PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. PROCEEDINGS THE ROYAL SOCIETY EDINBURGH VOL. XXVI. NOVEMBER 1905 to JULY 1906. EDINBURGH: PRINTED BY NEILL AND COMPANY, LIMITED. MDCCCCVII. CONTENTS. PAGE Office-Bearers, Session 1905-6, . ... 1 Some Electrical Measurements on Metals. By Charles E. Fawsitt, D.Sc., Pli.D. Communicated by Professor A. Crum Brown. Issued separately February 12, 1906, .... 2 The Tarpan and its Relationship with Wild and Domestic Horses. By J. C. Ewart, M.D., F.R.S. (With Three Plates.) Issued separately February 12, 1906, ..... 7 The Horse in Norway. By Francis H. A. Marshall, M.A. (Cantab.), D.Sc. (Edin.). (With Two Plates.) Issued separately February 12, 1906, . . . . .22 Notes on the Effect of Electric Oscillations (co-directional and transverse) on the Magnetic Properties of Iron. By James Russell. Issued separately February 8, 1906, . . .33 On a Theorem in Hypercomplex Numbers. By J. H. Maclagan- Wedderburn, Carnegie Research Fellow. Issued separately February 9, 1906, . . . . . .48 Library Aids to Mathematical Research. By Thomas Muir, LL.D. Issued separately February 14, 1906, . . .51 Bathy draco Scotise, Poisson abyssal nouveau recueilli par l’Expedi- tion Antarctique Nationale Ecossaise. Note preliminaire, par Louis Dollo, Conservateur au Musee royal d’Histoire naturelle, a Bruxelles. Presentee par M. R. H. Traquair, M.D., F.R.S. , Y.P.R.S.E. Issued separately March 29, 1906, . . 65 Some Further Results obtained with the Spectroheliometer. By Dr J. Halm. Issued separately March 29, 1906, . . 76 Preliminary Note regarding an Experimental Investigation into the Effects of Varying Diets upon Growth and Nutrition. By Chalmers Watson, M.D. ( From the Physiological Laboratory of the University of Edinburgh.) Presented by Professor E. A. Schafer, F.R.S. Issued separately February 22, 1906, . . 87 A Contribution to the Study of the Excretion of Allantoin in Thymus Feeding. By W. M‘Lachlan, M.D. ( From the Research Laboratory of the Royal College of Physicians, Edin- burgh.) Communicated by Dr D. Noel Paton. Issued separately March 29, 1906, ..... 95 VI Contents. On the Formation of Certain Lakes in the Highlands. By Dr Leon W. Collet, F. Swiss Geol. S., Assistant to Sir John Murray, K.C.B., and Dr T. N. Johnston, F.R.S.E. With a Note on Two Rock Basins in the Alps, by Dr Leon W. Collet. Issued separately April 16, 1906, . .107 On the Distribution of the Proper Fractions. By Duncan M. Y. Sommerville, D.Sc. Communicated by Professor Chrystal. Issued separately April 16, 1906, . . .116 On Vibrating Systems which are not subject to the Boltzmann- Maxwell Law. By Dr W. Peddie. Issued separately May 24, 1906, . . . ... 130 Some Experimental Results in connection with the Hydro- dynamical Theory of Seiches. By Peter White, M.A., and William Watson. Issued separately June 11, 1906, . .142 On the Methods of Standardising Suprarenal Preparations. By I. D. Cameron, M.B., D.P.H., Assistant to the Lecturer on Physiology, Edinburgh School of Medicine for Women. (From the Laboratory of the Royal College of Physicians , Edinburgh.) Communicated by Dr D. Noel Paton. Issued separately May 21, 1906, . . . . .157 Neobythites Brucei, Poisson abyssal nouveau recueilli par l’Expedition Antarctique Nationale Ecossaise. Note pre- liminaire, par Louis Dollo, Conservateur au Musee royal d’Histoire naturelle, a Bruxelles. Presentee 'par M. R. H. Traquair, M.D., F.R.S., V.P.R.S.E. Issued separately June 14, 1906, ........ 172 The Relation between Normal Take-up or Contraction and Degree of Twist in Twisted Threads. By Thomas Oliver, B.Sc. (Lond. & Edin.), Carnegie Research Fellow. Communicated by Dr C. G. Knott. Issued separately June 19, 1906, . .182 A New Form of Harmonic Synthetiser. By J. R. Milne, B.Sc., Carnegie Research Fellow. (With Plate.) Issued separately July 12, 1906, ....... 207 Preliminary Note on the Conductivity of Concentrated Aqueous Solutions of Electrolytes. By Professor J. Gibson. Issued separately August 29, 1906, ..... 234 Recherches sur la Glauconie. Par les Drs Leon W. Collet et Gabriel W. Lee, assistants cle Sir John Murray, K.C.B. Communique par Sir John Murray. (Avec 12 planches et 1 carte.) Issued separately August 30, 1906, . . . 238 Notes : — 1. On a Human Skeleton, with Prehistoric Objects, found at Great Casterton, Rutland. 2. On a Stone Cist containing a Skeleton and an Urn, found at Largs, Ayrshire. By Dr Robert Mutiro. With a Report on the Urn, by the Hon. John Abercromby ; and on the Skulls, by Professor D. J. Cunningham. Issued separately August 31, 1906, . 279 Note on a rare Dolphin ( Delphinus acutus), recently stranded on the Coast of Sutherland. By Sir William Turner, K.C.B., F.R.S. (With Plate.) Issued separately August 29, 1906, . 310 Contents. vii PAGE Contributions to the Craniology of the People of the Empire of India. Part III. : Natives of the Madras Presidency, Thugs, Veddahs, Tibetans, and Seistanis. By Sir William Turner, K.C.B. Title only , ...... 320 Note on the Smolt to Grilse Stage of the Salmon, with Exhibition of a Marked Fish recaptured. By W. L. Calderwood. Issued separately October 12, 1906, . . . . .321 Two Lecture Experiments in illustration of the Theory of Ionization. By Dr W. W. Taylor. Communicated by Professor Crum Brown. Issued separately October 12, 1906, . . 325 A Dietary Study of Five Halls of Residence for Students in Edinburgh. By I. D. Cameron, M.B., D.P.H. Communicated by D. Noel Paton, M.D. Issued separately November 9, 1906, 327 Further Study of the Two Forms of Liquid Sulphur as Dynamic Isomers. By Alexander Smith and C. M. Carson. Issued separately November 12, 1906, ..... 352 The Theory of Alternants in the Historical Order of Development up to 1860. By Thomas Muir, LL.D. Issued separately November 16, 1906, ...... 357 The Theory of Circulants in the Historical Order of Development up to 1860. By Thomas Muir, LL.D. Issued separately November 16, 1906, . . . . . . 390 Initiation of Deep-Sea Waves of Three Classes : (1) from a Single Displacement ; (2) from a Group of Equal and Similar Displacements ; (3) by a Periodically Varying Surface Pressure. By Lord Kelvin. Issued separately November 17, 1906, . 399 “ Scotia” Collections. On Echinorhynchus antarcticus , n. sp., and its Allies. By John Rennie, D.Sc., University of Aberdeen. Communicated by Wm. S. Bruce, Esq. (With a Plate.) Issued separately January 4, 1907,- ..... 437 Electrolysis through Precipitation Films. Part I. By W. S. Millar, M.A., B.Sc., Carnegie Research Scholar, and Dr W. W. Taylor. Communicated by Professor Crum Brown. Issiied separately January 4, 1907, ..... 447 Nematodes of the Scottish National Antarctic Expedition, 1902-1904. By Dr v. Linstow, Gottingen. Communicated by W. S. Bruce. (With Two Plates.) Issued separately January 4, 1907, .464 Scottish National Antarctic Expedition. “Scotia” Collections. Collembola from the South Orkney Islands. By George H. Carpenter, B.Sc., M.R.I.A., Professor of Zoology in the Royal College of Science, Dublin. (With Two Plates.) Communi- cated by William Evans, Esq. Issued separately January 14, 1907, 473 Statistical Studies in Immunity : The Theory of an Epidemic. By John Brownlee, M.D. (Glas.). Communicated by Dr R. M. Buchanan. Issued separately January 14, 1907, . . 484 Ylll Contents. PAGE On a Simple Way of Obtaining the Half-Shade Field in Polari- meters. By James Robert Milne, B.Sc., Carnegie Besearch Fellow. Issued separately January 14, 1907, . . . 522 On an Exception to a Certain Theorem in Optics, with an Application to the Polarimeter. By James Kobert Milne, B.Sc., Carnegie Kesearch Fellow. Issued separately January 14, 1907, 527 The Hessians of Certain Invariants of Binary Quantics. By Thomas Muir, LL.D. Issued separately January 16, 1907, . 529 The Sum of the r-line Minors of the Square of a Determinant. By Thomas Muir, LL.D. Issued separately January 16, 1907. 533 Obituary Notices, ..... 540 Meetings of the Royal Society — Session 1905-1906, .552 Abstract of Accounts for Session 1905-1906, 561 Index, . . ■ . . . 572 PROCEEDINGS ROYAL SOCIETY OF EDINBURGH. VOL. XXVI. 1905-6. The 123rd Session. GENERAL STATUTORY MEETING. Monday , 23 rd October 1905. The following Council were elected : — President. The Right Hon. Lord KELVIN, G.C.V.O., P.C., LL.D., D.C.L., F.R.S. Vice- Presidents. The Hoij. Lord M Taken, LL.D. | Ramsay H. Traquaie, M.D., LL.D., The Rev. Professor Flint, D.D. Robert Munro, M.A., M.D., LL.D. Sir John Murray, K.C.B., D.S.C., LL.D., D.C.L., Ph. D. , F.R.S. F.R.S., F.G.S. Alexander Crum Brown, M.D., D.Sc., F.R.C.P.E., LL.D., F.R.S. General Secretary — Professor George Chrystal, M.A., LL.D. Secretaries to Ordinary Meetings. Daniel John Cunningham, M.D., LL.D., D.C.L., F.R.S., F.Z.S. Cargill G. Knott, D.Sc. Treasurer— Philip R. D. Maclagan, F.F.A. Curator of Library and Museum — Alexander Buchan, M.A., LL.D., F.R.S. Ordinary Members of Council. Andrew Gray, M.A., LL.D., F.R.S. Robert Kidston, F.R.S., F.G.S. Diarmid Noel Paton, M.D., B.S.e., F.R.C.P.E. John Chiene, C.B., M.D., LL.D., F.R.C.S.E. John Graham Kerr, M.A. William Peddie, D.Sc. Leonard Dobbin, Ph.D. PROC. ROY. SOC. EDIN. — VOL. XXVI James Cossar Ewart, M.D.. F.R.C.S.E., F.R.S., F.L.S. Benjamin Neeye Peach, LL.D., F.R.S., F.G.S. James Johnston Dobbie, M.A.. D.Sc., F.R.S. George A. Gibson, M.A., LL.D. Johannes P. Kuenen, Ph.D. 1 2 Proceedings of Royal Society of Edinburgh. [sess. Some Electrical Measurements on Metals. By Charles E. Fawsitt, D.Sc., Pli.D. Communicated by Professor A. Crum Brown. (MS. received November 7, 1905. Read November 20, 1905.) There are many isolated records of facts showing that the physical properties of (pure) metals alter when the metals are subjected to hammering, rolling, heat-treatment, and other processes. Somewhat recently Beilby has shown * that the differences in all the various states of metals are explained by the fact that solids exhibit two distinct phases : these are the amorphous or vitreous, and the crystalline. The properties of a metal in any given state are due to the metal being made up either of one or other of these phases, or both. The amorphous or vitreous quality is produced by the various processes of “working” a metal; such are hammering, rolling, forging, or polishing. The crystalline condition is obtained by heat-treatment or annealing. The difference in the physical properties between a metal in the crystalline, and the same metal in the vitreous condition, is often very great, and it appeared to me that it might be worth while to examine the difference in potential which the two phases of a metal have, when placed in an electrolyte. When two metals are placed in an electrolyte, their potential with regard to the electrolyte is in general different, and the electro-motive force of the cell is easily measured. The E.M.F. of such a cell is given by the expression where T is the absolute temperature, nY and n2 the valency of the metals in the ionic condition, Px and P2 the solution pressure of the two metals, and pY and p2 the osmotic pressure of the ions of the two metals. If the solution pressure of two different phases * The Electro -Chemist ancl Metallurgist , 1904, ]>. 806. 1905-6.] Dr Fawsitt on Electrical Measurements on Metals. 3 of the same metal is different, then the expression for the E.M.F. -i 0*000198T , R becomes log10 — . n F2 It is a fact that two different “ kinds ” of the same metal, when placed in an electrolyte, give an E.M.F. Such results as have been recorded are however often contradictory, and for this, I think, two reasons can be given. In the first place, a very thin layer of oxide or other substance on the surface of a metal alters its potential very much ; and secondly, the potential of a metal in contact with an acid or an electrolyte which does not contain the ion of the metal is a very variable quantity. The following cases are, however, of considerable interest. Rolled copper is negative as compared with soft copper, and hammered copper is negative as compared with rolled copper.* Hard steel is negative against tempered steel, f Liidtke | found that finely-divided precipitated silver is negative against ordinary silver in a solution of silver nitrate. Gallium § is liquid at a temperature above 30° C., but also at a temperature below this if undercooled: a measurement of the E.M.F. of the solid against liquid gallium in solution of gallium sulphate showed that the liquid gallium is the negative pole. The liquid condition is a special case (kinetic) of the amorphous phase. These are instances of what I believe to be a general condition, namely, that the potential of the metal in the amorphous phase is negative, and in the crystalline phase positive, when the two kinds of metal are placed in a solution of a salt of the metal. The metal I have found most suitable for experimenting upon is silver. With regard to the electrolyte, the most suitable is a solution of a silver salt, as we are here dealing with an equilibrium between silver and silver ion. The concentration of silver salt in solution does not matter at all in theory, and within certain limits does not matter in practice. I have used J- normal, ^-normal, and T^o'juormal silver nitrate solutions with identical results. When the electrolyte is a salt of some other metal than silver, or if the concentration of silver ion is very small, some silver from the * Wiedemann, EleMrizitat, i. 723. t Wiedemann, ibid., i. 738. + Wied. Ann., 1893, 50, 678-695. § Wiedemann, EleMrizitat, i. 739. 4 Proceedings of Royal Society of Edinburgh. [sess. electrode dissolves ; and, owing to the variable amount which dissolves, the potential between metal and electrolyte is thus a variable quantity. The silver used was in the form of stout wire and was pure. Two silver rods which had received different treatment were inserted in silver nitrate solution, and this cell was then inserted along with a Weston cell in a circuit arranged for the measurement of E.M.F. by the compensation method.* The silver rods were first of all kept at a red heat for several hours. In this condition the surface of the rods had a crystalline frosted appearance, and the rods were quite soft. The cell Agi Ag -^^3 — Agn, where Ag:, Agn represent the two rods of silver, had no E.M.F. One of the rods (Agi) was then polished with emery-papers of different degrees of fineness, and then wiped with a clean cloth. The surface of the polished rod, when examined under the microscope, appeared smooth and unbroken, save for a few scratches, which are hard to avoid in polishing such soft material. The cell Agr — AgN03 — Agn now showed an E.M.F., the polished rod, Agx, being the negative pole. The polished metal has evidently a greater tendency to dissolve than the annealed rod. The average E.M.F. of the cell is 0‘013 volt. It is difficult to obtain always exactly the same value ; the smallest I have obtained is ’008 volt, and the largest '020 volt. The E.M.F. remains practically constant; if the cell be short- circuited for a day and then released, the E.M.F. rises in a few minutes from zero to within *002 volt of its former value. I have used sodium nitrate solution (t§-q) and sulphuric acid (•05 per cent.) as electrolytes instead of silver nitrate ; in both cases the E.M.F. was in the same direction and of approximately the same magnitude as in the case of silver nitrate, but the values obtained varied a great deal. Returning to the case where silver nitrate is used as an electrolyte, the polished rod of silver (Agx) was treated with nitric acid for a few seconds ; it was then washed with water, and warmed to about 200° C. for a short time. The vitreous skin was thus dissolved away, and the E.M.F. of the cell Agx — AgN03 — Agn was zero. * Ostwald -Luther, Physiko-chemischc Messungen, 2nd ed., p. 367. 1905-6.] Dr Fawsitt on Electrical Measurements on Metals. 5 Instead of removing the polished layer by nitric acid, the rod was next heated for several hours at a red heat ; on inserting in the cell again along with Agn the E.M.F. was found to be zero. Beginning again with the two rods in an annealed condition, one was then hammered (Agx) until reduced to the state of thin foil, which was quite elastic in comparison with the annealed rod. The E.M.F. of the cell Agj — AgNOg — Agn was '012 volt, Agj being the negative pole. This shows that polishing and hammering induce a similar change in the condition of the metal as measured by E.M.F. methods. On heating the hammered silver rod (Agj) red hot, it became soft again, and the E.M.F. of the cell was reduced to zero. Experiments on other Metals. Experiments on gold and platinum were carried out in the same manner as on silver. One rod was kept throughout in the annealed condition. Two rods of pure gold were annealed by heating for several hours at a red heat, and were then introduced into gold chloride solution (16 grams per litre). The E.M.F. of this cell was measured as in the case of silver : no matter how long the gold was heated it was not possible, to get the E.M.F. to zero as expected ; however, by short-circuiting it was possible to reduce the small E.M.F. to zero. The effect of polishing or hammering one of the gold rods wras such as to make it the negative pole of the cell ; the E.M.F. in this case is about twice as great as in the case of the silver cell, but the results of separate experiments varied so greatly that a definite figure can scarcely be given. The effect of dissolving the polished layer in aqua regia, or of annealing the hammered rod, was to reduce the E.M.F. of the cell to zero. AVith two rods of platinum in platinum chloride solution, similar results were obtained ; the polished or hammered rod is negative as measured against the annealed rod. Summary of Results. 1. If two rods of the same metal be inserted in a solution of a salt of that metal, and if one of the rods be in the soft or annealed 6 Proceedings of Royal Society of Edinburgh. [sess. condition, and the other in the hardened condition, then the hardened rod is the negatively, and the soft the positively, charged element of the cell. 2. Similar results are obtained by using other electrolytes than a salt of the particular metal, but their use is not to be recom- mended. 3. The results of these experiments corroborate previous experimental work and theory in connection with the existence of two phases of metals. I desire to express my thanks to the Executive Committee of the Carnegie Trust for the Universities of Scotland, for a grant to defray the expenses of this research. The University, Glasgow. ( Issued separately February 12, 1906.) 1905-6. Professor Ewart on the Tar pan. 7 The Tarpan and its Relationship with Wild and Domestic Horses. By J. C. Ewart, M.D., F.R.S., Regius Professor of Natural History, University of Edinburgh. (With Three Plates.) (Read November 6, 1905.) In December 1902, 1 communicated to the Society a preliminary note “ On a New Horse from the Western Islands,” 1 and six months later submitted the results of experiments made with a view to ascertaining whether Prejvalsky’s horse is a true wild species, or, as suggested by Flower and others, a chance hybrid between a kyang and an escaped Mongol pony. The new horse (now commonly known as the Celtic Pony) described in the first paper is characterised by a small head, large prominent eyes, short ears, and narrow nostrils ; by a long tail, mane, and forelock, and, during winter, by a thick, light yellow- dun woolly undercoat and a remarkable tail-lock (PI. II. 7) ; by having, like Prejvalsky’s horse, only 23 dorso-lumbar vertebrae, and also by the complete absence of callosities from the inner aspect of the hocks and from the region of the fetlocks — i.e. by the absence of the hind warts or chestnuts, and of the four ergots invariably present in typical specimens of the common horse. In speed and staying power, intelligence and docility, the Celtic pony takes after liigli-caste, fine-tempered Arabs. This new horse I provisionally named Equus caballus celticus ; but as it — apart from its coat — more profoundly differs from the common horse than either asses or zebras, it will probably be eventually regarded as a true species. The results of the experi- ments with the wild Asiatic ass submitted in the second paper 2 made it sufficiently evident that Prejvalsky’s horse is not a hybrid between a kyang (E. hemionus ) and a Mongol or other Eastern pony. Further observations have made it equally evident that while some of the horses found running wild amongst the Great Altai Mountains may count strayed domestic horses amongst 1 Nature , vol. lxvii. p. 239, 1903. 2 Proc. Royal Soc. Edin. , 1903, pp. 460-8. 8 Proceedings of Royal Society of Edinburgh. [sess. their recent ancestors, the majority of them probably have an unbroken chain of true wild ancestors. The existence in North- Western Europe of the Arab-like Celtic pony and in Central Asia of the long-headed Prejvalsky’s horse — forms quite distinct from the Equus caballus of Linnaeus, Gray, and other systematists — led me to feel less certain of the view provisionally adopted, well-nigh half a century ago, by Darwin, that all the existing races had descended “from a single dun- coloured more or less striped primitive stock.”1 The discovery, two years ago, in a remote part of the Western Highlands of Scotland of the remnant of a variety or species adapted for a forest life, made it impossible any longer to enter- tain the view that domestic breeds had all descended from a single post-glacial species. The conclusions arrived at during 1903 were in due time published in a paper entitled “The Multiple Origin of Horses and Ponies.” 2 In this paper I enumerated the chief characteristics of Prejvalsky’s horse and the Celtic pony, and indicated in what respects these types differed from the forest variety,3 which, 1 Animals and Plants under Domestication, vol. i. p. 65, 1875. 2 Trans. Highland and Agri. Soc. Scot., vol. xvi., 1904. 3 In a typical forest horse (PI. III. 9) the coat is of a dark yellow dun colour decorated by a broad dorsal band, remnants of stripes on the face, neck, shoul- ders, body, and loins, spots over the hind quarters and bars across the legs to a short distance below the knees and hocks, beyond which the legs are black ; the mane, forelock, and tail heavy, consisting of long dark coarse wavy hair — the tail having no tail-lock ; the hind as well as the front chestnuts large, prominent, and generally oval in form, and the fetlock callosities long and often curved ; the hoofs broad, rounded in front, and wide behind ; the head massive but well proportioned, the forehead broad with ridges extending from the prominent orbits towards the occipital crest, the profile convex from below the eyes to the level of the nostrils ; the upper lip long and prehensile, and the lower lip thick and often seen projecting beyond the upper ; the ears wide, of medium length, and usually carried upright ; the neck short and thick ; the shoulders straight, ending in broad flat withers ; the back hollow and long, owing to the presence of 24 dorso-lumbar vertebrae (18 dorsal and 6 lumbar) ; the hind quarters rounded so as to form a semicircle between the croup and the feebly-developed second thigh, with the tail inserted near the centre of the half circle ; tlie limbs short and strong with thick fetlock and knee- joints, the forelegs tied in at the elbow and back at the knee, the hind limbs straight and the hocks during action kept well apart. This horse is specially adapted for living in or near forests — for frequenting narrow paths, feeding on coarse grasses, leaves, twigs, and roots, and at need readily crossing swamps and clearing obstacles — by having prominent eyes, large teeth set in powerful 1905-6.] Professor Ewart on the Tarpan. 9 because it presents all the points of the common horse of Europe, I designated the E. caballus ty pious. I also in this paper mentioned that I regarded the Celtic pony as a member of a variety “which at a very remote period branched off from the main stem and possibly reached Europe and North Africa long before the advent of the Neoliths — to become the progenitors, not only of occidental, but also of African races,” 1 and I added that, apart from its coat, mane, and tail, it is almost identical with the smaller kinds of Arabs. While in this country an effort has been made to prove that the domesticated horses have had a multiple origin, the conclusion has been arrived at in the United States that several species of horses flourished in America at or about the beginning of the Glacial Epoch. In a recent address Professor Osborn states : “It was formerly believed, for example, that the modern horse had a single line of ancestors extending back into the Eocene period ; now it appears that in North America there were always four to six entirely different varieties of the horse family living contemporaneously, including slow-moving forest- living horses with broader feet, and very swift plains-living horses with narrow feet fashioned more like the deer.” 2 So much progress has been made during recent years in working out the origin and history of domesticated horses that the time has now come when enquiries may be profitably pursued along certain definite lines. In the first place (assuming that horses have had a multiple origin), enquiries should be instituted with a view to ascertaining as far as possible the characteristics of' the post-glacial species and varieties which have taken part in forming the present domestic races and breeds ; in the next place, enquiries should be instituted with a view to ascertaining to which of the lower Pleistocene species the more immediate ancestors of the living horses are most intimately related ; and in the third place, an attempt should be jaws, and broad hoofs, and by having a conformation eminently suitable for leaping and sufficient speed to enable it in times of danger to rapidly take cover in scrub or forest. The forest horse, though a clever leaper, lias no great speed, but given time and sufficient food, it can undertake long journeys. Though often timid and spiritless, he is intelligent and docile, moves well, and is capable of carrying heavy burdens. 1 Trans. Highland Soc., 1904, p. 259. 2 Science, N. S., vol. xxi., February 24, 1905. 10 Proceedings of Royal Society of Edinburgh. [sess. made to determine from which of the ancestral forms the various domesticated breeds have inherited their more striking characters, i.e. to ascertain to which ancestral types the Shire, Clydesdale, Percheron, and other heavy breeds, the Barb, Arab, Thoroughbred, Kattiawar, and other slender-limbed breeds, are indebted for their chief peculiarities. In this paper I shall not attempt to show that either Prejvalsky’s horse, the Celtic pony, or the Libyan variety recently described by Professor Bidgeway 1 is genetically related to pre-glacial species, or entitled to be regarded as an ancestor of one or more domestic breeds. Sufficient data for a discussion of this kind is not yet available. 1 propose now, by way of clearing the ground for the investiga- tions mentioned above, to enquire whether the Tarpan (long regarded as the wild progenitor of the common horse of Europe) deserves a place amongst the ancestors of living races and breeds. Up to a certain period the only horses in Europe were wild horses ; but in course of time the horse was domesticated and utilised for various purposes, and, as one area after another was settled, the districts suitable for herds of wild horses became gradually circumscribed, with the result that except in the wild wastes of the Gobi Desert and in the vicinity of the Great Altai Mountains true wild horses no longer exist. Where, and by whom, horses were first domesticated will prob- ably never be known ; but this much is certain that in Europe, and doubtless also in Asia, tame horses from time to time ran wild, either to join wild herds, or to give rise to feral herds, such as were once common in America. Had all domestic breeds sprung from a single wild ancestor, the individuals which ran wild would have been rapidly reabsorbed without in any way modifying the original wild stock — as tame rabbits are rapidly reabsorbed by the common wild rabbit. If, however, the escaped individuals had sprung from several perfectly distinct species, the result would be that, in addition to pure wild herds, one might have come across herds having a distinct infusion of tame blood, and also herds consisting of the mixed offspring of several domestipated breeds. 1 Origin and Influence of the Thoroughbred Horse, Cambridge, 1905. 1905-6. Professor Ewart on the Tarpan. 11 The multiple origin of the domesticated breeds being assumed, it follows that, in the case of the Tarpan herds once numerous in the east of Europe, it is necessary to enquire whether they con- sisted of a wild species, were the offspring of escaped domestic breeds, or were crosses between domestic and wild varieties. The first account of the Tarpan1 we owe to Gmelin , who came across a troop near Bobrowsk during his journey through Russia between 1733 and 1743. He describes them as mouse-coloured, with a short crisp mane ; the tail always shorter than in domestic horses, sometimes full, sometimes only furnished with short hair ; The Tarpan from Yogt and Specht's Natural History of Animals. This, like Hamilton Smith’s drawing, has the mane and tail of a young foal. In no adult horse, wild or tame, is the tail as short as in the conventional drawings of the Tarpan found in modern as well as old works on natural history. the legs dark from the knees and hocks to the hoofs ; and the head thick, with the ears sometimes long, sometimes short. Since this description appeared, some Continental naturalists have regarded the Tarpan as a true wild species ; others, like Dr Nehring, considered it the last survivor of the ancient prehistoric horses of Europe modified by an infusion of domestic blood ; while not a few agreed with Pallas that the Tarpan herds might very well be the offspring of escaped domestic horses. 1 By the Tarpan I mean the mouse-dun horse of Russian and other Continental naturalists, not the so-called “true” Tarpan of Hamilton Smith (Naturalists’ Library, vol. xii., 1841). 12 Proceedings of Royal Society of Edinburgh. [suss English naturalists have as a rule adopted the view of Pallas. Sir William Flower regarded the Tarpans of the Steppe country, north of the Sea of Azov, as the nearest approach to true wild horses, while Lydekker is inclined to believe that the Tarpan of Pallas might very well be the ancestral form of the common horse, E. caballus.1 Beddard, in support of this view, pointed out that in its general build and appearance the Tarpan is highly suggestive of the wild horses sketched by primitive man upon ivory.2 Notwithstanding all that has been written on the subject since Gmelin’s time, hippologists agree with Salensky that the relation- ship of the Tarpan with wild and domestic horses has not yet been cleared up.3 During the nineteenth century very little was done towards determining the systemic position of the Tarpan ; in fact, since the Tarpan was first described, the statements of one writer (for reasons which will appear later) have often contradicted those of another. But in 1866 a Tarpan foal was captured in the Zagradoffe Steppe on the property of Prince Ivatschubei, and reared by a domestic mare. When about eighteen years old this specimen was sent to the Moscow Zoological Garden, and eventually described in a paper published by Schatiloff. This, like Gmelin’s specimen, had a somewhat coarse head, was of a mouse colour, with legs black below the knees and hocks. The mane, however, instead of being short and crisp, as in Gmelin’s specimen, was 48 cm. (over 18 inches) in length and hanging to one side of the neck. Unfortunately the description of the tail of the Moscow specimen is somewhat meagre ; but as a full mane is invariably accompanied by a forelock and a full tail in the Equidae, it may be safely assumed that the tail resembled that of the common horse. As clearly realised some years ago by Gray of the British Museum, certain vestigial structures, known as callosities, warts, or chestnuts, are of considerable taxonomic value. Warts or chestnuts, as already mentioned, are present on both the fore and hind limbs of 1 Nature, vol. lxv. p. 103. 2 Beddard’s Mammalia, p. 241. 3 The chief papers on the Tarpan are mentioned in Salensky, Monograph on Prejvalsh/s Horse, St Petersburg, 1902. 1905-6.] Professor Ewart on the Tar pan. 13 the common horse, and they also occur on the hind as well as the fore limbs of Prejvalsky’s horse; while in the Celtic pony, as in asses and zebras, the hind chestnuts are completely absent. It is especially worthy of note that though the hind chestnuts were not invariably present in Tarpans (they were absent in a Tarpan described by Krymsch), they were present in the Moscow specimen. It thus appears that the Moscow Tarpan agreed in its colour with the specimens referred to by Gmelin and Pallas, but differed in the mane and tail, in both of which, as in its callosities, it resembled the common horse, E. caballus. Two Tarpan skeletons have been preserved — one in the Museum of the University of Moscow, the other in the Museum of the Academy of Sciences, St Petersburg. The chief point of interest about these skeletons is, that as in the kyang and Prejvalsky’s horse and in certain Arabs there are only 5 lumbar vertebrae. In having only 5 lumbar vertebrae these Tarpans differed from the common horse of Europe, at least from the forest variety E. caballus typicus, in which I have invariably found 18 pairs of ribs and 6 lumbar vertebrae. Erom this striking difference in the skeleton it follows that, even should the Tarpan turn out to be a true wild species, it cannot be regarded as the sole ancestor of the common horse of Europe. As to the skull of the Moscow skeleton, Czerski came to the conclusion that it has, on the one hand, all the characteristics of Oriental horses, while on the other it approaches the Scottish breed to which belongs the pony ; in other words, the skull of the Tarpan preserved in the Moscow Museum resembles that of the Celtic pony, and its near relative, the Libyan horse. The skull of the Tarpan in the St Petersburg Museum, as Salensky points out, resembles skulls of immature specimens of E. prejvalsldi, but the bones of the limbs and limb girdles are decidedly more slender, and have less pronounced muscular ridges than in the wild horse of Central Asia. It may here be mentioned that for over a century all the horses living in a wild state in Europe, which happened to be of a mouse- dun colour, seem to have been regarded as Tarpans. If these wild horses were the offspring of several varieties, it will 14 Proceedings of lioyal Society of Edinburgh. [sess. not be difficult to account for tlie remarkable difference in the characters between the Tarpan of Gmelin and Pallas and the Tarpan of the Moscow Zoological Garden. It is hardly necessary to point out that were we to find in one and the same herd adult horses with an erect or semi-erect mane and a short bushy tail, and others with a long flowing mane and a full tail, long enough to reach the ground, we should hesitate, even if they happened to be of the same colour, to regard them as intimately related ; and if in addition some of them retained, while others had completely lost, the hind chestnuts, we should unhesitatingly look upon them as belonging to two different varieties, if not different species. Such differences, coupled with a want of agree- ment in the number of the lumbar vertebrae or in the ribs, would make in favour of adopting the view that the herd in question consisted of the feral descendants of domesticated horses, or had resulted from the intercrossing of a true wild horse with members of one or more domesticated varieties. Seeing that herds of mouse-dun wild horses no longer occur in Europe, and have not during recent years been met with in even the most remote parts of Central Asia, it might perhaps be as- sumed that the Tarpan’s place in Nature must for ever remain a mystery. This was the conclusion I arrived at when my attention was first directed to the subject. But having ascertained that, by crossing carefully selected forms, remote types are sometimes restored in all their original purity, I thought it worth while to make some experiments. In the case of pigeons, by mixing the blood of two well-marked breeds (such as the owl and archangel breeds of fanciers) and crossing the mongrels with a white fantail, I at once obtained birds which closely resembled Columba livia , the recognised wild ancestor of all the tame pigeons. Bearing this and like experiments with zebras, dogs, rabbits, etc. in mind, I selected for my Tarpan experiments a mouse-dun Shetland pony mare, which seemed to me to be a blend of at least three varieties — in its head (PI. I. 1) it suggests the wild horse;1 in 1 Tlie wild horse ( E . prejvalsTcii ), like Grevy’s zebra (E. grevyi), has a very long head, the distance between the eye (inner canthus) and the nostril being decidedly longer than in a forest horse eight inches higher at the withers, and 1905-6.] Professor Ewart on the Tarpan, 15 its mane, tail, and trunk it takes after the forest variety (PI. III. 9) ; while in the limbs and hoofs it approaches the Celtic pony. This mare was crossed with a black Welsh pony, which belongs to an ancient British race and doubtless has in its veins not a little Celtic blood. The first foal, black like the sire but Celtic in make, failed to throw any fresh light on the question at issue ; it however sup- ported the view that, notwithstanding the large, heavy head, there was Celtic blood in the Shetland mare. Though in the first foal the Celtic blood prevailed, the second foal by the same sire has developed into an animal, now three years old, which, though bred in Scotland, will, I believe, be regarded by Continental naturalists as typical a Tarpan as ever roamed the Russian steppes (PI. I. 2, 3 ; PI. II. 5). This Scottish Tarpan, a mouse-dun with black points, has a distinct dorsal band (10 to 15 mm. in width) and faint bars above the knees and hocks, a somewhat heavy head, but a short body and well-formed limbs. The mane, of a light colour along each side but dark in the centre, is semi-erect, some of the hair arching to the right, some to the left, and some forwards between the ears to form an imperfect forelock. The mane, which relatively still longer than in the Celtic pony, while owing to the forehead being decidedly convex, from side to side as well as from above downwards, the eyes look outwards rather than forwards. Like the Celtic pony, Prej valsky’s horse is of a yellow-dun colour, with dark points and only vestiges of stripes — the dorsal band being narrow and the leg bars faint, especially during winter. Unlike the Celtic pony and the forest horse, the mane is upright during at least autumn and winter ; in spring it may be only semi-erect ; in young individuals out of condition it may, however, arch to one side of the neck. The distal end of the dock carries relatively few long hairs, the basal portion short hairs, while the middle section consists of hairs long enough to form a fringe around the hairs growing from the end of the dock (PI. II. 4). As in the forest horse there are four chestnuts and four ergots, but the hoofs are relatively longer and decidedly more contracted at the “ heels.” The ears are long and usually project obliquely outwards. In the skeleton it is especially noteworthy that there are only five lumbar vertebrse, and that owing to the sacrum being nearly horizontal the croup droops but little and the tail is set on unusually high, as in many Arabs. The description of the Asiatic “true ” Tarpan given by Hamilton Smith fits fairly well with Prejvalsky’s horse. This agreement between the wild horse now living in the Gobi and the Tarpan of the Tahtars has been specially dwelt on by Professor Ridgeway. It is, however, well to bear in mind that Hamilton Smith’s drawing of the Tarpan is about as unlike Prejvalsky’s horse as any drawing well could be. 16 Proceedings of Royal Society of Edinburgh, [sess. resembles that of zebra-horse hybrids, conforms to the description of the mane given by Pallas, but differs from the short crisp mane of Gmelin’s specimen, and still more from that of the Moscow Tarpan, which, it will be remembered, reached a length of 45 cm., and hung to one side of the neck. In the dun Shetland dam, the mane lies close to the right side of the neck, but never exceeds a length of 35 cm. In the Scottish Tarpan the mane, from 15 to 2 7 '5 cm. in length, is either nearly upright, or, as already mentioned, [■ G . A . Excart . A three-year-old wild horse (E. prejvctlskii) from the Great Altai Mountains, photographed September 1904 ; note erect mane and that the tail reaches the ground. arches outwards well clear of the neck (PI. II. 2, 3), whereas in a Fetlar (Shetland- Arab) pony of the same age, the mane reaches a length of 45 cm. and clings to the side of the neck.1 The tail 1 Whether the mane is long or short depends on two things : first, on the rate of growth of the hair ; and second, on how long the individual hairs persist. In the Celtic pony the mane hairs grow at the rate of nearly one inch per month ; in the forest variety they may persist until they reach a length of several feet ; even in Prejvalsky’s horse they may continue to grow until they are long enough to arch to one side of the neck, but eventually in the wild horse, as in zebras, these long hairs give place to short ones. 1905-6.] Professor Ewart on the Tar pan. 17 of the new Tarpan (PI. II. 5) is even more remarkable than the mane. The dock, which is 27*5 cm. in length, is furnished with three kinds of hair. The basal portion for 6 '5 cm. carries fine hair nearly circular in section, which, except in the part continuous with the dorsal band, is almost colourless ; the middle portion of the dock — about 13’75 cm. — carries thicker hair, slightly oval in section, with a thick cortex containing in some cases a considerable amount of pigment ; from the terminal part of the dock — about 7*5 cm. — spring coarse black hairs which are now long enough to reach the ground. These long hairs are oval in section, have a very thick cortex, and only a small central axis or medulla. The fine, short, light-coloured hairs (7 ’5-15 cm. in length) at the base of the tail form a conspicuous somewhat lozenge-shaped bunch (Pl. II. 5) ; the thicker hairs growing from the middle section of the dock reach a length of 30 cm. They emerge from under the light-coloured root hairs and expand to form a sort of fringe, from which escape the relatively few long black hairs of the distal part of the dock. In having a limited number of long hairs growing from the distal end of the dock, this cross-bred pony decidedly differs from the Celtic as well as from the forest types of horses. The interest of the tail in the Scottish Tarpan is not so much that it suggests a mule, as that it has a very striking resemblance to the tail of Prejvalsky’s horse (PI. II. 4). The only difference is that in the true wild horse the upper or light-coloured section of the tail is longer than in the Shetland- Welsh cross, which has, in fact, the kind of tail one would expect in a Prejvalsky hybrid in which the wild blood was dominant. I may here mention that I have had under observation for some time two imported Mongol ponies, and a half-bred Mongol colt. In both ponies and colt the tail is peculiar — is about intermediate between that of the restored Tarpan and a forest horse (PI. III. 9). Further, in the colt, the mane, though of the usual length, keeps clear of the neck.1 I may also mention that in a dark brown pony from the Outer 1 This condition of the mane is not unknown in cross-bred horses, and it was specially noticeable in two dun ponies, probably of Spanish descent, which I saw last winter in Mexico. PROC. ROY. SOC. EDIN. — YOL. XXVI. 2 18 Proceedings of Royal Society of Edinburgh. [sess. Hebrides with a massive head, and long mane like the Moscow Tarpan, the tail consists of three distinct sections ; and that in a mare (PI. III. 10) obtained by crossing an Arab with a yellow-dun Norwegian (fjord) stallion, and in a colt out of a half Arab mare by a similar stallion, the tail forcibly reminds one of the Tarpan obtained by crossing a Shetland mare with a Welsh stallion.1 A study of the mane and tail of the Shetland- Welsh cross, and of certain other crosses and breeds, strongly suggests that we must include amongst the ancestors of our domestic horses a species having a mane and tail such as we find in the wild horse still living in Central Asia. In the body hair and the foot-locks the Scottish Tarpan closely resembles the wild horse. Further, it resembles the wild horse in having a very short flank feather, but differs in having the face whorl (PI. I. 3) situated above the level of the eyes, as in the Celtic pony: in Prejvalsky’s horse as in the kyang this whorl lies well below the level of the orbits. In the Shetland mare the dorsal band is nearly as narrow as in the Celtic pony (PL II. 6) ; the right hind chestnut measures P5 cm. by *4 cm., while the left is only *5 cm. in diameter; the front ergots are absent, and the hind ergots are very small. In all these points the Shetland mare approaches the Celtic type. In the Scottish Tarpan the front ergots are small, the hind normal ; the front chestnuts are oval as in the wild horse, but decidedly smaller, while the hind chestnuts are only one-fifth the length of those in the wild horse. Finally, in the head, ears, form of the limbs and hoofs, the Tarpan-like Shetland- Welsh cross is as nearly as possible intermediate between a wild horse and a Celtic pony. Of the skeleton it is, of course, impossible to speak, but judging by the shortness of the trunk, the form of the head, and the conformation of the limbs, the probability is that there are only five lumbar vertebrae, as in the Moscow and St Peters- burg skeletons, and that the skull and limb bones resemble those of a young Prejvalsky horse. After very full consideration, Salensky came to the conclusion that the Tarpan is a type 1 Seeing that the mane and tail in various breeds, in the Old World and also in the New, often suggest the Tarpan, it may be inferred that the ancestors of the Tarpan were intimately related to the ancestors of some of the domesticated breeds. 1905-6.] Professor Ewart on the Tarpan. 19 specialised more to the side of E. caballus than to E. prejvalskii. Doubtless Salensky, in coming to this conclusion, was influenced not a little by the long mane and the full tail of the Moscow Tarpan. General Conclusions. When all the available facts are taken into consideration there seems no escape from the conclusion that the Tarpan, once common in the east of Europe, cannot be considered as a true wild species. Further, it may he assumed that the Tarpan herds were derived from at least three primitive stocks, viz. : (1) From a variety or species identical with or closely related to the wild horse ( E . prejvalskii) still surviving in Central Asia; (2) from a variety having the characteristics of the Celtic pony — E. c. celticus ; and (3) from a variety resembling the forest horse — E . c. typicus. It is only by assuming the multiplex origin of Tarpans that it is possible to account for some of them having a heavy head, long ears, a nearly upright mane, a mule-like tail, and five lumbar vertebrae, thus suggesting E. prejvalskii ; for others, wanting the hind chest- nuts and possessing a skull like that of certain Scottish ponies, thus suggesting E. c. celticus ; and for others having a thick head, full mane and tail, and hind as well as front chestnuts, thus suggesting E. c. typicus. By experiments now in hand I hope to settle what part Prej- valsky’s horse has taken in forming the Tarpan. If I succeed in showing that crosses between Prejvalsky’s horse, and either the forest, Celtic, or Libyan variety are practically identical with the cross between the Shetland mare and the Welsh pony stallion, I shall prove that at least certain of the domesticated breeds are indebted to Prejvalsky’s horse for some of their characteristics, and at the same time bring additional evidence in support of my view that domesticated races have had a multiple origin, and include plain as wTell as striped forms amongst their less remote ancestors — have not, in fact, as Darwin thought, descended from a single dun-coloured more or less striped primitive stock. [Towards the cost of this investigation contributions were received from the Carnegie Trustees, and from the Earl of Moray Research Fund of the University of Edinburgh. 20 Proceedings of Royal Society of Edinburgh. DESCRIPTION OF FIGURES. [SBSS. Plate 1. Fig. 1. — Head and shoulders of the mouse-dun Shetland dam of the Scottish Tarpan. In the form of the head, position of the eyes, and distance between the eyes and the nostrils this mare resembles Prejvalsky’s horse. A yearling filly, out of this mare by a Hebridean pony, instead of resembling a Tarpan, strongly suggests a Celtic pony. Figs. 2 and 3. — The mouse-dun Tarpan-like cross between the Shetland mare (fig. 1) and a black Welsh pony. This pony, though a cross, looks as if it belonged to an old-established race. It has a striking, well-formed, massive head, well-placed ears, full eyes, good quarters, and excellent limbs. The mane is, however, short and semi-erect, while the tail consists of three kinds of hair which differ in structure, thickness, colour, and arrangement. From photographs taken September 1905. Plate II. Fig. 4. — Hind quarters and tail of a three-year-old wild mare (E. prejvalskii) from a photograph taken in September 1905. The hind quarters and limbs are better formed than in the male wild horse (page 9) and the dorsal band is more distinct. In the upper part of the tail the hair, light in colour and relatively fine, o-rows obliquely outwards from the caudal portion of the dorsal band ; the hair of the middle part of the tail, darker and stronger than that of the root, lies nearly parallel with the dock and reaches to the level of the hocks; the hair of the tip, black, coarse and scanty, but long enough to reach the ground, emerges from within the hair forming the middle part of the tail. Like the hair of the mane, the light hair at the root of the tail is shed annually. Fig. 5. — Tail and hind quarters of the Scottish Tarpan from a photograph taken at the same time as fig. 4. As in the wild mare the hair of the tail consists of three portions. The basal portion only essentially differs from the corresponding portion in fig. 4, by being of less extent and lighter in colour, the middle portion is also lighter in colour and more plentiful than in the wild mare, while the hair growing from the end of the dock in the Tarpan very closely agrees in colour and amount with the terminal portion of the tail in Prejvalsky’s horse. Fig. 6. — Tail of a typical yellow dun Celtic pony, from a photograph taken in August. Fig. 7. — Tail of the same pony, from a photograph taken in January. In fig. 7 the tail-lock is seen at its maximum growth ; in fig. 6 a new crop of hair is growing to take the place of the long winter hairs which were gradually shed during the summer. gf*esee? [Yol. XXYI. c . 'Jfete Proc. Roy. Socy. of Edin.] Prof. J. C. Ewart. — Plate 1. [G. A. Ewart. Proc. Roy. Socy. of Edin. ] [Yol. XXYI. !. prejvalskii. 5. The Scottish Tarpan. 6-7. Celtic Pony, Summer and Winter. 8. Arab Mare. Proc. Roy. Socy. of Edin. ] [Vol. XXVI, Fig. 9. Fig. 10. [G. A. Ewart. Prof. J. C. Ewart. — Plate III, 1905-6.] Professor Ewart on the Tarpan. 21 Fig. 8. — The tail of a grey Arab taken at the same time as figs. 4 and 5. There is no marked difference between the middle and terminal parts of the tail as in the Tarpan, but a few hairs at the root of the tail are short and shed annually ; in the most proximal part of the tail this Arab agrees with a half-bred Celtic pony in my stud. Plate III. Fig. 9. — Photograph of a typical dark yellow dun forest horse from Western Ross-shire. The mane, forelock and tail, long and heavy, consists of strong wavy hair. The tail, which shows no vestige of a tail-lock, instead of looking a continuation of the vertebral column, looks as if it had been inserted between the rounded hips. This horse bears a close resemblance to the Gud- brandsdal breed of Norway, and it seems to have entered largely into the formation of the Norwegian Fjord horse. Figures of these Norwegian breeds will be found in Dr Marshall’s paper, p. 32 of this volume. Fig. 10. — Photograph of a cross between a grey Arab (fig. 8) and a light yellow dun Fjord stallion from Trondhjem. This cross has the long body of a forest horse, a small head, uniting the points of the Celtic and Libyan varieties, and a tail which in some respects resembles the Tarpan (fig. 5). In a cross out of a well- bred Connemara mare by a similar Fjord stallion there is a typical Celtic taillock, and the hind chestnuts are absent. A study of crosses obtained by Fjord stallions has led me to conclude that the more typical Fjord horses are a blend of the Celtic and forest varieties. ( Issued separately February 12, 1906.) 22 Proceedings of Royal Society of Edinburgh. [sess. The Horse in Norway. By Francis H. A. Marshall, M.A. (Cantab.), D.Sc. (Edin.), Carnegie Fellow and Lecturer on the Physiology of Reproduction in the University of Edinburgh. (With Two Plates.) (Read November 6, 1905.) Writers on the origin of the horse and its different breeds have been accustomed to refer to the horses of Norway as though they belonged to a single type. Thus Sanson, in his Zootechnie , includes the horses and ponies of that country in his sub-species Equus caballus hibernicus, to which he also refers the various ponies of the British Isles, the Breton in France, and the horses of Iceland and Sweden. The late Captain Maurice Hayes, in his well-known work on the Points of the Horse , refers collectively to Norwegian and Swedish horses as though they belonged to one natural group. Professor Ewart, in describing a typical repre- sentative of what he calls the Forest type, which, as he shows, differs essentially from the newly discovered “ Celtic pony,” alludes provisionally to the former as the “Norse horse,” because it is common in Norway. Moreover, Professor Ridgeway, in his recently published book on the Origin and Influence of the Thoroughbred Horse , appears to regard the Norse horse as repre- senting a single type which has undergone greater or less alteration through the introduction of Libyan or other foreign blood. As a result of a recent visit to Norway, during which I did not neglect opportunities for studying the native breeds of horses and ponies and their history, it has become evident to me — what indeed is well recognised by all horse-breeders in that country — that the Norwegian horses at the present day belong to two quite distinct types, which, howmver, have intermixed to a considerable extent. These two types are represented by the “pure” fjord horse * and the Gudbrandsdal horse. * It is doubtful whether any of the existing fjord horses are really pure, i.e. unaltered by admixture of Gudbrandsdal or foreign blood. 1905-6.] Francis H. A. Marshall on the Horse in Norway. 23 It will he remembered that Professor Ewart, in his original paper on the “ Celtic Pony,” read before this Society in 1902, described this animal as a small-headed horse with short ears, prominent eyes, slender legs, small joints, and a fringe of short hairs on the upper part of the tail, and without hock callosities (chestnuts on the hind limbs) or ergots (fetlock callosities)'. The most typical colour was light dun. The Celtic pony, as thus described, was stated to occur at the present day in certain of the Western Isles of Scotland, in the north of Ireland, and in the Faroes and Iceland. It was regarded as the living repre- sentative of a primitive small horse, whose range extended over North-West Europe, and whose fossil remains are found in the Brighton Elephant Bed. Living contemporaneously with this ancient Celtic pony, but having perhaps a far more extensive distribution, there was a robust, large-jointed “forest” horse, from which, according to the same authority, the cart-horses and other heavily built horses are largely derived. It will he well to state at the outset that I regard the Fjordhest as having been partly descended from the same stock as Professor Ewart’s “Celtic pony,”* and the Gudbrandsdal horse as repre- senting the “ forest ” or “ cart-horse ” type. The “pure” fjord horse is found in all the fjord districts of Western Norway . The largest specimens are said to occur in the Bomsdal and in the neighbourhood of Laerdalsoren. These may reach about thirteen hands high. The Fjordhest of Northern Norway is considerably smaller and more roughly coated. So far as I have observed, “ Celtic ” characters predominate in all the existing fjord horses. The forehead is flat, and the ears are relatively short, while the limbs are much slenderer, and the joints much smaller than in the Gudbrandsdal horse. The hoofs are generally wide at the heels, and almost circular in outline. Among the suggestions and regulations drawn up for the guidance of judges at the horse show at Lillehammer in 1857, * Stejneger appears to regard the Fjordhest, the Celtic pony, and also the Tarpan as representing one sub-species, but the evidence he adduces in support of this view is not very substantial. 24 Proceedings of Royal Society of Edinburgh. [sess. it was recommended that in the fjord horse “ the head should he comparatively small, with well-shaped and well-placed ears, the eyes should be large and the nostrils wide, and the neck must have a suitable arch, and he broad, hut it should be fine at its junction with the head.” “ The colour should he preferably light dun (horkede) or light brown (blak), with black mane and tail, the legs black below the knees and hocks, and the ear-tips black, with cross stripes on the knees and hocks, and with an eel-mark down the centre of the hack.”* This colour is prob- ably the commonest amongst fjord horses at the present day. Other colours — browns and greys — and a darker shade of dun, are not infrequent, and I have also seen mouse-coloured ponies. Black fjord horses are said to he extremely rare, and it is doubtful whether any pure ones exist. Nearly all the light-coloured ponies have a black dorsal stripe, and a large proportion of pure and partly bred fjord ponies (probably considerably over 50 per cent, of those which I saw) have cross stripes on the legs. Shoulder stripes are also occasionally present, at any rate in certain of the partly bred fjord horses. Professor Ridgeway, however, says that Dr Venn, F.R.S., and Mr J. A. Venn, who, on a visit to Norway in 1904, examined for him a large number of ponies at various coast towns, did not meet with a single instance of striping. In typical specimens of the Fjordhest, there is, in winter time, a well-marked caudal fringe or tail-lock. This is not shown in the pony in the photograph (fig. 1), which was taken in the month of August, since the fringe is shed in the summer time, leaving only a bunch of very short hairs in the upper part of the tail. I was informed, however, that in this particular individual a tail-lock consisting of hairs about six inches in length was present in the preceding winter. This pony, like several other fjord horses which I have seen, had no trace of ergots, but a small callosity was present on each hind leg. In numerous other fjord horses which I examined, the hock callosities were extremely small, being frequently scarcely larger than a pin’s head, but I only succeeded in finding a single case in which these vestiges had disappeared altogether. This was in a pony at Tonsaasen. In other characters, and more particularly in the length of * Gudbrandsdal Stud-book (see under Petersen). 1905-6.] Francis H. A. Marshall on the Horse in Norway. 25 the body, in the roundness of the quarters, and in being much “tied-in” at the elbow, the Fjordhest of to-day is frequently similar to the Gudbrandsdal and other horses which belong to the “forest” type. The Fjordhest has been much intercrossed with the Gudbrandsdal horse, and the great majority of the ponies seen in Christiania and the Norwegian towns, and those usually employed to draw the carrioles and stolkjerre, are partly bred animals, the fjord char- acters generally strongly predominating. I have found no positive evidence, however, of fjord horses having been crossed with any other breed than the Gudbrandsdal ; but since, as I shall show later, the latter breed has from time to time received infusions of foreign blood, it is probable that the fjord horse has been at least indirectly influenced, but probably to no great extent. The ponies used in the sledge races, which take place in winter at Bergen and Yossevangen, are stated to be pure-bred (according to modern ideas), and only such are eligible. In 1844 an attempt was made by the Norwegian government to establish a stud of fjord horses at Hjerkin, on the Dovre fjeld, with the object of improving the breed by judicious selection but without having recourse to intercrossing with foreign blood. Horses were advertised for, and it was announced in the advertise- ment that dun-coloured (borkede) animals would be given the preference. Next brown (brun) horses would be considered ; and lastly, yellow (gul) horses with white mane and tail would be accepted. Great difficulty was experienced in obtaining animals considered sufficiently good for the purpose, and in the end the scheme proved a complete failure. The stud was broken up, as wTas also a branch establishment at Foktuen (Dovre), at the end of 1858. The report on the experiment states : — “It is the usual experience that the Fjordhest does not thrive on coming to the mountains ; it is without exception attacked by strangles, and for the first couple of years is unfit for hard work. Subsequently it becomes enduring and strong in proportion to its size, but the general opinion in these parts is that it is in every respect far inferior to the Gudbrandsdal horse.” *. The third figure represents an “Udganger” pony or Nordlands- * Gudbrandsdal Stud-book (Petersen). 26 Proceedings of Royal Society of Edinburgh. [sess. hest. This type, which is now said to he extinct, is stated to have been in most points similar to the Fjordhest, from which, according to the reference to it in the preface to the Gud- brandsdal stud-book, it differed chiefly in being smaller and more roughly coated, being subjected to severer conditions of life. Probably the “ Udganger ” pony was almost purely “ Celtic ” in its characters.* The Gudbrandsdal horse, as will be seen from the figure, is very different from the fjord horse. The head is large and heavy, the legs are stout and the joints large, the distance between the nostril and the orbit is relatively appreciably larger than in the Fjordhest, and the lower lip frequently projects beyond the upper, as is sometimes seen in English cart-horses. The quarters are very much rounded. Hock callosities are invariably present so far as I have seen, and are generally well developed. Ergots are also present. There is no suggestion of anything of the nature of a tail-lock or fringe of short hairs on the upper part of the tail. The forehead is not generally flat, as in the Fjordhest, but has two ridges which meet below the forelock, and are sometimes very well marked. The Gudbrandsdal horse is also much bigger than the Fjordhest, and may be as much as sixteen hands high. The following is a translation of Lindeqvist’s description, written in the middle of the last century : — “ The Gudbrandsdal horse, by careful breeding and under peculiar local conditions, has developed into the noblest and most valuable representative of the Norwegian horses. It supplies excellent agricultural and artillery horses, and substantial, or even (as things go in Norway) handsome carriage horses.” In another place he wrote: — “The Gudbrandsdal horse has such a peculiar stamp that it is usually recognisable anywhere, and amongst a number of horses of different types. The most prominent specimens are about ten * Professor Ridgeway, who also publishes this figure, says that it represents the last individual of a small extinct breed from the Lofoden Islands (now stuffed in the Bergen Museum). As a matter of fact the “ Loftohest” in the Bergen Museum is a dark-coloured individual. Professor Ridgeway regards the animal here figured as a small horse of the heavy type and not a “ pony,” but I am unable to share this opinion. From inquiries at Messrs Knudsen’s of Bergen, who photographed the animal in question, I learn that this was done in the month of April, so that the pony would hardly be in its summer coat, as stated by Mr Ridgeway. 1905-6.] Francis H. A. Marshall on the Horse in Norway. 27 •quarters (157 centimetres) high,* are usually of brown or dark- brown colour, and are distinguished by their full, round forms ; pointed, shapely, and upright ears ; expressive physiognomy ; compact bone substance and firm muscles ; broad and strong knee and hock joints; sinewy legs, and thick, hard and tough 110018.” The chief defect is the “peculiarly short and unbending neck.” “This cannot be considered objectionable in the case of working horses, but for the better class of carriage horses, and especially for riding horses (for all of which purposes the Gudbrandsdal is used), this short and thick neck is an undoubted defect.” Lindeqvist regarded the Norfolk horse as representing the ideal which the Gudbrandsdal should emulate, but to which, however, it has never attained. Roller’s description, published in 1886, is very similar. The Gudbrandsdal horse belongs especially to the Hedemarken and Christiania amts, its breeding ground being the great valley from which it takes its name. Those from the lower parts are sometimes known as Doleheste, and those from the upper as Nordheste. The latter are described in the Gudbrandsdal stud- book, the first volume of which was published in 1902, as being the finer and the most sure-footed. Gudbrandsdal horses are bred also to a greater or less extent throughout all the south- eastern provinces, as well as in the Trondhjem’s amt and some •other places on the west coast. The attempt to divide Gud- brandsdal horses into two varieties has in recent years been practically ignored, for the large, heavy, heavy-legged Dolehest has gained the victory everywhere over the lighter breed, so that the former at the present day may be regarded as the typical representative of the Gudbrandsdal race. In former times Gudbrandsdalen was practically one huge stud. In 1729, according to Hiorthpy, there were 157 stallions and 1563 mares. The stallions used to fight for the possession of the mares, whom they also defended from wolves or other enemies. It appears, therefore, that formerly the horses were subjected to rigorous natural and sexual selection. Pontoppidan says : “ The Norwegian stallion shows much courage in fighting wolves and * The height is measured in Norway by tape measure, from over the withers to the hoof. 28 Proceedings of Royal Society of Edinburgh. [sess. bears, especially the latter, which he attacks, using his forelegs like a pair of drumsticks, and commonly being the master, which some people at Court doubted, until Statholder Wibe, in King: Frederick IV. ’s presence, made a trial with one of his carriage horses, which at Fredericksborg at once attacked a bear which was let loose, and left no life in it.” The mares, on the other hand,, are said to have been frequently killed by wolves and bears. As already mentioned, the horses of the Gudbrandsdal have from time to time received infusions of foreign blood. About a.d. 1040, nine Icelandic stallions were presented by King Einar.* In the race between Magnus Blinde, on horseback, and Harold Gille, on foot, in 1128, it is recorded that Magnus rode a very fast horse from Gothland. Also the Duke Skule, who fled from the battle of Oslo, when his horse was shot under him, was procured another from Gothland. These are isolated cases of importation. It is not, however, until a much later period that there is evidence of importation on any considerable scale. We may probably assume that the Danish and German troops, who were stationed in various country districts in the seventeenth century, brought with them Danish or other foreign horses, and some of these may have bred with the native horses. Indeed there is a record that, about 1650, in the district east of the Christiania fjord, the horses were mixed both as to size and colour, on account of crossing with Danish stallions brought by the mounted troops. It is also stated by a contemporary writer that horses were imported by a priest, at great expense, into North Gudbrandsdalen from Denmark, in the latter half of the seven- teenth century. There are also a few records in the Gudbrandsdal stud-book of instances of importation in the eighteenth century, and more numerous ones in the last century. Thus it is stated that Spanish stallions were brought to South-East Norway, while an English thoroughbred named “Odin” was brought from London, costing £257, and is said during the first four years to have served over a hundred mares. Since then various others have been imported, but it is unlikely that any of these exercised much influence over the Gudbrandsdal breed as a whole. * This, and certain of the information which follows, were derived from the Gudbrandsdal Stud-book. 1905-6.] Francis H. A. Marshall on the Horse in Norway. 29 In 1845 a government committee reported that it was almost impossible to find any horses which were beyond doubt of unmixed Norwegian descent. In curious contrast to this conclusion is the rule drawn up in 1872 for government horse shows outside the fjord districts, that only horses of pure Norwegian ancestry would be eligible. The Gudbrandsdal horse at the present day is usually either black or brown. I was present this year at the great annual horse sale which takes place for a week in August, at Lillehammer, where I had an excellent opportunity of seeing a large number of what are regarded as pure-bred Gudbrandsdal horses, and these were almost invariably either black or brown. Statsconsulent Borchgrevink told me that a pure Gudbrandsdal could not be dun or light coloured, the dark colour being now regarded as one of the essential characteristics of the breed. It is interesting to note, however, that there is every indication that the dark colour has resulted from the introduction of foreign blood, for there are numerous proofs that, in the early part of the eighteenth century, dark-coloured horses were the exception. Thus the Dragoon horses in the Gudbrandsdal in 1711, appear to have nearly all been light in colour. It is also recorded that of those in Hedemarken not more than 10 or 11 per cent, were brown or black. Pontoppidan says that the Gudbrandsdal horses were either yellow with black points, and a black “ eel ” down the back, or else were brown-grey or mouse-coloured. “ Black ones are very seldom seen — scarcely one in fifty.” Schytte, writing a little later, says that the best horses in Norway are the Gud- brandsdal, “especially the yellow ones with black legs and an ‘eel’ down the back.” It is to be noted that when the Gudbrandsdal and fjord horses are intercrossed, the offspring, although shaped much like the pure Gudbrandsdal horses, are very frequently light dun with stripes, but of course it is quite arguable that this character is derived solely from the fjord parent. Professor Bidgeway has pointed out that in Beowulf , which dates from the eighth century, the horses mentioned are dun or light-coloured, while at a later date, from the names given in the appendix to Sijmon’s Edda , we get indications of the existence 30 Proceedings of Royal Society of Edinburgh. [sess. of dark - coloured horses, which, like those mentioned in the Icelandic sagas, Professor Ridgeway regards as affording evidence of the importation of Libyan blood. He considers also that the existence of striped dun ponies points similarly to the conclusion that such an importation had occurred. The first of these con- clusions appears to me to be an extremely probable one, but, a& we have seen, black horses were very rare until far down into' the eighteenth century, so that it is unlikely that the infusion of dark Libyan blood had been at all considerable, and I cannot agree with Mr Ridgeway in the view that the striped dun ponies show any evidence of having had a Libyan origin. There can be little doubt that the two types of horses which occur in Iceland at the present day are derived respectively from the ancestors of the original fjord horse and from those of the unaltered Gudbrandsdal horse. One of the commonest and most typical colours among existing Icelandic horses is light dun with a dorsal stripe. I have searched the “ Landn&mabdc ” or record of the Viking settlements in Iceland.* and although there are numerous references to the importation of horses and other domestic animals, I can only find one to a horse’s colour. This is a reference to a stallion which was “apal-gr&r,” which is, I suppose, a shade of dun. But, as the saga of “Burnt Hjal” shows, horses of other colours than dun existed in Iceland in the tenth century, and practically all colours are represented there at the present time, though light dun is probably still the most frequent. Professor Ridgeway, in his recent work, has shown that there is Libyan blood in the ponies of the Hebrides and the north of Ireland, while in a paper published two years ago on the “ Horse in Iceland and the Faroes,” Mr Helson Annandale and I have pointed out that a large proportion of the Horse colonists in those islands had been living in the Hebrides or Ireland before they removed to the islands further north, and that it is thus extremely probable that the horses of the Faroes and Iceland are derived partly from the British Isles. So that it is not unlikely that the Icelandic and Faroe ponies have received infusions of Libyan blood through the importation of animals from the Hebrides and Ireland. In * See Yigfusson and York Powell, Origines Islandicce. 1905-6.] Francis H. A. Marshall on the Horse in Norway. 31 this way the red colour of many of the Faroe ponies may probably be accounted for. But since the “ Celtic ” ponies of the British Isles are not altogether dissimilar to the fjord horses, it is probable that they themselves have had a Scandinavian origin,* and been subsequently crossed with other horses of Libyan extraction. In view of these considerations, it seems to me by no means impos- sible that the primitive small-headed pony of North-West Europe (the horse of the Brighton Elephant Bed) was more closely represented in recent times by the dun-coloured Fjordhest (now probably extinct in its pure form except in Iceland) than by the red or dark-coloured Hebridean and Faroe ponies ; and I am disposed to agree with Professor Bidgeway that certain of the characteristics of the existing “ Celtic ” ponies (whether in the Hebrides, or in the Faroes, or in Iceland), including possibly the absence of hock callosities and ergots (which has been shown to be a Libyan character), may be due in part to an infusion of Libyan blood, f In conclusion, I wish to thank Professor J. C. Ewart for valuable suggestions and information, as well as to express my indebtedness to Lord Melville, H.B.M. Consul-General at Christiania, Mr Borchgrevink of the Norwegian Agricultural Department, Mr T. Townshend Somerville of Christiania, Mr Gran, jun., of Bergen, Dr Appelov of the Bergen Museum, Mr Johann Fleischer of Yossevangen, and others who rendered me assistance in my investigations in Norway. * This is also rendered very probable by the fact, mentioned above, that a number of the Norse colonists settled for some time in the British Isles. t Mr Lydekker, in a recent review of Professor Ridgeway’s “ Thoroughbred Horse” ( Nature , December 7th, 1905) expresses a doubt as to whether the “Celtic” and “forest” varieties, as well as E. przewalskii, should not be regarded as representing a single northern dun-coloured type, which is to be contrasted with a southern and eastern bay type, including the Barbs and Arabs. I am unable to share in the opinion that the first three varieties form one natural group. Representatives of the “forest” or cart-horse type are not 1 1 typically small animals with . . . tails often imperfectly haired at the base,” neither is the “Celtic” pony ordinarily large-headed and intractable in temper. (For Mr Lydekker’s views concerning the origin of the Barb and Thoroughbred, see Knowledge , August 1904.) 32 Francis H. A. Marshall on the Horse in Norway, [sess. REFERENCES TO LITERATURE. Dasent, The Story of Burnt Njal, Edinburgh, 1861. Ewart, “ The Multiple Origin of Horses and Ponies,” Trans. Highland and Agricultural Soc., 1904. Hayes, Points of the Horse , 3rd edition, London, 1904. Hiorth0y, Physisk og Ekonomisk Beskrivelse over Guldbrands- dalens Provstie i Aggerhuus Stift i Norge , med Robber e, Kjpben- havn, 1785. Roller, Handbok for Hdstvanner, Kristiania, 1886. Lindeqvist, Vor Tids bedste Husdyr-Racer, Kristiania, 1860. Indberetninger til Departementet for det Indie om haus Virksomhed for Husdyravtens Fremme i Aaret 1858, Kristiania, 1859. Marshall and Annandale, “The Horse in Iceland and the Faroes,” Proc. Camb. Phil. Soc., 1903. Petersen, Introduction to Stambog over Heste og Gudbrands- dalsk Rase , vol. i., Kristiania, 1902. Pontoppidan, Det forste Forsog paa Norges naturlige Historic , oplyst med Kaaberstykker , Kjpbenhavn, 1753. Ridgeway, The Origin and Influence of the Thoroughbred Horse , Cambridge, 1905. Sanson, Traite de Zootechnie , 4th edition, vol. iii., Paris, 1904. Schytte, Danmarks og Norges naturlige og politiske Forfatning, Kjpbenliavn, 1777. Stejneger, Den Geltiske Pony , Tarpanen ogfjordhesten, Naturen, 1904. Vigfusson and York Powell, Origines Islandicce, vol. i., bk. 1, “ Landnama-B6c,” Oxford, 1905. DESCRIPTION OF THE PLATES. Plate I. Fig. 1. “ Fjordhest,” Bergen, 1905. This animal, which was scarcely more than one-half “ Celtic ” in its characters, was regarded as having all the points of a pure Fjordhest. Fig. 2. Gudbrandsdal horse, Lillehammer, 1903. Plate II. Fig. 3. “ Udgangerhest,” or “ Nordlandshest,” Bodo, 1895. This pony was probably almost purely Celtic. Fig. 4. Hebridean pony. Figs. 1 and 3 have appeared in the Gudbrandsdal Stud-book, vols. i. and ii. Fig. 4 is from a block kindly lent by Professor Ewart. ( Issued separately February 12, 1906.) Proc. Roy. Socy. of JEdin.] [Vol. XXY1. Fig. 2. Mr Francis H. A. Marshall. [Plate I. Troe. Roy. Socy. of JEdin.] [Yol. XXVI. Fig. 3. Fig. 4. Mr Francis H. A. Marshall. [Plate II. 1905-6.] Electric Oscillations and Magnetic Properties of Iron. 33 Notes on the Effect of Electric Oscillations (co-directional and transverse) on the Magnetic Properties of Iron. By James Russell. (MS. received December 19, 1905. Read November 20, 1905.*) In the present communication I propose (1) to give some account of preliminary experiments in which the effects of co- directional and transverse oscillations upon the magnetic properties of iron are directly compared with each other and with the normal curves without oscillations. Numerous experiments have been made with co-directional oscillations alternately assisting and opposing the field ; also with oscillatory currents in iron wires longitudinally magnetised, giving rise to transverse oscillatory effects. I am not, however, aware that any experiments have previously been made in such a way that the effects of co-directional and transverse oscillations can be directly compared, without intro- ducing conditions so different as those which must obtain when oscillatory currents pass in the magnetic metal experimented with. I propose (2) to discuss the results thus obtained in the obvious bearing which they appear to have upon a few typical forms of magnetic detectors of electric waves. The supposition is made that the oscillations produced in the closed secondary of a small induction coil are not essentially different in their magnetic effects from the oscillations produced in wires by means of Hertz waves. Apparatus. One quality of sheet-iron with large hysteretic constant was used. The sheet was cut into two crosses of the same dimensions. An exploring coil wound diagonally round the central square of one of the crosses was in circuit with a ballistic galvanometer. The four arms of each cross, bent over at right angles to the * The discussion on magnetic detectors has been somewhat extended since date of communication. PROC. ROY. SOC. EDIN. — VOL. XXVI. 3 34 Proceedings of Royal Society of Edinburgh. [sess. central square, were slipped into four rectangular coils from opposite sides. The arms which thus overlapped were firmly clamped together. Each coil consisted of two independent wind- ings. The inner windings of the four coils were connected in series, and with the secondary terminals of a small induction coil. The outer windings were likewise connected in series, and with suitable resistances and a source of constant E.M.E. It is evident that connections could be arranged so that a unidirectional current in either winding would produce an induc- tion either co- directional or transverse, relative to the axis of the exploring coil. A rocker in each circuit conveniently effected this alteration. The constant E.M.E. connected through resistances with the outer windings was used, (1) to demagnetise the iron by reversals decreasing from a maximum, and (2) to produce the field (cyclic or otherwise) and the co-directional induction measured by the exploring coil. The induction coil in series with the inner windings supplied the oscillatory current which agitated the iron of the central squares, either in a co-directional or transverse direction in the sense indicated. Let this agitation be called either the co-direc- tional or the transverse oscillations, as the case may be (determined by the position of the rocker in circuit), or when this distinction is superfluous, simply the oscillations. This rotation of the direction in which the oscillations act through an angle of 90° throughout the two central squares involves no change in the ampere turns nor in the measured reluctance of the iron circuits. What it does change is precisely that, the effect of which it is desired to measure. Field Superposition. It is now absolutely necessary to discriminate between the order and manner in which oscillations and field are superposed the one upon the other. Two experimental methods were adopted : A. Oscillations were superposed upon constant field. B. A change of field was superposed upon oscillations per- manently acting. 1905-6.] Electric Oscillations and Magnetic Properties of Iron. 35 These conditions of field superposition are entirely dissimilar, and the recognition of this fact is of primary importance. It may be observed that if in the former case (A) the field is varied after the superposition of oscillations, the order of super- position passes into those under the B conditions, a change of field being then superposed upon permanently acting oscillations. It is evident that the converse does not hold. Experimental Methods under A conditions. First, after demagnetisation of the iron, a fixed maximum field is put on (by increasing reversals, to secure as far as possible symmetry about the zero of induction), and reversed twenty times. The plus change of inductions due to the twenty-first reversal is- measured. The co-directional oscillations are now superposed, and the plus induction change measured. Oscillations and field are now put off. Second, the iron is again demagnetised, and the same fixed maximum field put on in the same way as before and reversed twenty times. The plus induction change due to the twenty-first reversal is measured. A single step is now taken to any given point on the hysteresis loop, and the minus change of induction measured. The co-directional oscillations are nowr super- posed and the plus or minus reading taken. Oscillations and field are again put off. This second process is repeated for a sufficient number of points all round the loop. The whole process is repeated for transverse oscillations. Three curves result from the reduced galvanometer readings taken as above described. These are plotted in figs. 1 and 2, when the maximum cyclic induction values are (without oscilla- tions) B = 780 and B = 5620 respectively. The scale of fig. 1 is for both ordinates double that of fig. 2. The ordinates measuring induction are in C.G.S. units; the abscissae measuring field, in arbitrary units. Summary of Results under A conditions. At and near extreme cyclic values the superposition of oscillations produces for low values of field (see fig. 1) a relatively large increase of induction ; for higher values of field (see fig. 2), a 36 Proceedings of Royal Society of Edinburgh. [sess. relatively small increase of induction. For low values of field the increase is greater for co-directional than for transverse oscillations ; for high values of field, this relationship is reversed. After leaving cyclic extremes there are points when the field is decreasing where co-directional and transverse oscillations respectively produce neither an increase nor a decrease of induction. In low fields they are thrust from the cyclic extremes, in high fields, towards the cyclic extremes. When these points are passed, oscillations produce a decrease of induction, and this for all values of field is greater for co-directional than for transverse oscillations. This decrease passes into increase in the opposite sense, and the first conditions are reverted to at the other end of the cycle. In all cases the induction change is greatest when oscillations are superposed on an increasing field. For low fields this occurs 1 905-6.] Electric Oscillations and Magnetic Properties of Iron. 37 at or near cyclic extremes , where the slope of the curves is greatest. But as the cyclic field maximum is increased, the greatest induction change occurs at an earlier stage of the increasing field, where in this case also the curves are steepest. To so great an extent is this the case for co-directional oscillations, that the usual order of things is reversed and the up curve (increasing field) actually reaches higher induction values than the down curve (decreasing field). The curves for the complete cycle thus cross (forming three loops) where co-directional oscillations give rise to an induction the same in value and sign whether they are superposed on the decreasing or increasing field. It may be worth noting that at some maximum value of field between those given in figs. 1 and 2, these crossing points may coincide with the neutral point when the field is decreasing where superposed co-directional oscillations produce no induction change. The curves for transverse and co-directional oscillations given in figs. 2 and 3 must not he confounded with the usual hysteresis loops in the sense that the areas they enclose measure the energy loss during one complete cycle. They do not do so. They measure for any given value of field the instantaneous change of induction which takes place when oscillations — co- directional and transverse — are superposed at any and all stages of the normal hysteresis loop. Now suppose that after any instantaneous induction change has been measured, the field is made to vary by some small amount — say, by a decrement if the field had previously been decreasing — the induction change which now takes place is entirely different (see dotted arrow). Hysteresis or lag in the usual sense comes into full play, and one naturally passes to the conditions of field superposition where a cyclic field change may he regarded as superposed upon permanently acting oscillations. Experimental Methods under B conditions. First. After demagnetisation and twenty reversals of a fixed maximum field, the normal B - H hysteresis loop is determined by Ewing’s method of single steps from the fixed maximum to a sufficient number of points all round the loop. 38 Proceedings of Royal Society of Edinburgh. [sess. Second. The iron is again demagnetised and subjected to co-direction al oscillations, upon which the field at the same fixed maximum value is superposed. After twenty reversals of field, the hysteresis loop is determined as before. Third. The iron being again demagnetised, the same process is repeated for transverse oscillations and the corresponding measure- ments made. It is, of course, understood that in cases second and third the force sustaining the oscillations remains “ on ” and unaltered until the series of galvanometer readings has been completed. The above determinations were repeated for many field cycles, the maximum induction values at the extremes of each cycle ranging from a minimum of B = 20 to B= 12,000. Summary of Results under B conditions. Permeability * — See fig. 3, where the full line, dotted line, and dash line curves have the same signification as in figs. 1 and 2. For low values of field co-directional oscillations increase the permeability relative to the normal (i.e. without oscillations) to a greater extent than transverse oscillations. For higher values of field transverse oscillations, increase the permeability relative to the normal to a greater extent than co- directional oscillations. The crossing point of these curves occurs when the induction is about 5000 with oscillations. When the induction does not exceed a few hundreds without oscillations, the corresponding induction with transverse and co-directional oscillations is respectively about three and four times greater. When B = 20 without oscillations, these ratios become a little less. Retentivity. — When the field is reduced to zero from a cyclic maximum and the ratio of residual to maximum induction plotted, as in fig. 5, against maximum cyclic induction, the curve with transverse oscillations is higher within wide limits than that obtained with co-directional oscillations. These curves appear to coalesce, or even to cross, in higher fields. The normal retentivity curve lies above both. * After twenty reversals of field. 1905-6.] Electric Oscillations and Magnetic Properties of Iron. 39 It might thus appear that even under the B conditions the 'effect of oscillations is to reduce residual magnetisation. Such a statement, however, cannot he regarded as correct, because by- plotting against maximum induction one of the most important effects of oscillations has been eliminated, viz., that of increased permeability. In fig. 4 residual magnetisation is plotted against field, and we 0 5 500 5000 7500 10000 see that it is only at high values of induction — where per- meability is, so to speak, naturally eliminated — that the effect of oscillations is to reduce retentivity. For lower fields, and through- out a wide range, oscillations increase retentivity. A comparison with fig. 3 shows that the greater the permeability in the three cases, the greater is the residual magnetisation. The residual magnetisation curves with co-direction al and transverse oscillations also cross each other — as wTas found to be the case for permeability, — and greatly exceed in value the normal retentivity curve without 40 Proceedings of Royal Society of Edinburgh. [sess. oscillations, with the exception already mentioned, where the induction is great. Coercive Force. — For low fields of the order of a few hundreds, oscillations likewise increase coercive force, but with slightly higher fields this effect soon disappears, and thereafter oscillations decrease coercive force. Hysteresis Loss for Constant Induction. — When under normal conditions the maximum cyclic induction is of the order of one to five hundred, co-directional oscillations for constant maximum induction diminish the energy loss in the iron about four times, transverse oscillations about three times. But as the induction is increased this difference gradually lessens, and after B under normal conditions has reached 5000 it is apparent that transverse oscillations diminish the loss in the iron to a greater extent than co-directional oscillations. In all cases for constant induction, oscillations cause a diminution of the energy loss ; hut when the induction is high (say, B = 12,000), the diminution, although sufficiently w^ell marked, has become a relatively small effect. Hysteresis Loss for Constant Field. — When under normal conditions the maximum cyclic induction is of the order of a few hundreds, oscillations for constant maximum fields increase the energy loss in the iron about four times relative to the loss when no oscillations are acting. As the induction is increased to some thousands, the energy loss becomes very approximately the same with and without oscillations. When, however, the induc- tion is higher still — say, B= 12,000 — a sufficiently well-marked but relatively small decrease of hysteresis loss is caused by the oscillations. The energy loss for co-directional and transverse oscillations does not differ greatly relative to each other for constant maximum fields. The greater relative retentivity and the lower permeability at low fields (where the difference is so much greater than under normal conditions), under transverse relative to co-directioual oscillations are in harmony with this result. The curves exhibited are not here reproduced. Fig. 6, however, may be referred to, where the full line curve shows the normal hysteresis loop without oscillations, the dotted line curve, the greatly increased hysteresis loop with co-directional oscillations. They do not differ in type from each other. The phenomena of 1905-6.] Electric Oscillations ancl Magnetic Properties of Iron. 41 “ lag ” is equally well exhibited by both ; but for present purposes it is in my opinion also essential to state all the facts in terms of permeability (at cyclic extremes), of retentivity (when H = 0), and of coercive force (when B = 0), as has been done above. Magnetic Detectors of Electric Waves. The above experimental results have, in addition to their purely theoretical or physical aspect, an especial interest at the present time, in view of the fact that the magnetic properties of iron and steel have beeti utilised to detect electric waves in space. Magnetic detectors of the Rutherford type, either in their original forms or as they have been modified for continuous telegraphic work, illustrate what takes place when oscillations are superposed at one point of the normal hysteresis loop, viz., that where zero field has just been passed. The reduction of the magnetisation which occurs is sufficiently well understood. Other forms of detectors in which the adjustment and rates of motion of parts have been experimentally determined, seem to be less perfectly understood. If one may judge from the conflict of opinion which has arisen, and the anomalous results which have been obtained, magnetic detectors seem to have outrun very generally accepted theoretical knowledge. In view of the experimental results above arrived at, I propose now to discuss three forms of magnetic detectors of electric waves in the following order, viz., Marconi’s first form of detector, Marconi’s second detector, and the Ewing- Walter detector. First. In Marconi’s first apparatus a fixed core of iron or steel is used, long relatively to its diameter. The poles of a revolving horse-shoe magnet are, during each complete revolution, twice in close proximity to the ends of the core. It is thus continuously carried through a cycle between positive and negative maximum values. Wireless telegraphic signals are received at all stages of this cycle. The instrument is admittedly much less perfect than Marconi’s second form, but it is nevertheless interesting to examine its action. The broken curve of fig. 6 shows the effect of superposing and withdrawing oscillations at twenty-four approximately equi- 42 Proceedings of Royal Society of Edinburgh. [sess. distant points in succession during the determination of the hysteresis loop, the “ step-by-step ” method being in this case the only one available. To secure approximate symmetry about the zero of induction, the maximum cyclic value is first reached by reversals of field increasing from zero, the oscillations being “on.” After twenty reversals the oscillations are put off. The field is now decreased by the first step and the oscillations put on and off. The second step is next taken, followed by oscillations on and off as before. Three galvanometer readings are taken at each step ; the first reading measures the change of induction due to change of field, the second that due to oscillations, and the third any small change if any that may take place when the oscillations are put off. By proceeding in this manner the whole cycle may be completed. The dotted line curve is the hysteresis loop under the B con- ditions, i.e. when the field cycle is superposed upon permanently - acting oscillations. The full line curve is the normal hysteresis loop {i.e. without oscillations) determined by the step-by-step method for the same maximum values of field. 