straight and the tail is set on high up as in many desert Arabs. In the hybrid the croup slopes as in the Kiang and in many ponies, with the result that the root of the tail is on a decidedly lower level than the highest part of the hind quarters. Further, in the young wild 1902-3.] The Wild Horse (Equus prjevalskii). 467 horses I have seen the heels (points of the hocks) almost touch •each other, as in many Clydesdales, and the hocks are distinctly bent. In the hybrid the hocks are as straight as in well-bred foals, and the heels are kept well apart in walking. Another difference of considerable importance is, that while the wild horse neighs, the hybrid makes a peculiar barking sound remotely suggestive of the rasping call of the Kiang. The dun Mongol pony’s hybrid arrived five weeks before its lime, and, though perfect in every way, was short-lived. Only in one respect did this hybrid differ from the one already described. In the Exmoor hybrid the hock callosities are entirely absent ; in the Mongol hybrid the right hock callosity is completely wanting, hut the left one is represented by a small, slightly hardened patch of skin sparsely covered with short white hair.1 In zebra hybrids out of cross-bred mares the hock callosities are usually fairly large, while in hybrids out of well-bred pony mares the hock callosities are invariably absent. The Exmoor pony, though not as pure as the Hebridean and other ponies without callosities, has un- doubtedly a strong dash of true pony blood ; the Mongol pony is as certainly saturated with what, for want of a better term, may be called cart-horse blood. As I expected, there were no hock callosities present in the Exmoor hybrid. In the Mongol hybrid there was less evidence of hock callosities than I expected. From what has been said it follows that a Kiang-Mongol-pony hybrid differs from Prjevalsky’s horse (1) in having the merest vestiges of hock callosities ; (2) in not neighing like a horse ; (3) in having finer limbs and joints and less specialised hoofs ; (4) in the form of the head, in the lips, muzzle, and ears • (5) in the dorsal band ; and (6) in the absence even at birth of any suggestion of shoulder-stripes and of bars on the legs. While most of the zoologists who hesitated to regard Prjevalsky’s horse as representing a distinct and primitive type favoured the view that it was a mule, some asserted that it in no way essentially differed from an ordinary horse. The colts brought from Central Asia, they said, were the offspring of escaped Mongol ponies. 1 The presence of hair in the imperfectly-formed hock callosity of the Mongol hybrid, together with the presence of hair rudiments in the developing hock •callosity of the common horse, certainly lends very little support to the view held by some zoologists that the chestnuts of the horse are vestiges of glands. 468 Proceedings of Royal Society of Edinburgh. [sess. Others affirmed that they failed to discover any difference between the young wild horses in the London Zoological Gardens and Ice- land ponies of a like age. To test the first of these assertions, I, as already mentioned, mated the chestnut Mongol pony with a young Connemara stallion ; to test the second, I purchased last autumn a recently-imported yellow-dun Iceland mare in foal to an Iceland stallion. As I anticipated, the chestnut Mongol mare produced a foal the image of herself. This foal, it is hardly necessary to say, decidedly differs from the Prjevalsky colts recently imported from Central Asia by Mr Hagenbeck, and it as decidedly differs from the Kiang hybrids described above. The Iceland foal, notwithstanding the upright mane and the woolly coat, for a time of a nearly uniform white colour, could never be mistaken for a wild horse, and the older it gets the differences will become accentuated. If Prjevalsky’s horse is neither a Kiang-pony mule nor a feral Mongolian pony, and if moreover it is fertile (and its fertility can hardly be questioned), I fail to see how we can escape from the conclusion that it is as deserving as, say, the Kiang to be regarded as a distinct species. Granting Prjevalsky’s horse is a true wild horse, the question arises : In what way, if any, is it related to our domestic horses? It is still too soon to answer this question ; but I venture to think that should we by-and-by arrive at the conclusion that our domestic horses have had a multiple origin — have sprung from at least two perfectly distinct sources, — we shall probably subsequently come to the further conclusion that our big-headed, big-jointed horses, with well-marked chestnuts on the hind legs, are more intimately related to the wild horse than the small-headed, slender-limbed varieties without chestnuts on the hind legs ; that in fact the heavy horses found living in a semi- wild state and Prjevalsky’s horse have sprung from the same ancestors. [Note. — Towards the cost of the young Prjevalsky horse (without which this inquiry would have been difficult) the University con- tributed £100 from the Moray Research Fund.] {Issued separately August 8, 1903.) 1902-3.] Mr Win. Murray on Salmon in American Rivers. 