1905-6.] Electric Oscillations and Magnetic Properties of Iron. 43 The curves show the results fully. The effects of superposing oscillations, as nearly as possible uniform in intensity and frequency, at approximately equidistant points all round the cycle, are as follows first, the area enclosed is enormously increased in comparison with the normal hysteresis loop ; second, it is also greater than the hysteresis loop obtained when the oscillations are permanently acting (this is in accordance with what has been stated under “ Retentivity ” ) ; and third, the amount of instan- taneous induction change which occurs upon the superposition of oscillations depends entirely at what particular point of the cyclic field they are superposed. But the above conditions are not those of practical telegraphic work. If signals are to he received, the waves must be transmitted more or less irregularly. Tig. 7 shows what takes place when oscillations are superposed and withdrawn somewhat irregularly at various points during two complete cycles. The results are obvious. The induction changes now caused by the superposition of oscillations are determined not only by the particular point at which they are superposed, but by previous irregularities. Thus, if the irregularity be a pause in the signalling, the effect of succeeding signals may either be increased or decreased. If signals are not received when their effect is a maximum, the effect of succeeding signals may be decreased ; on the other hand, if they are not received when their effect is a minimum, the effect of succeeding signals may be greatly increased. Note, also, the limitation of the total induction change per cycle or half cycle to which the irregular receipt of wave signals may give rise. Marconi’s first form of magnetic detector appears to owe its sensitiveness to the fact that the instantaneous induction change which takes place when electric waves are superposed at many points of the field cycle is great ; but it is equally certain that it owes its imperfections to the fact that the field is varying, signals not being superposed at one definite point of the cycle. The function of the moving field, therefore, is not to increase the sensitiveness of the iron ; it merely renders the receipt of signals possible, by removing the iron from that particular cyclic position at which the immediately preceding signal was received, 44 Proceedings of Royal Society of Edinburgh. [sess. and which has been rendered unresponsive to a second wave signal of the same intensity and frequency. Second. — In Marconi’s second instrument, the “varying or moving magnetic field,” upon which the invention is based, and by means of which the magnetic material is supposed to become- sensitive to high frequency oscillations, is entirely departed from. A continuous hand of iron or steel is passed over the poles of two horse-shoe magnets in the following order: — one south pole, one double north pole, and finally one south pole. Let the direction of motion be from left to right. The magnetic circuits of the two horse-shoe magnets are in part completed in the moving hand y and if its section he small relative to that of the electromagnets, the magnetic induction in the hand will depend largely upon its permeability. The two magnetic circuits, however, are in opposite directions ; and it is evident that if change of permeability affected the circuits equally and in the same sense, no E.M.F. would be produced in a coil symmetrically placed between them. The hand as it approaches the double north pole from the left is passing through a field increasing to a maximum ; and as it leaves on the right, it is passing through a field decreasing from a maximum. We have, however, seen (figs. 1 and 2) that under these conditions the superposition of oscillations produces in all cases an increase of induction (increased permeability) which is greater with increasing than with decreasing field. In other words, the increase of induction in the hand is greater immediately to the left of the double north pole than it is to the right. This differential action is further increased by the fact that the moving hand distorts the field in the direction of motion, viz., to the right. Hence the receipt of oscillations will he recorded in the telephone connected with the exploring coil symmetrically placed in reference to the magnets hut unsymmetrically placed in reference to field. It is further evident that once wireless telegraphic communi- cation has been established and is proceeding with reasonable constancy, the increase of induction which takes place in the increasing field (to the left) tends towards a maximum, and renders negligible any further induction change (increase or decrease) taking place in the decreasing field (to the right). 1905-6.] Electric Oscillations and Magnetic Properties of Iron. 45 Hence in my opinion Marconi’s second instrument detects * the increase of induction which occurs when electric space waves are superposed upon field at or near a cyclic extreme, in precisely the same way as Rutherford’s original apparatus detected, or rather measured, a decrease at another point of the cyclic process. The function of the moving band appears to be twofold : — first, it supplies the hard iron or steel in a condition of lower permeability, in order that it may be raised to a condition of higher permeability when the signals are received from a distant station ; and second, it distorts the field in the direction of motion, the telephone thus tending to respond in a greater degree to signals received when the field is increasing near but not at the cyclic extreme. It has already been pointed out that for low field this is the most sensitive part of the cycle. Third. The Ewing-Walter form of detector is essentially different from those already discussed. It is based upon Ewing’s hysteresis tester, which measures the drag between field and iron when one of these is revolving. The best speed is stated to be from five to eight revolutions per second, while Marconi obtained good results at a speed of only half a revolution per second. The reason for this great difference is obvious. The Marconi instruments detect instantaneous induction change (A conditions), while the Ewing-Walter form integrates (B conditions). The inventors anticipated that the receipt of electric waves would be detected by a fall in the normal deflection, in accordance with what appears to have been the generally accepted but erroneous view that oscillations in all cases cause a diminution of the energy loss per cycle. A fall in the normal deflection was at first obtained when the oscillations may be supposed to have been co-directional oscillations, and thus to alternately assist and oppose the field. On experiments being continued, however, an increase in the normal deflection resulted, but in this case the oscillations were passed directly through the iron or steel core, thus giving rise to transverse oscillations. The former result — a decrease of energy loss, due to oscillations — is not in harmony with my experimental results for low fields. A probable explanation may be found in the fact that in my case * Or even measures, provided the signalling is not proceeding too rapidly. 46 Proceedings of Royal Society of Edinburgh. [sess. the magnetic circuit was completed wholly in the iron, and consequently the maximum induction at cyclic extremes depended entirely on the increased permeability due to the oscillations. It is just possible that in the earlier Ewing- Walter instrument, the induction in the iron rings depended upon their geometrical form and very little upon their permeability. If this were the case, the energy dissipated per cycle would be reduced. The later result, however — an increase of energy loss, due to oscillations — is fully in harmony with my experimental results. Messrs Ewing and Walter state ( Proc . P.S., vol. lxxii. p. 120) that the unexpected augmentation of hysteresis (loss) is probably to be ascribed to the oscillatory circular magnetisation increasing the range of longitudinal induction (permeability), and so indirectly increasing the energy loss in the iron or steel. This explanation, however, is not, taken by itself, altogether adequate. In the first place, it seems to imply that transverse (circular in this case) oscillations facilitate the magnetising process in some unexplained way not possessed by co-direction al oscillations. It affords no explanation why the first form of apparatus did not show increased hysteresis loss with co-direc- tional oscillations with low fields. In the second place, induction change due to oscillations (or even to mechanical vibrations) has generally been ascribed to a greater freedom on the part of the molecules to follow a changing field. If the field be increased, the induction will be increased (increased permeability); if the field be decreased, the induction will be decreased (decreased retentivity). Increased range of induction, therefore, due to oscillations, by no means implies, on the usually accepted views, increased energy loss per cycle. As a matter of fact, however, we have seen that all the more important magnetic properties of hard iron are modified by oscillations. For low fields increased permeability is associated with increased retentivity and also with increased coercive force ; for medium field, with increased retentivity but decreased coercive force; while for high fields increased permeability is relatively reduced and is associated both with decreased retentivity and decreased coercive force. But as these are precisely those factors which determine energy 1905-6.] Electric Oscillations amcl Magnetic Properties of Iron. 47 loss per cycle, it is evident that any apparatus which measures the drag between rotating field and core, or in any other way integrates the B - H cycle with respect to value and sign, will measure increase of energy loss for low fields, and decrease of energy loss for high fields ; and this irrespective of whether the oscillations are co-directional or transverse. Between these extremes, giving definite and opposite readings, uncertain results, or no results at all, may be anticipated from magnetic detectors of this kind. I acknowledge my indebtedness to the Royal Society of London for placing at my disposal a Government grant for the purposes of this investigation. The work is being continued. ( Issued separately February 8, 1906.) 48 Proceedings of Royal Society of Edinburgh. [sess. On a Theorem in Hypercomplex Numbers. By J. H. Maclagan-Wedderburn, Carnegie Research Fellow. (Read January 8, 1906.) Scheffers in the Mathematische Annalen, vol. xxxix., pp. 364-74, enunciates the following theorem : — If A is an algebra containing the quaternion algebra B as a subalgebra, and if A and B have the same modulus, A can he expressed in the form B C = A = C B, where C is a subalgebra of A every element of which is com- mutative with every element of B : in other words, if i1 , i2 , z3 , is a basis of B, it is possible to find an algebra C with the basis e-j , e2 , . . . ec , such that each of its elements is commutative with every element of B, and such that the elements eris(r= 1, 2 , ... c, s = 1, . . . 4) form a basis of A; and if a is the order of A, then a — 4c. The following is a short proof of this theorem. Let the basis of B be as usual 1, i,j, k where the laws of combination are the usual laws of quaternions. If x is an element of A, then x = x — ixi —jxj — kxk, x = ix + xi + kxj -jxk, x" —jx - kxi + xj + ixk , and xP = kx +jxi - ixj + xk are commutative with every element of B : further, x can be expressed in terms of x, ix", jx", and kxiv, in fact 4x = x - ix!' —jx" - kxP ; hence if C is the algebra of all elements of A which are commutative with every element of B, B and C satisfy the conditions required by the theorem. If V, Vi > 2/2 5 Vs are any elements of C, x = y 4-y-f + yJ + yfc can only vanish if y = yY = y2 = = 0, for if x = 0, then 4y = x - ixi - jxj - kxk = 0, and similarly y1 = y2 — y3 = 0 ; hence the order of A is four times the order of C. In addition to being much shorter than Scheffers’ proof, which occupies about ten pages, this proof has the advantage of being rational. The method used in the above proof may be regarded from two points of view. The first of these will be best understood from the following extension of it. Let B be the algebra generated by the two elements e1 and 1905-6.] On a Theorem in Hypercomplex Numbers. 49 e2 where e1n = e2n = e and e1e2 = ee2e1i e being the modulus of B and e a primitive nth. root of unity. If x he any element of an algebra A which contains B and has also e as its modulus, yt = ^ x~xxtxxr(t = 1, 2, . . . ri1) are commutative with every r= 1 element of B, if xY , x2, . . . represent epe^ (a, 1, 2 , . . . n); for n n e{e2 sVt = eie2^j € ~ apei ~ ae2 ~ ^xepe/ = 2, € " aP+a3e1r ~ ae2s ~ Pxtxepe/ a, /3 = 1 a, /3 = 1 = 2 e~aP~ ^re1 - ae2 ~ Px&ep+*ep+& = yte{e2s. hf ow if xr~l — e{e2, a, 0 = 1 n2 n n then 2e2"^ir“ae2Ve2^= unless <=1 a, 0 = 1 a, 0 = 1 r and s are multiples of n, i.e. unless xr = e ; hence n2x = y, ^ xflxr~1xtxxr = y,xtyt. It follows therefore as before r t t that x is contained in the algebra which is formed by taking the direct product of B and the algebra composed of those elements of A which are commutative with every element of B. It may be remarked here that the algebra B is, in the complex field, equivalent to the quadrate algebra epq(p, <2 = 1, 2, . . . n ) where epqeqr = epr , epqers = 0, if q 4= r and e = eu + e22 + . . . +enn; in fact, if we set e^ — + ee22 + . . . + ew enn , e2 = #j2 + e2g -!-...+ , then e p = e2n = e and eYe2 = ee2e1 . The second point of view is perhaps simpler. Let A be any algebra containing a subalgebra B which has the same modulus as A ; if A is expressible as the direct product of B and another algebra C, every element x of A can be expressed in the form x = ^,fr(x)xr where x1 , x2 , . . . xb is a basis of B and fr(x) r (r= 1,2, . . . b) are elements of C and are therefore commuta- tive with xx , x2 . . . xb. Suppose B is such an algebra that, if x = ^ £rxr is any element of B, $r(r= 1,2, . . . b) being scalar co-ordinates, it is possible to express these co-ordinates as rational functions of x and the elements of the basis, say £r = fr(x) : fr(x) is then necessarily linear in x and we may write fr(x) = ^ £pqxpxxq. p,q If y is any element of B, then yfr(x)=fr(x)y, or ]?£pqyxpxxq = P,q 4 PROC. ROY. SOC. EDIN. — YOL. XXVI. 50 Proceedings of Royal Society of Edinburgh. [sess. y, £pqxpxxqy for every yanda? belonging to B, therefore the matrix pq represented by £vo{yxv( )xq - xp{ )xqy) is identically zero. p, a Hence fr(x) is formally commutative with every element of B, and fr(x)x=xe = x where e is the modulus. If now x is any element of A, fr(x) is still commutative with every element of B and x— fr(x)xr , since A and B have the same modulus. Hence as before A can be expressed as the product of B and the algebra composed of those elements of A which are commutative with every element of B. If c is the order of this algebra, which as before will be denoted by C, the order of A is equal to be : for suppose if possible that y = yxxx + y2x2 + . . . + ybxb Bo, yx , y2 , . . . being b elements of C ; then fr{y) = 0, but fr(y)= ^ fr{yrxr) = yrfr(xr) = yr > r= 1 therefore yr = 0. For instance, let B be the quadrate algebra (epq) described above, n n then f pq(x) &tpXCqt ? @rsfpq(x) @rpXCqs =fPq{x)ers , and ^^fPq(x)epq t= i P, 9=1 n = eppxeqq x. p,q=l It is interesting to note in this connection that if A is a quadrate algebra of the above type whose order n2 = l2m2, it is expressible as the direct product of two quadrate algebras of orders Z2 and m1 m respectively. This may be shown by setting em— )m+t *=i i- 1 (P,q= 1, 2, . . . I ) for the algebra B and rjpq= V^etm+p>tm+q <= o (p,q= 1, 2 , . . . m) for C. epq and rjpq evidently have the proper laws of combination, and epm^_r> qm+s rjrs^p+i, q+i ~ ^-p+i,q-\-i*lrs • Hi general, if n=nln2 . . . nk, A can be expressed as the direct product of h quadrate algebras A1 , A2 , . . . Ak of orders nf, n22, . . . nk2 respectively. This theorem is the counterpart of one given by Clifford.* *■: American Journal , vol. i., pp. 350-58. ( Issued separately February 9, 1906.) 1905-6.] Library Aids to Mathematical Research. 51 Library Aids to Mathematical Research. By Thomas Muir, LL.D. (Read December 18, 1905.) (1) The aids which may fairly be expected from a library towards the prosecution of research, mathematical or other, are of two kinds. First, there is the aid given by furnishing the names of previous workers in the same field, accompanied by the names of the publications in which the results of their labours have been preserved or entombed. Second, there is the help in the oppor- tunity given of consulting the said publications themselves and of borrowing certain of them for lengthened study. In other words, what the scientific investigator wants from libraries is boohs and books about books. One of the two is often obtainable without the other, but in that case its value is immensely lessened. To the so-called “ordinary” reader, a library is a labyrinth which probably would give him all he needs if only he had a guide through its intricacies : to the specialist, on the other hand, with his methodic bibliographies and booksellers’ lists, it not unseldom presents itself as a lucky-bag in which the blanks are ultimately more in evidence than the prizes. It is the reader of the latter kind that a scientific society naturally wishes to assist, and it is desirable therefore that his difficulties and aspirations be known. All that has been done of recent years for the “ ordinary ” reader one notices not merely without a grudge, but with admiration : it must never be forgotten, however, that no country can afford to neglect the wants of the working specialist. After all, he it is who is the original source of supply for all readers, and it is therefore bis output which ought to be of the deepest concern. My purpose on the present occasion is to show what the situation is in this matter as regards the single subject of mathe- matics. There can be little doubt, however, that other subjects are in as bad a plight, and that the whole question of library aid is worth serious and prompt attention from all scientific men. 52 Proceedings of Royal Society of Edinburgh. [sess. (2) First, then, let us see what has been done during the last fifty years to supply mathematicians with bibliographical aids. In 1855 Dr Joseph Henry made his appeal to the British Associa- tion at Glasgow for the formation of a catalogue of philosophical memoirs referring to mathematics and physics, and the appeal did not fall unheeded. Before anything tangible, however, could result from it, the publication of Poggendorff’s Biograpliisch- literarisches Handworterbuch was begun (1858) in Germany, and partly filled the vacant place. The two volumes of it, alphabetically arranged according to authors’ names, were completed in 1863, and dealt with scientific papers published up to 1858. By the time of the appearance of the first part of “Poggendorff,” the seed sown by Henry had germinated, the Royal Society of London having taken it over from the British Association, and having formally resolved to prepare a catalogue of scientific memoirs beginning with the year 1800. The first volumes of this great undertaking, namely, those dealing with the period 1800-1863, were published at the rate of one volume a year, the last appearing in 1872. From the latter date, therefore, there were two books of reference available for the mathematician, the Handworterbuch and the Catalogue of Scientific Papers. The fact that they covered in the main the same period was not altogether a drawback, as even within that period they were often mutually supplementary. A more serious objection lay in the fact that neither of them had a subject-index, and that the more up-to-date of the two was almost ten years in arrears when its sixth volume appeared. About this time a very important fresh venture was made ; but, before speaking of it, it may be well to follow out the history of the other two. By 1879 two additional volumes of the Catalogue of Scientific Papers had been published dealing with the period 1864-1873, and by 1896 the next decade had been covered by three futher volumes : so that at the latter date there were avail- able eleven handsome quarto volumes devoted to cataloguing the scientific literature from 1800 to 1883. For a long time, on the other hand, no addition to the Handiobrterbuch had been made, and it seemed as if the Royal Society were to be left in possession of the field. Ultimately, however, fresh counsels must 1905-6.] Library Aids to Mathematical Research. 53 have prevailed with the publishers, for the year which gave birth to the Society’s eleventh volume also saw a beginning made with a new volume of “ Poggendorff.” This was completed in 1898, thirty-five years after the completion of the previous volume, the record being then brought up to the same point as the Catalogue had reached two years before. It was now the Royal Society’s turn to lag behind, for during 1902-1904 a fourth volume of the Handworterbuch was published, completing the record for the century, whereas all that the Royal Society has since printed is a twelfth volume dealing with the same period as the preceding eleven and supplementing them.* The fresh venture above referred to was the institution of the Jahrbuch iiber die Fortschritte der Mathematik, the first volume of which was published in 1871. It differed from its predecessors (1) in confining itself to mathematics alone; (2) in being an annual publication devoted to a year’s work, — the first volume, for example, dealing with the literature of the year 1868 ; (3) in being not a mere catalogue of titles, but giving also a short abstract of each paper’s contents ; (4) in being arranged according to subjects, and having two indexes, the one giving the mere titles arranged alphabetically according to authors’ names, and the other giving the same rearranged as in the body of the work. In the first volume the book proper occupied pp. 1-404, the first index (Namenregister) pp. 405-426, and the second ( Inhalts verzeich- niss) pp. ix.-xxxiv. In 1884 a miniature rival to the Jahrbucli was started at Stockholm, the Bibliotheca Mathematical giving lists of new publica- tions on mathematics and short bibliographical articles : but after three years it was altered in form and became devoted exclusively to the history of mathematics and to the bibliography of mathe- matics from the historic point of view. In April 1900 it assumed a third and more imposing form, still, however, restricting itself to the field of mathematical history. For our present purpose therefore it need not further concern us. A much more serious competitor took the field in 1893. This * The Royal Society has not ceased its work, and holds itself pledged to finish what it has undertaken, namely, to catalogue the scientific papers of the nineteenth century. 54 Proceedings of Royal Society of Edinburgh. [sess. was the Revue semestrielle des publications mathematiques , com- piled under the auspices of the Mathematical Society of Amsterdam. Its objects are almost exactly those of the Jahrbuch, but its plan is quite different. In the first place, its titles, accom- panied by short abstracts, are arranged according to countries , and within a country according to serials ; in the second place, since each title has prefixed to it the appropriate symbols indicating its subject according to the classification of the “ Congres inter- national de bibliographic des sciences mathematiques,” it can and does provide a subject-index in comparatively small space ; and in the third place, the abstract of a paper is, as a rule, given in the same tongue as the paper itself. Two parts appear in a year, each dealing with the literature of the half year ended three months previously — thus showing a marked promptitude as compared with the Jahrbuch , where the corresponding interval is not three months, but three years. In the first volume, embrac- ing the work of half a year, the book proper occupied pp. 1-96, the list of journals pp. 97-100, the subject-index pp. 100-109, and the index of authors’ names pp. 110-114. The cost of the book, it well deserves to be mentioned, is trifling, the subscription price for the yearly couple of volumes being only 8J francs. In the end of 1902 still another competitor appeared, namely, volume A of the International Catalogue of Scientific Literature , promoted by the Royal Society of London, and designed to be the cosmopolitan continuation in annual instalments of the nineteenth- century Catalogue of Scientific Papers above referred to. Its general object is the same as that of the Jahrbuch and Revue , save that abstracts are not given. A complete volume appears once a year, and contains (1) a list of the previous year’s writings arranged alphabetically according to authors’ names, and hence called “Authors’ Catalogue,” each title having appended to it a number indicating its subject and a number indicating its position on the list; (2) a “ Subject Catalogue,” in which the titles, printed anew, are arranged according to the order of their subject-numbers, and under any subject-number according to the alphabetical order of the authors’ names. The scheme of classification of subjects is neither that of the Jahrbuch nor that of the Revue. In the first volume the “Authors’ Catalogue” occupies pp. 47-111, and the 1905-6.] Library Aids to Mathematical Research. 55 “ Subject Catalogue” pp. 112-194, the last seven pages being taken up with a list of serials and the first 45 pages with the schedule of classification and an index to it, both printed in four languages. Such, then, is in brief the story of the cataloguing of mathe- matical writings during the last half century. The net result is seen to be (1) the production of two huge works of reference dealing with what we may call bygone literature, that is to say, in the main the literature of the nineteenth century ; * (2) the establishment of three annuals dealing with what we may call current literature, and aiming, all of them, at giving a methodically arranged list of the whole of each year’s literature within a comparatively short time of its appearance.' With this in evidence, who shall say that the mathematician is 'uncared for by bibliographers, cataloguers, and index-makers ?' The simple truth is, he is overburdened by their labours ; for, whereas either of the two former works and any one of the three latter, if complete, would suffice for his wants, he is compelled to use all of them if only as a means of detecting their several errors and omissions. (3) This being the state of matters • as regards “books about books,” our next inquiry naturally is, What is the situation as regards “books” themselves? Have the librarians and library managers successfully coped with the difficulties on their .side ? The mathematician who has turned up the Jahrbuch , the Revue , and the International A, and culled from them an array of “ references,” proceeds to realise the references at the counter of his library, and what is the result ? To obtain something like a definite answer to this, the following course has been adopted : — First of all, the field of inquiry was narrowed down to practicable dimensions. To have tabulated all the works referred to in these two dictionaries and three current year-books would have been an overwhelming task : if we restrict ourselves to mathematical serials, the work becomes quite manageable. That is what has been done ; and in order to make * A third venture might here have been referred to, namely, the Repertoire Mbliogrciphique des sciences mathematiques , a card catalogue following the system of classification adopted by the Revue. Between 1894 and 1905 fifteen hundred cards have been sent out, and the present rate of issue is only one hundred cards per year. 56 Proceedings of Royal Society of Edinburgh . [sess. the case still more simple and the argument from it more con- vincing, only one of the three catalogues (. International A) has been taken advantage of. That is to say, all the mathematical serials have not been tabulated, but only those which find a place in the List of Journals published in 1903 in connection with the International Catalogue of Scientific Literature. Such a list, in view of its relation to the Royal Society of London, and in view of the fact that an international council stands sponsor for it, must be a list free from any grave reproach. In the next place, the chosen serials having been tabulated, copies of the table were made and submitted directly or indirectly to the librarians of the more important appropriate libraries in Scotland and in London, for the purpose of ascertaining exactly which of the serials each library was at the moment receiving, and what back volumes of these it possessed. The whole of the information thus obtained it is not proposed at present to use ; the portion of most value and interest here and now is that concerning Edinburgh and Glasgow. (4) The test-list of serials is as follows : — Austria — 25. Archiv Mathematiky a Fysiky, .... (4 jahrl.), Praha. 290. Casopis pro Pestovani Math, a Fys., .... (5 H. jahrl), Praha. 304. Sbornik Jednoty Ccskych Math, v Praze, Praha. 207. Monatsheft fur Math. u. Phys., .... (zwanglos), Wien. Belgium — 8. Bull, period soc. beige de geometres (bimensuel), Anvers. 118. Mathesis. Recueil math (mensuel), Gand. Denmark — 11. Nyt Tidsskrift for Matematik, Kjobenhavn. Prance — 79. Annates sci. de Vec. norm. sup. (mensuel), Paris. 231. Bulletin de math, speciales (10 fois par an.), Paris. 244. Bulletin des sci. math, (mensuel), Paris. 322. A5 education mathematique . . . ., (bimensuel) Paris. 333. L’> enseignement mathematique . . . . , (mensuel), Paris. 382. L' inter mediaire des mathematiciens (mensuel), Paris. 395. Journal d,e V ecole polytechnique (annuel), Paris. 398. Journal des geometres (bimensuel), Paris. 401. Journal de math, pures et appliquees (4 fasc. par an.), Paris. 557. Nouvelles annates de math, (mensuel), Paris. 603. Bull soc. matb. de France (4 num. par an.), Paris. 719. Revue de math, speciales (mensuel), Paris. 1905-6.] Library Aids to Mathematical Research. 57 Germany — 76. Archiv der Mathematik u. Physik jahrl.), Leipzig. 167. Bericht des math. Vereinsd. Univ. Berlin (jahrl. ), Berlin. 217. Bibliotheca mathematica (| jahrl. ), Leipzig. 556. Mittheilungen d. math. Ges. zu Hamburg (1-2 H. jahrl.) Leipzig. 595 Journal f. d. reine u. angew. Math. (8 H. jahrl.), Berlin. 610. Jahrbuch uber die Fortschritte d. Math. (3 H. jahrl.), Berlin. 625. Jahresbericht d. deutschen Math.-Vereinigung (2-4 H. jahrl.), Leipzig. 776. Mathematische Annalen (J jahrl.), Leipzig. 1088. Verhandl. d. internat. Math. -Congresses, Leipzig. 1210. Zeitschrift fur Mathematik u. Physik (2 monatl.), Leipzig. 1211. Zeitschrift f. math. u. naturw. TJnterricht (8 H. jahrl.), Leipzig. 1310. Abhandl. zur Geschichte d. math. JViss. (zwangios), Leipzig. Holland — 2. Nieuw Ar chief voor Wiskunde, Amsterdam. 8. Wiskundige Opgaven, met de oplossingen . . . ., Amsterdam. 753. Revue semestrielle des publ. math. (2 fasc. par an.), Paris. Hungary — 10. Mathematikai es Physikai Lapok, Budapest. 11. Math, es Termeszettudomdnyi Frtesito, Budapest. 12. Math, es Termeszettudomanyi Koslemenyek, Budapest. Italy- 30. Bollettino cli bibliog. e storia delle sci. mat., Genova-Torino. 7. Annali di mat&matica, pura ed applic. , Milano. 85. Giornale di matematiche, . . . ., Napoli. 94. Le Matematiche pure ed applicate, Citta di Castello. 138. Rendiconti del circolo matematico, Palermo. 143. Periodico di matematiche, per . . . ., Livorno. 149. II Pitagora, Palermo. 157. Revue de mathdmatigues, Torino. 216. Supplemento al Periodico . . . ., Livorno. Japan- 38. Tokyo Sugaku Butsurigaku Kwai Kiji, Tokyo. Norway — 3. Archiv for Math, og Naturvidenskab , Kristiania. Poland — 37. Prace matematyczno-fizyczne (annual), Warszawa. 54. Wiadomosci matematyczne (once in 2 mos.), Warszawa. Portugal— 2. Jornal de sciencias math. e. astron., Coimbra. 11. Jornal de sciencias math., phys. e nat. , Lisboa. Russia— 19. Soobscenya .... (= Rapports .... soc. math, de Kharkov). 114. Math. Sbornik (=:Recueil math.), Moscow. 177. Zapiski .... ( = Mem soc. nouv. -Russie ....), Odessa. 58 Proceedings of Royal Society of Edinburgh. [sess. Sweden — 1. Acta mathematica , Stockholm. United Kingdom — 94. Proceedings of the Edinburgh Math. Soc., Edinburgh. 111. Math reprinted from Educ. Times, London. 262. Proceedings of the London Math. Soc., London. 316. Mathematical Gazette, London. 329. Messenger of Mathematics, Cambridge. 380. Quarterly Journal of Pure and Applied Math., London. United States — 16. American Journal of Mathematics, . . . ., Baltimore. 20. American Mathematical Monthly, Springfield, Mo. 23. Annals of Mathematics, pure and applied, Cambridge, Mass. 298. Bulletin of the American Math. Society, New York, N.Y. 336. Transactions of the American Math. Soc., New York, N.Y. It is seen that the number included in the list * is 67: if it were perfectly complete the number would be about 80. These 67 are apportioned among the different countries thus : — Austria 4, Belgium 2, Denmark 1, France 12, Germany 12, Holland 3, Hungary 3, Italy 9, Japan 1, Norway 1, Poland 2, Portugal 2, Russia 3, Sweden 1, United Kingdom 6, United States 5. Spain, it will be observed, does not appear : this is not because she publishes no mathematical serials, but probably because she has been slow in forming a “regional bureau” to work with the Council of the International Catalogue. In the second place, all serials like the Proceedings of the Royal Society of Edinburgh are excluded, being ruled out by the fact that the contents of such serials are not mainly mathematical. An afterthought, however, makes it doubtful whether the * The list lays itself open to criticism, but let him throw the first stone who knows the difficulties of organising the machinery necessary to produce not one year-book but the whole seventeen annual volumes of the International Scientific Catalogue. For one thing, the director of such an undertaking will find his numerous regional-bureaux at sixes and sevens as to where the line of exclusion is to be drawn, and consequently elementary journals will be inserted in one land and shut out in another. Then, of course, journals have an appreciable death-rate, and no list can be complete and accurate for long : that here used is the first drawn up. In view of what has been already accomplished, alike by the Council and by the Director, the manifest duty of all scientific men is to lend what aid they can towards now perfecting the Catalogue in detail. The number preceding the title of any serial is the consecutive number which distinguishes it in the above-mentioned List of Journals. 1905-6.] Library Aids to Mathematical Research. 59 NaehricJdeh of the Gottingen Society of Sciences ought not to have been inserted, as the mathematical papers of that society form a large proportion of those published in the Society’s mathe- matico-physical section. The serials of certain Russian physico- mathematical societies, being insufficiently known to me, may have been improperly excluded. Such matters, however, are of little moment, for the inclusions unjustly made will probably be more than balanced by the similar exclusions, the effect of an exclusion being to strengthen the argument and the effect of an inclusion not necessarily to weaken it. (5) The returns from the Edinburgh and Glasgow libraries when tabulated appear as follows : * — * Of those who have assisted me in obtaining details, I desire specially to thank Professor Gibson of Glasgow, and Mr Hardy, the worthy librarian of this Society. The London libraries which are of importance as regards mathematics are six in number, namely, those of the British Museum, Royal Society, Mathe- matical Society, University College, South Kensington Science, Patent Office. Some of these possess more serials than any one of the Scottish libraries ; but the same general state of affairs prevails. Overlappings repeatedly occur, serials possessed are frequently imperfect, and the full collection of the test- list cannot be furnished forth by all the libraries put together. The library which possesses representatives of the greatest number of serials is that of the Mathematical Society : unfortunately not more than half a dozen of the sets are complete, and the books are miserably housed. To provide proper accommodation, to complete the sets already represented, and to supply the missing sets, would be a splendid ambition to set before the little band of members resident in London. The serials contained in the British Museum, though somewhat fewer in number, are wonderfully complete and excellently cared for. Meanwhile it is desirable that a leaflet should be prepared containing a full list of serials, and showing which are obtainable in London, and where. For those which are more elementary, the library of the Mathematical Association might be utilised. [Table. 60 Proceedings of Royal Society of Edinburgh. [sess. 4-^ 445 CD CD a a o o o o Pi Pi a a o o P4 00 00 NOSOO >—1 CM CO CM t-H IOiH^Nm(M>QOOHNn®fflNNCOiaOOCOOOOHC INNrJC(NCOOOaO»OiOOr-iNCOHCQ05H(MNOOHrHT- HNIOIOCOCONOINCTK < a CO < P M' 3 O £ < Q f=H t" £ c O i. (1886) 1905-6.] Library Aids to Mathematical Research. 61 > a o o 00 O O H Ip/p^ ’ a a « 8 § « CM 0(1)0 -P-Poo a) © 'p/p^ a a > 05 03 ■p/p* a a o o 00 oo oo oo o o p^ P< Ph p p 03 where a represents the equatorial radius of the earth in km., <£' the geocentric latitude of Edinburgh, and S, t the declination and hour-angle of the sun, the positive sign denoting an increase in the distance of the observer from the sun. The annual term is expressed by 2ttA e T J 1 — e2 sin (0 -tt) = 2-7T. a e sin p.T Jl _e2 sin (0 — 7r), A being the mean distance of the earth from the sun, T the number of mean time seconds in the year, e the eccentricity of the earth’s orbit, O the true longitude of the sun and tt the longitude of the perihelion, and p the solar parallax (8,r,80). The small “ lunar ” term may be sufficiently expressed by + 0*014 sin(0 - <[ ) km. per sec., 0 and <[ being the longitudes of sun and moon. Turning to the measurements, I may first state that my computa- tions refer to the arithmetical means of the two distances, viz. — i[0+ + [«6]a + Hu + Hj. In other words, I have formed the means of the micrometer readings, J(aR + aA + cR + cA) for the two telluric lines, and + dA) for the solar lines, and taken the difference between the two. The screw value, expressed in wave-lengths, had to be determined on each day. It is based throughout strictly on the distance between the two telluric lines \bd\. Possibly the assumed distance between these two lines, 0*760 t.m., is still slightly 80 Proceedings of Royal Society of Edinburgh. [sess. wrong, but the error of this quantity is certainly small, and cannot sensibly affect the observed shifts. After the observations had thus been properly reduced, I pro- ceeded in the following way. My object being first to obtain the pure effect of the annual shift, I corrected for the diurnal and lunar displacement, using the above formulae, and then arranged the observations into groups of 25 to 30 single measurements accord- ing to the longitudes of the sun. This first collection of data is shown in Table I. Table I. © Distance in t.m. Weight. © Distance in t.m. Weight. 1901 1903 0 150 0-3904 1 179 0-3801 2 3 165 •3913 1 204 •3813 § 219 •3901 l 3 222 •3807 2 3 1902 1901^ o 341 •3764 1 338 •3609 1 14 •3693 1 17 •3579 1 93 •3794 1 68 •3655 1 161 •3859 1 100 •3729 1 208 •3853 1 176 •3809 1 189 •3809 1 1903 o 1905 348 •3621 1 3 o 32 •3629 2 3 71 •3649 1 64 •3631 1 82 •3655 1 75 •3644 1 93 •3671 1 147 •3738 o 3 115 •3725 f 201 •3747 o 3 Unfortunately, the low position of the sun, combined with the very unfavourable atmospheric conditions, rendered observations between November and February quite impossible. But in spite of these unavoidable gaps the effect of the annual displacement is very clearly shown in the following graph (fig. 2), which demonstrates the sinoidal character of the annual curves quite plainly. In accordance with the theoretical formulae, we find a 1905-6.] Dr J. Halm on the Spectro-heliometer. 81 '3^ • 2L- PROC. ROY. SOC. EDIN. — YOL. XXVI. 6 82 Proceedings of Royal Society of Edinburgh. [sess. maximum shift towards the red in spring, and the maximum displacement towards the blue in autumn. We also conclude from the curves that the amplitude of this annual displacement amounts to about 0*010 t.m., agreeing well with the shift required by theory, viz. 0*0105 t.m. A more minute numerical investigation will, however, be given later on. In looking at these curves, a most peculiar and, in my opinion, quite enigmatic feature will at once attract attention. One would naturally expect that the annual waves should proceed along a horizontal line. There is certainly no other motion in the line of sight which may account for the remarkable tendency of these curves to assume higher levels as we proceed from 1901 onwards. Nor is there much possibility of systematic errors in the measurements which might produce this progressive shift. No alteration, either in the instrument or in the method of observation, took place during the time from 1901 to 1903, when the shift was most pronounced. If it were due to any such cause, one would expect the change to be abrupt ; but in reality it has taken place gradually. I think this fact becomes much more obvious if we study the curve shown in fig. 3, which exhibits the curve of the distances of Table I., after the annual displacement has been eliminated. Instead of the expected grouping of the values along a horizontal straight line, we see a curve which steadily rises from its minimum value in 1901 to an almost stationary position between 1904 and 1905, indicating in its last branch, towards the end of 1905, a tendency to further elevation. I have satisfied myself that this enigmatic change affects chiefty the distances between the solar and telluric lines. If we investigate the distance between the two solar lines during the same interval of time, we find only feeble indications of a successive shortening and widening, the observed values ranging between 0*998 t.m. in 1902 and 0*995 t.m. in 1905. I think that even this much smaller difference is greater than its probable error, but no doubt it is insignificant if compared with the above fluctuation. I shall not venture upon hypothetical explanations of this singular phenomenon. What I should like to state here is simply the outcome of my observations. But if the shift is confirmed 83 1905—6.] Dr J. Halm on the Spectro-heliometer. by further observations, we have doubtless in it one of the most important phenomena of solar spectroscopy, the bearing of which on theoretical and practical questions cannot well be overestimated. I shall now enter somewhat more closely upon the numerical . 5 . "t\ TFV 0.363 4- 5 6 7 8 9 •370 1 1 3 4 5 6 7 a 9 •380 1 2 3 4 5 • 4 6 8 -0 2 4 -6 8 0 -2 4- 6 8 O 2 4 6 -8 0 2 4 6 8 1901 1902 <903 1904 1905 evaluation of the amplitudes and the positions of the turning- points in the annual curves. Since between 1903 and 1905 the shift, of which I have spoken, has certainly been small, we may for the present purpose limit our attention to this period only. I have again arranged the complete material into groups according to the sun’s longitude, but now no longer separating the single years. The new values for each group, comprising 30 single 84 Proceedings of Royal Society of Edinburgh. [sess. measurements, are shown along with their corresponding longitudes in the following Table II. Here again, of course, the diurnal and lunar shifts have been duly accounted for. Table II. © Distance in t.m. © Distance in t.m. 337 0-3611 o 88 0-3659 0 •3608 93 •3661 8 •3573 95 •3702 24 •3597 103 •3711 48 ■3625 120 •3757 61 •3616 151 •3762 65 •3645 170 •3792 70 •3650 176 •3802 73 •3675 183 •3801 77 •3658 197 •3808 81 •3670 201 •3798 83 •3650 214 •3809 If these values are represented by a simple harmonic of the form A + a sin (0 - a) and the values of A, a , and a are determined by the method of least squares, we find A = 0'3700, a= — 0’01048, and a=281°‘4, or, expressing a in km. per second : a= -0-4986 ± 0*0162 a = 281°-4 ± l°-9 Introducing in our formula 27r. a e sin p. T J 1 _e2 sin ( O - tt) the bestdaiown values of the constants, it assumes the numerical form : 0-499 sin (O -281°-3) Accidentally the agreement is perfect ; the assigned probable errors may, however, give an approximate measure of the accuracy of the values for a and tt obtained from the observations. Let us now investigate how far the measurements are capable of showing the diurnal shift. We correct the original dis- 1905-6.] Dr J. Halm on the Spectro-heliometer. 85 tances for annual and lunar shift, and collect the observations in groups according to the hour-angle of the sun. The diurnal displacement, as the formula shows, depends on the declination of the sun. We may assume, however, that in all the groups the factor cos 8 is the same, viz. 0’957. Table III. shows the observed distances for different hours of the day (true Edinburgh time). Table III. Hour. Distance in t. m. Hour. Distance in t.m. h. 21*71 0*3727 h. 0*45 0*3694 22*46 *3718 0*65 *3691 22*75 *3727 0*88 *3691 23*02 *3715 1*26 *3676 23*28 *3700 2*18 *3673 23*51 *3712 2*54 *3676 23*67 *3703 2*87 *3667 23*87 *3703 3*24 •3658 0*06 *3700 3*72 *3664 0*26 *3697 4*60 •3652 5*68 •3649 Representing these values by a formula A + b. sin we find for A and b the values A = 0*3700 t.m. and* b = — 0*00502 ± 0*00017 t.m. or in km. per sec. b= — 0*240 ± 0*008. The true value should be 0*245 km. per sec. Hence we find again an agreement within the limits of the probable error of the observed quantity. The observed annual and diurnal shifts are graphically represented in the figs. 4 and 5. The smooth curves indicate in each case the shifts due to the actual motions. Ho doubt a more strictly scientific method, but also a far more troublesome one, would have been to determine the two shifts simultaneously from the measurements. I am confident, how- ever, that the result would have been the same. The outcome of the preceding calculations seems to be interest- ing from two points of view. Hot only have the observations 86 Proceedings of Royal Society of Edinburgh. [sess. established two new cosmic proofs of the validity of Doppler’s al 4g0 i » • a I? q o v 0 tin O vin hm. J R. niO i ”7 k '-7 7 OOJ 7 7 \ OOfl. 7 7 V / — 7 -L- 004 7 223_ "T h t 002 / J M-'.j boi l// T j OQ 0 / / DO 1 / )0J. / K J 304 N L_ 70S 7 1/ . bo6 £ i * . r.JJ ~ j > ' f nnfi q 71 7 rjt»9 \ / / s / 010 / 7 r Oil r r T / 0 >, V p# 3 )o \| 0 * 0 5 3 • 7 0 -1 o V tO 1! il a 1* c ji 0 1 principle, but it seems also clearly shown that the Edinburgh spectro- heliometer possesses an extraordinarily high degree of accuracy, which promises well for the future measurements of the solar rotation, and also for the eventual final decision on the question as to whether those enigmatic shifts of the solar lines mentioned above have to be assigned to a real cause. ( Issued separately March 29, 1906.) ML *7 1905-6.] Effects of Varying Diets upon Growth and Nutrition. 87 Preliminary Note regarding an Experimental Investiga- tion into the Effects of Varying Diets upon Growth and Nutrition. By Chalmers Watson, M.D. From the Physiological Laboratory of the University of Edinburgh. Presented by Professor E. A. Schafer, F.R.S. (MS. received December 7, 1905. Read December 18, 1905.) The object of this research, which is still in progress, is to determine the influence of various diets on the growth and nutrition of animals. The diets employed have been — (1) uncooked horse flesh, (2) uncooked ox flesh, (3) rice boiled in water, (4) oatmeal porridge made with skim milk, (5) bread soaked in skim milk (the control diet). A known quantity of common salt was added to the rice and porridge foods, and an unlimited amount of water was given to all the animals except those fed on bread and milk. All the foods were supplied ad libitum. The chemical composition and heat values of samples of the various foods used were determined by Dr Andrew Hunter, and are given in the following table Dried Foods. Proteid. 4-5 cQ Carbohydrate. Ash. Physical Heat Values. Physiological Heat Values calculated. j Calculated. Observed. Horse flesh . 81-62 14-27 1-70 2*41 5506 5640 4772 Ox flesli 49-80 46-48 1-50 2*22 6967 6896 6519 Rice .... 7-65 1-08 88*76 2-57 4123 4098 4055 Porridge 16-11 8-24 72 15 3-50 4546 4610 4401 Bread and skim milk . 18-50 4-41 73-10 3-94 4341 4305 4176 The animals employed have been tame rats of various ages (in number over 300). 88 Proceedings of Royal Society of Edinburgh. [sess. The observations have been as follows : — (a) On very young rats newly weaned, the controls being taken from the same litter (10 litters). (b) On young rats set. 2 to 3 months (14 on each diet). ( c ) On adult rats (10 on each diet). (d) On castrated female rats * (17). Throughout the observations the bread and skim-milk diet was employed as the control diet ; it appears to be well adapted for the growth and nutrition of rats of all ages (see Charts I., II., III., IV.). Special attention was directed to the influence of the various diets on the supervention of pregnancy, on the growth and nutri- tion of the animals in the second generation, and on the recuperative power of animals under a normal diet that had deteriorated in consequence of having been fed for a time, in some cases for more than one generation, on an unphysiological diet. Results. Rice. — An exclusive diet of boiled rice arrests the growth of young rats. The animals always succumb eventually, although they may live for a few months. Adults lose weight rapidly, and die, as a rule, within two months. Castrated female rats tolerate the diet well, four out of five being alive and apparently in good health at the end of five months. Porridge. — This diet retards the growth and is speedily fatal to very young rats (see Chart I.) ; a similar result is observed in the case of young rats set. 2 to 3 months. Adult rats maintain their weight well for many months on this diet. It may be noted that the composition and heat value of the porridge is essentially the same as that of the bread and skim milk (see Table, p. 87). Horse Flesh. — In very young rats (newly weaned) a diet of horse flesh retards growth and is uniformly fatal within a few months (see Chart I.). Young rats set. 2 to 3 months exhibit a mixed result, the majority succumbing within a few months ; a minority live, thrive, and become pregnant. The power of lacta- * I am indebted to Dr F. H. A. Marshall and Mr W. A. Jolly for these animals. 1905-6.] Effects of Varying Diets upon Growth and Nutrition. 89' tion appears to be weakened, and the mortality in the progeny is high. Adult rats can live on horse flesh and maintain their weight. Ox Flesh. — An ox-flesh diet is tolerated by very young rats. better than horse flesh. About one-third of the animals failed to thrive, and succumbed within one or two months ; the remainder appeared to thrive, but their gain in weight was appreciably less than in the control, bread, and skim-milk fed subjects. None of these animals became pregnant. In young rats set. 2 to 3 months an ox-flesh dietary is conducive to rapid growth and development ; 90 Proceedings of Iioyal Society of Edinburgh. [sess. the animals may attain a greater weight than the controls. The animals become pregnant, hut, as in the case of horse-flesh fed rats, the mortality of the progeny in early life is high. 26l6l05 17/7 17/8 17f9 nJlQ/05 HO arms / cX 160 7 150 f 14-0 130 .. • 120 / / 110 f 100 / ' 90 / f ’ / 80 / 1 1 70 i 60 ]: 50 t 1 It 40 / 30 1 \ Bread £> (3/c/m Mi 1J^ Ox Flesli Chart II. — Comparison of ox flesh and bread and skim milk. The black line = average of 2 fed on bread and milk — 1 female which became pregnant. The dotted line = average of 4 fed on ox flesh — 2 females which remained non -pregnant, Two illustrative charts may he given. Chart I. shows that the ox-flesh fed animals succumbed within two months. A more usual result with this diet is illustrated in * On 27/9/05 the female rat gave birth to a litter of eight. 1905-6.] Effects of Varying Diets upon Growth and Nutrition. 91 Chart II., in which observation the ox-flesh animals lived and thrived for several months, but gained in weight more slowly and im- perfectly than the controls (bread and skim-milk fed) ; and although kept along with males, the females failed to become pregnant. Influence of a Meat Diet on the Second Generation. The following figures illustrate the high mortality in early life of the second generation of meat-fed animals Litters. Rats. No. alive at end of One Month. Percentage alive at end of One Month. Meat (ox flesh and horse 13 93 19 20 flesh) Bread and skim milk 14 97 82 84 The effects of a meat diet on pregnancy are further shown by the following history of one rat which had four litters : — Date. Diet. Number of Young. Result. Litter 1. April 22 Bread and skim milk 9 All lived and thrived. 9 J ; June 23 Horse flesh 9 All died within two months. „ 3. July 30 Horse flesh 6 All died within one month. „ 4. Sept. 9 Bread and skim milk 8 All lived and thrived. Observations on the Recuperative Powers. Observations were made on the recuperative power of animals which had been fed on an unphysiological diet for some time during the growing period and then transferred to a normal diet. Chart III. illustrates the recuperative power of three rats placed on a normal diet of bread and skim milk, after they had deteriorated in consequence of having been fed for six weeks on a diet of rice, horse flesh, and porridge respectively. sc/l/ll 9/z / tlzzijll 9jzi trlai/u 9lzi EOjt'llZ 92 Proceedings of Royal Society of Edinburgh . i A. Rice B. Horse Flesh C.Porridje Chart III. — To illustrate the recuperative power of rats under a normal diet (bread and skim milk). Rat A had been fed for six weeks on an exclusive rice diet, B on horse flesh, and C on porridge. The normal diet was begun in all three cases on the same day (June 12). The growth within the following five weeks is striking. 1905-6.] Effects of Varying Diets upon Growth and Nutrition. 93 In Chart IV. the recuperative power in the progeny of meat-fed parents is demonstrated. The animals (after weaning) were placed Chart IY. illustrates the recuperative power, under a normal diet (bread and skim milk), in the second generation of horse-flesh fed rats. The rats (four) were weaned on 20/8/05, and placed on an exclusive horse-flesh diet. On 4/9/05 two were transferred to a bread and skim-milk diet, the meat diet being continued with the remaining two. The latter succumbed within two or three weeks. on a horse-flesh diet for a period of two weeks, when two were transferred to a bread and skim-milk diet, with the results shown. 94 Proceedings of Royal Society of Edinburgh. Summary. I. The use of an excessive meat diet in rats induces a deteriora- tion in the health of the animals. This deterioration is shown in (a) an imperfect physical development, ( b ) a loss of reproductive power, ( c ) defective lactation, and (d) a high mortality in early life in the second generation of meat-fed subjects. II. The recuperative powers of these deteriorated animals, on a normal diet, is very striking. III. In animals deprived of their ovaries, the minimum amount of proteid required is less than in normal females. (. Issued separately February 22, 1906.) 1905-6.] Excretion of Allantoin in Thymus Feeding. 95 A Contribution to the Study of the Excretion of Allan toin in Thymus Feeding. By W. M ‘Lachlan, M.D. ( From the Research Laboratory of the Royal College of Physicians , Edinburgh.) Communicated by Dr D. Noel Paton. (Read January 8, 1906. MS. received January 12, 1906.) The establishment of the nature of the chemical relationship of the purin bodies, the diureides which form the chief end-product of proteid metabolism in birds and reptiles, with urea, the end- product in mammals, is not the least important of the valuable contribution of E. Eischer to physiological chemistry. The constant presence of diureides in the mammalian urine, and of urea in the urine of birds, shows that no hard-and-fast line exists between the metabolic processes in the two groups, and the fact that in the mammalian foetus an important end-product of meta- bolism is a diureide — allantoin — is a further proof of the close affinity of the tissue changes throughout the vertebrate series. Nor is this allantoin merely a product of foetal metabolism, for it has been found in small quantities in the urine of nearly all adult mammals in which it has been tested for. It was found in the urine of adult man by Ziegler and Hermann (1), and in that of cats, dogs, and rabbits, by Meissner (2). Pouchet (3) also found it in the urine of man, and in greater amount in the urine of pregnant women, as well as in diabetes insipidus and convulsive hysteria. Salkowski (4) found it in the urine of the ox to the extent of 0775 grm. per 1000 c.cm. I have found it in the urine of a rabbit to the extent of 0*14 per cent., and in the dog, upon a diet of oatmeal and milk, I have found, by the method afterwards to be described, the following amounts : — Per cent. 0T1 grm. 0*10 „ 0-05 „ 0-06 „ 3 3 Average, 0'08 96 Proceedings of Royal Society of Edinburgh. [sess. Not only is allantoin known to be normally present, but when disturbances in the metabolism are produced by drugs its amount may be increased. Pohl (5) failed to find allantoin in the tissues of the normal dog, but in hydrazine poisoning he found it in the liver and in traces else- where. On a few hours’ autolysis of the organs, allantoin appeared chiefly in the intestinal mucous membrane and in the liver. Hydroxylamine may cause the appearance of allantoin, but it does not appear with arsenic or phosphorus (Pohl) (5). In dogs diamido sulphate injected in doses of *05 grm. per kilo causes coma and death, with the presence of allantoin in the urine (Borissow) (6). Hydrazine sulphate does not cause the production of allantoin. In the urine of some animals, e.g. the dog, allantoin largely takes the place of uric acid, and the administration of uric acid has been said to lead to the excretion of allantoin, the uric acid being supposed to be oxidised and hydrated with the splitting off of C02— C5H4N403 + 0 + H20 = C4H6N403 + C02. Thus Salkowski (7) demonstrated its presence after feeding uric ucid to dogs. When 4 grammes were given on each of two suc- cessive days, 1-42 grammes of allantoin were recovered. Mendel and Brown (9) obtained a considerable yield of allantoin after feeding uric acid to cats. Poduschka (8), on the other hand, found that the administration of two grammes of urate of soda caused no increase in the allantoin of a dog’s urine. Swain (11) has failed to find a marked transformation of uric acid to allantoin. After the administration of 9 grammes of uric acid, only 1 gramme of allantoin appeared in the urine. Minkowski (10) showed that when liypoxanthin is given, 77 per cent, appears as allantoin, and that 9 methy-adenin also causes an increase of allantoin. He expresses the view that in the dog the metabolism of the purins still in nucleic acid more readily yields allantoin than the metabolism of the free purins. According to his observation adenin, the purin of the thymus, is not changed to allantoin, and Stadthagen (12) found that guanin, the purin of the pancreas, was not changed. 1905-6.] Excretion of Allantoin in Thymus Feeding. 97 Mendel and White (13) injected lithium urate in doses of 1 gramme into the veins of dogs, and they record the appearance of allantoin in the urine. The appearance was not constant, and the amount was small in all cases, being somewhat larger when the injection was made into the portal vein. When it is remembered that mere autolysis of the liver causes the production of allantoin (Pohl) (5), the significance of these experiments in indicating a production of allantoin for uric acid is decreased. It might be urged that the allantoin underwent further change in the organism, but the fact that allantoin, when injected, is nearly all recovered in the urine, invalidates such an explanation. The origin of allantoin from uric acid is, therefore, by no means satisfactorily established. The possible relationship of allantoin to uric acid and the purin bases tends to connect it with the metabolism of the nucleins, and evidence has been adduced by Mendel, Underhill, and White (14), that in the dog it is formed from this source. By injecting preparations of nucleic acid, prepared from different sources, into the blood vessels, intraperitoneally and subcutaneously in dogs, they caused the appearance of allantoin in the urine. The symptoms produced by these injections were fairly marked, and the possibility must be borne in mind that the appearance of allantoin maybe ascribed to a toxic action, just as the increased excretion of uric acid, observed by Loewi (15), is, after the subcutaneous injection of sodium nucleate, ascribed by them to a toxic effect. While the effects of administering the purin bodies, either free or still combined in nucleic acid, upon the production of allantoin is by no means distinct, the administration of thymus gland substance is always very pronounced. Minkowski and Cohn (10) discovered, independently, that allantoin may be excreted in the dog as an end-product in the metabolic processes which give rise in man to uric acid after thymus feeding, and an enormous rise in the excretion of allantoin in the dog under the administration of thymus gland was discovered by Salko^ski. This is generally explained as the result of the katabolism of the nucleins con- tained, but so far this theory has not been tested by actual experiment. PEOC. ROY. SOC. EDIN. — YOL. XXVI. 7 98 Proceedings of Royal Society of Edinburgh. [sess. Present Investigation. The object of the present investigation is to elucidate the question of the nature of the relationship of the excretion of allantoin to the administration of thymus gland. A. Methods. 1. Methods of Analysis. (A) Allantoin Estimation. The following methods were tried and rejected : — (a) Loewi’s method (16) — a method which depends upon the pre- cipitation of the nitrogenous compounds by means of mercurous nitrate without precipitating allantoin — is unsatisfactory owing to the fact that mercurous nitrate, if too acid, or if impure (containing mercuric nitrate) will also precipitate allantoin. ( b ) The second method — that of Moscatelli (17) — consists in the precipitation of allantoin with mercuric nitrate. The subsequent washing of the precipitate was found to result in considerable loss. Another cause of loss is the addition of too much ammonia, by which the allantoin nitrogen is set free. (c) The next method adopted was that of Poduschka (8), of which the following is a description : — A measured quantity of the urine is precipitated with basic lead acetate, and the excess of lead is removed from a definite volume of the filtrate by concentrated sodium sulphate solution. To a definite volume of the second filtrate 5-10 per cent, silver nitrate solution is added. This is filtered, and 1 per cent, dilute ammonia is added to the filtrate, and the precipitate is washed with 1 per cent, sodium sulphate solution until free from ammonia, and the allantoin is estimated by Kjeldahl’s method. The difficulty with this method is to render the precipitate free from ammonia. If the precipitate he hot free from ammonia, then the nitrogen of the added ammonia would be estimated as allantoin nitrogen. To render the precipitate ammonia-free it must he washed with 1 per cent, sodium sulphate until the alkaline reaction to red litmus 1905-6.] Excretion of Allantoin in Thymus Feeding. 99 has disappeared. The results with pure allantoin gave only a small fractional return, and with the washing the precipitate was. visibly disappearing. The presence of silver allantoin was demon- strated in the filtrate, and hence the method proved unreliable. It would therefore appear that the more one washes the precipitate, the smaller is the result, and the less one washes the precipitate, the nearer it approaches to a correct result, although not free from ammonia. To overcome this difficulty the following modification of Poduschka’s method was devised : — To 50 c.c. of the urine basic lead acetate was added, and the excess of lead was removed from the filtrate by concentrated sodium sulphate solution. Tins was filtered, and to the filtrate was added silver nitrate solution (5-10 per cent.), and again filtered, until no further pre- cipitate formed in the filtrate upon the addition of silver nitrate solution. Dilute ammonia wTas then added to the new filtrate, and the allantoin silver was precipitated and filtered. The precipitate and filter paper were then placed in an Erlenmeyer’s flask, with a pinch of oxide of magnesium, as in Morner and Sjoquist’s method, and a little water was added. This was placed over the steam-bath at 60° C. for f hour. By this modification the ammonia was liberated, and only the nitrogen of the allantoin was retained. Even if the allantoin undergoes a decomposition into urea, there is no loss of nitrogen. The flask and its contents were then left to cool, and the same flask, to avoid loss in transferring the contents to a Kjeldahl’s combustion flask, was used for oxidation. Pure sulphuric acid was then added, and the process continued as Kjeldahl’s method, by means of which the nitrogen was esti- mated and the allantoin equivalent was calculated. Deduction was made for the nitrogen in the filter paper and the sulphuric acid, and the loss in the several stages of filtration was measured and the final filtrate treated as an aliquot portion. The method of estimation by silver precipitation has been criticised by Salkowski (4) as one of the most difficult of reactions, but as carried out by me it has yielded satisfactory results. 100 Proceedings of Royal Society of Edinburgh. [sess. This modification has been tested with Merck’s allantoin, which itself was found to be pure by determining the nitrogen by Kjeldahl’s method. When dried in a desiccator, '5 grm. was found to yield *49 grm. The loss during each stage of filtration was calculated, and left .46 grm. to be recovered, as the final filtrate was estimated as an aliquot portion. The actual recovery was *46 grm. or 100 per cent. In another experiment 99*6 per cent, was recovered. ( B ) Other Estimations. The total nitrogen was estimated by Kjeldahl’s method. Methods of Experiment. For the experiments the animals used were a setter and retriever bitch — each of known weight — and the urine was collected into dilute hydrochloric acid. B. Results. The allantoin excretion was first found on a diet of oatmeal and milk, the quantities being 300 grm. oatmeal and 700 c.c. milk. Oatmeal contains *0212 per cent, purin nitrogen, and milk *002 grm. per litre, according to Dr Walker Hall. From this one finds that the diet of oatmeal porridge and milk contains *065 grm. purin nitrogen. Table I. Allantoin per diem upon oatmeal and milk diet. 1-140 grm. 0*901 „ 0*7)12 „ 0*683 „ Average, 0*809 ,, = 0*28 hh The animals were then fed upon weighed quantities of thymus gland or other substance to be studied. The thymus gland of the ox was procured from the butcher, and the fat determined in a sample. Having estimated the amount of fat, the animal to be experimented upon was fed upon a known weight of thymus gland, deduction being made for the amount of fat. 1905-6.] Excretion of Allanto'in in Thymus Feeding. 101 After raw thymus a marked rise in allanto'in excretion was observed. Table II. Observations. Allantoin in grammes per diem. Allantoin Nitrogen in grammes per diem. Food. Raw Thymus. 1. Setter 5-039 1-786 950 grm. 2. Setter 6*906 2-448 950 grm. 3. Retriever 3-698 1-311 680 grm. This simply confirms the results of Salkowski. Are the nucleins and pur ins of the thymus administered the source of the allanto'in excreted ? If this were so, simply boiling the thymus given to the animal, since this does not change the nucleins or purins, should not modify the yield of allanto'in. The question was tested in two ways. 1st. By estimating the excretion of allanto'in per 100 grm. of raw and per 100 grm. of boiled thymus gland administered. These results may be tabulated thus : — Table III. Raw Thymus. Observations. Allantoin. Allantoin per 100 grm. Thymus, less fat. Food. Per cent. Per diem. 1. Setter 2-65 5-039 •616 950 grm. 2. Setter 1-151 6-906 •690 950 grm. 3. Retriever bitch •617 3-704 •708 680 grm. Average, 5-2 •671 860 grm. 102 Proceedings of Royal Society of Edinburgh. [sess. Table III. —continued. Boiled Thymus. Observations. Allantoin. Allantoin per 100 grm. Thymus, less fat. Food. Per cent. Per diem. 1. Retriever bitch *314 1*480 •251 684 grm. 2. „ *431 1*568 •267 680 grm. 3- „ •466 3*032 •325 1080 grm. 4. I •311 4-977 •498 1160 grm. Average, 2-7 •335 901 grm. The effect of the raw thymus is thus double that of the boiled. 2nd. By studying the proportion of the nitrogen of the urine to the nitrogen in allantoin after feeding on raw, and after feeding on boiled, thymus gland. Table IY. Proportion of Total Nitrogen to Allantoin. Obser- vations. Total Nitrogen. Allantoin Nitrogen. Food. Per- cent. Per diem. Per cent. Per diem. Per cent, of total Nitrogen. Raw Thymus. 1. Setter 2-62 4-984 •94 1-786 *36*4 950 grm. 2. „ 2-35 14-159 •408 2-448 17-3 950 grm. ;i j /: Boiled Thymus. 3. ,, •946 12*3 •051 •663 5*3 790 grm. * The correctness of this figure is further indicated by the fall in the percentage of nitrogen in urea from about 84 to 57. The percentage of allantoin nitrogen after thymus feeding in the case of the setter is thus 36 -4 per cent., and 17*3 per cent, upon a diet of raw thymus gland, and 5 '3 per cent, in the case of boiled thymus gland. These observations appear to negative the view that the allantoin found is simply derived from the nucleins and purins of the thymus. 1905-6.] Excretion of Allanto'in in Thymus Feeding. 103 To what extent is the increased excretion of allanto'in after boiled thymus due to the free purin bodies ? To test this question, a watery extract of 750 grins, of boiled thymus was prepared and fed to the dog. This, of course, con- tained the free purins, but not the nucleo-proteids. The result was the appearance in the urine of 0’365 grm. of allanto’in per 100 grm. thymus — an amount which corresponds closely with that which follows boiled thymus. This tends to show that the free purins are the source of the allantoin after boiled thymus is given. The Part played by Nucleins in the Production of Allantoin. — Since, in feeding with boiled thymus, the increased formation of allantoin appears to be due to the purin bodies, it seemed of interest to investigate how far the formation of allantoin may be varied by varying the amount of nucleins in the food. For this purpose the proportion of nitrogen in allantoin to the total nitrogen excreted was investigated on the following diets — oatmeal and milk, liver, raw pancreas, boiled pancreas, and beef. Food. Total Nitrogen. Allantoin Nitrogen. Allantoin Nitrogen, per cent, of total Nitrogen. Per cent. Per day. Per cent. Per day. 1. Thymus raw, 950 grm. 2-62 4*9 •940 1-786 36-4 ,, 950 grm. 2-35 14*1 1 -408 2-448 17-3 2. Thymus boiled, 790 grm. 0-94 12-3 •051 •663 5-3 3. Raw pancreas, 1 200 grm. 2-0 5'4 •184 •496 »T ,, 1000 grm. 3-67 9*1 •520 1-300 14-2 4. Boiled pancreas, 730 grm. 1-7 6-3 •045 T62 2-5 ,, 780 grm. 3*8 14-0 •046 •165 1-1 5. Liver, 560 grm. 2-5 8*7 T57 •533 6'1 ,, 560 grm. 3-3 8-4 T47 •367 4-3 6. Beef, 200 grm. 4-0 8T *093 •186 2-2 7. Oatmeal 300 grin., Milk 700 c.c., 0-62 6*3 •040 •404 6-4 53 53 0-59 5-2 •036 •313 6-0 ” 1-3 3-54 •141 *364 10-2 104 Proceedings of Royal Society of Edinburgh. [sess. When fed on raw and boiled pancreas, the results were thus : — - Allantoin in grammes per 100 grm. pancreas. •49 . . Raw •24 . . . Raw *14 . . . Boiled The effect was slighter, but in the same direction as the thymus. Salkowski fed a small dog on five successive days upon If kgs. boiled pancreas. In the urine 3‘089 grm. allantoin, or 1*074 grm. nitrogen, with 40*7 grm. total nitrogen (2*64 per cent, of the total). Uric acid and purin bases showed no rise. Seven weeks later 2 kgs. pancreas caused only 1-058 grm. allantoin. It will thus be seen that in the dog, upon a purin-poor diet, about 6 per cent, of the total nitrogen is in the form of allantoin. The proportion remains unaltered when liver is administered, and when moderate amounts of raw pancreas are given, while it may rise temporarily to 14 per cent, when large quantities of raw pancreas are eaten. But, even then, the effect of these diets, rich in nucleins, was to produce at most a slight change in the proportion of allantoin nitrogen to the total nitrogen. Conclusions. These investigations seem to show : — 1. That allantoin is a normal constituent of the urine of the dog on all diets. 2. That the administration of raw thymus causes a great increase in the production of allantoin (*188 grm. allantoin nitrogen per 100 grm. thymus gland, or 36'4 per cent, of total nitrogen). 3. That boiled thymus produces a much less marked effect (*083 grm. allantoin nitrogen per 100 grm. thymus gland, or 5*3 per cent, of total nitrogen), and that therefore the increased formation of allantoin, after feeding with raw thymus, is not simply due to the nucleins and purins contained. 4. That liver and pancreas, although rich in nucleins, cause a rise of only -08 grm., and *08 grm. respectively of allantoin 1905-6.] Excretion of Allantovm in Thymus Feeding . 105 nitrogen, per 100 grin, of organ, or 5’2 and 11 ’6 per cent, of total nitrogen. 5. The evidence seems to indicate that the administration of raw thymus, and to a less extent of raw pancreas, exercises some specific action on the production of allantoin. To Dr Noel Paton, Superintendent of the Research Laboratory of the Royal College of Physicians, I am indebted for much advice and assistance during the progress of the research, and in the preparation of this paper. The expenses of this research were defrayed from Mr Francis- Mason’s donation to the Laboratory for the Investigation of the Ductless Glands. References. 1. Ziegler und Hermann, Archivf. Gynakologie , iii., 1871. 2. Meissner, American Journal of Physiology , vi. 3. Pouohet, Contrib. a la connais. des mat. extra, de V urine y Paris, 1880, xxviii., 37. 4. Salkowski, Zeitsch. f. phys. Chem.., Bd. xlii., p. 217, 1904. 5. Pohl, Archiv fur experimentelle Patliologie, xlvi. 367. 6. Borissow, Zeitschrift fur physiologische Chemie, xix., 499. 7. Salkowski, Berichte der deutschen chemischen Gesellschaft,. 1876, ix. p. 719. 8. Poduschka, Archiv filr experimentelle Pathologie und Phar- makologie , 1900, p. 65. 9. Mendel and Brown, American Journal of Physiology , 1900,, iii. p. 267. 10. Minkowski, Centralblatt fur innere Medicin , 1898, No, 19. 11. Swain, American Journal of Physiology, vol. vi. 12. Stadthagen, Archiv fur patliologische Anatomie, 1887, cix. p. 418. 13. Mendel and White, American Journal of Physiology , 1903,, viii. 14. Mendel, LTnderhill, and White, American Journal of Physiology, 1902, vii. p. 397. 15. Loewi, Archiv filr experimentelle Pathologie und Phar- makologie, 1901, xlv. p. 157. 16. Loewi, Archiv filr experimentelle Pathologie und Phar- mahologie, 1900, xliv. p. 20 17. Moscatelli, Zeitschrift fur physiologische Gliemie, 1889,, xiii. p. 203. 18. Walker Hall, Purin Bodies, pp. 46 and 49. 106 Proceedings of Royal Society of Edinburgh. [sess. Note by D. Noel Paton. Dr M‘Lachlan having been compelled to abandon the investiga- tion, I thought it well to determine the influence of feeding with lymphatic glands upon the production of allantoin. The abdominal lymphatics of the ox, freed from fat, were used. When given to the setter used in the previous experiments, they were vomited, and when next given, the dog refused to eat them. A fox terrier, weighing about 6 kilos, was consequently used. As I found it difficult to be sure that the ammonia is entirely got rid of from the silver allantoin precipitate by the method em- ployed by Dr M ‘Lachlan, I adopted the plan of passing a current of air through the Erlenmeyer’s flask, kept at 60° C. for several hours, till no change was produced on a piece of wet red litmus paper held in the flask. The following results were obtained : — Urine of Total N. per cent. Allantoin N. per cent. Allantoin N. per cent, of total. 24tli to 27th, after Pan- fO'06 creas 500 grm. . 1-00 \ 0*06 6 27th to 29th, Fast . 1-87 0-038 2 29th to 31st, after 325 grm. Lymph Glands . 0-549 0-059 10 It would thus seem that the administration of lymph glands causes an increase in the allantoin of the urine very little greater than that caused by feeding with pancreas, and that its effect is much smaller than that produced by thymus feeding. {Issued separately, March 29, 1906.) 1905-6.] Formation of certain Lakes in the Highlands. 107 On the Formation of certain Lakes in the Highlands. By- Dr Leon W. Collet, F. Swiss Geol. S., Assistant to Sir John Murray, K.C.B., and Dr T. N. Johnston, F.K.S.E. With a Note on Two Rock Basins in the Alps, by Dr Leon W. Collet. SCOTTISH LAKE SURVEY. Under the direction of Sir John Murray, K.C.B., F.R.S., D.Sc., LL.D., etc., and Laurence Pullar, F.R.S.E. (Read February 19, 1906.) When surveying the lochs of the Dee basin in the Highlands, we came across three interesting lochs on the formation of which we propose to deal in this paper. Loch Muick. Loch Muick (see fig. 1) lies at the head of Glen Muick, at a height of 1310 feet, on the property of His Majesty the King. On both sides of the loch the mountain slopes rise precipitously from the water’s edge, and reach a height of 2400 feet on the south-east and from 2326 to 3352 feet on the north-west. The rocks surrounding the lake are granite. The loch trends in a N.E. and S.W. direction, and is 2J miles in length, the maximum breadth being J mile at the north-eastern end. The maximum depth recorded is 256 feet, and the mean depth calculated from the volume of water is 111 '69 feet, approaching the half of the extreme depth. The ratio of maximum depth to the length is 46. That figure shows the importance of this basin, as in Loch Morar, the deepest loch in the British Islands (1017 feet), that ratio is 61, and the Lake of Geneva is 230 times longer than the maximum depth. Loch Muick is fed by numerous small burns and streams, the largest feeder being the Allt an Dubh-loch coming from the Dubh loch, which lies at the very head of the glen, at a height of •about 2100 feet. The Glas Allt flows into the loch on the north- 108 Proceedings of Boy al Society of Edinburgh. [sess. western shore near the south-western end, forming a big delta which is extending into the loch. The delta is wooded in contrast with the other shores, which are but scantily clothed with vegeta- tion. The Black Burn flows into the loch near the middle of the south-eastern shore, and does not form a delta, for it runs down a steep rocky slope. We found the deepest part of the loch where the mountain slopes on the opposite sides are steepest and the valley narrowest , this fact being an important one, as will be shown further on. Our soundings (87) show that the basin is of simple conformation, the bottom sloping on all sides towards the deepest part. Indeed the bottom of Loch Muick is a very flat one, as shown by the follow- ing figures : — Feet. Acres. Per cent. 0 to 50 170-41 31-06 50 „ 100 95-96 17-49 100 „ 150 68-59 12-51 150 „ 200 90-63 16-52 200 „ 250 113-37 20-67 over 250 9-60 1-75 548-56 10000 The contour lines of depth approach each other more closely along- the south-eastern shore , showing that the slope is steeper there than along the opposite shore. The valley or glen in which Loch Muick lies has been occupied in the Great Ice Age by a glacier, as is shown by its U-shaped cross-section. That shape is the most appropriate one for the glacier’s movements, according to Penck,* the well-known con- tinental authority in glacial matters. Loch Muick partakes both of the character of a rock basin and of a barrier basin. The barrier is the latest moraine thrown down by the glacier that once crept along the glen, and it serves still, as at first, to dam back the drainage. A small part of the barrier is above the level of the water, but by far the greater part is sub- * Dr A. Penck, “Glacial Features on the Surface of the Alps,” Geogr* Teacher, vol. iii. , p. 49, 1905. 1905-6.] Formation of certain Lakes in the Highlands. 109 merged, as shown on the sketch map by the irregular and stony bottom at the north-eastern extremity of the 50-feet area. We suppose the barrier to be represented by the slope from the north- eastern end of the loch to the 100-feet contour. Let us now discuss why we consider Loch Muick to be a rock basin. 1. If Loch Muick were not a rock basin, the frontal moraine which dams the drainage would be 200 feet high, a figure far too big for a small glacier, for we must not forget that this moraine was the last one deposited by the Muick glacier. The fact that we have never seen moraines 200 feet high, neither in actual glaciers nor at the head of glens, where one meets with moraines, 110 Proceedings of Royal Society of Edinburgh. [sess. prevents us from saying that Loch Muick is only a lake dammed up by an old frontal moraine. 2. Penck has shown that a mass of ice coming from a semi- circular head would he pressed into the diameter of the same circle, and in order to maintain a continuous movement, an increase of velocity would be necessary at this place, and this increased velocity must affect the bed of the glacier, until a sufficient depth is attained. How and then the increase of depth corresponds to the decrease of width in the glaciated valley. On the other hand, a sudden increase of width in a glaciated valley is often con- nected with a diminution in the depth of the trough. We think that, in the case of Loch Muick, precisely these conditions are present. 3. We pointed out that the submerged south-eastern slope was- far steeper than the opposite one. We may find an explanation of this in the fact that the glacier, being forced to change its course at that part, must have exerted a tremendous pressure on the south-eastern slope. We think we are justified, therefore, in saying that Loch Muick exhibits the characters of a rock basin and of a barrier basin. Loch Callater. Loch Callater is a small, narrow loch, lying in Glen Callater, about 5 miles to the south of Braemar, at an elevation of 1625 feet. Its length is 0‘84 mile, and the maximum breadth 0’20 mile. The maximum depth recorded is 29 feet. Loch Callater is a true barrier basin, dammed up by a frontal moraine. The burn flowing out of the loch has cut its way through the moraine. At the head of the loch is a large alluvial tract, which evidently at one time formed part of the lake, when the level of the water was higher, and before the burn had cut its way so deeply in the barrier. Indeed the loch is destined to disappear in the future. By and by the alluvial matter will fill up the head of the loch, and simul- taneously the burn will cut its way deeper through the barrier,, until ultimately the lake will be drained and converted into an alluvial plain, as has been shown by Sir Archibald Geikie and Professor James Geikie. 1905-6.] Formation of certain Lakes in the Highlands. Ill Loch Builg. Loch Builg (see fig. 2) lies in a short and narrow transverse valley, between Glen Avon and the head of the Gairn valley, at end, but it also contributes to the Dee, as water percolates through the moraine at the south end. As regards the formation of this loch, we believe that at the beginning of the later glaciation a lobe of ice passed northwards from the valley of the Gairn by Loch Builg, towards the Avon, which laid down a frontal moraine at its north end. Mr Hinxman, who mapped the area round Loch Builg, when consulted by Dr Horne, confirmed this view. How, to what is due the dam at the south end? After its regression, the glacier, increasing considerably, began again to creep down the mountain, now following the direction of the Gairn, laying down lateral moraines, of which one dammed the loch at the south end. 112 Proceedings of Royal Society of Edinburgh. [sess. The bottom of Loch Builg is divided into three basins, the more important one being in the middle of the loch. These basins are separated by two ridges, as shown on the sketch map. These shallowings might be caused by moraines on the floor of the lake, which seems to us a better explanation than to suppose that these basins are separated by rocky barriers ; it is impossible, however, to give a definite opinion on this point. When sounding, we found along the western shore a submerged talus one foot below the water level, about 12 feet broad, with a steep slope falling suddenly to 12 feet. Have we here an ordinary talus at the foot of a slope, or a submerged moraine ? It is impossible to say ; anyhow, the fact is worthy of record. Note on Two Rock Basins in the Alps. By Dr Leon W. Collet, F. Swiss Geol. S., Assist, to Sir John Murray, K.C.B. True rock basins are not rare at the head of many small mountain valleys or corries of France and Switzerland. Many flat alluvial plains above gorges in Switzerland, as well as in the Highlands of Scotland, were without doubt at one time glen lakes, as they are called by Sir Archibald Geikie, or true rock basins which have been filled up by the sand and the mud brought into them by their tributary streams. I propose in this short note to describe two of these interesting rock basins situated at an elevation of about 5500 feet on the mountains called by Swiss geologists “ Hautes Alpes Calcaires ” (high calcareous Alps). 1. Plain of Barberine.* The plain of Barberine, as shown in fig. 3, is an alluvial one, crossed in its length by a stream ; its cross-section is U-shaped. At the lower part of the plain the stream enters a gorge with steep slopes, i.e. with a V-shaped cross-section. The plain is an old true rock basin, which has been eroded by the glacier coming from the peaks above (where there are still hanging glaciers), and has * For the geological map of the region, see L. W. Collet, Mat, Carte Geol, Suisse , liv. xix., nouv. serie, 1904. 1905-6.] Formation of certain Lakes in the Highlands. 113 been filled up by detrital matter brought down the slopes by the streams. The plain is surrounded by Trias and Lias limestones. The gorge at the lower end is eroded in the gneiss, and on the slopes we can see beautiful “ roches moutonnees.” At first the gneiss resisted better the erosive action of the glacier than the limestones, and formed a rock barrier. When the glacier retreated, the stream cut its way through the rock barrier, making a V-shaped gorge. The glacier increasing again considerably, the ice found its way through the gorge scooped out by the stream. The gorge in Fig. 3. — Plain of Barberine. the gneiss may be due to river erosion during an interglacial period. The lake, now filled up, was due to glacial action in the softer rocks. 2. Lake op Yogealle. The small lake of Yogealle (see fig. 4) is one of the most beautiful examples of a rock basin I have ever seen. It is a corrie lake eroded by the hanging glacier out of the soft limestone called “Neocomien,” or Lower Cretaceous, the rocky barrier at the end of the loch being composed of a hard limestone called “Malm,” or Upper Jurassic. The photographs which accompany the present note speak for themselves, and show that we have here true rock basins. PROC. ROY. SOC. EDIN. — YOL. XXYI. 8 114 Proceedings of Royal Society of Edinburgh. [sess. Conclusions. — I have here dealt with two examples of rock basins or corrie lakes. Now, if small corrie glaciers have been able to erode rock basins, a fortiori the tremendous valley glaciers would erode rock basins. In my opinion the erosive action of a glacier is not only due to the pressure of the ice on its bed, but also to the action of the water running over the bed ; for the water will fill up the Fig. 4. — Lake of Yogealle. cracks and fissures of the bed, and, in certain seasons when freezing - and melting take place, will be a powerful agent of disintegration. It is a fact well known to mountaineers and climbers that the water running underneath the glaciers during the day freezes during the night at the end of the summer and in autumn, cutting off the supply of the streams originating therefrom. That fact is worth noting, because many antiglacialists say that a glacier is a protecting cover on the rocks. As a heading to chapter xvi. of his book Ice or Water , Sir 1905-6.] Formation of certain Lakes in the Highlands. 115 Henry H. Howorth gives this quotation from Lamplugh : “ This is the first glacier I have visited, and I brought away the impression that on the whole it was easier to give explanations of glacial phenomena before I had seen ice ! ” I am not of the same opinion, and I am sure that we can learn from the study of the present glaciers that there is a lateral action of the glacier against the walls of the surrounding cliffs, an erosive action which is small, I must say, but which cannot be denied. Now, when talking about the tremendous glaciers of the Great Ice Age which in Switzerland reached a thickness of 3000 feet, we are obliged to say that if the present glaciers, which are insignificant in comparison, have an erosive action, a fortiori those of the Great Ice Age must have been able to erode rocks. I have given in this short note facts of observation collected during my climbs in the Alps and during my geological studies in the field. I desire to thank Mr James Chumley for assisting me in preparing this paper for the press. Challenger Office, Edinburgh, January 1906. ( Issued separately April 16, 1906.) 116 Proceedings of Royal Society of Edinburgh. [sess. On the Distribution of the Proper Fractions. By Duncan M. Y. Sommerville, D.Sc. Communicated by Professor Chrystal. (MS. received 15th January 1906. Read March 19, 1906.) CONTENTS. Sec. Page Introduction . . .116 1. Normal Distribution . . 117 2. General Theorems on the Normal Distribution . 119 3. Method of writing down N.D 120 Tables. N.D. for */4>12 and */> 13 . . .121 4. Other “ even ” Distributions 122 5. Evenest Distributions by the Coalescing of Pairs of Classes . . . .124 6. Equations satisfied by the Weights of the Loaded Denominators . .124 Sec. Page N. D. is an Even Distribu- tion for Symmetrical Loading . . .125 7. Division into more than n Classes . . . .125 8. Division into fewer than n Classes . . . .125 Submultiple Division . 126- 9. Method of writing down any distribution . . 125 Table. Distribution of *'/ :J> 24 into 8 Classes . 127 10. General Equations for any Distribution . . .127 11, 12. Effect of omitting the Lower Denominators . 128 In statistical work which deals with integral variates, the data frequently appear in the form of ratios, or unreduced proper fractions, e.g. sex- and fecundity-ratios ; * and to facilitate com- parison these are arranged in classes, all the ratios falling within the same class being considered as equivalent. These classes, must, as far as possible, contain an equal number of the ratios. Further, in certain fields the different kinds of ratios do not all occur with the same frequency, that is, one denominator will occur more frequently than another, without any reference to the number of fractions having this denominator. In such cases the fractions must be distributed in such a way that, if to each fraction having a particular denominator p there be assigned a multiplier or weight fxp , in each class the sum of the numbers of * Cf. Karl Pearson, “ On the Inheritance of Fecundity in Thoroughbred Brood-mares,” Phil. Trans. A, vol. cxcii. pp. 294-296 (1898), where the evenness of various distributions of fecundity-ratios is discussed empirically. For this reference, and also for suggesting the problem, I am indebted to Mr David Heron, M. A., who is at present studying under Professor Pearson. 1905-6.] Distribution of the Proper Fractions. 117 the various fractions, each loaded with its respective weight, is as nearly as possible the same. The following paper is an attempt to find the evenest distribu- tion in any case that may arise. I. § 1. Consider all the proper fractions whose denominators do not exceed n. For shortness we shall denote any fraction with denominator p by */p, and by the assemblage of all the proper fractions whose denominators do not exceed n. . 0 1 0 12 The whole number of fractions i.e. -p -pi ->y> is equal to 2 + 3 + . . . + (n + 1) = ^n(n + 3). If the fractions n are distributed into n classes , 0/n to 1/n, 1/n to 2/n, . . ., (n— l)/n to n/n, and any fraction which falls between two classes is counted h in each of these two classes , each of the others being counted 1 in the class in which it occurs , then in each of the classes there will fall + 1) fractions, except in the extremes , ivhich contain n + Take any class, say that between — and ^ . If — is a n n p fraction belonging to this class, r n > — > p r+ 1 n ’ or < ns r + 1 ’ To each value of s there will correspond a certain number of values of p. Since p cannot be greater than n, it is evident that s cannot be greater than r+ 1, and if s — r+\, p must = w, giving the r _i_ i fraction , which is counted 1. n Let — denote the greatest integer less than — ; if — is an L r J r r integer, there will be a fraction — = — which is counted A : in & p n 2 this case we shall define f— j = — - A . 118 Proceedings of Royal Society of Edinburgh. [sess. OlfS 7ZS The number of integers lying between — and — - inclusive is then [fj - [jy . Giving s the series of values 1, 2, . . . , r, we find that the whole number S of fractions in this class is given by '] + [?]+-+[7]-{[m] + [rTl]+- + t?-.]} Suppose first that n is prime to r and to r + 1 ; then the remainders on dividing n , 2n , . . . , rn by r are 1, 2, . . . , r— 1,0 (though not necessarily in this order), and the sum of these is J r(r- 1); and as there is one integer in the series, we must sub- tract J, hence w ”1 T 2 n~\ + + Vm~\ _ n + 2n + + rn _ Jr(r - 1 ) _ 1 r J L r J L r J r r ' r r 2 i-r(r+i) Iii the same way, n being prime to r + 1, the remainders are 1, 2, . . . , r- 1, r, and there is no integer in the series, hence w 1 ^ d , ni 1 _n 2n rn |r(r + 1 ) r + 1 J Lr+lJ _r+lj r+ 1 r+ 1 r+ 1 r+ 1 n r+ 1 + 1 ) - i-r . Hence S - J = J n(r + 1 ) - \nr — £ n , or S = J(ft+l). Next, suppose that n and r contain a G.C.M. k , so that n = kn, r — kr\ and ri is prime to r \ the remainders, on dividing n, 2 w, . . . , rn, by r, i.e. k(n, '2n, . . . , rn) by kr\ are k(l} 2, 1, 0) ; &(1, 2, 1, 0) ; ... (k periods), 1905-6.] Distribution of the Proper Fractions. 119 the sum of which is A;2* J r\r'- 1), and there are here k integers, hence [±1+ • I — 1) 1^ L r J * L r J r * r kr 2 n r 1/ ,x 1,, n \ , lX 1 -r{r+\)-—hr = - . - r(r + 1) - — r as before : similarly if n is not prime to r+ 1. Hence in each class except the extremes there are \{n + 1) fractions, and in the first class there are the n fractions 0 0 ® and 1, n n 1 ’ 2 5 ' * which is counted J, i.e. altogether n + J. We shall refer to this distribution of the fractions into n classes as the normal distribution. If the series of improper fractions, positive and negative, be joined to the ends of the series of proper fractions, then the fractions -5-, . . . ; -i-, must each be counted only J, and the number of fractions in each class without exception is the same, i.e. 1). § 2. We shall now develop some theorems relating to the way in which the fractions */p are distributed throughout the classes. In any class there can occur , besides the limits , no two fractions For — , P ~s' = l. — =£ £ which is greater than — unless p = n and p p n If in any class */p and */n — p both occur , they must both be equal to one of the limits of the class. We have r .. s — -f* - n p 1 and l > _±_ > r±L n n — p n Therefore and — >r+ 1 ~ l' r n r+ 1 r ‘ n r + 1 120 Proceedings of Royal Society of Edinburgh. [sess. Suppose first that s + s >r + 1. Then n r+1 r+1 n which is impossible. Therefore s + s' r + 1, and similarly s + s' <^r. Hence s + s' must he equal either to r or to r + 1. If s + s' = r, n r nr r ’ p n n-p Similarly, if s + s' = r + 1, -IS y— - = — - — . p n n —p It follows from this that each class must contain either */P or */n — p, and if any class contains both , each of them occurs counted ^ ; for, leaving out the denominator n, there are n — 1 denominators, and only n - 1) fractions in the class. § 3. The classes in which any fraction * jp occurs in the normal distribution can be easily found. Divide n, 2 n, 3 n, ... by p ; let q2, q3, . . . be the quotients, and fv f2, /3, . . . the remainders. mi sn fs Then — = qs + — • p p j sn .. , , >?»+!. n p n Therefore the fraction — lies in the (qs+ l)th class. If fs = 0, — = q„ therefore = so that the fraction — lies A- in the p * pn p 2 gsth and \ in the (gs + l)th class. Also, if n, 2 n, ... he divided by n-p , and Qp Q2, . . . are the quotients and Ylt F2, . . . the remainders, then the Hence or 1905—6.] Distribution of the Proper Fractions. 121 § . — fraction will lie in the (Qs-t- l)th class, and the fraction */p will not lie in this class ; but if Fs = 0, lies h in the Qsth n -p and J in the (Qs+ l)th class, and — also lies \ in each of these classes, where s' = Q s- s. This gives a simple way of writing down the normal dis- tribution. It is convenient to write the classes horizontally, and the fractions *jp in columns under their respective denominators. The numerators s in any column must then occur consecutively from 1 up to p. We note also that the distribution is symmetrical about the middle horizontal line, so that, the upper half having been filled in, the lower half can be written down. In filling- in the right-hand half containing the larger denominators it is •easiest to determine the vacancies by dividing n, 2 n, ... by n-p. The other half can then be filled in in a complementary way ; or we may proceed vice versa, filling in the left-hand side first. The fractions which limit two classes may be distinguished by a bar. Thus the normal distributions for n= 12 and for n— 13 are represented thus : — I. 1 2 3 4 5 6 7 8 9 10 11 12 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 T 2~ 1 1 1 2 2 2 2 2T 3" I 1 2 2 2 3 3 3 3 4 1 2 2 3 3 4 4 4"_ I 2 3 3 4 4 5 5 6 1 2 3 4 4 5 5 6 TT Y 2 3 4 5 6 6 7 < 2 3 4 5 6 6 7 8 s V 3 4 5 6 7 8 9 9 TV 5 6 7 8 9 10 10 TT 1 2 3 4 5 6 7 8 9 10 11 1 1 1 2 I. Normal distribution for n even, =12. 1 22 Proceedings of Royal Society of Edinburgh . [: II. 1 2 3 4 5 6 7 8 9 10 11 12 13 0 0 0 0 0 0 0 0 0 0 0 0<>T 1 1 1 1 1 1 \ 1 1 2 2 2 2 *-g 1 2 2 3 3 3 1 2 3 3 4 4 \ 2 3 4 4 5 5 ‘V 12 3 4 5 6 \ 3 4 5 6 6 7 2 4 5 6 7 8 \ 3 5 6 7 8 9 4 5 7 8 9 10 1TTTT 6 7 8 9 10 11 alT^ 1 2 3 4 5 6 7 8 9 10 11 12 1213 II. Normal distribution for n odd, =13. We notice that when n is even the vertical middle column is JL l/y + 1 ) full : in fact, we always have the fraction f— or ' occurring \n \n in the class between — and , according as r is even or odd. n n And when n is odd the middle class contains, besides the limits, only the fractions numerically equal to J, for the number of these fractions is \(n - 1). § 4. We also observe that in each case the last column contains one (i.e. \ + J) fraction in each row except the first and last, which contain 1J, and the second last column contains one in each row. We therefore find the following additional “ even ” distributions : — (1) Into n- 1 classes. Consider only the fractions */^> (n - 1 ). These are distributed \n in each class except in the extremes, which contain n-\. The fractions */n then fall one in each class except the extremes, each of which contains 2 : viz., between —— and — we have n - 1 n - 1 — and — , between ^ and —^4 and so on. n n n - 1 n~\ n 1905-6.] Distribution of the Proper Fractions. 123 Hence the fractions */^>w are distributed J(?i + 2) in each of 3 the classes except the extremes, which contain n+ — . If the fractions are counted the extremes contain only ^(n + 3). (2) Into n + 1 classes. Introducing the fractions */n+ 1 , the fractions */lf>(n + 1) will be distributed n + 2) in each class except the extremes, which 3 contain n + . The fractions */n + 1 are distributed one in each class except the extremes, which contain ~ ; viz., between — and -J— we have --- and ( — — ^ and so on. Hence n+ 1 n+ 1 n+ 1 \n+V removing these we are left with \n in each class except the extremes, which contain n. If the fractions 0 1 1 ’ ’ ' * ’ T ’ are counted J, the extremes contain, like the other classes, \n. (3) Into n + 2 classes. Introducing the fractions */n + 1 and */n + 2, all the fractions *j^>(n+'2) are distributed h,(n + 3) in each class except the extremes, which contain n + -~- . Now the fractions */n + 2 are distributed one in each class except the extremes, which contain 3 . -p , and the fractions */n + 1 fall one in each class. Hence removing these we are left with \(n- 1) in each class except the extremes, which contain n. If the fractions , . . . are counted J, the extremes contain \n. Table II., if we strike out the last and the last two columns respectively, gives us the distributions of */^>12 and */^>ll into 13 classes. If we distribute the fractions into more than n + 2, or into fewer than n - 1 classes, we shall find that the middle class or classes will not have the same number of fractions as the others. 124 Proceedings of Royal Society of Edinburgh. [sess. § 5. In the normal distribution and in the three just considered, when we reckon the fractions at the extremes as 1, the number in the extreme classes is practically double that in the others ; but when we count these fractions as \ the number in all the classes is the same both for the normal distribution and for the distribution into n + 1 classes, and for the other two distributions the number in the extremes is J more than in the other classes. On the former convention, we may, if n is even, in the normal distribution make pairs of classes coalesce, starting with the second ; then there will he n + 1 in each of the \(n - 2) compound classes and n 4- J in the extremes. If n is odd, we may divide into n + 1 classes, and then make pairs of classes coalesce; then in each of the \(n — 1) compound classes and in each of the extremes there will be n fractions. These are therefore the evenest distributions on the assumption that all the fractions have equal weights. § 6. Let us now give weights to the different denominators, or consider the denominators not as occurring with equal frequency but with relative frequencies denoted by gp. We have now to €p = 1 or J, we have to find the distribution which makes this sum approximately the same in each class. It will be easier to proceed in the reverse way and find for any distribution what relations must hold between the coefficients gp in order that the sum in each class should he the same. We shall assume in what follows that for the fractions at the extremes e = J. (Without this assumption the results require considerable modification for the extremes.) Let us consider first the normal distribution. Taking any two classes, we have the equation II. reckon each of the fractions *jy not as 1, but as gp, and, if it is the limit of a class, as \gp. Then taking the sum ^ep/V w^ere i i [fj.n will not enter into this equation, since en = en = 1.] 1905-6.] Distribution of the Proper Fractions. 125 This may be written i(n— 1) or or J(w-2) 1 1 [If n is even pin will not enter, for ein = e'in= J.] {(ep ^ p)pp "f" {en—p f n—p)Pn—p} ^ • Now (§ 2} and the equation becomes \(n— 1) or l(n— 2) ^j(€P e p) (ftp Pn—p) 1 Now, by taking the classes in pairs, since the distribution is- symmetrical about the middle class or classes, we shall obtain J(' n — 1) or \{n — 2) distinct equations of this form, according as n is odd or even. But this is the number of the quantities^ /Xp - fxn_p , and since the equations are homogeneous and linear in these quantities they can only be satisfied by the quantities- f j.p - [xn_p all vanishing, i.e. for all values of p from 1 to n — 1 .* Hence the normal distribution will be an even distribution also in the case where the frequency curve for the denominators is symmetrical. § 7. Suppose now that we divide the fractions */^\>n into more classes than n, say into n + m, and consider the possibility of the sums of all the classes being equal ; it is the same thing as if we divide the fractions */ Ij> (n + m) normally and consider pp = 0 if p>n , and therefore also if pw to be divided into fewer classes than n, say into m, and let n = xm + y, where y s j> > m ^ ^ m then between the same limits there will be t + 1 fractions */p + tm For let p + tm be such a fraction, then (p + tm)r m {p + tm)(r+ 1) m or pr m >*' -tr efc±D+<. s= tr + s + t\ (^ = 0, 1, 2, . . t). But ii p^>m and no fraction *jp lies in this class, then s' = tr + ^ J + 1\ (f = 1, 2, . . .,t). Hence in the (t + l)th set we have each class exactly the same as the corresponding class in the first set, except for the addition of t fractions of each kind. It follows that if y = 0 all the classes will contain the same number of fractions, i.e. the fractions */%>n can be divided evenly into any submultiple of n classes. It follows also that they can be divided evenly into any submultiple of n 4- 1 classes, or any submultiple of n - 1 or n + 2 classes, though in the two latter cases the number of fractions in the extremes will be \ more than in the other classes. § 9. The distribution into any number of classes m 24 into 8 classes. 1905-6.] Distribution of the Proper Fractions . 127 III. Distribution of */^> 24 into 8 Classes. 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 17 18 19 20 21 22 23 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 2 3 1 1 1 1 2 2 2 2 3 3 3 3 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 1 2 3 4 5 6 1 I 2 2 2 3 3 4 4 4 5 5 6 6 6 1 3 3 4 4 4 5 5 5 5 5 6 6 6 7 7 7 6 6 7 7 7 8 8 8 3 6 9 2 3 3 5 6 6 8 9 9 4 4 5 5 6 6 7 7 7 7 8 8 9 9 10 10 | 8 8 9 9 10 10 11 11 i I 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 To II 12 9 10 10 11 11 12 12 13 5 6 6 # 7 8 8 9 10 11 11 12 12 13 13 14 3 4 5 8 9 10 13 14 15 5 10 15 11 12 12 13 14 14 15 16 6 7 7 8 9 9 10 11 12 13 13 14 15 15 16 17 2 3 4 5 6 8 9 10 11 12 14 15 16 17 18 3 6 9 12 15 18 13 14 15 16 16 17 18 19 7 8 9 10 10 11 12 13 14 15 16 17 17 18 19 20 4 5 6 7 11 12 13 14 18 19 20 21 7 14 2l 15 16 17 18 19 20 21 22 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ” 1 § 10. For a distribution of the fractions into m classes the equations become therefore AS F [Ap+m F • • • F ftp+xm = Pm-p F ASm-p F . . . F ? 128 Proceedings of Royal Society of Edinburgh. [sess. for p= 1, 2, . . \(m- 1) or \(m- 2), according as m is odd or even. The last term on the right will of course vanish if (x+\ )m-p>n, i.e. >mx + y, or if py. Neither will occur if y = 0; both will always occur if y = m - 1 . These equations are satisfied if yp = ym-P > . • . , pp+m = g2m-P , . . • If y = 0 or m - 1 we have a complete set of pairs, and none of the fs vanish, but in other cases certain of the /z’s will vanish. Thus for y — 1 or m — 2, pn = 0. Hence the distribution into m classes, m a submultiple of n or n+\, will still be even if the frequency curve for the denominators is periodically symmetrical with period m. §11. Let us consider now the effect of omitting certain of the lower denominators. We may for this purpose consider the assemblages */ */ ( xm - 1 ), */^> (xm + 1 ), */ Ij> (xm - 2), each distributed into m classes. The omission of a complete set or any number of complete sets will not affect the distribution. The omission of one less than a complete set or any number of complete sets will not affect the distribution. The omission of one more or two less than a complete set or any number of complete sets removes J more from the extremes than from the other classes. The results are indicated by the following table : These results on submultiple division require considerable modification as regards the extreme classes when the extreme fractions are reckoned 1 and not J. The number in the extremes is then not even approximately equal to or a multiple of the number in the other classes. § 12. One result which we have already obtained may be */ xm and */ ^> (xm + 1 ) and */^>(xm- 1) 2) Extremes \ less. Same in each. Same in each. Extremes J more. 1905-6.] Distribution of the Proper Fractions. 129 restated in this connection. The evenest distribution for the assemblage */^j>w where the lower ratios */^>m are to be omitted is found by dividing into n + m+1 classes (§ 7). When the extreme fractions are counted \ there are then n-m+1 ) in each class. If n + m + 1 is even, pairs of classes may now be made to coalesce, and now reckoning the extreme fractions as 1 there will be n — m + 1 in each class. This result is still true if the frequency curve is symmetrical from m + 1 to n. {Issued separately April 16, 1906.) PROC. ROY. SOC. EDIN. -YOL. XXYI, 9 130 Proceedings of Poyal Society of Edinburgh. [SESS. On Vibrating Systems which are not subject to the Boltzmann-Maxwell Law. By Dr W. Peddie. (MS. received February 23, 1906.) 1. It is well known that the Boltzmann-Maxwell Law, which asserts average equi-partition of kinetic energy amongst the various motional freedoms, finite or infinite in number, of suitably con- ditioned systems, meets with apparently insuperable difficulties in its application to actual gases. Even in the case of gases showing complicated spectra, and therefore possessing numerous vibrational freedoms, the actual values of the ratio of the specific heats never deviate much from values which should, according to the law, he limited to very simple systems. Possible modes of evading this result, such as that supplied by J. J. Thomson’s suggestion (Arch. Neerlandaises, 1900, Ser. II. t. v.) that individual molecules may not he concerned in radiation, are too problematical to give much relief. Lord Kelvin (Baltimore Lectures , Appendix B) says that Clausius’ theorem regarding the specific heats, “ taken in connection with Stokes’ and Kirchhoff’s dynamics of spectrum analysis, throws a new light upon what we are now calling a ‘ practically monatomic gas.’ It shows that, unless we admit that the atoms can he set into rotation or vibration by mutual collisions (a most unacceptable hypothesis), each atom must have satellites connected with it (or ether condensed into it or around it) and kept' (by the collisions) in motion relatively to it with total energy exceedingly small in comparison with the translational energy of the whole system of atom and satellites.” This asserts actual violation of the Boltzmann-Maxwell condition. Jeans’ recent suggestion (Nature, vol. lxxi., 1905) that the complete system must include the ether, and that radiation has hitherto prevented attainment, as between matter and ether, of that condition of statistical equilibrium, which the law requires, merely indicates a hypothetical mechanism necessitating Lord Kelvin’s conclusion, although it removes the law from direct attack on this side. 1905-6.] Dr W. Peddie on Vibrating Systems. 131 2. Rayleigh (> Scientific Papers, vol. iv., no. 253), in defending Maxwell’s argument, desires “some escape from the destructive simplicity of the general conclusion.” Kelvin {Balt. Led., App. B) speaks of the difficulty as one of the two “nineteenth century clouds over the dynamical theory of heat and light,” and concludes that the doctrine should be denied. He gives a number of tests tending to show considerable deviations from equi-partition of energy between translational freedoms and also between trans- lational and rotational freedoms. He also points out an, at least, ideal case in which this equi-partition is impossible. The object of the present communication is to determine cases in which equi-partition of vibrational energy cannot take place. 3. It is convenient to suppose that the motions are limited to one-dimensional space. The simplest case is that of two freedoms expressed, say, by the conditions $2 = ^2^1 — ^*2^2 * These give (r + r')^ = rk sin {nt + a) + rk ' sin {n't + a) , {r + /)£ 2 = A sin {nt + a) - A' sin ( n’t + a') , where n2 — aq — rb2 , n2 = a1 + rb2, the quantities r and - r being re- spectively the positive and negative roots of b2\2 + (a2 - a-l)\ — bl = 0. Hence we obtain 2 (r + /)2{m1i12 - m2£22} == n2(m1r'2 - m2)A2 + n2{m1r2 - w2)A1'2 where the bracket indicates an average taken over a time which is long relative to the longer of the two periods, and mx, m2, are the masses. If it is possible to determine conditions under which tbe quantity on the right-hand side of the equation is either always positive or always negative, the Boltzmann-Maxwell distribution of energy is impossible. Now the third law of motion necessitates the condition /;1m1 = 62m2, so that rV2 = m22/m12. Hence it is impossible to have both brackets on the right-hand side always of one sign. Thus, so far as this investigation goes, we can make no assertion as to the observance or non-observance of the Boltzmann-Maxwell Law, except in the case of numerical equality of the two roots, in which case it certainly is observed. 132 Proceedings of Boy al Society of Edinburgh. 4. It is worthy of note that the impossibility of obtaining a conclusion in the case of two freedoms depends on the observance of the third law of motion. If the third law be not observed in sub-atomic dynamics, there could be bi-periodic atoms exempt from the Boltzmann-Maxwell Law. There is no a priori reason why the third law should hold. Steady vibrations could take place until a collision, sufficiently violent to alter the equations of motion, occurred. At that stage, non-observance of the law could give rise to effects analogous to some of those made evident in the phenomena of radioactivity. 5. The next simplest case is that of three freedoms as typified by the equations £l = ^1^1 "k ^1^2 "h ^1^3 > ^2 = ^2^1 "k a2^2 4" ^2^3 > ^3 = C3^1 d* d&$2 + a3^3 • The third law of motion gives the conditions b1m1 — b'2m2i c1m1 = c3m3, d2m2 = d3m3; whence b\d2c3 = cf>2d3 (1) This is Tait’s condition ( Scientific Papers , vol. ii., art. cxx.), for the reality of the roots of the cubic u - x b , cn dc = 0 (2) u3 3 3 regarded as determining the non-rotated lines in homogeneous strain possible in matter. Indeed, the form of the above equations of motion exhibits the analogy between homogeneous three- dimensional strain and the vibrations of a three-period system. The problem of determining non-rotated lines in the one is the problem of determining fundamental periods in the other. Tait’s method of reducing the general differentially irrotational strain to a pure strain in the one is the method of reducing the general three-period system to an equal-mass system in the other. This will he farther discussed in § 9. Tait’s limitation of the strain problem to strain possible in matter corresponds to the subjection of the vibrating system to the third law of motion. We can still have the condition b1d2c3 = cf2d3 if we take, for example, 51m1 = &2m2, c3m3 = kc1mv 1905-6.] Dr W. Peddle on Vibrating Systems. 133 d3m3 = kd2m2, k 4= 1 . This implies denial of the third law of motion with reference to m3. More generally, the condition suits with blm1 = Jc1b2m2, c3m3 — k2c1in1 , d2m0 = k3d3m3, if k-Jc2k 3 = 1 . This implies denial of the law with reference to two masses at least, and asserts reality of the squares of the periods. Any other condition than klk2k3 = 1 makes two squares of periods imaginary. 6. The solution of the equations of motion, given above, is expressed by Aj sin (n^t + cq) A: /q A3 sin (?q£ + II A2 sin (?q£ + a2) 1 A2 , A = 1- ^ /^2 j A3 sin ( n3t + a3) 1 A3 1 A3 /q where - n x2 = aY + \^b2 + /^Cg, - — a1 + A 2b2 + g2c3, - n32 = tq + \3b2 + /a3c3, and A1} /q, etc., satisfy in pairs the equations Hai + b2X + C3J“) = bi + a2X + > I /3) g{a\ H" b0X + c3V) = iq + d2 A -f- a3g , f If we write the values of £15 g2, $3 in the form = a\ Al sin {n-f + cq) + a 2 A2 sin (w2£ + a2) + a'3A3 sin (n3t + a3) , $2 = ^'jAj sin ( nYt + aq) + b'2 A2 sin (w2^ + a2) + &3A3 sin ( n3t + a3) , £3 = c'1A1 sin (?q£ + oq) + c2A2 sin ( n2t + a2) + c3 A3 sin (n3t + a3) , we get, on integration over a period which is long relatively to the longest of the three fundamental periods, 2{£i2} = "WV + n2a'2 A2 + n2a2A2 , 2{^2} = w126/12A12 + n2b' 2 A2 + n2b' 2 A2 , 2{42} = Vc'i2Ai2 +w2V22A22 + n2d2A2 , where the brackets indicate time-averages, as before. Hence we have 2{mi£i2 ~ m242} = (%ft/i2 “ m2b\2)n^ Ax2 + (wqa'g2 - m2br 22)n22A22 + (mitt 32 — m2& 32)n32A32 , . . . . (4) with two other similar expressions. 7. From equations (3) we find that A is given by the cubic 134 Proceedings of Royal Society of Edinburgh. [sess. *^[^2 (^2^3 C'$aY) 4 ^3^2) ^3] + A2[&2(a^3 - JjCg) + (oq - a2)(b2d3 - c3a2 ) - d3(asb2 - c3d2) + c3{afaY - a2) + d2d3 - cxc 3)] + A[(flq — a^){afL3 - bYc^) — bfb2d?i — c3a2) + cfc\dz — af-^) — d^as(ai ~ a2) + d2dz ~ c2cs)] -W«A-W+#A-flsy]=0 .... (5) The roots of this equation being regarded as known, the first of equations (3) gives the corresponding values of g. The condition (1), for reality of the roots of the cubic (2), gives the single definite relation which subsists amongst the constants, and further limiting relations are imposed through the three equations of type (4). Tor, if the three quantities in brackets on the right-hand side of one of equations (4) be of like sign, there can never be equi-partitioning of energy between the masses with which that equation deals. The Boltzmann-Maxwell Law would, if it were applicable to vibrational freedoms obeying the generalised Hooke’s Law expressed by the equations of motion given in § 5, constitute a proof of the impossibility of satisfying all the above relations simultaneously. 8. To obtain a definite test, we may choose Xx= — 1, A2 = 0, *3 = 1; which give g1 = [{b2 - \) - (a1 - a2)]/ (c3 + dz), g2 = - bjd3, g3 = [(62 - + (a1 - a2)]/ (d3 - c3), and also a i~ ~ Uo 1 ® 2 = Mi 4 ^3 j a 3 = — g2 , b 1 = ^2 — /^3 5 ^ 2 = /^3 — Al ’ ^ 3 ~ Al — » Cl= 1 , C 2~ ~ > C3=1- Hence the inequalities are m1 > (H ~A3\2 > / A*”3 AiV > /Ai ~IH\ m2 = h < \ H ) > < VA3 + Ai/ » < l H ) m3 m2 ! I aJ a. CO Ito > < (h - hY AV ^3 ~ Aiy > < (/*i ~ A2)2j Vh : 9 > /Ai + /aA2 > 2 m1 C3 < /V » < l 2 j, < /V- If we take g1 + g3 = 2g2 , we get necessarily g3 — g1 = — 2(g2 — g3) = 2(g2 - gf, and the inequalities reduce to hj. (H ~ /bA2 b ] \ J 3 +6h -/•*)*. a 3 1905-6.] Dr W. Peddie on Vibrating Systems. 135 The values of the As being distinct, the values of the /x s are in general distinct. Also the periods, as given by — n2 — aY + b2\ + c3fx, are in general distinct, and can be made real by suitable choice of ar Reserving a1 for this purpose, and reserving b± to secure the first inequality, the remaining inequalities can be secured by reservation of d2 and cv which do not enter into the expressions for the g s. Similarly, distinctness of periods can be provided for by reservation of b2 and c3. The values of d3 and a2 are available to give each of /x2 and fx1 an infinity of values. But the reservations leave at disposal an infinity of values of each of av bv cv b2, d2, and c3. On the other hand, the assumed values of the As impose three relations amongst the constants. There is therefore a five-fold infinity of solutions possible, in eacli of which equi-partitioning of energy between any pair of masses is impossible. The values of the As being at disposal, there is really an eight-fold infinity possible in the general case of three masses. This may be regarded as the tri-dimensional case of a single mass under the action of forces which are non-isotropic with reference to the co-ordinates. If the inequalities become equalities, choice of one A fixes in general the remaining As and the /ms and the values of bv d2 , cv The fixing of the /xs gives three relations amongst constants, and the cubic for A gives three more. Thus, in general, there is only a single infinity of cases in which there is complete equi-partition of energy amongst three co-ordinates. 9. The extension to n dimensions of the usual process for finding the necessary relations which subsist amongst the nine constants of a homogeneous tri-dimensional strain, in order that the strain shall be pure, leads to the result that the determinant of the nth. order, whose roots determine the directions of non- rotated lines, must be axisymmetrical. The roots are, in this case, necessarily real. If we now apply the process of “ flyping ” to the medium thus strained, we get a homogeneous, pure, w-dimensional strain, the roots of whose determinant for non-rotated lines are proportional to the squares of the fundamental frequencies of vibration of a system of n equal masses, which are acted upon by linear systems 136 Proceedings of Royal Society of Edinburgh. [sess. of forces whose moduli are the n 2 constants which specify the original strain. Therefore that vibrating system has n real positive squares of periods. To pass to the case of systems of unequal masses, we have to extend to ^-dimensional space Tait’s process for the determination of non-rotated lines. The equality of action and reaction being postulated, any two constants, each of which is situated at the optical image of the position of the other relatively to the right- handedly downward diagonal of the determinant, bear to each other the inverse ratio of the corresponding masses. It follows at once that Tait’s condition (1), § 5, applies to every cubic minor situated on the right-handedly downward diagonal ; and that, in every square minor, and its image minor in the diagonal, the product of the right-hand diagonal terms of the one into the left-hand diagonal terms of the other is equal to the product of the left-hand diagonal terms of the one into the right-hand diagonal terms of the other. Thus, in the scheme below, a2ci^3 = a3c2^i> ^4C2^3 = ^3C4^2> = d^e^f^g^. . a2 • C1 C2 . d9 d, d* J 4 f 5 y 4 We may dispense with the special statement regarding cubic minors. If, for brevity, we use the term “image minors” with reference to any square minor and the minor situated at the image of its position with reference to the right-handedly downward diagonal, and if we use the term “cross-product” with reference to the product of the right-hand diagonal constituents of one minor into the left-hand diagonal constituents of the other, we can make the general statement that — The roots of an n- ic are real when the two cross-products of each pair of image square minors are equal. In the case of a diagonal cubic minor, one constant, e.g. c3 in the above scheme, is common to each cross-product, so that we 137 1905-6.] Dr W. Peddie an Vibrating Systems. get Tait’s condition. A diagonal square minor, e.g. al,a2,b1,b2, is its own image, and the cross-products are identical. 10. The ..(n - l)(n — 2)/2 square minors which are not self images give (n - l)(n - 2)/2 relations amongst the n 2 constants and leave a (n2 + 3n-2)/2 fold infinity of examples subject to restriction by the inequalities and the conditions that the periods shall be real. The n - 1 square minors which are their own images give, by the terms not lying on the main diagonal, the n — 1 ratios of the masses in each example. If we postulate observance of the Boltzmann-Maxwell condition, and so change the inequalities into equalities, the number of possible infinities of examples is much lessened. 11. In the preceding discussion, no conditions have been postulated for the purpose of ensuring that the centre of inertia of the n masses shall coincide with the origin. We shall see that, except when the periods are all coincident, the centre of inertia cannot lie at the origin. Hence we must presume the existence, at the origin, of a mass which is very large in comparison with the sum of the n masses. In any question regarding the total partitioning of energy in a system so constituted, the motion of this central mass must be considered. But the question of the partitioning of energy amongst the n satellites is not affected by the presence of the central mass. To see that we cannot dispense with this large mass at the origin if all the periods are not to be identical, we have to consider the various equations of the type £*A = [l,p]Ax sin {nYt + cq) + . . . + [n,p] An sin (nnt + an), where A = 1 X and [r, s] is the minor, of order n- 1, of the rth term in the sth column of A. Taking now the centre of inertia condition n 2 mp £p = 0, we get the necessary equations 138 Proceedings of Royal Society of Edinburgh. ?n1[],l] + m2[l,2]+ +mn[l,w] = 0 [sess. m-fnX\ + m2[w, 2] 4 4 mfn,ri] = 0. Uow, since the first column in A is composed of units, we have r=n y [r,s] = 0 for all values of s greater than unity. Therefore r= 1 ^£[r, l]-0; that isA =0. r= 1 But A = 0 is the condition that yd1 4 oq 4 d* 4 .... + = 0 , 7i2 4 Gq 4 d* C3/X2 + . . . . 4 tnT2 ~ 0 ) + aq 4 b2 \n + c%gn 4 . . . . + tnrn = 0 ; which means that nf, . . . , nn2, must have the common value n 2. 12. As a special example, leading to easily calculable numerical values which might serve for the specification of mechanical models, we may consider the scheme — 1 X fX T 1 2 1 2 1 1 0 3 Using these values in equations (3), together with the conditions for observance of the third law of motion, we get seven of the nine constants given in terms of the remaining two and the mass- ratios. The results are— 6t-i = (Xq 4 c,m, - m. ^3 j a2 ~ az d- m9 - uio , — -dQ m, mQ b\ — ~ 3^3 j ^2 ~ ~ 3~dz ) c'\= ~ 3~^3 > m, rrio 7 Co— ~ 3 — do , d., = —do. Since A is zero, the As in § 6 must be zero ; so that, in equations 1905-6.] Dr W. Peddie on Vibrating Systems. 139 (4), we must suppose that the divisor A is included in each A. The inequalities then reduce ’ to and must be negative. A four-fold infinity of examples is possible The Boltzmann-Maxwell Law holds in the three-fold infinity of cases obtained by changing the inequalities to equalities. 13. Although the case of a self-contained system of n masses vibrating in one-dimensional space ; or, say, of a system of k masses, where n— 3k, vibrating in three-dimensional space ; and not subject to the law of equi-partition, is of great interest, the other case in which a preponderating mass has to be placed effectively at the origin is of even greater interest as embodying Kelvin’s view, § 1, that “each atom must have satellites connected with it (or ether condensed into it or around it) and kept (by the collisions) in motion relatively to it with total energy exceedingly small in comparison with the translational energy of the whole system of atom and satellites.” When the mass of each satellite is. so small that the forces which act on it produce accelerations which are large in comparison with the accelerations to which the central mass is subject, communication of energy to the satellites may have little direct connection writh communication of energy to the central mass. And, in any case, there is no necessary observance of equi-partition of energy amongst the satellites, since the effects of collisions appear only in the squared coefficients such as Ax2, etc., in equation (4). The above results apply at once to the case of a luminous gas, provided that we postulate (1) that the time of description of the average free path is long relatively to the longest constituent period and to the time of impact ; (2) that the temperature is such that only a small proportion of collisions give rise to disintegration of the molecules or to change of their essential configurations ; (3) that the fractional loss of energy by radiation during free intervals is small. 14. The object of the investigation, as stated in § 2, is to determine systems in which equi-partition of energy, of vibrational type, cannot take place. And, as we have just seen, the systems "T > 1 _3 > I . m2 < 9 ’ m2 < J and the common value of - n x2, - n 22, - nB2, is 140 Proceedings of Royal Society of Edinburgh. considered present analogies to the molecular systems of luminous gases. We at least see, therefore, how it may he possible for gaseous systems to he free from subjection to the law of equi- partition of energy within their vibratory freedoms. But it seems also that the possible conclusions reach further. Rayleigh [Scientific Papers, vol. iv., no. 253), in verifying Maxwell’s conclusions, points out the severely restrictive character of Maxwell’s assumption. It implies that the system, between two successive attainments of any given phase, must have passed through all phases consistent with the equation of energy. Maxwell’s own words are: “The material points may act on each other at all distances, and according to any law which is consistent with the conservation of energy, and they may also be acted on by any forces external to the system, provided these also are consistent with that law. The only assumption which is necessary for the direct proof is that the system, if left to itself in its actual state of motion, will, sooner or later, pass through every phase which is consistent with the equation of energy. “Now it is manifest that there are cases in which this does not take place. The motion of a system not acted on by external force satisfies six equations besides the equation of energy, so that the system cannot pass through those phases, which, though they satisfy the equation of energy, do not also satisfy these six equations. “ Again, there may he particular laws of force, as, for instance, that according to which the stress between two particles is proportional to the distance between them, for which the motion repeats itself after a finite time. In such cases a particular value of one variable corresponds to a particular value of each of the other variables, so that phases formed by sets of values of the variables which do not correspond cannot occur, though they may satisfy the seven general equations. “But, if we suppose that the material particles, or some of them, occasionally encounter a fixed obstacle such as the sides of a vessel containing the particles, then, except for special forms of the sur- face of this obstacle, each encounter will introduce a disturbance into the motion of the system, so that it will pass from one un- disturbed path into another. The two paths must both satisfy the equation of energy, and they must intersect each other in the 1905-6.] Dr W. Peddie on Vibrating Systems . 141 phase for which the conditions of encounter with the fixed obstacle are satisfied, but they are not subject to the equations of momentum. It is difficult in a case of such extreme complexity to arrive at a thoroughly satisfactory conclusion, but we may with considerable confidence assert that, except for particular forms of the surface of the fixed obstacle, the system will sooner or later, after a sufficient number of encounters, pass through every phase consistent with the equation of energy.” These remarks of Maxwell seem to apply as conclusively to material systems constituted in the manner postulated in this paper. At least this is so if we presume that, in a collision, individual masses of the system may suffer independent impacts. If that view be correct, Maxwell’s assumption is satisfied while his conclusion is departed from. It seems that, before we can arrive at his conclusion, we must at least assume, when periodic motions are concerned, that no phase-cycle is to be continuously completed by any individual member of the system. Thus, in the liquid and isotropic solid states, equi-partition of energy amongst all freedoms may be possible, while in sparse gases it is impossible. But the statement, that “ any law of force which is consistent with the equation of energy ” can be postulated, is in general impossible because of the restricting assumption. In general the expression of the law involves relations amongst the momenta and the co-ordinates, which relations prevent the simplification that arises in the Boltzmann -Maxwell treatment, or in Jeans’s treat- ment, and leads to the law of equi-partition. The usual methods will then lead to results such as those obtained above. As a simple example, which includes Maxwell’s example as a special case, we find, in §5, the relations (&2£x + a2£2 + — (ai£i + ^1^2 + e^c* These relations, like the energy con- dition, subsist despite encounters and collisions. Similar conditions will hold with laws other than the generalised Hooke’s law. In this connection an interesting paper by Bryan, published in the volume of the Arch. Neerlandaises already referred to, § 1, would have been more fully quoted had I observed it before the preceding sections were written. In 1900, Bryan, by a different process, had indicated the preponderance of conditions for unequal partition {Issued separately May 24, 1906.) 142 Proceedings of Boyal Society of Edinburgh. [sess. Some Experimental Results in connection with the Hydrodynamical Theory of Seiches. By Peter White, M.A., and William Watson. (MS. received March 15, 1906.) In a paper read before the Royal Society of Edinburgh, in June 1905, on the Hydrodynamical Theory of Seiches, Professor Chrystal published a number of formulae from which could be calculated the periods and positions of the nodes of seiches in lakes of various shapes. The solutions of the most general problems, involving variations of the three dimensions, are there made to rest ultimately upon certain typical cases in which the element of breadth remains constant while the depth is some defined function of the length. In the paper referred to, formulae are given immediately applic- able to lakes of uniform breadth, whose longitudinal sections include concave and convex parabolae, quartic curves, and recti- linear shapes. The present paper owes its origin to a suggestion by Professor Chrystal, to find out how closely the values given by these formulae coincided with the results of actual experiments made with models to represent the typical cases. The experiments were carried out with water in a rectangular trough whose dimensions were — length, 15 ‘2 cm. ; breadth, 10 "5 cm.; depth, 12*5 cm. The curves characterising the special forms of the body of water to be experimented with were calculated and transferred to blocks of wood, which were cut and fitted into the trough, being weighted with lead to keep them in position. The experiments involved the determination of two classes of results — the periods of the different seiches excited and the positions of the nodes of these seiches. The nature of the problem will best be illustrated by reference to the particular case in which the water is in the form of a concave-parabolic curve. 1905-6.] The Hydroclynamical Theory of Seiches. 143 If the end of the trough is carefully raised and then lowered to its original position, the water moves in such a way as if the surface plane slid through a tranverse line across the centre. A seiche, or oscillatory motion of the entire body of water, is thus generated. The water rises and falls everywhere except at the central transverse line, which, from analogy with vibrating strings and air-columns in organ-pipes, is called a nodal-line. As in the case of vibrating strings, a node is a position of minimum transla- tional motion but of maximum rotational motion, so in a seiche there is minimum vertical motion hut maximum horizontal motion at a node. The liquid particles move backwards and forwards through a comparatively wide range across the central transverse line, in the direction of the seiche motion. In this present example of a uninodal seiche there is a nodal /7a. /• line in the middle of the vessel. The ends are positions of maximum vertical displacement, i.e. loops or ventral segments, and it is evident that exactly as in the case of open organ-pipes, the ends of a lake will always he ventral segments. If we compare the oscillating liquid with a stretched string, however, we find the circumstances reversed, since in the latter case the ends are always nodes ; and when the fundamental vibration is in existence, the centre is a loop. Taking advantage of the fact that at a node in a seiche, horizontal motion is a maximum, horizontal motion may he excited at other parts of the water-surface with the object of producing a seiche of a different period. Evidently, in the case referred to above, any such seiche will always have its nodes symmetrically situated about the central transverse axis. The problem, then, was to determine for any particular curve 144 Proceedings of Boyal Society of Edinburgh. [sess. the various possible periods of oscillation of the liquid and the positions of the corresponding nodal lines with reference to any fixed line ; also to determine for each curve the effect of altering the maximum depth of the liquid. The first experimental method to be devised was a means to excite the various seiches. Obviously oscillatory motion necessitates regularly-timed impulses. A method of imparting such impulses to the liquid by means of a weighted spiral spring f/G:2T. AT: AT: Wafer Line. r. n G Wire-gauze. FFFF Wood frame of pendulum KK Knife-edge supports. supporting wire-gauze. W Weight of pendulum. T Trough. SS Supports of Pendulum. is described on p. 612 of the paper by Professor Chrystal referred to above. This is not in all cases practicable. Another method that suggested itself was to make use of a pendulum with a sufficiently heavy bob to give it motive power. To this was attached a frame-work of wire gauze, which, besides being rigid, had the additional advantage of allowing the water to pass freely through it (figs. 2, 3). 1905—6.] The Hydrodynamicctl Theory of Seiches. 145 If the apparatus be so arranged that the pendulum is of such a length as to swing in the period of any particular seiche while the wire-gauze is situated at a nodal-line corresponding to the seiche in question, the latter is easily set up by causing the pendulum to oscillate. But while this is a most satisfactory method of supplying motive power to the oscillating system, when the period and the positions of the nodes are known, it is not without drawbacks while these latter are being investigated. For, what may not be altogether without interest, it was found that after the combined systems had been moving in apparent harmony, the water rising and falling synchronously with the pendulum, it was impossible to say whether the period had been the natural period of the seiche or a forced period differing somewhat from it. Occasionally when the pendulum was released from communication wTith the water, each system would gradually assume a new phase of oscillation, differing between themselves and from the original mutual period. PROC. ROY. SOC. EDIN. — VOL. XXVI. 10 146 Proceedings of Royal Society of Edinburgh. [sess. While the seiche may best be generated by motion transferred to the liquid in the neighbourhood of a node, the natural period can only be satisfactorily determined when the liquid is left to oscillate freely by itself. The principle of resonance enables us to overcome the difficulty involved in the ignorance of the true periods. If any motion except that approximating closely to a natural period of oscillation be imposed upon the liquid, it will not respond to any extent, and no seiche results. By communicating the motion by hand and regulating the impulses by the responsiveness of the water, the latter could be made to oscillate in some cases in as many as five or seven different periods. After the various seiches had been produced in this manner by repeated trials, varying the timing of the impulses and the position of application of these, the periods of the free oscillations of the water were observed and noted. The following tables show the relation between the calculated and the experimental results for the particular curves mentioned : — Tabulated Results. [A.ih — The experimental error-limit in the case of the periods tabulated, involves a correction of ± -02 seconds ; and in the case of the distance of the nodes from the centre, a correction of ± 2 cm.] N.B. — The references are to the Trans. Roy. Soc. Edin ., vol. xli., part iii. (No. 25). 1. Seiches in a concave symmetric complete parabolic lake [§ 27], N.B. — All linear measurements are in centimetres; the periods are given in seconds. Ar a 0_ cl A A [a =70 cm.; h =10*9 cm.] 1905-6.] The Hydroclynamical Theory of Seiches. 147 h 1 t2 t3 t4 I Position of Binode — . a obs. calc. obs. calc. obs. calc. obs. calc. obs. calc. obs. calc. 10-9 2-98 2’99 1-72 1-73 1-24 1*22 •98 •95 *80 *77 •567 •577 9-4 2-98 2-99 1-72 1-73 ... •566 •577 6‘2 2*98 2-99 1-72 173 •573 •577 2. Seiches in a concave semiparabolic lake [§ 34]. o a a \ct = 70 cm.; 7^ = 10*9 cm.] h T i T-2 Position of Uninode - • a X. Positions of Binodes — • a obs. calc. obs. calc. obs. calc. obs. calc. obs. calc. 10-9 1*73 1-73 •95 •95 *593 •577 •329 •340 •850 •861 9-4 1*72 173 *95 •95 •580 •577 •331 •340 •838 •861 6-2 1-74 1-73 •96 •95 •582 •577 •326 •340 •823 •861 3. Seiches in a convex symmetric parabolic lake [§ 37]. I, Tx 1 1 Position of Binode - . a lb ct obs. calc. obs. calc. obs. calc. 2-5 357 2*70 272 1-24 1-29 •444 •472 5-4 50 2-58 2-59 1-23 1-23 •457 •472 148 Proceedings of Royal Society of Edinburgh. [skss. Jc h ■ a Ti t2 obs. calc. obs. calc. 2*5 2-5 71-5 2*70 2*72 1-24 1-29 5 2-5 71-5 2-06 1-00 7 ’5 2-5 71*5 1-66 •84 10 2-5 71*5 1-50 •76 4. Seiches in a complete symmetric rectilinear lake [§ 49]. A1 a O a A h(x) = h(\ - ^ = 4-02 h 1 ■i t2 t2/t2 Position of Binode — . a obs. calc. obs. calc. obs. calc. obs. calc. 870 1 99 1-98 1-24 1-24 •622 •627 •560 *605 107 2-23 2*19 1-40 1*38 •625 •627 •554 •605 12-9 2-41 2-42 1*51 1-52 •626 •627 •577 •605 5. Seiches in a semi-complete rectilinear lake [§ 51]. a k 1905-6.] The Hydrodynamical Theory of Seiches. 149 h 1 Ta obs. calc. obs. calc. 10-5 1-42 1-43 •78 •78 11 1-50 1-49 •81 •81 11*4 1-52 1-51 •84 •83 11-8 1 '55 1-54 oo 05 •84 6. Seiches in a concave truncated quartic lake [§ 52]. Ar Q O a A [? = -t) 8/ n ' With the same approximation, the length of twisted thread will be 7 /, 7T2d2 , Let us now consider the case of twisting two single threads together. If two lengths, each = L , receive x turns of twist, and each single be considered to revolve without any motion amongst its constituent fibres, then the pitch of the twist where D is the diameter of the circle in which the axes of the singles revolve. D = d , the diameter of one of the single threads, if they are of the same size. The length of 2-ply thread produced from lengths L of the singles vs ^ - 7T2D2 V1-'™ = L 1 7r2B2x‘ 2L2 approximately. the contraction 7T2D2X2 2L 186 Proceedings of Royal Society of Edinburgh. [sess. The hypothetical condition that there will not be any relative motion amongst the fibres of the single threads will never be realised in practice. Again, let us consider the twisting of two lengths of untwisted single threads, each = Z, and whose diameter = d. Suppose that they receive n turns of twist each, and that in forming them inta a 2-ply thread, they are subjected to a further torsion of x turns. Then D of the general case = d. The length of twisted single thread as before Therefore the length of a strand in one turn of 2-ply thread C Fig. 2. — Take-up in 2-ply Threads. 7 7 r2eZ2 o or l- nl approx. 8 1 — L of the preceding example. x .*. the pitch of twist in 2 -ply thread the length of single thread x ' l .*. the length of 2-ply thread approx. the total contraction !) approx. 1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 187 We have directly, since the total contraction must be equal to the sum of the contractions due to the consecutive twisting operations of forming the single and double threads, But as these values will not differ by more than 1 or 2 per cent, in all cases which would arise in practice, we may safely take the first expression. This analysis takes no account of relative motion of the fibres in each thread. As the second twisting proceeds, the degree of twist Fig. 3 — Two-ply Thread with Open-band Twist both in Singles and in Double. in the singles will also vary. The result will be differently affected according as the second twisting is in the same direction as that which the singles possess (shown in fig. 3), or is in the opposite direction (shown in fig. 4). The former is scarcely ever used in practice, because of the rapid increase in hardness of the thread as the twist increases. Further, the friction between the surfaces of the fibres is not sufficient to prevent the elasticity of the thread from asserting itself in opening out the twist. The excessive degree of twist caused by both operations being in the same direction produces a thread of little use for cloth manufacture. Fig. 4 — Two-ply Thread with Open-band Twist in the Singles and Cross-band Twist in the Double. If the fibres have perfect freedom of motion, the total con- traction will be equal to that due to n + x turns in the single state plus that due to x turns in the double, if x be reckoned positive when in the same direction as n, and negative when in the opposite direction. In the latter case the opening out of the twist from the singles at first completely masks ^.2^72 --2T)2 Total contraction = n 2 + — — x2 . 8 1 2L i.e. the total contraction 188 Proceedings of Royal Society of Edinburgh. [sess. the contraction due to the second twisting. As a result, an elongation is really observed as twisting proceeds. The point of maximum length is, however, soon reached when the rate of con- traction due to the second twisting is equal to the rate of elongation due to the opening out of the twist from the singles. After this point the rate of contraction increases. A is the curve of contraction for the singles in the first twisting. A' ,, ,, ,, second direct twisting. A" ,, ,, ,, ,, inverse twisting after all the single twist is opened out. B is the curve of contraction for the 2-ply if no fibre rotation (inverse). C „ „ „ „ (direct). D ,, ,, ,, free ,, (inverse). The results are represented graphically in fig. 5. The ordinates represent contraction, and the abscissae turns of twist. 7T2d2 In the diagram is written as k. The ordinates of any point on curve D is obtained by measuring the vertical distance between curves A and B at the same abscissa, i.e. — + kx2. 4 The practical problem in connection with this subject does not involve the consideration of the length of the untwisted thread I, 1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 189 as the singles are always measured in the twisted condition. The “sliver,” as the untwisted woollen thread is technically called after it leaves the condenser or last machine in the carding process, is always drawn out about 50 per cent, in length on the mule before any twist is put in. But, in order to understand what is going on in the second twisting, it is necessary to introduce the hypothetical length l in the discussion. To compare the theory with the experimental evidence, the contraction should be referred to the length of the twisted single thread, and only that part of the contraction due to the second twisting need be considered. The contraction due to the twisting of the singles t r2d2 21 4 7 r2d2 n 2 „ 2L ‘ 4 practlcally- \ the contraction due to the second twisting; t r2d2 J (n + x)2 o n2 \ ~ZL \ “X" 4 j 8L x(hx + 2 n) . . •. the contraction = 0 when x = 0 or when x = — ■§ n , ,, is negative ,, x lies between the limits 0 and - 1 n , ,, is positive „ x lies beyond these limits. By differentiating the above expression, we get the minimum value for the contraction or the maximum value for the length of the 2-ply thread, V = dy dx ——x(5x + 2 n) 8L ^{lOx + 2 n) 8L d2 if we take — as a constant, which is not far wrong. JLj ~ will be zero, and y will have a minimum value when (XtC \0x + 2n = 0, i.e. when x = — n 5 190 Proceedings of Royal Society of Edinburgh. [sess. Substituting this value of x in the expression for the contraction, the minimum value of y may be found, 7t2c£2/5w2 ^min' = ~8L \25 7 rW2 rc2 7 rV2 n 2 8L * 5 °r 2L * 20 Summary of Results arising from the Theory of Free Fibre Rotation. (1) The 2-ply thread has the same length as the original length of each of its constituent singles when the turns of twist in Fig. 6. — Curves of Contraction during Second Twisting. A is the curve for the single twist. B ,, ,, 2-ply (inverse) if no fibre rotation. C „ „ „ (direct) „ „ D ,, ,, ,, (inverse) if free fibre rotation. E „ ,, ,, (direct) ,, the second twisting amount to two-fifths of the number which the singles possess, but in the inverse direction. (2) The 2-ply thread has its maximum length when the turns of twist amount to one-fifth the number of turns in the singles in the inverse direction. (3) The maximum elongation is equal to one-fifth part of the 1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 191 elongation which would result from opening out all the twist from one of the singles separately. (4) The 2-ply thread is longer than the singles until two- fifths of the number of turns in the singles are put on : when this number is exceeded the 2-ply thread is shorter than the singles. The latter is the important case in nearly all yarns used in cloth manufacture. (5) When the direction of twist is the same as in the singles, the contraction rapidly increases from the beginning. The assumption that is a constant is further from the truth at the point of maximum length than at any other point, because the diameter decreases throughout with increase of twist, while L is actually longer than at first. It will be useless, however, to attempt to find out by mathematical analysis how much this assumption will affect the results ; for there are other disturbing factors in operation of much greater importance. Prominent amongst these are — (1) When two threads are twisted they lose their cylindrical forms. As shown in my former paper on “ The Diameters of Twisted Threads,” communicated to this Society last year,* the circular eross sections are deformed into ellipses. The effect would make D, the diameter of the circle of axial revolution of the singles, = 2 b, where b is the semi-minor axis of the elliptical cross section. (2) In the above analysis the fibres which constitute the thread are considered as arranged parallel to the axis of the thread before twisting. This is only approximately true for worsted yarns, and very far from the truth for woollen yarns. This condition, though probably modifying the amount of the contraction, does not alter the general form of its expression. (3) In the initial stages of the second twisting, the fibres are free to rotate about the axes of the singles, and the experimental results are practically the same as those deduced from the above analysis. But the movements of the fibres become more restricted as the torsion proceeds, and long before the operation is completed * Proc. Roy. Soc. Edin ., 1905, vol. xxv., part vii. 192 Proceedings of Poyal Society of Edinburgh. [sess. each single thread revolves as a whole about the axis of the 2-ply thread without any relative motion amongst its fibres. For a twist composed of three or more singles, the analysis is on similar lines. When three singles of equal size are formed into a folded thread,, the diameter of the circle of axial revolution, D, would be = CD = BC sec 30c 2 JSd = 1*055 d. The total contraction for folded threads in general has been shown to be rfdf 8/ (n + x)2 + r2D2 2L 5 .*. the contraction in the formation of the 3-ply thread is (if l is taken = L) t r2d2 / / y“-8L l( n + x)2 — h •} _ 7r2d2/’2nx + x2 4x2 2L V TT2d2 "2ira /1 9 , ?z\ \Ef+ §)* + 2LVV3, d2x2 This expression is positive when x is positive, and when x is greater numerically than -~n in the negative direction. It is equal to zero when x = 0, and when x = - It is negative between the limits x = 0 and x = — 1905-6.] Mr T. Oliver on Tahe-up in Twisted Threads. 193 Differentiating as before, and equating to zero, dy 7r2(i2/19 n dx 2L \ 6 which vanishes when y has a minimum value, , 19 n A i. e. when — x + - = 0 6 2 3 „ x = 19w • Substituting this value of x in the expression for y gives the minimum value of the take-up or the maximum elongation, ^ r2d2f 3 2 3 « Vmin‘ = ‘2LW " ttH2 3 o ” 2L ' 76^ ’ i.e. T¥ of the elongation which would result from opening out all the twist from one of the singles separately. In general, if D = pd V = 7r^V2 nx + x2 2L +J?Z& y is zero when x — 0 and when x = - — _“h- _ n ip2 + 1 y is negative between these limits y is positive beyond „ ,, The maximum elongation = • -j— ~ — 8L \.p2 + 1 or — of the 4p2 + 1 elongation which would result from opening out all the twist from the singles separately, and it occurs when dy rr2d2 fn + x 0 9 \ a A 01 2l{— + 2PX] = ° i.e. when x = — — - . 4^2+1 It may be shown by geometry from diagrams similar to fig. 7 that p = J2 or 1*4 or a four-fold thread, 4 or 1*7 for a five-fold thread, J10-2 = 2 for a six- or seven-fold thread. PROC. ROY. SOC. EDIN., VOL. XXVI. 13 194 Proceedings of Royal Society of Edinburgh. [sess. Since normal folded threads with more strands than three are of little practical importance in cloth manufacture, it is unnecessary to discuss these in detail. If the origin of the graphs be shifted to the point of maximum elongation whose co-ordinates are n lc n 2 4p2 + 1 ’ 4 4p2 -f 1 , where h = 2L If X , Y he the co-ordinates of any point on the graph measured from the new origin, v n v+i y = Y- 4 if + 1 Substituting in the expression for the take-up, + j)*’} Y" l .TO ' M l(x-4^,)+("= + i)(x-cti)’ ! Keducing Y = Jc(p2 + i jX2 . Substituting the values of p in particular cases gives for 2-ply threads, Y = - /cX2 or 1 ’25/cX2 „ 3-ply „ 4-ply „ 5-ply Y =— *X2 or 1-58M2 12 Y = -kX2 or 2" 25/cX2 4 Y =37X~ v/5 /cX2 or 3-14/tX2. 20 - 4 J5 Since working out this theory my attention has been drawn by Professor Barker of Bradford Technical College to a long article in the September 1903 issue of the L’Industrie Textile , Paris, entitled “ Baccourcissement d’un hi par l’effet de la torsion et consequences pratiques que Ton peut en deduire ” (The con- traction of the thread by the effect of twist, and practical results which we can deduce from it), written by Professor Bartolome Amat, Tarrasa, Spain. He elaborates a theory which is quite at 1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 195 variance with experimental evidence. I give the opening sen- tences of his analysis, because it is in these I judge he goes wrong : “ Supposons, en effet, un cylindre de diametre d et longueur L et tachons de determiner le raccourcissement de ce fil par l’effet de la torsion. La ligne ab prendra la position bed quand L aura re9U un tour de torsion et nous aurons : ab = deb = ab' = L . De meme, l’on aura, pour deux tours de torsion, ab — a'eb — a"b' = L' . II s’ensuit que A' = L-L' = L± JL2 - 7r2d2 Translation : — “ Let us suppose a cylinder of diameter d and length L, and let us try to determine the contraction of this thread by the effect of twist. The line ab will take the position bed when L will have received one turn of twist, and we shall have ab = deb = ab' = L . Similarly, one wrill have for two turns of twist db = a'eb — a"b' = L' . It follows that A' = L - iMl ± ^\F^d2 Obviously, the positive sign to the radical gives an inadmissible solution, and, from the same principles as before, Professor Amat easily deduces that the contraction due to n turns of twist An — L — JL? - mr2d2 , which we may reduce to An — 17 ^ n approximately. 2L Differentiating this result, we find that dAn 7r2d2 dn 2L ^if the ratio be taken as a constant^ , 196 Proceedings of Royal Society of Edinburgh. [sess. i.e. the rate of increase of contraction relative to increase in the degree of twist is constant. This result at once appears suspicious. Anyone who has made even qualitative experiments would expect that the rate of increase in take-up will become much greater as the twist gets harder. The error which Professor Amat has dropped into is simply this : he asserts that ab = a"cb. Now ab is merely a geometrical contour of the cylinder which cuts across the fibres (see fig. 9), and is not a material line or \ \ \ \ -V Fig. 8. — Diagram illustrating Professor Amat’s Theory. fibre itself. The continuity of this line is not preserved as torsion proceeds. Therefore we cannot say that ab is equal to anything at a subsequent period of the torsion. It may be asserted, however, that since ab becomes ab for one turn, that it must be reduced to some shorter length a"b for two turns. That is true ; but if this procedure be adopted, acb will be equal to twice the hypothenuse of a right-angled triangle erected on half of ab, and not equal to the hypothenuse ab' of a right- angled triangle erected on ab, since a"cb is two turns of the 1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 197 spiral, and not one, as shown in fig. 8. Hence the solution becomes identical with the one I have given. We must now consider the experimental side of this subject. The apparatus shown in fig. 10, which I have used in this in- vestigation, was invented by Mr George R. Smith of Bradford for the purpose of the commercial testing of the strength and stretch of yarns. But as the amount of twist in the yarn under test may also be varied, I have found the apparatus, with only slight modification, admirably suited to my requirements in the present research. The thread A is stretched between two clamps B and C. C Fig. 10. forms the end of one arm of a bell-crank lever pi voted at G, whose other arm F carries a can E. Water may be run into E from a reservoir D, and thus the thread may be twisted under any tension. H is an adjustable counterpoise to balance E when can E is empty. The twist in the thread may be varied by turning the wheel M, the turns being indicated on dial T. The thread can be kept throughout the experiment at the same tension by turning wheel I, which communicates motion by a train of wheels to shaft Iv. K moves the clamp B so that the lever F always remains horizontal. The take-up in the length of the thread may be obtained by taking the difference of the readings on scale S. I took seven yarns at random — viz., 2/1 Os* cotton ; 2/12s; 2/1 6s, * 2/10s means two threads, each measuring 10 hanks per lb., twisted together. 198 Proceedings of Royal Society of Edinburgh. [sess. 2/24s, 3/24s crossbred worsted ; 2/48s Botany worsted ; 2/56 cut woollen — and tested them specially for this paper. If I had selected seven tests from my previous collection, I might have been tempted to select those which agreed best with the theory. The percentage contraction for each turn of twist per inch is shown in the following tables. Table I. — Percentage Contraction for Crossbred Worsted Yarns. Turns per inch. 2/1 2s. to os IfJ 2/24s. 3/24s. Open- band Cross- band. Open- band. Cross- band. Open- band. Cross- band. Open- band. Cross- band. 1 ■40 - -16 •44 - -20 •28 - *12 •20 - T1 2 1-00 - -16 •94 - *24 •66 - *18 •52 - T4 3 1*74 -•08 1-58 - -20 1-16 - *12 1-01 -•10 4 270 + •22 2-44 0 1-72 - *01 1-52 + -09 5 3-84 •58 3-32 + *28 2-38 + T4 2T4 •32 6 5*30 1’12 4-28 •60 3*06 •26 2-82 •68 7 6-74 1*76 5-50 1-08 3-82 •54 3-70 1T1 8 8-50 2-56 6*84 1-64 4-72 •86 4-72 1-52 9 10-40 3-42 8-18 2-32 5-70 1-22 5-82 2-12 10 12-50 4*54 9-84 3-04 6-78 1'62 7-01 2-79 11 14-60 5 ’90 11-62 3-84 7-96 2-10 8 '22 3-41 12 7-34 13*44 4*92 9*20 2-66 9*53 4-26 13 8-98 15-28 6-12 10-56 3-26 10-96 5-14 14 11-02 7 '50 12-04 3-96 12-38 6*16 15 13-04 8*92 13*40 4-72 7-24 Id 15-16 10-52 15*00 5*62 8-32 17 12-38 6-58 9-80 18 14-24 7-56 11-30 19 8-74 12-68 20 10-04 21 11-16 The direction of the twist at the top of each column is that of the folded thread. The twist in the singles is open-band. The numbers are also plotted on squared paper in fig. 11. The ordinates of the graph for each yarn represent percentage contraction ; and the abscissae, turns of twist. The forms of the graphs in fig. 1 1 suggest that if the origin of the diagram be shifted in each case to the point of maximum elongation, the equations to the graphs will probably be of the family y' = cxm. Let us consider any one of the curves, if its 1905-6.] Mr T. Oliver on Take-up in Tivisted Threads. 199 Table II. — Percentage Contraction for Miscellaneous Yarns. Turns per inch. 2/1 Os Cotton. 2/48s Botany Worsted. 2/56 cut Saxony Woollen. Open- band. Cross - band. Open- band. Cross- band. Cross- band. Open- band. 1 •68 - -60 •16 - -18 •34 - -20 2 1*74 - -84 •44 - -28 •66 -•36 3 2-90 - *94 •71 - -32 1-16 - '38 4 4-52 - -60 1-02 -•32 1-76 - -38 5 6-36 - -14 1-41 -•28 2-32 -*30 6 8-34 + *42 1-81 - -18 3-04 - -20 7 10-66 1-08 2-22 - -08 3-80 0 8 13-16 1-86 2-74 + •01 4-64 + •22 9 15-86 2-54 3*24 •12 5*44 •50 10 18*46 3-68 3-78 •32 6-58 *81 11 4-92 4-41 •84 7-64 1*12 12 6-26 4-98 1-08 8-88 1-56 13 7-76 5-60 1-32 10-26 2*02 14 9-36 6-31 1-54 11-78 2-50 15 11-36 7-06 1-88 3-16 16 13-40 7*70 2-20 . . , 3-72 17 15-40 8*56 2-61 4-36 18 1776 3-04 5*10 19 3-44 5-90 20 3*92 6-78 21 4-44 772 22 4-94 8-82 23 5*48 9-94 24 6-04 11-22 The twist in the cotton and worsted singles is open-hand, and cross-hand in the woollen single. maximum elongation be represented by a, and let this occur when b turns of twist are put on in the inverse direction. Then y = y + a x = x + b or x - b if we consider the inverse direction positive. If the general equation is y' = cx'm then y + a = c(x + b)m in the direction of the twist in the singles = c(x-b)m ,, inverse direction. To test if this relation is true, we plot on squared paper log (y + a) as ordinates, and log (x -f b) or log ( x - b) as abscissae. Fig. 12 shows the results plotted on a sheet of paper squared logarith- Turns of inverse Twist. Turns of direct Twist Fig. 11, 1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 201 202 Proceedings of Royal Society of Edinburgh. [sess. mically. In order to suit the scale of the logarithmic paper, the ordinates really represent logarithms of half the percentage -contractions. It is found in every case that a straight line can be drawn evenly amongst the plotted points for each thread, within reason- able limits of experimental error. The assumed law is therefore true, because if y + a — c(x ± b)m log {y + a) = log c + m log ( x ± b) Y = C +mX, which is an equation of the first degree, and therefore represents & straight line. Also Y and X are the ordinate and abscissae respectively of any point on the diagram in fig. 12, m is the slope of the line, C is the intercept which the line makes on the axis of Y, Y m — tan 0 = — — - (see fig. 13). The Y axis is not shown on the diagram, but C may be calculated 1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 203 by the consideration of the similar triangles ABC, EDO, shown in fig. 13. C ED AB Y , „ — = = — = = tan 0 or m, numerically. I DC BC X - 1 J Since C is measured downwards from line DX, the sign must he changed. . *. C or log c = - ml. From which e may he found by consulting tables of anti- logarithms. Table III. shows the v allies of C, c, and m for all the equations. Table III. Direct Twist. Inverse Twist. C or log c. 1 c m C or log c. 1 c m 2/10s Cotton -1*05 11*2 2-09 -1-065 11-6 1-93 2/48s Botany Worsted -1*63 43- 1'96 -1*79 62- 1-98 2/56s Cut Woollen -1-62 42* 2*15 -1*61 41* 2-03 i 2/1 2s Crossbred Worsted . -1*08 12* 2*06 -1*28 19- 2-08 2/16s -1*065 11*6 1*92 -1-325 21* 2-05 2/24s -1*21 16-2 1*91 -1*71 51- 2-15 3/24s „ „ . -1*23 17- 1-96 -1-29 19-5 1*96 Average m . 2-01 2 02 Table IV. Average Diameter of Singles. Turns fel per Inch in Singles. k or irhP 2L 'L' p. cent. *5! 20 p. cent. Max. Elong. from Expts. a p. cent. X 5 * © S *5? o & 3.2 « b 2/10s Cotton •016" 7-5 *126 •35 •94 1*5 3* 2/48s Botany •0065" 15* •021 •24 •32 3- 4* 2/56 Cut . •009" 17* •041 •58 •40 3-4 3-5 2/1 2s Crossbred •0145" 5-5 T03 T6 T8 1*1 1-6 2/16s •0125" 7 • •078 T9 •24 1*4 2' 2/248 „ *0101" 9* •049 T9 T8 1-8 2-2 3/24s •0101" 9* •049 T5* T4 1-4+ 1’4I '&*) Negative direction. 204 Proceedings of Royal Society of Edinburgh. [sess. Let us now compare the analytical and experimental results, more closely by plotting on squared paper the figures obtained for one of the yarns, — e.g. 2/1 2s worsted, as shown in fig. 14. The equation y or y + a = 5 k (x + g^2 is the relation when the fibres are perfectly free to rotate. The equation y = kx 2 is the relation when the fibres have no relative motion. Fig. 14. Curve Oc is symmetrical with the open-hand limb of the yarn curve about a vertical line through C. M is the point of maximum length for the condition of free fibre rotation. C is the point of maximum length for the actual yarn. It will now he instructive to summarise the main conclusions which may he inferred from a study of the analytical and experimental data set forth in the foregoing tables and diagrams. The following relations have been discovered to hold : 1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 205 Experimental: (1) y + a = c(x±b)m Analytical : when the fibres have perfect freedom of rotation about the axes of the singles during the second twisting. (3) y =kx*\ when the single threads revolve as elastic solids about the centre of the 2-ply thread without any relative motion amongst the fibres. The average value of m taken from Table III. is 2-02, and the extreme limits are 1*91 and 2T5. Therefore the analytical value deduced for m — i.e. 2 — may he considered as confirmed by experi- ment also. In the actual yarn, the fibres are neither absolutely free to rotate in the singles, nor absolutely constrained to move as a whole, so that some intermediate condition may be expected to hold good. On examining the open-band side of fig. 14 we find closely at first, but as torsion proceeds it moves away towards the curve y = kx 2, which it never reaches. The fibres, perfectly free for the first one or two turns, gradually become more constrained in their relative movements, though never becoming absolutely so. It will be seen from fig. 14 that the curve of the equation ceeds, and never differs much from the latter at any point. Considering now the cross-band side of fig. 14, we find that the yam curve differs widely from either of the curves corresponding to the analytical relations deduced. Much of the discrepancy, however, is due to the fact that (a) the turning-point of the graph of y = kx 2 is at 0; (b) the turning-point of the graph of y = ^x( 5x+ 2 n) is at M ; (c) the turning-point of the graph of the yarn equation is at C. The differences will be much less if the origins of the hypothetical graphs be transferred to C. Curve A positive although measured to the left hand from 0. This curve practically coincides with the yarn curve until the twist becomes this is the case. The yarn curve follows approaches closer to the yam curve as torsion pro- 2 in fig. 14 is the graph of y = — kx(5x - 3n), where x is taken 1 0 excessive. 206 Proceedings of Royal Society of Edinburgh. [sess. From Table IY. it may be seen that C, the point of maximum elongation, in every case lags behind the point M, where maximum elongation should occur if the fibres were perfectly free to rotate. This fact furnishes us with a clue to the cause of the difference between the actual behaviour of the yarn and the result deduced by analysis. The yarn fibres have acquired a 1 set ’ in one direction during the single twisting, which retards the contrac- tion in the second twisting until a later period. In concluding this paper, the author has pleasure in acknowledg- ing his indebtedness to the Carnegie Trust for the Universities of Scotland for the financial assistance which has enabled him to prosecute the research. ( hsued separately June 19, 1906.) 1905-6.] A New Form of Harmonic Synthetiser. 207 A New Form of Harmonic Synthetiser. By J. R. Milne, B. Sc., Carnegie Research Fellow. (With Plate.) CONT § 1. General Description ‘ . .208 § 2. Description of the Mechanism %ohich enables the Period of a Constituent Harmonic to be varied during motion 209 § 3. Description of the Mechanism which enables the Ampli- tude of a Constituent Harmonic to be varied during motion . . .210 § 4. Description of the Mechanism connected with the Pen and Paper .... 214 § 5. Certain Details of Construc- tion .... 215 § 6. Mathematical Investigation of the Deviation from its ENTS. | True Position of the Pen of an Instrument on the above Principle . .216 § 7. Expression for the Greatest Deviation of the Pen . 217 § 8. On the Necessity for finding the Deviation of the Pen with Exactness . . 21& § 9. Expression for the Average Deviation of the Pen . 227 § 10. Application of the Mathe- matical Results to the Present Instrument . . 230 § 11. Improvements suggested by the Ma thematical Theory .... 231 §12. Summary of the Paper . 232: The author has designed and constructed an instrument for the purpose of drawing the curve which is the summation of a number of simple harmonic curves ; and as the apparatus has been found to work very satisfactorily in practice, it seems worth while to publish a description of the machine, especially in view of its numerous applications. Various forms of harmonic synthetiser are already in existence,* hut there were two reasons which induced * The following papers may be consulted : — “Tide Predicting Machine in use at Nat. Ph. Lab.,” Great Trigonometrical Survey of India , vol. xvi. ; also Proc. of Institution of Civil Engineers, vol. lxv. pp. 1-70 (where also is given a description of Lord Kelvin’s Harmonic Analyser). ‘ ‘ On an Instrument for Compounding Vibrations,” Lord Rayleigh, Phil. Mag., p. 127, 1906. T. R. Lyle, “Preliminary Account of a Wave-Tracer and Analyser,” Phil. Mag., p. 549, Nov. 1903 ; and also ibid. p. 102, Jan. 1905 ; and ibid. p. 25, Jan. 1906. “A New Harmonic Analyser,” A. A. Michelson and S. W. Stratton, Phil. Mag., vol. xlv. p. 85, 1898. W. C. Baker, Note in Nature, p. 541, 28th Sept. 1905. The following references are to Harmonic Analysers: — “An Harmonic Analyser,” J. N. Le Conte, Phys. Rev., vol. vii. p. 27, 1898. “On a Simple ■208 Proceedings of Poyal Society of Edinburgh. [sess. the author to add another to their number. In the first place, it was desired to ascertain whether results of a high degree of accuracy could not be obtained from an apparatus of quite inexpensive design ; and in the second, to construct a machine in which both the amplitude and periodic time of each of the simple harmonic constituents might be varied at will while the instrument is in motion. § 1. General Description. The apparatus (see Plate, fig. 1) has a long narrow wooden base, from which spring uprights, carrying the horizontal axes of a succession of wheels arranged in a line, one after the other. The wheels are made of mahogany, with V-shaped grooves turned in their edges, and they are driven from a small electric motor by means of a single, endless leather belt of circular section. These wheels form the simple harmonic constituents of the machine : and they, or others geared to them, as will be explained later (see the section on change of period, and also that on change of amplitude), have each a pin fixed in the wheel at some distance from the centre, which stands out normally to the wheel, and has mounted on it a freely turning brass pulley with grooved edges. The base of the instrument is screwed down to a table from which uprights arise to carry a sort of light horizontal frame some three feet above. A length of jSTo. 38 silk-covered copper wire, with one of its ends fixed to this upper frame, is led vertically down and round one of the small eccentric pulleys just mentioned, and then returns up again to the frame (see fig. 7). Passing round a fixed pulley there, it once more descends, and passes round the brass pulley on the pin of another of the simple harmonic wheels, after which it returns back to the bracket ; and so on, until all the simple harmonic wheels have been included. Form of Harmonic Analyser,” G. V. Yule, Phil. Mag., vol. xxxix. p. 367, 1895. “Harmonic Analyser,” Geo. H. Rowe, Electrical World and Engineer , p. 587, 25th March 1905. Three papers by Prof. Sir W. Thomson, Proc. Roy. Soc., pp. 266, 269, 271, 1876. “On an Integrating Machine having a New Kinematic Principle,” Prof. James Thomson, Proc. Roy. Soc., vol. xxiv. p. 262, 1876. “ On a New Harmonic Analyser,” Prof. 0. Henrici, Phil. Mag., vol. xxxviii. p. 110, 1894. “Ueber Instrumente zur harmonischen Analyse,” 0. Henrici, Catalogue Munich Mathematical Exhibition, 1892-3. 1905-6.] A New Form of Harmonic Synthetiser . 209 The free end of the wire is finally Jed down to a “barograph” pen, which is mounted on a form of Watt’s parallel motion that compels it to move in a (sufficiently approximate) vertical straight line (Plate, fig. 2). In front of this arrangement an upright band of paper is made to slowly pass along in a horizontal direction,, and against this the spring carrying the “ barograph ” pen causes the latter to lightly press. The writing is found to be quite satisfactory, a good line is easily obtained, and there appears to be no appreciable stiction. § 2. Description of the Mechanism which enables the Period of a Constituent Harmonic to be varied during motion (see Plate, fig. 1). A simple harmonic element of variable period is constructed as follows. A wooden truncated cone, 6 inches long, 4 inches in diameter at the small end and 6 at the large, has fixed in it a steel rod which serves as axis, and rotates between the pointed ends of two screws fixed in uprights from the general base. A similar cone is arranged with its axis parallel to that of the first, but disposed so that its big end is opposite to the small end of the other. It is well known * that unless the belt employed to connect the two cones be crossed, it will only have the necessary length when in one position on the cones. In the present case the belt is not crossed, but a special tension arrangement is employed (see fig. 1). The belt, a leather one of circular section about T3g- of an inch in diameter, encircles the two cones B and C, and also passes round a deeply flanged guide pulley A. A weight hung from the end F of the lever DEF supporting the pulley draws the latter continually upwards, and keeps the belt taut. When in motion the position of the belt on the cones can be changed by sliding the lever D, and therefore the attached pulley, backwards or forwards along the pivot rod E. If then cone B be driven from the electric motor, and cone C carry the small eccentric pulley p round which the “summation wire” passes, it is clear that the periodic time of C can be varied at will relative to the rest of the machine by moving A. In practice the arrangement * Proof may be found in any work on Mechanism, e.g. in that by S. Dunkerley, p. 23 (edition 1905). PROC. ROY. SOC. EDIN., YOL. XXVI. 14 210 Proceedings of Royal Society of Edinburgh. [sess. has been found satisfactory : it allows the use of any period commensurable or incommensurable, and of the alteration of the period during motion. In fig. 2 will be seen a trace of a S.H.M. FiG. 1. — It has been found better to slightly alter the arrangement sketched above , and make the guide pulley A act on the upper side of the belt. Cf. the later photograph on Plate, fig. 1. in which the periodic time gradually increases, the lever D having been slowly slid along the axis E by hand as the machine was running. § 3. Description of the Mechanism which enables the Amplitude of a Constituent Harmonic to be varied during motion. The mechanism used to change the amplitude will be seen in the stereoscopic photograph, Plate, fig. 3. The further off vertical toothed wheel has rigidly attached to it a grooved wooden wheel round which passes the motor driving- belt. Motion is communicated to the nearer vertical toothed wheel through the intermediary of the “ crown ” wheel. The 1905-6.] A New Form of Harmonic Synthetiser. 211 Fig. 5 ( continued on p. 213). 212 Proceedings of Royal Society of Edinburgh. [sess. axle of this crown wheel can he moved round the common axis of the vertical side wheels and clamped in any required position by means of the set-screw and overhead arch, as shown. Each of the side wheels carries an eccentric pulley, and to these eccentric pulleys the “ summation wire ” descends. As the side wheels are of equal diameter (3 inches), they must revolve at the same speed ; and also the throw of their pulleys is made to he the same (J inch). Hence the resultant displacement of the summation wire is that due to two S.H.Ms. of equal period and amplitude. How the whole point of the device is this, that the relative phase of these S.H.Ms. can be altered at will by moving the axle of the crown wheel. That this is so may easily he seen by supposing one of the side wheels to be fixed, when it is apparent that if the axle of the crown wheel be unclamped and rotated through an angle a, the other side wheel will have to rotate through an angle 2a.* Mathematically we have then, ij being the displacement imparted to the summation wire, y = J cos t + J cos ( t + 2a), y = cos a . cos (t + a) ; . . . (1) that is, we obtain by means of this arrangement a S.H. dis- placement of the wire, the amplitude of which may be varied from 0 to 1 by turning the crown-wheel axle through an angle of 90°. In fig. 4 are shown two traces, each of a S.H.M., the amplitude of which was thus altered, the axis of the crown wheel having been slowly moved round by hand as the machine was running. If the above mechanism is to be used merely for setting the amplitudes before the machine is put in motion, it is quite satis- factory ; but if the crown-wheel axle be moved ivhen the machine is in motion, then in equation (1) a is no longer a constant, but becomes a function of the time; that is (1) no longer strictly represents a S.H.M. of periodic time r/t, and indeed in general it would not represent a S.H.M. at all. Of course in practice most likely it would only be wished to change the amplitude slowly, in which case the error would be very small. All error, however, can be removed by means of the following modification of * The matter is fully discussed in S. Dunkerley’s Mechanism , p. 97 (edition 1905). 1905-6.] A New Form of Harmonic Synthetiser. 213 Fig. 5 ( continued from p. 211). 214 Proceedings of Royal Society of Edinburgh. [sess. the device. If in making a change in the amplitude by this method it were arranged that the phase of one of the S.H.Ms. should be increased just as much as the phase of the other was diminished, then the periodic time would remain always invariable, for in that case y = cos (t + a) + J cos ( t - a), y = cos a . cos if. ..... (2) Now if the side wheels were each to he driven from the crown wheel, the latter being driven from the motor belt, then on moving the axle through an angle a each of the side wheels would he rotated through a ; and as these latter rotate in opposite directions, this means that one of them would have its phase increased by a, and the other its phase diminished by a. An arrangement of weighted jockey pulleys would take up the slack of the motor belt when the crown-wheel axle was shifted ; and it should he arranged to do so on both the advancing and retreating sides of the belt equally, as otherwise the crown wheel would on these occasions change its phase relative to the rest of the machine, which of course is inadmissible. § 4. Mechanism connected icith the Pen and Paper (see Plate, fig. 2). The paper used is obtained in long rolls, and is 4 inches wide. In the instrument the roll turns loosely on a fixed upright pin, and the paper is allowed to unwind of itself. First of all it passes round an upright revolving drum driven by a belt from a “stepped” pulley on the machine. Two rubber-shod wheels press the paper band against the drum and ensure its gripping the latter. The electric driving motor runs at a greater or less speed from instant to instant according to the amount of work it is doing, and therefore it is necessary that the motion of the hand of paper should he produced, not by an independent clock-work arrangement or like device, but by something connected with the general mechanism, so as to vary similarly in speed. The rate of the drum relative to the other mechanism may be varied by using the several steps of the pulley, that is to say, the horizontal scale of the trace may be altered at will. The vertically moving pen is so placed that it writes on the paper soon after it leaves the winding drum, and the strip then 1905-6.] A Neiv Form of Harmonic Synthetiser. 215 passes along some 18 inches to the receiving drum. The latter is caused to turn by means of a weight which is sufficiently heavy to make it wind up the paper and keep it taut as it passes from the one drum to the other, but is too small to interfere at all with the action of the winding drum, which alone controls the rate of motion. The object of having a space of 18 inches between the pen and the receiving drum is simply to allow a considerable number of the waves traced out to be seen simultaneously by the operator. This is always convenient, but it is of course specially so when any progressive change in the wave-form is being studied, or when it is desired to observe the effect produced by gradually altering the period or amplitude of one of the constituent harmonics. § 5. Details of Construction. It may be well to mention certain points of detail. The pulleys employed are made of brass, they are carefully turned up, and have a diameter of about an inch. The axle of each pulley is pointed and runs between the hollow ends of two screws, an arrangement which allows good adjustment to be easily made, without shake on the one hand or stiffness on the other, and which also has the useful effect of minimising friction. There are no slides anywhere throughout the machine : each moving part turns on pivots, even the rectilinearly moving pen, and no doubt to this fact is due the smooth and successful work- ing of the apparatus ; which has thus justified the hope that, if properly designed, a simply made instrument of the kind would be quite satisfactory. It has been found that the use of a leather belt of circular section, engaging in the V-shaped grooves of wooden wheels, which are nowhere less than 4 inches in diameter, has obviated any trouble that might arise from slipping. The plan adopted of having only one main driving belt, with a simple arrangement to keep the tension constant, gives no trouble, and is a great simpli- fication. The small German-made motor employed to drive the apparatus is sent out by the makers furnished with a pulley which is driven from the armature shaft by a worm gear, and the use of this pulley at one stroke gives the necessary reduction to a slow speed. 216 Proceedings of Boy al Society of Edinburgh. [sess. The whole machine runs without attention, and can safely he left to itself in cases where for any reason a prolonged tracing is required — as for example when the constituent S.H.Ms. have been set to incommensurable periodic times ; in which case of course the resulting trace never quite repeats itself, but is always taking new forms. [A machine was constructed by the author in which pendulums were employed to give the S.H.Ms. ; but although the machine was satisfactory in many respects, it was abandoned, chiefly because of the fact that the pendulums necessarily ran down, and so rendered the instrument unsuitable for any sort of prolonged work.] As a further example of work done by the instrument, and to show the interesting character of the composite curves, the traces shown in fig. 5 have been reproduced. § 6. Mathematical Investigation of the Deviation from its True Position of the Pen of an Instrument on the above Principle. We now come to discuss the most important consideration in connection with such an instrument as the present — its accuracy. It hardly needs to be pointed out that a principle of construction possessed of great merits from the point of view of simplicity, smoothness of working, and so on, would nevertheless be of little value were the curves produced by the apparatus embodying it liable to appreciable errors of form. After being satisfied by actual trial that any suggested design of synthetiser is satisfactory on the mechanical side, there must always remain the essential question : “ In view of the principle adopted, what will be the accuracy of the trace produced ? ” Accordingly we must now go on to discuss the pretensions of the present apparatus from this point of view. It is a nice question whether in such instruments it is more important to minimise the greatest deviation or to minimise the average deviation. If the data of the resultant curve are to be deduced from measurements of it at a number of different places, then it would be desirable to have the average deviation as small as possible ; but if some single property of the resultant curve is to be ascertained, e.g. the point at which it cuts the time axis, 1905-6.] A New Form of Harmonic Eynthetiser. 217 then the greatest deviation should be minimised, because of the chance that the measurements are being made just at the placer where the greatest deviation occurs. Hence in the case of any given instrument it is desirable to know the value both of its greatest deviation and also of its average deviation. § 7. Expression for the Greatest Deviation of the Pen. In fig. 6 let S be an eccentric pin fixed in the rotating wheel A and let SRQ be a perfectly flexible wire turning abruptly round an infinitely thin pin R, its end Q being constrained to- 0 Fig. 6. For the sake of clearness , in each of the above figures the size of the wheel has been greatly exaggerated relative to the total height. For the same reason , in fig. 7 the three pulleys are slioivn without flanges. move in the horizontal line RQ. If Q have a sufficient force applied to it to keep the wire always taut, then as the wheel revolves, Q will describe approximately the S.H.M. •which i& produced by resolving the circular motion of the pin S in a vertical direction. Calling e the deviation at any time t ( i.e . the distance of the actual position of Q from its proper position at any time t), and writing for shortness RO = Z and OS = r, we have, e = RS - RM = + r 2 - 2 Ir cos t) -l + r cos t . (3) de lr sin t ^ . dt J(l2 + ?-2 — 2/r cos t) 1 Sm 218 Proceedings of Pay al Society of Edinburgh. [sess. hence the turning values of e occur when t — o or 7 r, or when t = cos-1 (r/2Z). The two former values of t give the minimum values of the deviation ; the latter value of t gives the maximum value of the deviation, which on substitution of this value for t is found to be e = r2/(2Z). In the present machine r is equal to J inch, and l to about 30 inches, and so the value of e in the above formula becomes of an inch. One point remains. If fig. 7, which represents the actual arrangement of one of the harmonic wheels of the latter, he compared with fig. 6, it can he shown without difficulty that the motion of the free end of the wire Q in fig. 7 (P being supposed fixed) will b.e almost exactly double * the motion of the free end of the wire Q in fig. 6. On the other hand, however, there must he set against this that, in order to restrict the motion of the pen within the limits imposed by the width of the paper band, it is as a rule necessary to arrange a pulley between the end of the summation wire and the pen, on the principle employed in grandfathers’ clocks, so that the pen, like the clock-weight, has a motion only half as great as that of the wire. The net result of the atpove is that our formula for the deviation remains as before. § 8. On the Necessity for finding the Deviation of the Pen with Exactness. At first sight it might seem that a deviation such as the above is wholly negligible, but in reality this is far from being the case. The above deviation is that of only one of the S.H. wheels of the instrument ; and it must be remembered that the greatest deviation of a trace compounded of n S.H. constituents will be n times the above quantity, for every now and then the deviations of all the different S.H. wheels will fall together with the effect of producing a total deviation in the position of the pen equal to their sum. Of course, as regards the present apparatus, which possesses but * The diameter of the small pulleys is 1 inch, which is also the distance between the upper pair ; and these are both distant about 30 inches from the lower pulley, whose circle of rotation is \ inch in diameter. Such being the dimensions of the machinery, we are probably led into no appreci- able error by having to substitute the above approximate statement for the exact one, which is too complicated to be used. 1905-6.] A New Form of Harmonic Synthetiser. 219 three S.H. wheels in all, the greatest deviation cannot be more than zT o an in°h i but the important thing here is not the consideration of a temporary model, but is rather to ascertain whether the mechanical principle employed would be satisfactory in the case of a complete instrument possessing a large number of constituents. As that means a large value for unf it becomes essential to ascertain the value of the deviation of each S.H. wheel with precision. How, in the mathematical reasoning employed above, to obtain an expression for the value of this deviation there may lurk a fundamental fallacy. In order to measure the deviation it is necessary to take some particular S.H.M. as a standard for comparison : and the S.H.M. which we choose for this purpose should be that to which the actual motion is most nearly akin. Unless, therefore, the S.H.M. that was employed in our former discussion can be shown to be the particular one which satisfies this condition, the result arrived at, being deduced from mistaken premises, must be misleading. In order to investigate this point, let ABODE in fig. 8 be the curve that would be drawn by the recording arrangement of Plate, fig. 2, were the latter actuated by the harmonic wheel of fig. 6 ; it being assumed that the wire has been sufficiently lengthened to allow its free end to come down over a pulley at Q, and be attached to the pen, which we arrange vertically underneath. Consideration of fig. 6 will show that AD must be symmetrical about the central line FC ; and that it will, except at the points A, C, and E. everywhere lie above the curve AGCHE, which represents the S.H.M. obtained by resolving in a vertical direction the circular motion of the pin S in fig. 6. e in equation (3) gives the difference between the ordinates to the two curves respectively ; and this difference we saw will be a maximum at the point £ = cos-1 (r/2Z) ; i.e. (as may easily be shown) at the point where ABODE cuts the time axis. How, were AGCHE to be bodily moved any short distance upwards, keeping it parallel to itself, then the length of its ordinates Avould all be altered by the same amount ; and therefore the maximum dis- crepancy between the two curves would still occur at t = cos-1 (r/2l) ; and therefore the value of the maximum discrepancy would be 220 Proceedings of Royal Society of Edinburgh. [sess. reduced. That is to say, without any further proof it is clear that AGCHE cannot he that particular S.H. curve to which the actual trace most closely approximates. Plainly the proper procedure in this problem is first of all to find the particular S.H. curve to which the trace ABODE most nearly conforms, and then to ascertain the greatest difference between the two curves, which quantity will be the required maximum deviation of the trace. Fig. S. — To keep the two curves sufficiently apart in the above figure , the deviation of the curve ABODE has been very greatly magnified, by the device of drawing the latter curve as if the length of 1 in the machine were only 2*5, instead of 30. The distance between the fixed point R in fig. 6 and the pen which we have supposed to be attached to the free end of the wire led down from Q, if measured up to Q and along QE, is given by the expression m - J(l2 + r2 - 2 rl cos t) . . . (4) where l = RO, and r = OS as before, and where m is the total length of the wire from the pen to the point S. The distance from R, also measured up to Q and along QR, of an imaginery point which moves with any S.H.M. whatever in the same vertical line as the pen, is given by the expression w + p cos (qt + d) .... (5). 1905-6.] A New Form of Harmonic Synthetiser. 221 Therefore the distance (or “ deviation ”) at any time t between the actual pen and the imaginary moving point of (5) is given by the equation, e =p cos ( qt + a) + w - [m — J(l2 + r2 - 2rl cos t) } . (6). But the motion of the pen given by (4) has a periodic time 27 r, and is, we know, very nearly S.H. ; and so the comparison S.H.M. must also have a periodic turn 27r, as otherwise the two would eventually get out of step. Therefore in (5) we may put q= 1. Further, it can be shown that the actual and comparison motions Fig. 9. — In the above figure the curve APB SC is the same as the curve ABODE in fig. 8, i.e. it has been drawn as if the length of 1 in the machine were but 2‘5, instead of 30. must be cophasal. In fig. 9 let the curve APBSC represent the actual trace of the pen ; and let DQETF be a S.H. curve, cophasal and isoperiodic with APBSC, but of any amplitude. There will be some point on the time axis for which the difference between the two curves is a maximum PQ. Because of the symmetry of both curves about HK, there must be an equal maximum difference ST between them at an equal distance from HK on the other side. Now suppose the curve DQETF to be shifted bodily a short distance to the right (say) ; and let it now cut the lines PQ, ST in the points Q' and T# ; then ST' will be 222 Proceedings of Royal Society of Edinburgh. [ses?.. less than ST, but PQ' will be greater than PQ. Therefore the maximum discrepancy that now exists between the curves APBSC and DQETF must be greater than before ; for if PQ' be still the maximum discrepancy, it is greater than PQ ; and if PQ' be not now the maximum discrepancy, then the latter — whatever it be — is greater than PQ,' and hence a fortiori greater than PQ. Therefore the maximum discrepancy between the actual motion and an isoperiodic S.H.M. of any amplitude is always least when the two- motions are cophasal. As this result holds for any amplitude of the comparison it must hold for that particular one which approaches most nearly to the actual motion. Therefore we may put a = 0 in equation (5) ; which thus finally becomes iv + p cos t ; and writing w — m=v . . . . . (7) (6) then becomes e =p cos t + v+ J (l2 + r2 - 2 rl cos t). In this equation e, the error of motion, is a function of three- independent variables, namely t, jy, and v. First of all to deal with the occurrence of t in the above equation. Reference to fig. 6 will show that whatever succession of valuer e assumes as S in the course of its rotation passes down from its highest to its lowest point, e will again assume in reverse order as S returns up once more to the highest point. That is to say, we need only investigate the values of e, where t ranges from 0 to tt (inclusive), all other values of t giving mere repetition in tho values of e. Then again we are only concerned to find the largest error of motion committed, and this must occur at the time when 0e/0Z = O .... (8) But de . , rl Sin t dt 1 v/(/2 + r2 - 2rl cos t) hence from (8) either sin t — 0 or 7r, ^ 1905-6.] A New Form of Harmonic Syntheliser. 223 That is, the largest error of e will occur at the time t — 0, or t = 7r,. or t = cos-1 If it occur at the time t — 0, its value w \ rl p2J will be p + l-r + v = tl (say); if at the time t = tt, its value will be — p + i + r + v = e2 (say) ; /l2 + r2 rl\ and if at the time t = cos 1 A( — — - — ) its value will be, ri p2 rl l2 + r 2p 2^- P + v = H (say). • (io> (11) (12). value given by (12), namely, t — cos 1 J Which of these three is greatest depends on the values assigned top and v; and our object is of course so to assign the values (i.e. determine the comparison S.H.M.) that for the chosen values the greatest of the three functions (10), (11), and (12) — whichever that turns out to be — shall be as small as possible. Before proceeding to this it should be pointed out that for values of prl/(l-r) the time at which e has the 'l2 + r2 rl\ . . NT ~f2f 18 lmagmary; and therefore for such values of p (12), although itself real, has no physical meaning, and is to be ignored. It will be convenient to regard each of the functions (10), (11), and (12) as being a surface, determined by the two independent variables p and v. Taking then three mutually perpendicular axes forp, v, and e respectively, we have a system of three surfaces, consisting of two planes and an hyperbolic cylinder. Fig. 10 represents the section of these surfaces by the p - e plane.* If any other section be taken parallel to the p - e plane, and at a distance - v from it, then, because of the manner in which v enters into the equations, fig. 10 will equally represent this other section, * The termination of the hyperbolic branch at its points of contact with the two straight lines, i.e., at the points the abscissae of which are respectively 2^— and -2l— is simply because, as explained above, (12) is physically meaningless beyond these limits. As negative values of p are also meaningless in the present connection, all that lies on the left of the € axis has been omitted from the figure. 224 Proceedings of Royal Society of Edinburgh. [sess. provided the axis of p be moved upwards parallel to itself through -a (^stance = v. The problem may now be restated thus : it is required to find that point in the p - v plane whose ordinate to the most distant of the three surfaces is a minimum. This, a problem in three dimensions, may be reduced to a problem in two only, by expressing it in the following form, which, Fig. 10. — In drawing the above figure, 1 teas taken — 2 and r = l , for the sake of clearness. Such a combination of values would never of course represent an actual case. from the foregoing discussion of fig. 10, can easily be seen to be its equivalent : find that position for the p axis of fig. 10, supposed moved parallel to itself, and that value of p , such that the ordinate •of the latter to the most distant of the three curves (the hyperbola and the two straight lines) is a minimum. A little consideration will show that the solution is as follows. Move up the p axis through such a distance that it bisects the shortest vertical line that can be drawn, terminated at one end by 1905-6.] A New Form of Harmonic Synthetiser. 225 the hyperbola and at the other by the line RK or KT as the case may be : and take for the value of p the abscissa of the ordinate on which the shortest vertical line lies. It may easily be ascertained by subtracting (10) and (11) respectively from (12) that RKT comes nearer to the hyperbola at its apex K than anywhere else, the distance from the one to the other being measured vertically. Accordingly the p axis must be made to pass through the middle point of KW, and because KW has a length equal to r2/(2Z), as may readily be proved, while the length of UK is Z, therefore The abscissa of the point K is r , hence p — r ..... (14) When t = 0, Q (fig. 6) is at its greatest distance from R, and RQ = m - (Z - r). Simultaneously the distance from R of the imaginary point given by (tv + r cos t) is iv + r. Hence, calling S0 the distance at the time t = 0 between Q and the imaginary point, we have, S0 =m -l - w. But from (7) and (13) m-w = l + —; 4 1 When t = 7r, and Q is at its least distance from R, we have similarly, The meaning of the above is, that the time axis of the curve that would be traced out by our imaginary point on the strip of paper passing along at right angles to the line of motion of the point, would be parallel to the time axis of the curve traced out by the pen on that paper; but the time axis of the point would be situated at a distance r2/(4l) above the time axis of the pen (meaning by the latter a horizontal line drawn along the paper half-way between the highest and lowest positions of the pen). PROC. ROY. SOC. EDIK, YOL. XXYI. 15 226 Proceedings of Royal Society of Edinburgh. [sess. We have then in this way completely determined the S.H.M. to which the actual motion is most akin ; and we find it to he of the same period and phase (p. 221), and amplitude (equation 14), hut situated with its time axis at a distance r2/(4Z) above the time axis of the actual motion. Now this means that if we are to regard the actual curve as an attempt on the part of one of the constituents of the instrument to draw this particular S.H. curve, and are to estimate the deviation of the actual curve accordingly, then when in using the instrument in practice we measure on the paper any ordinate of the trace produced by the pen, it must be done from a line drawn in in the position of the hypothetical time axis of the comparison S.H. curve, and not in the position of the time axis of the actual trace. In the above argument we have been considering only one of the constituent harmonics of the instru- ment ; but the quantity r2/(4Z) has the same value for all the harmonics; and hence when in practice a number are employed to draw a compound curve, the rule still applies. The true harmonic motion with which the actual motion is to be compared having been found, we now go on to ascertain the greatest discrepancy between the two, the “ deviation ” of the actual motion. This can easily be done by means of fig. 10, for the ordinates to the three curves from any point in that figure give the “ turning values” of the deviation function e for the values of p and v corresponding to the point. Now it will be remembered that the values of p and v we have been led to choose correspond to the point U on the y axis when the latter is moved up to bisect WK. Hence the deviation has the same numerical value, ±|WK= ±r2/(4Z), at each of its three turning values. This then is the value of the maximum deviation ; and it will occur five times -in each complete revolution, namely, at the times t = 0, t — cos 1 \ l2+r2 H — ),* t ~ 7T, Z=COS_1 -l IV and rl pL rl pl t — 27t. The first three of these times are given by (9), and occur during the down stroke ; the existence of the last two, on the up stroke, is inferred at once from symmetry. * The angle is to be taken between 0 and -k. t The angle is to be taken between -k and 2 ir. 1905-6.] A New Form of Harmonic Synthetiser. 227 § 9. Expression for the Average Deviation of the Pen. By the average deviation of the actual trace we mean the average value of the discrepancy between it and some S.H. curve chosen as a standard for comparison. In fairness this latter curve should, as was pointed out before, be that particular one which is most like the trace ; but the criterion of likeness is not to be the magnitude of the greatest difference between the two curves, as in the case discussed above, but the magnitude of the average difference. We have first of all then to ascertain the constants of the comparison curve, which — in general at least — will not be identical with the comparison curve determined before ; although the two comparison curves will have this in common, that both must be isoperiodic and cophasal with the actual trace. As regards isoperiodicity, what was said in the former case applies equally now ; as regards cophasality, the proof is simpler than before. It is as follow's. Consider any S.H. curve differing in phase from the trace by an amount a, and again by an amount — a, in both cases the average error of the trace, measured against this S.H. curve as a standard, would be the same. Here then is a function (the average deviation) which has equal values for values of its argument (the phase difference) that are equal in magnitude but opposite in sign. It follows therefore that when the phase difference is zero, the average deviation must be a maximum or a minimum. In this case obviously it will be a minimum — which proves our theorem. Accordingly, as in the former case, the com- parison harmonic must have the form (w + p cos t) ; and the difference between it and the trace at any time t will be e = \w +p cos t\-\m- J(l 2 + ?’2 - 2 Ir cos t)\ say, e=p COS t + V+ J(ol- 2/3 COS t), . . . -(lb) where v — w — m, a = l2 + r2, and /3 = Ir. . . . -(16) The value of the average deviation, e' say, will be, * o and to complete the determination of the comparison S.H. we 228 Proceedings of Royal Society of Edinburgh. [sess. must find the particular values of p and v which minimise e'. As e may change sign between 0 and 7 r, it is necessary to write the above integral cos \-m+n) -cos \-m-ri) it ' = ~fdt. +f l (18) edt 4- J edt cos cos ~\-m-n) where m and n have the following values. Ascertain the points at which e changes sign by putting e equal to zero in equation (15), which leads to p2 cos2 t 4- 2 pv cos t + v2 = a - 2/5 cos t. pv + [3 + J(ap2 4- 2 (3vp + /32) . cos t = or, say, where, p* pl cos t = - m + n , pv + (3 , and n= J(ap 2 4- 2 (3vp 4- (32) (19) P“ p“ From (18) by partial differentiation, remembering that cos_1( — m ± n) wdien substituted for t in (15) makes e equal to zero, we have — de dp df 2 dp y II V I IV J \j\JO y — u V — I V J It = - | f cos t dt - J cos t dt 4- j cos t dt | ^ 0 cos_1(~™'+w) J cos -1(-m-n) ~ { s/[l - ( - m + n)2] - ^[1 - (m + »)2] | At a minimum value of e', d e /dp must be zero. But there are only two ways in which this can occur, either m = 0, or n = 0. In the latter case it can be shown that de /dv = 1 ; hence the only way to minimise e' is to make m = 0. With this value of m, by partial differentiation of (18) we have 5-1' 1 dt 0 J cos-1 n J cos-i(-?i) 0e dv 7 r df_2 dV 7T cos 1 n ,.cos k - n) it ’ = l{ (ldt - fldt + fldt i ’ IT * H " nr\a 1 nn * r»r\c 1/ m \ ' | cos 1 n - cos 1( - n) 4- ^ | Now 0 and therefore when both de/dp = 0 and de/dv = 0, we have m = 0 .... (20) -1 7T cos = 4 (21) 1905-6.] A New Form of Harmonic Synthetiser. 229 In any actual instrument the dimensions are such that powers of r/Z greater than the second may be neglected, hence we have from (20) and (21) with (16) (and 19) This completes the determination of the comparison S.H., and, precisely as in the former case, it may be shown that it is a curve having its time axis depressed by a distance - r2/(4Z) below the time axis of the actual curve. The displacement that should accordingly be given to the time axis when drawing in the latter on the paper has already been explained. The interpretation of equation (22) is this. To make the average error as small as possible, the curve drawn by any one of the wheels of the instrument, the amplitude of the eccentric pin of which, according to the present theory, is r, ought to be regarded as a slightly inaccurate representation of a true S.H. curve, the amplitude of wdiich is not r, but r^l — The value of the average deviation can now be found by putting the values of p and v into equation (17) ; or, what comes to the same thing, by putting the values for m and n given by (20) and (21) into equation (18). There results, TT TT TT