469 Statistical Evidence regarding the Influence of Artificial Propagation upon the Salmon of the American Rivers. By William Murray. Communicated by Dr D. Noel Paton. (Read July 6, 1903.) The materials for this paper upon the Influence of Artificial Pro- pagation on the Salmon of the American Rivers have been gathered from an examination of the salmon fishery statistics of Canada and the United States of America, where salmon culture has been ex- tensively pursued. The rivers examined include five principal rivers, or districts of rivers, in Canada, in which salmon fry have been liberated, — the Penobscot, the only river in eastern America which has now an appreciable run of salmon, the Columbia and Sacramento upon the Pacific coast. The salmon bred in the Columbia and Sacramento are of course members of the Pacific varieties, and, as such, differ from the fish as known here. Of the varieties hatched, the Chinook or Quinnat is the chief, and this fish has been planted in some of the other and smaller rivers of the Pacific coast, but the difficulty of obtaining sufficient statistical information has made the inclusion of these streams in this review inadvisable. The statistics con- sidered cover a long period of years, varying from twenty to nearly thirty. Returns for consecutive years are not collected in the United States, but for each fourth or fifth year, as the demands upon the time of the central department permit. In Canada, on the other hand, the returns are made yearly, and upon some streams extend backward over a very long period, forming a most valuable informative collection of salmon fishery precedents. The statistics, which have been studied with the object of discovering what, if any, has been the effect of artificial culture upon the productive powers of these rivers, seem, as a whole, to throw up two main conclusions : — (1) Amongst the rivers where salmon culture has been estab- lished, those have maintained or increased their yield where re- strictive regulations regarding netting and close times have been enforced, combined with facilities for natural reproduction. 47 0 Proceedings of Royal Society of Edinburgh. [sess. (2) There is no example of the establishment or maintenance of a commercial salmon fishery upon any river in North America which has depended for its yield upon artificial culture, unsupported by restrictions upon netting, or by accessible spawning grounds. According as either one or other of the two latter factors has been neglected, the yield of the river has declined. There are eight or nine salmon hatching stations in Canada. Three of them supply the province of Nova Scotia and the river St John in New Brunswick. The work of these three hatcheries will not be included in this review, as special circumstances have rendered the compilation of exact statistics in respect of them more difficult than ordinary. Their inclusion would make no difference to the results discovered in this paper. With these exceptions salmon culture by the State in Canada has been confined to the five rivers or districts which will now be discussed. Private efforts seem to have been few and far between, and are of little account. The Canadian Government has always supplemented its fish cultural work by a vigorous administrative policy. A system of licenses has long been established. Netting stations have been reduced wherever advisable and possible, nets are confined to tidal waters, and the upper waters protected and rendered accessible to salmon. The proper way in which to judge of the columns of figures representing the yield of the river and the plantation of fry in the following tables may be open to question. In the writer’s opinion the comparison of one year with another is very misleading. Our knowledge of fish life is far too fragmentary to allow of a pro- ceeding which assumes so much. Lt seems better to compare the averages of succeeding periods of four years’ duration each; and the following tables, dealing with the Fraser, the Ristigouche, the Miramichi and the Saguenay rivers and districts, have been pre- pared with this object in view. It should be noted that in order to decide how far the output of fry has affected the yield, the average output of fry for any four- year period in the tables can be compared with the average yield for the immediately succeeding four-year period ; that is to say, the average for the period ending 1882 should be contrasted with the average yield for the period ending 1886, and so on to the end of the table. 1902-3.] Mr Wm. Murray on Salmon in American Rivers. 471 Fraser River and District. Table showing the Output of Canned Salmon, Number of Fry Planted, and the Length of Drift-nets in use within the Fraser River and District, so far as affecting Canada, during a period of Years, (a) Year. Output of Canned Salmon ( b ). Average. Number of Sockeye Fry Planted. Average. Total length of Drift-nets. Average. Average Catch per Yard of Net. Number of Canneries. 1876 lbs. (c) 511,056 lbs. yds. yds. lbs. 3 1877 3,090,576 44,040 5 1878 5,044,880 2,882,*170'2 114,580 79,310 36 8 1879 2,423,520 65,600 7 1880 2,023,440 105,240 7 1881 6,840,768 124,400 8 1882 9,561,972 5,249^425 205,600 215,780 125,210 ii 13 1883 5,265,648 12 1884 1,844,976 210,770 6 1885 4,301,616 ? 