or Am = 2 Jbxm + by . . . (4). The significance of maximum specific conductivity is reserved for discussion in connection with the experimental investigations relating to this part of the subject. Meantime it may be stated that whenever an electrolyte is sufficiently soluble to give a solution of maximum specific conductivity, the maximum falls 1905-6.] Prof. Gibson on Aqueous Solutions of Electrolytes. 237 within the range of concentration for which the linear relationship holds. It is important to notice that as y = 2L and = \s, the adoption of unit of mass instead of unit volume does not affect the numerical statement of the relationships which have been established for dilute solutions; for when s=l, as is practically the case in dilute solutions, y coincides with rj and with A. To prevent confusion, the old units, viz., rj, m, A will be referred to as volume units ; and the new units, y, T, and A^, as mass units. (Issued separately August 29, 1906.) 238 Proceedings of Royal Society of Edinburgh. [sess. Recherches sur la G-lauconie. Par les Drs Leon W. Collet et Gabriel W. Lee, assistants de Sir John Murray, K.C.B. Communique par Sir John Murray. (Avec 12 planches et 1 carte.) (MS. received May 28, 1906. Read June 4, 1906.) I. Introduction. Depuis Murray et Renard deux travaux importants sont venus cnrichir la bibliographie de la Glauconie : le memoire de MM. Calderon et Chaves, “ Contribuciones al estudio de la Glauconita,” et l’oeuvre magistrate de M. L. Cayeux, Etude micrographique des terrains sedimentaires. Murray et Renard avaient etudie specialement la Glauconie actuelle ; MM. Calderon et Chaves baserent leur synthese de la Glauconie actuelle sur une analyse de Glauconie des roches sedimentaires ; M. L. Cayeux limita strictement ses belles recherches a la Glauconie des roches sedimentaires et n’attaqua pas le probleme de sa genese. Apres avoir termine notre etude des “ Concretions phosphatees de 1’ Agulhas Bank,5’ nous fumes invites par Sir John Murray a entreprendre une sorte de “ mise au point ” de la question de la Glauconie. En effet, Giimbel en 1886, puis Murray et Renard en 1891, virent dans la Glauconie un silicate ferri-potassique ; pour MM. Calderon et Chaves la Glauconie, au contraire, est un silicate ferropotassique. Qui avait raison ? De plus M. L. Cayeux signala des caracteres tres speciaux dans la Glauconie de ses roches sedimentaires, caracteres qui etaient inconnus dans la Glauconie actuelle, une etude comparative se justifiait done. Comment remercier Sir John Murray, qui nous chargea de cette etude que lui seul pouvait nous permettre d5 entreprendre, grace a sa vaste erudition et aux collections qu’il accumule au Challenger Office depuis si longtemps. Les nombreuses heures passees en contact avec un tel maitre, le createur de l’Oceanographie, seront pour nous un souvenir indelebile. Rous devons egalement de sinceres remerciements a M. le 1905-6.] Recherches sur la Glauconie. 239 Prof. L. Cayeux, de Paris, qui nous communiqua tres aimablement des coupes de Glauconie pour les comparer aux notres, et. qui s’interessa d’une fa^on tres speciale a nos recherches. MM. Arnold Heim, du laboratoire de Geologie de l’Universite de Zurich ; Chs. Jacob, du laboratoire de l’Universite de Grenoble ; Buxtorf, du Musee de Bale • Teall, directeur du H.M. Geological Survey a Londres ; H. B. Woodward, a Londres, ont aimablement repondu a nos demandes d’echantillons, qu’ils reQoivent ici l’expression de notre gratitude. M. Georges West, du “Lake Survey,’’ a bien voulu se charger des microphotographies, qu’il revive ici nos sinceres remerciements. Dans le present travail, M. le Dr G. W. Lee s’est specialement occupe de la question purement mineralogique, M. le Dr L W. Collet de la question geologique. La partie chimique ainsi que les conclusions ont ete faites conjointement. II. Glauconie en tant que Mineral. Habitus. — On sait depuis longtemps que la Glauconie se presente sous trois formes differentes : comme produit de remplissage de coquilles de foraminiferes et comme grains prenant part a la formation, des sables verts et autres roches sedimentaires glau- conieuses. La troisieme forme est la Glauconie dite pigmentaire qui impregne la roche, tel un enduit vert du a la presence d’une multitude de paillettes submicroscopiques. Comme nous le verrons plus tard, ces deux dernieres categories etant derivees sans aucun doute de la premiere, nous pouvons les grouper sous le nom de Glauconie second air e. 1. “ Moules glauconitiques .” — C’est l’etude des “ glauconitic casts ” des naturalistes du Challenger qui doit nous donner la clef du probleme de la genese de la Glauconie. L’interpretation de cette forme affectee par la Glauconie sera l’objet d’un chapitre special, et nous nous limiterons pour le moment a une courte description des moules memes. Les moules typiques, dont le diametre est en general infchieur a 1 mm., sont vert fonce, et quelquefois pourvus d’un noyau jaune ou brun. La partie verte a toutes les proprietes de la Glauconie telle que les mineralogistes la comprennent : l’indice 240 Proceedings of Royal Society of Edinburgh [sess. de refraction, la birefringence et la structure cryptocristalline, sont les memes que cbez la Glauconie des gres verts, etc. D’autre part le noyau brun est opaque et amorphe. Nous attirons l’attention du lecteur sur le fait que jusqu’a present il n’a point ete rencontre de moules a centre vert et Peripherie brune, cela bien entendu tant que la coquille du foraminifere est conservee. La presence d’un noyau brun entoure d’une zone de Glauconie parfaite ne peut s’expliquer que par un processus chimique sur lequel nous reviendrons au chapitre traitant de la genese de la Glauconie. Glauconie en grains. — La plus grande partie de la Glauconie mentionnee ou . etudiee par les geologues appartient a cette categorie ; elle est le constituant le plus caracteristique des gres verts des assises geologiques et des Sables et Boues Verts des mers actuelles, et il est a peu pres hors de doute qu’elle est due au remaniement des moules glauconitiques. Comme ce chapitre n’est qu’une entree en matiere, il est inutile que nous revenions sur la description de ces grains, dont on trouvera la description detaillee, accompagnee de figures, dans les travaux de Murray et Renard, et de M. Cayeux. Glauconie pigmentaire. — La teinte verte de nombreux echan- tillons lithologiques sedimentaires et des mers actuelles, tels que concretions phosphatees, etc., est due a la presence de Glauconie a l’etat finement divise, appelee par M. Cayeux Glauconie pigmentaire ; elle est due ou bien a une precipitation chimique, ou bien au depot tranquille de Glauconie preexistante amenee par trituration a l’etat pulverulant. Dans certains cas elle peut remplir des coquilles de foraminiferes, donnant ainsi naissance a des moules dont Torigine est facile a mettre en lumiere lorsqu’on peut voir la Glauconie incluse se propager sans interruption dans le ciment englobant les coquilles de foraminiferes. Les moules ainsi formes sont, par opposition a ceux cites plus haut, de faux moules, et n’ont pas d’importance au point de vue de la genese de la Glauconie. Pro'prietes physiques. Poids specifique. — Selon M. Lacroix, la valeur de la densite de la Glauconie oscille entre les chiffres de 2'2 a 2*3 ; M. Rosen busch donne le chiffre de 2*30. Pour M. Spurr elle 1905-6.] Becherches sur la Glauconie. 241 depasserait souvent 3, et atteindrait dans nil echantillon qu’il cite la valeur 3 *634. Nous devons dire que, bien que la Glauconie se montre souvent capricieuse dans ses proprietes, la densite est une constante physique troy importante pour qu'il soit permis d’admettre une telle variation. Un mineral dont la densite est 3-6 ne peut etre de la Glauconie, meme si l'on considerait la Glauconie comme une famille ou groupe de mineraux, et non pas comme une espece ; et meme dans les limites de groupes tels que ceux des Amphiboles ou des Pyroxenes, oil la nature des bases varie encore plus que ce n’est le cas chez la Glauconie, on ne trouve pas une pareille variation dans la valeur de la densite. Ce mineral est si difficile a separer du quartz au moyen des liquides lourds, que les chiffres donnes par M. Lacroix et M. Rosenbusch sont sans aucun doute les vrais. Durete. — La Glauconie est assez tendre pour etre ecrasee sous l’ongle ; ce pen de cohesion est un argument en faveur de la genese purement mecanique de la Glauconie pigmentaire (les cas d’epigenie dont nous parlerons plus loin devant etre attribues a un depot par solution), et explique la forme generalement arrondie des grains. Structure. — La Glauconie ne se presente que rarement sous forme de cristaux proprement dits ; chaque grain ou moule consiste en un aggregat de cristaux lamellaires microscopiques dont le diametre ne depasse en moyenne guere 0*003 mm., et dont les axes d’elasticite n’ont pas une orientation commune de sorte que le grain ou moule, quoique d’apparence homogene en lumiere ordinaire, offre des plages diversement illuminees entre nicols croises. M. le Prof. Cayeux a decrit une variete bien cristallisee de Glauconie, dont il a eu l’amabilite de nous envoyer quelques echantillons aux fins de les comparer a d’autres de la collection de Sir John Murray, et provenant du cretace d’lrlande, de Folkestone et du nummulitique du Santis en Suisse. Cette variete est fortement polychroique, ng. vert, np. jaune pale, et s’eteint a 0°. De toutes les sections etudiees, celle qui etait le plus pres d’etre parallele a ng.-np. avait une birefringence voisine de la valeur 0*030, birefringence notablement plus elevee que celle des particules 6lementaires des grains a structure cryptocristalline. Ces cristaux montrent en outre un clivage PROC. ROY. SOC. EDIN. — YOL. XXYI. 16 242 Proceedings of Royal Poddy of Edinburgh. [sess. plus ou moms parfait, parallele a l’axe ng. Cette variete est tres rare et n’a pas encore ete rencontree en voie de formation dans les mers actuelles, de sorte que nous sommes portes a Penvisager comme etant due a une action metamorphique. Proprietes optiques. La couleur de la Glauconie pure et non attaquee est verte, vert-olive, et, dans le cas du Gault de Saxonnet en Savoie, d’un beau vert presque bleu. L’intensite de la couleur varie passable- ment, et semble diminuer avec l’age du mineral : des echantillons de Glauconie du Cambrien et du Carbonifere d’Angleterre sont vert-d’eau extremement pale, presque incolore ; cela est evidemment du au metamorphisme, car ici la couleur pale n'est pas due au melange mecanique d’une grande quantite d’argile claire, comme cela est le cas pour les moules glauconitiques vert-pale recueillis par le Challenge r. Polychrdisme. — Nous avons deja parle de cette propriete caracteristique de la variety bien cristallisee ; elle existe aussi chez la Glauconie ordinaire, mais il faut pour l’observer que les particules elementaires aient une taille suffisamment grande pour que Ton puisse isoler l’une d’elles au moyen du diaphragme oculaire, avec le concours d’un fort grossissement. Nous avons observe que lorsque les particules elementaires ont un diametre inferieur a O’OOS mm., elles sont trop enchevetrees les unes dans les autres pour qu’il soit possible d’observer la moindre trace de polychroisme. Ce dernier est le meme que pour la variete bien cristallisee c.a.d. ng. vert, np. jaune pale. A notre connaissance cette propriete n’avait pas encore ete remarquee dans la Glauconie ordinaire des mers actuelles. Birefringence. — Meme avec les plus forts grossissements il n’est pas possible d’agrandir suffisamment les particules elementaires pour y observer les figures d’interference en lumiere convergente. On ne peut done dire avec certitude si les particules qui polarisent le plus haut dans l’^cbelle de Newton appartiennent a la section principale ng.-np. ou a une section d’orientation quelconque ; cependant comme dans le nombre immense de particules formant les grains ou les moules nous n’en avons jamais rencontres dont la 1905-6.] Recherches sur la Glauconie. 243 birefringence depasse le chiffre 0'020, ce chiffre doit etre celui de la birefringence maxima, car il serait inadmissible qu’aucune de ces myriades de particules lie fut parallele a la section ng.-np. La Glauconie ordinaire differe done en outre de la Glauconie en grands cristaux par sa birefringence notablement moins elevee ; cette derniere s’abaisse meme d’une maniere frappante dans les echantillons du Cambrien et du Carbonifere que nous avons etudies, tout comme la couleur propre du mineral, la meme action meta- morphosante qui fait disparaitre la couleur abaissant la birefringence. Systeme cristallin.— M. Lacroix eonsidere la Glauconie comme monoclinique, et la place dans le groupe des chlorites ; Tangle des axes varie de 30° a 40°, et peut meme s’approclier de 0° ; sauf erreur, ces donnees ont ete etudiees sur la variete en grands cristaux, et nous les avons retrouvees dans les echantillons ayant servi a notre etude : une section de Glauconie du cretace d’lrlande est presque rigoureusement uniaxe, de signe optique negatif, la figure d’interference etant de toute nettete, tandis que d’autre part la meme variete cristallisee, provenant du Santis, est franchement biaxe ; mais les sections observees n’etant pas rigoureusement perpendiculaires a la bisectrice aigiie, nous n’avons pu proceder a une mesure d’angle. Quant a la Glauconie ordinaire on n’a pas les donnees necessaires pour etablir d’une maniere absolue le systeme cristallin auquel elle appartient, mais il est tout-a-fait legitime de la placer comme l’autre variete, dans le groupe des chlorites, et de la considerer •comme monoclinique. Relations entre la Glauconie et ses diverses manieres d'etre. Les sondages entrepris durant ces trente dernieres annees ont montre que la Glauconie existe le plus souvent sous forme de grains formant un des elements constituants des boues et sables verts ; 4es moules sont moins frequents, et la Glauconie pigmentaire n’est bien observable que dans le cas de depots consolides. Nous avons mene de front l’etude de la Glauconie des terrains sedimentaires et ■ celle des mers actuelles, de fagon a avoir une bonne base de comparaison ; nous avons ainsi trouve que, en regie generale, la differenciation est poussee plus loin dans la Glauconie sedimentaire • que dans la moderne. 244 Proceedings of Royal Society of Edinburgh. [sess. Differ enciation cliez les grains glauconitiques modernes. — Une observation digne de remarque est que les grains cimentes montrent une differenciation beaucoup plus avancee que les grains libres des sables et boues, aussi l’ample moisson de concretions phosphatees recueillie par les bateaux du “ Department of Agriculture of the Cape of Good Hope ” sur 1’ Agulhas Bank nous a-t-elle permis de faire de nombreuses observations sur ce sujet, car il ne faut pas oublier que les concretions phosphatees sont, avec les nodules de manganese, les principaux depots marins trouves tout formes a l’etat compact. Les phenomenes de differenciations observes sont: 1°. Une difference dans l’intensite de la couleur au sein d’un seul et meme grain. Les grains de Glauconie de certaines concretions phosphatees de l’Agulhas Bank ont leur partie mediane d’un vert tres pale, tandis que la partie externe a la couleur vert fonce propre au mineral ; le passage de la teinte claire a la teinte foncee se fait graduellement, sans aucune transition brusque. En general c’est le contraire qui est le plus frequent : le centre est plus fonce que la peripherie, avec, comme dans le cas precedent, passage graduel. Souvent aussi le grain est entoure d’une tres mince gaine, visible aux forts grossissements seulement, et que l’examen entre nicols croises montre etre isotrope ; a cette game succede quelque- fois une nouvelle zone, de couleur verte, plus homogene et moins cryptocristalline que le noyau, et pourvue de clivages radiaux. Entre nicols croises cette zone se comporte a la fagon des spherolites en donnant le phenomene de la croix noire. Les fibres formant cette couche externe ont uti allongement positif, et leur birefringence est de O'Ol 1 ; en outre, comme la forme de cette couche est quelconque, on ne peut attribuer cette derniere a la pseudomorphose de la calcite de coquilles de foraminiferes. II peut etre interessant de noter que son indice de refraction est inferieur -a celui de la Glauconie du noyau interne. C’est aussi a un phenomene de differenciation qu’il nous semble devoir attribuer la formation, dans certains grains, de plages dues a la soudure de particules ayant la meme orientation ; ces plages cessent d’etre cryptoeristallines et ont une teinte de 1905-6.] Becherches sur la Glauconie. 245 polarisation uniforme, bien que souvent elles aient des extinctions roulantes prouvant que leur homogeneite n’est pas parfaite. Dans les grains de Glauconie sedimentaire on retrouve tous les exemples de differenciation cites ci-dessus avec en plus la variete bien cristallisee et clivee dont nous avons deja parle, et qui n’existe pas chez la Glauconie moderne. Quant aux moules glauconitiques proprement dits, ils sont presque completement exempts de ces phenomenes, et les exemples observes peuvent se rapporter a deux cas : l’un du a une difference dans la cristallisation, l’autre a la presence de mineraux etrangers. Le premier consiste en la presence, dans quelques rares cas seulement, d’un anneau de forme tres reguliere situe a l’interieur du moule, et du a l’arrangement de particules ayant la meme birefringence, de sorte qu’entre nicols croises on voit au centre du moule un anneau illumine d’une teinte vive et uniforme. Ce cas a ete observe sur des moules d’un sable vert recueilli par le Challenger a la station 164, pres de Sydney. Dans le materiel de cette meme station nous avons rencontre des moules renfermant des taches foncees et opaques, que leurs contours montrent etre dus a la presence de mineraux englobes dans le remplissage glauconitique, et devenus meconnaissables par suite d’une action chimique leur ayant fait subir une sorte de digestion plus ou moins complete (v. PI. I.). Ce point est interessant a noter, en ce qu’il indique la presence d’un agent chimique puissant, qui pourrait jouer un role dans la genese des moules glauconitiques. Ce que nous avons dit dans le chapitre precedent nous amene a parler ici de moules non exclusivement glauconitiques, et que nous pourrions appeler moules imjparfaits , et qui marquent un acheminement vers la formation des moules glauconitiques. Les depouilles de foraminiferes deposees sur le fond de la mer sont toutes designees a etre tot ou tard remplies par la matiere minerale tenue en suspension par l’eau, de sorte qu’une coupe microscopique, faite dans un sable vert ou une boue a globigerines prise a une profondeur ne depassant pas la limite d’environ deux mille metres, ofFrira tous les termes de passage entre des moules simplement argileux, et des vrais moules de Glauconie. 246 Proceedings of Royal Society of Edinburgh. [sess. Ainsi dans une meme preparation on pourra voir que certain es coquilles de foraminiferes ne sont qu’en partie rem plies par une matiere argileuse gris-clair, d’autres sont completement reraplies par cette meme matiere; continuant cet examen, toujours dans la meme preparation, on verra que certains de ces moules, qui sont ce que Sir John Murray a appele “ imperfect casts,” sont teintes de brun par l’adjonction d’une certain e quantite de fer, et, la pro- portion de fer augmentant, on trouvera, comme terme extreme de la serie, des moules brun-fonce, completement opaques. Ces derniers sont facilement attaquables par les acides sulfurique et chlorhydrique, lentement a froid et rapidement a chaud, en laissant un residu de silice ; done, si les moules bruns etaient formes d’un melange mecanique d’argile et de limonite, les acides, attaquant plus rapidement la limonite que l’argile, laisseraient cette derniere sous forme d’un residu facile a reconnaitre. C’est pourquoi il nous semble permis de considerer les moules bruns comme essentiellement formes d’un silicate de fer, tandis que les moules gris sont essentiellement alumineux. Le microscope permet d’aller plus loin encore dans l’etude des differentes phases de l’elaboration d’un moule de Glauconie : le meme preparation supposee ci-dessus contient certains moules bruns dont la peripherie montre une manifeste transformation en Glauconie, c.d.d. qu’au lieu d’etre brune, amorphe et opaque, elle devient verte , cristalline, et translucide en lame mince. On peut ainsi faire defiler sous l’objectif toute une gamme de moules dont la partie verte l’emporte de plus en plus sur la partie brune, et il est evident que la premiere est plus recente que la derniere, car il serait absurde d’attribuer le noyau brun de ces moules a un phenomene de decomposition, cette derniere partant toujours de la peripherie. La transformation se fait tres graduellement, et il n’y a pas de transition brusque entre les parties brunes et vertes ; la limite comprise entre ces dernieres occupe une surface reguliere, de sorte que la metamorphose ne se fait pas irregulierement le long de prolongements ou diverticules, sauf dans les cas ou la nature de la coquille ou son etat de conservation auront permis ii l’agent de transformation d’operer selon une surface irreguliere. Yu l’importance que nous avons accordee a ce sujet, le lecteur pensera peut-etre que les moules glauconitiques forment une part 1905-6.] Beclierclies sur la Glauconie. 247 considerable des depbts marins ; il n’en est cependant rien, ce qui pourrait etre nne objection a l’hypothese que les grains de Glauconie modernes et des assises geologiques ont tous passe par le stade de moules glauconitiques. A cela nous pouvons repondre comme suit : Le processus de la formation de la Glauconie est evidemment tres lent, et si le nombre des moules glauconitiques est infini par rapport a celui des grains, le temps qui s’ecoule est suffisant pour permettre a un nombre donne de moules de se transformer mecaniquement en grains, en meme temps que de nouveaux moules se reforment, et ainsi de suite. Des lors, cette hypothese n’a rien que de tres admissible, pour peu que Ton songe que si le processus n’agit pas par grands efforts et par grandes masses, il se repartit sans interruption durant des temps infinis. Nousrenvoyons au chapitre traitant de la genese de la Glauconie les diverses fagons possibles d’interpreter le mecanisme selon lequel la silice, le fer, la potasse et l’eau se combinent pour former la Glauconie, et aborderons maintenant l’etude de la troisieme maniere-d’etre de ce mineral, la Glauconie pigmentaire. C’est de nouveau dans les depdts compacts que Ton trouve de bons exemples de Glauconie pigmentaire, se pretant a l’etude microscopique des diverses fagons dont elle se presente. Il est tres probable qu’elle existe aussi au sein des depdts meubles, mais etant dans ce cas a l’etat de boue impalpable noyee dans le reste du sediment, son manque d’individualite empecherait que Ton puisse en faire une etude utile. Ce que M. Cayeux a dit pour la Glauconie pigmentaire des terrains sedimentaires, nous l’avons retrouve dans celle des con- cretions pbosphatees de l’Agulhas Bank, et nos dernieres recherches nous permettent de completer ce que nous avons ecrit a ce sujet dans notre precedent travail (11). La Glauconie pigmentaire, qui est toujours melangee a de la calcite trituree ou a du phosphate de cliaux, est ou bien uniformement repandue dans la roche, ou bien plus ou moins individualisee sous forme de taches, ou enfin peut remplir certaines cavites, telles que des loges de foraminiferes. Dans le premier cas elle ne fait que communiquer une teinte verte au ciment de la concretion, et ce dernier est en proportion trop 248 Proceedings of Royal Society of Edinburgh. [sess. considerable pour que la Glauconie qui y est diffusee puisse reagir sur la lumiere polarisee. Quand elle se presente sous forme de taches le centre est quelquefois suffisamment riche en Glauconie pure pour pouvoir reagir sur la lumiere polarisee, de sorte qu’entre nicols croises on voit des couleurs de polarisation qui diminuent d’intensite du centre a la peripherie, ou le ciment l’emporte sur la Glauconie comme dans le premier cas. Nous attirons l’attention sur ce fait qui prouve l’identite de la Glauconie pigmentaire et de la Glauconie ordinaire, bien que la premiere ne puisse etre directe- ment etudiee pour les raisons deja indiquees. Si nous repetons ce que nous avons deja dit a propos de la Glauconie pigmentaire remplissant des coquilles de foraminiferes, c’est pour mettre en garde l’observateur, qui pourrait confondre ces faux moules avec les veritables moules glauconitiques. La confusion n’est pas possible si Ton opere entre nicols croises avec un bon grossissement, car on voit alors les particules de Glauconie noyees dans le ciment, qui est isotrope s’il est argileux ou phos- phatique, et montre les teintes de polarisation irisees de la calcite, s’il est dd a la trituration de cette derniere. Enfin, en regie generale, la couleur de ces faux moules n’est pas aussi foncee que celles des veritables, et en plaques minces leur relief n’est pas aussi considerable. Avant de trancher en faveur de l’une ou de l’autre des hypotheses touchant l’origine de cette maniere d’etre de la Glauconie, nous voulons parler ici de la Glauconie ejyigenique , avec laquelle elle pourrait avoir certains rapports. Comme la Glauconie epigenique a eite etudiee tres en detail par M. Cayeux (8) dans certaines roches sedimentaires, il est interessant de signaler le fait que divers cas d’epigenie de ce mineral se produisent de nos jours au sein des mers. Ce phenomene affecte 1° des elements mineraux organises, tels que la calcite de coquilles de Foraminiferes, et 2° des mineraux detritiques, tels que le quartz et les feldspaths. Le centre de la figure PL III. est occupe par un moule glauconitique de Globigerine dont la coquille, quoique fraiche et bien conservee, est impregnee de mouchetages (paraissant noirs sur la photographie) de Glauconie parfaitement typique. La figure PL YI. represente un cas plus complexe : la masse generale de la preparation est teintee en vert- 1905-6.] Reclierches sur la G-lauconie. 249 clair par de la Glauconie pigmentaire, tandis que la section de coquille au milieu a sa calcite completement remplacee par de la Glauconie epigenique vert-fonce. Enfin, la figure PI. VIII. donne un exemple d’epigenie sur un grain de quartz ; les mouche- tages et les barres noires bien visibles sur cette photographie sont dus a la presence de Glauconie ne se distinguant en rien de la Glauconie ordinaire, et ici l’epigenie ressemble absolument a ce que l’on observe dans le cas des roches sedimentaires, bien que chez ces dernieres le phenomene se soit souvent produit apres la con- solidation de la roche, comme l’a inontre M. Cayeux. Maintenant, la Glauconie pigmentaire et la Glauconie epigenique sont-elles dues au depot du produit de la trituration de Glauconie preexistante ? Ou bien sont-elles dues a un depot de Glauconie tenue en solution par un agent quelconque? Ou enfin faut-il chercher l’explication dans la metamorphose d’un silicate de fer brun, comme dans le cas des moules bruns se changeant en moules verts ? La premiere de ces hypotheses ne peut s’appliquer qu’a la Glauconie pigmentaire, car on ne pourrait s’expliquer le remplace- ment epigenique par Taction mutuelle de deux corps a l’etat solide, c.a.d. de Glauconie toute formee d’une part, et d’une coquille ou d’un mineral, selon les cas, d’autre part. L’intervention d’un dissolvant permet de trouver une explication assez plausible aux phenomenes d’epigenie, mais ne pourrait guere s’expliquer dans le cas de la Glauconie pigmentaire. En effet, dans le premier cas, on comprend assez bien que la Glauconie tenue en solution arrive a s’infiltrer dans les pores les plus intimes d’une coquille ou d’un mineral, et arrive meme a les remplacer complete- ment par pseudomorphose, dans le cas oil ce meme dissolvant aurait dissous du meme coup les mineraux epigenises, tout en deposant a leur place la Glauconie qu’il tenait en solution. On congoit que d’autre part l’explication ci-dessus convienne moins bien au cas de la Glauconie pigmentaire, car il semble qu’un precipite doive se deposer en couches homogenes et compactes, et non pas sous forme de paillettes noyees dans un ciment ; en outre les formes bizarres qu’affectent souvent les trainees de Glauconie pigmentaire evoquent a l’esprit la notion d’une matiere pulverulente que de petits remous ou tourbillons forcent k se deposer dans des endroits determines. La troisieme hypothese nous a ete suggeree par quelques 250 Proceedings of Royal Society of Edinburgh. [sess. observations, dont un exemple est donne par la figure PL IX. Le centre de la figure est occupe par la section d’une coquille dont la calcite est completement remplacee par une matiere ferru- gineuse brun-fonce. Le remplacement est ici indubitable, mais a l’encontre de ce qui se passe chez les moules bruns et verts, on ne peut dire si la matiere brune est posterieure ou anterieure a la Glauconie, cette derniere epigenisant des fragments de coquilles poreuses concurremment avec la matiere brune, mais seulement selon une surface plane, ce qui ne permet done pas de trancher la question (nous avons observe plusieurs exemples de ce phenomene sur des echantillons de nodules phosphates jaunes de la station 7, Agulhas Bank, par 238 m. de profondeur). Formation des grains de Glauconie. — Si nous plagons ici l’etude de cette question, e’est que nous pourrons avoir a recourir a ce qui a ete dit dans les chapitres precedents pour arriver aux hypotheses les plus vraisemblables. C’est un fait bien connu que la Glauconie se presente le plus sou vent sous forme de grains arrondis, generalement ovoi'des. Cette forme reguliere a toujours intrigue les geologues, et nom- breuses sont les hypotheses auxquelles ils out eu recours pour l’expliquer, ainsi que la taille meme des grains, qui depasse souvent de beaucoup celle qu’auraient les produits du remaniement pur et simple de moules glauconitiques. Les di verses hypotheses exprimees au sujet de la genese des grains de Glauconie peuvent grosso modo se ramener a deux : une ecole, avec Murray et Eenard en tete, admettent que les grains derivent des moules, apres brisure de la coquille et accroissement subsequent par apport de matiere glauconieuse ; l’autre ecole, representee surtout par M. Giimbel, attribue aux grains une genese toute differente et independante de la matiere organisee. Nous renvoyons pour plus de details au travail original de M. Giimbel, et nous contenterons de dire ici que si les grains etaient dus au depot de Glauconie sur l’enveloppe de bulles gazeuses, avec remplissage subsequent, par intussusception, de la coque ainsi formee, il n’y aurait pas de raisons pour que la Glauconie seule aie une telle attraction pour les bulles de gaz, et dans ce cas il n’y aurait rien d’etonnant a voir maintes especes minerales, telles que la pyrite, le manganese, la phillipsite, etc., obeir a la meme loi. 1905-6.] Becherches sur la Glauconie. 251 Sans apporter une preuve decisive a l’appui de la premiere maniere de voir, nous croyons cependant devoir nous y ranger en modifiant. legerement la maniere dont Murray et Renard expliquent le- processus. Comme eux, il nous senible qu’il est parfaitement logique d’admettre que les grains derivent originellement de moules, mais sans toutefois nier qu’il soit possible que le moule continue a croitre, pour ainsi dire, en faisant eclater sa coquille et en perdant ainsi sa forme primitive, nous pensons qu’il nous sera permis d’offrir une explication un peu difierente, qui nous a ete suggeree par le grand role que paraissent jouer la Glauconie pigmentaire et la Glauconie epigenique. En effet, supposons que des moules aient perdu leur coquille et se trouvent places dans un endroit oil l’une des variet^s de Glauconie, pigmentaire ou epigenique, soit justement en train de se deposer ou de s’elaborer. Or, puisque cette Glauconie peut souder entre elles des particules de calcite et impregner les fissures de fragments de quartz, etc., pourquoi ne pourrait-elle souder entre eux des fragments de moules h Cela suppose, le roulement et le frottement avec les autres corps solides deposes au fond de la mer donneront a cet agglomerat la forme arrondie et l’aspect luisant si caracteristiques des grains de Glauconie. Nous voyons de suite les objections que l’on pourrait faire a cette maniere de voir, objections du reste deja formulees par M. Giimbel ( Ueber die Natur . . . etc., p. 435): l’examen d’une preparation contenant des grains de Glauconie, ne revele au microscope la presence ni du ciment ni de lignes de suture. Avant d’y repondre nous prions le lecteur de ne pas perdre de vue la plus caracteristique de toutes les proprietes physiques de la Glauconie : c.a.d. que cette derniere est cryptocristalline. Cela pose, supposons de nouveau deux ou plusieurs fragments de moules ; chacun de ces fragments est compose d’une multitude de particules diversement orientees. Maintenant, le ciment glauconieux qui vient souder les fragments de moules, est lui aussi forme de particules diversement orientees, de sorte que cette espece d’hetero- geneite dans la structure intime, des fragments comme du ciment, doit done masquer toute ligne de suture. Associations minerales.—Ce sujet etant important dans l’histoire de la Glauconie, nous comptons en traiter ici, bien que Murray et;. 252 Proceedings of Royal Society of Edinburgh. [sess. Renard d’une part, et M. Cayeux de l’autre, lui aient deja consacre plusieurs pages. Les mineraux qui accompagnent la Glanconie dans ses divers gisements peuvent etre ou bien detritiques ou bien secondaires, e.a.d. s’etre formes in situ au fond de la mer. Des mineraux detritiques le quartz est celui qui joue de beaucoup le plus grand role ; il va de soit qu’il se rencontre en toutes pro- portions, et dans le cas des sables verts il est le principal constituant de ce sediment. Il se presente en grains dont le diametre varie d’une fraction de centieme de mm. a 1 et meme 2 mm., avec une moyenne assez constants de 0*3 mm. Ce n’est que rarement qu’ils sont roules et arrondis ; en general leurs aretes sont tranchantes ou seulement emoussees. Ils sont le plus souvent libres, et dans le cas de concretions phosphatees, isoles au milieu du ciment ; nous n’avons observe qu’exceptionnellement des grains de quartz reunis entre eux au moyen d’un ciment siliceux : ils prennent alors en lumiere parallele l’apparence de larges plages qui se resolvent entre nicols croises en une mosaique a elements diversement orientes. Les autres mineraux detritiques jouent un role tout-a-fait subordonne ; le Zircon et la Tourmaline sont pour ainsi dire toujours presents, mais au nombre de quelques grains seulement par coupe. Les Feldspaths, en cristaux plus petits que les grains de quartz, appartiennent, comme nous l’avons fait remarquer dans un travail precedent, aux Plagioclases, et, chose digne de remarque, leur composition ne s’ecarte en general guere de celle d’un Labrador plus ou moins basique. La parfait etat de fraicheur de ces cristaux prouve qu’ils sont remarquablement refractaires a l’action des agents chimiques de l’eau du fond des mers, ce qui est juste le contraire dans le cas des feldspaths potassiques : ces derniers sembleraient done avoir ete completement dissous avant d’avoir pu arriver a la zone de sediments ou la Glauconie se forme. En effet, dans toutes les coupes 4tudiees nous n’avons observes qu’un ou deux cristaux pouvant se rapporter a l’Orthose, et encore ces derniers etaient-ils completement kaolinises. Le Mica aussi est fortement decompose ; il perd toute birefringence et prend un aspect terreux, et ne se reconnait qu’a sa forme. 1905-6.] Recherches sur la Glauconie. 253 Depuis Murray et Renard les geologues sont a peu pres tous d’accord pour admettre que les Feldspaths et les Micas potassiques fournissent, en se decomposant, la potasse necessaire a la formation de la Glauconie; or, ces mineraux terrigenes ne pouvant etre transports jusqu’a la zone de la “Red Clay,” on comprend alors que ces regions abyssales soient pauvres en potasse, et partant depourvues de Glauconie. Outre les especes minerales enumerees plus haut, on en rencontre d’autres telles que Rutile et Hornblende, mais en proportion mininie, et variant avec la nature des cotes voisines du gisement glauconieux. Les matieres minerales secondaires associees a la Glauconie sont surtout representees par le Carbonate et le Phosphate de Chaux. Les fragments et particules de Calcite sont toujours dus a la destruction d’organismes calcaires ; cela est indubitable dans le cas des gros fragments, oil la structure organique est souvent reconnaissable, et quant aux particules, qui sont un des constituents des boues d’une part, et du ciment de certaines con- cretions d’autre part, leur origine organique ne peut non plus faire l’object d’aucun doute. En effet, du Carbonate de Chaux precipite chimiquement par des actions lentes prendrait une forme cristalline reguliere, tandis que ces paillettes, examinees aux forts grossissements, montrent des formes irregulieres et dechiquetees. Sans vouloir revenir ici sur la question du Phosphate de Chaux, nous tenons cependant a appuyer sur ce fait qu’il n’est jamais individualise sous forme de cristaux ; avec ou sans le concours de particules de Calcite, il sert de ciment aux mineraux detritiques et a la Glauconie englobes dans les concretions phosphatees. Enfin, nous pouvons placer Ja Pyrite parmi les mineraux secondaires accompagnant la Glauconie, mais comme elle s’y trouve frequemment a l’etat d’inclusions, nous developperons cette question dans le chapitre suivant. Inclusions minerales. — Tous les mineraux caracteristiques du sediment ou la Glauconie a ete recueillie peuvent se trouver au milieu de grains ou de moules de cette derniere a l’etat d’inclusions. Chez les moules les inclusions sont plutot rares et naturellement ■254 Proceedings of Royal Society of Edinburgh. [sess. (Tune taille ne depassant pas le diametre des orifices qui leur ont •donne passage. Au contraire, cliez les grains les inclusions de quartz, calcite, etc. peuvent avoir un volume qui depasse quelque- fois le quart de celui du grain rneme. Nous tenons specialement a attirer l’attention sur ce fait, qui est un argument en faveur de l’explication que nous avons donnee touchant la formation des grains de Glauconie marine. Si, partant de la notion que cette derniere est engenrlree dans les loges des Foraminiferes, l’on trouve des inclusions trop volumineuses pour qu’il leur eut ete possible de penetrer dans ces loges, c’est que ces elements etrangers auront ete englobes en meme temps que les fragments de moules par le ciment glauconieux que nous faisons intervenir dans le cliapitre traitant de la genese des grains. De telles inclusions sont assez frequentes, et si l’on envisage leur origine comme nous venons de le faire, cela n:a rien que de tres naturel, et prouve que la Glauconie peut fonctionner comme un ciment qui se fusionne sans traces de differenciation avec les elements qu’il empate. Ainsi, l’objection 4levee par M. G umbel, c.a.d. que les grains de Glauconie ne montrent rien dans leur structure rappelant un ciment ou des lignes de suture, doit done tomber. Comme il est inutile de redonner une description complete des diverses inclusions minerales, cela ayant deja ete fait par Murray et Renard pour la Glauconie marine, et par M. Cayeux pour la Glauconie sedimentaire, nous nous arreterons seulement a celles que nous con- siderons comme les plus importantes. On sait que la Pyrite et la Magnetite se rencontrent parfois dans les grains et moules de Glauconie ; la premiere est soit en cristaux bien definis, soit sous forme d’une poussiere disseminee dans la Glauconie ; dans ce dernier cas il est difficile de la reconnaitre au moyen de sa couleur en lumiere reflechie, et pour la distinguer de la Magnetite nous avons utilise l’inegale action des acides nitrique et chlorhydrique sur ces deux mineraux. Quant a la Magnetite, nous ne l’avons pas observee sous forme de cristaux ; tous les exemples que nous en connaissons sont en petits amas informes. Cette presence de mineraux essentiellement ferrugineux au sein de la Glauconie pourrait faire croire qu’il y a entre eux un rapport 1905-6.] Reclierches sur la G-lauconie. 255 de consanguinite et que l’un procede de l’autre ; toutefois nous pensons qu’il n’en n’est pas ainsi. En effet, quand la pyrite se presente sous forme de cristaux ces derniers ne presentent aucun exemple de corrosion ou digestion, et la Glauconie qui est en contact immediat avec eux ne differe en rien du reste de la masse, de sorte que si la Glauconie avait emprunte son fer a la Pyrite, clle devrait etre plus foncee, plus birefringente, etc., au voisinage de cette derniere qu’ailleurs. Le meme raisonnement s’applique au cas de la Magnetite. Une chose frappante est que ces inclusions de Pyrite et de Magnetite sont beaucoup plus frequentes chez la Glauconie des terrains sedimentaires que chez la Glauconie actuelle, du moins en ce qui concerne les echantillons, tres nombreux du reste, que nous avons eus entre les mains. II nous semble que dans le cas de la Glauconie sedimentaire il faut admettre avec M. Cayeux que cette derniere, meme au sein d’une roche consolidee, jouit d’une mobilite considerable qui permet l’introduction de ces mineraux. Pour ce qui est de la Glauconie actuelle on a affaire ou bien a de simples inclusions, ou bien, dans le cas ou les cristaux de Pyrite sont bien delimites et ne paraissent pas avoir subi Taction des agents de transport, a une genese simultanee de la Glauconie et de la Pyrite, ce qui confirmerait les vues de Murray et Renard sur le role du soufre dans la production de la Glauconie. Decomposition de la Glauconie. — C’est un fait bien connu que sous certaines conditions mal connues la Glauconie est instable et se transforme en une matiere ochreuse. Le manque de con- naissances regardant la nature de l’eau des boues marines fait que Ton ne sait pas quel est l’agent auquel cette decomposition est due ; quant a la Glauconie des roches sedimentaires, dans certaines localites ou elle se trouve en grande abondance, elle peut se decomposer sur une vaste echelle. En fait, deux monographies ont ete ecrites sur des gites de minerai de fer devant leur origine a cette cause. La premiere est celle de M. K. Glynka (17) sur certains gisements de Russie ; la seconde, due a M. Spurr (40), traite des roches ferriferes de la chaine de Mesabi, Minnesota. Ces deux auteurs sont d’accord pour attribuer aux eaux telluriques Taction decomposante, mais sans specifier quel est Tagent actif 256 Proceedings of Royal Society of Edinburgh. [sess. qu’elles contiennent, bien que d’apres une experience de M. Glynka, de la Glanconie aie perdu, apres avoir ete plusieurs mois en contact avec de l’eau et de la neige : CaO, K20, MgO et Fe203, le produit final etant une argile ferrugineuse (la Glauconie de M. Glynka differant beaucoup par sa composition de la Glauconie marine). Dans les cas etudies par M. Spurr la decomposition va encore plus loin, le produit final etant des couches alternativement formees de silice et de limonite. II semblerait done ressortir de cela que la decomposition est surtout due a la perte de cet element important : la potasse. Une chose qui prouve que les eaux doivent contenir un principe special pour operer cette decomposition, e’est que la Glauconie se comporte a le point de vue d’une fagon qui varie avec les localites ou elle a ete recueillie, cela aussi bien pour la Glauconie marine que pour la Glauconie des terrains sedi- mentaires. Ainsi la Glauconie analysee par nous et recoltee par le U.S.S. Tuscarora par 38° 32' de lat. ]Sr. et 123° 24' de long. W., est d’une fraicheur parfaite, tandis que dans un echantillon de sable vert identique a tous les points de vue, drague par 39° 16' de lat. N. et 124° 43' de long. W., la Glauconie montre de manifestes traces de decomposition, affectant presque tous les grains de l’echantillon. Uous avons effleure ce sujet pour §tre complet, mais, nous tenons a le repeter, on ne pourra en donner une explication satisfaisante que lorsqu’on aura obtenu et analyse l’eau meme qui impregne le sediment. III. Composition chimique. Comme on l’a reconnu depuis longtemps, la Glauconie est un silicate hydrate de fer et de potasse, contenant certaines quantites d’alumine, cliaux, magnesie et soude. De toutes les analyses citees par Giimbel, Dana, Hintze, Lacroix, etc., il n’y en a pas deux qui concordent ; cela provient evidem- ment du fait que la Glauconie ayant une densite variant de 2 '2 a 2*8, il est pour ainsi dire presque impossible d’operer sur un materiel vraiment pur, surtout exempt de Quartz. Dans le tableau ci-contre nous donnons le resultat de 8 analyses recentes qui sont bien loin de concorder entre-elles ; neanmoins elles valent la peine d’etre discutees. 1905-6.] Becherches sur la G-lauconie. 257 No. SiO2 A1203 Fe20» FeO CaO MgO K20 Na20 H20 Total. 1 56-62 12-54 15-63 1-18 1-69 2-49 2-52 0-90 6-84 100 41 2 50-85 8-92 24-40 1-66 1-26 3-13 4-21 0-25 5-55 100-23 3 51-80 8-67 24-21 1-54 1-27 3-04 386 0-25 5-68 100-32 Glauconie des ! mers actu- 4 55-17 8-12 21-59 1-95 1-34 2-83 .i 336 0-27 5-76 100-39 elles. 5 2774 13-02 39-93 1-76 1-19 4-62 0-95 0-62 10-85 100-68 6 46-90 4-06 27-09 3-60 0-20 0-70 6-16 1-28 9-25 99-24 7 40-00 13-00 16-81 10-17 1-97 1-97 8-21 2-16 6-19 100-48 \ 1 Glauconie des i y roches s6di- 8 52-86 7-08 7-20 19-48 tr. 2-90 2-23 tr. 8-43 100-18 \ mentaires. No. 1-5, Analyses du Challenger (34). No. 6, Glauconie de F Agulhas Bank, Giimbel (20). No. 7, Glauconie d’Antrim, Hoskins (23). No. 8, Glauconie de French Creek, Knerr et Schoenfeld in Dana (13). Les six premieres analyses ont ete faites sur de la Glauconie des mers actuelles, les deux dernieres sur de la Glauconie de roches sedimentaires. Le materiel utilise pour les analyses du Challenger etait loin d’etre pur ou plutot homogene. Les moules glauconitiques des organismes calcaires sont de differentes couleurs, suivant l’etat plus ou moins avance de la formation de la Glauconie. Murray et Kenard (34) ont distingue des moules blancs, gris, jaunes, vert pale et vert fonce. A notre avis les moules blancs, gris, jaunes, vert pale doivent etre envisages comme de la Glauconie en voie de formation, les moules vert fonce etant seals formes par de la Glauconie typique. Le No. 1 contenait : 65 % de moules blancs, gris et quelques jaunes. 20 % ,, vert pale. 11 % ,, vert fonce. 14 % de mineraux et organismes siliceux. Le No. 2 contenait : 15 % de moules blancs, gris et jaunes. 35 % ,, vert pale. 45 % „ vert fonce. 5 % de mineraux et organismes siliceux. PEOC. ROY. SOC. EDIN. — YOL. XXYI. 17 258 Proceedings of Royal Society of Edinburgh. [sess. Le No. 3 contenait : 10 % de moules blancs, gris et jaunes. 25 % ,, vert pale. 60 % ,, vert fonce. 5 % de mineraux et organismes siliceux. Le No. 4 contenait : 30 % de moules blancs, gris et jaunes. 40 % ,, vert pale. 20 % ,, vert fonce. 10 % de mineraux et organismes siliceux. Aucune description du materiel du No. 5 n’est donnee, mais il etait probablement compose de Glauconie decomposee, comme nous le verrons plus loin. Les analyses du Challenger , specialement le No. 1, montrent une forte teneur en Silice, sauf le No. 5, ou nous avons probablement un cas de decomposition, et ou, comme nous le verrons dans le cbapitre de la decomposition, la Silice est balancee par une forte augmentation de fer ferrique. Le fer ferreux dans ces analyses varie de 1"18 a 1 ‘95, tandis que le fer ferrique varie de 15*63 a 39*93. Dans les deux analyses de Glauconie de roches sedimentaires la teneur du fer ferreux aug- mente considerablement et atteint 19*48 % dans le No. 8, et 10*17 % dans le No. 7, cette derniere analyse ayant ete faite sur de la Glauconie triee et vert fonce. Nous voyons done une grande difference entre la composition de la Glauconie des mers actuelles et celle des roclies sedimentaires. Ayant eu la bonne fortune de rencontrer par mi les importantes collections du Challenger Office le plus pur echantillon de Glauconie actuelle qui ait jamais ete trouve, nous en avons fait une etude microscopique tres detaillee, accompagnee d’une analyse quantitative. Ce materiel fut drague en 1873 par le U.S.S. Tuscarora a une profondeur de 317 metres au point lat. N. 38° 32', long. W. 123° 24'; il se composait de grains vert fonce sans aucune trace de decomposition, et de grains de Quartz qui ont et6 separes au moyen d’un puissant electro-aimant. 1905-6.] Becherches sur la G-lauconie. 259 L’analyse nous a donne les resultats suivants : SiO2 = 47-46 % Fe203 = 30-83 % A1203 = 1-53 % FeO = 3-10% MgO = 2-41 % K20 = 7-76 % H20 = 7-00 % Total = 100-09 % L’analyse qui concorde le mieux avec la notre est celle de Giimbel (No. 6) ; la forte teneur en Silice et en Alumine des analyses du Challenger s’explique par le fait de la presence de moules blancs et gris dans la substance analysee, ces moules etant en grande partie constitues par de l’argile. La presence de CaO et de Na20 avait deja ete attribute, avec raison, par plusieurs auteurs a des impuretes, ce que confirme notre analyse qui pourra desormais servir de type. La G-lauconie etant un mineral se formant aujourd’hui sur le fond des mers, ce ni est pas la Glauconie des roches sedimentaires , qui certainement a subi des transformations , quHl faut etudier your connaitre la genese de cet inter essant mineral* Dans leur interessant travail, “ Contribuciones al Estudio de la Glauconita ” (6), MM. Calderon et Chaves donnent une synthese de la Glauconie qui malheureusement est basee sur les resultats d’une analyse de Pisani, faite a un moment ou dans la Glauconie on dosait tout le fer a l’etat ferreux ; erreur due, comine l’a fait remarquer Giimbel (20) deja en 1886, a la couleur verte de ce mineral. La question de la synthese de la Glauconie est done loin d’etre resolue, et e’est de la Glauconie actuelle qu’il faudra desormais partir et non de la Glauconie de roches sedimentaires comme l’ont fait MM. Calderon et Chaves. Nous aurons l’occasion de revenir sur cette synthese dans un chapitre special. IY. Formation de la Glauconie. Absence et presence de la Glauconie dans les depots mar ins. — Avant d’attaquer le probleme de la formation de la Glauconie * Voir aussi Collet et Lee, Comptes-Rendus Acad. Sciences , Paris , Avril 1906. 260 Proceedings of Royal Society of Eclinburyh. [sess. nous voulons essayer de discuter l’absence ou la presence de ce mineral parmi les depots marins. Comme nous l’avons vu precedemment, la Glauconie est generalenient presente dans les Boues Bleues, mais ne peut, en aucune facon, en etre consideree comme caracteristique. Grace a un travail de grande valeur de Murray et Irvine, intitule “ On the Chemical Changes which take place in the Composition of the Sea-water associated with Blue Muds on the Floor of the Ocean” (36), nous pouvons desormais comprendre que la Glauconie se rencontre en petite quantite dans les Boues Bleues. D’apres ces savants les reactions dont les Boues Bleues sont le theatre sont les suivantes : la matiere organique en decomposition reduit les sulfates, en solution dans l’eau de mer, en sulfures, qui sont subsequemment decomposes par l’anhydride carbonique precedemment forme ; ainsi le soufre de Vacide sulfurique present comme sulfate de chaux ou de magnesie dans l’eau de mer est retire et fixe dans la Boue a l’etat de Sulfure de Fer (FeS), tandis que l’anhydride carbonique prend la place de l’acide sulfurique et une certaine quantite de Bicarbonate de Chaux est forme, proportionnellement au Soufre extrait de l’eau de mer. Ces reactions, confirmees par des experiences de laboratoire, peuvent se representer comme suit : (1) RSO4 + 2C = 2C02 + BS, oil R est un metal alcalino-terreux. (2) RS 4- 2C02 + H20 =-■ H2S + RC03C02. (3) RS + RC03C02 + H20 = 2RC03 + H2S. Cet Hydrogene sulfure rencontrant de l’oxyde ferrique present dans la couche superficielle de la Boue donnera lieu a la reaction suivante : (4) Fe203 + 3H2S = 2FeS + S + 3H20. De cette fagon une partie du Soufre est fixee dans la Boue comme sulfure de fer. S’il n’y a pas assez de fer dans la Boue pour fixer tout l’Hydrogene sulfure forme en (3), ce dernier s’echappe dans l’eau de mer, et rencontrant de l’oxygene, s’oxydera pour donner de l’acide sulfurique et de la un sulfate (RSO4). 1905-6.] Recherches sur la Glauconie. 261 Dans la Mer Noire (33), on nous avons en quelque sorte une exageration des phenomenes se passant dans les Boues Bleues, nous ne trouvons aucune trace de Glauconie, tout le fer etant precipite a l’etat de sulfure, l’Hydrogene sulfure etant en exces sur ce dernier metal. M. Chaves (9), continuant ses investigations synth 4tiques au sujet de la Glauconie, decrit dans une note intituiee “Con- tribuciones a la sintesis de los silicatos ferriferos por via humeda ” les produits resultant de Taction durant vingt mois d’un silicate de soude sur du sulfure ferreux prepare artificiellement. Ce savant obtint : 1°, formation de cristaux de sulfate de soude; 2°, depot de grains blancs insolubles dans l’eau, solubles dans l’acide nitrique en laissant un residu blanc de silice pulverulente ; la dissolution contenant du fer ; 3°, formation d’un depot gris verdatre, pulverulent, cristallin sous le microscope, soluble dans l’acide nitrique et offrant les memes proprietes que le precedent. Ces experiences demontrent qu’on peut obtenir un silicate de fer en partant du sulfure et d’un silicate de soude, le' sulfure de fer done n’a pas meme besoin d’etre oxyde en sulfate pour etre combinable a la silice. Le sulfure de fer des Boues Bleues semble incapable de former du silicate de fer ; la silice en solution ne doit pas manquer, e’est done l’Hydrogene sulfure qui doit empecher toute reaction entre le sulfure de fer et la silice en solution. Des experiences de laboratoire nous ont egalement montre que l’Hydrogene sulfure empeche toute reaction entre le sulfate de fer et le silicate de potasse. Dans les descriptions des depots du Challenger nous voyons que beaucoup de Boues Bleues contiennent des moules glauconitiques parfaits et d’autres en voie de formation. Dans la description du materiel de la station 167 (p. 87) nous relevons ce qui suit : “This deposit contains a great many Glauconite grains, which are mostly irregular in form, but would appear to have been at one time perfect casts of Foraminifera and other organisms. In some cases the transition can be traced by microscopic examination.” Plus loin dans la description du materiel de la station 209 (p. 103) nous lisons : “ This seems to be a green mud in process of formation.” 262 Proceedings of Royal Society of Edinburgh. [sess. Nous ne devons pas oublier qu’il n’y a pas de limite marquee entre les Boues Bleues et les Boues Vertes, et que les unes passent aux autres insensiblement comme du reste cela se produit pour tous les depots marins. Nous pourrons done dire avec beaucoup de certitude que la formation de la Glauconie dans les Boues Bleues depend de la quantite de matiere organique presente dans le depot et par Id dV Hydrogene sulfur e. La matiere organique est-elle en faible quantite, le sulfure de fer devient combinable ou transform- able en sulfate, et nous avons alors une Boue Verte avec grande quantite de moules glauconitiques et de grains de Glauconie. La Glauconie est presque absente dans les Vases a Globigerines; sur 118 echantillons du Challenger , 13 seulement contiennent ce mineral en de tres faibles proportions, a l’exception des stations 140 et 166, ou Von approche du Continent. Parmi les analyses de Vases a Globigerines du Challenger nous en trouvons 5 seule- ment avec une relativement forte teneur en oxyde ferrique (variant de 6*16 % a 20*93 %) ; ce sont celles du materiel des stations 12, 16, 17, 293, 176, qui contiennent toutes des nodules de manganese ou de la palagonite, ce qui nous prouve que le fer s’est depose sous des conditions tout autres que dans le cas des Boues Bleues. Comme Font fait remarquer Murray et Benard, on ne trouve pas de Glauconie dans les Boues Bouges des cotes du Brezil et de la Mer Jaune oil la limonite amenee de Finterieur des terres par l’Amazone, F Orinoco et le Yang-tse-Kiang est en quantite con- siderable. Ce fait de totale absence de Glauconie est tres etonnant, car il y a certainement assez de matiere organique dans ces Boues pour produire des reductions; neanmoins nous ne pourrons expliquer d’une fagon satisfaisante l’absence de la Glauconie dans ces depots que lorsque nous pourrons etudier Veau de mer associee aux depots marins , etude qui seule pourra nous renseigner sur les reactions chimiques qui ont lieu an fond de la mer. Nous saisissons l’occasion de recommander cette etude aux stations marines et aux expeditions futures en cherche de nouveautes ; le problem e sera difficile, il faudra inventer de nouveaux appareils, rnais les resultats recompenseront les efforts faits. En resume, nous pouvons dire que le fait de la presence ou de Fabsence de la Glauconie dans les u depots terrigenes ” est 1905-6.] Becker ches sur la G-lauconie. 263 intimement lie a la faible ou forte presence de matiere organique ; cette derniere, comme nous l’avons vu, donnant lieu a d’importantes reactions. Les “ depots pelagiques ” ne contiennent pas de Glauconie parce qu’ils manquent d’un des constituants importants, la potasse. La Glauconie est done une caracteristique des depots terrigenes en general et des Boues Vertes et Sables Verts en particulier. Formation de la Glauconie. — Depuis l’explication de la for- mation de la Glauconie, donnee par Murray et Renard dans le volume des “Deep-Sea Deposits” des Reports du Challenger (34), un seul travail traitant de la synthese de ce mineral est venu enricliir la bibliographic. II est du a la plume autorisee de MM. Calderon et Chaves (6) de Madrid, et est intitule “ Contribuciones al Estudio de la Glauconita.” Comme nous l’avons deja fait remarquer dans un precedent chapitre, ces savants se sont bases sur une analyse de Pisani de Glauconie de Villers sur Mer ; le fer fut dose entierement par cet analyste comme fer ferreux , comme du reste on le faisait a cette epoque ou Ton croyait, comme l’a fait remarquer Giimbel, que la couleur verte de la Glauconie impliquait du fer a l’etat ferreux. Basant malheureusement leurs ingenieuses recherches sur cette analyse de Pisani, MM. Calderon et Chaves reussirent a obtenir un silicate ferroso-potassique en faisant reagir du sulfate ferroso- potassique sur du silicate de potasse en presence d’un reducteur. Dans le silicate ainsi obtenu le rapport de l’oxyde ferreux a la silice etait de 20 '4 a 50* 1, ce qui concordait avec l’analyse de Pisani, ou ce rapport est de 20*1 a 54*1. Puis partant de l’idee que la Glauconie est un silicate ferroso-potassique necessitant pour sa conservation l’influence d’un milieu reducteur, ces savants pensent que les agents reducteurs a la faveur desquels la Glauconie s’est formee sont d’une part la matiere gelatineuse qui existe en suspension dans l’eau de mer, d’autre part la substance organisee qui existe dans les chambres de foraminiferes et d’autres organismes.* * “ Ahorabien ; siendo la Glauconita un silicato de protoxido de origen submarino, necesita indispensablemente para su conservacion la influencia de un medio reductor, que podria ser la substancia sarcodica, mas propicia para seme jaute papel que cualqniera otra, enj’a existencia, ademas, nos serias dificil imaginar. Creemos, por tanto, que la materia gelatinosa qui existe en suspension en las aquas, y sobre todo la substancia organizada misma que 264 Proceedings of Royal Society of Edinburgh. [sess. Comme nous l’avons vu dans le chapitre traitant de la “ Com- position chimique,” la Glauconie des mers actuelles doit etre envisagee non comme un silicate ferroso-potassique mais comme un silicate ferrico-potassique. La belle synthese de MM. Calderon et Chaves ne nous parait pas etre applicable au cas de la Glauconie des mers actuelles. Revenons a ^explication donnee par Murray et Renard dans les Reports du Challenger : ces auteurs supposent que la matiere organique renfermee dans la coquille ainsi que celle que contient la boue transforme l’oxyde de fer en sulfure. Ce sulfure peut etre oxyde en hydrate, le soufre etant en rneme temps mis en liberte serait oxyde en acide sulfurique, que servira a decomposer la fine boue pour former de la silice colloide. Cette silice pourra alors se combiner avec l’hydrate de fer pour former un silicate. Une etude tres approfondie des differentes especes de moules recueillis par le Challenger nous ont amene a modifier legerement cette explication fondamentale, en distinguant dans la formation de la Glauconie trois differentes phases : 1. Le premier stade dans la formation de la Glauconie est represente par les moules gris, composes exclusivement d’argile, c.a.d. de silicate d’alumine. 2. Les diverses nuances de moules bruns represented divers stades dans le remplacement de l’alumine de l’argile par le peroxyde de fer, comme une analyse faite sur des moules bruns nous l’a montrd. En outre, ces moules — et nous attirons l’attention sur ce fait — ne contiennent pas trace de potasse. Done le second stade est represente par un silicate ferrique provenant d’une elimination progressive d’alumine et de son remplacement par de l’oxyde ferrique. Les moules brun clair contiennent encore de l’alumine, les moules bruns en contiennent tres peu. 3. Le troisieme stade est celui de la glauconitisation , si l’on nous passe ce mot, des moules brun fonce. En effet, nous avons decrit au chapitre des “ Proprietes physiques” la marche de ce processus ; il nous reste maintenant a l’expliquer. Les vrais moules glauconitiques sont un silicate ferrico-potassique hydrate, tandis que elanaba las camaras de los foraminiferos o las cavidades de otros pequenos seres provistos de esqueleto, hau sido, sire duda alguna, los agentes reductores a favor de los cuales se ha consolidado el silicato de protoxido de hierro y potasio.” ] 905-6.] Eecherches sur la Glauconie. 265 les moules brans, comme nous venons de le voir, ne contiennent pas de potasse. La transformation en Glauconie est par conse- quent connexe de V introduction de la potasse et aussi probablement de V entree de Veau de constitution. Tout ce qui vient d’etre dit est base sur des faits d’observation et non sur des vues de l’esprit. Quelles sont les reactions chimiques qui perniettent a la potasse de se combiner au silicate ferrique ? Nous ne pouvons actuellement que poser la question, car il serait oiseux de fatiguer le lecteur en donnant la liste des essais infructueux faits en vue de transformer des moules brans en moules verts. Nous devons evidemment tenir compte du facteur temps , et peut-etre pression ? Absence de la Glauconie dans les lacs. — Parmi les differents problemes qui se presentent dans l’etude des depots marins et lacustres, celui de l’absence de la Glauconie dans les boues des lacs n’est pas un des moins interessants.* En effet, l’absence de ce mineral ne peut etre attribute ni a une raison bathymetrique, ni a une proportion insuffisante de fer, silice et potasse, ces elements constitutes de la Glauconie se trouvant dans l’eau des lacs. Depuis les belles decouvertes de Senft et de Julien, il nous semble que c’est a l’action des acides organiques (creniques, humiques, etc.) que nous devons faire appel. Julien (25) a montre que des silicates etaient entierement solubilises sous forme de sels a radical acide complexe (silico-azo- humate, etc.) dans Veau douce par Taction des dits acides organiques. Il crut pouvoir appliquer cette notion a la formation de la Glau- conie (p. 363, op. cit.\ mais comme l’ont montre Murray et Irvine (35) (p. 240, note), ces acides organiques une fois au contact avec l’eau de mer sont decomposes et precipites. C’est a la presence des acides du groupe de l’acide humique, depuis longtemps reconnus dans l’eau des lacs, que l’absence de la Glauconie dans les depots lacustres nous semble devoir etre attribute. Comme l’a montre Julien, dans les lacs le fer est soluble quand il est sous forme de silico-azo-humate ou de crenate ferreux. Une * Cette absence de la glauconie dans les lacs est indiscutable depuis les travaux du Lake Survey of Scotland, entrepris sous la direction de Sir John Murray et L. Pullar. 266 Proceedings of Royal Society of Edinburgh. [sess. oxydation vient-elle a se produire, le fer est precipite sous forme d’oxyde ferrique ou de limonite, avec production (p. 346, op. cit.) d’acide silico-azo-humique soluble et incombinable avec l’oxyde ferrique precipite. Le fer en solution dans beau, grace a la presence des acides organiques, ne peut done se combiner a la silice pour donner un silicate ferrique, etant par oxydation immediatement precipite a l’etat d’oxyde. Les depots ochreux du Loch Ness que nous avons eu l’occasion d’etudier nous paraissent devoir etre attribues a cette cause. Y. La Glauconie et les Concretions phosphatees. Comme nous l’avons rnontre dans notre etude des “ Concretions phosphatees de 1’ Agulhas Bank” (11), la Glauconie est souvent associee aux concretions phosphatees. La Glauconie et les concretions phosphatees se forment actuelle- ment sur le fond des mers, existe-t-il une relation entre ces deux formations au point de vue de leur genese? Cette question se pose naturellement quand on etudie les depots marins, et nous croyons etre maintenant en mesure d’y repondre negativement. Les concretions phosphatees sont pour ainsi dire V image du fond dans lequel on les rencontre , ce qui prouve bien leur formation in situ. Ce fond est-il un Sable Vert, comme dans le cas de 1’ Agulhas Bank, les concretions phosphatees contiendront de la Glauconie en grande abondance; est-il une Boue a Globigerines formee non loin du continent mais en eau profonde (3475 metres pour un des echantillons du Challenger), la concretion sera entierement formee de Globigerines avec mineraux detritiques mais sans Glauconie. Comme le Dr Lee (11) l’a rnontre, la Glauconie apparait dans les concretions phosphatees sous deux etats differents : 1°, a l’etat de grains arrondis a contours tranches qui font partie du nodule au meme titre que le Quartz et les autres mineraux, c.a.d. ont ete formes avant la formation du nodule ; 2°, a l’etat de pigment, ce dernier etant manifestement posterieur au depot des autres mineraux. Cette Glauconie pigmentaire doit provenir d’un brassage du depot, brassage qui a occasionne la pulverisation des grains de Glauconie. Ce brassage du fond peut etre cause par un courant, e’est a dire qu’il n’est pas necessaire de faire intervenir une force 1905-6.] Recherches sur la Glauconie. 267 tres grande, car il ne faut pas oublier que la Glauconie actuelle s’ecrase facilement sous l’ongle. La Glauconie ainsi pulverisee en particules microscopiques peut etre tenue en suspension dans l’eau, et les conditions de mouvement venant a s’arreter elle pourra etre precipitee sur le fond ou remplira des fissures de concretions. Un excellent exemple de ce mode de formation nous est donne dans une des concretions de 1 ’Agulhas Bank (voir fig. 3 et p. 870, op. cit .), ou nous voyons une ligne* noire de \ mm. d’epaisseur qui n’est autre qu’un depot de Glauconie pigmentaire, comme Font prouve nos dernieres recherches. Le Dr Collet (11) faisait remarquer Fan dernier que' la partie brillante et non recouverte d’organismes des concretions phos- phatees devait se trouver dans la vase, tandis que la partie recouverte d’organismes et de couleur grise etait exposee dans l’eau. Quelques unes d’entre les grosses concretions des stations 11 et 12 etaient entierement recouvertes d’organismes, ce qui prouverait non que ces grosses concretions aient ete changees de place sur le fond de la mer, mais plutot un changement dans le niveau du fond. !, changement gui peut provenir de faction plus ou moins forte des courants. Les concretions phosphatees de F Agulhas Bank nous autorisent done a admettre l’idee d’un brassage du fond ayant occasionne la pulverisation de la Glauconie en grains. M. Gosselet, dans son interessant memoire sur “ La Sedimenta- tion de la Craie ” (18), fait remarquer que la couleur verte des bancs durcis et des galets est due a de la Glauconie. “Ce n’est pas, comme on pourrait le croire, un simple placage a la surface du calcaire. La Glauconie penetre dans la roche jusqua une certaine profondeur,” ecrit ce dernier auteur. Plus loin ce savant fait remarquer l’analogie de Fenveloppe glauconieuse avec le vernis brun phosphate des nodules roules de la craie dure dans la craie phosphatee, et admet que Fun et l’autre ont pris naissance pendant le roulis par les courants. II nous parait exister une certaine relation entre l’enduit glauconitique des bancs durcis de M. Gosselet et la Glauconie pigmentaire de certaines concretions phosphatees, Glauconie qui, comme nous l’avons demontre, est le resultat d’un brassage ou d’un roulis du a des courants. 268 Proceedings of Royal Society of Edinburgh. [sess. Nous ne voulons pas terminer ce chapitre sans relever cette phrase importante de M. Gosselet dans son memoire sur les “ Observations geologiques faites dans les Exploitations de Phosphate de Chaux” (19): “Or les couches de craie phosphatee sont au milieu de la craie blanche ; elles alternent avec les couches de craie blanche, elles passent lateralement a la craie blanche. La consequence logique est que la craie blanche n’est pas un depot de mer profonde. C’est une conclusion importante, si on songe que l’idee de voir dans la craie un depot de mer profonde, est un des arguments en faveur de l’instabilite des continents et contre la permanence des oceans.” Nos etudes des concretions phosphatees et de la Glauconie des mers actuelles nous permettent de confirmer les theses de M. Cayeux (8) et de M. Gosselet sur la question si discutee de la formation de la Craie. Des depots renfermant des concretions phosphatees et de la Glauconie ne sont pas des depots de mer profonde. VI. Distribution de la Glauconie dans les Mers actuelles. En 1856 Bailey, apres une etude des echantillons collects par le Comte Pourtales le long de la cote atlantique de l’Amerique du Nord, fut le premier a faire remarquer que la Glauconie se formait actuellement en plusieurs endroits sur le fond des mers actuelles. De 1873 a 1876 le Challenger trouva de la Glauconie dans les depots de mers profondes (deep-sea deposits), et Murray et Renard (34), dans leur celebre volume des “Reports” du Challenger , indiquerent la repartition de cet interessant mineral dans les depots des mers actuelles. Nous trouvons en effet la Glauconie dans les Boues Bleues (Blue Muds), les Boues Vertes (Green Muds), les Sables Verts (Green Sands), qui appartiennent a ce que Murray et Renard ont appele les Depots Terrigenes (Terrigenous Deposits) formes dans des eaux profondes en dehors de la ligne de 100 fathoms (183 metres). La Glauconie fut egalement rencontree dans les Boues a Globi- gerines, mais en faible quantite, car sur 118 echantillons du Challenger 13 seulement contenaient de la Glauconie. 1905-6.] Recherches sur la Glauconie. 269 (a) Glauconie dans les Boues Bleues. Murray et Renard appelerent Boue Bleue, un depot qu’on rencontre en eau profonde entourant le continent. La couleur bleue est due a la presence de matiere organique et de sulfure de fer. On trouve la Glauconie dans les Boues Bleues sous deux aspects differents : a l’etat de moules (casts) d’organismes calcaires et a l’etat de grains parmi les particules minerales. La loi suivante peut etre formulee : La Glauconie est generale- ment presente dans les Boues Bleues, mais ne peut en aucune /agon en etre consideree comme caracteristique , etant en quantite minime comparativement a celle des Boues Yertes et des Sables Verts. II serait trop long d’enumerer ici chaque point ou la Glauconie a ete trouvee dans des Boues Bleues, mais nous dirons neanmoins qu’elle fut rencontree assoeiee a des concretions phosphatees dans les draguages du Challenger entre les lies Falkland et le Rio de la Plata. Ce mineral fut egalement trouve a quatre stations diffe- rentes dans des Boues Bleues a proximite des glaces antarctiques (env. 65° S.), et Sir John Murray (32) ecrivait en 1894: “Its presence in the Blue Muds of the far south is therefore most suggestive of an Antarctic Continent.” Parmi les boues collectees par la recente expedition ecossaise de la Scotia (39), la Glauconie est rare et se trouve seulement a l’etat de moules de foraminiferes. Dans les echantillons de boues envoyees a Sir John Murray (33) par le Prof. N. Andrusson, collectees dans la Mer Noire par les expeditions russes en 1890 et 1891, il n’y avait pas trace de Glauconie. Nous discuterons plus tard l’importante question de la presence et de l’absence de la Glauconie dans certains depots. (b) La Glauconie dans les Boues Vertes et les Sables Verts. La Glauconie en grains et en moules d’organismes calcaires est la matiere caracteristique des Boues Yertes et des Sables Verts. On rencontre aussi dans ces depots une matiere amorphe verte, organique car elle devient noire apres avoir ete chauffee sur la lame de platine et laissant ensuite une cendre coloree en brun par de l’oxyde de fer. 270 Proceedings of Royal Society of Edinburgh. [sess. Les Sables Verts different des Boues Vertes par leur aspect granuleux, du a line plus petite quantite de matiere amorphe ; generalement on les rencontre dans des eaux moins profondes que les boues. Le tableau suivant donne une idee de la difference qui existe entre les Boues Bleues, les Boues Vertes et les Sables Verts. Composition moyenne des Boues Bleues, des Boues et Sables Verts. D’aprEs Murray et Renard. Boues Bleues. Boues Vertes. Sables Verts. f Foraminifferes p61a- 7-52 14-59 21-00 Carbonate giques de Especes vivant sur le 175 2-94 15-00 Chaux. , fond i { Autres organismes 3-21 7-99 1378 12-48 25-52 49-78 1 rOrganismes siliceux . 3-27 13-67 8-00 R6sid\i J Min6raux 22-48 27-11 30-00 1 Partie fine (fine wash- 6177 33-70 12-22 I l ings) 87-52 74-48 50-22 100- 100- 100- Les Boues Vertes et les Sables Verts sont presque toujours developpes le long des cotes escarpees et exposees, ou aucun grand fleuve n’apporte des matieres. detritiques dans la mer , a une pro- fondeur rtexcedant generalement pas 2000 metres. Quand il y a une grande quantite d’hydrate ferrique dans un depot, comme le long des cotes du Brezil, ou quand les depots sont principalement formes d’elements detritiques fluviatiles, la Glauconie est generalement absente ou tres rare. Depuis l’expedition du Challenger , presque toutes les collections de depots rapportees par les expeditions ou les bateaux de differents Services ont ete examinees au Challenger Office, sous la direction de Sir John Murray, par MM. James Chumley et Robert Dykes; il nous parait utile de dire quelques mots sur la distribution actuelle des Boues Vertes et des Sables Verts, attenau que nous disposons de documents entierement inedits. Ocean Atlantique Nord. 1873. Le Challenger (34) trouva des Boues Vertes et des Sables Verts le long des cbtes du Portugal et de FEspagne du 1905-6.] Becherches sur la Glauconie. 271 point lat. 38° 31' N., long. 9° 31' W., an point lat. 36° 25' N., long. 8° 12' W. Ils proviennent, sauf deux exceptions, de profondeurs inferieures a 1000 fathoms (1830 metres). 1880. Les memes formations furent rencontrees par le U.S. Coast Survey Steamer Blake (1) le long de la cote des Etats Unis entre le Cap Hatteras et la latitude 31° 48' N., a une profondeur variant de 91 a 183 metres; c’est a dire sur le bord continental du Gulf Stream sur la ligne de separation entre les sables siliceux et les fonds calcaires, a l’endroit oil le courant etait le moins rapide. Ca et la ces memes Sables Yerts furent dragues en eau plus profonde sous le courant meme. Nous trouvons associes avec ces depots des nodules de manganese et des concretions phosphatees. Un autre gisement fut trouve par le Blake au point lat. 21° 2' N., long. 74° 44' W., devant Cayo de Moa, a une profondeur de 2842 metres. 1883. Le S.S. Dacia , croisant dans l’Atlantique Nord, dragua ces memes depots au point lat. 31° 48' 30" N., long. 10° 5' W., par 400 metres. 1886. Le S.S. Buccaneer trouva des Sables Yerts le long de la cote occidentale de l’Afrique aux points suivants : lat. 6° 9' N., long. 10° 55' W., par 82 metres; et lat. 5° 5' 6" N., long. 4° 00' 7" W., par 121 metres. Ocean Indien. 1887. Comparativement aux autres oceans, 1’Ocean Indien a ete etudie beaucoup plus tard, et les premieres collections de depdts furent faites par le Capitaine J. P. Maclear, R.N., a bord de H.M.S. Flying Fish (29). Des Boues Yertes et Sables Yerts furent collectes aux points suivants : Prof. Lat. S. Long. E. 631 m. 10° 42' 124° 52' 1500 m. co CO o o 124° 13' 1027 m. 11° 7' 121° 52' 1531 m. 11° 5' 121° 50' 2652 m. 7° 3' 103° r 3512 m. 7° 3' 102° r 272 Proceedings of Royal Society of Edinburgh. [skss. 1889. En cette annee de nombreuses adjonctions furent faites a nos connaissances des depots de cet Ocean par les collections rapportees par le Capitaine Pelham Aldrich dn H.M.S. Egeria{ 30), et par le Capitaine A. Carpenter du H.M.S. Investigator. Les collections de depots faites a bord des bateaux posant des cables le long de la cote orientale d’Afrique ont ete egalement etudiees au Challenger Office, et nous pouvons maintenant donner une idee assez juste de la distribution des depots a Glauconie dans cet Ocean en disant qu’on les rencontre le long de la cote E. d’Afrique, le long des cotes AY. et S. d’Australie, a une profondeur generale- ment inferieure a 1830 metres (1000 fathoms). Agulhas Bank. 1873. La Glauconie est tres remarquablement rejnesentee dans les depots de l’Agulhas Bank, comme cela a ete demontre par Murray et Renard. 1875. L’expedition allemande de la Gazelle rencontra les memes formations sur ce meme point. 1898-99. L’expedition recente de la Valdivia rapporta de r Agulhas Bank des Sables Yerts et des concretions phosphatees. L’an dernier nous avons decrit une importante collection de Concretions phosphatees provenant de depots dragues sur 1’ Agulhas Bank par les bateaux du “Department of Agriculture of the Cape of Good Hope,” prouvant que la Glauconie et les concretions phos- phatees sont egalement distributes sur toute l’etendue du banc en dehors de la ligne de 100 fathoms (183 metres). Ocean Pacifique. 1873. Le Commandant Geo. E. Belknap trouva des Boues Yertes et des Sables Yerts tandis qu’il etudiait la cote AY. de l’Amerique du Nord a bord du U.S.S. Tuscarova. Les collec- tions furent envoy ees a Sir John Murray et furent etudiees a nouveau de 1899-1901 au Challenger Office. Parmi ces collections nous trouvons 4 echantillons de Boues Yertes et 26 echantillons de Sables Yerts provenant de 30 stations differentes du point lat. 44° 54' H., long. 125° 13' W., au point lat. 32° N., long. 118° 26' AY., a des profondeurs inferieures a 1830 metres. Proc. Roy. Socy. of Edin. ] [Vol. XXVI. Plate I. Sable vert recueilli par le Challenger a la Station 154 par 750 m. Le centre de la figure est occupe par un moule glauconitique contenant des inclusions opaques. DrsJLeon W. Collet et Gabriel W. Lee. Proc. Roy. Socy. of fidin.] [Vol. XXYI. Plate II. Boue a Globigerines provenant de la Station 176 ( Challenger ). La coqnille occupant le centre de la figure est en partie remplie par une matiere argileuse brune, et constitue uu moule imparfait. Drs Ij&on W , Collet et Gabriel W. Lee, Proc. Eoy. Socy. of EdAn. ] [Vol. XXVI. Plate III. Le moule occupant le centre de la figure est brun fonce, mais ses contours, c.a.d. la partie en contact avec la coquille, montrent un mince lisere de Glauconie. En outre les taches noires qui criblent la calcite de la coquille, sont vertes en lumiere naturelle et sont formees de Glauconie typique. (Station 164.) Drs LtiQN W. Collet et Gabriel W. Lee. Proc. Roy. Socy. of Edin. ] [Vol. XXYI. Plate IV. Moules glauconitiques laves a l’acide. Celui du centre de la figure contient encore un noyau brun de silicate de fer amorphe ; les autres sont parfaits, et certains d’entre eux renferment des inclusions de quartz et autres mineraux (Station 164, Challenger.) Drs L£on W. Collet et Gabriel W. Lee. Proc. Roy. Socy. of Edin.] { Vol. XXVI. Plate V. Dans cette coquille (grossissement = 74 diam. ) les loges de couleur claire sont glauconitiques, tandis que les autres, brun-fonce, ne montrent pas encore de transformation en Glauconie. Ce monle est contenu dans un nodule phosphate de l’Agulhas Bank. Drs LkoN W. Collet et Gabriel W. Lee. Proc. Roy. So 'y. of Eclin.] [Vol. XXVI. Plate YI. Nodule phosphate dont le ciment est teinte en vert-clair par de la Glauconie pigmentaire. La calcite de la coquille au centre de la figure a disparu, remplacee par de la Glauconie epigenique. Drs LhoN W. Collet et Gabriel W. Lee. Proc. Roy. Socy. of Edin.] [Vol. XXVI. Plate VII. Nodule phosphate. La ciment est teinte en vert par de la Glauconie pigmentaire et se propage a l’interieur de coquilles de Foraminiferes, con- stituant ainsi de faux moules. Drs Leon W. Collet et Gabriel W. Lee. Proc. Roy. Socy. of Edin.] [ Vol. XXVI. Plate VIII. Exemple de Glauconie epigenisant un grain de quartz, au sein d’un nodule phosphate. Drs Leon W. Collet et Gabriel W. Lee. Proc. Roy. Socy. of EdinP\ [Yol. XXVI. Plate IX. Au centre de la figure, section de coquille dont la calcite est pseudomorphosee par une matiere ferrugineuse amorphe. Dus Leon W. Collet et Gabriel W. Lee. Proc. Roy. Socy. of Eclin. ] [Vol. XXV r, Aspect general des grains de Glauconie dans un nodule phosphate. Plate X. Dus Leon W. Collet et Gabkiel W. Lee. Proc. Roy. Socy. of Eclin ] [Vol. XXVI. Plate XI. Nodule phosphate. Grain de Glauconi.e a centre plus pale que la masse generale, et entoure d’une zone externe isotrope. Dus Leon W. Collet et Gabriel W. Lee. Proc. Boy. Socy. of Edin. ] [ Vol. XXVI. Plate XII. Exemple de differenciation chez la Glauconie. La partie interne du grain au centre de la figure est cryptocristalline, tandis que la zone externe est fibro-radiee et a une orientation optique unique. Cretace d’Antrira (Irlande). Drs Leon W. Collet et Gabriel W. Lee. 1905-6.] Recherches sur la Glauconie. 273 Trois echantillons provenant des stations suivantes sont particulierement interessants : - Prof. 774 m. Lat., 39° 02' K, long., 124° 09' W. „ 232 m. „ 39° JY, „ 124° W. „ 317 m. „ 38° 32' K, „ 123° 24' W. Ils sont presque totalement composes de grains de Glauconie avec quelques foraminiferes et quelques mineraux, specialement du Quartz. Sans aucun doute ce sont les specimens les plus purs de Sables Yerts qui aient ete jamais trouves, s’ils se trouvent bien a l’etat dans lequel ils furent dragues; il se pourrait que les echantillons aient ete passes an tamis avant d’avoir ete envoyes. Quoiqudl en soit, ce material 4tant si pur nous en avons fait une etude tres detaillee, comme on le verra plus loin. 1874. Le Challenger dragua des Sables Yerts devant Sydney a 6 stations differentes. Puis il rencontra les memes formations entre Cape York et Arrou Island aux points suivants : Lat. 9° 36' S., long. 137° 50' E. „ 8°56'S., „ 136° 5'E. De Arrou Island an Japon des Boues Yertes furent collectees a une station dans la mer d’Arafura et a 2 stations aux points suivants : Lat. 5° 41' S., long. 134° 4*3' E. „ 5° 26' S., „ 133° 19' E. entre Arrou Island et Banda. De Samboangan a Manille a 3 stations : Lat. 12° 43' 1ST., long. 122° 9' E. „ 12° 46' AT., „ 122° 10' E. „ 9°26'K, „ 123° 45' E. les memes depots furent dragues. Le long de la cote du Japon, la Glauconie fut trouvee dans les Boues Yertes a 4 stations et pour la derniere fois dans le Pacifique par le Challenger , car la Glauconie n’a jamais ete trouvee dans les grands fonds au milieu de cet Ocean. 1878. Le U.S.S. Tuscarora trouva de nouveau la Glauconie a une station lat. 23° 35' 1ST., long. Ill0 57' W., par 908 metres. 1882-83. Le U.S.S. Enterprise rencontra ces memes depots pres la Chatham Island dans le Sud. PROC. ROY. SOC. EDIN. — VOL. XXVI. 18 274 Proceedings of Royal Society of Edinburgh. [sess. 1889. Lorsque A. Agassiz (2) etudiait la cote W. de l’Ame- rique du Nord a bord de U.S.S. Albatross , il dragua des Boues Yertes avec de nombreux moules de Glauconie au point lat. 31° 3' 30" JST., long. 117° 40' 15" W., par 1566 metres. Ce nouveau point situe entre les deux sondages extremes sud du Tuscarora, semble montrer que les Sables Verts s’etendent sur la cote W. de l’Amtrique du Nord de la lat. 44° 54' a la lat. 23° 35' X. 1891. Etudiant la cote W. de l’Amerique Centrale, A. Agassiz, toujours a bord de Y Albatross, trouva 2 gisements de Sables Verts devant Mariato Point, non loin de la cbte aux points suivants : lat. 6° 35' IV., long. 81° 44' W., par 914 metres; et lat. 6° 30' N., long. 81° 44' W., par 1280 metres. 1896. Le H.M.S Dart dragua de semblables formations sur la cote E. d’Australie aux points suivants : lat. 23° 6' 30" S., long. 152° 16' 30" E., par 145 metres; et lat. 23° 9' 40" S., long. 152° 19' E., par 165 metres. 1901. Le S.S. Britannia dragua des Boues Yertes a 2 stations au nord de la Nouvelle Zelande : lat. 34° 36' 32" S., long. 173° 35' 37" E., par 329 metres ; et lat. 34° 31' 42" S., long. 173° 34' 30" E., par 307 metres. 1904. Comme les echantillons de Sables Yerts collects en 1891 par A. Agassiz (3) a bord de Y Albatross furent perdus, deux draguages furent faits sensiblement aux memes points, et les Sables Yerts furent retrouves. Pour etre complet il nous faut citer les localites ou le U.S.S. Nero (16) trouva des Boues Yertes, tandis qu’il etudiait la route pour la pose d’un cable entre les Etats Unis, les Philippines et le Japon. Plusieurs des determinations comme Boues Yertes du Nero nous paraissent etre en disaccord avec la description type donnee par Murray et Benard, et nous semblent devoir etre attributes a des Boues Bleues. Ces depots furent dragues a Dingola Bay (Luzon) a 14 stations differentes a des profondeurs variant de 188 metres a 4272 metres'? Devant Yokohama, a 21 stations differentes; toutes, sauf 5, en dessous de 1800 metres. Ces dernieres localites se rapprochent de celles oil le Challenger en 1874 dragua les memes formations. 1905-6.] Reclierches sur la Glauconie. 275 Mediterranee. D’apres Sir John Murray (31) la Glauconie a et4 trouvee a de grandes profondeurs dans la Mediterranee a l’4tat de grains et de monies de foraminiferes. En 1901 S.A.S. le Prince de Monaco, a bord de la Princesse Alice , dragua devant la cote du Maroc a la station suivante : lat. 33° 59' 30" N., long. 10° 33' W., par 851 metres, ce que le Prof. Thoulet appelle : Vase sableuse tres calcaire glauconieuse. Aujourd’hui d’apres ce que nous savons nous pouvons dire que la Glauconie est representee en grande quantite dans ce que Murray et Renard out appele les Boues Yertes et les Sables Yerts, et dans les localites suivantes : le long des cdtes de la Caroline et de la Floride sur les bords des Boues Bleues ; le long des cotes du Portugal et de l’Espagne ; le long des cotes S. et E. d’Afrique ; le long des cotes W., S. et E. d’Atistralie ; dans quelques mers de l’Archipel Indien ; le long de la cote E. du Japon; le long de la c6te W. de l’Amerique du Nord ; le long de la cote W. de Chatham Island ; en quelques points sur la cote W. d’Afrique. “ Challenger ” Office, Edinburgh, Mai 1906. Liste des Ouvrages consultes. (1) Agassiz, Alexander, “Three Cruises of the Blake” Bull. Mus. Comp. Zool ., Cambridge , U.S.A. , vol. xiv., 1888. (2) Agassiz, Alexander, “ General Sketch of the Expedition of the Albatross from February to May 1891,” Bull. Mus. Comp. Zool ., Cambridge , U.S.A., vol. xxiii., Ho. 1, 1892. (3) Agassiz, Alexander, “ On the Progress of the Albatross Expedition to the Eastern Pacific,” Am. Jour, of Scien., vol. xix., Eeb. 1905. (4) Bailey, J., “ On the Origin of Greensand and its Formation in the Oceans of the Present Epoch,” Am. Jour. Sc., 2i6me S., vol. xxii., 1856; et Proc. Boston Soc. of Nat. Hist., vol. v., 1856. (5) Bischof, G., Elements of Chemical and Physical Geology , vol. iii., 1859. 276 Proceedings of Royal Society of Edinburgh. [sess. (6) Calderon, S., et Chaves, F., “ Contribuciones al Estudio de la Glauconita,” An. Soc. espan. de Hist. Nat., vol. xxiii., t. iii., Madrid, 1894. (7) Cayeux, L., “Notes sur la Glauconie,” An. Soc. Geol. Nord. t. xx. p. 380, Lille, 1892. (8) Cayeux, L., Contribution a V Etude micrographique des Terrains sediment air es, Lille, 1897. (9) Chaves, F., “ Contribuciones a la Sintesis de los Silicatos ferriferos por via humeda,” An. Soc. espan. de Hist. Nat., vol. xxiv., Actas, p. 157, Madrid, 1895. (10) Chaves, F., “Estudio sobre las pseudomorfosis de proceso quimico,” An. Soc. espan. de Hist. Nat., t. xxviii., Madrid, 1899. (11) Collet, L. W., et Lee, G. W., “Les Concretions phos- phatees de l’Agulhas Bank, avec une note sur la Glauconie qu’elles contiennent,” Proc. Roy. Soc. Edin., vol. xxv., part x., 1905. (12) Corse and Baskerville, Am. Ch. J., xiv. 627, 1892. (13) Dana, The System of Mineralogy, sixth edition. (14) Dana, First Appendix of the sixth edition of Dana’s System of Mineralogy. (15) Delesse, Lithologie des Mers de France et des Mers prmcipales du Globe, Paris, 1871. (16) Flint, J. M., “A Contribution to the Oceanography of the Pacific,” Bull. U.S. National Plus., No. 55, Washington, 1905. (17) Glynka, K., Der Glauhonit, seme Enitstehung, sein cliemischer Bestand und die Art und Weise seiner Verwitterung , St Petersbourg, 1896. (18) Gosselet, J., “Observations sur la sedimentation de la Craie,” An. Soc. Geol. Nord, t. xxxi., p. 63, Lille, 1902. (19) Gosselet, J., “Observations geologiques faites dans les exploitations de phosphate de chaux en 1901,” An. Soc. Geol. Nord, t. xxx. p. 208, Lille, 1901. (20) Gumbel, V., “Ueber de Natur und Bildungsweise des Glaukonits,” Sitz. d. Math. Pliys. Clas. d. k. Akad. Wiss. Munchen, 1886. (21) Gumbel, V., “ Ueber die Grlinerde von Monte Baldo,” Sitz. Ber. A. K. Munchen, xxvi. 545, 1896. (22) Heddle, “Chapters on the Mineralogy of Scotland,” Trans. Roy. Soc. Edin., vol. xxix., part i., 1878-79. 1905-6.] Recherches sur la Glauconie. 277 (23) Hoskins, A. Percy, “ On Glauconite from Antrim,” Geol. Mag., July 1895. (24) Irving, A., “Organic Matter as a Geological Agent,” Proc. Geol. Assoc., vol. xii., 1892. (25) Julien, A., “On the Geological Action of the Humus Acids,” Proc. Amer. Assoc. Adv. of Sc., vol. xxviii., 1879. (26) Lacroix, A., M'ineralogie de la France et de ses colonies, 1893. (27) Murray, John, “Report on the Specimens of Bottom Deposits,” Bull. Mus. Comp. Zool., Cambridge, U.S.A., vol. xii., No. 2, 1885. (28) Murray, John, “On Seas and Estuaries about North Britain,” Phil. Soc. Glasgow , 1886. (29) Murray, John, “On some Recent Deep-Sea Observations in the Indian Ocean,” Scot. Geog. Mag., Nov. 1887. (30) Murray, John, “ On Marine Deposits in the Indian, Southern, and Antarctic Oceans,” Scot. Geog. Mag., Aug. 1889. (31) Murray, John, “The Maltese Islands, with special reference to their Geological Structure,” Scot. Geog. Mag., vol. vi. p. 449, 1890. (32) Murray, John, “The Renewal of an Antarctic Explora- tion,” Geog. Jour., Jan. 1894. (33) Murray, John, ei On the Deposits of the Black Sea,” Scot. Geog. Mag., Dec. 1900. (34) Murray and Renard, Deep-Sea Deposits. Challenger Reports, 1891. (35) Murray, John, and Irvine, Robert, “ On Silica and Siliceous Remains of Organisms in Modern Seas,” Proc. Roy. Soc. Edin., vol. xviii., 1891. (36) Murray, John, and Irvine, Robert, “ On the Chemical Changes which take place in the Composition of the Sea-water associated with Blue Muds on the Floor of the Ocean,” Trans. Roy. Soc. Edin., vol. xxxvii., part ii., 1893. (37) Natterer, C., “ Chemisch-geologische Tiefsee Forschung,” Geog. Zeit., V. Jahrg., Leipzig, 1899. (38) Peake, R. E., and Murray, Sir John, “Deep-Sea Sound- ing Expedition in the North Atlantic during the Summer of 1899,” Supp. Paper of the Geog. Jour., 1901. 278 Proceedings of Boyal Society of Edinburgh. [sess. (39) Pirie, J. H. Harvey, “ Deep-Sea Deposits of the South Atlantic Ocean and Weddell Sea,” S cot. Geog. Mag., vol. xxi., 1905. (40) Spurr, J. E., “The Iron-hearing Rocks of the Mesabi Range,” Geol. and Nat. Hist. Surv. of Minnesota, Bull. No. X., Minneapolis, 1894. (41) Tizard and Murray, John, “Exploration of the Faroe Channel during the Summer of 1880 in H.M.S. hired ship Knight Errant ,” Proc. Roy. Soc. Edin., May 1882. (42) Thoulet, J., Echantillons d’eaux et de fonds provenant des campagnes de la “ Princesse- Alice.” Camp. Prince Albert Pv, fas. xxii., Monaco, 1902. (43) Zirkel, Lehrbuch der Petrographie, 1894. ( Issued separately August 30, 1906.) p [Vol. XXVI. Proc. Roy. Socy. of Edin ] [Vol. XXVI. L. W. Collet et GW Lee. 1905-6.] Human Skeleton , with Prehistoric Objects. 279 Notes: — 1. On a Human Skeleton, with Prehistoric Objects, found at Great Caster ton, Rutland. 2. On a Stone Cist containing a Skeleton and an Urn, found at Largs, Ayrshire. By Dr Robert Munro. With a Report on the Urn, by the Hon. John Abercromby ; and on the Skulls, by Professor D. J. Cunningham. (Read March 19, 1906. MS. received May 18, 1906.) I. The following is an extract from a letter dated November 18th, 1905, which I received from Y. B. Crowther-Beynon, Esq., F.S.A., Hon. Secretary of the Rutland Archaeological and Natural History Society : — “ I should be most sincerely grateful to you if you could give me the benefit of your opinion on the skull of which I send some photos (fig. 1). I have been comparing it with those illustrated in your ‘Fossil Man’ chapter in Prehistoric Problems , and it seems to me that it is not without interest. I am no craniologist or anatomist, and can bring no scientific knowledge of that kind to bear on the matter.” In replying to Mr Crowther-Beynon’s letter,* I stated that it would be impossible to form an opinion having any scientific value from photos alone ; but that, if he sent the skull to Edinburgh, Dr Cunningham, Professor of Anatomy in the University of Edinburgh, who makes a special study of physical anthropology, would examine the specimen and report on its special character- istics. Along with the skull, my correspondent sent the following graphic and lucid account of the position and circumstances in which the specimen was found, as well as of the objects supposed to have been associated with it : — “In August 1905, some quarrymen in the employ of Mr Woolston of Stamford at a freestone quarry situated at Great Casterton, on the extreme eastern border of Rutland, struck into a fissure or swallow-hole (‘gull’ in the local phraseology) in the rock which was filled with clay. In the course of removing the clay a discovery of human bones was made at a depth of about 17 feet 6 inches from the original surface-level. The fissure was 280 Proceedings of Royal Society of Edinburgh. [sess. funnel-shaped, narrowing to some 20 inches at the point where the skeleton lay (see Section, fig. 2), and the clay was set hard, a fact which rendered the removal of the hones a matter of difficulty. Unfortunately the find was not reported at once, and consequently there is an absence of any accurate notes either as to the exact disposition of the skeleton in the fissure, or the attendant circumstances generally. It would appear, however, that the body lay on the back, with the limbs in a contracted position above,— -the appearance being that of a body which had become Fig. 1. — Skull and Stone Axe. jammed, by its fall, in the narrow space of the fissure. Before the bones were all extracted the mass of hardened clay enclosed by the sides of the fissure suddenly fell, some of the men having a narrow escape of being crushed by the downfall. It was in this mass of fallen material that the worked objects described later on were found, and it will be obvious, therefore, that it is difficult to state accurately where and at what level in the fissure the various relics had lain before the fall of the clay, so that we are thus deprived of valuable evidence as to their association with the skeleton. Fig. 2. — Section through fissure. 282 Proceedings of Royal Society of Edinburgh. [sess. “ My inquiries made on the spot as soon as I heard of the find, though not till after the removal of all the remains, elicited little information of a definite kind. The men were pretty positive that there were no symptoms of there having been any opening or passage into the fissure except from above. “The find consisted of the following antiquarian objects: — “ (1) Human remains consisting of skull and a quantity of bones, but not the complete skeleton. The teeth in the jaws were of a strong type and much worn down with use. “ (2) Polished hornstone celt of late Neolithic type, measuring 4 inches in length, 2J inches in width at lower end, and 1J inches at upper end, with a maximum thickness of J inch. This is a well- wrought implement, with a finely ground edge and polished all over. It was found in the mass of fallen clay near the rock face, thus showing that it had originally lain low down in the fissure, and consequently at or near the level where the skeleton lay. The clay in the fissure had hardened into a ‘ pillar,’ so to speak, and fell outward at full length ; so that the respective distances of the relics from the base of the fallen mass would approximately indicate their original positions in the fissure. “ (3) A stone rnuller or triturating stone, having one flat surface of an irregular oval shape, measuring about 5 or 6 inches in length and 4 inches in width. Its position in the clay is uncertain. “ (4) Four fragments of pottery, pronounced by Mr Wright, Curator of the Colchester Museum, and Professor Boyd Dawkins, to be mediaeval. Their position in the clay is also uncertain. “(5) Three pieces of sandstone slabs, about J inch thick, bear- ing evidence of having been used as tools, apparently for shaping bone or horn implements. One piece, measuring 3 by 2 inches, has a groove, \ inch in breadth, running across the shorter diameter, which has the peculiarity of being deep and well- defined at one edge, while towards the other it becomes gradually narrower and shallower. The other two fragments have each a semicircular groove in one of their margins, showing evidence of friction. The position of these objects in the clay is also undetermined. “(6) Some quantity of decomposed wood, black and soft with age, was found among the fallen clay.” 1905-6.] Human Skeleton , with Prehistoric Objects. 283 Having carefully considered the conditions under which the above-described relics were discovered, I do not think we can legitimately associate any of the worked objects with the skeleton. There can be no doubt that the stone axe, the muller, and the grooved rubbing-stones were tools used by people of the Neolithic Age, and, being deposited in the clay at a higher level than the skeleton, we are entitled to assume, a fortiori , that the latter also belonged to the Neolithic Age. The skull appears to be similar to those described by Professor Boyd Dawkins from the sepulchral caverns and tumuli of North Wales as belonging to the dark, long-headed Iberians, of whom we shall have something to say later on. (See Early Man in Britain , chap, ix.) II. The Largs skull came into my hands in the following manner : — Happening to be at Largs on the 26th January 1906, I heard various rumours of the discovery of a stone grave, con- taining a human skeleton, which had been made a few days pre- viously on the estate of Haylee, the property of C. J. C, Douglas, Esq. While pondering over the best way of obtaining precise information on the matter, Mr Douglas and Mr Fryers, architect, called at my house to see if I would accompany them in making an inquiry into the details of the discovery. So we at once started on the business. In the Skelmorlie Mausoleum within the old kirkyard we were shown by Mr Paton a series of red sand- stone flags, some seven or eight in number, of which the walls of the cist had been constructed, as well as the covering-stone, broken into many fragments. We then drove to the site of the discovery, and finally to the office of the master of works, where were pre- served the remains of the skeleton and a solitary piece of pottery — the rest of the vessel having crumbled into small fragments at the time of its removal from the cist. The cist was uncovered close to the hedge bounding the east side of the Irvine road, while digging a drain from the new cottages now being erected on the ground opposite to May Street, and immediately below the site of a great chambered cairn which, upwards of a century ago, stood near Haylee House, and of which some of the stones of the chamber still remain in situ. The cist lay lengthways across the drain, the cover being 2 feet below the surface of the road, but, as was pointed out by one of the workmen, the latter was consider- 284 Proceedings of Royal Society of Edinburgh. [sess. ably below the present level of the adjoining field. As already mentioned, the walls of the cist were constructed of red sandstone flags set on edge, sometimes forming a double row, in which case the intervening crevices were said to have been stuffed with clay. The cover, which had to be broken for removal, was a massive block of conglomerate. Both these kinds of rock are common in the district. The dimensions of the cist were 4J feet long, 21- feet wide, and 2 feet deep. The body was apparently in a sitting posture, but it, especially the skull, as well as the urn, had been badly damaged by the breakiug up of the cover-stone. The outcome of our consultation over these interesting memorials of the past was to authorise me to procure a report on the portion of the urn from the Hon. John Abercromby, who has made a special study of the ornamentation and chronological range of this class of sepulchral pottery ; and to submit the skull to Professor Cunningham, to see if its fragmentary condition would permit of a report being made on its anatomical characteristics. General Remarks. From Dr Cunningham’s report it will be seen that these two skulls are almost typical specimens of the cranial conformation of two different races who formerly inhabited the British Isles, the one dolichocephalic and the other brachycephalic. As early as 1850, Sir Daniel Wilson maintained, as the result of an investigation of the craniolos;ical materials then available, that the earliest British people were characterised by markedly elongated and narrow skulls, to which he gave the name kumbeceplialic , and that after a time a brachycephalic people appeared on the scene, who, though still practising the simple methods prevalent in the Stone Age, were to some extent acquainted with the use of bronze ( Prehistoric Annals of Scotland , vol. i. p. 253). Through the researches of Bateman (Tew Years ’ Diggings , etc.), Thurnam and Davis ( Crania Britannica ; Mem. Anthrop. Soc., vols. i. and iii. ), Busk ( Journ . Etlmol. Soc. Loudon , 2nd S., vol. vi.), Greenwell and Bolleston ( British Barrows ), and others, archaeologists have been long conversant with the fact that, as a rule, the crania found in the chambered cairns of Wiltshire, Somerset, Gloucester, and some 1905-6.] Human Skeleton , with Prehistoric Objects. 285 adjacent localities were dolichocephalic ; hut, on the other hand, that both forms Avere found, almost in equal proportions, in the round barrows and other graves of the Bronze Age. Although Dr Thurnam’s aphorism, “ Long barroAvs, long skulls ; round barrows, short skulls,” is not strictly accurate, it undoubtedly conveys an important ethnological fact, which is thus stated by Professor Rolleston : — “No skull from any long barrow, that is to say, in no skull undoubtedly of the Stone Age, examined by us, has the breadth been found to bear so high a relation as that of 80 : 100 of the length.” The more recent discoveries of human remains in the Oban caves ( Proc . Soc. Antiq. Scot., vol. xxix. p. 410), and in the chambered cairns of Arran {ibid., vol. xxxvi. p. 74 et seq.), also lend support to the same view. A new and wider significance has been given to the above generalisation by the Hon. John Abercromby in a paper com- municated by him to the Anthropological Section of the British Association held at Belfast in 1902, and published in the Journal of the Anthropological Institute for the same year (vol. xxxii. p. 373), in Avhich he advocates the hypothesis that the beaker, or, as it was formerly called, “ drinking-cup,” is the oldest Bronze Age ceramic in Great Britain, and that it Avas an imported type from Central Europe, by Avay of the Rhine valley. In the discussion Avhich folloAved the reading of this paper, Dr T. H. Bryce made the following remarks : — “Not the least interest- ing feature of Mr Abercromby’s valuable paper is the Avay in which his conclusions conform Avith the general trend of the evidence derived from the study of skull forms. Wherever the beaker has been found in this country associated with human remains, the skull has been brachycephalic in proportions, and the region from Avhich he derives this ceramic is Avithin the area of the ‘ Alpine ’ broad-headed type ” {ibid., p. 396). Since then, Dr Bryce {Proc. Soc. Antiq. Scot., vol. xxxix. p. 418) has tabulated the records of some tAvelve cist-interments Avhich disclose this relationship betAveen beakers and brachycephalic skulls. To these may be added another example from Duns, in Berwickshire {ibid., vol. v. pp. 240, 279), Avhich, together with the Largs specimen, make fourteen in all. It must not, however, be forgotten that beakers are not exclusively confined to short, 286 Proceedings of Royal Society of Edinburgh. [sess. •cists, as they have been found in cairns, barrows, stone circles, etc. Nor had the brachycephalic people in Scotland a monopoly of the short-cist mode of sepulchre, as one of Sir Daniel Wilson’s kumbecephalic skulls was found in a short stone cist at Cockenzie, East Lothian ( Prehistoric Annals , vol. i. p. 238). Nor, indeed, is the association of beaker and brachycephalic skull within a short cist an absolute rule, as a cist at Broomend, Aberdeenshire, contained a beaker and a couple of skeletons, one of which had a skull with a cephalic index of 78 (ibid., vol. vii. p. 113). A few exceptions would not, however, invalidate the general deduction suggested by Mr Abercromby’s paper. The culture of the brachy- •cephalic immigrants who surged from the Alpine regions into Western Europe, and ultimately entered Britain by way of the Bhine, would doubtless become more or less affected by that of their predecessors, and vice versa. That this has been the case is proved by the finding of skeletons of both races in the round barrows of England and in the Bronze Age burials of Scotland. Thus, Dr Garson describes seven skeletons found in a round barrow in Yorkshire, the cephalic index of which varied from 65*5 to 79'6, or an average of 74*7 (Journ. Anthrojp. Instil., vol. xxii. p. 8). Of 17 skulls from Bronze Age interments in Scotland, examined by Sir William Turner, 12 were brachycephalic and 5 dolichocephalic (Prehistoric Scotland, p. 455). These statistics conclusively prove that the brachycephalic had found their way to Scotland in considerable numbers, so much so that in some places they appear to have been more numerous than the dolicho- cephalic,— a result which may perhaps be explained on the supposi- tion that the latter had fled or retired to the higher grounds as the former advanced over the country. With regard to the ethnology of Ireland, Sir William Wilde expressed the opinion that two races existed simultaneously in that country, viz. a long-headed, dark, Irish stock on the west of the Shannon, and a fair-haired, globular-headed stock on the north-east of that river. But this view has not been corroborated bv subsequent researches. So far as I know, the opinion of Pro- fessor Huxley, published forty years ago, still holds good. “As the evidence stands at present,” writes the Professor, “ I am fully disposed to identify the ancient population of Ireland with the 1905—6.] Human Skeleton, ivith Prehistoric Objects. 287 * long-barrow ’ and ‘ river-bed ’ elements of the population of England, and with the long-headed or * kumbecephalic ’ in- habitants of Scotland ; and to believe that the ‘ round-barrow ’ or Belgic element of the Britannic people never colonised Ireland in sufficient numbers to make its presence ethnically felt ” (Pre- historic Remains of Caithness , p. 127). The fact that the beaker type of sepulchral ceramic has very rarely, if at all, been found in Ireland, may be accounted for on the supposition that the Continental brachycephalic were later in entering that country, or perhaps that they found their way to it by a different route from those who entered Britain by way of the Rhine valley. Anyhow, the rarity of both the beaker and the brachycephalic skull in the prehistoric burials of Ireland is a remarkable coincidence, and supplies fresh evidence in support of the above exposition of Irish •ethnology by Professor Huxley. Without entering on further argumentative details, the following propositions may be accepted as a fair summary of the ethnic •elements, so far as these have been determined by modern research, which have helped to mould the physical characters of the highly mixed population now inhabiting the British Isles — but, of course, altogether apart from the influence of the environment, which, as a modifying influence on racial characters, may have been very potent. (1) Anthropological researches have shown that during the Neolithic Age a long-headed race, of short stature but strong physique (average height 5 feet 5 inches), and who buried their •dead in rudely constructed stone chambers, had spread over the whole of Western Europe, from the Mediterranean to the south of •Scandinavia. Tacitus informs us that he identified the Silures, a people then occupying South Wales, as Iberians, on account of their swarthy complexion and curled hair (Agricola, xi.). The inference that these Silures were the direct descendants of the primitive long-headed people was not unreasonable, more •especially as by that time the eastern parts of Britain had been taken possession of by successive waves of Gaulish and Belgic immigrants from the Continent — thus causing the earlier in habitants to recede more and more westwards. And if this be so, it follows that the long-headed man of the Chambered Cairns of 288 Proceedings of Poycd Society of Edinburgh. [sess. Britain, Ireland, France, as well as many other parts of the Continent, had a swarthy complexion, with dark hair and eyes, like so many people still inhabiting the more secluded parts of these localities. (2) The incoming brachycephali were taller than the dolicho- cephali already in possession of the country, a statement which is proved by actual measurements of skeletons ( average height 5 feet 8 inches). Although they have been described by many modern writers as “light in hair and complexion” ( British Barrows , p. 636), there does not appear to be any archaeological evidence to support the assertion. The mistake seems to have arisen from inadvertently applying to the Bronze Age brachycephali qualities which were undoubtedly applicable at a later period to the Celts of history. The former buried their dead in short cists and round barrows, and carried with them a knowledge of bronze. While these two early races (the dolichocephali and brachycephali) were living together, apparently in harmony, the custom of disposing of the dead by cremation spread over the land — a custom which was introduced from the Continent, and had its origin probably in the strong religious elements of the time, as it was practised by both. (3) At a considerably later period, but not many centuries prior to the occupation of Britain by the Romans, there was another Continental wave of immigrants, generally regarded as an offshoot of the “ Glalli ” of classical authors, and probably the Belgae of Csesar, who introduced the industrial elements of the civilisation known in this country as “Late Celtic.” These newcomers differed radically from the former so-called Celtic invaders in having dolichocephalic heads — a statement which is supported by archaeological evidence, as, for example, a skull found in a characteristic late Celtic tumulus at Arras, Yorkshire, was described by Dr Thurnam as having a cephalic index of 7 3 *7. They were a branch of the Celts of history, whose very name at one time -was a terror in Europe, and by classical writers they are described as very tall and fierce-looking, with fair hair, blond complexion, and blue eyes. (4) The next and last of the great racial elements which entered into the ethnic composition of the British people of to-day were 1905-6.] Human Skeleton, with Prehistoric Objects. 289 the successive Teutonic invasions from Germany, Denmark, and Scandinavia, all belonging to a tall, blond, dolichocephalic people who existed in Central Europe from time immemorial — possibly the descendants of the Neanderthaloid races of Palaeolithic times. We have made no reference to the Roman occupation as a factor in British ethnology, because the Romans were a mere ruling caste, who, although they introduced new arts, industries, and customs into the country, kept themselves aloof from the natives, and did not, as a rule, intermarry with them. So that when they finally abandoned Britain they left its inhabitants racially un- affected, much as would be the case if the British were now to retire from India. To-day we hunt for remains of military roads, camps, accoutrements of war, and other relics of their civilisation, but of their skeletons we know very little, and of their British offspring nothing at all. But there is another standpoint from which these skulls have a special interest to the British people of to-day, viz. — What has been the subsequent fate of the primitive races they represent? Can these highly differentiated skull-forms, so strikingly illustrated by the specimens now before us, be still traced among our modern populations? If not, has the assimilation of the two races with one another, and with subsequent immigrants, so equalised their early cranial peculiarities as to be no longer recognisable ? From the researches of modern writers on physical anthropology, as stored up in various anthropometric documents, it appears that there is great uniformity in the cephalic index among the present inhabitants of the British Isles. Thus, Dr Beddoe gives, as the result of the measurement of seven groups of Scotchmen in various stations of life, 7 5 ‘5 as the highest average cephalic index, and 74*2 as the lowest ( Anthropological History of Europe , p. 104). Mr Ripley, in his Races of Europe (p. 304), thus writes : — “Wherever heads have been measured, whether in the Aran Islands off the west coast of Ireland, the Hebrides and Scottish Highlands, Wales and Cornwall, or the counties about London, the results all agree within a few units. These figures, noted upon the localities where they were taken, are shown upon our little sketch map on page 304. It will be observed at once that the indexes all lie between 77 and 79, with the possible exception PRO(J. ROY. SOC. EDIN. — YOL. XXVI. 19 290 Proceedings of Royal Society of Edinburgh. [sess. of the middle and western parts of Scotland, where they fell to 76.” Notwithstanding this uniformity of craniological types now prevalent among our populations, it is not a remarkably rare occurrence to meet with a specimen of the dolichocephalic skull which has survived to historical times, with apparently little deviation from its primitive normal characters, such as that from a modern graveyard in Aberdeenshire, referred to by Professor Cunningham (p. 294 et seg.). On this point the following remarks by M. de Quatrefages are worth quoting : — ‘ ‘ In passing through the Copenhagen Museum, I was struck by the Neanderthal characters presented by one of the crania in the collection ; it proved to be that of Kay Lykke, a Danish gentleman, who played some part in the political affairs of the seventeenth century. M. Godron has published the drawing of the skull of Saint Mansuy, Bishop of Toul, in the fourth century, and this head even exaggerates some of the most striking features of the Neanderthal cranium. The forehead is still more receding, the vault more depressed, and the head so long that the cephalic index is 69 *41. Lastly, the skull of Bruce, the Scottish hero, is also a reproduction of the Canstadt type ” ( Human Species , p. 309). But all this merely proves the strong tendency to survivalism which seems to be peculiar to dolichocephalism. On the other hand, we seem to possess fewer traces of the brachy cephalic skulls among our populations of the present day, a fact which is more remarkable inasmuch as the descendants of the “Alpine” broadheads are still the predominating race among the modern populations of France, Belgium, Italy, and Germany. It would seem that brachycephalism was a mere mushroom growth of the Neolithic period, for evidence of it is not, as far as I know, to be found in Palaeolithic times. If so, it may be a comparatively unstable factor in the organic evolution of man, and may be paralleled with the fact that a domestic animal when allowed to run wild quickly reverts to the general wild stock and assumes its primitive characteristics. What, then, becomes of the general opinion held by so many of our foremost ethnologists, that these tall, round-headed invaders of our country in the Early Bronze Age were the true Celts of history % If we accept the affirmative of this problem, it would appear as if the brachycephalic skull has become so modified, in the course of some two thousand years, by cross-breeding, etc., as to come within the category of a 1905-6.] Human Skeleton , ivith Prehistoric Objects. 291 well-filled dolichocephalic skull. To this conclusion there is only one alternative, viz. that the' race has gradually died out, on the principle of the extinction of the unfittest, having had to give way to the superior vitality of the long-headed race, who were much longer acclimatised to the country. The cephalic index is, however, only one of the factors which ethnologists make use of in their investigations. Stature, colour of hair and eyes, and even language, especially when fossilised in place-names, supply important evidential materials, not to mention the incidental references of classical authors to the proto-historic inhabitants of Europe. Now, as regards all these physical features, there is a sufficient diversity to be seen among the present inhabitants of this country to suggest a thorough blending of all the racial factors which has to be accounted for in discussing British ethnology. The effect of cross-breeding on the colour of the hair and eyes is difficult to be determined, as the child may sometimes strongly resemble the father and sometimes the mother, or sometimes neither, but may revert, on the principle of atavism, to the type of a more remote ancestor. That, however, a blend in pigmentation ultimately takes place in the course of many generations, and has taken place, is undoubted. Possibly the hazel and grey eyes, now so commonly met with, may be in- termediate shades between the dark Iberian and the blue-eyed later Celts, or Galli, or Belgians. It is a remarkable fact that the so- called “ Celtic fringe ” of to-day — i.e. the highland and more inaccessible regions in which the Celtic languages have survived longest — almost coincides with the scattered geographical areas where the dark Iberian people still form the majority of the population. Now, if the language of this dark, long-headed race was not Celtic, we have a striking instance of the instability of language as a racial character. Eor it would appear as if the descendants of this primitive race, within the historic period, had captured and appropriated the entire heritage of the renowned Celts of Europe as regards language, tradition, and civilisation ; while the modern representatives of the latter, so far as concerns Britain, are absolutely lost among, and undistinguishable from, the modern Teutons. Whatever may be the inherent value of these general remarks, 292 Proceedings of Royal Society of Edinburgh. [sess. they bear evidence of the difficulties encountered in the prosecu- tion of anthropological researches, and of the absolute necessity of subjecting every new discovery, whether it be a fragmentary human skeleton or a relic of man’s handiwork, to a minute examination at the hands of experts, such as we have had on the present occasion. One of the greatest drawbacks to physical anthropology is the difficulty of associating the facts of craniology with those of the other racial characters on which ethnology is founded. Skeletons do not reveal to us anything of the colour of the hair, eyes, or skin of the individual who owned them ; nor of the language they spoke, nor of the religious ceremonies they enacted, nor of the implements, weapons, ornaments, and clothing by means of which they fulfilled their destinies in the organic world. One of the most puzzling problems transmitted to us by classical writers is that they describe two early European races, one short and dark, and the other tall and fair, both of which were dolichocephalic. That brachycephalic immigrants entered France from somewhere to the east at the dawn of the Neolithic period, while the tall dolichocephalic race still lived in the country, there is abundant evidence to show. These latter I am inclined to regard as the descendants of the Palaeolithic people of Europe, who had acquired their fair skin, hair, and eyes during their struggles for existence against the severe conditions of life imposed upon them by the Ice Age. But as to the brachycephalic hordes who ultimately pushed their way into Britain, and introduced the Celtic language, which subsequently became the prevailing speech of the British Isles, I am absolutely at a loss to account either for their origin or racial characteristics, beyond the fact that they possessed round-headed and mentally capacious brain-cases. Note on the Fragments of a Beaker from Largs. By the Hon. John Abercromby. The fragment in question has an extreme length of 20 cm., and a chord of the circumference measures 13*3 cm. When whole, the beaker must have had a maximum diameter of about 17 cm., a height of about 22 ‘9 cm., and it seems to have belonged to type /3, i.e. ovoid cup with recurved brim. Although such a height 1905-6.] Stone Cist containing Skeleton and Urn. 293 is unusual, it occurs with two beakers of the same type from Court Hill, Dairy, Ayrshire, from Largie, Poltalloch, Argyleshire, and on a beaker of type y, i.e. low-brimmed cup, from Collessie, Fife. The tallest on record is from Somersham, Hunts, and measures as much as 27 -4 cm. The clay is fairly well levigated, and contains only a few small stones. Externally the fragment presents a reddish-brown colour. The fracture shows on the outside half of the thickness a reddish-chocolate colour, which becomes darker and blacker towards the inner surface of the vessel. The outside surface is fairly smooth. As is usual in this class of ceramic, most of the ornament is executed with a narrow instrument, such as a notched slip of hone or wood, leaving small rectangular holes, separated by a narrow septum. But, owing to the shortness of the lines, the horizontal line-chevron fringe bordering each ornamented band is made with a blunt point. The ornament is quite normally disposed in horizontal bands, alternately plain and ornamented. Here we have two of the former and three of the latter. In the central band the principal motive of ornament consists of two parallel line-chevrons, spaced, with their opposite angles united by vertical lines. This motive seems to be a special northern development, as it is found from Boss-shire to Staffordshire, but not further south. The only new motive is the vertical fringe of short horizontal strokes |=E H| which breaks the uniformity and continuity of the belt of ornament of two of the ornamented bands. A similar fringe is found on a beaker of type /3 from Gian yr Avon, Denbighshire. So far as I judge, this fragment belonged to a beaker that may be placed about the middle of the Beaker period. Report on Two Crania, submitted by Dr B. Munro for Examination. By D. J. Cunningham, M.D., F.B.S. The specimens which have been placed in my hands by Dr Munro may be respectively designated the Rutland cranium and the Largs cranium, from the districts in which they were found. They belong to two very different types — the Rutland specimen being long, narrow, with prominent brows and sloping forehead, whilst 294 Proceedings of Royal Society of Edinburgh. [sess. the Largs cranium is round and lofty, with a straight vertical forehead. Rutland Cranium. The Rutland specimen consists of the calvaria alone, and even that is slightly damaged. The facial and basal parts of the skull are gone, or only represented by a number of small fragments, of which a portion of the lower jaw is alone of any value for re- construction purposes. The calvaria, evidently that of a male, possesses certain strongly pronounced characters which give it a striking individuality. These are — (1) a marked projection of the supraorbital part of the frontal hone, due to expansion of the frontal air-sinuses ; (2) a constriction of the cranium behind the orbits, leading to consider- able narrowing of the forehead at this point; and (3) a strong backward slope of the frontal plate of the frontal bone. It is a type of skull with which the anatomist is not unfamiliar. A calvaria described many years ago by Sir William Turner, and now in the Anatomical Museum of the Edinburgh University, presents somewhat similar characters. This may be regarded as a comparatively speaking modern specimen, as it was found while digging the foundation of Gordon’s Hospital in Aberdeen, an institution which is built on the site of the Blackfriars Monastery.* Another skull obtained from a “ Sambaqui ” in Santos in Brazil, and described by Nehring, may also be said to show corresponding features.! Amongst the Australians a similar type of cranial contour is likewise sometimes met with. Such skulls are not infrequently called Neanderthaloid, on account of the forehead and eyebrow regions presenting some resemblance in general contour to the corresponding parts of the famous Neanderthal cranium, but in other respects they stand upon so much higher a plane that such a term is misleading and inappropriate. * Additional Note on the Neanderthal Skull, by William Turner, M.B., Quarterly Journal of Science, 1864, October, p. 758. t “ Menschenreste aus einem Sambaqui von Santos in Brasilien unter Ver- gleichung der Fossilreste des Pithecanthropus erectus Dubois,” by A. Nehring, Verhandlungen der Berliner anthropologischen Gesellschaft, November 16, 1895. 1905-6.] Report on Two Crania. 295 Cephalic Index. — The length and breadth measurements of the three skulls referred to above are the following : — Maximum Length. Maximum Breadth. Cephalic Index. Rutland specimen 188 138 73'4 Aberdeen ,, ... 195 150 76*9 Brazil , , ... 183 142 77-6 These figures might lead one to place the Rutland skull, which is dolichocephalic, in a different category from the other two specimens. Its low cephalic index in comparison with the higher mesaticephalic indices of the Aberdeen and Brazil skulls is a feature which cannot be ignored. Still, it must be remembered that the maximum antero-posterior diameter of the cranium is composed of two factors of altogether different moment and significance, and in crania with large inflated frontal air-sinuses the cephalic index loses much of its importance as a differential character. When the depth of the frontal air-sinus is omitted from the calculation of the index, the Rutland and Aberdeen skulls enter the brachycephalic list, although they are still differentiated with some sharpness by this character ; unfortunately, in the case of the Brazil specimen, we have not the figures necessary for the calculation of the index. Maximum Length of the Brain case. Maximum Breadth. Cephalic Index. Rutland skull 172 138 80*2 Aberdeen , , 178 150 84*3 Projection of the Supraorbital part of the Frontal Bone.— As is well known,* the projection in this region may be due to expansion * See Logan Turner, Accessory Sinuses of the Nose, Edinburgh, 1901 ; also Zuckerkandl, Normale unci Patliologische Anatomie der Nasenhohle und ihrer pneumatischen Anhdnge, 1 Bd. 2 Auflage, 1893. 296 Proceedings of Royal Society of Edinburgh. [sess. of the frontal air-sinus, or to a large extent to a deposit of bone, as in the case of certain Australian skulls. In both the Rutland and the Aberdeen specimens the supraorbital projection is due to the former cause, and not to any massing of bone in this neighbour- hood,— the front wall of the sinus in both cases being not more than 4 mm. thick. Schwalbe has taught us to examine carefully and critically the contour of the supraciliary region and its relation to the margin of the orbital opening.* In some cases the supraciliary ridge is fused with the upper part of the orbital rim, and the result is an arcuate continuous projection overhanging the orbital opening. This is the case in the chimpanzee, gorilla, pithecanthropus, and Neanderthal skulls, and in the crania of certain Australian aborigines. In other cases the inner part of the supraciliary ridge is fused with the inner part of the upper portion of the orbital rim, whilst its outer part stands above and apart from the margin of the orbit, being borne upwards, as it were, by the supraorbital nerve, which seems to have some effect in producing this condition. There are many races which show this type of supraciliary projection, and it is not infrequent to meet with it in the modern European skull. Both the Rutland and the Aberdeen skulls fall within this group. Probably in the European skull it is most usual to find the supraciliary ridge standing quite apart, in its whole length, from the orbital margin. This may be regarded as a third type of supraciliary contour. The narrowing of the Cranium behind the Orbits. — The minimum frontal diameter of the Rutland skull, determined between the temporal ridges, measures 91 mm. It is a difficult matter to deal with this diameter in such a way as to arrive at a proper con- ception of its true worth from a comparative point of view, and yet the information gained by the eye is sufficient to show that it is a factor of considerable importance. The absolute measure- ment as well as the various indices which have been devised may altogether fail to give expression to its proper value and to convey to the mind its right significance. * “Studien iiber Pithecanthropus erectus, Dubois," by G. Schwalbe, Zeit- schrift fur Morphologie und Anthropologie, Band i. Heft 1. 1905-6.] Report on Two Crania. 297 In a table given by Schwalbe which deals with 352 skulls, there are no less than 66 specimens with a minimum frontal diameter below 91 mm., and included amongst the latter there are two modern European skulls. The majority of these instances in which this diameter attained so small a dimension occurred in such races as the Yeddahs and Australians, in some of which it reached the exceedingly low dimension of 81 mm. From his investigation into this character, Schwalbe comes to the conclusion that it is not always in the lowest races that the lowest postorbital breadth of cranium is found, nor yet is it in the highest races that the diameter attains its maximum. In the Brazil skull the minimum frontal diameter was even less than in the Rutland specimen. It measured 88 mm. In the Aberdeen skull, on the other hand, in conformity with its larger size, the minimum frontal diameter is 97 mm. With the view of pushing the comparison still further, outline tracings of the norma verticalis of the Rutland and Aberdeen crania were taken by means of the American periglyph ; these were then reduced to a common standard of size by photography and superimposed. The result was very instructive, because it became evident that the contour lines of the two specimens, not only in the postorbital region, but also throughout the entire extent of the tracings, were very similar. The fronto-parietal index has been employed to express the relative degree of postorbital constriction. In calculating this index the maximum breadth of the skull is taken as 100 and compared with the minimum frontal diameter. The calculation is made in the following manner : — Minimum frontal diameter x 100 Maximum breadth Virchow had considerable faith in this index, and employed it in connection with his study of the cranium of Pithecanthropus,* but I am in complete agreement with Schwalbe in the view that it gives no true information regarding the point at issue. The more variable factor, viz. the parietal breadth, is taken as the standard of comparison, and it is not surprising, therefore, that in narrow- * “ Ueber Pithecanthropus erectus, Dub.,” Zeitschrift f. Ethnologic, October 1895, Heft 6. 298 Proceedings of Boyal Society of Edinburgh. [sess. headed or dolichocephalic races such as the Australians the index should on the average he higher than in the European. This index, therefore, affords no proper conception regarding the degree of postorbital constriction. Another and a better method is to compare the biorbital diameter ( i.e . the measurement between the outer margins of the extremities of the two external angular processes of the frontal bone) with the minimum frontal diameter thus : — Minimum frontal diameter x 100 Biorbital diameter Fronto-Parietal and Biorbital Indices. Min. Front. Dia. Fronto-parietal Index. Biorbital Index. Rutland specimen 91 61-5 89-2 Aberdeen , , 97 64*6 88-2 Brazil , , 87 61*2 75 This table illustrates what is said in regard to the value of these two indices. The fronto-parietal index would appear to indicate that the degree of postorbital constriction is equal in amount in the Rutland and Brazil specimens, and that both of these are relatively more constricted in this region than the Aberdeen specimen. An ordinary inspection by the eye is sufficient to show that this is not the case, and that the figures of the biorbital index give a much more accurate idea of the degree of cranial narrowing behind the orbits. The most notable feature in the Brazil skull is the narrowing in this region. In Schwalbe’s table xviii. there is no human skull which presents so low a biorbital index as the Brazil specimen. The average biorbital index for the natives of Alsace is 91*8 for the males and 94 for the females, although there are individual cases in which it sinks as low as 82’9, 86*7, and 87T. In the light of this information, therefore, whilst it is clear that the postorbital constriction in the Rutland skull constitutes a marked feature, we cannot say that its degree is exceptional, even amongst modern European crania. 1905-6.] Report on Two Crania. 299 Inclination of the Frontal Bone. — The slope of the frontal plate of the frontal bone must always he regarded as possessing a very considerable degree of anthropological interest. As we descend from the higher to the lower races, the general tendency towards an increasing degree of backward slope of the forehead becomes manifest. It has already been stated that the obliquity of the frontal region of the calvaria constituted a striking characteristic of the Butland specimen. Fig. 1. — The Rutland Cranium. Tracing of the mesial longitudinal arc obtained by the American periglyph. Reduced by one-half. Several methods may be adopted for the purpose of expressing more or less accurately the degree of frontal inclination. Three of these which were followed by Schwalbe in his study of the cranium of Pithecanthropus may in the first instance be applied to the Rutland specimen. The first step in each of these methods consists in obtaining an accurate tracing of the mesial longitudinal arc of the cranium. This may be done by the American periglyph, and upon the tracing a base line should be drawn from the inion to the centre or most prominent point of the glabella. (1) Schwalbe’s first method consists in dropping a perpen- dicular from the bregma so as to intersect the base line at right 300 Proceedings of Royal Society of Edinburgh. [sess. angles. If the frontal arc were relatively of the same length in all skulls, the point of intersection of the base line would move forwards and backwards as the frontal bone becomes elevated and depressed. This is the principle on which Schwalbe proceeds, but the results of the method are very unreliable. There are two sources of error : ( a ) as everyone knows, the relative length of the frontal part of the longitudinal arc is a variable quantity in different races, and even in different individuals of the same race ; (b) and further, increase in degree of verticality of the frontal Fig. 2. — Aberdeen Cranium. Tracing of mesial longitudinal arc obtained by the American periglyph. Reduced by one-half. region does not depend so much on the erection of the whole frontal bone as on an increase on its degree of curvature. An index of frontal inclination formed on data ascertained by this method may be obtained as follows : — Part of base line in front of perpendicular from bregma x 100 Length of base line (2) A second method consists in drawing a straight line from the most prominent part of the glabella (i.e. anterior end of the base line) to the most prominent part of the frontal curve, and measuring the angle formed by it and the base line. 1905-6.] Report on Two Crania. 301 This may be called the frontal angle', and whilst it is calculated to give a fair general idea of the slope of the forehead, it must not be forgotten that the varying degree of prominence of the glabella constitutes a disturbing element and introduces a different factor into the result. In skulls such as those under consideration, where the supraorbital projection is great, the general slope of the frontal bone is exaggerated through the pushing forward of the lower end of the frontal line. Fig. 3. — Australian Cranium — low type. Tracing of mesial longitudinal arc obtained by American periglyph. Reduced by one-half. (3) A third method of estimating the slope of the forehead which has been employed by Schwalbe consists in drawing a line from the most prominent part of the glabella to the bregma (bregma line), and measuring the angle which it forms with the base line. This is not a satisfactory plan, because it altogether leaves out of count the degree of forward bulge of the frontal bone, and, as already mentioned, this is quite as important a factor in determining a vertical forehead as the general inclination of the bone. None of these methods, therefore, are altogether trustworthy, although in all probability the frontal angle gives the best results. 302 Proceedings of Boy al Society of Edinburgh. [sess. Conjoined with this, however, the degree of frontal curvature should always he determined. This can he done hy dropping a perpendicular on the hregrna line from the point of maximum frontal curvature, and comparing the length of such a line with the length of the hregrna line thus : — Length of frontal perpendicular x 100 Length of hregrna line The following table gives the results obtained hy the applica- tion of these different methods in the case of the three skulls under consideration. Frontal Inclination. Index showing position of Bregma perpendicular on Base Line. Frontal Angle. Angle of Bregma Line. Index of Frontal Curve. Rutland specimen . 29*4 73*5° 59° 13 Aberdeen , , 38*9 69° 51° 137 Brazil , , 33*6 o O 54° 12 If we examine the indices which show the position of the hregrna perpendicular on the base line, it becomes evident that no just conception can he obtained from them as to the slope of the forehead. Schwalbe gives the average index for the inhabitants of Alsace as 30 -5 ; for the negro as 32*1 ; and for the Kalmuck as 32*8. In the Rutland skull the index fails to bring out the degree of frontal inclination ; in the Aberdeen skull it exaggerates it, seeing that the frontal section of the longitudinal arc is un- usually long ; whilst in the Brazil skull the index, compared with those furnished by Schwalbe, probably gives a tolerably true idea of the condition. The frontal angle affords better information on this matter. Amongst living races of men Schwalbe only found one skull with an angle as low as 73°. Had his investigation extended over a wider range, he would probably have found many more with a irontal angle at least as low. For the natives of Alsace he 303 1905-6.] Report on Two Crania. obtained an average angle of 93'7° for the females and of 91 *4° for the males. The angle of the bregma line fails altogether to give a true result in the case of the three skulls under consideration. As will be shown later, the frontal slope of the Brazil and Aberdeen skulls are almost identical as regards degree and quality, and yet the bregma angle separates them in this point widely from each other. The index of the frontal curve brings out satisfactorily the amount of forward bulge of the frontal bone. In the three specimens it is very nearly the same, and must be regarded as being exceptionally low. In a brachycephalic Irish skull I found the index 22*7, whilst in a dolichocephalic Scotch skull it was 18*6. From the various tests applied to the frontal region of the Rutland skull it becomes apparent that the forehead is unusually low and receding, although not in so great a degree as in the Brazil and Aberdeen specimens. Height of the Calvaria. — The base of the skull being absent, it is necessary to estimate the height of the calvaria by measuring the length of a line drawn from the most distant point of the longitudinal arc to the base line, and in such a direction as to cut the latter at right angles. An index of calvaria height can be estimated thus : — Calvaria height x 100 Length of base line This method gives excellent results, seeing that the measurements from which the index is calculated deals with that part of the cranium which holds the great brain, or that portion of the cranial cavity which is subject to the largest amount of racial and individual change. Index of Calvaeia Height. Base Line. Height. Index. Rutland skull .... 180 100 55-5 Aberdeen . 195 98 50*3 Brazil, ,, 51-6 304 Proceedings of Royal Society of Edinburgh. [sess. These are low indices, but more particularly is this the case in the Aberdeen and Brazil specimens. The Rutland skull stands in this respect midway between these and the modern European skull. According to Schwalbe, the natives of Alsace have a calvaria height index of 59*8 ; in a dolichocephalic Scottish skull it was 58T ; and in an Australian skull of low type it was 52 1. When tracings of the longitudinal arcs of the crania in question are reduced to a common length of the base line (i.e. diameter between glabella and inion), and superimposed, a very effective Fig. 4. — Cranial outlines superimposed. 1. Female Gorilla. 2. Pithecanthropus erectus, Dubois. 3. Neanderthal (tracing taken from photograph by Schwalbe). 4. Aberdeen Cranium (Sir William Turner). 5. Brazil Cranium (Nehring). 6. Eutland Cranium (Munro). demonstration of the relative height of each is obtained, and the quality of the curvature in each case becomes evident. In the accompanying figure the numbers 4, 5, and 6 indicate the outlines of the Aberdeen, Brazil, and Rutland crania respectively. The greater height of the Rutland specimen (6), which is also shown in the index, is manifest, whilst the close manner in which the 305 1905-6.] ' Report on Two Crania. outlines of the Aberdeen and the Brazil crania follow each other is very remarkable. In instituting this comparison between the three crania in question, it should be noted that only in one, viz. the Brazil specimen, is the face preserved — and this only partially. Still, enough remains to show that the Brazil skull was distinctly prognathic. It is most unlikely that the Aberdeen skull possessed the same degree of this character ; but it should be noted that there is no proof one way or another. In the case of the Rutland specimen it is safe to say that it was not prognathic. A fragment of the front part of the body of the lower jaw of this skull has been preserved, and in this the chief distinguishing features are (1) the vertical sockets for the incisor and canine teeth, and (2) the very pronounced mental prominence — a prominence which extends outwards in a ridge-like manner beyond the incisor portion of the jaw, and which terminates on each side in a marked tubercle. Largs Skull. The evidence which the Hon. Mr Abercromby has advanced to show that the beaker urn belongs to the most remote period of the Bronze Age has been the means of stimulating an increased degree of interest in the human remains which have been found associated with this form of ceramic. Dr Bryce * has gathered together the records of twelve such crania, all found within the Scottish area, and each singly within a closed short cist, under conditions similar to those under which the Largs specimen was discovered. These crania exhibit a remark- able uniformity in almost all essential details, and one cannot help concluding that they are derived from a very homogeneous and distinct race. The Largs specimen conforms in a striking manner with this type. A very casual examination is sufficient to show, notwithstanding its damaged condition, that in it we have a combination of definite characters seldom, if indeed ever, encountered in association with each other in modern crania. * “Notes on a Human Skeleton found in a Cist with a Beaker Urn, and on the Cranial Form associated with that type of Ceramic,” Proc. Soc. Antiq. Scot., May 8, 1905. See also Prehistoric Human Slcelctons found at Merthyr Mawr, Hepburn, Cardiff, 1905. PROC. ROY. SOC. EDIN. — YOL. XXVI. 20 306 Proceedings of Iioyal Society of Edinburgh. [sess. The specimens which were obtained from the Largs short cist consisted of — (1) an imperfect cranium ; (2) a portion of the right upper jaw ; and (3) the lower jaw, — all clearly belonging to the same individual. The cranium is that of an adult male. The face (with the exception of the jaw-bones mentioned) and likewise the part of the floor of the cranium around the foramen magnum are absent : fortunately the basion is in position. From the mid-parietal Fig. 5. — Largs Cranium. Tracing of mesial longitudinal arc obtained by the American periglyph. Reduced by one-half. region forwards the whole of the left lateral, left frontal, and left parietal portions of the wall are gone, but the right part of the roof and of the frontal bone as low as the right supraciliary ridge are present, and thus afford the means of approximately fixing the position of the glabella. The external occipital protuberance is strongly marked, and coincides with the posterior occipital point when the specimen is held in proper position. Above the inion the posterior wall ascends with a steepness which is remarkable, and the back V\5 r 1905-6.] Report on Two Crania. 307 of the head in consequence presents a peculiar flattened appear- ance. The supramastoid crest stands out with an unusual degree of strength, and immediately below there is at the base of the mastoid process and immediately behind the ear-hole a deep depression. The various measurements which were possible are given in the following table. Two of the series of skulls dealt with by Dr Bryce, both of which have been described by Dr Low, of Aberdeen, are included in the table for purposes of comparison. Largs Cranium. Stoneywood Cranium (Dr Low). Persley Cranium (Dr Low). Maximum length . 175 (ap. ) 169 188 Maximum breadth 148 156 160 Cephalic index 84 *6 (ap.) 92*3 85 Height 138 (ap.) 133 146 Height index .... 78*8 (ap.) 78*7 77*7 Height-breadth index 95 ’1 (ap. ) 85*2 91*2 Frontal angle 90° (ap.) 97° (ap.) 90° (ap.) Angle of glabello-bregma line 65° (ap.) 62° (ap.) 65° (ap.) Frontal Curve index 16*5 (ap. ) 21*9 (ap.) 20 (ap.) Length of glabello-inial line 1 (base line) . . . / 175 (ap.) Maximum height of Calvaria . 105 Index of Calvaria height 60 (ap.) From the above figures it will be noted that the Largs cranium not only presents a high degree of brachycephaly, but also a very high altitudinal index. In the Scottish crania examined by Sir William Turner this index presents an average of 70*9, a small proportion only being hypsicephalic, as is the case with the three crania included in the above table, as well as almost all the other crania which belong to this type (see Bryce’s table). Outline tracings were taken of the longitudinal arc of the photographs of Dr Low’s Stoneywood and Persley specimens, which are given by Dr Bryce. These were then enlarged until the base line (glabello-inial line) in each attained a length equal to that of the photograph of the Largs cranium. The three figures thus obtained have been superimposed in the accompanying illustration, with the view of bringing out the similarity of type exhibited in the quality of the cranial arc. 308 Proceedings of Royal Society of Edinburgh. [sess. Too much, reliance, however, cannot be placed on this figure. It only gives an approximate result, seeing that the glabella and the median frontal outline were absent in the Largs specimen, and that the inion in the photographs of the other two specimens could not be determined with absolute certainty. It may be well to mention that the Persley and Stoneywood crania were not chosen for the purpose of this comparison because, of Dr Bryce’s Fig. 6. — Superimposed cranial outlines. Photograph, Largs specimen (Munro). Continuous line, Persley cranium (Low). Dotted outline, Stoneywood cranium (Low). series, they seemed to present the greatest degree of similarity to the Largs cranium, but because it appeared to me that in these photographs I could determine with the best chance of accuracy the position of the inion. The fragment of upper jaw belonging to the Largs cranium calls for no special remark. The sockets show that all the molar teeth had been in place ; and further, that the incisor teeth had been implanted vertically. In the case of the lower jaw, the only part absent is the right 1905-6.] Report on Tiuo Crania. 309 ramus. It is quite a commonplace jaw, with nothing to distin- guish it from the modern mandible. The three molars of the left side and the first of the right side are present. There is little difference between these teeth in so far as the size of the crowns is concerned, but they are all much worn down, and each presents a more or less flat, even surface at the summit. Both canines and the first premolar on the right side are also present, and consider- ably ground down on the crowns. The incisors are absent, but the vertical character of the sockets, in conjunction with the corre- sponding character in the upper jaw, clearly shows that there was no tendency towards prognathism. The whole lower jaw is somewhat lightly built in comparison with the size of the molar teeth. (. Issued separately August 31, 1906.) 310 Proceedings of Royal Society of Edinburgh. [sess. Note on a rare Dolphin ( Delpliinus acutus ), recently stranded on the Coast of Sutherland. By Sir William Turner, K.C.B., F.R.S. (With Plate.) (Read June 4, 1906. MS. received June 29, 1906.) Through the courtesy of my friend, the Rev. Dr Joass, of Golspie, I received early in April of this year (1906) a specimen of a female dolphin, which had been stranded two days previously on the beach, about half a mile to the east of Dunrobin Castle, where a streamlet enters the sea. The animal, which was carefully secured and packed by a keeper and a gardener of the Duke of Sutherland, reached the Museum in excellent order, with the markings on the skin well seen, and the cuticle not abraded. The short pointed beak, the well-defined yellowish and white band, extending for some distance along the side of1 the body, and the dimensions of the animal, enabled me, without difficulty, to distinguish the species to be Deljihinus acutus (Gray), or, as he subsequently called it, Lagenorhynckus leucopleurus , the white-sided dolphin. As this dolphin has seldom been captured on our coasts, I took the opportunity to have it photographed and to write a tion of its characters. The principal measurements were as follows : — descrip- Extreme length along midline of back, 6 ft. From tip of beak to anterior margin of dorsal fin, 29 in. „ „ to anterior border of flipper, . 14 j) Length of flipper in straight line, 10 33 „ of attached border of flipper, 4 33 ,, of attached base of dorsal fin. 11 33 Height of dorsal fin, 6-5 33 Width of tail, ...... 15 33 Girth in front of dorsal fin, .... 35-75 Girth at root of tail, ..... 7-5 33 Length of lower jaw, ..... 12-1 33 The head ended in front in a pointed tip, and the lower jaw scarcely projected beyond the upper. One and three-quarter inch Proc. Roy. Socy. of Edin.] [Vol. XXVI. Sir Vm, Turner Fig. 2. — Ventral surface of Delphinus acutus. The parallel markings near the umbilicus were artificial, and had been made by a sharp instrument. ( These and the other figures are reproduced from Photographs by Mr John Henderson.) 1905-6.] On Delphinns aeutus from Coast of Sutherland. 311 behind the tip a groove was seen, which extended backwards on each side of the beak for 5 inches, and marked off a narrow ledge below and in front of the convexity of the head. From this groove the head, at first somewhat flattened laterally, ascended with a gentle curve to the blow-hole on the top of the head. The lateral flattening of the head contributed to give the pointed form anteriorly. Behind the blow-hole the back was rounded from side to side and the body had the greatest girth. The dorsal fin was sickle-shaped and projected vertically from about the middle of the back. Behind this fin the back was keeled, and preserved this character as far as the tail ; the width of the body also gradually diminished and became flattened laterally. The belly was also flattened in the greater part of its extent, but was keeled for a short distance in front of the tail. The caudal fin consisted of two horizontal flattened flanges separated by a mesial notch, and its posterior border was concave. The flipper was flattened on its two surfaces ; the anterior border was convex, the posterior a little concave, the tip pointed. The palpebral fissure was If inch behind the angle of the mouth ; the auditory meatus, so minute as to be seen with diffi- culty, was If inch behind the palpebral fissure. The mouth-slit was 8f inches long. No hairs were seen on the skin of the beak, or on that covering the symphysis of the mandible. The blow- hole, If inch wide, was 7 inches from each eye, crescentic in shape, with the concavity directed forwards. The dolphin was a female, and the genital fissure was 4f inches long ; on each side a narrow mammary fissure f inch long was situated. A small anal orifice was placed If inch behind the genital fissure. The umbilicus was 13 inches in front of the same fissure. Colour. — The beak was glossy black to the tip and sides of the upper jaw ; the dorsum and the sides of the head, the back and upper part of the side of the body, the upper surface of the tail as far as its concave border, the dorsal fin and both surfaces of the flipper were also a rich black. A long light-coloured band, pointed at its anterior end, situated on the side of the body, commenced below the dorsal fin, broadened as it passed backwards, and at about half its length was 2f inches wide, when it narrowed for some distance and terminated behind, 9 inches in front of the 312 Proceedings of Royal Society of Edinburgh. [sess. posterior border of the tail, in a bulbous expansion. In its anterior part the upper half of this band was yellowish-brown and the lower half was white, but posteriorly it was entirely yellowish-brown. The belly was white from near the tip of the lower jaw to 4 inches behind the genital fissure, on each side of which was a longitudinal greyish-black patch. Between the white belly and the band on the 'Side of the body the colour of the skin was greyish -black, and a slender band similarly coloured extended forward from the base of the flipper to the angle of the mouth. The ventral surface of the tail was greyish- white. Skeleton. — The length of the female, 6 feet, was less than that of a male, 8 feet 3 inches, described by Dr Duguid,* a difference due partly to sex and in part to the ossification in the former being in- complete, as the epiphysial plates were not fused with the bodies of the vertebrae. The Dunrobin specimen, therefore, was not adult, but in the stage of growth which may be called adolescent. The spinal column was 54f inches long. When examined with the vertebrae undisturbed in their natural position, the vertebral formula was cervical 7, dorsal 15, lumbo-caudal 58, in all 80, which is less than the number 82 said to have been present in some specimens of this species, and materially below the 88 to 92 vertebrae found in Delphinus albirostris. The 1st, 2nd, and 3rd cervicals were fused in their bodies, neural arches, spines and transverse processes into a relatively massive bone. Intervertebral discs were present between the bodies of the other cervicals, which were flattened, and the whole series formed a short, stunted, compressed neck. The dorsals had relatively long spines and articulated with fifteen pairs of ribs. The first five pairs were jointed with the bodies and transverse processes of the corresponding vertebrae ; the remaining ten pairs with the free ends of the last ten dorsal vertebrae. The lumbo-caudals diminished in size before they reached the tail, and the transverse processes ended at the 39th post- costal vertebra ; in the tail itself fifteen vertebrae were repre- sented by only the bodies. The tips of the lumbar and dorsal * Ann. and Mag. Nat. Hist., vol. xiv., 3rd series, p. 133, 1864. 1905-6.] On Delphinus acutus from Coast of Sutherland. 313 spines and the tips of the transverse processes were partly cartilaginous. The spinal cord ended opposite the 5th post-costal vertebra. Twenty-five chevron bodies were counted : the 1st began in line with the 20th post-costal vertebra; it and three succeeding chevrons were small, and the two lateral halves were not united mesially to form a ventral spine. The last chevron was associated with the 45th post-costal vertebra; it and the four chevrons immediately anterior, though small, possessed each a stunted ventral spine. The intermediate chevrons were considerably larger and with well-marked ventral spines. The sternum, 5J inches long, consisted of three segments, and formed an elongated, flattened bone. The manubrium was the widest ; its anterior border was recurved and a pair of cornual pro- cesses projected from it; the 2nd piece was about half the width of the manubrium ; the 3rd piece was narrower and bifid at its free end. Five pairs of ribs articulated with the sternum and their sternal segments were ossified. From the 6th to the 10th rib the corresponding sternal segment was ossified and the segment belonging to the rib in front was overlapped by the corresponding segment of the rib immediately behind ; in the respiratory movements one was permitted to play upon the other. The last five ribs were floating and their free ends were tipped with cartilage. These arrangements, together with the articulations of the last ten pairs with only the transverse processes of the dorsal vertebrae, gave great mobility to the chest walls and permitted the full expansion of the lungs which takes place when the animal dives. The pelvic bones were almost straight and slender, 2 ”2 inches long; the ends were cartilaginous. The length of the head in a straight line from the occipital condyls to the tip of the beak was 14'3 inches (375 mm.). The length of the beak from the notch in the maxillary bone to the tip of the premaxillaries was 7T inches, being in proportion to the length of the head about 1 to 2. The transverse diameter of the base of the beak between the two maxillary notches was 3 '8 inches. The interzygomatic breadth of the skull was 7 '6 inches. The malar bone was a long, very slender style. The two pterygoids met in the mesial plane behind the palate, their suture was continuous 314 Proceedings of Royal Society of Edinburgh. [sess. with the mid-palatal suture, and their posterior borders formed a continuous transverse edge. The hyoid apparatus consisted of the customary elements. The teeth were conical and pointed, small in size, and the longest projected only 8 mm. beyond the gum; the six most anterior teeth in the upper jaw and the three most anterior teeth in the mandible had not cut the gum. The dental formula was §y. The mucous membrane covering the hard palate was perfectly smooth. The tongue was free at the tip. The anterior nares were not symmetrical, the right being larger than the left ; the mes-ethmoid was prolonged into the medio- Fig. 3. — Dorsal view of the skull of Delphinus acutus. rostral cartilage. The tympanic bullae were bilobed ; the outer was the larger of the two lobes and smooth. The bulla was 32 mm. long and 19 mm. in greatest breadth. The petrous bone was 31 mm. long and 19 mm. broad.* Systematic writers on the Cetacea have attached importance in the discrimination of the species of Dolphins to the relation between the length of the skull, measured in a straight line, from the occipital condyl to the tip of the beak, and the length of the beak itself. I have accordingly, for purposes of comparison, taken * See my paper on the ‘‘Sperm Whale,” in Proc. Roy. Soc. Edin., vol. xxiv. p. 430, 1903, for measurements and characters of the tympanic and petrous bones in the Cetacea. 1905-6.] On Delphinus acutus from Coast of Sutherland. 315 these measurements in the skulls of several species of dolphins in the University Museum : — Length of Skull. Length of Beak. Breadth of base of Beak. Delphinus acutus , . ,, albirostris, ,, delphis, . ,, tursio, . 14 -3 inches 17 „ 17*8 ,, 21*5 „ 7*1 inches 8*5 „ 11 *4 ,, 11*25 ,, 3 “8 inches 5-8 „ 3-8 „ 5-9 „ It will be seen that in D. acutus , D. albirostris , and D. tursio the beak was about half the length of the entire skull, but in D. delphis the proportion of the beak to the length of the skull was materially greater. Although the skulls of albirostris and delphis were of almost the same length, the difference in the breadth of the base of the beak was 2 inches ; whilst acutus and delphis had the same breadth of beak, the skull of acutus was 3 inches shorter; in D. tursio again, though the skull was 4J inches longer than in albirostris, the beak had almost the same breadth at its base. D. acutus had not reached its full size, and possibly the proportions may be somewhat altered in the adult. Skeleton of Flipper. — The Scapula was plate-like ; the acromion was flattened and strongly projecting ; the spine was a slender ridge ; the prsespinous fossa was a narrow surface in comparison with the post-spinous. The Humerus, 2 inches long, was thick and stunted and the epiphyses were fused with the shaft. The Radius was 2'4 inches long and 1*5 inch broad ; the distal epiphysis was distinct from the shaft. The Ulna was 1*7 inch long and 1*1 inch broad; the distal epiphysis was distinct from the shaft and the olecranon process was moderate in size. The Manus consisted of carpus, metacarpus, and phalanges ; it was pentadaetylous. The carpalia were arranged in two rows. The proximal row consisted of radiale, intermedium and ulnare, and at the ulnar border a plate of cartilage represented an un- ossified pisiform. Two bones were readily recognised in the distal row ; one articulated with the carpal ends of the 316 Proceedings of Royal Society of Edinburgh. [sess. metacarpals of annularis and medius, another with the carpal ends of the metacarpals of medius and index. A third hone was situated distal to the radiale, and close to the radial border of the proximal epiphysis of the metacarpal of the index. The question arose, Could this be the metacarpal of the pollex? hut as it was in the same transverse plane as the distal carpalia, and, like the other carpals, showed no sign of an epiphysis, it was presumably a third distal carpal, and represented the carpal element of the pollex. Each digit had its metacarpal bone. That of the pollex was in the same transverse plane as the shaft of the other metacarpals ; it was slender and elongated, with a cartilaginous prolongation at both its proximal and distal ends ; there was no sign in the cartilaginous Tig. 4. — Radiograph of the flipper of I). acutus, showing the bones and the centres of ossification, reduced to about one-third. prolongation of an epiphysis or a centre of ossification which nould be regarded as even the rudiment of a bony phalanx. The index was the longest of the digits, and they diminished in length from it to the minimus ; each of these four had a metacarpal bone, which possessed in the index, medius and annularis a proximal and a distal epiphysis, whilst in the minimus only a proximal epiphysis had a centre of ossification. The index had eight phalanges, three of which had proximal and distal epiphyses ; the fourth had only a proximal, and the ter- minal four each showed only a small ossific centre without epiphyses. The medius had five phalanges, of which two had proximal and distal epiphyses, and the third had a faint trace of a proximal. The annularis had an ossific centre for each of three phalanges in the right manus but only two in the left. The minimus had a cartilaginous 1905-6.] On Delphinus acutus from Coast of Sutherland. 31 7 rod but no centre of ossification in it.* The nranus, owing to the animal not being adult, presented an interesting study of the progress of ossification in this species of Dolphin. The following formula represents the osseous elements of the nranus : — Pol. lx Me An Mi Phal. ? 8 5 3 2 Met. 1 1 1 1 1 Ox C2 and C3 C4 and C5 radiale intermedium ulnar e Radius Ulna Owing to the absence of ossific centres the phalangeal formula, in the pollex and minimus cannot be stated in this specimen. The species Delphinus acutus was established in 1828 by J. E. Gray, from the examination of a skull from Orkney, now in the Museum at Leyden, which skull he figured in the Zoology of the voyage of the Erebus and Terror , Plate 12. Other specimens were subsequently obtained from the Faroes and the coasts of Norway (Rasch) and Holland, which have been at various times named Delphinus eschrichtii (Schlegel), Delphinus leucopleurus (Rasch), Lagenorhynchus leucopleurus (Gray), Lagenorhynchus acutus (Flower), but their identity with D. acutus is now recognised. In 1835 Robert and Frederick Knox obtained from Orkney a female dolphin, which measured 6 feet 5J inches.! They named it Delphinus tursio , and they prepared the skeleton. The vertebral formula was C7, D 15, LC59 = 81. The cranial cavity was opened and the brain removed. Dr J. E. Gray stated (' Catalogue of Whales , p. 274, 1866) that this specimen was in the Museum of the University of Edinburgh, and he assigned it to the species Lagenorhynchus leucopleurus. As the museum referred to by him might have been the Natural History Museum of * I may refer to my account of the Anatomy of Sowerby’s Whale, Journ. Anat. and Phys., Oct. 1885, and to my description of Balcenoptera rostrata, Proc. Roy. Soc. Edin., Feb. 1892, for a critical examination of the con- stitution of the manus in the Cetacea. t Measurements of the skull, skeleton and other characters of this dolphin are given in Knox’s Catalogue of Anatomical Preparations illustrative of th& Whale , Edinburgh, 1838 ; also in Proc. Linnean Soc., Zoology, 1857, p. 67. 318 Proceedings of Boy al Society of Edinburgh. [sess. the University, which was transferred to the Government in 1854, and is now incorporated in the Royal Scottish Museum, I have examined, along with Dr Traquair, the Keeper of the Natural History section, the specimens in the collection, hut could not identify Knox’s specimen. A skeleton marked Tursiops tursio is there; it is about 10 feet long, and is stated to have come from the Firth of Forth; its cranium was 2025 inches long and the beak was 9 -75 inches. Obviously the skeleton of tursio now in the Museum was not, either as regards dimensions or habitat, that of the dolphin prepared by the brothers Knox. The Anatomical Museum of the University has long possessed a skull, the skull-cap of which had been taken off and the brain removed ; the mandible was absent. Wires were attached to the basis cranii, which showed that it had at one time been articulated to the spine, but the rest of the skeleton was no longer attached to it and could not be found, neither was the skull labelled with .a name nor the locality where it had Ipeen obtained, so that no mark existed to enable one to identify it. The dimensions of the skull were, length 14*9 inches, length of beak 7*7, breadth of beak 3*9, breadth of skull 8 inches. Its dimensions were slightly greater than in the Dunrobin dolphin, though the ossification in it was also incomplete. Its appearance both in profile and in the dorsal view of the beak was in close correspondence with the Dunrobin specimen. The proportion of beak to skull, the configuration of the maxillse and premaxillae, the form of the pterygoids, their relation to the palate, the want of symmetry of the anterior nares, the continuation of the mes-ethmoid into a medio-rostral cartilage, and the shape and size of the bi-lobed tympanic bullae, closely resembled each other in the two specimens. The teeth in shape and size were identical in both skulls, and twenty- nine were counted on one side projecting through the gum ; as the most anterior tooth was a little distance behind the tip of the beak, several were in all likelihood concealed in the dried gum. From its resemblance in size, form, and proportions, I concluded that it also was the skull of a Delpliinus acutus. From the brain cavity having been opened, and from the evidence of its having at one time belonged to an articulated skeleton, I think that possibly it was that of Knox’s Orkney specimen obtained in 1835. 1905-6.] On Delphinus acutus from Coast of Sutherland. 319 In 1858 a school of dolphins was driven ashore at Scalpa Bay, near Kirkwall, one of which, a male, measured 8 feet 3 inches in length, and along with a specimen 7 feet 2 inches was described by Dr Duguid.* From a drawing of the animal which illustrated Dr Duguid’s memoir, it was recognised by Dr Gray as identical with the dolphin to which he was then applying the name Lageno- rhynclius leucopleurus. Mr Moodie Heddle had in his possession f a drawing of a male dolphin killed at Scalpa in 1858, doubtless one of the school previously described by Dr Duguid, and he further stated that three specimens ran ashore at Melsetten in 1886. That this species frequented the Orkney and Faroe seas, and that it had been occasionally found on the opposite shores of the North Sea, has been satisfactorily established. Evidence of its recognition on the coast of the mainland of Scotland is not, however, so definite, though it should be stated Messrs Harvie- Brown and Buckley have recorded that the Rev. N. Macpherson saw a dolphin lying on the pier at Ardrishaig, Argyleshire, which, from a comparison with the figure in Bell’s British Quadrupeds he regarded as of this species, but no description or measurements were given.; That I am now able to state without doubt that Delphinus acutus is an occasional visitor to the coast of the mainland, I owe to the Rev. Dr Joass, whose interest in the natural history and archaeology of the northern counties and whose courtesy to his fellow-workers are so greatly appreciated by men of science in Scotland. The skeleton will be added to the series of skeletons of the Cetacea in the Anatomical Museum of the University. * Ann. and Mag. Nat. Hist., vol. xiv., 3rd series, op. cit. t Vertebrate Fauna of | the Orkney Islands, by J. A. Harvie-Brown and T. E. Buckley, Edinburgh, 1896. X A Fauna of Argyle and the Inner Hebrides, Edinburgh, 1892. ( Issued separately August 29, 1906.) 320 Proceedings of Royal Society of Edinburgh. [sess. Contributions to the Craniology of the People of the Empire of India. Part III. : Natives of the Madras Presidency, Thugs, Veddahs, Tibetans, and Seistanis. By Sir William Turner, K.C.B. (Proceedings, June 4, 1906.) This Memoir is printed in extenso in the Transactions of the Society, vol. xlv., part i., 1906. 1905-6.] The Smolt to Grilse Stage of the Salmon. 321 Note on the Smolt to Grilse Stage of the Salmon, with Exhibition of a Marked Fish recaptured. By W. L. Calderwood. (MS. received July 13, 1906. Read July 13, 1906.) In tracing the various stages of the salmon’s growth, precise information has been difficult as to the length of time occupied between the first descent to the sea as a smolt and the first return to the river as a grilse. The difficulty of attaching to so small a fish as a salmon-smolt a suitable mark for subsequent identifica- tion has, in the main, been responsible for this lack. The average smolt is 5J to 6 inches long (14-15 cm.), and weighs from 1 to 2 ounces. In the next stage with which we are familiar the fish may weigh 3 to even 10 lbs. It is clear, therefore, that any mark attached to the smolt must not only be sufficiently small and light for a fish only a few inches long to carry without incon- venience and injury, but must be adaptable to the rapid and great increase of growth which takes place in the sea. In the past a method of marking has been repeatedly resorted to which, though having the merit of simplicity, is not really reliable ; I refer to the method of fin-cutting, especially and commonly the cutting or the removal of the adipose fin. The famous Stormontfield experiments, reported upon by Buist in 1867, and repeated in other localities by other observers since that date, were conducted, so far as the study of migratory movements went, almost exclusively by fin-cutting. These experiments have been chiefly responsible for the belief that the smolt which enters the sea in greatest numbers in the spring, returns to fresh water as a grilse in two or three months, i.e. during the immediately succeeding summer. The Duke of Bedford’s experiments in Devonshire were conducted by fin-cutting, and are held to show a different con- clusion. Only three cases are, I think, on record in which the smolts have been marked by the attachment of a foreign substance, and recapture effected. These cases come from the PROC. ROY. SOC. EDIN., YOL. XXVI. 21 322 Proceedings of Royal Society of Edinburgh. [sess. Tweed ( Tweed Salmon Reports , 1868). The marking was done in 1854, 1855, and 1857, and in each case the recapture of the fish as a grilse was fully a year after the date of marking. In the spring of 1904 I commenced experiments in smolt- marking at the Cunninghaugh Ponds, near Fochabers, the property of the Duke of Richmond and Gordon. The method first used was the attachment of a small silver disc to the operculum by means of a split pin passed through the operculum from beneath ; but so many of the marks were torn out, by the yielding of the delicate bones of the gill cover, within a few weeks, that the method was abandoned. I next used, on the Tay, a simple piece of silver wire, which I passed through the skin of the back close to the adipose fin, and formed into a loop by twisting the ends together. The method seemed fairly satisfactory, but the wire used was rather heavy, and interfered somewhat with the balance of the little fishes as they swam away. Next year (1905) the Tay Salmon Fisheries Company, under Mr P. D. Malloch’s super- vision, took up the marking of smolts, and a wire was employed of lighter weight — so thin that it could be easily cut with scissors. The wire was passed through the dorsal fin close to the anterior border and a short distance above the base of the fin rays, and was formed into a loop or loose ring and the ends snipped off. Mr M‘Nicol had charge of the operations, and succeeded in marking 6500 smolts in the spring of 1905. No recaptures were made in 1905. This summer (1906) a considerable number of those smolts have been recaptured as grilse. In each case the recapture has been made in the Tay estuary, wdiere the marking was originally conducted. The particulars of the first five are as follow : — 1st June 1906, grilse weighing 26th 28th „ 3rd July ,, 5 5 4th 5 J 2 lb. 15 oz. 4 „ 8 „ 4 „ 12 „ 3 „ 4 „ 5 8 „ The specimen exhibited is the second on the list, caught on 26th June, and weighing 4 lb. 8 oz. It is a male fish with rudimentary testes, and measures to the fork of the caudal fin 24 inches 1905-6.] The Smolt to Grilse Stage of the Salmon, 323 (61 cm.) ; the greatest depth is 4 inches (10'2 cm.), and the depth of the caudal peduncle 1J inches (3*7 cm.). These recaptures, then, confirm the belief that the smolt does not return as a grilse the same year as it descends, but rather after a year or fully a year has elapsed. The development shown on the scales of grilse, as well as the capture of the Galway Dorsal fin of grilse, caught 1st June 1906, marked as a smolt in May 1905. (Photograph by P. D. Malloch, Perth.) specimen which I exhibited to the Society about a year and a half ago, all give evidence that the first normal return of a smolt to fresh water takes place when the fish, as a grilse, is three to three and a half years old, and after it has been fully a year in the sea. From other observations it seems certain that all grilse do not, however, enter fresh water, but that many fish pass the grilse stage in*the sea and return for the first time to fresh water as small spring salmon, four years of age. 324 Proceedings of Royal Society of Edinburgh. [sess. With regard to the other wired grilse which have been taken during the fishing season of 1906, I may add that their weights ranged to a maximum of 9 lb., and that when further time has elapsed in which the possible return of those fish may be noted, additional particulars will be published. ( Issued separately October 12, 1906.) 1905-6.] Dr W. W. Taylor on The Theory of Ionization. 325 Two Lecture Experiments in illustration of the Theory of Ionization. By Dr W. W. Taylor. Communicated by Professor Crum Brown. (MS. received July 2, 1906. Read July 2, 1906.) The two experiments, which, so far as I am aware, have not hitherto been described, illustrate two of the conclusions drawn from the theory of ionization. The first is that the degree of ionization of a solution of an acid is diminished by addition of a salt of the acid. The experiment differs from many of those employed for the purpose in that it demonstrates the increase in concentration of the un-ionized acid, and not the diminution of activity as an acid, i.e. decrease in concentration of H*. A solution of egg-albumin is added to a solution of nitric acid which is so dilute that no coagulation of albumin occurs; a saturated solution of potassium nitrate is then added to the mixed solution, and coagula- tion immediately takes place. The potassium nitrate solution causes no coagulation when the nitric acid is not present. That un-ionized nitric acid causes the coagulation is shown by the fact that H*, K", N 03', and un-ionized potassium nitrate do not do so under the same conditions. The second experiment illustrates the proposition that, on addition of a weak acid to a solution of a salt of a strong acid, some of the strong acid is displaced by the weak acid. Saturated solution of potassium nitrate, added to an albumin solution, causes no coagulation ; a solution of acetic acid likewise causes no coagulation when added to the albumin solution. When the two are added to the albumin, coagulation takes place immediately. That in this case, also, the precipitation is due to the formation of un-ionized nitric acid is shown by the facts that H", K’, N03', C2H302', and un-ionized potassium nitrate do not cause it. A separate experiment with potassium acetate, the only other sub- stance formed in the reaction, proves that the precipitation is not 326 Proceedings of Royal Society of Edinburgh. [sess. caused by it. Un-ionized nitric acid must have been formed to a sufficient extent to coagulate the albumin. This experiment avoids the complication which ensues when a weak acid acts upon a salt of a strong acid with formation of an insoluble salt of the weak acid, e.g. the action of hydrogen sulphide on many metallic chlorides. Chemical Laboratory, University of Edinburgh. ( Issued separately October 12, 1906.) 1905-6.] Miss I. D. Cameron on A Dietary Study. 327 A Dietary Study of Five Halls of Residence for Students in Edinburgh. By I. D. Cameron, M.B., D.P.H. Communicated by D. Uoel Paton, M.D. (MS. received June 18, 1906. Read June 18, 1906.) CONTENTS. PAGE PAGE I. Dietary Standards — (Edinburgh, York, and Yoit, etc 327 Dublin) 332 Atwater .... 328 (c) Japanese studies 334 Chittenden .... 328 III. Present Investigation Modifications for age, sex, and Results 335 etc. . . 331 IV. Com parison with other II. Previous Dietary Studies— Studies of Students’ (a) Atwater’s American work 331 Dietaries . 346 (&) British investigations V. Summary of Results 350 I. Dietary Standards. In considering the steps leading up to the present dietary standards, it is necessary first to mention the work of Yoit and Pettenkofer. They constructed tables representing “ exchange of material ” ; these were based on the weight of the animal experimented on, and the amount of food taken, considered in relation to the work done and heat developed. These results may be expressed in three ways : (a) As energy, and stated as Calories. ( b ) In terms of the contained nitrogen and carbon. (c) In relation to the nutritive materials, as proteids, fats, and carbohydrates. The method of investigation originated by Yoit has been followed by Playfair in England and by Moleschott in Italy. Their results do not differ widely. The general result is that the daily requirement of an average man at moderate work is : proteids, 120 grammes ; fats, 60 grammes; and carbohydrates, 500 grammes. When expressed as energy, this represents 3130 calories. Given in another form, it is equivalent to 20 grammes of nitrogen and 320 grammes of carbon daily. 328 Proceedings of Royal Society of Edinburgh. [sess. Atwater developed this method of investigation. The numerous reports submitted by him to the United States Department of Agriculture give his results along three lines of study : (1) With a respiration calorimeter ; (2) Chemical analyses of food-stuff's ; and (3) Dietaries of representative classes of the community. He suggests as a dietary standard: proteids, 125 grammes; fats, 125 grammes ; and carbohydrates, 450 grammes per man per diem. The noticeable difference from the Voit standard is the increase in fats and the diminution in the amount of carbo- hydrate. Atwater specially emphasises the fact that no allowance is made for differences in digestibility, and that personal idiosyn- crasy is not considered. Professor Chittenden of Yale has recently carried out an elaborate investigation into the amount of food required in health. These researches are published in book form, Physio- logical Economy in Nutrition. Chittenden’s interest in the subject was aroused by the contention of Mr Horace Fletcher that a high standard of health could be maintained on a low proteid intake. Chittenden’s criticisms of the former work on this subject are : (1) that the chemical analyses are not accurate, and (2) that the amount of food ordinarily taken is by no means necessarily the amount required. To eliminate possible fallacies in the work, observations were made over several months with men of different classes and nationalities engaged in different work. The three classes of men studied were : (1) professional men, (2) soldiers, and (3) college athletes. The quantity of food taken was gradually reduced : not only proteids, but fats and carbohydrates were diminished. The body-weight fell while the change of food was being made, but it soon became stationary. The nitrogen equilibrium was then tested, and a slight plus balance was found. Systematic tests of strength showed an increased vigour, and the subjects of the experiment professed to be in improved health. The amount of proteid taken in these cases was only about oue-third the ordinarily accepted standard. Chittenden concludes his report by pointing out that, if health 1905-6.] Miss I. D. Cameron on A Dietary Study. 329 can be maintained on one-third the usual proteid allowance, then the additional proteid adds enormously to the wear and tear of the tissues. It is an incubus instead of a help in the bodily economy. The discussion of the amount of proteid required by young tissues is not touched by Chittenden. It is reasonable to suppose that more proteid is required during growth, as proteid is the “ muscle-builder.” The power of resistance shown during a long, exhausting disease by tissues nourished for a considerable time on a low proteid intake is also a point of interest. Chittenden’s work does not enter on these points, and was not intended to do so. The influence of this recent work on proteid requirements is seen in the Report of the Departmental Committee on Vagrancy, issued this year. The daily food allowance for casual wards and labour colonies is 70 grammes of proteid, while the total energy value of the diet is 3000 calories. The ration may, however, be supplemented from pocket-money given for industry. So far, the history of the present dietary standards (expressed as proteids, fats, and carbohydrates) has been considered ; and by what methods — at first empirical and later experimental — they have been arrived at. But other standards may be used. The food required may be stated in terms of the contained nitrogen and carbon. The energy value expressed as heat also serves as an important standard. Lavoisier showed that the final change in the food in the body was a process of oxidation. Frankland decomposed different articles of food, and estimated the amount of heat liberated in the process of disintegration. The unit employed is the Calorie, i.e. the amount of heat required to raise 1 kilogramme of water 1 degree Centigrade. In estimating the heat value of proteid, it is necessary to remember that the final product of its metabolism is urea, and allowance must be made for this incomplete oxidation. Riibner * has estimated that 1 gramme of proteid gives on combustion 4T Calories ; 1 gramme of carbohydrate also gives 4'1 ; while 1 gramme of fat yields 9 '3 Calories. Taking into consideration more accurate chemical analyses and more recent * Zeitscbr. f. Biol., xxi. (1885), p. 337. 330 Proceedings of Royal Society of Edinburgh. [sess. knowledge as to the digestibility of different foods, as well as working with an improved calorimeter, Atwater and Bryant * suggest 4*0 as the factor for proteids and carbohydrates, and 8*9 for fats. Rubner’s estimate is the one generally used. The Calorific value cannot he taken as a rigid means of comparison between two dietaries, hut it is none the less of great value. In order to perform severe muscular work, a diet of high energy value must he consumed. Since carbohydrates and fats are the main source of energy, at first sight it seems as if an increase in these nutrients would meet the requirements. But most people find that digestive disturbances are set up by a high fat and carbohydrate diet, and the increased amount of food must of necessity be proteid. A practical application of this point is seen in Dunlop’s Report to the Prison Commissioners.! The diet of certain convicts was of an energy value of 3928 Calories, and the waste was great. There was no waste when the diet was reduced to 3517 Calories, hut the prisoners lost weight, and complaints were rife. When 74 grammes of bread were added to the ration, the diet was of 3707 Calories. There was no further loss of weight, the complaints ceased, and the waste was inconsiderable. The dietary standards fixed by different authorities may be tabulated : — Proteids, grammes. Fats, grammes. Carbo- hydrates, grammes. Calories. Voit . 