189,200 6 1886 4,758,576 4,042*704 ? (d) 1,0*0*0,000 232,920 212,167-2 19 11 1887 6,182,688 2,405,000 350,850 12 1888 3,677,568 3,870,000 282,520 12 1889 14,789,856 4,046,500 254,200 16 1890 11,742,600 9,098*178 5,540,000 3,96*5,375 298,880 296,612-2 30 17 1891 8,527,552 3,603,000 244,810 22 1892 4,277,552 22,763,380 5.600.000 5.764.000 252,580 22 1893 355,900 26 1894 17,451,172 13,549,414 6,300,000 5,3*1*6,750 503,900 339,297-2 39 28 1895 20,780,170 6,390,000 528,000 31 1896 18,016,544 6,393,000 803,800 35 1897 42,197,516 (e) 12,682,780 5.928.000 5.850.000 709,400 792,900 43 1898 23, 419, *252-2 6,165,250 708,525 33 (a) Most of the figures given in this table are to be found in the U.S.F.C. Report for 1899. The averages are not in the Report referred to. (&) Sockeye salmon form by far the greater part of this output ; in quite recent years some cohoes have been canned ; in 1899 Humpback and Dog salmon were first used. (c) From 25 per cent, to 50 per cent, ought to be added to the weights given in this column to reach the weight of the fish when captured. The loss in weight to the fish during the process of canning is seldom less than from 30 per cent, to 40 per cent. (d) During the years 1885-86 some 6,000,000 sockeye eggs seem to have been procured, but the number of fry planted in the Fraser each year does not seem to have much exceeded 1,000,000. (e) 9,600,000 lbs. also given, U.S.F.C.R. for 1899, p. 328. Fraser. — The yield in this river, as in the other three taken under this heading, has, especially in recent years, greatly increased. None the less, a study of the table will show how difficult, if not im- possible, it is to arrive at the true relation, supposing there is one, between an increasing yield and an increased output of fry. Writers on salmon culture are fond of claiming that results are first apparent “in the fifth year” after the planting of the fry. If this test be applied to the Fraser table it will be found that, beginning with the year 1887, the output of fry in that year and the increased 472 Proceedings of Royal Society of Edinburgh. [sess output during the following years has in each case been followed in the fifth year either by a decline or by an increase in the yield so large as to outnumber (in numbers of fish) the total quantity of fry planted in the parent year. If the fourth or sixth year be assumed to be the year in which the first effects of artificial culture will be apparent, results very similar await the calculator, showing that in only four cases out of the twenty-four examined can artifi- cial culture upon the widest assumptions claim even a modest interest in an increase or decline. Ristigouche and District. This table shows : Catch of Salmon upon Ristigouche District and Fry liberated there, (a) Year. Ristigouche. Yield in lbs. Average. Output of Fry. ib) Average. 1879 838,961 1,470,000 1880 357,890 1,600,000 1881 351,629 700,000 1882 364,735 478,303 *3 (c) 1,400,000 1,292*500 1883 446,117 (c) 300,000 ! 1884 476,699 • •• 940,000 1885 574,538 • • • 660,000 1886 794,664 573,009-2 1,380,000 820,' 0 00 1887 710,275 1,500,000 l 1888 662,788 1,720,000 1 1889 676,380 1,280,000 1890 669,319 679,' 690-2 2,326,000 1,70*6* 500 1891 656,195 1,530,000 1892 799,615 1,240,000 1893 1,487,226 (d) 883,000 1894 1.560,470 1, 125*876 1,018,000 1,168* 750 1895 1,490,102 2,885,000 1896 1,703,166 990,000 1897 712,376 2,100,000 1898 697,400 1,150,761 1,185,000 1,790*000 (a) The Ristigouche column of yields is compiled from the returns for the counties of Gloucester and Ristigouche, and from those for the Ristigouche division in Quebec. Where the divisions made in the later reports do not correspond with those of the earlier, a like course has been followed to that described in the Gaspe table. Ristigouche district includes salmon caught in the Nepissiguit and Jacquet. ( b ) Pisciculture in the Ristigouche district began in 1875, averaging about <600,000 fry per annum. (c) Estimated, as no details of distribution for these years are available. (d) Or 783,000. 1902-3.] Mr Wm. Murray on Salmon in American Rivers. 473 Saguenay River and District. Table showing the Catch of Salmon and Output of Fry in Saguenay River and District during a period of Years, (a) Year. Yield in lbs. (&) Average in lbs. Output of Fry. (d) Average. 1871 . (c) 34,620 1872 . 33,120 1873 . 24,810 1874 . 24,820 29,342-2 1875 . 9,810 60,000* 1876 . 28,300 150,000 1877 . 23,620 1,180,000 1878 . 27,460 22,2*97-2 707,000* 524*250 1879 ( a ) 18,490 1,210,000* 1880 . 6,580 .1,107,000* 1881 (6) 5,840 300,000* 1882 . 9,690 10,150 600,000* 80*4*250 1883 (c) 12,510 995,000* 1884 . 10,810 985,000 1885 . 22,580 720,000 1886 . 14,790 15,172-2 1,627,000 1,081*750 1887 . 16,720 880,000 1888 . 24,600 835,000 1889 r 37,900 1,580,000 1890 . 60,966 35,048 1,670,000 1,241,250 1891 . 64,940 1,300,000 1892 . (e) 48,000 624,000 1893 . . ' 68,780 2,060,000 1894 . 82,400 66,030 1,975,000* 1,489*750 1895 . 76,680 2,060,000 1896 . 146,820 2,500,000 1897 . 46,100 3,082,000 1898 (d) 108,400 94,500 (d) 1,950,000 2, 39*8*000 (a) Compiled from the Canadian Fishery Reports. ( b ) The yields are those returned for the Saguenay division only ; possibly a proportion of the salmon caught in the county of Charlevoix should be added to the table in order to include the whole district which may have been influenced by artificial culture in the Saguenay. The proportion, however, cannot be struck, and the county of Charlevoix returns, being only about 2000 lbs. per annum, would not, if included, materially influence the result of the table. (c) The yields for the years 1871-80 are to be found C.F.R. for 1881, p. 142, in numbers of fish. It is plain from C.F. Reports for 1884-5 that the