118 56 500 3054 Riibner 127 52 509 3091 Playfair 119 51 531 3139 Moleschott . 130 40 550 3160 Atwater 125 125 450 3520 It is interesting to compare these with the results of Chittenden’s experiments, in which the proteid varied from 44-50 grammes daily, with a Calorific value of 1550-3000 from the entire food. * Connecticut Storrs Station Report, p. 73. t See also “ Food Requirements of Various Labour,” Scottish Medical and Surgical Journal, 1901. 1905-6.] Miss T. D. Cameron on A Dietary Study. 331 These standard dietaries are constructed to represent the requirements of an average man doing a moderate amount of muscular work. The amount required varies with the work done, with sex, age, weight, and climate. Personal idiosyncrasy in the matter of food is also a matter of common knowledge. Atwater gives a woman’s requirements in food as 0*8 of a man at moderate labour. A boy of 14-16 years of age requires the same amount as a woman; a girl of 14-16 is regarded as requiring 0*7 of a man. A child under 2 years requires 0'3 of a man. Atwater derived these factors from Camerer’s * work on energy requirements per unit of work at different ages. Konig gives practically the same ratio in suggesting 118 grammes of proteid, 56 of fat, and 500 of carbohydrate for a man ; and 92 grammes of proteid, 44 fat, and 400 carbohydrate for a woman. It is important to note that, in these estimates, no allowance is made for differences in the digestibility and absorbability of different kinds of food. II. Previous Dietary Studies. Many studies of the actual diet consumed by different classes have been made both in America and in Europe. Atwater and his co-workers in America have investigated the dietary of different classes in the United States, and they have collected a large number of similar studies. These, of course, do not touch the question of the desirability or necessity for the ingestion of food in such quantities or in these nutritive proportions. Atwater did certainly suggest a standard, as has been already stated. But his dietary results are simply presented as the actual food on which the people live and work. They are taken as fairly representative of different classes of the community, of various social grades, with different customs as regards food. No doubt, the amount of food taken is largely the result of habit ; but it must, to some extent, be founded on what experience has taught. Atwater f emphasises the fact that Americans eat more than people of the same social position in Europe. This applies especially to the working classes, and he explains on this ground * Vierodt’s Daten u. Tabellen, 1888, p. 7. t “ Foods, Nutritive Yalue and Cost,” Farmer’s Bulletin , 23. 332 Proceedings of Royal Society of Edinburgh. [sess. what he regards as the greater working power of the American. As illustrative of the wide range of difference in the studies collected in America,* a low and a high dietary may be quoted. A seamstress, whose diet was studied by Playfair, consumed daily 53 grammes of proteid, 33 grammes of fat, and 316 grammes of carbohydrate (energy value, 1 820 Calories). A Californian student football team used 270 grammes of proteid, 416 grammes of fat, and 710 grammes of carbohydrate per man per day (energy value, 7885 Calories). It is needless to remark that this huge amount was taken during training. Jaffa, who reports this study, says that “ the study seems, on the whole, to warrant the conclusion that the team was overfed.” A number of American middle-class dietaries, including studies of men and women university students, will be referred to later. The dietary of public institutions has been worked out in a number of cases. Aitchison’s Investigations into the Diet of a Scotch Workhouse and Smith’s Report of Dietaries of Lunatics and Workhouses are examples of this kind of work. A Royal Commission was appointed to inquire into prison dietaries, and their report was presented to the Houses of Parliament in 1878. The diet of soldiers was the subject of a Royal Commission inquiry, and a report on the subject was submitted in 1889. This contains suggestions for the improve- ment of the quality of the diet, and a comparison with the army dietary in other European countries. The British allowance was shown to be greater than that of the Continental armies. A soldier’s daily allowance was then made 113 grammes of proteid, 38 grammes of fat, and 482 grammes of carbohydrate (Calories, 2793). De Chaumont considered this diet deficient, and thought that the allowance ought to be increased, especially in the case of the younger soldiers. More recent work along the lines of the comparison of the * Report to the U.S, Commissioners on Fish and Fisheries , 1888 ; and Nutrition Investigations at the Californian Agricultural Experimental Station , 1900. 1905-6.] Miss I. D. Cameron on A Dietary Study. 333 actual food with the standard requirements is found in Dr Craufurd Dunlop’s Report on Dieting of Pauper Lunatics in Scotland and in his Report on Prison Dietaries. The diet of the working classes in Edinburgh was studied by Drs Noel Patou, Dunlop, and Inglis.* This work was carried out under the auspices of the Town Council of Edinburgh, on the recommendation of the Committee of Public Health. Fifteen families (ninety-five individuals) were studied. These varied from the well-to-do working-class household to those whose total income was less than 20s., and who were not in permanent employ- ment. The average result was a daily amount of 107 *7 grammes of proteid, 88*4 grammes of fat, and 479*4 grammes of carbohydrate (Calories, 3224) per man. One old woman subsisted on the meagre allowance of 46'1 grammes of proteid, 33'7 of fat, and 151 *3 grammes of carbohydrate (Calories, 1124), expressed per man per day. Rowntreef carried ont an investigation into the social condition of the working classes in York. He discusses the general conditions under which these people live. A special study was made of their food. As a result of his work, he drew up a poverty scale — a table of the minimal amount per week on which “merely physical efficiency” could be maintained. His allowance per man per week for food on this scale is 3s. Of the families studied in York, the daily amount per man was 89 grammes of proteid, 79*9 of fat, and 385 5 of carbohydrate (Calories, 2685). These York dietaries were compared with Atwater’s standard, and 29 per cent, were deficient in proteid, while there was 23 per cent, deficiency in fuel value. Unfortunately, there is an error in Rown tree’s work. The percentage of carbohydrate in flour is entered at 57*1 per cent, instead of 75*1 per cent., and as flour enters largely into the food, the total error is considerable. Lumsden made an inquiry into the food-supply of the employees of Messrs Guinness in Dublin. His results are published in hook * A Study of the Diet of the Labouring Classes in Edinburgh. Otto Schulze & Co. t Poverty : A Study of Town Life. London, 1901. 334 Proceedings of Royal Society of Edinburgh. [sess. form.* The average food “as purchased” per man per day in the seventeen families was : 98 grammes of proteid, 89*8 of fat, and 467*7 grammes of carbohydrate. When the coefficient of digesti- bility of the different foods is allowed for, this represents 85*35 grammes of proteid, 86*2 of fat, and 447’6 of carbohydrates. Lumsden’s investigation had the advantage of being carried out for a number of weeks. It is noteworthy that only three of these families had a proteid allowance equal to Atwater’s standard. The others were 21*5 per cent, below the requirements. Lumsden says of four families in particular : “All the families are living under the poverty line, and one will rather expect to find a lamentable state of want : however, strange to relate, these families live a happy, contented existence ; the children are well kept and particularly healthy-looking.” Numerous Japanese dietaries have also been studied. They have recently been published in English.! Tahara sug- gested that Japanese, owing to their smaller stature, required less than average Europeans and Americans as their daily allowance. His conclusion was, that the Japanese required 96 grammes of proteid, 20 grammes of fat, and 450 grammes of carbohydrate (Calories, 2380) ( Bullet . Imp. Sanit. Lab. Tohio , 1887, No. 2). Riibner’s factor for energy value is stated to be inapplicable to Japanese food, as it differs from that of Europeans in digestibility. For vegetable proteid, 3*56 was the factor used, and 4*45 for animal proteid. Nearly 400 different dietary studies have been made in Japan. All classes of the community were studied, but no studies of women were made. The period of observation varied from three days to a year. The general result was that the Japanese were found to be well nourished. They are not vegetarians to anything like the extent that they are supposed to be. Rice is a most important article of diet, but the well-to-do classes take meat and fish to a considerable extent. The vegetarianism is more from economy than from principle, except in the case of strict Buddhists. * An Investigation into the Income and Expenditure of Seventeen Brewery Families , and a Study of their Diets, 1905. + “A Digest of Japanese Investigations on the Nutrition of Man,” by Kintaro Oshima, Bulletin 159 of the U.S. Department of Agriculture, 1905. 1905-6.] Miss I. D. Cameron on A Dietary Study. 335 Cows are scarce in Japan ; dairy products are expensive, and can only be obtained by the wealthy classes. The Japanese eat less than Europeans, but their body- weight is considerably less. The studies of the diet of Japanese students will be considered later. Studies of the actual food consumption of communities in this country have been confined, so far as I am aware, to these investigations of the dietary of the working classes and of public institutions. There is an impression, which is steadily gaining ground, that the average amount of food consumed by the middle classes, where the factor of cost is not all-important as with labourers, is much greater than even the usual dietary standards. III. Present Investigation and Kesults. In order to determine the actual food consumed, a study was made of five residences for students in Edinburgh. Many of these students, especially those studying medicine, have a considerable amount of physical exercise while at work. A number also engage in athletics ; but, taken all round, one cannot say that students have what, as purely physical exertion, would be regarded as a moderate day’s work. The food require- ments of mental work have been studied by Atwater* with great care. A man was confined in a respiration calorimeter. The bodily waste was not found to be increased when the subject of the experiment diligently studied a German scientific book. It has been suggested that, since the nervous system contains 8 per cent, of fatty material, much fat is necessary in mental work.f Leaving the difficulty of digestion of fats out of con- sideration, the absence of appreciable bodily waste during hard study makes this view untenable. Other popular theories are that fish and phosphorus are direct mental stimulants, but these are not founded on scientific facts. The truth seems to be that easily digested food is best for those engaged in brain-work. The digestion of a heavy meal entails an expenditure of nervous energy, and the blood-supply to the brain is interfered with by the increased supply to the digestive organs. * U.S. Department of Agriculture, Bulletin 44, 1897. t Referred to by Yeo, in Food in Health and Disease. 336 Proceedings of Royal Society of Edinburgh. [sess. The method employed in making the present studies was that adopted by Atwater and his colleagues. A detailed account of this is given by Bryant in “ Some Results of Dietary Studies in the United States ” (reprint from the Yearbook of the Depart- ment of Agriculture for 1898). An inventory is made of all the food in the house. Everything is weighed accurately on a tested balance. Each article of food as it is purchased is weighed and added to the inventory. At the end of the study, everything is again weighed. The amount actually used is then calculated. But all the food is not actually consumed. The “ refuse ” (that is, the inedible part) is allowed for in the chemical analyses, and so in the determination of calorific value. A certain amount of the food is also inevitably wasted. All the “ waste ” (that is, nutritive material which has not been actually consumed) is also sorted out, and it is carefully weighed and deducted from the food used. In this way the actual con- sumption during the period of study is arrived at. The cost of the food is also taken into account, and the expenditure on this item per man per diem is calculated. In giving the cost of the “waste,” an estimate was obviously all that could be given. A note is also made when anyone is absent from a meal, and the presence of guests is also taken into consideration, in order to make the results as accurate as possible. The analyses used were chiefly taken from Atwater’s “ Chemical Composition of American Food-Stuffs ” (. Bulletin 28 — revised edition — of the U.S. Department of Agriculture). Konig’s Ghemie der menschlichen Nahrungs und Genussmittel , Wynter Blyth’s Foods , their Composition and Analysis , Noble and Firth’s Text- book of Hygiene , Mitchell’s Flesh Foods , and Hutchison’s Food and Dietetics were also consulted. Several analyses made by Dr Craufurd Dunlop in the Laboratory of the Royal College of Physicians, Edinburgh, were also used. In two cases, galantine and haggis, the quantities used were so small that an analysis was not made, but an estimate was made from their composition. The present studies were conducted for one week — during the month of February in one case, and March in the others. In Studies A, B, C, and D the students in residence were men. All the servants were women, and they were allowed for by 337 1905-6.] Miss I. D. Cameron on A Dietary Study. Atwater’s estimate — a woman is regarded as eating 08 times as much as a man. In Study E the students were women, so each member of the household was calculated on the 0*8 basis. As the nutritive value of beverages and condiments is small, and their composition varies somewhat, they have not been included in the general results. The amount expended on these two items has been calculated separately. In two of the studies — A and D — the cost of beer is included in the beverages. In the other three studies, beer is not considered. Five halls of residence were studied, by the kind co-operation of the housekeepers, wrho undertook the arduous work of keeping accurate accounts of the amounts purchased, and seeing that the waste was collected. The studies represent collectively the dietary of 1129 men for one day. Allowance in each case was made for absence from meals, and for the presence of guests. .Study A = 239*4 men for one day ( 31 men, 8 women). Men Women Total B = 198*8 C= 316*4 D= 207*2 E = 167*6 ( 24 „ 6 ( 39 „ 9 ( 25 „ 7 ( 30 women.) 1129*4 (119 men, 60 women). The figures given in brackets above are those in residence for the week, without allowance for guests and absence. The others are with the allowance made. (A) General Statistics of Studies. Table I. Proteid, grammes. Fat, grammes. Carbo- hydrate, grammes. Calories. Cost in pence. Men for one day. Study A 146*41 160*57 531*33 4303*05 16*7 239*4 „ ' B . 121*89 106*58 527*55 3663*89 15*5 198*8 „ c . 134*43 137*25 496*79 3864*41 14*3 316*4 „ D • 154*66 146*11 507*50 4073*46 15*8 207*2 „ E . 161*67 139*54 494*92 3989*73 13*6 167*6 This table shows how comparatively closely all these studies PROC. ROY. SOC. EDIN., YOL. XXVI. 22 338 Proceedings of Royal Society of Edinburgh. [sess. follow the average. Study E is, however, highest in proteid, while B is considerably lower than the others. Study B is, again, noticeably lowest in fats, but is of high ■carbohydrate value, and closely approaches the average of energy value. In Study A, the large amount of fat probably accounts for the slightly higher cost. The high energy value in D results from the large amount of fat associated with proteid and carbohydrates, both of which are over the average in these studies. The amount of proteid in each one of these diets is in excess of the amount allowed in any dietary standard, the average exceeding the 125 grammes daily of Atwater by 18 grammes, and the 130 grammes of Moleschott by 13 grammes. When, however, this proteid intake is compared with Chittenden’s results, it is seen how greatly his requirements are exceeded. If the highest proteid value in his work — soldiers on 55 grammes per man per day — is taken, and compared with the average in the present — 143 grammes, — we find the proteid ingested in the latter case is more than twice and a half the amount in the former. The smallest amount of proteid taken exceeds 55 grammes by as much again — the highest proteid is just short of three times 55 grammes. In Study E, the students were healthy, active, young women. On contrasting this with the other studies, one is struck with the high proteid value of their diet (expressed per man per day). The fat just touches the average, and the carbohydrate falls below it. Although general conclusions cannot be drawn from one study, it is interesting to see that this is in direct opposition to the popular idea that women consume relatively less proteid •than men, and more carbohydrate. (B) Proportion of Animal and Vegetable Proteid. Bubner* considers that in a properly balanced diet the animal should exceed the vegetable proteid. These studies show the proportion that Biibner thought important, and give practically the percentage — 60 per cent. — that he considered * Zeitschrift f. Biologie, KF., Band iii., 1885, p. 374. 1905-5.] Miss I. JD. Cameron on A Dietary Study. 339 desirable. The average shows that 63 per cent, of the total proteid is animal, while 37 per cent, is vegetable. It is notice- able that Study E, with the highest total proteid — 161 grammes — has the lowest amount of proteid of vegetable origin. The animal proteid is 70 '4 per cent, of the total. In Study B, the animal is more nearly approximated to the vegetable, being 58 ‘4 per cent, of the total proteid. Table II. Showing amount of animal and vegetable proteid. Animal. Vegetable. Total. A 89-39 57-02 146-41 B 71-29 50-60 121-89 C 82-92 51-51 134-43 D 96-34 58-32 154-66 E 113-73 47*94 161-67 Total 453-67 265-39 719-06 Average . 90-73 53-07 143-81 This result — the greater amount of animal proteid present — is in contrast to what was found in the Edinburgh labourers’ diet. As the result of fifteen investigations, Drs Noel Paton and Dunlop* found that the animal proteid was only 44*9 per cent, of the total proteid. (C) Energy Intake. If Table I. is again referred to, the high calorific value of the food is noticed. This is due in great measure to the large amount of fat present. In B, the comparatively small amount of fat is compensated for, as regards energy value, by the high carbohydrate. In each one of the studies, however, the energy value is considerably in excess of even Atwater’s liberal standard of 3520 Calories. (D) Cost of Diet. The average cost per man per day is here 15T pence (Table III.). In this, the cost of beverages and condiments is not included, as beer is only allowed for in two studies. The relative costs of * Op. cit. 340 Proceedings of Royal Society of Edinburgh. [sess. “ beverages and condiments ” are consequently not comparable, but, omitting stimulants, the average cost would be about 0‘7 pence. Of the total expenditure on animal and vegetable food, 66 per cent, is for animal food. Table III. Cost per man per day, in pence. On Animal Food,* pence. Vegetable Food, pence. Total, pence. Beverages and Condi- ments, pence. Grand Total, pence. A . 11T 5*6 16-7 1*7* 18-4 B . 10-2 5-3 15-5 1-0 16*5 C . 9-7 4*6 14*3 0-8 15T D . 9*6 6-2 15-8 1-7* 17*5 E . 9*9 37 13-6 07 14-3 Total . 50-5 25-4 75-9 81-8 Average 10*1 5-0 15T 16-36 * Includes beer. In the following table, the amount of animal and vegetable food and the energy value for this expenditure is given : — Table IY. Animal and vegetable food per man per day. Proteid, grammes. Fat, grammes. Carbo- hydrate, grammes. Energy Value, calories. Cost, pence. Animal . 90-73 126-20 38-78 1708-86 10-1 Vegetable 53-07 11-80 472-83 2270-04 5-0 Total 143*80 138-0 511-61 3978-90 15*1 The above table shows again the points already referred to, — the excess of animal over vegetable proteid and the greater amount of money expended on animal food. The average return per penny expended on animal food is 9 grammes of proteid, 12 ’6 grammes of fat, and 3*8 of carbohydrate, with an energy value of 170 Calories: a penny spent on vegetable food gave 10’6 grammes of proteid, 2 3 grammes of fat, and 94’5 grammes of carbohydrate, with an energy value of 454 Calories. 1905-6.] Miss I. D. Cameron on A Dietary Study. 341 The following table gives a comparison of present study with Edinburgh labourers’ family diet as regards return for one penny. Table Y. Proteid, grammes. Fat, grammes. Carbo- hydrate, grammes. Calories. Animal r Present study . 9 1 12*6 1 3-8 170 Labourers 11 19T 3-4 | 235 Vegetable. Present study . 10*6 2-3 94-5 454 Labourers 23-0 4'1 167-9 836 The consumption of food-material in grammes per man per day is shown below. Table YI. A. B. C. D. E. Average and Total. Beef, veal, and mutton 181-4 [ 149-17 177-24 233-76 238*82 980-39 (196-07) Pork, lard, etc. 58-0 25*04 12-25 28*4 22-55 146-24 (29-24) Poultry and game 387 26-47 19-31 18-29 21-69 124-46 (24-89) Fish, etc. . 78-6 83-27 101-75 95-90 114-04 473-56 (94-71) Eggs 62-7 55-0 59-2 41 T 31-83 249 -83 (49-96) Butter 47-9 43-83 58-66 60-91 49-30 260-60 (52-12) Cheese 36-0 14-66 7-33 6-96 4-74 78-69 (15-73) Milk 771-0 697-32 752-24 869-22 784-97 3874-75 (774-95) Total animal . 1241*9 1094-76 1188*24 1354-54 1167-94 6047*38 (1209-47) Cereals 437-05 374-90 467-05 380-07 451-67 2110-74 (522-14) Sugars and starches 175-8 232-70 169-95 158-41 159-02 895-98 (179-19) Yegetables 528-8 37871 364-04 743-49 467*70 2482-74 (496-54) Fruits 106-9 87-20 21-64 59-50 46*07 321-31 (64-26) Total vegetable 1248-55 1073-51 1022-68 1341-47 1124-46 5810-67 (1162-13) Total food 2490-45 2168-27 2211T6 2696-01 2292-40 11858-29 (2371-65) 342 Proceedings of lioyal Society of Edinburgh. [sess. In Tables VII. and VIII. individual articles of food have not been selected for estimation of cost and to calculate the relation to the total diet. In working-class diet, there are certain staple foods, such as potatoes and bread, which bulk largely in the food each day, and so may be compared in different households. In these studies it was thought that a better classification would be made by taking classes of food, e.g. cereals and fruits, and this has been done in the two following tables. Table VII. Cost of various food-materials per man per day. A. B. C. D. E. Total. Average. Beef, etc . 4-0 4*6 3*6 4*2 4-6 21-0 4-2 Pork, etc. IT 0*5 0*2 0-5 OT 2-7 0-5 Poultry, etc. . 0-8 0-6 0-6 0*3 0*05 235 0-47 Fish 0*6 0-7 0*7 0-6 1-0 3*6 07 Eggs 1*3 0*9 1*0 0-6 0-5 4*3 0-8 Butter 1-4 IT 1-6 1*7 1*4 7-2 1*6 Cheese 0-08 0’04 0T 0T 0T 0*42 0-08 Milk 1-9 1-8 1-9 1*6 1-9 9'1 1*8 Cereals 2-5 2-2 2-4 2-0 1-6 107 2T Sugars and starches . 1-5 1-9 IT 1-3 0-9 67 1*3 Vegetables 1-0 0-6 1-0 2’4 0-8 5*8 IT Fruits 0-6 0-6 0T 0-5 0-4 2*2 0-4 Table VIII. Percentage of total food-material. A. B. C. D. E. Beef, etc. 7-4 7-3 8*0 8-7 9-6 Pork, etc. 2-5 1-2 0-5 IT 1-0 Poultry, etc. . 1-7 1-3 0-8 0-8 0-9 Fish ■'.... 3-2 3-8 4-6 3-6 4-9 Eggs .... 2-6 2-6 2-6 1-6 1-4 Butter .... 1-9 2-0 2-6 2-2 2-1 Cheese . . . 0-2 0-7 0-3 0-2 0-2 Milk .... 30-5 31-7 34-2 32-2 33-1 Cereals .... 17-5 17-2 21-3 14-0 18-6 Sugars, etc. 7*0 10-6 7-7 5-9 6-9 Vegetables 21-2 17-4 16-5 27-5 19-3 Fruits .... 4*3 4-2 0-9 2-2 2-0 100*0 lOO’O 100-0 100-0 1000 343 1905-6.] Miss I. D. Cameron on A Dietary Study. (E) Total Animal and Vegetable Foods. Table IX. A, grammes. B, grammes. c, grammes. D, grammes. E, grammes. Animal . 1241*9 1094 76 1188*24 1354-54 1167-94 Vegetable 1248-55 1073*51 1022*68 1341-47 1124-46 Total . 2490-45 2168-27 2210-92 2696*01 2292-40 In Tables VIII. and IX. it is seen how closely the diets are related in the percentage of the different food-materials taken. Table IX. shows that the total food ingested varied only from 2168 to 2696 grammes per man per day. There is also great similarity in the relative amount of individual groups of food-stuffs. This is more marked, as one would expect, in the animal food- materials. Cereals, fruits, etc., are more likely to vary with chance circumstances, such as the menu for the particular week of study. This is seen in Table VIII. most markedly with regard to fruits. In Study A, fruit is 4’3 per cent, of total food consumed ; while in Study C 0'9 per cent, of fruit is present, with the high cereal figure of 2 1 *3 per cent. (F) Waste. Tables X. and XI. show that the waste varied very considerably. The proteid waste varied from 3 5 per cent, to 8 per cent, of the proteid purchased. A similar variation is seen in the fat waste — from 5*6 per cent, to 11 per cent, of the total fat. The high proteid and fat waste are found together as they represent the waste in animal food-stuffs — beef and bacon, etc. The percentage of unused carbohydrate is lowest as 0*5 per cent, and highest as 10*2 per cent. In this case, the carbohydrate waste is chiefly in bread and potatoes. The cost of the waste materials can only be given approxi- mately. The lowest estimate is 0 31 pence per man per day, and the highest is associated with high proteid and fat waste, and is given as 1*2 pence. Stated in another way, the price of waste material varies from 2-4 per cent, to 7 per cent, of the total money expended on food. 344 Proceedings of Royal Society of Edinburgh. [sess. This statement of waste cannot be taken as more than an indication of the average amount of food purchased which is not consumed. In the different studies very different food -materials were used, and from a study of only one Table X. Waste per man per day. Proteiil, grammes. Fat, grammes. Carbo- hydrate, grammes. Fuel Value (Calories). Cost (approxi- mately), pence. A. Animal . 3'31 7*41 o-o 61-46 0-3 Vegetable 3-27 0-18 13-33 69-73 o-oi Total . 6*58 7-59 13-33 131-19 0-31 B. Animal . 4*21 9-07 o-o 101-61 0 76 Vegetable 0-3 0*02 2-01 9 65 o-o Total . 4-51 9-09 2-01 111-26 0-76 C. Animal . 576 11-97 0-02 134-61 0-9 Vegetable 3 33 0-36 21-56 105-39 o-i Total . 9-09 12 33 21-58 240 00 1-0 D. Animal . 7*4 8 17-52 0-37 195-11 0 7 Vegetable 575 0-72 32-64 164 08 0-5 Total . 13*23 18-24 33-01 359-19 1-2 E. Animal . 3-13 73 o-o 80-72 1 0-3 Vegetable 9-16 1-08 50-68 255-38 0-4 Total . 12 29 8-38 50-68 336 10 07 week accidental differences are emphasised. Bearing in mind the difference between “refuse” and “waste,” as previously defined by Atwater, one sees that increased consumption of certain articles of food tends to increased waste. An example is the increased fat waste associated with a large consumption of ham and bacon, while eggs have practically no waste. Becognising 1905-6.] Miss I. D. Cameron on A Dietary Study. 345 this, the waste is best expressed as an average. This is found to be 9T4 grammes of proteid, 11 T2 grammes of fat, and 26 T 2 grammes of carbohydrate per man per day. This represents 2 34 ‘54 Table XI. Waste as percentage of total food purchased. Proteid. Fat. Carbo- hydrate. Fuel Value. Cost. A. Animal . 2-1 4-4 0*0 1*6 1*8 Vegetable 2-1 21 2*4 1*5 0*6 Total . 4*2 6*5 2*4 3*1 2*4 B. Animal . 3’3 7-9 0*0 2*8 4*8 Vegetable 0*2 0*0 0*5 0*2 0*0 Total . 3*5 7-9 0*5 3*0 4*8 0. Animal . 4*5 7*9 0*1 3*3 5*8 V egetable 2-3 0*6 4*2 2*5 0-6 Total . 6-8 8-5 4*3 5*8 6*4 D. Animal . 4*6 10-6 o-i 4*7 4*1 Vegetable 3-4 0*4 6*0 3-6 2-9 Total . 8-0 11-0 6*1 8*3 7*0 E. Animal . 1*7 4*9 o-o 1-9 2*1 Vegetable 5-2 0*7 10-2 5*9 2*9 Total . 6 '9 5*6 10*2 7*8 5*0 Calories, and costs 0*79 pence. A comparison of the waste of this case with that in American college studies will be given later. This waste cannot reasonably be compared with that in working- elass households with children, as in these cases the younger members of the household are frequently fed on what remains from the meals of their elders, and scraps can naturally be utilised in a way that is impracticable in better-class houses. 346 Proceedings of Royal Society of Edinburgh. [sess. IV. Comparison of Present Results with other Similar Dietary Studies. These results may be compared with the diet of the poorer labouring classes in Britain. The rural diets were compiled from the Board of Trade returns ; the urban are the result of the investigations already referred to. Rural Diets (grammes per man per day). England (Eastern Counties). Scotland (Northern Counties). Ireland. Proteids 100 124 98 Fats .... 76 81 57 Carbohydrates 578 570 586 Calories .... 3480 3601 3337 Urban Diets. Edinburgh (Noel Paton and others). York (Rowntree). Dublin (Lumsden). Proteids 107 89 98 Fats .... 88 80 90 Carbohydrates 479 386 468 Calories .... 3228 2685 3107 There is greater interest, however, in a comparison with the dietary of students in other countries. A number of studies have been made in America and in Japan. Atwater and Bryant* have tabulated the diet of American university boat crews in training, and give as the average of six studies of crews and one individual study of a captain the following figures: proteid, 155 grammes; fat, 177; carbohydrate, 440; Calories, 4085 per man per day. Other college athletes’ results are : — College football team, Connecticut ( Connecticut Storrs Station Report , 1891) : proteid, 181 grammes; fat, 292 grammes; carbohydrate, 577; Calories, 5740, and a Californian college football team (“Nutrition * Bulletin 75, U.S. Department of Agriculture, “Dietary Study of Uni- versity Boat Crews.” 1905-6.] Miss I. D. Cameron on A Dietary Study. 347 Investigations at the Californian Agricultural Experiment Station,” Bulletin No. 84, U.S. Department of Agriculture), with 270 grammes of proteid, 416 grammes of fat, and 710 grammes of carbohydrate, representing 7885 Calories. Comparison with American and Japanese Students’ Dietaries. Proteid, grammes. Fat, grammes. Carbohydrate, grammes. Calories. Average of 16 men’s clubs 105 147 465 3705 Average of women’s clubs 101 139 414 3402 Average of 2 Japanese men’s clubs 98 15 440 3320 Average of Edinburgh studies 143 138 511 3978 These men were “in training,” and so the results are not a good means of comparison. Other American college studies, expressed per man per day, are as follows : — Proteid, grammes. Fat, grammes. Carbo- hydrate, grammes. Calories. Average 5 clubs (Tennessee)* * * § 103 3820 Men ,, 5 ,, (Conn. )f ,, 127 3880 ,, 5 ,, (Maine) + ,, 159 5440 ,, 3 ,, (Missouri) § ,, 107 3920 ,, of 16 men’s clubs Women’s club (Lake Erie 105 147 465 3705 85 144 401 3330 College) || Women’s club (Chicago Uni- versity) IT Women’s Club (Middletown, 135 128 476 3685 105 160 330 3270 Connecticut) ** Women’s Club (Fargo, North 80 124 450 3325 Dakota) || j * “Nutrition Investigations in the University of Tennessee,” Bulletin No. 53, U.S. Department of Agriculture, t Connecticut Storrs Station Report , 1896. + U.S. Department of Agriculture, Bulletin 37. § Same, Bulletin 31. || Same, Bulletin 91. IT Review of Reviews, 1896. ** Connecticut Storrs Station Report , 1894. 348 Proceedings of Royal Society of Edinburgh. [sess. The Japanese men students’ dietary * is also expressed per man per day : — Proteid, grammes. Fat, grammes. Carbo- hydrate, grammes. Calories. Medical students (poor 74-4 6-0 479*2 2419 class) Medical students (good 86-0 13*2 333-6 1845 class) Good-class students other study) (an- 109-8 17*8 546-3 2796 Several other results are given, hut they are not quoted, as waste material is not considered. The average result for the present studies is: proteid, 143 grammes; fat, 138; carbohydrate, 511 ; Calories, 3978. With the single exception of the fats, this is higher than the American work, although the high energy value of fat makes the total calorific results closely resemble each other. The disparity between the American and the Edinburgh dietaries is most noticeable in the women’s studies. The Edinburgh women students consumed a greater amount of proteid and carbohydrate, and their food had a higher Calorific value. The Japanese proteid and fat value, especially the latter, are noticeably small, and are both the result of the low animal intake, but it is necessary to remember the small stature of the Japanese. In connection with this comparison between British and American dietaries, it is noteworthy that Atwater refers to the American of the same social position as being more abundantly fed than the Briton. He believes that the American is more energetic and able, and he attributes this to the more abundant food-supply. His conclusions are hardly borne out by the investiga- tion into the proteid supply of labourers in Edinburgh. These men have as much proteid daily as the labouring class in Philadelphia and New York, although their supply is less than people of the same condition in Chicago.! If we accept Chittenden’s view, the additional proteid is an evil instead of an advantage. * Opt cit. t See Bulletin 21 and Bulletin 46, U.S. Department of Agriculture. 1905-6.] Miss I. D. Cameron on A Dietary JStudy. 349 Comparison of Percentage Waste of Nutrients with that in American College Clubs. Proteid, per cent. Fat, per cent. Carbo- hydrates, per cent. Calories. Women’s clubs (3) * . 13*8 10-2 6 7 9-1 Men’s clubs (14) 16-8 19-2 10-9 14-9 Edinburgh students . 5-8 7-9 4-7 5-6 * References previously given. A comparison of waste cannot be made strictly, as the amount is bound to vary with many circumstances ; but so far as it goes it tends to show that the housekeeping in Edinburgh is more economical than in college residences in America. Of course it is impossible to compare the American and Edinburgh students’ results as rigidly representing the same class.. Students, even in the same university, are by no means similarly situated as regards the amount of money which can be expended on living generally, and on food in particular. It would be absolutely erroneous to conclude that the average amount spent by students in Edinburgh or in other Scotch university towns in any way approaches the expenditure given in these studies. Dire necessity limits the expenses of many Scotch students. The fairly close correspondence in the studies given is of no little interest. With increased expenditure on food, the nutritive value of food does not necessarily rise. This has already been shown in Table V. The tendency is probably not so much to increase the quantity of food eaten as to get a better quality, and to use more expensive articles of diet. If the question of cost has not to be considered, there can be no possible objection to this. But in the great majority of cases, all over the country, the struggle for existence is so keen that money spent needlessly on food cripples the funds for other requirements. A knowledge of true economy in food is then of the highest importance. A study of the food-materials used shows that on the whole they were those ordinarily used in middle-class dietaries, and 350 Proceedings of Royal Society of Edinburgh. [skss. so it is perhaps not too much to assume that these results may he taken as fairly representative of the diet of this class, giving as they do the food consumed by over a thousand men in one day. The expenditure also approximates to the average middle-class expenditure. The only accurate account that I could obtain of the expenses of a private well-to-do family has been put at my disposal by a lady who has kept a detailed account of her household for five years. This works out at 11s. 4d. per man per week, but in this no allowance is made for the presence of guests and for special food in time of illness. Most people, when asked, give 10s. a week as an average middle- -class food allowance, and the average in the present study is 9s. 2d. per week. This lower figure may be due partly to co- operation in diet, with the possibility of contract prices ; but as the data for comparison are uncertain, it is, unfortunately, impossible to emphasise this important point. V. Summary of Results. 1. The diet of five halls of residence was studied. This represents the food of 11 29 ’4 men for one day. 2. The average amount taken per man per day was : proteids, 143 grammes; fats, 138 grammes; carbohydrates, 511 grammes; with a fuel value of 3979 Calories. 3. In all the studies the proteid value is high. The animal proteid is 63 per cent, of the total proteid. 4. The average cost per man per day (exclusive of beverages and condiments) is 15T pence. Sixty-six per cent, of this is ■expended on animal food. 5. The amount of nutritive material per penny is much lower than that given in the study of the diet of Edinburgh labourers’ families. 6. The waste varies considerably. The approximate cost of waste was from 2’4 per cent, to 7 per cent, of the total money spent on food. 7. The proteids and carbohydrates are higher than in the American college studies. As the fats are lower, the Calorific 1905-6.] Miss I. D. Cameron on A Dietary Study. 351 value is about the same. The amount of food taken is greatly in excess of that in Japanese students’ dietaries. 8. The waste is only about one-half of that in the American -studies of college residences. 9. The expenditure on food of 9s. 2d. per man per week seen in this study is probably about the average spent on a middle-class dietary. {Issued separately November 9, 1906.) 352 Proceedings of Royal Society of Edinburgh. [skss. Further Study of the two Forms of Liquid Sulphur as Dynamic Isomers. By Alexander Smith and C. M. Carson. (MS. received July 13, 1906. Read July 13, 1906.) (Abstract.) When sulphur to which after recrystallisation no air has had access is melted, or when ammonia is led for a few minutes through ordinary sulphur after it has been melted, the two forms of liquid sulphur (yellow, mobile SA, and brown, viscous S^) adjust them- selves very rapidly to those proportions which are in equilibrium at that temperature to which the liquid may have been raised. The adjustment occupies but a few moments. When specimens thus prepared are then chilled by plunging into water, the reversion of the to S\ is equally rapid, and therefore the product is wholly brittle, crystalline, monoclinic sulphur. This behaviour is observed whether the liquid has been heated at, say, 155°, where the amount of at equilibrium is 7*2 per cent., or at 448°, where the amount is at least 34 per cent. Sulphur which has been exposed to the air since recrystallisation, when melted, reaches a condition of equilibrium with measurable slowness at the lower temperatures. Thus at 155° the proportion of Sjt at the end of an hour is only 6 '8 percent., and not until nearly two hours have elapsed does it reach 7*2 per cent. Leading a few bubbles of sulphur dioxide or of hydrogen chloride through melted sulphur which since recrystallisation has never been exposed to the air confers upon it the same slowness in reaching equilibrium. Specimens of ordinary sulphur, and of sulphur treated with sulphur dioxide, therefore, when chilled do not lose their content of SM by reversion. Hence practically the whole amount present at equilibrium at a given temperature may be supercooled and obtained after extraction of the mass eventually as amorphous sulphur. Thus the proportions at equilibrium at any temperature may most quickly and accurately be measured by leading in ammonia 4905-6.] Study of Liquid Sulgghur as Dynamic Isomers. 353 during the heating, to accelerate the adjustment, and then using sulphur dioxide before chilling, in order to retard the reverse change. All the gases named above act as simple catalytic agents. Iodine retards the adjustment to equilibrium also, hut it likewise acts as, a second component in the system and displaces markedly the equilibrium. Two parts of iodine to one hundred of sulphur at 150° increase the by 7 per cent., and at 448° by 30 per cent. The two liquid forms of sulphur, SA and S^, are to he classed as “dynamic isomers.” The freezing-point of the former is 119*25° (these Proceedings , voL xxiv. (1902), p. 300); that of the latter is unknown. In presence of ammonia the freezing-point is 114*5°. This is the temperature at which the two liquid forms of sulphur are in equilibrium with one another and also with solid, monoclinic sulphur. It is the triple point or so-called “natural freezing- point.” At this point the proportions of SA and in the liquid are 96*3 and 3*7 per cent, respectively. It has not been possible to observe any freezing-point and simultaneously measure the proportion of (by chilling ; these Proceedings , vol. xxiv. (1902), p. 299), below 112*45°, where the proportion of was 5*3 per cent. In the figure, which is drawn to scale, these facts are shown diagrammatically. A is the freezing-point of SA (119*25°, — 0), D the natural freezing-point (114*5°, SjU=3*7 per cent.), PROC. ROY. SOC. EDIN., YOL. XXYI. 23 354 Proceedings of Royal Society of Edinburgh. [sess. B is the freezing point of S^, and its exact location is unknown, and G is the eutectic point (also unknown.) The curve DG in the diagram is the line showing the pro- portions of Sa. and in the liquid at various temperatures, and extends from the natural freezing-point to the boiling-point. Only the lower part is shown. The general form of this curve was determined by measurements at intervals of ten degrees, which were given in a previous paper (these Proceedings, vol. xxv. (1905), p. 590, par. 2). The proof that there were two distinct forms of liquid sulphur, given in another paper (these Proceedings, vol. xxv. (1905), p. 588), however, suggested the possibility that there might be a transition point at which a sudden change from little to much took place, with formation of a new phase composed mainly of viscous sulphur. The dilatometric experiments given in the last- mentioned paper showed that this point, if it existed, must be situated at 160’0°, the point of minimum coefficient of dilatation. In order to ascertain whether any such sudden increase in the proportion of occurs, a new series of observations in the neighbourhood of 160° has now been made, and the exact form of the curve in this region has been determined. These measure- ments show that, when equilibrium has been reached, the pro- portion of at 155° is 7 ‘7 per cent, and at 165° 15 ’5 per cent., and that the change in concentration, degree by degree, between these points is continuous. It follows, therefore, that although there are two markedly different forms of liquid sulphur, these forms are miscible in one another to such an extent that no separation into phases occurs when the system is in equilibrium. At all events, the new observations do not afford any evidence that there is a separation. The two phases can be observed only when the system is cooling and is in an unstable condition. In view of the conclusion just mentioned, there was at least one anomaly which required explanation, namely, the marked absorption of heat and fall in temperature at, or just above, 160°, which are observed when liquid sulphur is heated continuously. The phenomenon is so strongly suggestive of a transition point, accompanied by the formation of k new phase, that it requires separate elucidation. A large number of experiments, of which a few illustrations only are here given, served to clear up the difficulty. 1905-6.] Study of Liquid Sulphur as Dynamic Isomers. 355 When ordinary liquid sulphur is heated at the rate of about two degrees per minute, the thermal effect asserts itself, with simultaneous sudden access of viscosity, at 167°. The same degree of viscosity is attained with slow heating at 160°, without thermal effect, and when equilibrium is reached at this tem- perature the proportion of is 10*7 per cent. Apparently, when the heating is rapid the proportion of lags behind that required for equilibrium, and the proportion necessary to give the marked viscosity has not accumulated until 167° has been reached. When the same experiment is continued by allowing the specimen of sulphur to solidify and the material is then remelted and heated rapidly once more, the viscosity and thermal effect now supervene at 163°. Evidently the cooling and solidification, although they have destroyed a part of the Sja, have not destroyed it all. Hence during the second heating the proportion of is always somewhat larger than during the first heating, and thus the thickening occurs sooner and at a lower temperature. When the sulphur is cooled only to 130° before being heated for the second time, the loss of is even less, and with rising temperature the mass is still nearer to the equilibrium condition at each temperature. Under these circumstances, therefore, the viscosity begins gradually at 159°, and no noticeable thermal effect whatever is observed. These inferences were confirmed by measurements of the proportions of present at various stages of the rapid heating. With sulphur melted at 121° and heated at the rate of two degrees per minute, the following proportions of were found : — Temperature . 121° 154° 156° 162° 165° 167° Per cent. 2° per min. 0-04 5-4 5-7 6-4 7-5 10-3 Per cent. Sm, equilibrium 3-75 7-5 8*0 13-5 15*5 16-7 At 167° the sample was taken just as the great viscosity was setting in. In the lower line the percentages of at the same temperatures, when equilibrium has been reached, are given for comparison. It will be seen that with rapid heating the pro- portion is from 2 to 6 per cent. less. Finally, the changes in concentration going on at 167°, during the process which manifests itself in heat absorption and rapid increase in viscosity, were studied by chilling various samples 356 Proceedings of Royal Society of Edinburgh. [sess. at different stages of the thickening, hut always at the same temperature (167°). The percentages of S ^ found were : — 10-5 11-6 12*9 14*0 15*1 The rapid adjustment of this system, when once 167° has heen reached, to a state nearly approaching that required for equilibrium, involving as it does an acceleration of the endo-thermal action SA->S/x, thus offers a sufficient explanation of the sudden absorption of heat at this point. ( Issued separately 'November 12, 1906.) 1905-6.] Dr Muir on the Theory of Alternants. 357 The Theory of Alternants in the Historical Order of Development up to 1860. By Thomas Muir, LL.D. (MS. received July 2, 1906. Read July 18, 1906.) My last communication in reference to the history of alternants dealt with the period 1795-1841 ( Proc . Roy. Soc. Edin., xxiii. pp. 93-132). The present paper continues the history up to the year 1860, but in addition contains an account of three writings belonging to the previous period, namely, by Murphy (1832), Binet (1837), and Haedenkamp (1841). Murphy (1832, Nov.). [On elimination between an indefinite number of unknown quantities. Transactions Cambridge Philos. Soc., v. pp. 65-76.] Murphy’s third example in illustration of his method is the set of equations 1 “I" X-^ "I- + 1 + 2x1 + 2%2 + 1 + 3aq + 32x 2 + 1 + nx1 + n2x 2 4- . + xn — 0 + 2nxn = 0 + 3nxn — 0 + nnxn = 0 which he neatly and easily solves, giving the value of xm, and thus in effect evaluating 1 1 1 . . . . 1 1 . . . . 1 1 1 . . . . 1 1 2 22 . . . 9m-l 2^+i e yn 2 22 . . . . 2n <-r 1 3 32 . . . . 3m_1 3m+1 . . . . 3n 4- 3 32 . . . . 3n 1 n n2 . . . . nm~x %m+1 . . . . nn n n2 . . . . nn His connection with our subject is thus seen to be similar to Prony’s. It should be carefully noted, however, in passing, that Prony’s 358 Proceedings of Royal Society of Edinburgh. [sess. set of equations is not the same as Murphy’s, the determinant of the one being conjugate to that of the other.* When the use of determinants is debarred or avoided, this difference is far from unimportant, — a fact which might readily be surmised from the present instance, since Murphy’s mode of procedure, though strikingly effective upon his own set, is quite inapplicable to Prony’s. It should also he observed that the solution of Murphy’s set is not essentially different from the solution of the familiar interpolation-problem to determine , a2 , . . . , an , so that aY + a2x + asx2 + . . . + anxn~l or y may have the values y1, y2, . . . , yn when x has the values xx , x2, . . . , xn respectively , — a problem which had been solved in one way by Newton (1687), in another way by Lagrange (1795), and in a third way to a certain extent by Cauchy (1 8 1 2). f Binet (1837). [Observations sur des theoremes de Geometrie, enoncees page 160 de ce volume et page 222 du volume precedent. Journ. (de Liouville) de Math., ii. pp. 248-252 : or, in abstract, Nouv. Annates de Math., v. pp. 164, 165.] The main object of this short paper of Binet’s was to draw attention to the fact that a. theorem regarding homofocal surfaces * The two sets of equations are Cir s% + . +.«”*„ =ur | r=n \ r= 1 . . (I) ar1xl + + . ■ £ r=n \ r=l . ; . . (J) The former is substantially the interpolation-problem which goes back to Newton, and which may therefore for distinction’s sake be associated with his name : the latter being first found solved by Lagrange (Recherch.es sur les suites recurrentes . . . Mem. de Vacad. de Berlin , 1775, pp. 183-272 ; 1792, pp. 247-299 : or (Euvres completes, iv. pp. 149-251 ; v. pp. 625-641) may be called Lagrange’s set,, provided we remember that he also gave a solution of the other. The first to deal with both of them in more or less general form by means of determinants was Cauchy (1812):° but in saying so, a mental reservation must be made in view of Cramer’s mode (1750) of Continuing Newton’s work. t Newton, Principia, lib. iii. lemma v. : also Aritlimetica Universalis , probl. lxi. Lagrange, Journ. de Vec. polyt., ii. cah. 8, 9, pp. 276, 277 : or (Euvres completes, ,vii. pp, 285, 286. Cauchy, Journ. de Vec. pclyt., x. eah. 17, pp. 73, 74 : or Euvres completes, 2e ser. i. 1905*6.] Dr Muir on the Theory of Alternants. 359 which Lame had just published was originally given by Binet in 1811. He thus has occasion to say that the form under which he had considered the equation of homofocal surfaces was a2 . b 2 r B4" K-A + K- B ' K.-C where a, b, c are the co-ordinates of any point on the surface, A, B, C are positive constants such that A>B>C, and K is a quantity which may be of, any magnitude greater than C. And as Lame had obtained expressions for the co-ordinates in terms of three values given to K, Binet intimates that many years before he had not only done the same but had extended the solution to the case of n equations. It is this purely algebraical problem which is of interest to us, and fortunately Binet gives it in full. Taking the set of equations in the form a K-A = 13 + K-b + K-C+; ■ . . =1, + K1-B + K1-C+ * ' . . =1 , b c + k2-b+k2-g+ ' ' . . =1, he introduces, for temporary purposes, two functions, E(flj), f{x\ the former being (x - A)(x- B)(# — C) ..... and therefore of the wth degree, and the latter being any integral function of a degree less than n. He then recalls the fact that f(x) -r F(#) can be partitioned into n fractions having x — A , x - B , x — C , . . . for denominators, the result as given by Euler being m _ /(A) . /(B) /(C) 1» - (x - A)/ (A) {x - By'(B) (x - G)f(C) Substituting successively K, , K2, . . . for x' in this, a set of equations is obtained from which it is seen that the solution of the set -^+^+^+ K-A K-B K-C F(K) ’ a ■+.,» -| ,r .|. K, - A Kj-B K, -C ‘ • F(Kj) ’ 360 Proceedings of Royal Society of Edinburgh. [sess. g I i c i _ /(k8) k2-a k2-b k2-c ' ' ' ' r(K2)’ /(A) _ /(A) a-F(A) (A - B)(A - C)(A - D) . . . , /q>)_ m F(B) (B-A)(B-C)(B-D) . . . Now the set of equations here solved is more general than that with which we started, the latter being the particular case of the former where /(K) /(K,) F(K) F(Ki)“ To effect this specialisation it is only necessary to make the arbitrary function /(a;) equal to (x - A)(x - B)(x - C) - (x - K)(x - KjXa - K2) or equal to F(aj)-f(a?) say; (where, be it observed, the condition as to the degree of f{x) is fulfilled); for then, since f(ar) vanishes when £C = K, Kx, K2, . . . . we have /(K) = F(K), /(Kj) = F(Kj) , /(K2) = F(K2) , . . The corresponding change in the values of the unknowns is easily made : for example, in the case of a , we have only to substitute F(A)-f(A), — or, what is the same thing, -f(A), — for /(A) in the numerator, the result being (A-K)(A-Ki)(A-K2) (A-B)(A-C) .... We have thus as Binet’s theorem : — The solution of the set of equations + + /q £q /?2 b1 + + ^2 Pi t>2 ft 2 ^2 A + .••• + h ~Pn Xl . ^2 , .^3 , , - *n ■ bn-f3,bn-f32 + bn-/3s+ •" ' • ‘ +bn-pn~L .1905-6.] Dr Muir on the Theory of Alternants. 361 _ (ft-ZftCft-M (ft -6.) *1- (ft - ft) — (ft -ft)’ (ft-ft)(ft-M (ft-ft) 2~ (ft -ft) . . . . (ft -ft)’ the binomial factors of the numerator in the case of xr being got by subtracting from (3r all the b’s in succession , and the similar factors of the denominator by subtracting from (3r all the other (3Js. Remembering that Binet had originally been an expert in working with determinants, it is not a little curious to note that he did not compare with these expressions for x1 , x2 , x3 , . . . . the expressions in terms of determinants, viz. — «*! = i(^-ft)-1-- i (^2 — A2) 1 • • -Pi -ft)"1 • •<&>- ft)'1 fti-ftnvftr1-- (^-ftm-ft)-1.. ..ft -ft)-1 ..(ft-ft)-1 l (ft,- ft)-1.. • •(ft-ft)? (ft-ftftXft-ft)-1.. ..(ft- ft)-1 Had he done so he would undoubtedly have reached a result which was not brought to light until four years later by Cauchy. Haedenkamp (1841). [Ueber Transformation vielfacher Integrale. Crelle’s Journ ., xxii. pp. 184-192 *] The transformation referred to in the title has its origin in a special equation of the nth- degree in y , viz. — X1 X2 _[_ .... -p Xn — 1 • a!~y «2 -y ' an~y and, as Haedenkamp gives the values of x1 , x2 , . . . , xn in terms of the n roots yx , y2 , . . . , yn of this equation he may of course be viewed as having solved the set of linear equations — * See also Crelle’s Journ., xxv. pp. 178-183 (1842), and Grunert's Archiv d. Math. u. Phys., xxiii. pp. 235, 236 (1854). 362 Proceedings of Royal Society of Edinburgh. [sESS. X1 + X2 + . . t . + Xn — 1 i «1 — 2/l «2“2/l ' an-Vi Xl + X2 + + Xfl _ X ' a2~y2 ’ an-y2 which Binefc had, explicitly dealt with four years before. Borchardt (1845, Jan.). [bTeue Eigenschaft der Gleichung, mit deren Hiilfe man die secularen Stor ungen der Planeten bestimmt. Crelle’s Journ ., xxx. pp. 38-45 : Gesammelte Werke , pp. 3-13.] For the present this paper is onty noteworthy as containing the square of the difference-product in the form of a determinant of the particular type soon after to be named (z) — 0, . . Crelle’s: Journ., xxx. pp. 157-165.] Although the subject of alternating functions is incidentally dealt with in Rosenliain’s paper (p. 161), nothing of importance occurs. The identity * * A still better form for tlie right-hand number is r=m+ 1 i • • • > ®s) . j ^m+2 > . • • , Un) s=l , 2, .... m 1905—6.] Dr Muir on the Theory of Alternants. 363 r—m+l , ; . . , rt Vl > «2 > • • • » an) = T(ai » «2 > • • ■ >0 • SKm » a«+2 I • • • » «nj • n(Or “ as) s=l ,2 m appears in the form alt a2i . . an) = Tl{aXi a2, . . . , am) • n(am+1, am+2, . . , , where the mode of denoting the rectangular array of differences cannot be commended. Sturm (1845), Terquem (1846). [Cours d’analyse de PEcole Poly technique, 4to, lithogr., Paris.*] [Sur la resolution d’une certaine classe d’equations a plusieurs inconnues du premier degre. Nouv. Annettes de. Math., v. pp. 67-68, 162-165.] Employing the method of “undetermined multipliers ” Sturm here supplies the want left by Prony, namely the solution of ar1x1 + a^2+ • • • +arnxn==l)r (r = 0 , 1 , 2 , . . . , n - 1) The said method may be generally described as making the solu- tion of a set of n equations dependent on the solution of a set of n — 1 equations, the latter set being related to the former in having its determinant conjugate to a primary minor of the determinant of the other set. Thus the given set being aqaq + d2x 2 + a3x3 + a4x4 = a5" hxi + \X2 + \X2 + b4X4 = h5 ^ e i^i e^x 2 -I- c3x3 -I- C4x4 = c3 d4x4 + d2x2 + d3x3 + d4x4 = d5j where the suffixes are seen to run twice from 1 to n. Another identity, just as worthy of note, is r=vi+ 1 , .... n f *(«! , «2 an) = tHai • TUflr - as) . (’(% , am+i , . . , an). 6‘— 1 , 2 , . . . , m - 1 The one is exemplified by the partition a2 -ax a3-a4 a4- a 1 a5 - ax a6 - ax $3 CL 2 CL 4 CL 2 CL^ CC^ CC^ ”■ CL 2 : a4 - as a5 — a3 a6 — a3 a 5 — a4 ~ a4 a6 ~ a5 > the other when instead of this the right- to-left dotted line is made to separate the third row of differences from the second. The former is that to which we have drawn attention when dealing with Jacobi’s memoir of 1841. * Notdhe posthumous book with this title edited by Prouliet and published in 1857. 364 Proceedings of 'Royal Society of Edinburgh. [sess. we conclude therefrom that the equation + XbY + jiuq 4- vd^)xl + (a2 + Xb2 4- pc2 + vd2)x2 + = a5 + Xb5 + gc5 + vd5 holds for all values of X, p, v ; and in order to obtain the value of Xj we have to solve the set cl2 + b2X 4- c2/x + dp/ — 0 CIq 4“ &gA. + Cg/A 4" dp/ = 0 i ) cl ^ 4* b^X + c^/x 4- dp/ = 0 J where the determinant of the coefficients of the unknowns is the conjugate of the complementary minor of a1 in | a^b2c3d^ |. With this fact in view, and along with it the nature of the relation of Murphy’s set to Prony’s, it will he readily seen that both sets appear in Sturm’s procedure. Terquem follows Sturm, and extends his method to the set of n equations x1 + 0.x2 4- x3 4- . . . + xn = b0y a1x1 + l.x2 +a3x3+ . . . + anxn = bY a\xx 4- 2 a1x2 + a \ x3 4- . . . 4 -a2nxn = b2 > a\xx 4-3 apc2 4- a\x3 4- . . . 4 - a3nxn= b3 where the coefficients of x2 are the differential-quotients of the corresponding coefficients of x{. The possibility of this solution rests on selecting x2 as the first unknown to be determined, and on the set being thus reducible to one of the previous type. Cayley (1846, Aug.). [Note sur les fonctions de M. Sturm. Journ. ( de Liouville) de Math., xi. pp. 297-299 : Collected Math. Papers , i. pp. 306-308.] The functions referred to, which are really Sylvester’s substi- tutes * for Sturm’s functions, are introduced in the form — * Sylvester. On rational derivation from equations of existence, Philos. Mag., xv. (1839), pp. 428-435: Collected Math. Papers, i. pp. 40-46. Sturm. Demonstration d’un theoreme d’algebre de M. Sylvester. Journ. (1 de Liouville ) de Math., vii. (1842), pp. 356-368. 1905-6.] Dr Muir on the Theory of Alternants. 365 / (x) = (x - a\)(x - - a3) . . . (x - an) f1(x) = 'Z(x-a2)(x-a3)(x-a4) .... f2(x) = i:,(a1-a2)2.(x-a2)(x-ai) .... ffx) = 2(«! - «2)2(fl2 - a3)2(a3 - ai)2-(x -aA) ... . P2 fm(x) —fix) . _ a^(oc - • ' • > ttm) = » a3 > • • • » am) ' ' {x- a1)(x — a2) . . . (x - am ) x — aY t\al , as , . . . , am) + x- a2 so that the coefficient of x~r is seen to be • £2(^2 ’ ^3 5 ■ • • j ^m) ^2 £2(^1 j a3 3 • • * J ®m) P and therefore to be 1 «1 a2 . . a™ 2 «; 1 (-)”-1 1 «2 a\ • • . . a™-2 «rl 1 < • • • • <~2 c1 Multiplying both sides by P and performing the requisite sum- mation we find that the coefficient of x~r in fm(x) — f(x) is sQ % ... sm_ 2 sr-l s2 ... Sm_ i sr or Yr_i say, 1 • • • ^2m— 3 ^r+m— 2 3 where is the sum of the qth powers of all the a’s ; in other words, that * It may be noted in this connection that C*(a 1 , '(«&) - ( - )n_*C2(«i , «2 3 • • • >aK-l 3 3 • • • 3 On)- if cf){x) = (x- %) (x -a2) . . . (x- an). 366 Proceedings of Royal Society of Edinburgh. [sess. = ar?rvM + + It only remains now to multiply by f(x) in the form xn +p2xn~2 - obtaining fm(x) = xn~m-Ym_1 + + a;»-j*-2(Vm+1 -joiym+p2 Ym_!) + and then to condense the coefficients, — an easy operation, since all the V’s are identical save in their last columns : for example ^m+ 1 "b ]?2^ m— 1 50 S1 • • • • 2 51 \ *• • • Sm- 1 Sm- 1 Sm • • • • S‘2m-i Swi+1 — P\sm +i^2sm-l Sm+ 2 —]?lsm+ 1 + P^m, ^2 m _Pl^2m-l "b-^2®2m— 2 Chelini (1846). [Determinazione geometrica in coordinate ellittiche . . . . Raccolta sci. di Polomba, ii. pp. 109-113, 126-131; see also v. pp. 227-263, 333-374.] Grunert (1847). [Vollstandige independente Auflosung der n Gleichungen der ersten Grades . . . Archiv d. Math. u. rhys., x. pp. 284-302.] The equations are A1 + A2ar + A3ar + . . . + An ar — CLr (v — 1,2,..., ?i) that is to say, are of the type to which Murphy’s belong, and with which a problem in interpolation is connected ; and the solution, rather tardily reached (p. 301), is m K(as > , a3 , a4 > * * • > an) (al- - a2)(al - a3)(al - a4) • • « ‘ (al -a«) a3 , a4 , > an) a (a2“ " al)(a2 ~ as)(a2 “ a4) . . . (a2- „ \2 an ) m K(a,, an-l) ^ (an - a1)(an - a2)(an - a3) . . . , (an - a^) n ’ m where by K is denoted “die ??ite Klasse der Kombinationen ohne Wiederholungen.” Rosenhain (1849). [Auszug mehrerer Schreiben .... liber die hyperelliptischen Transcendenten. bTo. IV. Crelle’s Journ ., xl. pp. 347- 360.1 In the course of an investigation regarding the relation between two Abelian integrals Rosenhain is brought up against the determinant ^.11 1 l\ ~ ai a2 In- 1 an-l already dealt with by Cauchy in 1841, and afterwards known as u Cauchy’s double alternant.” The multiple integrals in question have to suffer transformation of the variables, and as a pre- liminary it is ascertained that the Jacobian V +'’£) dt2 and + 'dtl dt2 ^ ~dx1 ‘dx2 dxn_ i dtn- 1 U dxn_i = c.2, -1 = D-Z 1 ^1 ' ^2 ^2 t1 a 1 t2 a2 In- 1 — an- 1 1 In- 1 ~ an- 1 where C and D are specified functions of the a’s and t’s. From this by multiplication it follows that i v± 1 . -L_ _j. r= j_, ( t j cq t2 (an-i) • 2 ± 1 al ^2 — a2 (2) its degree in any one of the a’s or f s; (3) the sign of any one of its terms. The exact words are — “Der Beweis der obigen Formel ergiebt sich durch die Betrachtung, dass (« d . 0(a2) . • . 0(«n-i) is arrangeable as a square array of binomial factors, being in fact, save as to sign, the product of all the denominators in the double alternant, and is thus seen to be symmetrical with respect both to the a’s and to the f s. If therefore we multiply each row of the alternant by the product of the denominators of the row, or each column by the product of the denominators of the column, we multiply the alternant by ( - l)(n-1)^-2)0(a1) . 0(a2) . • • Jc -t- X 1 c + y {c + y)\/c + y 1 c + y \Jc + y 1 c + z ( c + z)\Jc + z 1 c + z \Jc+z = 1 A A - (a-c) 1 y v? j say, ' 1 V v 2 1 C C2 /j. A I • {yv + v£+{y.-(a-cj) • V v2 c H PROC. ROY. SOC. EDIN. — YOL. XXVI. 24 370 Proceedings of Royal Society of Edinburgh. [s-kss. 1 (a-c)( 1+^-2) (a-cf^-' + t3) 1 (a - c)( 1 + rj~‘2) (a - c) f (i?-1 + 77“3) 1 (a-c)( 1+r2) (a-c^r + O , thence into (a-cyt*v~T3 and finally into -(a-c)rVT £3 £3 + £ £2+l rj 3 77s + 77 rj1 + 1 C3 £3 + £ £2+i , I £2 { £2 (£ + ?? + £- £>?£)• Brioschi (1854). [La teorica dei determinanti, e le sue principali applicazioni, yiii + 116 pp., Pavia: French translation by Edouard Combescure, ix + 216 pp., Paris, 1856 : German translation by Schellbach, vii + 102 pp., Berlin, 1856.] Brioschi devotes the 9th section of his text-book (pp. 73-84) to “ determinanti delle radici delle equazioni algebriche,” viewing the difference-product and its allies as arising when the roots of the equation xJ1 + An_!a;n-1 + An_zXn~2 + .... + Apr + A0=0 are substituted for x, and the values of A„_j , An_2 , , are to be determined from the n equations thus resulting. His proof, obtained in this way, that the common denominator of the A’s is resolvable into binominal factors is not of consequence. It is more important to note that, as an alternative, he proceeds “facendo uso di sole proprieta dei determinanti,” obtaining in the first place 1 1 . . . . 1 al a2 . . an «i ~ a2 ■ a2 a3 ■ • a«-i “ a« a" 1 2 tt2 *• . . an 2 2 ttl — ft2 2 2 a2 ~ a3 . . a" 71-1 2 — a 71 n— 1 ai n— 1 a2 . . n- 1 . a n 71 — 1 n— 1 ai “a2 71 — 1 71— 1 a2 ' a3 71— 1 •* “»-! 71—1 — an from which he removes the factors oq - a2, a2 - a3, . . . ; then 1905-6.] Dr Muir on the Theory of Alternants. 371 repeating the first set of operations he removes the factors cq - «3, a2 — a4, . . . , and so on. After this an application is made to the solution of a set of linear equations which differs from Prony’s set by having z°, zl, z2, ... in place of z0, zlt z2, . . . , and where therefore, as Cauchy in 1812 had pointed out, the numerators of the unknowns, as well as the common denominator, are resolvable into binominal factors. The determinants in s0, , s2 , . . . , got by multiplication, are also given. The remaining pages (77- 84) contain illustrations. Joachimsthal (1854, May). [Bemerkungen liber den SturnTschen Satz. Crelle’s Journ ., xlviii. pp. 386-416.] In the course of his investigations Joachimsthal evaluates (§ 5) the determinant where sq =x\ +x\-r x\. Using the fact that by reason of the trinomial elements the determinant is partitionable into twenty - seven determinants with monomial elements, he shows next that all of the twenty-seven except six vanish ; that the six contain the common factor (* - xl)(x - xl)(x - XS) ■ (*3 - Xl)(X3 - xtt)(x2 - xl) i that the aggregate of the cofactors is x2x2 - x\xx + x\Xi - x\xz + x lx3 - x\x2 or 0*3 ~ •rl)(;*'3 “ X2)(X2 ~ Xl) > and that therefore finally the given determinant is equal to y, S) 3 a + y + S a2 + y2 + S2 a + y +8 a2 + y2 + S2 a3 + y3 + 83 1 a2 + y2 + S2 a3 + y3 + 83 a4 + y4 + 84 Borchardt (1855, March). [Bestimmung der symmetrischen Yerbindungen vermittelst ihrer erzeugenden Funktion. . Monatsber . . . . Akad. d . Wiss. zu Berlin , 1855, pp. 165-171 : Crelle’s Journ ., liii. pp. 193-198 : Gesammelte Werke , pp. 97-105.] The generating function in question is 1905-6.] Dr Muir on the Theory of Alternants. 375 or T say, the sign of summation being meant to indicate that of the two series of elements the one is to remain unaltered and the other is to be permitted in every possible way. The development of this function according to descending powers of t , t1 , t2 , . . . , tn leads to those simplest types of integral symmetric functions of a , ax , a2 , . . . , an which originate by permutation from a single product of integral powers of the said variables. The determina- tion of such functions is thus reduced to the problem of trans- forming T so as to have no longer occurring therein the single elements a , ax , a2 , . . . , an , but instead those combinatory sums of them which are the coefficients of the powers of 2 in the development of (z - a)(z - a-^)(z - a2) . . . (z- an) or f(z) say. Without further preparatory statement the announcement is made that the solution is readily reached when the relation of T to the determinants y ±_L • — •••• t (X Ctj y + J__ . 1 1 tn - a, or A , 1 (*n-02 or is known, namely, the relation D = T-A. D, In proof of this relation it is pointed out that {M ■ fih) • f(h) • • • /(<»)}2-D being an integral alternating function both with respect to the elements t , tY , t2, . . . , tn and with respect to the elements a , ax , a2 , . . . , an is exactly divisible by the two difference- products 5 t^ 1 t2 , . . . , tn ) , II(a , a] , a2 , . . . , a , and that although we cannot with equal promptness tell the remaining factor, we are able to determine it from knowing a sufficient number of its special values, namely, those values got by putting each t equal to one of the a’s. Since the number of ways in which the n + 1 a’s can be taken when repetitions are allowed is (n + l)n+1, this gives us (w+l)n+1 values, of which, however, only two are different, namely, the value (- l)in{n+1) • f\a) • /'(af - f'(a2) . . . /'( an) obtained in the n !. cases where all the a’s used are different, and the value 0 obtained 376 Proceedings of Royal Society of Edinburgh. in every other case. The determination, we are told, can he made by using an extension of Lagrange’s interpolation-formula, the outcome of the work being y\ _ m / _ i un(n+i) H(^ , , . • • , C) ■ U(a , , a2 , . . . , CLn) f\t) -Ah) -/(y aq which, of course, gives us D = T-A. This relation having been established, Borchardt then proceeds in a line or two to use it for the main purpose of his paper. As the determinant D, he says, arises out of the determinant by performance of successive differentiation with respect to all the variables f , t1, t2, . . . tn , there is obtained at once an alternative expression for T , namely, T _ / _ 1 y.+i /(0 -/(n+ 1) ; and as it is exactly divisible by the difference-product of the u Js, which is of the order \n(n + 1), it follows that Y = A (ult u2, . . ., un+l) • Y1 where Yx is a function whose order-number is not greater than n2. Noting now that the other form of the resultant, namely [1,2,. . . , n+ 1] , can by addition be transformed into U A]A2 . . . A?l where U cannot contain any of the differences of the A s , and in 380 Proceedings of Royal Society of Edinburgh. [sess. its order-number cannot exceed n(n + 1) - n i.e. n 2, Joachimsthal concludes that Y1 and U can only differ by a factor dependent on the a’s. He thus has the two results J • A\A\ . . . A2 = A(uy , u2 , . . . , un+1) • Y1 and V, = { • U = ( • AXA2 . . . An[l , 2 , . . . , n+ 1] where £ is a rational function of the a’s : and by combining the two there is deduced J MU n _j_ i j . , z^2 , . . . , un+1) ' -A]A2 * * * ^n At this stage, we are told, the investigation rested for five years until the publication, in 1855, of Borchardt’s paper in the Berlin Monatsbericht. Taking a hint from this, Joachimsthal, in order to determine £, multiplied both sides of this result by the product of all the denominators occurring in the diagonal of J, and then put ul = - aY , u2= - a2 , . . . , un= - an. The left-hand side was thus changed into 1 0 .... 0 0 0 1 .... 0 0 0 0 .... 1 0 I 1 I ! («, + «a+,)2 («2 + un+1f ' ‘ (a„ + un+1f or 1 ; the second factor of the right-hand side, being equal to ( fa j — Un+ 1)(^2 — ^n+l) * * * fan ~ ^n+ 1) • ^fa\ ) ^2 > * * * > ^ n )) was changed into l)n(a1 + un+1)(a2 + Un+1) . . . fan + un+1) . (-iy-n(n~1)A(a1,a2, . . . , an) and the third factor [1,2, . . . , ?z+ lJ/AjAg . . . An into a fraction with the numerator 1 and with the denominator (a1- a2)(a1- a3) . . . (a1-an) fal + un+l) . (a2 - a1)(a2 - a3) . . . (a2 - an) (a2 + un+1) . (a3 - a1)(a3 - a2) . . . (a3-an) fa3 + un+1) . (an - a1)(an - a2) . . . (an-an_1 )fan + un+l) or < - l)in(n_1)A(a1 , a2 , . . . , an)2 • (al + \+i)(fl2 + %) . . . fan + un+1). 381 1905-6. Dr Muir on the Theory of Alternants. The result of the whole change was therefore (-ir 1 = £ A (a1 , a2 , . . . , an) ’ whence it followed that £ = ( — 1) ?A(oq } a2 f ... , an ) ) and so the longed-for result was reached J = (- ■ ■ ■ .“.+!>[! ,.,n+l\ AjA.j . . . A„ Thereupon additional results come with a rush. First we are told that in a similar manner the determinant got from J by changing the second power in the denominator of every element into the first power * is found equal to / _ ^(^1 ’ ^2 ’ ‘ ' • ’ ^n) • ^(ul ; U2 > • • • > ^n+l) V ' A1A2 . . . An Then “ E combinatione aequationum prod it det. j f 1 1 1 1 1 ! (a1 + uf ’.(«2+“)2’ ' ( an + u )2’ J det. 1 f J 1 1 i [ i + u 5 ? * a2 + u an + u ) =[i. + .!]■ u=u1, =u2> = : Un+ 1 Faciendo un+1= quantitati infinite magnae, aequatio in relationem a cl. Borchardt inventam transit, scilicet in det. j | 1 1 1 i 1 (aY + u)2’ ( a2 + u f ’ (i an + u )2 \ det. -1 ( 1 1 L i \ ax + u ’ a2 + u ’ u=u{ , =u2 , = . ’ an + u 1 . . , Un 2 1 (a1 + u1)(a2 + u2) . . . {an + Bellavitis (1857, June). [Sposizione elementare della teorica dei determinant!. Mem. . . . Istituto veneto . . . , vii. pp. 67-144.] Bellavitis reaches the subject of the difference-product in § 47 of his exposition, and his proof of the results dealt with in the * Previous suggestions of such a determinant appear in Binet’s paper of 1837 and Joachimsthal’s of 1854. 382 Proceedings of Royal Society of Edinburgh. [skss. preceding year by Prouhet is his own and interesting. Denoting the equation whose roots are ax , a2 , . . . , an by xn -p-^x71-1 +p2xn~2 - = 0 and the difference-product of the roots by II , he multiplies both sides of the identity (x-a1)(x-a2) . . . ( x-an ) = xn -p^-1 +p2xn~2 - .... by II ; and as the result on the left-hand side is evidently * the difference-product of al , a2 , . . . , an , xl he obtains | a\a\ . . . <"V| = (xn-f>1xn~1+ )n. It only remains then to equate like powers of x and there results | a\a\a\ .... <;X I = ^n, I « • • • I - ftH, | a\a\a\ .... I = P»H. He points out also that as an alternative to this we may begin with \a\a\a\ . . . arf1xn\, express it as a determinant of the next lower order, remove the factors (x - oq) , (x - a2) , . . . , (x - an ) , change the product of these into xn —pxn~x 4- . . . . , and then •equate coefficients of like powers of x as before. Multiplying again by n he has of course | a\a\a\ . . . | . n = (xn~p-ixn~1+ . . . )n2, and by changing n on the left into 1 1 1 . . 1 0 «2 as . . . • an 0 a\ ‘ 2 a 2 a\ • • < 0 n— 1 71—1 a2 n — 1 a, • • ■ ■ 0 0 0 0 . . 0 1 and twice using the multiplication-theorem there is obtained See footnote to page 365. 1905-6.] Dr Muir on the Theory of Alternants. 383 so sx • . • • Sn_i 1 So Si • • • • Sn_i Si s2 • • . . Sn X = (xn — ppcn 1 + . . • •) Si s2 • • • . «« sn— 1 «» • ■ < . . S‘jn—2 xn~l s« Sw+i • • ■ ■ • s2n- 1 xn Sn— 1 Sn • . • • S2n — 2 a result already reached by Joachimsthal, and which by the equatement of like powers of x gives “ i coefficient p espressi da rapporti di determinant di nesimo grado.” Betti (1857, June). [Sur les fonctions symetriques des racines des equations. Crelle’s Journ , liv. pp. 98-100.] Betti recalls Borchardt’s result of the year 1855, namely, that the symmetric function 2 1 X1 z n where xl , x2 , . . . , xn are the roots of the equation 0 = xn — pps71'1 + p2xn~2 — . , =f(x) say, is the coefficient of 2+1> . . . t~ in the develop- ment of / JL ± a r n(<.)} t -(al + l)^ _(®2 + l) -(«»+!) { ^(h. 5 ^2 > • • • J In) } £-(®l+l)/ _(<*2+l) /-(t*n+l) 1 v 2 * * ' n where the notation u^ed is sufficiently explained by saying that in accordance with it the coefficient of xr in the expansion of Fhc) is denoted by {*(*>}*■ Baltzer (1857). [Theorie und Anwendung der Determinanten, vi + 129 pp., Leipzig: French translation by J. Houel, xii + 235 pp., Paris, 1861.] The section (§ 12) dealing with the “Product aller Differenzen von gegebenen Grossen ” belongs to the second part of Baltzer’s text-book, that is to say, the part concerning “applications.” It occupies eleven pages, those devoted strictly to alternants being the first three (pp. 50-53). At the outset he establishes the determinant form for the difference-product P(cq , a2 , . . . , an) : then he gives two determinant-forms for P(cq , a2 , . . . , aB) . P^ , /32 , . . . , (3n ) : passes thence to the persymmetric determinants in s0 , , s2 , . . . : and finally gives Cauchy’s evaluation of the double alternant | (cq - /I-l)-1 (a2 - yS2)_1 . . . (an - /?n)_1 j . The applications, which come next, concern the solution of Lagrange’s set of linear equations, Sylvester’s transformation of a binary quantic into canonical form, and the discussion of the equality of two roots of the equation a/-t-a„_/'1-l- . . . + a0=:0, or say/(#) = 0, viewed in con- nection with what he calls the “determinant” of the equation, although Sylvester’s use of the word “discriminant” is explained a page or two later. Under this last head an interesting transformation falls to be noted. Calling the roots of the said equation cq , a2 , . . . , an , and taking the determinant which is the square of their difference- product, namely, 1905-6.] Dr Muir on the Theory of Alternants. 385 so S1 • • Sn-1 S1 #2 • V sn-l • • S2n-2 or Z say, he substitutes for it a determinant of the (2 n - 2)th order 1 0 0 . . . . 0 0 0 .... 0 0 1 0 . . . . 0 0 0 .... 0 0 0 1 . . . . 0 0 0 .... 0 0 0 0 . . . . £0 S2 .... sn-1 0 0 0 . . • • *i S2 S3 .... sn 0 *0 % • ; • • Sn- 3 «n- -2 Sn-1 .... S2n_4 *0 si s2 . . • • Sn_ 2 «»- -1 Sn • • • . $2n- 3 S1 S2 s3 . . • • Sn- 1 ^ n Sn+ 1 • • • • S2n-2 the first n- 2 row’s do not contain an s. and the following contain all the s’s in descending order from right to left, beginning with gn_j in the last place of the (n - l)111 row, with sn in the last place of the nth row, and so on. He then multiplies every row by an, and performs the operations which we may indicate by ar cob + — ^col, an cob dcol 2 + ?2=5col1; a„ cob -xcoL + jlJco12 + — — -col^ , thus obtaining an ®n- 1 &n- 2 0 an M'n—l 0 0 an 0 an* 0 ansi + fl„_i s0 anS0 ansx + an_xsQ ans2 + an_1s1 + an_2s0 ans i anS2 + an_xsx cins3 + an_xs2 + an_2sx PROC. ROY. SOC. EDIN. — YOL. XXYI. 25 386 Proceedings of Royal Society of Edinburgh. [sess. and by using Newton’s relations nan = ans0 , (n-l)an^ = ansL + an_1s0 , (n — 2)an_2 = ctns2 + an_1s1 + ci,n_2s0 , (n — 3)an_s = (XmS3 + + &n_2Sl + ®n-3S0 > the elements of the last n rows of the right-hand determinant, we are told, can be so changed that in each there will occur only one of the a! s and that in the first power. Thereupon the conclusion is formally announced that the -determinant with which we started can be expressed as a rational integral function of the (’2n- 2)th degree in the quantities «o ^n— 1 " 5 5 * ' * J 5 an an an and that the said function becomes homogeneous on multiplication by The actual result is not given, but in the second edition (1864) it is stated to be an «n-l an_ 2 . . . . 0 «n- 1 - - - - 0 0 an .... 0 nan - - - - nan (n - 2)an_2 . . . . an- 1 2an_2 3an_s . . . . “eine Determinante (2n - 2)ten Grades, bei welcher die m-2 ersten und die m — 1 folgenden Zeilen in Bezug auf die nicht verschwindenden Elemente ubereinstimmen.” Part of the object which Baltzer had here in view was to establish the relation between two forms of the discriminant of the given equation ; namely, that obtained by squaring the determinant-form of the difference-product and that obtained as the eliminant of the equations f\x) = 0 , nf(x) - xf\x) fi 0 , or the equations (anxn + an_{xny + . . . +aQyn) = 0 1905-6.] Dr Muir on the Theory of Alternants. 387 Now a glance at the final determinant suffices to show that it is not the eliminant sought, there being in it three types of rows, whereas the two equations giving rise to the said eliminant being both of the (n — 1 )th degree, the coefficients of the one must occur in as many rows of the eliminant as the coefficients of the other. Further, since the coefficients of the equation f'(x) = 0 are seen to occur in their full number of rows, and those of the other equation in the last row only, it is therefore the first n - 2 rows that need to be changed. The set of operations requisite to effect this is n • rowx - row2n_3 , n • row2 - row2n_4 , n • rown_2 - rown . Brioschi (1857, Oct.). [Sullo svillippo di un determinante. Annali di Mat., i. pp. 9-11.] Brioschi enunciates without proof the proposition that the eveii-ordered determinant 1 1 l 1 1 1 ®1 - Oi - «i)2 x1 - a2 (*1 - a2f • ' '*i - an {xx - «n)2 1 1 l 1 1 1 x2 — aq ( - x2 - — a2 )2 x2 — an (x2 -<)2 1 1 l 1 1 1 x2n - (X2n - - 6^2 (x2n a2f x2n an (X2n - an )2 which is seen to be a function of 2 n %’s and n a’s, is equal to / n4(ax, a2, . . . , an) • n(aq , %2, ... , x2n) [ > <£>,) • tf(a2) . . . (x) = (x - x^)(x — x2) . . . (x — x2n). He then obtains similar expressions for the principal minors, namely, (1) for the cofactor of any element in an odd-numbered column, and (2) for the cofactor of any element in an even-numbered column, his procedure being to express the minor in question in terms of determinants like, the original but of the order 2n—2 and then to make the substitutions which are thus rendered possible. 388 Proceedings of Royal Society of Edinburgh. SESS. Prouhet (1857, Nov.). [Questions 410, 411. Nouv. Annales de Math., (1) xvi. pp. 403, 404; xvii. pp. 187-190.] By reason of the existence of the identity 2s-1 cos *a = cos sa + s cos (s- 2)a + 1) cos (s-4)a+ . . . it is clear that the determinant cos na0 cos (n — \)a0 cos (n - 2)a0 . . . cos 0.a0 cos naY cos (n— l)ax cos ( n - 2)ax . . . cos O.aj cos na2 cos (n - l)a2 cos (n - 2)a2 . . . cos 0.a2 COS ?lan cos ( n — l)an COS ( n - 2)an . . . cos 0.an may he transformed into ()n— 1 COSMa0 o n-2 cosn_1a0 9«-3 COSn_2a0 . . , . COS°a0 2n— 1 COS’1^ e)n—2 cos”-1^ 9n-3 COS”-2^ . . . COS0^ 2»-i COSna2 On— 2 cos”-1^ 9n-3 COS,l_2a2 . . . cos0a2 c)n— 1 COSnan 2n~2 COSn-1an 9«— 3 COSn-2an . . . cos°an by increasing the 1st column by multiples of the 3rd, 5th, 7th, ... , the 2nd column by multiples of the 4th, 6th, 8th, . . . and so forth. In this way there is deduced the result ^ _ 2*ra(«-i) _ j) } where \ is the first determinant, and D is the determinant got from by changing the multipliers of a0 , , a2 , . . . into indices of powers of cos a0 , cos a2 , cos a2, ... . Again by using the identity sin (s + 1 )a — - sin a{ 2scossa • coss_2a + q • coss“4a + . . . } on every element of the determinant sin ( n 4- l)a0 sin na0 . . sin a0 sin ( n + 1)0^ sin nax . . , . sin sin (n + l)an sin nan . . , . sin an it is seen that the factors sin a0 , sin a3 , . . . . can be removed from the rows in order, and that the determinant so produced is 1905-6.] Dr Muir on the Theory of Alternants. 389 simplifiable into a multiple of D : so that there is obtained the second result A 2 = 2*n(M“1) • sin aQ sin oq . . . sin an • D . The two results are Prouhet’s, who set them for proof by others. Salmon (1859). [Lessons introductory to the modern higher algebra, xii + 147 pp., Dublin.] The determinant form of the difference-product and the determinants in s0 , Sj , . . . are given, but merely as illustrative examples of the general subject. Liouville (1846).* [Sur une classe d’equations du premier degre. Journ. (de Liouville ) de Math ., xi. pp. 466-467 ; or Nouv. Annales de Math., vi. pp. 129-131; or Archiv d. Math. u. Phys., xxii. pp. 226-228.] The set of equations referred to is that dealt with by Binet in 1837. Chelini and Liouville arrived at a new solution, much simpler than Binet’s, and related to that used by Murphy in 1832 in solving other sets of linear equations. LIST OF AUTHORS whose writings are herein dealt with. 1832. Murphy page . 357 1854. Brioschi PAGE . 370 1837. Binet . . 358 1 1854. JOACHIMSTHAL . 371 1841. Haedenkamp . 361 1854. Brioschi . 373 1845. Borchardt . 362 1855. Borchardt . . 374 1845. Rosenhain . 362 1856. Prouhet . 376 1845. Sturm . . 363 1856. ScHEIBNER . . 377 1846. Terquem . 363- 1856. JOACHIMSTHAL . 377 1846. Cayley . 364 1857. Bellavitis . . 381 1846. Chelini . 366 1857. Betti . . 383 1846. Liouville . 389 1857. Baltzer . 384 1847. Grunert . 366 1857. Brioschi . 387 1849. Rosenhain . 367 1857. Prouhet . 388 1850. Mainardi . 368 1859. Salmon . 389 1853. Cayley , 369 * This should have been inserted under but not separate from ‘ 1 Chelini (1846),” the proper joint heading being “ Chelini and Liouville (1846).” {Issued separately November 16, 1906.) 390 Proceedings of Royal Society of Edinburgh ■ [sess. The Theory of Circulants in the Historical Order of Development up to 1860. By Thomas Muir, LL.D. (MS. received July 9, 1906. Read July 13, 1906.) So far as mathematical writers have as yet noted, a set of equations of the type alxl + a2x 2 + . . . + anxn — u j , a nsc1 4- aYx2 + . • • +VA = %, '>n-lXl + anX 2 + • . . + an_ - 2x1 + 2x2 = u1-u2, — 2x2 + 2x8 = u2 — u8 , > -2xn_1 + 2xn = un_1-un. ) D’apres ce qui precede, le determinant A' du nouveau systeme sera ( - l)”- hi A. Mais, d’un autre cote, en comparant A' au determinant A" du systeme x1 + x2 + . . . + X3 ~ -*1 d~ x2 = X2 + XS ~ Xn—1 d* Xn = on a A' = (n — 2)2n_1A". Enfin, d’apres le n° 13, et en observant que les quantites - A2 , A2 - A3 , . . . . ont ici change de signe A " = n . * This result is reached in a way different from Catalan’s by performing on A' the operation rowj + (?t-l)row2 + (%-2)row3 + . . . + rown , separating out the factor n, and then showing that the resulting determinant is ( — l)n—1A . 392 Proceedings of Royal Society of Edinburgh. [sess. On d4duit, de ces diverses formules A = (w — 2)( - 2)n_1.” This result, which at a later date would have been written C(-l, 1, 1, 1) = (»-2)(-2)»-1, and which, we may point out in passing, could also he reached by the operations row1 + row2+ . . . +rown, remove factor to - 2 , rown - rown_! , rown_x - rown_2 , .... is then attempted to he generalised (§18) by withdrawing the restriction as to the number of negative units in a row. The reasoning, however, seems to have been incautiously conducted, the extension arrived at being C( - 1 , -1, 1, l)„-r = (»-2p)(-2)"-1, where the number of consecutive negative units in the first row is p, and the number of positive units n-p. Catalan then passes (§19) to the consideration of the similar circulant whose first row consists of p consecutive positive units followed by to— p zeros, separating the investigation into two parts, (1) the case where p and to have a common factor other than unity, (2) where they are mutually prime. In the former case he shows that the equations which have the circulant in question for determinant are “ ind6terminees ou incompatibles ” : in the latter case he shows that the equations are determinate. He thereupon goes on to supplement the information in the second case by proving that the circulant is equal to p : he omits, however, any similar proof that in the first case the circulant is zero. Lastly, he attacks the general circulant, or, as he calls it, “le determinant du systeme CL-^ CLcy (f/g • • • a2 a3 a4 ... «3 «5 ‘ ‘ ' a2 an a1 a2 ... 1905-6.] Dr Muir on the Theory of Circulants. 393 The procedure, however, is rather perverse, the theorem of § 13 being forced into service. This gives A =(-l)"-1l A', n where A ' is the determinant of the system s ax -a2 a2 — a3 . . . . (in_ i - a, s a2 -as i CO a4 . . . . an ~ai s aB -a. a4“ a5 . . . . Oj - a( s an — rtj aY - a2 an_2 — an_x , after which A ' js is partitioned into determinants with monomial elements, and certain more or less evident reductions made. The result is “ Le determinant du systeme propose s’obtiendra en multipliant al + a2 + ... +an (i.e. s) par le determinant du systeme ax - a2 a2 a3 . . . . a„ _i - a, a to 1 a3 - «4 • • . . an , - a- e e8 an . . an 3 _ a theorem which afterwards came to be written in the form C(flh , a2 , . . . , an) = (a1 + a2 + ... + an) • V{al-a2 , . . . , an_x-an , an-a4 , . . . , an_3-«n_2) the symbol P(cc, y , z,w, v) being used to stand for the “per- symmetric ” determinant x y z y z w z w v Spottiswoode (1853). [Elementary theorems relating to determinants. Re-written and much enlarged by the author. Crelle’s Joum ., li. pp. 209- 271, 328-381.] In the section (§ xi.) which did appear in the first edition, and which bears the title “ Miscellaneous instances of determinants,” 394 Proceedings of Royal Society of Edinburgh. [sess. the following is given (p. 375), being the fourth of the said instances : — “ Let 1 , q , i2 , . . . , in be the n+ \ roots of the equation xn+1-l = 0, then, whatever be the values of A , A1 , A2 , . . . , An = ( A + A1 + . . . + An)( A + ix A1 + . . . + i\ An) • (A + fnAT+ . . . + inAn).} A A, • ■ , . A„ A, A2 . . .. At An A,.. ■ • An_x No word of proof is added: probably the result was reached by Sylvester’s “ dialytic ” method of elimination. But however this may be, it should be noted that resolvability into linear factors soon came to be looked on as the fundamental property of the circulant. It has to be noted that Spottiswoode makes a slip in omitting the sign-factor (- l)^(n_1) from the right-hand member; and that he writes his determinant in such a way as to have it per- symmetrie with respect to the principal diagonal, whereas Catalan wrote his so as to have it persyinmetric with respect to the secondary diagonal. Putting C' for the functional symbol in the former case we have C (a1}a2, . .., O = (-lp-^-^.C \alia2i ..., an) . If therefore Spottiswoode had followed Catalan’s mode of writing, his result would have been strictly accurate. Cremona (1856). [Intorno ad un teorema di Abel. Annali di sci. mat. e fis.y vii. pp. 99-105.] To prove the theorem of Abel referred to in the title, Cremona starts by establishing three lemmas, the first of which is Spottis- woode’s theorem regarding circulants. Taking any n quantities «] , «2 » * ' • ’ an-\ and denoting $q -f- a^cf -}- a^a2r -f- . . . + an_i (P ^ by 0r 395 1905-6.] Dr Muir on the Theory of Girculants. where ar stands for ar and a for a primitive root of the equation xn - 1 = 0, he multiplies the determinant «o «1 a2 . , • • «»- 1 cq «2 a3 * . . a0 a2 a3 a4 • . . ax or D say, «»-i ao aY . • • Un-2 by the determinant 1 1 1 .. 1 1 ai a2 • • an- 1 1 «; • • -Li or A say, 1 anfr ar1 • • c; 5 and obtains a; product-determinant from whose columns, he says, the factors 01 . , 02, . • * 5 0n may be removed in order, so that their results 1 1 1 . . 1 1 2 n— 1 an- 1 <-l * • • • an_! DA = Of 2 . . . on 1 an- 2 2 an-‘2 * • n— 1 • • %-2 1 2 71 — 1 aL . • . . a, — Of) 2. • • • 0n • (-1)! n(n- 1)^ ^ and .*. D = ( -l)1 n(n— 1) Of) % • • • 0n . The proof, which is said to be due to Brioschi, is not improved in neatness by introducing the conception of a primitive root, nor by writing the root 1 in a different form from the other roots. The second lemma concerns the differential-quotient of D with respect to any variable of which the a’s are functions. Denoting this differential -quotient by D', and by Dr the determinant got from D by substituting for each element in the rth column the differ- ential-quotient of that element, Cremona of course has at once D = Dj + D2 + . . . + Dn . As, however, Dj here can be shown by translation of a number of rows and the same number of columns to be equal to any one of the D’s following it, there results 396 Proceedings of Royal Society of Edinburgh. [sess. V — nD i = »Da = The third Lemma is to : the effect that the quotient of the determinant mo do qxd . • • • qn- *dn —2 m1 d did q2d2 . . . . gn-\dn -1 mn _fn~l dn-ldn~ 1 do • . • qn-sdn -3 by d is a rational function of dn. By multiplying the 2nd, 3rd, 4fch, . , . . columns by dn, dr> i—l Jn— 2 . . . respectively, and then dividing the corresponding rows by d , d2, d*, ... , . respectively, there is obtained m0 q0dn (x,Z, t) / 2 (qfx 1 \ Vpfb-^) -gt2z e 4P2 (137), where p= s/(z2 + x2) , and ^ = tan 1(x/z), and in all of §§ 1-31, this notation and — £ was consistently used, with — £ to denote, when positive, upward displacement of the water (represented by upward ordinates in the drawings). In the two curves of § 4, fig. 1, that which has its maximum PROC. ROY. SOC. EDIN. — VOL. XXVI. 26 402 Proceedings of Royal Society of Edinburgh, [sess. 1905-6.] over 0 represents (137), for t = 0. The other curve of fig. 1, with positive and negative ordinates on the two sides of 0 , represents (137), with — {RD} instead of {RS}. The symbols {RS} and {RD} were introduced in § 3 above; {RS} to denote a realisation by taking half the sum of what is written after it with ±i, and {RD} to denote a realisation by taking — of the 2c formula written after it minus — of the same formula with + 1 Jii changed into - t . A new curve in which the ordinates are numeri- cally equal to fj2~dx ^he or(^nates °f the second of the old curves of fig. 1, is now given in the accompanying diagram, fig. 33 ; and close above it the first of the old curves of fig. 1 is reproduced, with ordinates reduced in the ratio 2^/2 to 1 , for the sake of comparison with the new curve. This new curve represents the more convenient initiational form referred to in the title of the present paper. Its equation, found by taking t = 0 in (139) or in (144) [most easily from the imaginary form of (139)], is as follows : 0) = 1 '2J2 V(p+^)(2z- ) p3 . . (138). § 101. The original derivation of the new particular solution, (which we shall call \j/,) from the primary (136), as indicated in § 100, is shown by the following formula : , M) = {RI)} d_ -1 dx + ix) -g# e4(z+ta;) \ -g&z € 4p2 (139), where p = J(z2 + x2), and x = tan \x/z) . An equivalent formula for the same derivation, which will be found more convenient in §§ 135—157 below, is as follows : ^(£,M) = {RS} 1 d2 -1 g dt2 J(z + lx) - gt 2 e4(z+ia;) = -1 df gJ2 dt 2 $(x,z, t) (140). o UJ'dvr-CAbru [sess. 1905-6.] Lord Kelvin on an Initiational Form. 405 The equivalence of (139) and (140) is easily proved by remarking that by (133) and (135), dF _ dF dx dz g dt 2 (141), and therefore d {ED} 1 dx J(z + lx) ~9t 2 l (?. £4(^)={ES} 1 a -1 -gt 2 g dt 2 J(z + lx) e4 (z+LX) (142). § 102. Look now to fig. 33, and see within how narrow a( space, say from x — — 2 to x + 2 , in the new curve, the main initial disturbance is confined, while in the old curve it spreads so far and wide that at x — ± 20 it amounts to about *16 of the maximum disturbance in the middle, and according to the law of inverse proportion to square root of distance, which holds for large values of x for the old curve, at ^ = 80 it would still be as much as '1 of the maximum. The comparative narrowness of the initial dis- turbance represented by the new curve, and the ultimate law of decrease according to x"'i (instead of x~i for the old curve) are great advantages of the new curve in the applications and illus- trations of the theory to be given in §§ 135-157 below. § 103. Kemark also that the total area of the old curve from — cc to + cc is infinitely great, while it is zero for the new curve. Eemark also that the potential energy of the initial disturbance, being igjdx[£(x, 1,0)P .. . . . (143), is infinitely great for the old curve, while for the new it is finite. § 104. Equation (139) may be written in the following modified form, which is more convenient for some of our interpretations and graphic constructions : cos A . (144),. where - |X - tan" . (145). § 105. The main curves, which for brevity we shall call water- curves in the accompanying six diagrams of fig. 34, represent the 406 Proceedings of Royal Society of Edinburgh. [sess. surface displacements according to our new solution if/(x , z, t) for the six values of t respectively, 0, J^/tt, n/it, ^ Jit , 8^/7 r. The formulas are simplified by taking g = 4. This is merely equivalent to taking as our unit of length half the space descended in one second of time, by a body falling from rest under the influence of gravity. For simplification in the writing of formulas we take z = 1 for the undisturbed level of the water-surface. The subsidiary curves, explained in § 107 below, are called argument- curves, as they represent the argument of the cosine in (144). § 106. One exceedingly curious and very interesting feature of these curves is the increasing number of values of x for which the displacement is zero as time advances, and the large figures, sixteen and sixty-four, which it reaches at the times, ijir and 8^/73-, of the last two diagrams. These zeros, for any value of t, are given by the equation A = (2;+1)tt/2 (146). § 107. Notwithstanding the highly complicated character of the function represented in (145), the zeros are easily found by tracing an argument-curve, with A as ordinate, and x abscissa (as shown on the ir-positive halves of the six diagrams on two different scales chosen merely for illustration, not for measurement), and drawing parallels to the abscissa line at distances from it representing - f-7r , - J-7T , J73- , -|7t , -§-73- , etc. A parallel at distance - £73- is an asymptote to each of the argument-curves, and is shown in diagrams 2, 3, 4, on one scale of ordinates. The parallel corre- sponding to distance ^-7 r is shown in the fifth and sixth diagrams, on the smaller scale of ordinates used in their argument-curves. § 108. The first diagram shows zeros at x— ± J3 , of which that at x = — ^3 is marked 1 . In the second diagram the argument- curve indicates zeros for the - and - J7 r parallels, which are seen distinctly on the water-curve. The zero corresponding to the - J7 r parallel was formed at the origin at the time when \gfi was equal to 3, that is, when t was 1/^/2, or *707. It is a coincidence of two zeros for ^-positive and cc-negative. Diagram No. 3 shows that, shortly before its time, a maximum has come into existence in the argument-curve, which still indicates only two zeros. These are marked by crosses. 1905-6.] Lord Kelvin on an Initiational Form. 407 Diagram No. 4 shows that, in the interval between its time and the time of No. 3, two zeros of the water-curve for ^-positive have come into existence. These and the corresponding zeros for x-negative are seen distinctly on the water- curve; and their indications for ^’-positive are marked by four crosses on the argument-curve. Diagram No. 5 shows that, between its time and that of ISTo. 4, twelve fresh zeros have come into existence on each side of 0 Z, one pair of which is indicated for example on the argument-curve by the parallel ^7 r. Nine only out of all the sixteen zeros on either side are perceptible on the water-curve. The seven imperceptible zeros, on each side, all lie between x = 0 and x — ±\. Diagram No. 6 shows that, between its time and that of No. 5, forty-eight fresh zeros for ^-positive have come into existence, one pair of which is indicated by the parallel ^fLr. Fourteen only out of all the sixty-four zeros on each side are perceptible on the water-curve. Thirty-one of the fifty imperceptible zeros on each side lie between x= 0 and x= ± 1. § 109. After the time l/J’2, the zeros originate in pairs on the two sides of the origin* (^-positive and ^-negative) : those on the positive side by the two intersections of one of the parallels corresponding to (2^ + l)7r/2 with the argument-curve. The maximum of the argument-curve travels slowly in the outward direction towards x=l as time advances to infinity. At times 4^/7 r and SJtt, of diagrams 5 and 6, it has reached so close to x= 1 that this point has been regarded as the actual position of the maximum, both for the purpose of drawing the curve, and for the determination of the total number of zeros. § 110. Each zero which originates according to an intersection on the outward side of the argument-curve travels outwards with increasing velocity to infinity, as time advances. Each of the others of the pairs of zeros, that is to say, each zero originating according to an intersection on the inward side of the argument- curve, travels very slowly inwards with velocity diminishing to nothing as time advances to infinity. Thus the motion of the * If we continue the argument-curve to the side of the origin for ai-negative, we must include large negative values of i in (146) : but for simplicity we have confined the argument-curve to positive values of x. 408 Proceedings of Royal Society of Edinburgh. [sess. water in the space between x = -1 and x — +: 1 becomes more and more nearly an increasing number of inward travelling waves, with lengths slowly diminishing to zero ; and, as we see by the exponential factor in (144), with amplitudes and with slopes also slowly diminishing to zero : as time advances to infinity. § 111. The semi-period of one of these quasi standing waves is, 9 2 as we find from (139), approximately equal to when the time is so far advanced that J- gt 2 is very great in comparison with p. Thus we see that the period is infinite at the origin. This agrees with the history of the whole motion at the origin, which, as we see by putting x = 0 in (139), with 2=1 and p = 4,, is expressed by the formula The motion of the water in the space between x = — 1 and x = + 1 is of a very peculiar and interesting character. Towards a full understanding of it, it may be convenient to study the simplified approximate solution which the realised part of (139) gives when \ gt 2 is very large in comparison with p. § 112. The outward travelling zeros on the two sides, beyond the distances ± 1 from the origin, divide the water into con- secutive parts, in each of which it is wholly elevated or depressed. These parts we may call half-waves. They travel outwards with ever-increasing length and propagational velocity. Each of the half-waves developed after t = Jtt , as it travels outward,, increases at first to a maximum elevation or maximum depression, and after that diminishes to zero as time advances to infinity. § 113. It is interesting to trace the progress of each of the zeros in the intervals between the times of our six diagrams. This is facilitated by the numbers marked on several of the zeros in the different diagrams. Thus, confining our attention to the left-hand (147). (148), 1905-6.] Lord Kelvin on Groups of Deep-Sea Waves. 409 side of fig. 34, we see in diagram 1 a single zero numbered 1. The future zeros are to be numbered in the order of their coming into existence, 2 ; 3 , 3 ; 4 , 4 ; . . . ; 10,10; . . . ; 33 , 33 ; . . all in pairs after zero 2. Thus diagram 2 shows zero 1 considerably- advanced leftwards (that is, outwards) ; and zero 2 beginning its outward progress. Diagram 3 shows zeros 1 and 2 each advanced arther outwards, 1 farther than 2. Diagram 4 shows all the zeros which have come into existence at time These are zeros 1 and 2, both farther outwards than at time Ji r, and a pair, 3,3, which have come into existence shortly before the time The outer of these two travels outwards and the inner inwards. Some time later 4 , 4 come into existence between 3 and 3 : later still 5 , 5 come into existence between 4 and 4. In diagram 5, zero 1 has passed out of range leftwards : but we see distinctly the outward zeros 2, 3, 4, 5, 6, 7, 8, 9, and indications of the inward zeros 9,8. The whole train of zeros for time 4 Jtt, .shown and ideally continued - to the middle by numbers, is 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3; sixteen in all. Zero 3 has passed out of the range of diagram 6, but we see in it distinctly the outward zeros 4,5,.... 12, and an indication of the pair 33 , 33 , which has come into existence before the time 8 Jit. The whole train of zeros for time S^Ar, indicated by numbers, is 1 , 2 , .... 32, 33, 33, 32, .... 4, 3; sixty- four in all. (2) Illustrations of the Indefinite Extension and Multiplica- tion of a Group of Two-Dimensional, Deep-Sea Waves Initially Finite in Kumber. §§ 114-117. § 114. The water is left at rest and free, after being initially displaced to a configuration of a finite number of sinusoidal mountains and valleys — five mountains and four valleys ; in the diagrams placed before the Society. The initial group of waves, shown in diagram 1, of fig. 35, is formed by placing side by side, at distances equal to 2 (taken as unity), nine of the curves of diagram 1, fig. 34, alternately positive and negative. Diagrams 2 and 3, of fig. 35, are made by corresponding superpositions of Initial group of five elevations and four depressions emerging as two groups travelling in opposite directions. [sess. 1905-6.] Lord Kelvin on Groups of Deep-Sea Waves. 411 the curves of diagrams 5 and 6, of fig. 34. Thus what, according to the known law of deep-sea periodic waves (§ 19 above), would be definitely and precisely the wave-length, if the numbers of crests and hollows were infinitely great, would he 2 ; and as we are taking g — 4, the period would be Jtt, and the propagational velocity would he 2/^/7 r. § 115. Immediately after the water is left free, the disturbance begins analysing itself into two groups of waves, seen travelling in contrary directions from the middle line of the diagram. The perceptible fronts of these two groups extend rightwards and left- wards from the end of the initial single static group, far beyond the “hypothetical fronts,” supposed to travel at half the wave- velocity, which (according to the dynamics of Osborne Reynolds and Rayleigh, in their important and interesting consideration of the work required to feed a uniform procession of water-waves) would be the actual fronts if the free groups remained uniform. How far this if is from being realised is illustrated by the diagrams of fig. 35, which show a great extension outwards in each direction far beyond distances travelled at half the “ wave- velocity.” While there is this great extension of the fronts outward from the middle, we see that the two groups, after emergence from co-existence in the middle, travel with their rears leaving a widening space between them of water not perceptibly disturbed, but with very minute wavelets in ever-augmenting number following slower and slower in the rear of each group. The extreme perceptible rear travels at a speed closely correspond- ing to the “halfwave-velocity,” found by Stokes as exactly the group-velocity of his uniform succession of groups, produced by the interference of two co-existent infinite processions of sinusoidal waves, having slightly different wave-lengths. § 1 16. Our fairly uniform rear velocity is illustrated in diagrams 1 and 3, of fig. 35. In diagram 1, R indicates the perceptible rear of the component group commencing its rightward progress at t = 0. In diagram 3, R shows the position reached at time 8 7r (eight periods) by an ideal point travelling rightwards from the R of diagram 1 at a speed of half the wave-velocity. This R of diagram 3 corresponds to a fairly well-marked perceptible rear of the rightward travelling group. 412 Proceedings of Royal Society of Edinburgh. [sess. Look now to F, F, F, in the three diagrams of fig. 35, and /, /, in diagrams 2 and 3. In diagram 1, F marks a perceptible front for the rightward travelling component group. In diagrams 2 and 3, F, / show ideal points travelling rightwards from it at speeds respectively, the half wave-velocity, and the wave-velocity. We see a manifest wave-disturbance far in advance of F, F ; and very small but still perceptible wave-disturbance in front of/,/. Thus the perceptible front travels at speed actually higher than the wave-velocity, and this perceptible front becomes more and more important relatively to the whole group with the advance of time, as we may judge from fig. 9 of § 20 above. § 117. It is interesting to see by these diagrams how nearly the hypothetical group-velocity is found in the rears : while the fronts advance with much greater and with ever-increasing velocity. The more elaborate calculations and graphical constructions of §§ 20-29 above led to corresponding conclusions in respect to the front and rear of a procession, given initially as an infinitely great number of regular sinusoidal waves travelling in one direction. The diagrams, figs. 9 and 10, showed respectively, at twenty-five periods after a sinusoidal commencement, a front extending forward indefinitely, and a perceptible rear lagging scarcely two wave-lengths behind a point, travelling from the initial position of the rear at a speed of half the wave-velocity. (3) The Initiation and Continued Growth of a Train of Two-Dimensional Waves due to the Sudden Commence- ment of a Stationary, Sinusoidally Varying, Surface- Pressure. §§118-158. § 118. A forcive consisting of a finite sinusoidally varying pressure is applied, and kept through all time applied, to the surface of the water within a finite practically limited space on each side of the middle line of the disturbance. In the beginning the water was everywhere at rest and its surface horizontal. The problem solved is, to find the elevation or depression of the water at any distance from the mid-line of the working forcive^ and at any time after the forcive began to act. 1905—6.] Lord Kelvin on the Growth of a Train of Waves. 413 §119. As a preliminary (§§119-126) let us consider the energy in a uniform procession of sinusoidal waves, in a straight canal, infinitely long and infinitely deep, with vertical sides. If the water is disturbed from rest by any pressure on its upper surface, and afterwards left to itself under constant air pressure, we know by elementary hydrokinetics that its motion will be irrotational throughout the whole volume of the water : and if, at any subsequent time, the surface is brought to rest, suddenly or gradually, all the water at every depth will come to rest at the: instant when the whole surface is brought to rest. This, as we know from Green, is true even if the initial disturbance is so violent as to cause part of the water to break away in drops : and it would be true separately for each portion of the water detached from the main volume in the canal, as well as for the water remaining in the canal, if stoppage of surface motion is made for every detached portion before it falls back into the canal. § 120. Because the motion of the water is irrotational, we have c=dF . where v 2 is eliminated by (157). § 125. Thus we see that the kinetic energy per wave-length, and the potential energy per wave-length, are each equal to the work done per period by the water on the negative side, upon the water on the positive side, of any vertical plane perpendicular to the length and sides of the canal. Thus we arrive at the remarkable and well-known conclusion that in a regular pro- cession of deep-sea waves, the work done on any vertical plane is only half the total energy per wave-length. This is only half enough to feed a regular procession, advancing to infinity with abruptly ending front, travelling with the ivave- veloeity v. It is exactly enough to feed an ideal procession of regular periodic waves, coming abruptly to nothing at a front travelling icith half the “ wave-velocity ” v\ which is Osborne 416 Proceedings of Royal Society of Edinburgh. [sess. Reynolds’ * important contribution to the ideal doctrine' of “group-velocity.” § 126. The dynamical conclusion of § 125 is very important and interesting in the theory of two-dimensional ship-waves. It shows that the approximately regular periodic train of waves in the rear of a travelling forcive, investigated in §§ 48-54 and 65-79 above, cannot be as much as half the space travelled by the forcive, from the commencement of its motion ; but that it would be exactly that half-space if some modifying pressure were so applied to the water-surface in the rear as to cause the waves to remain uniformly periodic to the end of the train ; without, on the whole, either doing work on them, or taking work from them. A corresponding statement is applicable to our present subject, as we shall see in §§ 156, 157 below. § 127. Go back to § 118; and first, instead of a sinusoidally varying pressure, imagine applied a series of impulsive pressures, each of which superimposes a certain velocity-potential upon that due to all the previous impulses; and let it be required to find the resulting velocity-potential at any time t, after some, or after all, of the imjDulses. Consider first a single impulse at time t -q ; that is to say, at a time preceding the time t by an interval g. Let the velocity-potential at time t, due to that single impulse applied at the earlier time t -q, be denoted by CV(jj, z, q) (168). According to this notation the instantaneously generated velocity- potential is CV(£, z, 0), and the value of this at the bounding surface of the water is CV(x, 1, 0). Hence, by elementary hydro- kinetics, if I denotes the impulsive surface-pressure, we have 1= -C Y(x, 1, 0) (169). §128. Considering now successive impulses at times preceding the time t , by amounts qY, q2, .... q^, and denoting by S(x,z,t) the sum of the resulting velocity-potentials at time t , we find S(a;, a, t) = Oft(x, z. , qY) + C2V(:r, z, q2) + .... . CiY(^, z, ft) (170). Supposing now the impulses to be at infinitely short intervals of * Nature , August 1877, and Brit. Ass. Report , 1877. 1905-6.] Lord Kelvin on the Growth of a Train of Waves. 417 time, we translate the formula (170) into the language of the integral calculus as follows : S (x,z,t)=f dqj(t-q)Y(x,z,q) . . . (171), *'o where f(t-q ) denotes an arbitrary function of (t - q), according to which the surface-pressure, arbitrarily applied at time (t — q)y is as follows : n(t-q)= -f(t-q)V{x,l,Q) . . . (172). Hence the pressure applied to the surface at time t , denoted by n(ir, 1, t), is as follows : n(a,i,*H* 1,0). (173). § 129. The solution (170) or (171) gives the velocity-potential throughout the liquid which follows determinately from the dynamical data described in §§ 127, 128. From it, by differentia- tions with reference to x and z, and integrations with respect to t , we can find the displacement components £, £ of any particle of the liquid whose co-ordinates were x , 2 when the fluid was given at rest. But we can find them more directly, and with consider- ably less complication of integral signs, by direct application of the same plan of summing as that used in (170), (171). Thus if, instead of Y (x , z , q) in (171), we substitute -^-Y(x,z,q), ax and again — -Y(# , z , q), we find £ and £. And if we take az Jdq^V(x,z,q) and ) dq^-V(x,z,q) . . (174) in place of Y(xyz,q) in (171), we find the two components £, £ of the displacement of any particle of the fluid. Confining our attention to vertical displacements, and using (179) below, we thus find £(x,z,t) = -[ dqf(t-q)~Y(x,z,q) ■ ■ (175). (JJo . d1 § 130. To illustrate the meaning of the notation and analytical expressions in (171), (173), (175), take the simplest possible PKOC. ROY. SOC. EDIN. — VOL. XXVI. 27 418 Proceedings of Royal Society of Edinburgh. [sess. example, f(t — q) = 1. This makes II the same for all values of t ; and (173) becomes n= -Y(x, 1 , 0) (176); and by integration (175) becomes £(^M) = y[v(a,M)- V(a,z,°)] . . (177). Putting now in this z — 1, and using (176), we find 1_ 9 L V(*,M)-V(*,1,0) 1 J y Y(aj,l,#) + n (178). The interpretation of this, as t increases from 0 to oo , is that the sudden application and continued maintenance of a pressure — V(a:,l,0) over the whole fluid surface, initially plane and level, produces a depression, £, which gradually increases from 0, at t = 0, to its hydrostatic value II jgt at t= oo . The gradual subsidence of the difference from the static condition, as time advances from 0 to go , is illustrated by the diagrams of fig. 34, for the case in which we choose for Y(x , 1 , 0) the i f/(x , 1 , 0) of .§§ 100-104 above. § 131. To understand thoroughly the meaning of Y(x,z, q) as defined in § 127 ; remark first that it is the velocity-potential of a possible motion of water, under the influence of gravity, with no surface-pressure, or with merely a pressure uniform over its infinite free surface. This is equivalent to saying that Y(x,z,q) fulfils the equations d2Y d2Y ^ , dY d2 Y dx2 + dz2 ’ an ^ dz dq2 . . (179). Secondly, remark that at the instant q = 0, there is no surface displacement; hence Y(x , z , q) is the velocity-potential at time q, due to an instantaneous impulsive pressure, — Y(x, 1 , 0), applied to the surface of the fluid at rest and in equilibrium, at time ^ = 0. iSow, allowing negative values of q, think of a state of motion from which our actual condition of no displacement, and of velocity- potential equal to Y(x , z , 0), would be reached and passed through when q passes from negative to positive. It is clear that the 1905-6.] Lord Kelvin on the Growth of a Train of Waves. 419 values of Y (x , z , q) are equal for equal positive and negative values of q. Hence, when q = 0, we have f-Y(x , z > 0) = 0 dq . (180). § 132. Consideration of the V (x , z , q), defined in § 127, which allows Y(x, 1 , 0) to he any arbitrary function of x, but requires dY/dq to be zero when ^ = 0, suggests an allied hydrokinetic problem: — to find W fulfilling (179) with W in place of Y ; and, at time q = 0, having W = 0 and dW/dq any arbitrary function of x. We assume, as is convenient for our present purpose, that for large values of x Y(x , z , 0) '= 0, and W(cr , z , 0) = 0 . . . (181). This implies that for all values of x and z, large or small, but for large values of q, Y(x , z , q) = 0, and W(a; , z , q) = 0 . . . (182), § 133. In the Y-problem the initiational condition is: — displace- ment zero and initiational velocity virtually given throughout the fluid as the determinate result of an arbitrarily distributed im- pulsive pressure on the surface. In the W-problem the initiational condition is : — the fluid held at rest with its surface kept to any arbitrarily prescribed shape by fluid pressure, and then left free by sudden and permanent annul- ment of this pressure. Without going into the question of a complete solution of this (Y, W) problem for any arbitrary initiational data, we find a class of thoroughly convenient solutions in a formula origin- ally given in the Proceedings of the Royal Society of Edinburgh , January 1887 ; republished in the Phil. Mag., February 1887 ; and used in § 3 and § 99 above. We may now write that formula in the following comprehensive realised expression for Y or W :— {KS} or {RD}yTh’+* -gv 4(2+1*) dfdx^z* J(z + lx) = Y(x,z, t ), when i is even ; = W (x,z , t), when i is odd. (183). 420 Proceedings of Royal Society of Edinburgh. [sess. By using (179) we may, instead of (183), take the following as equally comprehensive : — {RS} or {RD}(a + Bd\£ l dx/dt J(z + lx) -g*8 4(2+ 1*) = Y(x , z , t), when i is even : = W(x , z , t), when i is odd. (183'). § 134. Going hack to (171) and (175), remark that integration by parts gives j‘dqf(t - q)fv(x ,Z,t) =/( 0)V(* ,Z,t) -f(t)V (x ,Z, 0) + / dgf(t-q)V(x, z, q) . (184). *'6 This shows that if by quadrature or otherwise we have calculated the velocity-potential S(^,2,^), as given by (171), we can find the vertical component displacement of any particle of the liquid by (175), without farther integration. The formula (184) also shows how by successive integrations by parts we can reduce jodqf(t-q)~V(x,z,q) .... (185) to the primary integral S(a? ,ztt)t as expressed in (171). §135. Going back now to §§ 128, 127, 118: to make the applied forcive a sinusoidally varying pressure put = (186); which, by (173), makes n(a , 1 , 0 = - > 1 > 0) • • • (187). And now let us arrange to fully work out our problem for two cases of surface distribution of pressure, corresponding to the two initiational forms , if/ , described in §§96-113 above. For this purpose take, with the notation of § 101, 1 d2 Y(x,z,t) = cf>(x,z,t) ; or Y (xiz,t) = if/(x, z,t)= ^#(«,M). (188).. For brevity we shall call these two cases case <£ and case t }/. Thus, in these Cases (171) and (175), expressing respectively the 1905-6.] Lord Kelvin on the G-rowth of a Train of Waves. 421 velocity-potential at, and the vertical component displacement of, any point of the fluid at any time, become § 136. The illustrations in figs. 36, 37, 38 are time-curves in which the ordinates have been calculated hy continuous quad- rature from one or other of the four formulas (189), (190). § 137. The curves - in fig. 39, being space curves in which the ordinates are vertical component displacements of the water- surface, are therefore pictures of the water-surface (greatly exaggerated in respect to slopes of course), and may he shortly named water-surface curves. Their ordinates have been calculated by an analytical nrethod described in § 151 below. They cannot be calculated continuously for successive values of x by the method of continuous quadratures ; if that were the method employed, the value of the ordinate for each value of x would to the particular value of t for which the water-surface is repre- sented by the curve. The values of t chosen for fig. 39 are respectively it, (i + 1/S)t, (^ + 2/8)t, (z'+3/8)t, (^ + 4/8)r, where i is any very large integer, and r denotes 2 77-/00, the period of the In all our illustrations we have taken w = which makes r = 2 Jtt, and, with g == 4 as in § 105, makes the wave-length X — 8. §138. In figs. 36 and 37, all the curves Correspond to cos io(t - q) in ' the formulas. In fig. 38, all the curves corre- spond to sin o)(t — q) in the formulas. S = dq a>(t-q)(x,z,q)-, need to be calculated by an independent quadrature ^ varying surface-pressure to which the fluid motion considered is due. Figure 36. Unit of ordinates’ scale. Figure 37. Unit of ordinates* scale. 424 Proceedings of Royal Society of Edinburgh, [sess. 1905-6.] In fig. 39, the inscriptions of times correspond to cos w(t - q) in the formulas. The same curves, with the inscriptions altered to (i + 2/8)r, (i + 3/8)r, (i+ 4/8)r, (^ + 5/8)r, (i + 6/ 8)r, correspond to sin , and -•5 for case if/, of which the former is 2-83 times the latter. The semi-amplitudes of the subsequent variations of velocity-potential shown in the first and third curves are -954 for case and ‘318 for case if/, of which the former is 3'00 times the latter. § 140. The first, and third, and fifth, curves of fig. 37 show, at a distance of one wave-length from the origin, the complete history of velocity-potential and of surface displacement through all time from the beginning of application of pressure to the surface. The very approximately accurate sinusoidality of each of these three curves through periods 6, 7, 8, shows that the continuation through endless time is in each case sinusoidal. In remarkable contrast with the initial agreement between S^(0 , 1 , t) and S^(0 , 1 , t) , to which we alluded in § 139, we find very instructively a remarkable contrast between S^(8 , 1 , t) and S<^(8 , 1 , 7)- throughout the whole of the first period. Remembering that in a liquid of unit density the pressure is equal to minus the rate of augmentation of the velocity-potential per unit of time, and remark- ing that the displacement £^(0 , 1 , t) is, as is shown in its curve, very nearly zero throughout the first period, and that £^,(0 , 1 , t) is Figure 38. Unit of ordinates’ scale. Figure 39. Unit of vertical scale. 1905-6.] Lord Kelvin on the Growth of a Train of Waves. 427 certainly still more nearly zero throughout the first period, though we have no curve to represent it, we see that the negatives of the tangents of the slopes in the curves for S^(8 , 1 , t) and S^(8 , 1 , t) re- present very nearly the values of the applied surface-pressures during the whole of the first period.* Look now to fig. 33 ; see how near to zero is if/(8 ,1,0), and how far from zero is <£(8 ,1,0); and we see dynamically how it is that S^(8, 1 , t) is very nearly zero throughout the first period, and S<^(8 , 1 , t) is very far from zero, and is somewhat near to being sinusoidal. § 141. We have also a very instructive comparison between £$(8 , 1 , t) and S^(8 , 1 , t). In the case, for values of x as large as 8, or larger, we approach somewhat nearly to the case of a sinusoidally varying uniform surface-pressure over an infinite plane area of water, in which there would he no surface displace- ment, and the pressure at and below the surface would be at every instant equal to the applied surface-pressure plus the gravitational augmentation of pressure below the surface. Thus we see why it is that, with a great periodic variation of applied surface-pressure, at a; =8, there is scarcely any rise and fall of the surface level there, until after a period and a half from the beginning of the motion, as shown in the curves for £^(8,1, t). § 142. The second, fourth, and sixth, curves of fig. 37 represent the arrival of three classes of disturbance, , £^ , S^, , at x= 32, four wave-lengths from the origin. If the front of the disturbance travelled at exactly the wave-velocity, the disturbances of the different kinds would all commence suddenly at the end of period 4. In the cases of S^,(32 , 1 , £) and £(32 , 1 , t) the diagram shows that they are quite imperceptible at the end of period 4, and begin to he considerable at the end of period 8, which would be the exact time of arrival if there was a definite “ group- velocity ” equal to half the wave-velocity. The largeness of S^,(32 , 1 , t), approximately uniform throughout the first four periods, is explained in § 141. Its gradual augmentation through periods 5, 6, 7, 8, depends on the wave propagation of disturb- ances from the origin, as shown for S^(32 , 1 , t) and £^(32,1 ,t) in the second and fourth curves. * Remember that downward ordinates in all the curves of figs. 36, 37, 38, 39, correspond to positive values of the quantities represented. 428 Proceedings of Royal Society of Edinburgh. . [sess. § 143. The £4,(0 , 1 , t) curve of fig. 38 may be compared with the curve of the same designation in fig. 36. They differ because of a quarter period difference in the phase of commencement of the disturbing pressure, which commences suddenly at its maximum for all the curves of fig. 36, and commences at zero for all the curves of fig. 38. If the , S$. curves for initiating pressure commencing at zero were drawn, they would differ from the first and third curves of fig. 36 in being at the commencement tangential to the line of abscissas, instead of being inclined to it in the positive direction, as shown in fig. 36. The £ curves are all initially tangential to the line of abscissas, but the tangency is only of the first order in fig. 36, while it is of the second order in fig. 38. § 144. The third and fourth curves * of fig. 38 show the whole history for the points, x = 0 , and x = A. , of the surface displacement expressed by the formulas ^(x , 1 , t) = - f dq sin (o(t - 2)-ife , 1 , q) , (191), 9J0 ay which expresses the surface displacement due to surface-pressure expressed by H(x , 1 , t) = - sin wt i]/(x , 1 , 0) . . . (192). The fifth curve of fig. 38 shows the history, after period 3, to almost half a period after period 9, of the disturbance at the place x = 32. The disturbance has not yet become sinusoidal, but would certainly become almost exactly sinusoidal after a few more periods. § 145. In fig. 39, two sets of five curves show, for case and case i(/, the periodically varying water-surface on each side of the middle, at any long enough time after the beginning of the motion, to give a regular regime of sinusoidal vibration as far as two or three wave-lengths on each side of the middle. The third curve in each case is a curve of sines. The first curve represents the surface at the beginning of. a period from it to (i + l)r. The fifth curve, being the first curve inverted, re {>resents the water- surface at the middle of the period. The other two curves may * The scale of ordinates of the third, fourth, and fifth cui ves ‘of fig. 38 is double that of the first and second, indicated on the figure. 1905-6.] Lord Kelvin on the Growth of a Train of Waves. 429- be described as components of the first and third, according to the following formula : £(x , 1 , t) = P sin wt - Q cos wt . . . (193), where P= - A cos (194), and Q is a continuous transcendental function of x, having equal values for ±.r, expressed by (195) for positive or negative values of x, exceeding a wave-length. For x positive, Q = - A sin 2-irx/X ; for x negative, Q = + A sin 2ttx/X (195), where A denotes the semi-amplitude of the vibration, at any time long enough after the beginning, and place far enough from the middle of the disturbance, to have very approximately sinusoidal motion. The determination of the transcendental function Q, and the calculation of A, for both P and Q, will be virtually worked out in § 151 below. § 146. We have now an exceedingly interesting and suggestive analysis of the circumstances represented in fig. 39. Consider separately the two motions corresponding to P sin wt alone, and to - Q cos wt alone. The motion P sin wt , if at any instant given from x = • — oc to x — 4- oo , would continue for ever, as an infinite series of standing waves, without any surface-pressure. Hence our application of surface-pressure is only required for the Q- motion : and if this motion be at any instant given from x= - oc to x = + oc , it will go on for ever, provided the pressure — cos wt , 1 , 0) is applied and kept applied to the surface. §147. The plan of § 1 46 may be generalised as follows: — Displace the water according to the formula (193) with P omitted, and with Q any arbitrary function of x for moderately great positive or negative values of x, gradually changing into the formula (195) for positive and negative values outside any arbitrarily chosen length MON (MO not necessarily equal to 0 H). Find mathematically the sinusoidally varying surface-pressure, F(.e) cos wt , required to cause the motion to continue according to this law. Superimpose, upon the motion thus guided by surface- pressure, the motion -A cos 27t^/A sin (o^, which needs no surface- pressure. In the motion thus compounded, we have equal 430 Proceedings of Royal Society of Edinburgh, [sesb, sinusoidal waves travelling outwards in the two directions beyond M N (semi-amplitude A) : and, in the space M H, we have a varying water-surface found by superimposing on the motion P sin c ot an arbitrary shape of surface, varying sinusoidally according to the formula — Q cos a >t. § 148. A curiously interesting dynamical consideration is now forced upon us, The P-component of motion needs, as we have seen, no surface-pressure. The Q-component of motion is kept correct by the surface-pressure F(a;) cos cot , which, in a period, does no total of work on the Q-motion ; but work must be done to supply energy for the two trains of waves travelling outwards in the two directions. Hence this work is done by the activity of the surface- pressure upon the P-component of the motion. § 149. Another curious question is forced upon us. Our solution of §§ 135-145 has given us determinately and unambigu- ously, in every variety of the cases considered, the motion of every particle of the water throughout the space occupied. The synthetic method of quadratures which we have used could lead to no other motion at any instant due to the applied surface- pressure ; but now, in § 147, we have considered a Q-motion alone, kept correct by the applied surface-pressure. Would this motion be unstable? and, if unstable, would it in a sufficiently long time subside into the motion expressed in the determinate solution of §§ 135-145? The answer is Yes and Ho. At any instant, say at t = 0 , let the whole motion be the Q-component alone of § 148. Let now the surface-pressure, F(a?) cos c ot , be suddenly commenced and continued for ever after. It will, according to §§ 135-145, produce determinately a . certain compound motion (P , Q) which will be superimposed upon the motion existing at time t = 0 ; and this last-mentioned motion, given with its infinite amount of energy distributed from x='—cq to x = + oo , and left with no surface-pressure, would clearly never come approximately to quiescence, through any range of distance from 0 on the two sides. Thus we see that, though the Q- motion alone of § 148 is essentially unstable, the condition of the fluid does not subside into the determinate solution of §§ 135-145. It would so subside, if it were given initially only through any finite space however great, on each side of 0. In fact, any given 1905-6.] Lord Kelvin on the Growth of a Train of Waves. 431 distribution of disturbance through any finite space however great on each side of 0, left to itself without any application of surface- pressure, becomes dissipated away to infinity on the two sides ; and leaves, as illustrated in §§ 96-113, an ever-broadening space on each side of 0, through which the motion becomes smaller and smaller as time advances. § 150. It remains only to look into some of the analytical details concerned in the practical working out of our solutions (189), (190). Taking cos a >(t — q) in the formulas, and taking case (x ,z,q); and Q= 1 dq sin wqc^{x , z , q) (197). Jq -A) When P and Q have been thus found by quadratures, for all values of t, and any particular value of x, by integration by parts on the plan of § 134, we readily find, without farther quadratures, or integrations, expressions for the seven other formulas included in (189), (190). § 151. Let us first find P and Q for t — co. Using the exponential form for <£ , given by (137), we find P={RS}^/— J dq cos coqe~mg2 ; and Q = {RS}^/— j dq sin o)qe~mq2 (198) where m = -^g/(z + ix). Hence, according to an evaluation given by Laplace in 1810,* wTe find, taking g = 4, -W2 P={ES}A/|e4” (199). The definite integral for Q is a transcendent function of w and m, not expressible finitely in terms of trigonometrical functions or exponentials. By using the series for sin c oq in terms of (a >#)2i+1, and evaluating I dq q2i+1e~q2 by integrations by parts, we find the * o following convergent series for the evaluation of Q, for t = ca ; and g = 4 : — Memoires de PInstitut, 1810. See Gregory’s Examples, p. 480, 432 Proceedings of Royal Society of Edinburgh. [sess. 1 CO 1 ( <*> ^ B 1 ; i j ( m \ 72 __ V™ 2 . 1 . 31 1 + 22 . 1 . J. o1 l Jm) 1 23 . 1 . 3 . 5 . Jm) 4- etc. X_ K/W®_ 3 (ufpY (200), ^j2 2 2 13 2 ^ + 22 1 3 5 2 ^ (oijpy 7 2M.3.5.7C0S¥X+etC' where, as in §§ 100-113 above, p = J(z2 + x2), and x = tan-1 (x/z). This series converges for every value of wjp however great. But for values of wx/p greater than 4, it diverges to large alternately positive and negative terms before it begins to con- verge. The largest value of wjp for which we have used it is o)Jp = 5 ’03, corresponding to a; = 8, and requiring, for the accuracy we desire, twenty-one terms of the series. But for this value of c ojp and for all larger values, we have used the ultimately divergent series (208), found in expressing analytic- ally, not merely for t = co as in (198), (199), (200), but for all positive values of t great and small, the growth to its final condition when t — oo , of the disturbance produced by our periodically varying application of pressure to the surface of the water initially (t = 0) at rest. The curve for ir in fig. 39 has been actually calculated by (200) for values of x up to 8, and by the ultimately divergent series for values of x from 5 to 10. The agreement between those of the values which were calculated both by (200) and by the ultimately divergent series (208), was quite satisfactory : so also was the agreement between values of Q found by quadratures for x = 1 and x = 8, with values found by (200) for &•= 1 and by (208) for a? = 8. It is also satisfactory that the values of P found by quadratures, for x=l, and x = 8, agreed well with their exact values given by (199), for t= go . § 152. Going back now to the expressions (197) for P and Q, we see that, by an obvious analytical method of treatment, we can reduce them, and therefore (§ 150) all our other formulas, to expressions in terms of a function defined as follows : — dcr e “ 0-2 (201), 1905-6.] Lord Kelvin on the Growth of a Train of Waves. 433 a function well known to mathematicians* through the last hundred and fifty or two hundred years, in the mathematical theory of Astronomical Refraction, and in the theory of Probabili- ties. I have taken E as an abbreviation of Glaisher’s f notation “Erfc,” signifying what he calls “Error Function Complement,” which he uses in connection with his name “Error Function,” defined by Erf (\/7€‘M K Jmt ~ ‘2 fn) + E( + ‘2 i). (205); Q= jelZ K - E( 2Efe)" (206). § 153. Remark first in passing that, when Jmt is infinitely great in comparison with 2^'-3 (modulus understood if cr2 is imaginary), and after that it diverges, the true value being intermediate between the sum of the con- vergent terms and this sum with the first term of the divergent series added. The proper rule of procedure to find the result with any desired degree of accuracy, is to first calculate by the ulti- mately divergent series, and see whether or not it gives the result accurately enough. If it does not, use the convergent series (207), which, by sufficient expenditure of arithmetical labour, will certainly give the result with any degree of accuracy resolved upon. § 155. As a guide, not only for numerical calculation, hut for judging the character of the desired result without calculation, it is convenient to find the moduluses of the three complex arguments of the function E, in (205), and (206). They are as follows : — § 156. The- very interesting questions regarding the front of the procession of waves in either direction, of which we have found illustrations in figs. 36, 37, 38, and which we had under consideration in §§ 11-31, 114-117 above, are now answerable in a thoroughly satisfactory mathematical manner, by aid of the formulas (205), (206), (209), (210), (211). When, in the arguments (210); (211). 1905-6.] Lord Kelvin on the Growth of a Train of Waves, 435 of E, in (205), and (206), Jmt is very great in comparison with ayfijm, the two added terms in (205) are approximately equal, and (206) is reduced approximately to its last term ; and all the solu- tions (189), (190), become approximately sinusoidal, in respect to t. This is the case when t is very great in comparison with unity, and in comparison with del . M'F a.rla,ne &. Erskine ,Libh.Edinl 1905-6.] Electrolysis through Precipitation Films. 447 Electrolysis through Precipitation Films. Part I. By W. S. Millar, M.A., B.Sc., Carnegie Research Scholar, and Dr W. W. Taylor. Communicated by Professor Crum Brown. (Read July 13, 1906. MS. received October 31, 1906.) In a paper on the aluminium anode by one of us, in con- junction with Inglis,* it was pointed out that if the suggested theory of the aluminium anode is correct, a precipitation film of aluminium hydroxide should diminish the conductivity of salt solutions to very different extents depending upon the ions present, and that, e.g., the diminution in the case of potassium chloride might be expected to be slight, in the case of potassium bromide somewhat greater, and in the case of potassium sulphate to be very considerable. Preliminary experiments to measure these differences directly were not successful, almost certainly, as was stated at the time, because of the difficulty of freeing the pre- cipitation membrane from the concomitant soluble impurities. The experiments described in the present paper conclusively prove that the anticipated effect does exist ; and, though the quantitative agreement is fairly good, we hope to further improve the method in certain respects and so obtain exact measurements. That diaphragms may affect ions to a different extent was recognised by Hittorf, when he discarded the use of parchment paper or other diaphragms in his classic investigations on the migration ratios of ions. Tammann t found that in some cases osmotic pressure membranes offered considerable resistance to the passage of the ions which form the membrane, but that generally the resistance offered by the membrane was very small. Morse,]; in the preparation of osmotic pressure membranes by electrolysis, observed that an efficient film offered a very great resistance to the ions which form the film. * Taylor and Inglis, Phil. May., March 1903. t Tammann, Zeit. f. Physikal. Chem., vi. p. 237, 1890. t Morse, Amer. Chem. Journal , xxix. p. 173, 1903. 448 Proceedings of Roycd Society of Edinburgh. [sess. Apparatus and Method. As it is obviously necessary to have a certain amount of rigidity of support for the precipitation film, the form of apparatus must be adapted to the support selected. In the early experi- ments various supports were tried, viz., filter paper, hardened filter paper, filter paper impregnated with a solution of hard gelatine, and several kinds of porous earthenware. For these preliminary experiments a Kohlrausch U cell, with large electrodes and a narrower tube connecting the two limbs, was employed. It was cut into two equal parts, the edges were ground true, and a brass coupling joint was cemented on in such a way that the diaphragm could be mounted, liquid-tight, between two rubber washers. This type of cell was found to be quite satisfactory, and was employed in most of the subsequent experi- ments. Filter paper, hardened and unhardened, was found to be quite useless, nor was the gelatinised filter paper much more satisfactory. Incidentally, one or two of the results showing the effect of filter paper and of gelatinised paper on the conductivity of solutions are included later on. A fairly hard and compact porous earthenware was found to give the most consistent results. It also had the advantage over more open material that the diaphragms could be ground very thin, and were less liable to fracture when mounted between the washers. The same diaphragm was used as often as possible ; it was cleaned each time by prolonged boiling with a mixture of potassium chlorate and concentrated nitric acid, with hydro- chloric acid, and repeatedly extracted with boiling water. It was then allowed to soak for a long time in the solution which was to be measured. In a few of the experiments a different type of cell was employed. It was a modification of the Arrhenius type, but the electrodes were much smaller, and the upper one was separated from the other by a wide glass tube, the lower end of which was closed by porous earthenware cemented on water-tight. This form of cell had the disadvantage of not being so easily set up and renewed as the other one ; the porous plate was much more troublesome 1905-6.] Electrolysis through Precipitation Films. 449 to clean, owing to the cement; hut, on the other hand, the diaphragm was always visible, and there was no metal joint through which leaks or short-circuits might take place. This cell is subsequently referred to as the unsymmetrical cell. The results obtained in the two very different types of cell were generally in close agreement. Up to the present, the measure- ments have been made by means of alternating current and telephone only. The principal difficulty in this case appears to be the purification of the film. When a film is formed by the interaction of two reagents (precipitants), a soluble salt is also formed at the same time, and this is probably enclosed within the film. To remove this enclosed salt by washing is extremely difficult, if not altogether impossible, there being, in addition, the danger of rupturing the film or of washing the colloid film away altogether when the amount of electrolyte is much reduced. The only feasible way of removing the enclosed salt seemed to be by electrolysis with direct current, and experiments in this direction rendered it probable that the salts can be removed in this way, but that, at the same time, the film itself appeared to undergo changes when subjected to this method of purification. Evidence in support of this will be found further on. Finally, the following procedure was adopted as almost entirely avoiding the difficulties mentioned above. If a comparison is to be made of the effect, say, of a film of aluminium hydroxide on Cl', Br', and S04" respectively, in the first instance a fairly strong solution of ammonium chloride is placed in each division of the complete cell, and bridge readings are taken until the resistance is constant. The solution is then removed, and in the one division is placed a solution containing ammonium chloride and aluminium chloride, in the other division a solution containing ammonium chloride and ammonia. The concentration of ammonium chloride on each side is the same as before ; the concentrations of the pre- cipitants are equivalent and small in comparison with that of the ammonium chloride. Thus almost the whole of the conductivity is due to the ammonium chloride, and the change of conductivity, owing to decrease of concentration of the precipitants by formation of the film, will be so small as to be negligible. PROC. ROY. SOC. EDIN. — YOL. XXVI. 29 450 Proceedings of Royal Society of Edinburgh. [sess. Measurements were repeatedly made on the ammonium salt solutions, and on the mixed solutions at the beginning and at the end of an experiment, and in every case the above assumptions were verified. Immediately after the cell is set up, bridge readings are taken at small intervals of time. The conductivity first of all increases slightly, which may be a temperature effect, then slowly falls over a long period, and finally becomes practically constant. The maximum is taken as the correct initial value, and the constant end value as the final one. When the final value is reached, the mixed solutions may be replaced by the original solution of ammonium chloride. The whole of the experiments were made at constant temperature, generally 25° C. Parallel experiments are then carried out, for bromide, with solutions of ammonium bromide, aluminium bromide, and. ammonia ; and for sulphate, with ammonium sulphate, aluminium sulphate, and ammonia. In some of the experiments the solutions of the ammonium salts were of equivalent concentration, whilst in others they were of the same specific conductivity; the con- centrations of the precipitants were always equivalent. By this method the only salt produced by the formation of the film is the salt which is under investigation ; it is, therefore, unnecessary to attempt to remove it, and all that is required is that equilibrium between the concentration of the salt in the film and in the solution be attained. Results. Before giving, in the form of tables, the results of the measure- ments, we may give in detail some observations which may be of interest. 1. Effect of filter paper and of gelatin on the conductivity of solutions. A. Potassium chloride solution. (1) No diaphragm, conductivity *00422 mho. (2) Hardened filter paper, con- ductivity . . *00419 „ 99*3 per cent.. (3) Gelatinised filter paper, conductivity . . *00406 „ 96*2 „ 1905-6.] Electrolysis through Precipitation Films. 451 B. M/12 Ammonium sulphate solution. (1) No diaphragm, conductivity *00971 mho. (2) Filter paper, conductivity . *00967 „ 99*6 per cent. (3) Gelatinised paper, con- ductivity . . *00954 „ 98*2 „ Attempts in the first case to form a serviceable film of aluminium hydroxide, in the latter case of chromic hydroxide, completely failed. 2. Aluminium hydroxide film. Ammonium sulphate solution. The following is the complete record of the measurements made in one experiment, and may be taken as typical of them all. A. Ammonium sulphate solution, 0*15 mol. per litre. Time in Bridge reading Resistance Conductivity Minutes. in mm. Box. in mhos. 508*5 550 Electrolyse for five minutes each way with ten-volt direct current* replace with fresh solution — 508*2 550 Electrolyse three minutes each way — 508*3 450 *001879 B. T5m (NH4)2S04 + *033m A12(S04)3 in one limb. *15m (NH4)2S04 + *2m NH3 in other limb. 0 509-5 550 3 511*5 )> 6 511*8 9 512*0 12 511*7 5) 15 511*2 18 510*4 >> 21 509*0 >> 24 507*2 >> 27 505*5 *001908 452 Proceedings of Royal Society of Edinburgh. [sess. Time iD Minutes. Bridge reading in mm. Resistance Box. Conductivity in mhos. 30 5034 550 33 502-0 5? 36 500*7 33 41 498*4 5) 46 497-2 5) 51 475-0 53 56 494-0 33 1080 462-0 550 •001561 Readings were then taken every hour and remained constant within 1 mm. The ten-volt direct current was then passed through the cell for ten minutes, the electrode in the limb containing aluminium sulphate being the anode. The solutions were then thoroughly stirred up. 0 448 600 •001353 3 456 33 6 458 33 9 460 33 15 462 33 21 466 33 33 470 33 48 472 33 88 473 33 137 472 33 185 471 33 220 471 33 1250 479 33 •001532 The reason for electrolysing with a direct current and the effects produced by it will be discussed later. In the tables of results which follow, the first column indicates the cell in which the experiment was made ; this is desirable, as the dimensions of the cells employed differed to a considerable extent ; * the second column gives the initial conductivity in mhos ; * Tt should be noted that a cell marked A in one table is not necessarily the same as cell A in another table. The cells frequently broke at the brass joints, and new parts were fitted on. 1905-6.] Electrolysis through Precipitation Films. 453 the third column gives the final conductivity in mhos ; and the fourth column is the percentage ratio of the final to the initial conductivity. Aluminium Hydroxide Films. 1. Ammonium chloride. 1 mol. NH4C1 + \ mol. NH3 ; i mol. NH4C1 + -fc mol. A1C13. I. II. III. IY. A •001426 •001357 95*2 B 1231 1192 96-8 C 1252 1198 95*8 X 1473 1401 95*3 Y 1122 1078 96-1 Z 1084 1018 939 Mean 95-5 2. Ammonium bromide. mol. NH4Br3 + Jinol. XH3; Jmol.XH4Br + 2V mol. AlBr, I. II. III. IY. A •001423 •001331 93*5 1455 1338 92*0 1437 1319 91-7 B 1449 1348 93-0 1426 1327 93-1 1466 1324 90*3 1409 1331 94-5* X 1176 1097 93*3 Y 1159 1104 95 3* Z 1192 1144 96 0* Mean 92*5 454 Proceedings of Royal Society of Edinburgh. [: 3. Ammonium sulphate. (a) £ mol. (NH4)2S04 + i mol. NH3 ; £ mol. (NH4)2S04 + mol. A12(S04)3. I. II. III. IY. A •001892 001546 81*7 1908 1561 81-8 1976 1565 79-2 B 1805 1449 80-3 1773 1437 81-1 1705 1431 839 1805 1448 80*2 Mean 81-2 mol. (NH4)2S04 + xo mol. NH8;imol.(NH4) 2so4+1 A12(S04)3. I. II. III. IY. X •001431 •000980 68*5 Y 1487 1026 69-0 Z 1530 1096 71*6 Mean 69-7 (e) £ mol. (NH4)2S04 + i mol. NH3 ; £ mol. (NH4)2S04 + ^ mol. A12(S04)3. I. II. III. IY. A •000799 •000629 79-7* B 797 564 70-8 786 582 74-1 815 537 65-9 C 691 490 70-9 697 478 68-6 Mean 70'1 1905-6.] Electrolysis through Precipitation Films. 455 Chromic Hydroxide Films. 1. Ammonium chloride. I mol. NH4C1 + i mol. NH3; ± mol. NH4C1 + mol. CrCl3. I. II. III. IV. A •001811 •001434 79-2 1774 1417 79*9 1893 1572 83*0 B 1598 1212 75*9 1637 1345 82*2 1644. 1452 88*4* 1677 1412 84*2 C 1611 1378 85*5* * 1624 1400 86*2* 1573 1378 87*6* 1566 1378 88-0* Mean 80-7 2. Ammonium sulphate. i mol. (NH4)2S04 + i mol. NH3 ; J mol. (NH4)2S04 + ^ mol. Cr2(S04)3. I. II. III. IY. B •000944 •000516 54-7 1037 500 48-2 916 414 45*2 935 454 48*6 929 308 33*2* C 897 455 50-7 921 341 37-0* 1004 471 46-9 950 450 47*4 Mean 48*8 456 Proceedings of Royal Society of Edinburgh. [sess. 3. Potassium sulphate. J mol. K2S04 , H-Jmol. KOH; ^ mol. K2S04 + f-Q mol. Cr2(S04)3. I. II. III. IV. A •002101 •001041 49-5 2325 869 37*4* 1940 1000 51*5 B 2174 1217 56-0 C 1604 845 52-6 1595 770 48*3 1721 802 46*6 Mean 50-8 4. Sodium sulphate. mol. Na2S04 + i mol. NaOH ; | mol. Na2S04 + mol. Cr2(S04)3. I. II. III. IV. A *001700 •000628 36-9 1738 831 47*6* 1827 708 38*8 1857 869 46-8* B 1906 654 343 1900 667 351 1857 858 46-3* C 1327 591 44-5* 1344 524 39*0 1331 640 48T* Mean 36*8 (Mean of *, 46*7.) Before proceeding to the discussion of these results, we may briefly describe certain isolated experiments which have been made. A. Temperature Coefficient. The unsymmetrical cell, already referred to, was set up with solutions containing J mol. (NH4)2S04 + \ mol. NH3, and i mol. (NH4)2S04 + * mol. Cr2(S04)8. 1905-6.] Electrolysis through Precipitation Films. 457 A very satisfactory film was formed, and the conductivity was determined at various temperatures. The temperature coefficient was calculated from the formula JL K,-Kg K9, t- 25 in which and K25 are the conductivities at temperatures t and 25° C. The results are as follows : — Time. Temperature. Conductivity in mhos. c. 12.0 p.m. 25*0° C. •001523 ... 12.30 „ 12-0 „ 835 •0348 1.0 „ 12-3 „ 869 338 2.0 „ 0-3 „ 544 260 3.30 „ 16-2 „ 1105 312 4.30 „ 25*0 „ 1782 . . . Following morning 25*0 „ 1517 B. Effect of a chromic hydroxide film made in sulphate solutions on the other salts. In this series of experiments solutions of ammonium chloride, ammonium bromide, and of ammonium sulphate were prepared so as to have the same specific conductivity ; the concentrations were *25 mol. NH4C1, ‘25 mol. NH4Br, and about T7 mol. (jST[4)2S04. A film of chromic hydroxide was then formed in the unsymmetrical cell exactly as described in the previous experiment. When the final conductivity had been constant for several days, the mixed solutions were replaced by the T7 mol. (NH4)2S04 solution, wrhich was allowed to remain until the readings were constant. This solution was then replaced by a fresh portion of the ammonium sulphate solution, but the readings were found to be the same as before. The ammonium sulphate solution was similarly replaced by the ammonium chloride solution, and the chloride by the ammonium bromide solution ; and, in conclusion, the bromide solution was washed out with the ammonium sulphate 458 Proceedings of Royal Society of Edinburgh. [sess. solution, and readings were taken with the sulphate. The following results were obtained : — Date. Solution. Conductivity in mhos. Ratio to Conductivity of NH4C1. July 6 (NH4)2S04 •00339 '76 „ 8 nh4ci •00445 1-00 9 NH4Br •00410 •92 „ 9 (NH4)2S04 •00370 •83 Unfortunately the cement of the tile gave way before the last measurements on the sulphate solution were made. The sulphate solution had only been in the cell for ninety minutes, whereas, in the case of all the other solutions, the cell was left for about twenty-four hours before the final readings were taken. The same series of experiments was also made in one of the (J cells. The results were similar in every respect. The ratios of the specific conductivity of the ammonium solutions of chloride, ammonium bromide, and ammonium sulphate were U00 : U073 : 1-035 ; with the chromic hydroxide film the ratios were TOO : '92 : '765. C. Effect of a chromic hydroxide film on solutions of sodium ammonium d -tartrate, and of sodium ammonium racemate. Solutions of sodium ammonium c?-tartrate and of sodium ammonium racemate were prepared of such a concentration that they had approximately the same specific conductivity as the solutions employed in the experiments described in section B. The exact ratio of the specific conductivity of the tartrate solution to that of the racemate solution was 1 '000 : 1 '046. A film of chromic hydroxide was formed with sulphate solutions in one of the (J cells, and a series of measurements was taken with the three solutions exactly as in the previous experiments. With the film, the ratio of the conductivity of the .). The feet in the present species have, as in I. Beselsii , untoothed claws and no tenent hairs; the apex of the smaller claw is, however, drawn out in a slender process, which is not so fully developed in the northern species. The mouth-parts of Isotoma Brucei show several interesting peculiarities. The mandibles are remarkably narrow and parallel- sided at the apex (fig. 3), and exhibit two very prominent acuminate processes at the hind dorsal corner of the grinding area (fig. 3, mo.). The maxilluke (fig. 4, mxl.) have prominent apexes, armed with several stout curved bristles ; the spines, arranged in series along the inner edge of the basal region of the maxillula, are elongate and sharp-pointed. In the structure of the maxilla (fig. 4, mx.) I. Brucei differs from other species of Isotoma, and indeed from members of its family generally, by the slender and elongate form of the “ head ” (compare the typical semi-globose “head” of Cryptopygus, fig. 19). The head in the present species is composed of a strongly chitinised dorsal lobe or “galea” (fig. 4, ga.), terminating in three prominent teeth. Ventral to this, and protruding beyond it, is a delicate falcate lamella (fig. 4, la.'), fringed with long delicate hairs, while a smaller lamella, also fringed with fine hairs (fig. 4, la ."), lies internal to the galea. The palp (fig. 4, pa.} carries six prominent bristles, the most distal being inserted on a long acuminate process. The stipes of the maxilla (fig. 4, sti.) articulates with the cardo (fig. 4, car.), which is itself in connection, as usual, with the supporting “foot” and ligament (fig. 4, pd. lig .) of the tongue (fig. 4, tin.). Several at least of these characteristic features of the jaws in 1. Brucei may also be detected in its northern ally I. Beselsii. The form of the retinaculum in 1. Brucei, as seen from the side, is shown under high magnification in fig. 8. 47 6 Proceedings of Iioyal Society of Edinburgh. [sess. Isotoma octo-oculata, Willem, var. gracilis, nov. PL II. figs 9-12. Length 1 *5 mm. Differs from the type by the short sub-crescentic ridge surrounding the post-antennal organ (fig. 10, p. a. o .) and the slender mucro of the spring, with its anterior dorsal tooth pointed and prominent (fig. 12). In this latter character the present variety agrees with that described from Kerguelen by Enderlein (1903). Localities. — Laurie Island, on cliff and moss 200 feet, one speci- men, 18th December 1903 ; Saddle Island, one young specimen, 4th February 1903. As only two specimens can be detected in the collection, this is presumably a scarce species in the South Orkneys. The type-form was described by Willem (1902) from insects collected on the shores of Gerlache Strait, between Danco Land and neighbouring islands,* and a sub-species, Kerguelensis , has since been described, as mentioned above, by Enderlein. In the form of its mucro our insect agrees with the latter, from which, as well as from the type-form, it may be readily distinguished by the short and relatively broad post-antennal organ. According to Enderlein’s figure, how- ever (1903, taf. xxxvi. fig. 66), the organ is broader and shorter in the var. Kerguelensis than in the type (Willem, 1902, pi. iv. fig. 11). Cryptopygus crcissus, sp. nov. PL II. figs. 13-23. Length 2 mm. Post-antennal organ elongate, crescentic. Six ocelli on each side of head (fig. 16). Feet with two tenent hairs, not clubbed at the tip, and with untoothed claws (fig. 17). Spring with mucro one- third length of dens, bearing two slight teeth, a terminal and a dorsal (fig. 21). Colour very deep blue-violet, almost black in adult specimens. Localities. — Saddle Island, innumerable specimens, 4th February 1903; Laurie Island, two specimens in moss on cliff 200 feet, 18th December 1903. The remarkable genus Cryptopygus, showing affinities to Anuro- phorus and to Isotoma, was erected by Willem (1902) for a new species of springtail ( C . antarcticus ) found in numbers on the shores * Recorded also by Wahlgren (1906) from South Shetland, Graham Land and Paulet Island. 1905-6.] Collembola from the South Orkney Islands. 477 of Danco Land and the neighbouring islands.* The present species from the South Orkneys is very closely related to Willem’s insect, differing chiefly in having only twelve ocelli (instead of fourteen), and in the comparatively short and stout mucro of its spring. In the adult C. crassus the six ocelli on either side are arranged in an anterior triangular group of three and a posterior curved row of three (fig. 16), (the fourth ocellus, nearest to the centre of the head, which is present in C. antarcticus, being here absent). In the very young C. crassus the six ocelli are more closely grouped (fig. 15). These very young individuals (fig. 13), only *5 mm. in length, have the violet pigment mottled over their bodies, contrast- ing strongly with their almost black parents. The springs of these young — especially the dentes and mucrones — are shorter and stouter than those of the adults (figs. 21, 22, 23). Among the adults the males may be distinguished from the females (fig. 14) by their more slender form and more elongate feelers (fig. 16). In the male these have the terminal segment half as long again as the third, while in the female there is no appreciable difference in length. The antennal organ consists of a single papilla at the extreme tip of the terminal antennal segment (fig. 16, a. o.).f In the excessive reduction of the hindmost abdominal segment (fig. 14, abd. vi.), retracted and almost hidden in a depression of the genital segment (fig. 14, abd. v.), C. crassus agrees closely with G. antarcticus as described and figured by Willem. This character gives the name to the genus. Examination of the mouth-parts of C. crassus (figs. 18-20) shows that they conform to the type usual in the Collembola. The mandible is very slender at the tip, which bends markedly towards the centre of the head (fig. 18, ap.) and ventralwards (fig. 20); the apical teeth are feeble and close together. On the outer edge of the mandible, opposite the grinding surface (figs. 18, 20, mo.), is a characteristic prominent shoulder (fig. 18, hu.\ and the conical process (for attachment of a rotatory muscle) on the dorsal aspect of the base of the mandible (fig. 20, pro.) is also prominent. * Recorded by Wahlgren from South Shetland, Graham Land, Paulet Island, and South Georgia. t Cryptopygus cinctus, newly described by Wahlgren (1906) from Tierra del Fuego and East Falkland, has, like C. crassus , only twelve ocelli, and no clubbed hairs on the feet. It is, however, variegated in coloration when adult. 478 Proceedings of Royal Society of Edinburgh. [sess. The inaxillulae (fig. 19, mxl.) are simple in form, with a few minute bristles at the tip or their inner faces ; the arm (fig. 19, hr.) which supports the maxillula is bent and irregularly furcate in shape. It is connected by a ligament (fig. 19, lig.) with the outer framework of the maxillula, this being itself continuous with the inner chitinous rod of the maxilla (fig. 19, rh. int.), as explained by Folsom (1899) for Orchesella. The maxillae (fig. 19, mx.) are of the typical Collembolan form ; the palp, however, is remarkable on account of the production of its tip into a tongue-shaped process bearing four bristles, and the excessive development of the long proximal bristle and its papilla (fig. 19, pa.). Distributional Notes. As mentioned in the introduction to this paper, the existence of identical or of nearly allied species of Collembola on widely separated areas may be regarded as strong evidence for ancient land connections between those areas. Many recent writers on zoological geography have expressed belief in a former extension of the Antarctic continent, wide enough to connect with America, Africa, and Australia. A full discussion of the problem has recently been given in Ortmann’s valuable paper (1904, pp. 310- 324, with map, pi. xxxix.) on the Tertiary invertebrate fauna of Patagonia, and there can be no doubt that the trend of modern speculation is against the doctrine of the permanence through past ages of the great ocean basins of the present day, as upheld in the classical writings of Darwin and Wallace. Hutton, who many years ago suggested the Antarctic continent as a former means of communication between Australia and Patagonia, and subsequently withdrew the hypothesis in favour of a trans-Pacific continent, has now re-affirmed his former belief (1905), laying special stress on the Collembola of South Victoria Land as evidence for the former connection of that remote region with the northern continents. From the facts established in the present paper, further support for the ancient extension of Antarctica may be readily drawn. The existence of the genus Cryptopygus and of the species Isotoma octo-oculata on the South Orkneys as well as on Danco 1905-6.] Gollembola from the South Orkney Islands. 479 Land, together with the presence of the Isotoina on Kerguelen, point to the former existence of extensive land tracts south of the American continent, with connection, either by way of Antarctica or of South Africa, to Kerguelen. It cannot indeed be inferred from the distribution of these springtails that there was at any one period a continuous land surface from Patagonia and Graham Land to Kerguelen. But it can hardly be denied that the insects must have travelled overland, though the land connections may have varied in extent, and become broken at different points during different periods. The bathymetrical work of the Scotia Expedition, as set forth by Bruce (1905), demonstrating a con- tinuous bank, less than 2000 fathoms beneath the surface of the South Atlantic, stretching eastwards from the South Orkneys towards South-East Africa, makes the former existence of one such land-tract the more credible. And the geological structure of the South Orkneys leaves no doubt that they must be regarded as strictly “ continental ” islands. Similarly, the “Kerguelen plateau,” as mapped by the explorers of the Valdivia (Schott, 1902), renders in the highest degree probable the former union of Kerguelen with Antarctica ; and a connection thence to South Africa is not impossible of acceptance. If, as we believe, these springtails — apparently members of a typically Antarctic fauna — owe their presence on the islands that they now inhabit to a former extension of the Antarctic continent, they must be of a considerable geological age. Ort- mann (1904) considers that the greatest extension of Antarctica existed in the Cretaceous and Eocene eras. Hutton (1905) argues for the Jurassic as the period of most extensive land in southern regions. We may safely conclude that Cryptopygus and Isotoma octo-ocidata have survived throughout the Tertiary epoch at least, with comparatively little change of structure. The affinities of Isotoma Brucei open up a problem of even greater interest. It is closely allied, as we have seen, to I. Beselsii , a springtail which has been found in Spitzbergen, Jan Mayen Island, Scotland (shores of the Firth of Forth), Greenland (Polaris Bay), and Massachusetts. We cannot doubt that this affinity points to a former connection between the Antarctic continent, of which the South Orkneys once formed part, and the 480 Proceedings of Royal Society of Edinburgh. [sess. northern continents. The presumption seems that this connection was by way of America, and the distribution of some allied springtails supports this presumption.* The common European Isotoma palustris , Muller, occurs both in North and South America; and Schaffer (1897) has described an Isotoma—/. obtusicauda — from Valparaiso, closely allied to two peculiar northern species, I. crassicauda, Tullberg, and I. litter alis, Della Torre. These last-mentioned insects come nearer than any other species of Isotoma to I. Brucei and I. Beselsii , agreeing with them in the evident position of the spring on the fourth abdominal seg- ment, but differing in the absence of prominent teeth on the mucrones. We find, therefore, that these groups of springtails, considered until a few years ago characteristically Arctic and sub- Arctic, are represented in the Andean sub-region of South America, in Tierra del Euego, and in the distant South Orkney Islands. Must I. Brucei , with its northern affinities, be regarded as an older or a newer member of the South Orcadian fauna than the distinctively Antarctic species that share its present home 1 Northern species, at or beyond the southern limits of the present American continent, must be either comparatively recent immi- grants— Pliocene or later — or else carry us back to early Mesozoic times ; for the existence of some sea-channel across America, checking migration from north to south, during the Cretaceous and Early Tertiary periods, is generally admitted. Von Jhering, for example, lays stress (1891) on the faunistic distinction between southern and northern South America, and suggests the existence in Secondary and Early Tertiary times of two continents — an “ Arclii- plata ” connected with Antarctica, and an “ Archicguyana ” con- nected by an Atlantis with West Africa. Now it seems unlikely that I. Brucei can be a late Tertiary immigrant into the Antarctic regions. The necessary connection of the South Orkneys with Patagonia can hardly have lasted late enough. And the group to which the species belongs is a primitive group even of this com- paratively primitive genus and order. In these insects, as mentioned above, the spring evidently belongs to the fourth abdominal segment, whereas in most species of the genus and * Which receives unexpected confirmation from Wahlgren’s discovery (1906) of I. Beselsii in Tierra del Fuego. 1905-6.] Collembola from the South Orkney Islands. 481 family it is apparently borne on the fifth. Willem (1900 ) has shown, however, that in reality it always belongs to the fourth. Thus we see that in the group of I. Brucei an ancient character has been retained, and the shore-haunting habit of all the species belonging to it is another mark of high antiquity. It seems probable, therefore, that I. Brucei is older than the typically Antarctic species ; and that, for the land connections over which its ancestors travelled, we must go back to early Secondary times. ) It is startling to conclude that these frail insects of the far north and the remote south, now separated by thousands of miles of land and sea and ice, have passed through so great a length of geological time with such slight structural deviation from their common progenitors. REFERENCES. 1905. W. S. Bruce, “Bathymetrical Survey of the South Atlantic Ocean and Weddell Sea,” Scot. Geog. Mag ., August 1905. 1899. G. H. Carpenter and W. Evans, “The Collembola and Thysanura of the Edinburgh District,” Proc. R. Phys. Soc. Edin ., vol. xiv., 1899, pp. 221-266, pis. v.-viii. 1902. G. H. Carpenter, “ Insecta Aptera,” in Report on the Col- lections of Natural History made in the Antarctic Regions during the Voyage of the “ Southern Cross,” pp. 221-3, pi. xlvii., London (British Museum), 1902. 1903. G. Enderlein, “Die Insekten und Arachnoideen der Kerguelen,” in Wissenschaftliche Ergebnisse der deutschen tiefsee Expedition auf dem Dampfer “ Valdivia ,” 1898-9, vol. iii. pp. 199-248, taf. xxi.-xxxvii., Jena, 1903. 1899. J. W. Folsom, “The Anatomy and Physiology of the Mouth-parts of the Collembolan, Orchesella cincta, Linn.,” Bull. Mus. Comp. Zool. Harvard, vol. xxxv., 1899, No. 2. 1905. F. W. Hutton, “Ancient Antarctica,” Nature, vol. lxxii., 1905, pp. 244-5. 1891. H. von Jhering, “On the Ancient Relations between New Zealand and South America,” Trans. Proc. N. Zeal. Inst., vol. xxiv., 1891, pp. 431-445. 1904. A. Ortmann, Reports on the Princeton University Expedi- tions to Patagonia, 1896-1899, vol. iv., “ Palaeontology,” Part 2, “ Tertiary Invertebrates. ” pp. 45-332, pis. xi.-xxxix., Princeton, 1904. PROC. ROY. SOC. EDIN. — VOL. XXVI. 31 482 Proceedings of Royal Society of Edinburgh. [sess. 1897. C. Schaffer, “ Apterygoten,” Hamburger magalhaensische Sammelreise , Hamburg, 1897. 1900. C. Schaffer, ‘‘Die arktischen und subarktischen Collem- bola,” Homer and Schaudinn’s Fauna Arciica , pp. 237-258, Jena, 1900. 1902. G. Schott, “ Ocean ographie und maritime Meteorologie,” in Wiss. Ergeb. Exped. “ Valdivia ,” Jena, 1902. 1906. E. Wahlgren, “ Antarktische und subantarktische Col- lembolen gesammelt von der schwedischen Siidpolarexpedition,” Wissens. Ergebn. der schwed. Siidpolarexpedition, 1901-3, vol. v., 1906. 1900. V. Willem, “Recherches sur les Collemboles et les Thysanoures,” Mem. Cour. Acad. Roy. Sci. Belg ., vol. lviii., 1900. 1902. V. Willem, “Collemboles: Resultats du voyage de S.Y. Belgica en 1897, 1898, 1899, sous le commandement de A. de Gerlache de Gomery,” Rapports scientifiques , Anvers, 1902. DESCRIPTION OF PLATES. Plate I. Fig. 1. Isotoma Brucei , side view, x 40. Fig. 2. Do. left side of head, dorsal view, showing ocelli, feeler, antennal organ (a. of and post-antennal organ (p. a. o.), x 100. Fig. 3. Do. right mandible, ventral view, x 200 ; ap., apical teeth ; mo., grinding surface. Fig. 4. Do. right maxilla ( mx .), ventral view, shown in as- sociation with the tongue (tin.), and the right maxillula (mxl.) ; ga., galea ; la! , ventral lamella ; la.”, inner lamella ; sti., stipes ; car., cardo ; pa., palp ; pd., foot of tongue ; lig., ligament of tongue. The left maxilla and the left half of the tongue are removed to expose the left maxillula (mxl!), and its supporting arm, hr., x 200. Fig. 5. Do. hindmost leg, showing claws, x 200. Fig. 6. Do. dorsal view of spring, x 200. Fig. 7. Do. dens and mucro of spring, side view, x 250. Fig. 8. Do. retinaculum, side view, x 600. Plate II. Fig. 9. Isotoma octo-oculata , var. gracilis, side view, x 40. Fig. 10. Do. left corner of head, showing ocelli, post- antennal organ (p. a. o.), feeler, and antennal organ (a. o .), x 100. Fig. 11. Do. hindmost foot, with claws, x 200. Fig. 12. Do. tip of dens with mucro, side view, x 300. Fig. 13. Cryptopygus crassus, young specimen, x 40. Fig. 14. Do. adult female, side view, x 40. Vol . XXVi. Proc. Roy. Soc. Edm. CARPENTER : ANTARCTIC COLLEMBOLA- Plate I. Gr.K-.C. del. MEa/flane &,Erslne year B. — Table giving the Constants of the Theoretical Curves 1. Smallpox . London N.W. to Cases 7-4279 7-6882 196-6696 •1442 S.E. 2. „ Liverpool . E. & VV. ■ 4-1247 2-6121 66-1562 •0972 3. „ 4. Relapsing Glasgow N. & S. 1-6609 2-1135 13-0490 •9771 E. & W. 1-5224 •2124 9-7323 •01025 Fever 1 905-6.] Studies in Immunity : Theory of an Epidemic. 521 CORRESPONDING TO THE COURSES OF EPIDEMICS. TYPE IV. 02 d md, 1 r V a No of Diagram. Remarks. 4-583 •14798 •6527 6-8217 •9470 5-0493 I. 4-5526 •2346 1-2859 8-9635 3-2622 3-5332 II. 4-2209 8-2141 4-4240 •71213 6-8129 17-1334 18-931 6-1659 IV. 7-9384 •9448 5-5113 9-6667 105-175 •50655 V. 'These epidemics are 1 almost identical al- 4-5863 •4685 3-7369 13-9478 11-7934 4-4195 III. 1 though the max. of 5-7329 •5946 4-0726 11-6988 | the first was in May 1 and the second in v January. 3-9677 •0824 •4712 9-4378 •8711 51049 VII. 5-1578 - -3342 1-3523 10-0932 4-9054 2-7823 VI. 3-5593 •1530 1-2434 14-2556 1-7394 10*1675 i 3-4183 - -0905 •8825 17-4959 •8133 18-986 3-1718 •0479 •9715 38-551 2-5513 14-601 IX. 3-1997 •1181 2-2574 36-2085 5-8743 13-94 X. 4-01274 - -11288 •62727 9-1136 •7625 7-4959 VIII. 3- 3046 4- 2688 - -0897 1-17172 24-1147 3-2589 8-6703 VIIlA. 3-35995 •4328 4-3664 - -2172 1-25287 9-53494 3-55635 3-35904 xi. 3-11622 •01688 •47918 54-7740 1-4512 18-0862 XII. 3-36441 •29294 4-87785 31-3251 10-6226 XV. 2-99658 XIII. Normal curve. 4-1111 - -0318 •1663 8-4564 •405896 3-4727 XIV. 3-8333 •06834 •42576 10-45943 •76214 4-7257 4-44641 - '12539 •60384 7-6313 1-2489 3-6896 3-7514 •34452 5-7297 31-26136 3-94458 4-5408 XX. 3-38995 5-2256 *2482 1-2067 7-7248 3-3742 2-7633 xvin. 6'5863 •5151 3-5450 11-7664 4-8054 •86803 XIX. 4-5083 •34823 2-01485 9*5718 4-05043 4-76246 XXIII. 7-08272 •32812 1-27706 5*78413 2-68123 2-75496 XXII. TYPE I. 02 r mi m2 «2 d 3-2638 15-6768 XXI. 2-7998 10-0166 2-1776 5-8390 3-708 9-942 •6224 XVI. 2-7010 3-33101 1-7606 6-6703 1-7773 6-7345 •5824 CORRESPONDING TO THE DISTRIBUTION of Epidemics in Cities. r d md V a 3-5645 20-8514 •4269 4-8780 9-1465 11-1204 XXVI. Type IV. 3-8885 7-3258 Type IV. 4-7303 Type VI. 4-0435 8-8507 •0440 •2388 • 61.27 3-451 Type IV. 520 Proceedings of Royal Society of Edinburgh. [sbss. A.— Table of the Constants of the Theoretical Curves type IV. Disease. Locality. Date. Cases 01 Deaths. r Unit of Time. M2 * ft 1. Miliary Oise . . 1821 Cases 1 week 7-0285 -4-0867 226*366 *0479 2. Plague 1665 Deaths 4 weeks 1*7758 1*3117 14*3663 *3072 3. ,, 6-1454 3-0194 123-7395 •0393 4. Cholera . 1832 1 week 9-3849 120-1387 *6223 5. „ Exeter . 1832 Cases 2 weeks 3-5318 10*1086 98-9950 2-3411 1891 Deaths 2-4678 3*08748 27*9962 •6342 7. „ 1891-2 2*3093 30-5716 1*2217 8. Smallpox . Warrington . 1743 ” 1 month 3*1149 *7893 38-4964 *0206 9. „ Boston, U.S.A. 1721 1-3010 -1-2993 8*7323 ■7636 10. Glasgow 1784 1 1 week 7-8888 2*8044 1-5082 •0161 11. ; Gloucester . Cases 21-9069 - 5-3486 1640-46 •00272 12. „ . , Severe cases 4 weeks 5*7644 *1938 105*39 •00196 13. „ London Deaths | 5*6492 1-4905 102*115 *01322 14. „ 1902 Cases 1 6-9321 -2-4550 192-8322 •01794 15. „ . | Orsett Union ” 3-3240 - -68839 36-5115 *0129 16. ■ , Liverpool . (local epi- demic) 1901 2 weeks 2-5540 *9005 27*8201 *0487 Sheffield . 1887-8 7*2097 - *68647 174*6607 *12574 18. Measles . Glasgow 1808 Deaths 1 month 1-7039 1-13235 12*7061 *25920 19. Zymotic 1890-1903 ” 2 weeks 6-0870 •21329 115-4617 *00020 20. „ . London 1854-1903 4-5056 2-9978 68-2995 *09825 21. „ Manchester . 1878-1887 Cases 1 month 1-4458 ■04423 6*2549 Islington Deaths 1-62106 - *16702 10-8032 •006 23! \\ \ 1857-62 Cases (1-5) 4 weeks 2-3797 •47904 217134 *01703 24. „ Cases ' ages 1 2*10781 - -90387 19 7548 *08724 25. Enteric Coventry . 1900 LJ 1 week 1*76744 1*95543 11*71872 *3471 26. Rotherham . 1892 5-27047 5-14338 94-1654 *17811 27. Scarlet Halifax 1880-1 2 weeks 1-35199 - 1-13994 9*55176 *52582 28. „ 6 . Thorshavn . 1873-4 1 month 1*83684 3-28100 22-22123 1-7371 Glasgow 1 day 3-08965 3*18762 43-03616 •34451 Wimbledon . 2 days 1-92734 2-60173 26-30963 *94548 TYPE I. M2 M3 M4 * 31. Enteric Fever 32. Yellow Fever 33. Zymotic Diarrhoea Maidstone . Demerara Islington . . 1900 1840-1 1857-62 Cases Cases under 3 days 1 month 12-0281 3-6612 1-45193 26-7769 -3-0409 *57488 472-168 37-5459 4*61391 ■4121 *1883 •10797 B.— Table giving the Constants of the Theoretical Curves 1. Smallpox . 2. „ j 4. Relapsing Fever London Liverpool Glasgow N.W. to S.E. E. & W. N. &S. E. & W. T 7-4279 4-1247 1*6609 7*6882 2*6121 2*1155 *2124 196-6696 66-1562 13-0490 97323 *1442 *0972 •9771 ■01025 1905-6.] Studies in Immunity : Theory of an Epidemic. 521 CORRESPONDING TO THE COURSES OF EPIDEMICS. TYPE IV. 02 d rnd - • No of Diagram. Remarks. 4-583 *14798 •6527 6-8217 •9470 5*0493 I. 4-5526 4-2209 4-4240 4-5863 5 7329 111 Pi 6-8129 5*5113 8-9635 8-2141 17*1334 9*6667 13*9478 11-6988 18 931 106-175 11-7934 61659 *50655 4-4195 :: II. IV. V. III. /These epidemics are | almost identical al- J though the max. of | the first waB in May I and the second in 3-9677 5-1578 •0824 - -3342 *4712 1-3523 9-4378 10-0932 •8711 4-9054 61049 2-7823 VII. VI. V. January. 3-5593 3-4183 3-1718 •0479 1-2434 *8825 •9715 14-2556 17-4959 38*551 1-7394 •8133 2*5513 10*1675 14-601 IX. 31997 4-01274 3- 3046 4- 2688 *1181 - -11288 - -0897 2-2574 ■62727 1-17172 36-2085 9-1130 24-1147 5-8743 •7625 3-2589 13*94 7- 4959 8- 6703 X. VIII. VIIlA. 3-35995 4*3664 3*11622 •4328 - -2172^ 1*25287 *47918 9-53494 54-7740 3*55635 1*4512 3-35904 18*0862 " xi. XII. 3*36441 2-99658 4-1111 •29294 - *0318 •06834 4-87785 •42676 31*3251 8-4564 10*45943 ■405896 ! •76214 10*6226 3*4727 4-7257 XV. XIII. XIV. Normal curve. 4-44641 - *12539 •60384 7-6313 1-2489 3-6896 3-7514 ■34452 5-7297 31-26136 3-94458 4*5408 XX. 3-38995 5-2256 *2482 1*2067 7*7248 3*3742 2*7633 xviii. 6- 5863 4*5083 7- 08272 *5151 •34823 •32812 1-27706 11-7664 9*5718 6*78413 4-8054 4-05043 2-68123 •86803 2-75496 XIX. XXIII. XXII. TYPE I. ft r in. <*2 d 3-2638 15-6768 XXI. 2-7998 10-0166 2-1776 5*8390 3-708 9-942 *6224 XVI. 2 7010 3*33101 1-7606 6-6703 1*7773 6*7345 *6824 CORRESPONDING TO THE DISTRIBUTION of Epidemics in Cities. r d - a 3-5645 20-8514 ■4269 4-8780 9-1466 11*1204 XXVI. Type IV. 3- 8885 4- 7303 4-0435 7*3258 8*8507 •0440 *2388 3*451 Type IV. Type VI. Type IV. 522 Proceedings of Royal Society of Edinburgh. [sess. On a Simple Way of Obtaining the Half-Shade Field in Polarimeters. By James Robert Milne, B.Sc., Carnegie Research Fellow. (Road July 16, 1906. MS. received October 5, 1906.) Summary. The half-shade effect in polarimeters is usually obtained, either by the well-known method of Laurent, or else by the more recent method of Lippich.* In the former a quartz plate is employed to give the necessary rotation to one-half of the beam of polarised light propagated through the instrument ; in the latter, a Xicol prism additional to the polariser serves the same end. It occurred to the author that the required effect might be obtained very simply by merely interposing a glass plate in the beam of light, so that half the beam traversed it, in an oblique direction. It follows at once, from Fresnel’s laws of the intensity of refracted light, that this will produce a slight rotation of the vibration-direction in the traversing half of the beam. In practice the method is found to give very good results.! Theory of the Method. Let a parallel beam of plane polarised light proceeding in the direction 0 Z (fig. 1) meet the glass plate 0 Q R V as shown. Let 0 P represent the light vibration both in direction and amplitude ; and let the angle Q 0 Y be the angle of polarisation for glass. On resolving 0 P along 0 X and 0 Y, the latter component will be transmitted through the glass with undiminished amplitude, but the former will have its amplitude 0 S reduced to (say) 0 S'. * For a description of the latter, see, for instance, Landolt’s “ Das optische Drehungsvermogen. ” t The author afterwards learned that the same principle of rotation by selective reflection had already been applied to the polarimeter, although in a different manner, by Professor Poynting. See A Method of Making a Half- Shadow Field in a Polarimeter by two Inclined Glass Plates, by J. H. Poynting, Sc.D., F.R.S. ; B. A. Report , p. 662, 1899 ; also Catalogue of the Optical Convention, p. 224, 1905. 1905-6.] Obtaining the Half -Shade Field in Polarimelers. 523 On recombining the two components, the new direction of light vibration is OP', giving an angle of rotation of POP'. The magnitude of this angle depends on the angle of inclination Q 0 Y of the plate : and on the direction of the light vibration (defined by the angle POX say). As regards the effect of the first of these, it is clear that if the angle be made less than the angle of polarisation, then the component along 0 Y will suffer loss as well as the component along O X (though, as may easily be shown, always to a less degree), and the angle POP' will be diminished. As regards the effect of the second factor, it appears from the diagram that P 0 P' is a maximum when P 0 X is in the neighbour- hood of 45°,* and that POP' gradually declines to zero as POX proceeds towards either 0° or 90°. To obtain the value of the maximum rotation let QOY = 57°i P 0 X = 45°, and OS = l; then if p be the fraction of the light energy polarised in the plane of incidence which is reflected from the front surface of the plate, by Fresnel’s formula, p = sin2 (£-?•) = sin2 [57|°-(90o-57J°)] p = sin2 25° = 0 1786. But the same fraction of the incident light is reflected from the back surface as from the front, hence the amplitude after trans- mission through both surfaces is given by OS'= 1— p = 0*821. |tan P' 0 Y = 0 S' ; P0P' = 45°-39io = 5i°. * More exactly 42° ‘2. But therefore 524 Proceedings of Royal Society of Edinburgh. [sess. This rotation is more than is usually required in a polarimeter, hut it can he reduced to any desired degree; either hy turning the glass plate about the axis OX, so as to decrease the angle QOY, or by altering the vibration-direction OP of the light, so as to change the angle POX. It will be noticed that this device has the advantage over that of Laurent, that the same plate may be used with any colour of light. In illustration of the point, it may be mentioned that the writer is at present using the arrangement described later in a spectro-polarimeter, where it is giving very satisfactory results. The Method in Practice. It is obvious that there are various ways in which this half- shade method might be applied in practice. The author experi- mented with a number of different schemes, and was led to the conclusions now to be briefly indicated. In any apparatus of this kind, it is first of all requisite that the two halves of the field shall not be separated by a dark band, but shall be brought perfectly in Contact. This condition may be fulfilled by grinding off the top edge of the inclined glass plate so that it makes only a very small angle with the direction of the light rays. The diagram shows the position of the glass relative to the Fig. 2. — The broken line indicates the path of one of the multiple reflections. polariser. The rays of light fall on the plate at the polarising angle, say 57 J°. The upper surface AB is opposite to the middle of the Nicol, and is very slightly inclined downwards from B to A. This ensures that there will be no gap between the edge B and the upper half of the beam. On the other hand, there can be no gap 1905-6.] Obtaining the Half -Shade Field in Polarimeters. 525 between the edge B and the lower half of the beam, because the rays which enter AD are, on account of their refraction, more inclined upwards than the surface A B. The rays which strike the surface A B, whether internally or externally, are reflected by it harmlessly aside out of the way, as may easily be shown. At C, where the first multiple reflection emerges from the glass, a change of intensity occurs, the two portions of the surface, B C and C E, appearing of different brightness to the eye. This makes it necessary to screen off all the surface beyond C, and to use the portion B C only. The explanation of the change of intensity lies in the fact that beyond C the light emerges elliptically polarised, being a combination of rectilinear vibrations rotated to different ex- tents, due to the direct and to the retransmitted rays respectively. Now it might be supposed that the necessity for limiting the field could be avoided by using an exceedingly thin plate of glass so that C would sensibly coincide with B. This would mean, how- ever, that the light proceeding from each point of the surface would be polarised in a slightly elliptical manner (which is attested in practice by the fact that it is impossible with any position of the analyser to obtain complete extinction over the rotated half of the field). Now it may be shown from the mathematical theory of the polarimeter, that in such circumstances the sensitiveness of the instrument must be somewhat impaired. It can be shown, how- ever, that the ellipticity is very slight, and in practice the author has found that very good results may be obtained by this method ; and that with nothing more elaborate in the way of apparatus than a microscope cover-glass cemented to a piece of cork, fixed slantwise behind the polariser. For the better plan of the thick plate there is necessary a piece of plain parallel glass, optically unstrained, and having one edge ground off at the proper angle, and subsequently polished. A piece of plate glass can be selected which is sufficiently good to fulfil all requirements. The thickness of the plate is determined by the desired area of the field — it can easily be proved that in fig. 2 the connection between the dimensions of B C and D E is, 2 BC = -DE. The plate is to be securely mounted behind the /X 526 Proceedings of Royal Society of Edinburgh. [sess. polarising Nicol as shown in the figure, and so arranged that the Nicol can be rotated without disturbing the glass (or else vice versa). The object of such relative rotation is to provide a means of altering the angle P 0 X of fig. 1, and therefore the angle POP' as explained already; and therefore the sensitiveness of the half- shade field. The alternative method of effecting such a change, by altering the angle QO Y, cannot be employed, because it would require the angle BAD of fig. 2 to be capable of alteration; which of course is impossible. With this type of rotator a “ triple-field,” a form that is preferred by some observers to the more usual double one, can of course be easily arranged by the provision of a second glass plate mounted on the opposite side of the Nicol. The middle part of the beam of light then passes unaffected, while both its side portions are rotated to an equal extent in the same direction. The author desires to express his best thanks to Professor MacGregor for the opportunity of carrying out the necessary experimental work in the Physical Laboratory of Edinburgh University. ( Issued separately January 14, 1907.) 1905-6.] Exception to a Certain Theorem in Optics. 527 On an Exception to a Certain Theorem in Optics, with an Application to the Polarimeter. By James Robert Milne, B.Sc., Carnegie Research Fellow. (Read July 16, 1906. MS. received October 15, 1906.) [Abstract.] There is a well-known law in geometrical optics, that the “ intrinsic luminosity " of the image formed by any lens system whatever is the same as the intrinsic luminosity of the object. An exception, however, which seems not to have been pointed out before, exists in the case of polarised light, based on the fact that by the agency of a double-image prism two light rays polarised in directions mutually perpendicular may be combined into one ray, which carries the total energy of both. In this way an intrinsic luminosity of image can be attained which is twice as great as that of the object. Now, because the iris opening of the eye is of a fixed size, the D Fig. 3. only way of increasing the brightness of a retinal image is to increase its intrinsic luminosity; hence it is in connection with images formed in the eye that the above principle has its chief interest. The application to the case of the polarimeter is shown diagramatically in fig. 3. A is the polarising Nicol ; B the tube for the liquid ; C a Nicol half-covering the field as in Lippich’s half-shade device (or instead, there may be used the half-shade 528 Proceedings of Royal Society of Edinburgh. [sess. device of the preceding paper), and so placed that the division of the field is vertical ; D, a quartz plate giving a rotation of 90° to the lower half of the field ; E, a double-image prism of such angular strength that the upper and lower halves of the field are superposed in the eye of the observer ; and F, the usual telescope, focussed on the dividing edge of C. E acts as analyser for both the upper and lower halves of the field ; for their respective plans of polarisation, having been made mutually perpendicular by D, are symmetrically disposed to the two vibration-directions of the double-image prism respectively. Measurements are made by rotating, not E, but A. The advantage of this arrangement is that it produces a field of view, the illumination of which is twice that of the ordinary field. A much reduced half-shade angle can therefore be em- ployed, from which there results, of course, a corresponding increase in the accuracy of the instrument. (. Issued separately January 14, 1907.) 1905-6.] Hessians of Certain Invariants of Binary Quantics. 529 The Hessians of Certain Invariants of Binary Qnantics. By Thomas Muir, LL.D. (MS. received August 11, 1906. Read November 5, 1906.) (1) The cubinvariant J of a binary quartic being a b c ace + 2 bed - ad? - b2e - c3 or bed its Hessian, H(J), is c d e e -2d c . - 2e 2d 2c -2b e 2d - 6c 2b a - 2d 2c 2b -2a c -2b a Performing on this the operations c-col3 — e-col5 , c-col4 + 2 ^4) "b (^1 > ^2 ’ ^3 » ^4) *b (^1 j ^2 ’ ^3 j ^4) “b (fi 5 ^2 ’ ^3 ’ ^4) This being equal to 2(fll >a2>a3’aAai’a2>a3’ a±) + 22(ai’ tt2> «3> «4 iS &1 > b2 1 &3> &4)> let us attend to the terms in it which only contain letters with the suffix 1. Of these there are manifestly under the first 2 n 2 h 2 r 2 (12 5 ) Uj\ 5 and under the second S 2 afj-L , 2 axc,x , 2«1c?1 , 2^^ , 2 51c?1 , 2^^ : so that the aggregate is (a1 + b1 + cx + df2. Similarly the sum of the terms in which only the suffix 2 occurs is ( a2 + b2 + c2 + d2)2 ; and so as to the other suffixes. Further, there are no terms involving a variety of suffixes : consequently we have as the full result (cq + \ + cx + f)2 + (a2 + b2 + c2 + d2)2 + (a3 + b3 + c3 + d3)2 + (a4 + &4 + c4 + d4)2 ; 534 Proceedings of Royal Society of Edinburgh. [sess. and the general theorem — The sum of the elements of the square of a determinant of the nth order is expressible as the sum of n squares , each of ivhich is the square of the sum of the elements of a column of the original determinant. (2) It is easily seen what change is necessary when, in squaring, the multiplication is performed in column-by-column fashion. When row-by-column multiplication is used, the result is no longer a sum of squares but is a sum of binary products each of which has for its first factor the sum of the elements of a row, and for its second factor the sum of the elements of a correspondiug column. Denoting by Rr the sum of the elements of the rth row, and by Cr the sum of the elements of the rth column, we see there- fore that the sum of the elements of A2 = Cf + C22 + . . . -1- Cn2 when A2 = A x rrA , = ~Rf + R22 + . . . 4- Rn2 when A2 = A x CCA , = R1C1 + R2C2 + . • . + RnCw when A2 = A x rcA . (3) Turning now to the 36 two-line minors of A2 we see that they are 11 12 11 13 11 14 1 12 13 13 14 12 22 , 12 23 5 12 24 , ' 22 23 , , 23 24 11 12 11 13 13 14 13 23 , 13 33 , 33 34 11 12 13 14 14 24 , . . . 34 44 13 23 1 33 34 14 24 , . . . 34 44 the array being of course axisymmetric. The first of the 36 is CL i $2 ^3 ^ a1 a2 as a4 or row2 ! \ \ h h \ b2 b3 bt row2 the second, which occurs twice, is rowj row4 1 row2 r0W3 1905-6.] The r-line Minors of the Square of a Determinant. 535 and so on. Therefore, just as in the previous case, the sum may be expressed in the very contracted form j rowj + rowi + row4 + r0W2 + row2 + r0Ws 1 row2 row3 row4 lrow3 row4 row4 J 1 rowr I2 + 2^! rowr rowr, 1 row, | row, ! rows, or where r, s is any pair of the integers 1, 2, 3, 4, and r\ s any other pair. Falling back, however, on the lengthier form, it is next seen that each of the 36 parts of it is expressible as the sum of six products ; for example, the first j* ^ a* j* f = laAI2 + laA!2 + l«AI2 + IVsl2 + i“AI2 + \asbt? ■ so that altogether we have 216 products of pairs of two-line minors of the original determinant. Keeping an eye on those having only the suffixes 1, 2 we find under the first 2 | afb 2 p + | a4c2 12 + | aft2 12 + | \c2 |2 + 1 hxd2 |2 + | c^d2 |2 , and under the second % 2{|a152!-|a1c2| + I^AI'K^I + and therefore in all {! and making an evident specialisation we have 1 1 1 1 «1 a2 2 1 «1 a3 2 1 a2 a3 «lVs I2 = 1 K ^2 + 1 h + 1 b2 h 1 ci C2 1 C1 C3 1 C2 C8 Now the left-hand member here is equal to the sum of the signed primary minors of | a1b2c3 12 ( Proc . Roy. Soc. Edin ., xxiv. pp. 387-392) ; and the right-hand member is equal to (a, + b3 + c3y + (a2 + b2 + c2y + (a, + b, + c,)*. That 4 unsigned ’ may legitimately be substituted for 1 signed ’ is made evident on bordering with 1 , - 1 , 1 instead of l , 1 , 1. (5) In the theorem of § 3 it is the sum of all the r-line minors that we are concerned with : there is, however, an equally important theorem when we confine ourselves to the coaxial minors. It is — The sum of the coaxial r-line minors of A2 is equal to the sum of the squares of all the r-line minors of A. No formal proof need he given in view of what has come to light in proving the other theorem. Merely as an illustration we may note that when A = | afb2c3dA \ and r — 2 we have the sum of the two-line coaxial minors, 11 121+ 11 13 + 11 14 + 22 23 j + 22 24 + 33 34 12 22 13 33 14 44 23 33! 24 44 34 44 Ro Yo I <*i Vs \ I al/^2V3 I 1905-6.] The Y-line Minors of the Square of a Determinant. 537 al a2 «8 a 4 2 + \a1 a2 az ai p+ \ b3 h 1 tq C2 Gz CJ = | af2 12 + 1 afi 3 p + | af, p + 1 a2b3 p + | a2b 4 P + 1 azb 4 p ' + 1 a^2 P + 1 ttjCg P 4- v + + \ and the sum of the two-line coaxial minors is I ai^2 I ' I ai^2 I d- | a1bs | • | a^g j + j a2b3 | • j a2/33 | d* | aic2 I " I al72 I d- I aYC g | • | cqyg | + j a2Cg I • I a2y3 j d“ I ^\c2 I * I PiY2 I d- | fiiC3 | • | f31 y3 1 + | b2c3 j • | fS2 y3 1 . (7) The ultra-symbolical expressions used in §§1,3 suggest that a freer use of non-quadrate arrays might be advantageous. We might, for example, use them as elements of a determinant, thereby arriving at such identities as = KVel + NaVeU («1 , ai ) (aS » a,i) Qi,h) Os > h) (h,h) (Cj , C2) (Cg , C4) (C5 , Cq) 538 Proceedings of Royal Society of Edinburgh. [sess. Pi P2 Pi i 7i 72 7s I » C1 C2 Ci C1 C2 C3 C1 C2 H Pi P2 Pi 7l 72 73 \ \ h d-^ d^ dg d-^ c?2 dg 6^2 6^2 ^3 ft ft ft 7i 72 7s S2 S3 They might even be used as such alongside of quantitative elements in the same determinant ; and there is at least one case of this where the advantage is most striking : indeed it is not too much to say that we are thus enabled to express with fulness and accuracy a famous theorem of Binet’s which up till now has remained unformulated. Binet in 1812 ( Journ . de V Ec. polyt., ix., cah. 16, p. 284) says, “ On verifie aisement les formules suivantes 'Zab' = ^al,b - %ab, %ab'c” = ^aZb^c + 2 ^abc - ^ a%bc - ^b%ac - 'Sc'Sab, 'Zab'c'd'" = 'Xd%b%c%d - ttabcd - %d%b^cd - ^a%c%bd - %d%d%bc - %b%c%ad - 'Ib'^d’^ac - '%c%d%ab + ^ab^cd + 2 ac%bd + ’Sadlbc + ’2%a^bcd + 2 '^b'Xcda + 2 ^c%dab + 2 %d%abe, %ab'c"d'"e!"' = %a%b%cZd%e + .... \ h \ ^2 &3 ft ft CO oa. 7i 72 73 O O'1 to hi % C2 ci ci C2 ci 1 C1 C2 ft ft ft 7i 72 7s 1 \ \h h h 6j b. 2 ^3 'CO to Pi 7i 72 73 1 *1 & 2 ^3 1 The law of formation of the right-hand members was left un- divulged: and probably Bellavitis in 1857 was the first to draw attention to the fact that the said members bear a wonderful resemblance to the final expansions of axisymmetric determinants {Sposizione element are . . . . §91); but he only got so -far as to say that in order to complete Binet’s fourth instance “lo sviluppo del determinante simmetrico ” of the fifth order must first be found, and then certain arbitrary changes made therein. With 1905-6.] The Y-line Minors of the Square of a Determinant. 539 the use of single-line arrays as elements all difficulty vanishes, the identities then becoming aY + a2 + ... + an (ax , a2 , • . • , an) (alta2i . . . , an) \ + b2 + ... +bn a1 + a2 + ... + an (a1 , a2, . . . , an) (a1 , a2 , . . . , an) (Pit b2 , . . . , bn ) b-^ b2-\- ... + bn , b2 , . . . , bn ) i5 ^2 5***5 Cn) (^i j ^2 ’ ' ’ • > ^1 "•* ^2 d" ••• d" Cn = 2 I «AC3 I » and so on. (. Issued separately January 16, 1907.) OBITUARY NOTICES. Professor A. W. Williamson. By Professor A. Crum Brown. (Read January 8, 1906.) Alexander William Williamson was bom at Wandsworth, May 1st, 1824. He studied chemistry under Gmelin in Heidelberg and under Liebig in Giessen, where he graduated as Ph.D. In 1848 he studied mathematics under Comte in Paris. In 1849 he was appointed Professor of Practical Chemistry in University College, London ; and in 1855, in addition, Professor of Chemistry. He was elected Fellow of the Royal Society of London in 1855, and was Foreign Secretary of the Society from 1873 to 1889, and Vice-President in 1889, 1890. He was President of the British Association in 1873. He was elected Hon. Fellow of this Society in 1883. In 1887 he resigned his chair and retired to Haslemere, where he died May 6th, 1904. Williamson’s chemical work was not great in quantity, but was of the very highest importance, and his name will always remain in the history of chemistry in the list of the great leaders. Berzelius gave H20 as the formula of water, but the duplicity of the hydrogen in this formula was deduced from physical considerations only, and was not used to explain any chemical phenomena. The chemical unit of hydrogen was to Berzelius and his followers the “ equivalent ” H2 and not the atom H, and they wrote hydrochloric acid and ammonia H2C12 and U2H6, until a special symbol, a barred letter, was invented for the equivalent in the case of each element the equivalent of which consisted of two atoms. It was Williamson who brought to light the chemical meaning of the 2 in H20, and showed that these two atoms are not permanently tied together, but are each separately united to the one indivisible atom of oxygen. By the action of potassium on alcohol one-sixth of the hydrogen Obituary Notices. 541 of the alcohol is removed and a compound formed containing potassium in the place of this hydrogen. It occurred to Williamson that if this potassium alcohol were treated with the halogen compound of a hydrocarbon radical the potassium and the halogen would unite, and a new and more complex alcohol be formed containing the hydrocarbon radical in place of the potassium, and therefore in place of the hydrogen which had been removed by the action of the potassium. This idea was the foreshadowing of a very important method of synthesis, but the result showed that it was not applicable in this case. By acting on potassium alcohol with ethyl iodide Williamson obtained indeed potassium iodide, but the other product was not a new alcohol but common ether. He at once saw the explanation of this. He regarded alcohol not as a compound of ether and water, the common view at that time, but as an intermediate substance, not Ae20,H20, but AeHO, in which the ethyl and the hydrogen are independently united to the one atom of oxygen, and he recognised that it is this hydrogen and not hydrogen of the C2H5 that is replaced by potassium. “ Thus C H C H alcohol is 50, and the potassium compound is and by acting upon this by iodide of ethyl we have C^50 + C2H6I = IK + ^Ao. . . . Alcohol is therefore water in which half the hydrogen is replaced by carburetted hydrogen, and ether is water in which both atoms of hydrogen are replaced by carburetted hydrogen, thus : C2H5Q C2H5q „ HU’ H U’ C2H5a But as the formation of ether in this way could be explained on the supposition that alcohol is a compound of ether and water, and potassium alcohol a compound of ether and oxide of potassium, half of the ether produced being that united with oxide of potassium and the other half coming from the action of the oxide of potassium on the ethyl iodide, Williamson devised and carried out a crucial experiment. By acting on potassium ethylate with methyl iodide and on potassium methylate with ethyl iodide he obtained in both cases the same product, which was not a 542 Proceedings of Royal Society of Edinburgh. C TT mixture of the two ethers but an intermediate ether V^rr50. In Ori3 a similar way he prepared the intermediate amyl-methyl and amyl-ethyl ethers. He then goes on to use these principles to ex- plain the ordinary process for preparing ether, giving the sequence of actions now familiar to every student of chemistry, and show- ing experimentally that three ethers are formed when a mixture of two alcohols is distilled with sulphuric acid. Williamson’s work on etherification was published in a paper read before the Chemical Section of the British Association at its meeting in Edinburgh in 1850 and printed in the Philosophical Magazine . It is in this paper that the following striking passage occurs : — “ Before quitting the subject of setherification I would wish to add a few words on an application which naturally enough suggests itself of the fact to which the process is here ascribed. I refer to the transfer of homologous molecules in alternately opposite directions, which, as I have endeavoured to show, is the cause of the continuous action of sulphuric acid in this remark- able process. It may naturally be asked, why do hydrogen and carburetted hydrogen thus continuously change places'? It cannot be from any such circumstance as superior affinity of one molecule over another, for one moment sees reversed with a new molecule the transfer effected during the preceding one. Now, in reflecting upon this remarkable fact, it strikes the mind at once that the facility of interchange must be greater the more close the analogy between the molecules exchanged; that if hydrogen and amyl can replace one another in a compound, hydrogen and ethyl, which are more nearly allied in composition and properties, must be able to replace one another more easily in the same compound ; and that the facility of interchange of hydro- gen and methyl, which are still more similar, will be still greater. But if this be true, must not the exchange of one molecule for another of identical properties be the most easily effected of all 1 Surely it must, if there be any difference at all ; and if so, the law of analogy forbids our imagining the fact to be peculiar to hydrogen among substances resembling it in other respects. We are thus forced to admit that, in an aggregate of molecules of any compound, there is an exchange constantly going on between the elements Obituary Notices. 543 which are contained in it. For instance, a drop of hydrochloric acid being supposed to be made up of a great number of molecules of the composition C1H, the proposition at which we have just arrived would lead us to believe that each atom of hydrogen does not remain quietly in juxtaposition with the atom of chlorine with which it* first united, but, on the contrary, is constantly changing places with other atoms of hydrogen, or, what is the same thing, changing chlorine.” The observed facts of balanced actions of double decomposition led Williamson to this view, and it is interesting to note that the observed facts of electrolysis led Clausius quite independently to a sotnewhat similar hypothesis about seven years later. A paper read before the Chemical Society of London, June 1851, contains, besides further details as to the preparation, analysis, and vapour density of the new intermediate ethers, a very important note on the constitution of acetone, and an account of an intermediate ketone, with a very clear statement of the constitution of these bodies and a forecast of the general method of preparing aldehydes afterwards independently discovered by Limpricht and by Piria. In a paper in the Chemical Gazette , 1851, he points out the analogy between ether and the anhydrous monobasic organic acids, then unknown, but soon afterwards discovered by Gerhardt, who obtained them by a process perfectly analogous to that used by Williamson for the preparation of the ethers. As Williamson had thus, in 1850, established the “water type” on a secure experimental basis, so, in 1854, he extended similar reasoning and demonstration to the case of sulphuric acid, and showed how dibasic acids and their derivatives can be referred to the double type of water. In a paper communicated to the Royal Society of London he writes : — “ An atom of nitric acid, being eminently monobasic, is, as we have already shown, represented in the monobasic type gO by the formula in which peroxide of nitrogen (N02) replaces one atom of hydrogen. In like manner, hydrate of potash go) is obtained by replacing one atom of hydrogen in the type by its equivalent of potassium ; and nitrate of potash by a 544 Proceedings of Royal Society of Edinburgh. simultaneous substitution of one atom of hydrogen by peroxide of nitrogen, the other by potassium. Sulphuric acid is formed from two atoms of water H 0 H Ho Hu ; one of hydrogen from each is removed, and the two replaced by the indivisible radical S02. The series Sulphuric acid Acid sulphate of potash Neutral sulphate of potash H SO H O 2Q ’ H so2g. K U K SO K 0 20 explains itself.” He then describes the action of pentachloride of phosphorus on sulphuric acid : — “ Confining my remarks for the present to the case of sulphuric acid, whose decomposition is doubtless typical of that of other bibasic acids, I may state as the result of numerous experiments with the most varied proportions of pentachloride and acid, performed on a scale of considerable magnitude, that the first action of the pentachloride consists in removing one atom of hydrogen and one of oxygen (empirically peroxide of hydrogen) from the acid, putting in an atom of chlorine in their place, and 0, which is strictly intermediate be- 2 Cl tween the hydrated acid and the final product S02C12 formed by a repetition of the same process of substitution of chlorine for peroxide of hydrogen. The existence and formation of this body, which we may call chloro-liydrated sulphuric acid, furnishes the most direct evidence of the truth of the notion, that the bibasic character of sulphuric acid is owing to the fact of one atom of its radical S02 replacing or (to use the customary expression) being equivalent to two atoms of hydrogen. Had this radical been divisible like an equivalent quantity of a monobasic acid, we should have obtained a mixture , not a compound , of the chloride with the hydrate, — or, at least, the products of decomposition of that mixture.” In another paper in the same volume of the Proceedings we find the following : — “ According to the results of recent researches in the constitution of salts and the method thence introduced of H forming the compound Obituary Notices. 545 explaining chemical reactions, it is equally correct to represent such a reaction as that of hydrochloric acid on hydrate of potash, as consisting in an exchange of hydrogen of the one for potassium of the other, or of chlorine in one for peroxide of hydrogen in the other. In Mr Kay’s researches, as described in the following brief outline, this notion has obtained very striking illustrations ; for he has obtained a peculiar body in which the chlorine of chloroform is replaced by peroxide of ethyle by the action of chloroform on three atoms of ethylate of sodium, which product may be equally well conceived to be a body in which the hydrogen of three atoms of alcohol is replaced by the tribasic radical of chloroform. According to the older theories of the capacity of saturation of salts, this compound would contain a tribasic modification of formic acid, for it has the same relation to formic ether as a so- called tribasic phosphate has to a monobasic one.” It will be seen from the examples referred to that the leading principle of Williamson’s work was the then quite novel idea of the “ atomic value ” of radicals, which is exactly what we now call their valency, and that his favourite method was the forma- tion of intermediate substances as a guide to a knowledge of the constitution of the bodies between which they lie. This principle and this method led in Williamson’s hands to important develop- ment and simplification of chemical theory, and they still bear good fruit. Williamson’s influence on the progress of chemistry is not to be measured only by the work done directly by him ; every chemist who had the privilege of being his friend knows how much of his clear, intelligent knowledge of chemistry is due to Williamson. PEOC. ROY. SOC. EDIN. — VOL. XXVI. 35 Samuel Pierpont Langley, Secretary to the Smithsonian Institution, Honorary Fellow of the Royal Society of Edinburgh, 1902-6. By Dr W. Peddie. Through the death of Samuel Pierpont Langley this Society has lost one of the most eminent of its distinguished Foreign Members, and Science has lost one of the great leaders who have placed America in the front rank of the nations which concern them- selves with the advancement of knowledge. The announcement of his death came as a surprise ; for, although he had passed the threescore-and-ten limit, his powers for work were so entirely untouched as to justify the hope that many years of useful labour still lay before him. The work which he actually performed was so colossal, and some of it so recent, that years may necessarily pass before all its results are fully made public. Born at Roxbury, in Massachusetts, on the 22nd day of August 1834, Langley received his general education at Boston High School. Leaving the school in 1851, he took up the study of civil engineering and architecture, and subsequently practised these professions until he had succeeded in acquiring means which made him independent of routine work. From that time onwards his employments coincided more fully with his mental inclinations. Astronomy had attracted him powerfully since the days of his childhood; so the years 1864 and 1865 found Langley visiting the chief observatories of Europe, and making acquaintance with its scientific societies, many of which were in subsequent years to bestow upon him their highest honours. In 1865 he became assistant astronomer at Harvard College Observatory. In 1866 he was appointed assistant professor of mathematics at the United States Naval Academy. In 1867 he became director of the Alleghany Observatory at Pittsburg, a post which, along with the professorship of astronomy and physics at Pennsylvania, he held until, in 1887, he was appointed Obituary Notices. 547 Assistant Secretary, and soon afterwards Secretary, to the Smithsonian Institution. This connection continued uninter- ruptedly until his death on the 27th of February this year. The variety of his successful employments bears eloquent wutness to the magnitude of his mental equipment : engineer, architect, mathematician, physicist, astronomer, and administrator by profession, he was also a successful writer, a student of art and of archaeology. The great characteristic of Langley’s work is its pioneer nature. Problems of like type to problems already solved had no attraction for him. New problems which presented no special difficulty in their solution were passed by. The problem whose difficulties were such that others had failed to solve it, the problem whose difficulties were such that no other had attempted to attack it — these were the problems which Langley attacked and mastered ; and his attack was conducted almost with impatience. He never sat down beforehand to perfect a method of procedure ; he began at once on what he believed to be the likeliest' lines, and perfected his method as he proceeded. A subject which had once attracted Langley attracted him always. Questions arising in his earliest work appeared again in work which was uncompleted at the time of his death. All his investigations arose naturally, as all great investigations do, in the course of daily labour. It would serve no useful purpose to enumerate them here. It seems better that a mere indication of their nature and extent should be given, along with a fresh expression of this Society’s appreciation thereof. Between the years 1870 and 1877 Langley’s attention was devoted to the question of the structure of the solar disc and the radiation of heat from its various portions. The results were published in a series of papers during that period. It is found that, the more perfect are the atmospheric conditions for observa- tion, the more closely do present-day results agree with Langley’s early drawings. The practical aim of all his work is well in- dicated by a paper, in that series, on the direct effect of sun-spots on terrestrial climates. Another, and perhaps the most distinctive, branch of his work was that which dealt with the distribution of energy in the solar 548 Proceedings of Royal Society of Edinburgh . spectrum. Finding the thermopile, which at that time was the most delicate instrument available for his object, far too sluggish in its indications, he devised the bolometer, an instrument which, to this day, has no superior, and only one equal, in such work. By its aid he pushed the investigation of the solar spectrum into previously unexplored regions in the infra-red radiations. A natural extension of that work led to the mapping, by Langley and Very, of the lunar energy spectrum. This was a work of immensely greater difficulty, because of the slight differ- ence between the temperatures of the source and of the surround- ings of the instrument, and also because of atmospheric absorption. The results led to the conclusion that the temperature of the moon’s surface is not much above 0° C. A further extension was made to terrestrial sources of radiation, the mapping being pushed more than twice as far into the long wave-length region as had been found possible with solar radiation. In the process, the dispersive power of rock-salt was carefully determined. In 1892 Langley immensely improved his bolometer by making it an automatic, self-registering instrument, and the investigation of solar radiation was pushed as far into the infra-red region as it had been carried with terrestrial sources. Another distinctively great piece of work was that on the effect of the earth’s atmosphere in absorbing solar radiation, and on the determination of the solar constant. The value found for the constant must be regarded as at least a good first approximation. After he became Director of the Smithsonian Institution, Langley founded the Smithsonian Astro-physical Observatory and arranged its work primarily for the purpose of determining the natural influences having a direct bearing on climate and life. A part of its work, not completed at the time of his death, dealt with the question whether or not the solar radiation was variable to an extent sufficient to affect the earth’s climate, and whether or not the effects were predictable. The results already indicate an affirmative answer to the former part of the question. Another line of work, in which the daring nature of Langley’s attack on unsolved problems is well exhibited, is that on the problem of aerial navigation. His papers on Experiments in Obituary Notices. 549 Aerodynamics and the Internal Work of the Wind commanded wide attention. He made successful models of flying machines ; and, although the launching of his actual airship was unattended by success, unimpeachable photographic evidence showed that the failure was not in the vessel but in the launching apparatus. This Society cannot do other than endorse the strong simple words of the resolution come to by the representatives of the great Institution whose work he so long and so ably controlled — that the scientific world is indebted to Mr Langley for the invention of important apparatus and instruments of precision, for numerous additions to knowledge, more especially for his epoch- making investigations in solar physics, and for his efforts in placing the important subject of aerial navigation upon a scientific basis. Rev. George Matheson, D.D., LL.D., F.R.S.E. By Rev. James Lindsay, D.D. (Read November 5, 1906.) It is to me a real, though melancholy, satisfaction to utter what Ijischylus calls a few “ posthumous words in praise of a divinely good man” — hriTVjxfiiov a Ivor h r dvSpl Oeiip* Dr George Matheson became a Fellow of this Society in 1890. One of the most valuable features of the Royal Society of Edinburgh is its recognition of literary distinction as well as of Scientific eminence, even though nothing has yet been done to differentiate and develop its literary resources after the manner of the Royal Society of Canada. To this literary side of the Society Dr Matheson belonged. Yet he was not without a keen interest in scientific theories such as those of Tyndall, Spencer, Darwin, and Comte. Indeed, such hooks as Can the Old Faith live ivith the New ? and The Psalmist and the Scientist were, at the time of publication, highty useful attempts at some reconcile- ment of science with religion. Born at Glasgow in 1842, he became M.A., with philosophical honours, at the University there in 1862, and B.D. in 1866. He held ministerial charges at Innellan and St Bernard’s, Edinburgh, in both cases with distinguished success. In 1879 the degree of D.D. was conferred upon him by Edinburgh University, that of Aberdeen bestowing upon him LL.D. at a later period. In 1881 he held the Baird Lectureship; and in 1899 he was appointed to the Gifford Lectureship in Aberdeen, which, however, he declined. Blind from his youth, Dr Matheson’s intellectual interest and Miltonic courage won a supreme conquest. His literary industry was astonishing, his mental energy great and unceasing. He thought rapidly : truth came to him in intuitive flashes. Of volumes he published almost a score, and, in addition, many magazine articles. His work was varied in character and contents — historical, doctrinal, apologetical, exegetical, devotional, and * Agamemnon, 1547. Obituary Notices. 551 poetic. So much poetic charm and vital individuality went to the making of his best prose work, that his fame was carried to the ends of the English-speaking world. His genius was religious , but it was religious genius — the genius of insight and unique performance. Passing from his work to his personality, one must emphatically say that the man was more than his work, his character greater than his performance. The chief feature of that character was an essential nobility of mind, in which respect Dr Matheson stood far above the level of most distinguished men. He was withal genial and companionable to a high degree. Suddenly, but peacefully, he was, on the 28th of August, withdrawn into the mystic, eternal shadows. Brave in spirit and strenuous in endeavour to the last, his life remains an example and an inspiration. Meetings of the Royal Society — Session 1905-1906. The 123rd Session. Monday, 2%rd October 1905. General Statutory Meeting. Election of Office-Bearers, p. 1. FIRST ORDINARY MEETING. Monday, Qth November 1 905. Professor Crum Brown, LL.D., F.R.S., Vice-President, in the Chair. The following Communications were read : — | 1. Preliminary Note on the Conductivity of Concentrated Aqueous Solutions of Electrolytes. By Professor J. Gibson, p. 234. 2. The Tarpan and its relation with Wild and Domestic Horses. By Professor J. C. Ewart, F.R.S. ( With Lantern Illustrations.) p. 7. 3. The Horse in Norway. By F. H. A. Marshall, M.A., D.Sc. { With Lantern Illustrations.) p. 22. 4. Elimination in the case of equality of Fractions whose Numerators and Denominators are linear functions of the Variables. By Thomas Muir, LL.D. Trans., vol. 45, p. 1. SECOND ORDINARY MEETING. Monday, 20 tin November 1905. The Hon. Lord M£Laren, LL.D., Vice-President, in the Chair. The following Communications were read : — 1. Some Further Results obtained with the Spectro-heliometer. By J. Halm, Ph.D. p. 76. 2. Observations on the Normal Temperature of the Monkey and its Diurnal Variation, and on the Effect of Changes in the Daily Routine on this Variation. By Sutherland Simpson, M.D., D.Sc., and J. J. Galbraith, M.D. Communicated by Professor Schafer, F.R.S. Trans., vol. 45, p. 65. 3. Notes on the Effect of Electric Oscillations (co-directional and transverse) on the Magnetic Properties of Iron. By Mr James Russell. p. 33. 1905-6.] Meetings of the Society. 553 4. Some Electrical Measurements on Metals. By Charles E. Fawsitt, D.Sc„, Ph.D. Communicated by Professor Crum Brown. p. 2. No other Candidate having been nominated, Dr R. M. Ferguson was unanimously re-elected the representative of the Society on the Governing Body of George Heriot’s Trust. Mr Robert Mathieson, F.C.S., was balloted for, and declared duly elected a Fellow of the Society. THIRD ORDINARY MEETING. Monday , 4dh December 1905. The Rt. Hon. Lord Kelvin, G.C.V.O., etc., President, in the Chair. The following Communications were read : — 1. The Development of the Skull and Visceral Arches in Lepidosiren and Protopterus. By W. E. Agar, BA. Communicated by Professor J. Graham Kerr. Trans., vol. 45, p. 49. 2. Perturbations in Longitude of Neptune by the Hypothetical Planet. By Professor George Forbes, F.R.S. 3. Exhibition of two Lantern Slides of Zoological Interest. By Professor D. J. Cunningham, F.R.S. FOURTH ORDINARY MEETING. Monday , 18£7t December 1905. Dr R. H. Traquair, F.R.S., Vice-President, in the Chair. The following Communications were read — 1. Library Aids to Mathematical Research. By Thomas Muir, LL.D. p. 51. 2. Preliminary Note regarding an Experimental Investigation into the Effects of Varying Diets upon Growth and Nutrition. By Dr Chalmers Watson. Communicated by Professor Schafer, F.R.S. p. 87. The remaining Papers in the billet were postponed till next ordinary meeting. Mr Wm. Speirs Bruce, Dr Thomas James Jehu, Dr Wm, Thomas Ritchie, and Alexander Durie Russell, B.Sc., were balloted for, and declared duly elected Fellows of the Society. 554 Proceedings of Royal Society of Edinburgh. [sess. FIFTH ORDINARY MEETING. Monday , 8th January 1906. Professor Crum Brown, LL.D., F.R.S., Vice-President, in the Chair. The following Communications were read : — 1. Obituary Notice of Professor Alexander W. Williamson, F.R.S. By Professor Crum Brown, p. 540. 2. Bathy draco Scotice, Poisson abyssal nouveau recueilli par 1’ Expedi- tion Antarcticpie National Ecossaise. Note preliminaire, par M. Louis Dollo, Conservateur au Musee royal d’Histoire naturelle, a Bruxelles. Presentee par M. le Dr. R. H. Traquair, F.R.S. p. 65. 3. Influence of Thymus Feeding on Allantoin Excretion. By Dr W. M‘Lachlan. Communicated by Dr Noel Paton. p. 95. 4. On a Theorem in Hypercomplex Numbers. By J. H. Maclagan Wedderburn, M.A. p. 48. SIXTH ORDINARY MEETING. Monday , 22 nd January 1906. The Hon. Lord M£Laren, LL.D., Vice-President, in the Chair. The following Communications were read : — 1. On a Form of Initiational Disturbance more convenient than that of §§ 3-31 of previous Papers on Waves. By the Rt. Hon. Lord Kelvin, President, p. 398. 2. Illustrations of the Indefinite Extension and Multiplication of a Group of Two-dimensional Deep-Sea Waves, Initially Finite. By the Same. p. 409. 3. On the Initiation and Continued Growth of a Train of Two- dimensional Waves due to the Sudden Commencement of a Stationary Periodically Varying Forcive. By the Same. p. 412. Mr John Bennett Carruthers, Mr Henry O’Connor, C.E., Mr Fraser Story, Mr Gilbert Thomson, M.A., C.E., Dr Dawson F. D. Turner, F.R.C.P.E., Dr Robert Alexander Fleming, F.R.C.P.E., the Rev. Samuel M. Johnston, B.A., and Dr Duncan Scott Macnair, H.M.I.S., were balloted for, and declared duly elected Fellows of the Society. 1905-6.] Meetings of the Society. 555 SEVENTH ORDINARY MEETING. Monday , bth February 1906. Professor Crum Brown, F.R.S., Vice-President, in the Chair. The following Communications were read — - 1. The Relation between Normal Take-up or Contraction and Degree of Twist in Twisted Threads. By Thomas Oliver, B.Sc. Com- municated by Dr C. G. Knott, p. 182. 2. Some Experimental Results in Connection with the Hydro- dynamical Theory of Seiches; with Experiments. By Peter White, M.A., and Mr W. Watson. Communicated by Professor Chrystal. p. 142. EIGHTH ORDINARY MEETING. Monday , 19 th February 1906. Dr R. H. Traquair, F.R.S., Vice-President, in the Chair. The following Communications were read : — 1. On the Elevation of the Boiling Point of Aqueous Solutions of Electrolytes. By the Rev. S. M. Johnston, B.A. (With Lantern Illustrations.) Trans ., vol. 45, p. 193. 2. On the Formation of certain Lakes in the Highlands. By Dr L£on W. Collet and Dr T. N. Johnston. With a Note on Two Small Rock Basins in the Alps, by Dr Collet, p. 107. 3. On the Methods of Standardising Suprarenal Preparations. By Dr Isabella Cameron. Communicated by Dr Noel Paton. p. 157. Lt.-Col. Arthur Frederick Appleton, P.R.C.V.S., Dr Thomas Wm. Dewar, F.R.C.P., the Rev. Alexander Moffat, M.A., B.Sc., Mr Herbert Watkins Pitchford, F.R.C.V.S., and Dr Caleb Williams Saleeby were balloted for, and declared duly elected Fellows of the Society. NINTH ORDINARY MEETING. Monday , 5 th March 1906. Professor Crum Brown, F.R.S., Vice-President, in the Chair. The following Communications were read : — 1. The Igneous Geology of the Bathgate and Linlithgow Hills. Part II. — Petrography. By J. D. Falconer, M.A., D.Sc. Com- municated by Professor Geikie, F.R.S. Trans., vol. 45, p. 133. 556 Proceedings of Royal Society of Edinburgh. [sess. 2. The South Orkney Collembola of the Scottish National Antarctic Expedition. By George H. Carpenter, B.Sc., M.R.I.A., Professor of Zoology in the Royal College of Science, Dublin. Communicated by Mr William Evans, p. 473. 3. The Turbellaria of the Scottish National Antarctic Expedition. By J. F. Gemmill, M.A., M.D., and R. T. Leiper, M.B., Ch.B. Com- municated by Sir John Murray, K.C.B. 4. Scottish National Antarctic Expedition. — On Echinorhynchus antardicus , n. sp., and its Allies. By John Rennie, D.Sc. Com- municated by Mr William S. Bruce, p. 437. TENTH ORDINARY MEETING. Monday , \Wi March 1906. The Hon. Lord MDaren, Vice-President, in the Chair. The following Communications were read : — 1. On the Distribution of the Proper Fractions. By Dr D. M. C. Sommerville. Communicated by Professor Chrystal. p. 116. 2. Notes : — (1) On a Human Skeleton, with Prehistoric Objects, found at Great Casterton, Rutland ; (2) On a Stone Cist containing a Skeleton and an. Urn, found at Largs, Ayrshire. By Dr Robert Munro. With a Report on the Urn by the Hon. John Abercromby, and on the Skulls by Professor D. J. Cunningham, p. 279. 3. A New Form of Harmonic Synthetiser. By Jas. R. Milne, B.Sc. p. 207. Professor Frank Watson Dyson, M.A., E.R.S., Mr Alexander Taylor Innes, M.A., Mr John Patrick Fair Bell, F.Z.S., Dr Edward David Wilson Greig, B.Sc., Dr Thomas Coke Squance, and Mr James Stuart Thomson, F.L.S., wmre balloted for, and declared duly elected Fellows of the Society. ELEVENTH ORDINARY MEETING. Monday , 1th May 1906. Dr R. H. Traquair, F.R.S., Vice-President, in the Chair. The following Communications \\rere read : — 1. On Vibrating Systems which are not subject to the Boltzmann - Maxwell Law. By Dr Wm. Peddie. p. 130. 2. On the Superposition of Mechanical Vibrations upon Magnetisation, 1905-6.] Meetings of the Society. 557 and Conversely, in Iron, Steel, and Nickel. By Mr James Russell. Trans., vol. 45. 3. Neobythites Brucei , Poisson abyssal nouveau recueilli par 1’Ex.pedition Antarctique Nationale Ecossaise. Note Preliminaire par M. Louis Dollo, Conservateur au Musee royal d’Histoire naturelle, a Bruxelles. Presentee par M. le Dr. R. H. Traquair, F.R.S. p. 172. 4. The Nematodes of the Scottish National Antarctic Expedition. By Dr von Linstow. Communicated by Mr W. S. Bruce, p. 464. 5. A Pfaffian Identity and related Vanishing Aggregates of Deter- minant Minors. By Dr Thomas Muir. Trans., vol. 45, p. 311. TWELFTH ORDINARY MEETING. Monday , 21s£ May 1906. Held in the Egyptian Hall, 75 Queen Street. The Hon. Lord M‘Laren, LL.D., Vice-President, in the Chair. At the request of the Council, M. Teisserenc de Bort gave an Address on “ Meteorologie de T Atmosphere Libre.” FIRST SPECIAL MEETING. Monday , 28 th May 1906. Sir John Murray, K.C.B., Vice-President, in the Chair. The following Communications were read : — 1. Life in Reservoirs in relation to the Water Supply of Towns. By Mr James Murray. 2. The Rotifera of Scottish Lochs. By Mr James Murray. Trans. , vol. 45, p. 151. 3. Scottish National Antarctic Expedition. — Tardigrada of the South Orkneys. By Mr James Murray. Trans., vol. 45, p. 323. 4. The Temperature of the Fresh- water Lochs of Scotland, with special reference to Loch Ness. By E. M. Wedderburn, M.A. Trans.,. vol. 45. 558 Proceedings of Royal Society of Edinburgh. [sess. THIRTEENTH ORDINARY MEETING. Monday , 4 th June 1906. Dr R. H. Traquair, F.R.S., Vice-President, in the Chair. The following Communications were read : — 1. Recherches sur la Glauconie. Par les Drs LIson W. Collet et Gabriel W. Lee, assistants de Sir John Murray, K.C.B. Com- munique par Sir John Murray, p. 238. 2. Note on a rare Dolphin ( Delphinus acutus ) recently stranded on the Coast of Sutherland. By Sir William Turner, K.C.B. p. 310. 3. Contributions to the Craniology of the People of the Empire of India. Part III. — Natives of the Madras Presidency, Thugs, Veddahs, Tibetans, and Seistanis. By Sir William Turner, K.C.B. Trans., vol. 45, p. 261. 4. Interpolation for a Table of Fractions, with a Notice of Synthetic Division and its Use. By Dr James Burgess, C.I.E. 5. On the Length of the Normal Chord of a Conic. By Professor Anglin. 6. The Hydroids of the Scottish National Antarctic Expedition. By James Ritchie, M.A. Communicated by Mr W. S. Bruce. Trans., vol. 45. 7. Professor D. J. Cunningham will exhibit a Photograph by Mr W. E. Ward of the Salmon in the Corrib River, Galway. FOURTEENTH ORDINARY MEETING. Monday, 18 th June 1906. Dr Munro, Vice-President, in the Chair. The following Communications were read : — 1. A Dietary Study of Five Halls of Residence for Students in Edinburgh. By Dr Isabella Cameron. Communicated by Dr D. Noel Paton. p. 327. 2. On the Theory of Epidemics. By Dr John Brownlee. Com- municated by R. M. Buchanan, M.B. p. 484. 3. The Plant Remains in the Scottish Peat Mosses. Part II. — The Scottish Highlands. By Francis J. Lewis, F.L.S. Communicated by Professor James Geikie. Trans., vol. 45, p. 335. Mr Frank A. Newington, Memb. Inst. C.E., Mr William 0. Vandenbergh, Dr Daniel E. Anderson, and Dr David Ellis were balloted for, and declared duly elected Fellows of the Society. 1905-6.] Meetings of the Society . 559 FIFTEENTH ORDINARY MEETING. Monday , 2nd July 1906. Professor Cram Brown, LL.D., Vice-President, in the Chair. The following Communications were read : — 1. On the Use of Soluble Prussian Blue in investigating the Reducing- power of Animal Tissue. By Dr D. Fraser Harris. 2. The Viscosity of Solutions. Part I. By C. Ranken, B.Sc., Carnegie Research Scholar, and Dr W. W. Taylor. Communicated by Professor Crum Brown. Trans., vol. 45. 3. Two Lecture Experiments in Illustration of the Theory of Ionisation. By Dr W. W. Taylor. Communicated by Professor Crum Brown, p. 325. SECOND SPECIAL MEETING. Friday, \Wi July 1906. Dr R. H. Traquair, F.R.S., Vice-President, in the Chair. The following Communications were read : — 1. Obituary Notice of S. P. Langley, Secretary, Smithsonian Institu- tion. By Dr Peddie. p. 546. 2. The recent Epidemic of Trypanosomiasis in Mauritius : its Cause and Progress. By Dr Alex. Edington and Dr J. M. Coutts, 3. Note on the Smolt to Grilse Stage of the Salmon, with exhibition of a Marked Fish recaptured. By Mr W. L. Calderwood. p. 321. 4. The Effect of Precipitation Films on the Conductivity of Electrolytes. Part I. By W. S. Millar, B.Sc., Carnegie Research Scholar, and Dr W. W. Taylor. Communicated by Professor Crum Brown, p. 447. 5. The Theory of Alternants in the Historical Order of Development up to 1860. By Dr Thomas Muir. p. 357. 6. The Theory of Circulants in the Historical Order of Development up to 1860. By Dr Thomas Muir. p. 390. 7. On the Length of a Pair of Tangents to a Conic. By Professor Anglin. 8. Further Study of the Two Forms of Liquid Sulphur as Dynamic Isomers. By Professor Alexander Smith and Mr C. M. Carson. p. 352. 560 Proceedings of Royal Society of Edinburgh. [sess. SIXTEENTH AND LAST ORDINARY MEETING. Monday , 16^ Jidy 1906. The Hon. Lord M'Laren, LL.D., Vice-President, in the Chair. The following Communications were read : — 1. Limnographic Apparatus and Measurements on Loch Earn. By Professor Chrystal. Trans., vol. 45, p. 361. 2. Preliminary Limnographic Observations on Loch Earn. By Mr James Murray. Communicated by Professor Chrystal. (Apparatus and Lantern Illustrations were shown.) Trans., vol. 45, p. 361. 3. A Note on the Polarimeter. By J. R. Milne, B.Sc. (An Instrument was shown.) p. 522. 4. Spectroscojiic Observations of the Rotation of the Sun. (Further Communication.) By Dr J. Halm. 5. A Monograph on the General Morphology of the Myxinoid Fishes, based on a study of Myxine. Part II. — The Anatomy of the Muscles. By F. J. Cole, B.Sc. Communicated by Dr R. H. Traquair, F.R.S. Trans., vol. 45. 1905-6.] Abstract of Accounts. 561 AB STRACT OF THE ACCOUNTS OF THE LATE PHILIP ROBERT DALRYMPLE MACLAGAN, ESQ., As Treasurer of the Royal Society of Edinburgh. SESSION 1 905-1906. . ACCOUNT OF THE GENERAL FUND. CHARGE. 1. Arrears of Contributions at 1st October 1905 £160 13 0 2. Contributions for present Session : — 1. 163 Fellows at £2, 2s. each £342 6 0 135 Fellows at £3, 3s. each 425 5 0 £767 11 0 2. Fees of Admission and Contributions of sixteen new Resident Fellows at £5, 5s. each 84 0 0 3. Fees of Admission of ten new Non- Resident Fellows at £26, 5s. each 262 10 0 Interest received — Interest, less Tax Annuity from Edinburgh, and £372 14 2 District Water Trust, less Tax... 49 17 6 422 11 8 4. Society’s Transactions and Proceedings sold Ill 15 0 5. Annual Grant from Government 300 0 0 6. Residue payable to the Society from the Estate of the late Henry Dirck 33 12 2 Amount of the Charge £2142 12 10 PROC. ROY. SOC. EDIN. — YOL. XXYI. 36 562 Proceedings of Royal Society of Edinburgh. DISCHARGE. 1. Rent of Society’s Apartments for Year, less Tax <£285 2. Insurance, Gas, Electric Light, Coal, Water, etc. : — Insurance £9 19 6 Gas 1 5 4 Electric Light 2 17 11 Coal 8 3 0 Water 2 2 0 Income Tax 15 0 0 3. Salaries : — General Secretary £100 0 0 Librarian 150 0 0 Do. Special Allowance 75 0 0 Assistant Librarian 45 0 0 Doorkeeper 12 0 0 Office Keeper 35 0 0 Treasurer’s Clerk 25 0 0 4. Expenses of Transactions : — Neill & Co., Ltd., Printers £311 5 7 M‘Farlane & Erskine, Lithographers 38 2 0 Y. J. Pentland, do. 27 6 0 Alex. Ritchie & Son, do. 8 5 0 J. Bartholomew & Co., do. 16 0 0 Hislop & Day, Engravers 21 8 0 Orrock & Son, Bookbinders 74 15 0 5. Expenses of Proceedings : — Neill & Co., Ltd., Printers £510 1 8 Hislop & Day, Engravers 31 16 6 M'Farlane & Erskine, Lithographers 6 10 0 6. Books, Periodicals, Newspapers, etc. : — Otto Schulze & Co., Booksellers £116 15 7 James Thin, do. 53 12 4 R. Grant & Son, do. 7 7 6 Kegan Paul & Co., do. 2 5 4 Bell & Bradfute, do. 0 15 6 International Catalogue of Scientific Literature 17 0 0 Robertson & Scott, News Agents 7 11 6 Egypt Exploration Funds, Subscription 3 3 0 Ray Society, do. 110 Palaeontographical Society, do. 110 Orrock & Son, Bookbinders 28 2 6 39 442 497 548 238 [sBSS. 0 0 7 9 0 0 1 7 8 2 15 3 Carry forward £2050 12 9 1905-6.] Abstract of Accounts. DISCHARGE — continued. 563 Brought forward £2050 12 9 7. Other Payments : — Neill & Co., Ltd., Printers, General Account £98 2 3 Williams & Norgate, Publishers 73 14 3 R. Blair & Son, Confectioners 29 13 6 Orrock & Son, Bookbinders 10 9 6 Lantern Exhibitions, etc., at Lectures... 13 4 3 Lindsay, Jamieson & Haldane, Auditors 6 6 0 National Telephone Co 7 6 6 Petty Expenses, Postages, Carriage, etc. 44 11 1 283 7 5 8. Irrecoverable Arrears of Contributions written off.. 2 2 0 9. Arrears of Contributions outstanding at 1st October 1906:— Present Session <£118 13 0 Previous Sessions 79 16 0 198 9 0 Amount of the Discharge £2534 11 2 Amount of the Charge £2142 12 10 Amount of the Discharge 2534 11 2 Excess of the Discharge £391 18 4 Floating Balance in favour of the Society at 1st October 1905 £246 3 5 Excess of the Discharge as above 391 18 4 Floating Balance due by the Society at 1st October 1906 £145 14 11 Being — Accounts included in the Discharge, but not paid until after 1st October 1906 £775 9 3 Less Balance due by the Union Bank on Current Account £599 3 6 ,, Dividends included in the Charge but not paid into Bank until after 1st October 1906 30 10 10 629 14 4 £145 14 11 564 Proceedings of Royal Society of Edinburgh. SESS. II. ACCOUNT OP THE KEITH FUND To \st October 1906. CHAKGE. 1. Balance due by the Union Bank at 1st October 1905 £56 12 10 2. Interest Eeceiyed : — On £896, 19s. Id. North British Kail- way Company 3 per cent. Debenture Stock for year to Whitsunday 1906, less Tax £25 11 4 On £211, 4s. North British Eailway Company 3 per cent. Lien Stock for year to Lammas 1906, less Tax 6 0 4 31 11 8 £88 4 6 DISCHAEGE. Nil. Balance due by the Union Bank at 1st October 1906 85 4 4 Dividend Warrant, uncashed at do 3 0 2 £88 4 6 1905-6.] Abstract of Accounts. 565 III. ACCOUNT OF THE NEILL FUND To Is£ October 1906. CHARGE. 1. Balance due by the Union Bank at 1st October 1905 £30 16 2 2. Interest Received : — On £355 London, Chatham and Dover Railway Company 4J per cent. Arbitration Debenture Stock for year to 30th June 1906, less Tax 15 3 6 £45 19 8 DISCHARGE. Nil. Balance due by the Union Bank at 1st October 1906 38 7 11 Dividend Warrant, uncashed at do 7 11 9 £45 19 8 566 Proceedings of Royal Society of Edinburgh. IV. ACCOUNT OF THE MAKDOUGALL-BRISBANE FUND To \st October 1906. CHARGE. 1. Balance due by the Union Bank of Scotland at 1st October 1905 : — On Deposit Receipt £135 0 0 On Current Account 24 18 4 £159 18 4 2. Interest Received : — On £365 Caledonian Railway Company 4 per cent. Consolidated Preference Stock No. 2 for year to 30th June 1906, less Tax 13 17 4 £173 15 8 DISCHARGE. Nil. Balance due by the Union Bank of Scotland at 1st October 1906 : — On Deposit Receipt £135 0 0 On Current Account 31 17 0 £166 17 0 Dividend Warrant, uncashed at 1st October 1906 . 6 18 8 £173 15 8 1905-6.] Abstract of Accounts. 567 V. ACCOUNT OF THE MAKERSTOUN MAGNETIC METEOROLOGICAL OBSERVATION FUND To ls£ October 1906. CHARGE. Sum on Deposit Receipt with the Union Bank of Scot- land at 1st October 1905 £197 2 5 DISCHARGE. Nil. Above Sum on Deposit Receipt with the Union Bank of Scotland at 1st October 1906 £197 2 5 568 Proceedings of Royal Society of Edinburgh. VI. ACCOUNT OP THE GUNNING- VICTORIA JUBILEE PRIZE FUND To ls£ October 1906. (Instituted by Dr R. H. Gunning of Edinburgh and Rio de Janeiro.) CHARGE. 1. Balance due by the Union Bank of Scotland at 1st October 1905 <£17 7 10 2. Interest received on £1000 North British Railway Company 3 per cent. Consolidated Lien Stock for year to Lammas 1906, less Tax 28 10 0 £45 17 10 DISCHARGE. Nil. Balance due by the Union Bank of Scotland on Current Account at 1st October 1906 £31 12 10 Dividend Warrant, uncashed at do 14 5 0 £45 17 10 1905-6.] Abstract of Accounts. 569 STATE OP THE FUNDS BELONGING TO THE ROYAL SOCIETY OP EDINBURGH .4 6- at ls£ October 1906. 1. GENERAL FUND— 1. £2090, 9s. 4d. three per cent. Lien Stock of the North British Railway Company at 89f per cent., the selling price at 1st October 1906 £1873 11 7 2. £8519, 14s. 3d. three per cent. Debenture Stock of do. at 90 J per cent., do 7710 6 9 3. £52, 10s. Annuity of the Edinburgh and District Water Trust, equivalent to £875 at 176J per cent., do 1544 7 6 4. £1811 four per cent. Debenture Stock of the Caledonian Railway Company at 119J per cent., do 2168 13 5 5. £35 four and a half per cent. Arbitration Debenture Stock of the London, Chatham and Dover Railway Company at 123 per cent., do 43 1 0 6. Arrears of Contributions as per preceding Abstract of Accounts 198 9 0 £13,538 9 3 Deduct Floating Balance due by the Society as per preceding Abstract of Accounts 145 14 11 Amount £13,392 14 4 Exclusive of Library, Museum, Pictures, and Furniture of the Society’s Apartments at the Royal Institution. 2. KEITH FUND— 1. £896, 19s. Id. three per cent. Debenture Stock of the North British Railway Company at 90 J per cent., the selling price at 1st October 1906... £811 14 10 2. £211, 4s. three per cent. Lien Stock of do. at 89f per cent., do 189 5 9 3. Balance due by the Union Bank of Scotland (£85, 4s. 4d.), and uncashed dividend warrant in hand (£3, 0s. 2d.) 88 4 6 £1089 5 1 Amount 570 Proceedings of Royal Society of Edinburgh. [sess. STATE OF FUNDS — continued. 3. NEILL FUND— 1 . £355 four and a half per cent. Arbitration Debenture Stock of the London, Chatham and Dover Rail- way Company at 123 per cent., the selling price at 1st October 1906 £436 13 0 2. Balance due by the Union Bank of Scotland (£38, 7s. lid.), and uncashed dividend warrant in hand (£7, 11s. 9d.) 45 19 8 Amount £482 12 8 4. MAKDOU GALL-BRISB ANE FUND— 1. £365 four per cent. Consolidated Preference Stock No. 2 of the Caledonian Railway Company at 1 1 3 J per cent., the selling price at 1st October 1906 £414 5 6 2. Sum on Deposit Receipt with the Union Bank of Scotland 135 0 0 3. Balance due by do. on Current Account (£31, 17s.), and uncashed dividend warrant in hand (£6, 18s. 8d.) 38 15 8 Amount £588 1 2 5. MAKERSTOUN MAGNETIC METEOROLOGICAL OBSERVATION FUND— Sum on Deposit Receipt with the Union Bank of Scotland at 1st October 1906 £197 2 5 6. GUNNING-VICTORIA JUBILEE PRIZE FUND— Instituted by Dr Gunning of Edinburgh and Rio de Janeiro — 1. £1000 three per cent. Consolidated Lien Stock of the North British Railway Company at 89| per cent., the selling price at 1st October 1906 £896 5 0 2. Balance due by the Union Bank of Scotland (£31, 12s. 10d.), and uncashed dividend warrant in hand (£14, 5s.) 45 17 10 Amount £942 2 10 Edinburgh, 15th October 1906. — We have examined the six preceding Accounts of the Treasurer of the Royal Society of Edinburgh for Session 1905-1906, and have found them to be correct. The securities of the various Investments at 1st October 1906, as noted in the above State- ment of Funds, have been exhibited to us. LINDSAY, JAMIESON & HALDANE, Auditors. 1905-6.] Abstract of Accounts. 571 VIDIMUS of ESTIMATED INCOME of THE GENERAL FUND FOR SESSION 1906-1907. 1. Interest : — On £8519, 14s. 3d. Railway Debenture Stock at 3 percent £255 11 10 On £2090, 9s. 4d. Railway Lien Stock at 3 per cent 62 14 4 On £1811 Railway Debenture Stock at 4 per cent. 72 8 8 On £35 Railway Debenture Stock at 4J per cent... Ill 6 £392 6 4 2. Annuity from the Edinburgh and District Water Trust 52 10 0 £444 16 4 Deduct Income Tax at Is. per £ 22 4 10 £422 11 6 3. Annual Contributions : — Of 162 Fellows at £2, 2s. each £340 4 0 Of 137 Fellows at £3, 3s. each 431 11 0 771 15 0 4. Annual Grant from Government 300 0 0 5. Sales of Society’s Transactions 30 0 0 Total Estimated Income, £1524 6 6 Exclusive of Fees of Admission and Contributions of New Fellows who may be admitted during the Year. INDEX. Abercromby (The Hon. John). On an Urn found at Largs, 285, 292. Allantoin Excretion in Thymus Feed- ing, by W. M'Lachlan, 95-106. Alternants. The Theory of Alter- nants in the Historical Order of Development up to 1860, by Thomas Muir, 357-389. Analyser, Harmonic. See Synthe- tiser, 207-233. Anglin (A. H. ). On the Length of the Normal Chord of a Conic ( Title only), 558. On the Length of a Pair of Tangents to a Conic ( Title only), 559. Antarctica, Evidence for Former Extension of, from Distribution of Collembola, by G. H. Carpenter, 473-483. Batliydraco Scotice, Poisson abyssal nouveau, par L. Dollo, 65-75. Boltzmann-Maxwell Law, On Vibrat- ing Systems which are not subject to the, by W. Peddie, 130-141. Bort (Teisserenc de). Meteorologie de 1’ Atmosphere Libre ( Title only), 557. Brown (A. Crum). Obituary Notice of Alexander William Williamson, 541. Brownlee (John). Statistical Studies in Immunity. The Theory of an Epidemic, 484-521. Burgess (James). Interpolation for a Table of Fractions, with a Notice of Synthetic Division and its Use {Title only), 558. Calderwood (W. L.). Note on the Smolt to Grilse Stage of the Salmon, with Exhibition of a Marked Fish recaptured, 321-324. Cameron (I. D.). On the Methods of Standardising Suprarenal Prepara- tions, 157-171. A Dietary Study of Five Halls of Residence for Students in Edinburgh, 327-351. Carpenter (G. H.). Scottish National Antarctic Expedition. “Scotia” Collections. Collembola front the South Orkney Islands, 473-483. Carson (C. M.)and Alexander Smith. Further Study of the Two Forms of Liquid Sulphur as Dynamic Isomers, 352-356. Casterton, Human Skull and Pre- historic Relics found at, by Prof. D. J. Cunningham, 279-309. Circulants. The Theory of Circulants in the Historical Order of Develop- ment up to 1860, by Thomas Muir, 390-398. Collembola, New Species of Isotoma and Cryptopygus from the South Orkneys, by G. H. Carpenter, 473-483. Collet (Leon W.) and T. N. Johnston. On the Formation of certain Lakes in the Highlands, 107-115. Collet (Leon W.) et Gabriel W. Lee. Recherches sur la Glauconie, 238- 278. Coutts (J. M.). See Edington (Alex. ). Crowther-Beynon (V. B.). On Pre- historic Relics at Casterton, 279. Cunningham (D. J.). On Human Skulls from Casterton and Largs, 293-309 Exhibits two Lantern Slides of Zoological Interest, 553. Exhibits Photograph, by Mr W. E. Ward, of the Salmon in the Corrib River, Galway, 558. Determinants, Minors of Square of, by Thomas Muir, 533-539. Dietary Study of Five Halls of Residence for Students in Edin- burgh, by I. D. Cameron, 327-351. Dollo (Louis). Batliydraco Scotice, Poisson abyssal nouveau recueilli par l’Expedition Antarctique Na- tional Ecossaise, 65-75. Neobythites Brucei, Poisson abyssal nouveau recueilli par FExpedition Antarctique Nationale Ecossaise. Note preliminaire, 172- 181. Dolphin ( Delphinus acutus). Note on a Specimen recently stranded on Coast of Sutherland, by Sir Wm. Turner, 310-319. Doppler’s Principle. Two new Cosmic Proofs of its Validity 572 Index. 573 obtained with the Spectro-heiio- meter, by J. Halm, 76-86. Echinorhynchus antardicus, n. sp., and its Allies, by John Rennie, 437- 446. Edington (Alex.) and J. M. Coutts. The Recent Epidemic of Trypano- somiasis in Mauritius : its Cause and Progress ( Title only), 559. Electric Oscillations and Magnetic Properties of Iron, by Janies Russell, 33-47. Electrolysis through Precipitation Films, by W. S. Millar and W. W. Taylor, 447-463. Electrolytes, Preliminary Note on the Conductivity of Concentrated Aqueous Solutions of, by Prof. J. Gibson, 234-237. Electromotive Force of Metals, Measurements of the, by Charles E. Fawsitt, 2-6. Epidemic, Theory of an, by John Brownlee, 484-521. Ewart (J. C. ). The Tarpan and its Relationship with Wild and Domestic Horses, 7-21. Fawsitt (Charles E.). Some Electrical Measurements on Metals, 2-6. Ferguson (Dr R. M.). Elected Society’s Representative on George Heriot’s Trust, 553. Forbes (George). Perturbations in Longitude of Neptune by the Hypothetical Planet ( Title only), 553. Fractions, Proper, On the Distribu- tion of the, by D. M. Y. Sommer- ville, 116-128. Gemmill (J. F.) and R. T. Leiper. The Turlellaria of the Scottish National Antarctic Expedition {Title only), 556. Gibson (John). Preliminary Note on the Conductivity of Con- centrated Aqueous Solutions of Electrolytes, 234-237. Glauconie, Recherches sur la, by Drs Collet et Lee, 238-278. Growth, Effects of varying Diets upon, by C. Watson, 87-94. Halm (J.). Some Further Results obtained with the Spectro-helio- meter, 76-86. Spectroscopic Observations of the Rotation of the Sun {Title only), 560. Harmonic Curves, A New Instrument for drawing the Sum of a number of Simple, by J. R. Milne, 207- 233. Harris (D. Fraser). On the Use of Soluble Prussian Blue in investi- gating the Reducing-power of Animal Tissue {Title only), 559. Hessians of Invariants of Binary Quantics, by Thomas Muir, 529- 532. Highlands. Formation of Lochs Muick, Builg, and Callater, by L. W. Collet and T. N. Johnston, 107-115. Horse. The Tarpan and its Relation- ship with Wild and Domestic Horses, by J. C. Ewart, 7-21. Horse, The, in Norway, by Francis H. A. Marshall, 22-32. Human Skeletons at Casterton and Largs, by Dr R. Munro, 283, 293. Hydrodynamical Theory of Seiches, Some Experimental Results in connection with the, by William White and Peter Watson, 142-156. Immunity, Statistical Studies in. Theory of an Epidemic, by John Brownlee, 484-521. Invariants, Hessians of, by Thomas Muir, 529-532. Ionization, Two Lecture Experiments in Illustration of the Theory of, by W. W. Taylor, 325-326. Kelvin (Lord). Initiation of Deep- Sea Waves of Three Classes : (1) from a Single Displacement ; (2) from a Group of Equal and Similar Displacements ; (3) by a Periodi- cally Varying Surface-Pressure, 399-436. Lakes, Formation of, in Scottish Highlands and in the Alps, by L. W. Collet and T. N. Johnston, 107-115. Langley (S. P ), Obituary Notice of, by Dr Wm. Peddie, 546-549. Largs, Bronze Age Burial at, by Prof. D. J. Cunningham, 279-309. Lecture Experiments in Illustration of the Theory of Ionization, by W. W. Taylor, 325-326. Lee, Gabriel W. See Leon W. Collet. Leiper (E. T.). See Gemmill (J. F.). Library Aids to Mathematical Re- search, by Thomas Muir, 51-64. 574 Index. Lindsay (Rev. James). Obituary Notice of the Rev. Dr George Matheson, 550-551. Linstow (0. von). The Nematodes of the Scottish N ational Antarctic Expedition, 1902-1904, 464-472. M ‘Lachlan, W. A Contribution to the Study of the Excretion of Allantoin in Thvmus Feeding, 95-106. Maclagan-Wedderburn, J. H. See Wedderburn, J. H. Maclagan. Magnetic Detections of Electric Waves, by James Russell, 33-47. Magnetic Properties of Iron and Electric Oscillations, by James Russell, 33-47. Marshall (Francis H. A.). The Horse in Norway, 22-32. Matheson (the Rev. Dr George), Obituary Notice of, by the Rev. James Lindsay, 550-551. Meetings of the Society during Session 1905-1906, 552-560. Metals, Electrical Measurements on, by Charles E. Fawsitt, 2-6. Millar (W. S.) and W. W. Taylor. Electrolysis through Precipitation Filins, 447-463. Milne (Jas Robert). A New Form of Harmonic Synthetiser, 207- I 233. On a Simple Way of Obtain- ing the Half-Shade Field in Polari- meters, 522-526. On an Exception to a Certain Theorem in Optics, with an Appli- cation to the Polarimeter, 527-528. Muir (Thomas). Library Aids to Mathematical Research, 51-64. The Theory of Alternants in the Historical Order of Develop- ment up to 1860, 357-389. The Theory of Circulants in the Historical Order of Develop- ment up to I860, 390-398. The Hessians of Certain In- variants of Binary Quantics, 529- 532. The Sum of the r-line Minors of the Square of a Determinant, 533-539. Munro (Robert). On a Human Skeleton found at Casterton, 279, and at Largs, 283, — 279, 283. On Earlv British Races, 284-292. Murray (James). Life in Reservoirs in Relation to the Water Supply of Towns ( Title only), 557. Nematodes of the Scottish National Antarctic Expedition, 1902-1904, by O. von Linstow, 464-472. Neobythites Brucei , Poisson abyssal nouveau, par L. Dollo, 172-181. Nutrition, Effects of varying Diets upon, by C. Watson, 87-94. Obituary Notices. Williamson (Alexander William), 540-545. Langley (S. P.), 546-549. Matheson (Rev. Dr George), 550-551. Office-Bearers, Session 1905-1906, 1. Oliver (Thomas). The Relation between Normal Take-up or Con- traction and Degree of Twist in Twisted Threads, 182-192. Optics. On a Simple Way of Obtaining the Half-Shade Field in Polarimeters, by James Robert Milne, 522-526. On an Exception to a Certain Theorem in Optics, with an Application to the Polarimeter, 527-528. Papers, List of, read during Session 1905-1906, 552-560. Peddie (Wm.). On Vibrating Systems which are not subject to the Boltzmann-Maxwell Law, 130-141. Obituary Notice of S. P. Langley, 546-549. Polarimeter. On a Simple Way of Obtaining the Half-Shade Field in Polarimeters, by James Robert Milne, 522-526. On an Exception to a Certain Theorem in Optics, with an Application to the Polarimeter, by James Robert Milne, 527-528. Precipitation Films, Electrolysis through, by W. S. Millar and W. W. Taylor, 447-463. Races, Early British, by Dr R. Munro, 284-292. Rennie (John). On Echinorhynchus antarcticus, n. sp., and its Allies, 437-446. Russell (James). Notes on the Effect of Electric Oscillations (co-direc- tional and transverse) on the Mag- netic Properties of Iron, 33-47. Salmon Smolt ascends Rivers as Grilse fully a year after Descent to Sea, by W. L. Calderwood, 321-324. Seiches, Some Experimental Results Index. 575 in connection with the Hydro- dynamical Theory of, by Peter White and William Watson, 142-156. Smith (Alexander) and C. M. Carson. Further Study of the Two Forms of Liquid Sulphur as Dynamic Isomers, 352-356. Sommerville (D. M. Y.). On the Distribution of the Proper Frac- tions, 116-128. South Orkney Islands, Collembola from, by G. H. Carpenter, 473-483. Spectro - heliometer, Some Further Results obtained with the, by J. Halm, 76-86. Sulphur, Further Study of the Two Forms of Liquid, as Dynamic Isomers, by Alexander Smith and C. M. Carson, 352-356. Sun. Some Further Results obtained with the Spectro-heliometer, by J. Halm, 76-86. Suprarenal Preparations, On the Methods of Standardising, by I. D. Cameron, 3 57-171. Synthetiser, A Hew Form of Harmonic, by J. R. Milne, 207-233. Take-up or Contraction in Twisted Threads, by Thomas Oliver, 182-192. Tarpan and its Relationship with Wild and Domestic Horses, by J. C. Ewart, 7-21. Taylor (W. W.). Two Lecture Experiments in Illustration of the Theory of Ionization, 325-326. Taylor (W. W.) and W. S. Millar. Electrolysis through Precipitation Films. 447-463. Thymus Feeding, Excretion of Allan - tom in, by W. M ‘Lachlan, 95-106. Turner (Sir Wm.). Note on a rare Dolphin ( Delphinus acutus ), recently stranded on the Coast of Sutherland, 310-319. Contributions to the Crani- ology of the. People of India. Part III. — Natives of the Madras Presidency, Thugs, Veddahs, Tibetans, Seistanis ( Title only), 320. Twisted Threads, Take-up or Con- traction in, by Thomas Oliver, 182-192. Urn or Beaker of the Bronze Age, by the Hon. J. Abercromby, 292-293. Vibrating Systems which are not subject to the Boltzmann-Maxwell Law, by W. Peddie, 130-141. Watson (Chalmers). Preliminary Note regarding an Experimental Investigation into the Effects of varying Diets upon Growth and Nutrition, 87-94. Watson, William. See White, Peter. Waves. Initiation of Deep-Sea Waves of Three Classes : (1) from a Single Displacement ; (2) from a Group of Equal and Similar Displacements ; (3) from a Periodi- cally Varying Surface-Pressure, by Lord Kelvin, 399-436. Wedderburn, J. H. Maclagan. On a New Theorem in Hypercomplex Numbers, 48-50. White (Peter) and William Watson. Some Experimental Results in connection with the Hydro- dynamical Theory of Seiches, 142-156. Williamson (Alexander William), Obituary Notice of, by Professor A. Crum Brown, 540-545. PRINTED BY NEILL AND CO., LTD., EDINBURGH. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SESSION 1905-6. No .1.1 VOL. XXVI. [PP. 1-64. CONTENTS. PAGE Office-Bearers, Session 1905-6, 1 Some Electrical Measurements on Metals. By Charles E. Fawsitt, D.Sc., Ph.D. ( Communicated by Pro- fessor A. Crum Brown), .... 2 {Issued separately February 12, 1906.) The Tarpan and its Relationship with Wild and Domestic Horses. By J. C. Ewart, M.D., F.R.S. (With Three Plates), ..... 7 ( Issued separately February 12, 1906.) The Horse in Norway. By Francis H. A. Marshall, M. A. (Cantab.), D.Sc. (Edin.). (With Two Plates), . ,22 {Issued separately February 12, 1906.) [ Continued on page iv of Cover. EDINBURGH : Published by ROBERT GRANT & SON, 107 Princes Street, and WILLIAMS & NORGATE, 14 Henrietta Street, Coyent Garden, London. MDCCCCYI. Price Three Shillings and Sixpence. REGULATIONS REGARDING THE PUBLICATION OF PAPERS IN THE PROCEEDINGS AND TRANS- ACTIONS OF THE SOCIETY. The Council beg to direct the attention of authors of communications to the Society to the following Regulations, which have been drawn up in order to accelerate the publication of the Proceedings and Transactions, and to utilise as widely and as fairly as possible the funds which the Society devotes to the publication of Scientific and Literary Researches. 1. Manuscript of Papers. — As soon as any paper has been passed for publication, either in its original or in any altered form, and has been made ready for publication by the author, it is sent to the printer, whether it has been read or not. The ‘ copy ’ should be written on large sheets of paper, on one side only, and the pages should be clearly numbered. The MS. must be easily legible, preferably typewritten, and must be absolutely in its final form for printing ; so that corrections in proof shall be as few as possible, and shall not cause overrunning in the lines or pages of the proof. All tables of contents, references to plates or illustrations in the text, etc., must be in their proper places, with the page numbers left blank ; and spaces must be indicated for the insertion of illustrations that are to appear in the text. 2. Illustrations. — All illustrations must be drawn in a form im- mediately suitable for reproduction; and such illustrations as can be reproduced by photographic processes should, so far as possible, be preferred. Drawings to be reproduced as line blocks should be made with Indian ink (deadened with yellow if of bluish tone), preferably on fine white bristol board, free from folds or creases ; smooth, clean lines or sharp dots, but no washes or colours should be used. If the drawings are done on a large scale, to be afterwards reduced by photography, any lettering or other legend must be on a corresponding scale. If an author finds it inconvenient to furnish such drawings, the Society will have the figures re-drawn at his expense ; but this will cause delay. When the illustrations are to form plates, a scheme for the arrange- ment of the figures (in quarto plates for the Transactions, in octavo for the Proceedings) must be given, and numbering and lettering indicated. 3. Proofs. — In general, a first proof and a revise of each paper will be sent to the author, whose address should be indicated on the MS. If further proofs are required, owing to corrections or alterations for which the printer is not responsible, the expense of such proofs and corrections will be charged against the author. All proofs must, if possible, be returned within one week, addressed to The Secretary , Royal Society , Mound , Edinburgh , and not to the printer. [ Continued on page iii of Cover. iii To prevent delay, authors residing abroad should appoint some one residing in this country to correct their proofs. 4. Additions to a Paper after it has been finally handed in for publication, if accepted by the Council, will be treated and dated as separate communications, and may, or may not, be printed immediately after the original paper. 5. Brief Abstracts of Transactions Papers will be published in the Proceedings, provided they are sent along with the original paper. 6. Separate Issue of Reprints ; Author’s Free and Additional Copies. — As soon as the final revise of a Transactions paper has been returned, or as soon as the sheet in which the last part of a Proceedings paper appears is ready for press, a certain number of separate copies or reprints, in covers bearing the title of the paper and the name of the author, are printed off and placed on sale. The date of such separate publication will be printed on each paper. The author receives fifty of these reprints free, and may have any reasonable number of additional copies at a fixed scale of prices which will be furnished by the printer, who will charge him with the cost. To prevent disappointment, especially if the paper contains plates, the author should, immediately after receiving his first proof, notify to the printer the number of additional copies required. 7. Index Slips. — In order to facilitate the compilation of Subject Indices, and to secure that due attention to the important points in a paper shall be given in General Catalogues of Scientific Literature and in Abstracts by Periodicals, every author is requested to return to the Secretary along with his final proof a brief index (on the model given below), of the points in it which he considers new or important. These indices will be edited by the Secretary, and incorporated in Separate Index Slips, to be issued with each part of the Proceedings and Transactions. MODEL INDEX. Schafer, E. A. — On the Existence within the Liver Cells of Channels which can be directly injected from the Blood-vessels. Proc. Roy. Soc. Edin., vol. , 1902, pp. Cells, Liver, — Intra-cellular Canaliculi in. E. A. Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp. Liver, — Injection within Cells of. E. A. Schafer. Proc. Roy. Soc. Edin., vol , 1902, pp. IV CONTENTS. Notes on the Effect of Electric Oscillations (co-directional and transverse) on the Magnetic Properties of Iron. By James Russell, ..... ( Issued separately February 8, 1906.) On a Theorem in Hypercomplex Numbers. By J. H. Maclagan-Wedderburn, Carnegie Research Fellow, . {Issued separately February 9, 1906.) Library Aids to Mathematical Research. By Thomas Muir, LL.D., ...... ( Issued separately February 1 4, 1906.) PAGE 33 48 51 PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SESSION 1905-6. NO. II ] VOL. XX VI. [Pp. 65-128. CONTENTS. PAGE Bathydraco Scotise, Poisson abyssal nouveau recueilli par l’Expedition Antarctique Rationale Ecossaise. Note preliminaire, par Louis Dollo, Conservateur au Musee royal d’Histoire naturelle, a Bruxelles. [Presentee par M. R. H. Traquair, M.D., F.R.S., V.P.R.S.E.), . 65 {Issued separately March 29, 1906.) Some Further Results obtained with the Spectroheliometer. By Dr J. Halm, . . . . . 76 ( Issued separately March 29, 1906.) Preliminary Note regarding an Experimental Investigation into the Effects of Varying Diets upon Growth and Nutrition. By Chalmers Watson, M.D. [From the Physiological Laboratory of the University of Edin- burgh.) (. Presented by Prof. E. A. Schafer, F.R.S.), . 87 {Issued separately February 22, 1906.) [ Continued on page iv of Cover. EDINBURGH : Published by ROBERT GRANT & SON, 107 Princes Street, and WILLIAMS & NORGATE, 14 Henrietta Street, Covent Garden, London. MDCCCCYI. Price Three Shillings. KEGrU LATION S REGARDING THE PUBLICATION OF PAPERS IN THE PROCEEDINGS AND TRANS- ACTIONS OF THE SOCIETY. The Council beg to direct the attention of authors of communications to the Society to the following Regulations, which have been drawn up in order to accelerate the publication of the Proceedings and Transactions, and to utilise as widely and as fairly as possible the funds which the Society devotes to the publication of Scientific and Literary Researches. 1. Manuscript of Papers. — As soon as any paper has been passed for publication, either in its original or in any altered form, and has been made ready for publication by the author, it is sent to the printer, whether it has been read or not. The £ copy 5 should be written on large sheets of paper, on one side only, and the pages should be clearly numbered. The MS. must be easily legible, preferably typewritten, and must be absolutely in its final form for printing ; so that corrections in proof shall be as few as possible, and shall not cause overrunning in the lines or pages of the proof. All tables of contents, references to plates or illustrations in the text, etc., must be in their proper places, with the page numbers left blank ; and spaces must be indicated for the insertion of illustrations that are to appear in the text. 2. Illustrations. — All illustrations must be drawn in a form im- mediately suitable for reproduction; and such illustrations as can be reproduced by photographic processes should, so far as possible, be preferred. Drawings to be reproduced as line blocks should be made with Indian ink (deadened with yellow if of bluish tone), preferably on fine white bristol board, free from folds or creases ; smooth, clean lines or sharp dots, but no washes or colours should be used. If the drawings are done on a large scale, to be afterwards reduced by photography, any lettering or other legend must be on a corresponding scale. If an author finds it inconvenient to furnish such drawings, the Society will have the figures re-drawn at his expense ; but this will cause delay. When the illustrations are to form plates, a scheme for the arrange- ment of the figures (in quarto plates for the Transactions, in octavo for the Proceedings) must be given, and numbering and lettering indicated. 3. Proofs. — In general, a first proof and a revise of each paper will be sent to the author, whose address should be indicated on the MS. If further proofs are required, owing to corrections or alterations for which the printer is not responsible, the expense of such proofs and corrections will be charged against the author. All proofs must, if possible, be returned within one week, addressed to The Secretary , Royal Society , Mound , Edinburgh, and not to the printer. [' Continued on page iii of Cover . To prevent delay, authors residing abroad should appoint some one residing in this country to correct their proofs. 4. Additions to a Paper after it has been finally handed in for publication, if accepted by the Council, will be treated and dated as separate communications, and may, or may not, be printed immediately after the original paper. 5. Brief Abstracts of Transactions Papers will be published in the Proceedings, provided they are sent along with the original paper. 6. Separate Issue of Beprints; Author’s Free and Additionai Copies. — As soon as the final revise of a Transactions paper has been returned, or as soon as the sheet in which the last part of a Proceedings paper appears is ready for press, a certain number of separate copies or reprints, in covers bearing the title of the paper and the name of the author, are printed off and placed on sale. The date of such separate publication will be printed on each paper. The author receives fifty of these reprints free, and may have any reasonable number of additional copies at a fixed scale of prices which will be furnished by the printer, who will charge him with the cost. To prevent disappointment, especially if the paper contains plates, the author should, immediately after receiving his first proof, notify to the printer the number of additional copies required. 7. Index Slips. — In order to facilitate the compilation of Subject Indices, and to secure that due attention to the important points in a paper shall be given in General Catalogues of Scientific Literature and in Abstracts by Periodicals, every author is requested to return to the Secretary along with his final proof a brief index (on the model given below), of the points in it which he considers new or important. These indices will be edited by the Secretary, and incorporated in Separate Index Slips, to be issued with each part of the Proceedings and Transactions. MODEL INDEX. Schafer, E. A. — On the Existence within the Liver Cells of Channels which can be directly injected from the Blood-vessels. Proc. Boy. Soc. Edin., vol. , 1902, pp. Cells, Liver, — Intra-cellular Canaliculi in. E. A. Schafer. Proc. Boy. Soc. Edin., vol. , 1902, pp. Liver, — Injection within Cells of. E. A. Schafer. Proc. Boy. Soc. Edin., vol , 1902, pp. IV CONTENTS. A Contribution to the Study of the Excretion of Allantoin in Thymus Feeding. By W. M ‘Lachlan, M.D. (From the Research Laboratory of the Royal College of Physicians , Edinburgh .) (Communicated by Dr D. Noel Paton), ..... {Issued separately March 29, 1906.) On the Formation of certain Lakes in the Highlands. By Dr Leon W. Collet, F. Swiss G-eol. S., Assistant to Sir John Murray, K.C.B., and Dr T. N. Johnston, F.R.S.E. With a Note on Two Rock Basins in the Alps, hy Dr Leon W. Collet, {Issued separately April 16, 1906.) On the Distribution of the Proper Fractions. By Duncan M. Y. Sommerville, D.Sc. (Communicated by Prof. Chrystal), ...... {Issued separately April 16, 1906.) PAGE 95 107 116 PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SESSION 1905-6. No. III.] YOL. XXVI. [Pp. 129-192. CONTENTS. PAGE On Vibrating Systems which are not subject to the Boltz- mann-Maxwell Law. By Dr W. Peddie . .130 ( Issued separately May 24, 1906.) Some Experimental Results in connection with the Hydro- dynamical Theory of Seiches. By Peter White, M.A., and William Watson, .... 142 (. Issued separately June 11, 1906.) On the Methods of Standardising Suprarenal Preparations. By I. D. Cameron, M.B., D.P.H., Assistant to the Lecturer on Physiology, Edinburgh School of Medicine for Women. {From the Laboratory of the Royal College of Physicians , Edinburgh.) Communicated by Dr D. Noel Paton, . . . . .157 ( Issued separately May 21, 1906.) [ Continued on page iv of Cover, EDINBURGH : Published by ROBERT GRANT & SON, 107 Princes Street, anS WILLIAMS & NORGATE, 14 Henrietta Street, Covent Garden, London. MDCCCCVI Price Three Shilli: REGULATIONS REGARDING THE PUBLICATION OF PAPERS IN THE PROCEEDINGS AND TRANS- ACTIONS OF THE SOCIETY. The Council beg to direct the attention of authors of communications to the Society to the following Regulations, which have been drawn up in order to accelerate the publication of the Proceedings and Transactions, and to utilise as widely and as fairly as possible the funds which the Society devotes to the publication of Scientific and Literary Researches. 1. Manuscript op Papers. — As soon as any paper has been passed for publication, either in its original or in any altered form, and has been made ready for publication by the author, it is sent to the printer, whether it has been read or not. The ‘ copy ’ should be written on large sheets of paper, on one side only, and the pages should be clearly numbered. The MS. must be easily legible, preferably typewritten, and must be absolutely in its final form for printing ; so that corrections in proof shall be as few as possible, and shall not cause overrunning in the lines or pages of the proof. All tables of contents, references to plates or illustrations in the text, etc., must be in their proper places, with the page numbers left blank ; and spaces must be indicated for the insertion of illustrations that are to appear in the text. 2. Illustrations. — All illustrations must be drawn in a form im- mediately suitable for reproduction ; and such illustrations as can be reproduced by photographic processes should, so far as possible, be preferred. Drawings to be reproduced as line blocks should be made with Indian ink (deadened with yellow if of bluish tone), preferably on fine white bristol board, free from folds or creases ; smooth, clean lines or sharp dots, but no washes or colours should be used. If the drawings are done on a large scale, to be afterwards reduced by photography, any lettering or other legend must be on a corresponding scale. If an author finds it inconvenient to furnish such drawings, the Society will have the figures re-drawn at his expense ; but this will cause delay. When the illustrations are to form plates, a scheme for the arrange- ment of the figures (in quarto plates for the Transactions, in octavo for the Proceedings) must be given, and numbering and lettering indicated. 3. Proofs. — In general, a first proof and a revise of each paper will be sent to the author, whose address should be indicated on the MS. If further proofs are required, owing to corrections or alterations for which the printer is not responsible, the expense of such proofs and corrections will be charged against the author. All proofs must, if possible, be returned within one week, addressed to The Secretary , Royal Society , Mound , Edinburgh , and not to the printer. [ Continued on page iH- of Cover. Ill To prevent delay, authors residing abroad should appoint some one residing in this country to correct their proofs. 4. Additions to a Paper after it has been finally handed in for publication, if accepted by the Council, will be treated and dated as separate communications, and may, or may not, be printed immediately after the original paper. 5. Brief Abstracts of Transactions Papers will be published in the Proceedings, provided they are sent along with the original paper. 6. Separate Issue of Reprints; Author’s Free and Additional Copies. — As soon as the final revise of a Transactions paper has been returned, or as soon as the sheet in which the last part of a Proceedings paper appears is ready for press, a certain number of separate copies or reprints, in covers bearing the title of the paper and the name of the author, are printed off and placed on sale. The date of such separate publication will be printed on each paper. The author receives fifty of these reprints free, and may have any reasonable number of additional copies at a fixed scale of prices which will be furnished by the printer, who- will charge him with the cost. To prevent disappointment, especially if the paper contains plates, the author should, immediately after receiving his first proof, notify to the printer the number of additional copies required. 7. Index Slips. — In order to facilitate the compilation of Subject Indices, and to secure that due attention to the important points in a paper shall be given in General Catalogues of Scientific Literature and in Abstracts by Periodicals, every author is requested to return to the Secretary along with his final proof a brief index (on the model given below), of the points in it which he considers new or important. These indices will be edited by the Secretary, and incorporated in Separate Index Slips, to be issued with each part of the Proceedings and Transactions. MODEL INDEX. Schafer, E. A. — On the Existence within the Liver Cells of Channels which can be directly injected from the Blood-vessels. Proc. Roy. Soc. Edin., vol. , 1902, pp. Cells, Liver, — Intra-cellular Canaliculi in. E. A. Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp. Liver, — Injection within Cells of. E. A. Schafer. Proc. Roy. Soc. Edim, vol , 1902, pp. IV CONTENTS. PAGE Neobythites Brucei, Poisson abyssal nouveau recueilli par l’Expedition Antarctique Nationale Ecossaise. Note preliminaire, par Louis Dollo, Conservateur au Musee royal d’Histoire naturelle, a Bruxelles. Presentee par M. R. H. Traquair, M.D., F.R.S., V.P.R.S.E., . 172 (. Issued separately June 14, 1906.) The Belation between Normal Take-up or Contraction and Degree of Twist in Twisted Threads. By Thomas Oliver, B.Sc. (Lond. & Edin.), Carnegie Research Fellow. Communicated by Dr C. G. Knott, . .182 (. Issued separately June 19, 1906.) S'Ofa^i K PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SESSION 1905-6. No. IV.] VOL. XXVI. [Pp. 193-320. CONTENTS. PAGE A New Form of Harmonic Synthetiser. By J. R. Milne, B.Sc., Carnegie Research Fellow. (With Plate), . 207 {Issued separately July 12, 1906.) Preliminary Note on the Conductivity of Concentrated Aqueous Solutions of Electrolytes. By Prof. J. Gibson, ...... 234 {Issued separately August 29, 1906.) Recherches sur la Glauconie. Par les Drs Leon W. Collet et Gabriel W. Lee, assistants de Sir John Murray, K.C.B. Communique par Sir John Murray. (Avec 12 planches et 1 carte), . . . 238 {Issued separately August 30, 1906.) [ Continued on page iv of Cover. EDINBURGH : Published by ROBERT GRANT & SON, 107 Princes Street, and WILLIAMS & NORGATE, 14 Henrietta Street, Covent Garden, London. MDCCCCVI. Price Seven Shillings. I S OCT 1 Sb REGULATIONS REGARDING THE PUBLICATION OF PAPERS IN THE PROCEEDINGS AND TRANS- ACTIONS OF THE SOCIETY. The Council beg to direct the attention of authors of communications to the Society to the following Regulations, which have been drawn up in order to accelerate the publication of the Proceedings and Transactions, and to utilise as widely and as fairly as possible the funds which the Society devotes to the publication of Scientific and Literary Researches. 1. Manuscript of Papers. — As soon as any paper has been passed for publication, either in its original or in any altered form, and has been made ready for publication by the author, it is sent to the printer, whether it has been read or not. The ‘ copy ’ should be written on large sheets of paper, on one side only, and the pages should be clearly numbered. The MS. must be easily legible, preferably typewritten, and must be absolutely in its final form for printing ; so that corrections in proof shall be as few as possible, and shall not cause overrunning in the lines or pages of the proof. All tables of contents, references to plates or illustrations in the text, etc., must be in their proper places, with the page numbers left blank ; and spaces must be indicated for the insertion of illustrations that are to appear in the text. 2. Illustrations. — All illustrations must be drawn in a form im- mediately suitable for reproduction ; and such illustrations as can be reproduced by photographic processes should, so far as possible, be preferred. Drawings to be reproduced as line blocks should be made with Indian ink (deadened with yellow if of bluish tone), preferably on fine white bristol board, free from folds or creases ; smooth, clean lines or sharp dots, but no washes or colours should be used. If the drawings are done on a large scale, to be afterwards reduced by photography, any lettering or other legend must be on a corresponding scale. If an author finds it inconvenient to furnish such drawings, the Society will have the figures re-drawn at his expense ; but this will cause delay. When the illustrations are to form plates, a scheme for the arrange- ment of the figures (in quarto plates for the Transactions, in octavo for the Proceedings) must be given, and numbering and lettering indicated. 3. Proofs. — In general, a first proof and a revise of each paper will be sent to the author, whose address should be indicated on the MS. If further proofs are required, owing to corrections or alterations for which the printer is not responsible, the expense of such proofs and corrections will be charged against the author. All proofs must, if possible, be returned within one week, addressed to The Secretary , Royal Society , Mound , Edinburgh, and not to the printer . [ Continued on page iii of Cover. Ill To prevent delay, authors residing abroad should appoint some one residing in this country to correct their proofs. 4. Additions to a Paper after it has been finally handed in for publication, if accepted by the Council, will be treated and dated as separate communications, and may, or may not, be printed immediately after the original paper. 5. Brief Abstracts of Transactions Papers will be published in the Proceedings, provided they are sent along with the original paper. 6. Separate Issue of Reprints; Author’s Free and Additional Copies. — As soon as the final revise of a Transactions paper has been returned, or as soon as the sheet in which the last part of a Proceedings paper appears is ready for press, a certain number of separate copies or reprints, in covers bearing the title of the paper and the name of the author, are printed off and placed on sale. The date of such separate publication will be printed on each paper. The author receives fifty of these reprints free, and may have any reasonable number of additional copies at a fixed scale of prices which will be furnished by the printer, who will charge him with the cost. To prevent disappointment, especially if the paper contains plates, the author should, immediately after receiving his first proof, notify to the printer the number of additional copies required. 7. Index Slips. — In order to facilitate the compilation of Subject Indices, and to secure that due attention to the important points in a paper shall be given in General Catalogues of Scientific Literature and in Abstracts by Periodicals, every author is requested to return to the Secretary along with his final proof a brief index (on the model given below), of the points in it which he considers new or important. These indices will be edited by the Secretary, and incorporated in Separate Index Slips, to be issued with each part of the Proceedings and Transactions. MODEL INDEX. Schafer, E. A. — On the Existence within the Liver Cells of Channels which can be directly injected from the Blood-vessels. Proc. Roy. Soc. Edin., vol. , 1902, pp. Cells, Liver, — Intra-cellular Canaliculi in. E. A. Schafer. Proc. Roy. Soc. Edin, vol. , 1902, pp. Liver, — Injection within Cells of. E. A. Schafer. Proc. Roy. Soc. Edin., vol , 1902, pp. IV CONTENTS. Notes : — 1. On a Human Skeleton, with Prehistoric Objects, found at Great Casterton, Rutland. 2. On a Stone Cist containing a Skeleton and an Urn, found at Largs, Ayrshire. By Dr Robert Munro. With a Report on the Urn, by the Hon. John Aberoromby ; and on the Skulls, by Professor D. J. Cunningham, ...... (Issued separately August 31, 1906.) Note on a rare Dolphin (Delphinus acutus ), recently stranded on the Coast of Sutherland. By Sir William Turner, K.C.B., F.R.S. (With Plate), ( Issued separately August 29, 1906.) Contributions to the Craniology of the People of the Empire of India. Part III. : Natives of the Madras Presidency, Thugs, Veddahs, Tibetans, and Seistanis. By Sir William Turner, K.C.B., ( Title only.) PAGE 279 310 320 PROCEEDINGS Ck OF THE ROYAL SOCIETY OF EDINBURGH. SESSION 1905-6. No. V.] YOL. XXVI. [Pp. 321-432. CONTENTS. PAGE Note on the Smolt to Grilse Stage of the Salmon, with Exhibition of a Marked Fish recaptured. By W. L. C ALDER WOOD, . . . . . .321 (. Issued separately October 12, 1906.) Two Lecture Experiments in illustration of the Theory of Ionization. By Dr W. W. Taylor. Communicated by Professor Crum Brown, . . . 325 {Issued separately October 12, 1906.) A Dietary Study of Five Halls of Residence for Students in Edinburgh. By I. D. Cameron, M.B., D.P.H. Communicated by D. Noel Paton, M.D., . 327 ( Issued separately November 9, 1906.) [■ Continued on page iv of Cover . EDINBURGH : Published by ROBERT GRAFT & SON, 107 Princes Street, and WILLIAMS & FORGATE, 14 Henrietta Street, Covent Garden, London. Price Six Shillings. REGULATIONS REGARDING THE PUBLICATION OF PAPERS IN THE PROCEEDINGS AND TRANS- ACTIONS OF THE SOCIETY. The Council beg to direct the attention of authors of communications to the Society to the following Regulations, which have been drawn up in order to accelerate the publication of the Proceedings and Transactions, and to utilise as widely and as fairly as possible the funds which the Society devotes to the publication of Scientific and Literary Researches. 1. Manuscript of Papers. — As soon as any paper has been passed for publication, either in its original or in any altered form, and has been made ready for publication by the author, it is sent to the printer, whether it has been read or not. The ‘ copy ? should be written on large sheets of paper, on one. side only, and the pages should be clearly numbered. The MS. must be easily legible, preferably typewritten, and must be absolutely in its final form for printing ; so that corrections in proof shall be as few as possible, and shall not cause overrunning in the lines or pages of the proof. All tables of contents, references to plates or illustrations in the text, etc., must be in their proper places, with the page numbers left blank ; and spaces must be indicated for the insertion of illustrations that are to appear in the text. 2. Illustrations. — All illustrations must be drawn in a form im- mediately suitable for reproduction; and such illustrations as can be reproduced by photographic processes should, so far as possible, be preferred. Drawings to be reproduced as line blocks should be made with Indian ink (deadened with yellow if of bluish tone), preferably on fine white bristol board, free from folds or creases ; smooth, clean lines or sharp dots, but no washes or colours should be used. If the drawings are done on a large scale, to be afterwards reduced by photography, any lettering or other legend must be on a corresponding scale. If an author finds it inconvenient to furnish such drawings, the Society will have the figures re-drawn at his expense ; but this will cause delay. When the illustrations are to form plates, a scheme for the arrange- ment of the figures (in quarto plates for the Transactions, in octavo for the Proceedings) must be given, and numbering and lettering indicated. 3. Proofs. — In general, a first proof and a revise of each paper will be sent to the author, whose address should be indicated on the MS. If further proofs are required, owing to corrections or alterations for which the printer is not responsible, the expense of such proofs and corrections will be charged against the author. All proofs must, if possible, be returned within one week, addressed to The Secretary , Royal Society , Mound , Edinburgh , and not to the printer. [ Continued on page iii of Cover. Ill To prevent delay, authors residing abroad should appoint some one residing in this country to correct their proofs. 4. Additions to a Paper after it has been finally handed in for publication, if accepted by the Council, will he treated and dated as separate communications, and may, or may not, be printed immediately after the original paper. 5. Brief Abstracts of Transactions Papers will be published in the Proceedings, provided they are sent along with the original paper. 6. Separate Issue of Reprints ; Author’s Free and Addition a i Copies. — As soon as the final revise of a Transactions paper has been returned, or as soon as the sheet in which the last part of a Proceedings paper appears is ready for press, a certain number of separate copies or reprints, in covers bearing the title of the paper and the name of the author, are printed off and placed on sale. The date of such separate publication will be printed on each paper. The author receives fifty of these reprints free, and may have any reasonable number of additional copies at a fixed scale of prices which will be furnished by the printer, who will charge him with the cost. To prevent disappointment, especially if the paper contains plates, the author should, immediately after receiving his first proof, notify to the printer the number of additional copies required. 7. Index Slips. — -In order to facilitate the compilation of Subject Indices, and to secure that due attention to the important points in a paper shall be given in General Catalogues of Scientific Literature and in Abstracts by Periodicals, every author is requested to return to the Secretary along with his final proof a brief index (on the model given below), of the points in it which he considers new or important. These indices will be edited by the Secretary, and incorporated in Separate Index Slips, to be issued with each part of the Proceedings and Transactions. MODEL INDEX. Schafer, E. A. — On the Existence within the Liver Cells of Channels which can be directly injected from the Blood-vessels. Proc. Boy. Soc. Edin., vol. , 1902, pp. Cells, Liver, — Intra- cellular Canaliculi in. E. A. Schafer. Proc. Boy. Soc. Edin., vol. , 1902, pp. Liver, — Injection within Cells of. E. A. Schafer. Proc. Boy. Soc. Edin., vol , 1902, pp IV CONTENTS. PAGE Further Study of the two Forms of Liquid Sulphur as Dynamic Isomers. By Alexander Smith and C. M. Carson, . . . . .352 (. Issued separately November 12, 1906.) • The Theory of Alternants in the Historical Order of Development up to 1860. By Thomas Muir, LL.D., 357 ( Issued separately November 16, 1906.) The Theory of Circulants in the Historical Order of Development up to 1860. By Thomas Muir, LL.D., 390 {Issued separately November 16, 1906.) Initiation of Deep-Sea Waves of Three Classes : (1) from a Single Displacement ; (2) from a Group of Equal and Similar Displacements ; (3) by a Periodically Vary- ing Surface Pressure. By Lord Kelvin, . . 399 {Issued separately November 17, 1906.) PROCEEDINGS OF THE ROYAL SOCIETY OE EDINBURGH SESSION 1905-6. No. VI. VOL. XXVI. [Pp. 433-575. CONTENTS. PAGE “ Scotia ” Collections. On Ecliinorliynchus antarcticus, n. sp., and its Allies. By -John Bennie, D.Sc., University of Aberdeen. Communicated by Wm. S. Bruce, Esq. (With a Plate), ( Issued separately January 4, 1907.) 437 Electrolysis through Precipitation Films. Part I. By W. S. Millar, M.A., B.Sc., Carnegie Besearch Scholar, and Dr W. W. Taylor. Communicated by Professor Crum Brown, .... ( Issued separately January 4, 1907.) 447 Nematodes of the Scottish National Antarctic Expedition, 1902-1904. By Dr v. Linstow, Gottingen. Com- municated by W. S. Hruce. (With Two Plates), {Issued separately January 4, 1907.) [ Continued on page iv of Covey 464 EDINBUBGH : Published by ROBERT GRANT & SON, 107 Princes Street, and WILLIAMS & NORGATE, 14 Henrietta Street, Covent Garden, London. MDCCCCYII. FEB 18 190? I ^131 Price Six Shillings. REGULATIONS REGARDING THE PUBLICATION OF PAPERS IN THE PROCEEDINGS AND TRANS- ACTIONS OF THE SOCIETY. The Council beg to direct the attention of authors of communications to the Society to the following Regulations, which have been drawn up in order to accelerate the publication of the Proceedings and Transactions, and to utilise as widely and as fairly as possible the funds which the Society devotes to the publication of Scientific and Literary Researches. 1. Manuscript of Papers. — As soon as any paper has been passed for publication, either in its original or in any altered form, and has been made ready for publication by the author, it is sent to the printer, whether it has been read or not. The ‘ copy ’ should be written on large sheets of paper, on one side only, and the pages should be clearly numbered. The MS. must be easily legible, preferably typewritten, and must be absolutely in its final form for printing ; so that corrections in proof shall be as few as possible, and shall not cause overrunning in the lines or pages of the proof. All tables of contents, references to plates or illustrations in the text, etc., must be in their proper places, with the page numbers left blank ; and spaces must be indicated for the insertion of illustrations that are to appear in the text. 2. Illustrations. — All illustrations must be drawn in a form im- mediately suitable for reproduction; and such illustrations as can be reproduced by photographic processes should, so far as possible, be preferred. Drawings to be reproduced as line blocks should be made with Indian ink (deadened with yellow if of bluish tone), preferably on fine white bristol board, free from folds or creases ; smooth, clean lines or sharp dots, but no washes or colours should be used. If the drawings are done on a large scale, to be afterwards reduced by photography, any lettering or other legend must be on a corresponding scale. If an author finds it inconvenient to furnish such drawings, the Society will have the figures re-drawn at his expense ; but this will cause delay. When the illustrations are to form plates, a scheme for the arrange- ment of the figures (in quarto plates for the Transactions, in octavo for the Proceedings) must be given, and numbering and lettering indicated. 3. Proofs. — In general, a first proof and a revise of each paper will be sent to the author, whose address should be indicated on the MS. If further proofs are required, owing to corrections or alterations for which the printer is not responsible, the expense of such proofs and corrections will be charged against the author. All proofs must, if possible, be returned within one week, addressed to The Secretary , Royal Society , Mound , Edinburgh , and not to the printer. [' Continued on page iii of Cover. iii To prevent delay, authors residing abroad should appoint some one residing in this country to correct their proofs. 4. Additions to a Paper after it has been finally handed in for publication, if accepted by the Council, will be treated and dated as separate communications, and may, or may not, be printed immediately after the original paper. 5. Brief Abstracts of Transactions Papers will be published in the Proceedings, provided they are sent along with the original paper. 6. Separate Issue of Reprints; Author’s Free and Additional Copies. — As soon as the final revise of a Transactions paper has been returned, or as soon as the sheet in which the last part of a Proceedings paper appears is ready for press, a certain number of separate copies or reprints, in covers bearing the title of the paper and the name of the author, are printed off and placed on sale. The date of such separate publication will be printed on each paper. The author receives fifty of these reprints free, and may have any reasonable number ofi additional copies at a fixed scale of prices which will be furnished by the printer, who will charge him with the cost. To prevent disappointment, especially if the paper contains plates, the author should, immediately after receiving his first proof, notify to the printer the number of additional copies required. 7. Index Slips. — In order to facilitate the compilation of Subject Indices, and to secure that due attention to the important points in a paper shall be given in General Catalogues of Scientific Literature and in Abstracts by Periodicals, every author is requested to return to the Secretary along with his final proof a brief index (on the model given below), of the points in it which he considers new or important. These indices will be edited by the Secretary, and incorporated in Separate Index Slips, to be issued with each part of the Proceedings and Transactions. MODEL INDEX. Schafer, E. A. — On the Existence within the Liver Cells of Channels which can be directly injected from the Blood-vessels. Proc. Roy. Soc. Edin., vol. , 1902, pp. Cells, Liver, — Intra-cellular Canaliculi in. E. A. Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp. Liver, — Injection within Cells of. E. A. Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp. IV CONTENTS. G PAGE Scottish National Antarctic Expedition. “ Scotia ” Collec- tions. Collembola from the South Orkney Islands. By George H. Carpenter, B.Sc., M.R.I.A., Professor of Zoology in the Royal College of Science, Dublin. (With Two Plates.) Communicated by William Evans, Esq., . . . . . .473 ( Issued separately January 14, 1907.) Statistical Studies in Immunity : The Theory of an Epidemic. By John Brownlee, M.D. Glas. Com- municated by Dr R. M. Buchanan, . . . 484 {Issued separately January 14, 1907.) On a Simple Way of Obtaining the Half-Shade Eield in Polarimeters. By James Robert Milne, B.Sc., Carnegie Research Eellow, . . . .522 {Issued separately January 14, 1907.) On an Exception to a Certain Theorem in Optics, with an Application to the Polarimeter. By James Robert Milne, B.Sc., Carnegie Research Eellow, . . 527 {Issued separately January 14, 1907.) The Hessians of Certain Invariants of Binary Quantics. By Thomas Muir, LL.D., . . . .529 {Issued separately January 16, 1907.) The Sum of the r-line Minors of the Square of a Deter- minant. By Thomas Muir, LL.D., . . 533 {Issued separately January 16, 1907.) Obituary Notices, . . . 540 Meetings of the Royal Society — Session 1905-1906, . 552 Abstract of Accounts for Session 1905-1906, . . 561 Index, . . .572